Research Article On Relations between One-Dimensional ...downloads.hindawi.com/journals/amp/2015/652026.pdf · Research Article On Relations between One-Dimensional Quantum and Two-Dimensional

Research ArticleOn Relations between One-Dimensional Quantum andTwo-Dimensional Classical Spin Systems

J Hutchinson J P Keating and F Mezzadri

School of Mathematics University of Bristol Bristol BS8 1TW UK

Correspondence should be addressed to F Mezzadri fmezzadribrisacuk

Received 31 August 2015 Accepted 1 November 2015

Academic Editor Pierluigi Contucci

Copyright copy 2015 J Hutchinson et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

We exploit mappings between quantum and classical systems in order to obtain a class of two-dimensional classical systemscharacterised by long-range interactions and with critical properties equivalent to those of the class of one-dimensional quantumsystems treated by the authors in a previous publication In particular we use three approaches the Trotter-Suzuki mapping themethod of coherent states and a calculation based on commuting the quantum Hamiltonian with the transfer matrix of a classicalsystemThis enables us to establish universality of certain critical phenomena by extension from the results in the companion paperfor the classical systems identified

1 Introduction

Mappings between statistical mechanical models have pro-vided new pathways to compute thermodynamic proper-ties of systems which were previously intractable [1ndash3] Inparticular critical phenomena in 119889-dimensional quantumsystems have been investigated by mapping them to (119889 +1)-dimensional classical systems for which there are betterdeveloped techniques such asMonteCarlo simulations [4 5]For example one well known connection is that betweenthe one-dimensional 119883119884119885 model and the two-dimensionalzero-field eight-vertex model namely the Hamiltonian ofthe quantum model and the transfer matrix of the classicalmodel have the same eigenvectors Baxter [1] found theground state energy for the 119883119884119885 model by first findingthe partition function of the eight-vertex model and thenshowing that the quantum Hamiltonian is effectively thelogarithmic derivative of the transfer matrix for the classicalsystem

In this paper we exploit these quantum to classical (QC)mappings for the opposite reason to take advantage of knownground state critical behaviour in a general class of quantumspin chains with long-range interactions to determine thefinite temperature critical properties of an equivalent class ofclassical spin systems

In a companion paper to appear in [6] we computed thecritical exponents 119904 ] and 119911 corresponding to the energygap correlation length and dynamic exponent respectivelyfor a class of quantum spin chains establishing universalityfor this class of systems We also computed the groundstate correlators ⟨120590119909

119894120590119909

119894+119903⟩119892 ⟨120590119910

119894120590119910

119894+119903⟩119892 and ⟨prod119903

119894=1120590119911

119894⟩119892for this

class of systems when translation invariance is imposedThese correlators were found to exhibit quasi-long-rangeorder behaviour when the systems are gapless with a criticalexponent dependent upon the system parameters

The class of quantum spin chains studied in [6] consists of119872 spin-12 particles in an external field ℎ with aHamiltonianquadratic in Fermi operators given by

H =

119872

sum

119895119896

(119860119895119896119887dagger

119895119887119896+120574

2119861119895119896(119887

dagger

119895119887dagger

119896minus 119887

119895119887119896)) minus 2ℎ

119872

sum

119895=1

119887dagger

119895119887119895 (1)

where the 119887119895s are the Fermi operators satisfying the usual

Fermi commutation relations

119887dagger

119895 119887

119896 = 120575

119895119896

119887dagger

119895 119887

dagger

119896 = 119887

119895 119887

119896 = (119887

dagger

119895)2

= (119887119895)2

= 0

(2)

The measure of anisotropy 120574 is real with 0 le 120574 le 1 thematrix 119860

119895119896must be Hermitian and 119861

119895119896antisymmetric both

Hindawi Publishing CorporationAdvances in Mathematical PhysicsVolume 2015 Article ID 652026 18 pageshttpdxdoiorg1011552015652026

2 Advances in Mathematical Physics

containing only real entries without loss of generality andperiodic boundary conditions 119887

119872+119895= 119887

119895are assumed We

can think of119860119895119896

and 119861119895119896

as band matrices whose thickness1determines the length of the spin-spin interaction

This model can be diagonalised [7] so that

H = sum

119902

10038161003816100381610038161003816Λ

119902

10038161003816100381610038161003816120578dagger

119902120578119902+ 119862 (3)

with the dispersion relation |Λ119902| determined bymatrices119860

119895119896

and 119861119895119896 the 120578

119902s are Fermi operators and 119862 is a constant

In [8 9] Keating andMezzadri restricted theHamiltonian(1) to possess symmetries corresponding to the Haarmeasureof each of the classical compact groups 119880(119873) 119874+

(2119873)Sp(2119873) 119874+

(2119873 + 1) 119874minus

(2119873 + 1) and 119874minus

(2119873 + 2) enablingthe calculation of |Λ

119902| using techniques from randommatrix

theory This corresponds to a symmetry classification of spinchains similar to that introduced for disordered systems byAltland and Zirnbauer [10ndash12] These symmetry propertieswere encoded into the structure of the matrices119860

119895119896and 119861

119895119896

as summarised in Appendix A For example when restrictedto 119880(119873) symmetry2 [8 9]

Λ119902= 4(Γ +

119871

sum

119896=1

(119886 (119896) cos 119896119902 + 119894119887 (119896) sin 119896119902))

= 4 (119886119902+ 119894119887

119902)

(4)

where 119871 = [(119872 minus 1)2] with [ ] denoting the integer part ofits argument The real and imaginary parts of (4) are

119886119902= Γ +

119871

sum

119896=1

119886 (119896) cos 119896119902

119887119902=

119871

sum

119896=1

119887 (119896) sin 119896119902

(5)

Furthermore

Γ =1

2

119886 (0) if 119872 is odd

119886 (0) + (minus1)119897

119886 (119872

2) if 119872 is even

119902 =2120587119897

119872 119897 = 0 119872 minus 1

(6)

In general the symmetry constraints were achieved using realfunctions 119886(119895) and 119887(119895) even and odd functions of Z119872Zrespectively to dictate the entries of matrices119860

119895119896and 119861

119895119896 as

reported in Appendix AExploiting the formalism developed in [8 9] enabled us

to compute the critical properties of this class of spin chains[6] demonstrating a dependence of the critical exponents onsystem symmetries and establishing universality for this classof quantum systems

Having established universality for the above class ofquantum spin chains in [6] we now make use of QCmappings to obtain a class of classical systemswith equivalent

critical properties establishing universality for this class ofclassical systems as well by extension This is our main goal

There is no systematic technique to construct a classi-cal 119889 + 1-dimensional lattice system from a quantum 119889-dimensional one there is no alternative but to develop an adhoc approach for each case This is usually a major challengeFurthermore such a mapping is not unique However overthe years severalmodels have been proved equivalent Suzuki[3] introduced a powerfulmethod based onTrotterrsquos formulaAnother technique exploits the fact that if the quantumHamiltonian commutes with the transfer matrix of a classicalsystem then they are equivalent This idea was used bySuzuki [2] to prove the equivalence between the generalisedquantum 119883119884 model and the two-dimensional Ising anddimer models Krinsky [13] showed that the eight-vertex freefermion model with an electric field is equivalent to theground state of the 119883119884 model in the presence of a magneticfield Peschel [14] demonstrated that the quantum119883119884modelcan be mapped to Ising type models with three differentfrustrated lattice structures Lifting the translational invari-ance Minami [15] proved the equivalence of the 119883119884 modelto a class of two-dimensional Ising models with nonuniforminteraction coefficients Igloi and Lajko [16] showed that thequantum Ising model with site-dependent coupling param-eters in a transverse magnetic field is equivalent to an Isingmodel on a square lattice with a diagonally layered structureSome of these systems are not translation invariant but theyall have nearest neighbour spin-spin interactions to ourknowledge there is no system with long-range interactionsfor which classical-quantum equivalence has previously beenproved

A quantum and a classical system are equivalent if theirpartition functions are the same such a correspondencehowever is not unique as different classical systems canbe equivalent to the same quantum system We will hereadopt the following different approaches to map the partitionfunctions of the quantum spin chains (1) onto those of ageneral class of two-dimensional classical systems

(i) The Trotter-Suzuki formula (Section 2)(ii) The method of coherent states (Section 3)(iii) The simultaneous diagonalisation of the quantum

Hamiltonian and the transfer matrix for the classicalsystem (Section 4)

2 Trotter-Suzuki Mapping

This approach was developed by Suzuki [3] who applied theTrotter product formula

119890119860+119861

= lim119899rarrinfin

(119890119860119899

119890119861119899

)119899

[119860 119861] = 0 (7)

to map the partition function for a 119889-dimensional quantumsystem to that for a (119889 + 1)-dimensional classical one Inparticular he applied it to the partition function of a 119889-dimensional quantum Ising model in a transverse magneticfield mapping it to that of a (119889 + 1)-dimensional classicalIsingmodel [3] He then proved the equivalence of the critical

Advances in Mathematical Physics 3

properties of the ground state of the quantum system and thefinite temperature properties of the classical system

Here we harness this technique to supply us with a classof two-dimensional classical systems with critical propertiesequivalent to those of the ground state of the quantum spinchains (1) Like the original quantum system the classicalcounterparts are also able to possess symmetries reflectedby those of the Haar measure of each of the differentclassical compact groups3 enabling the dependence of criticalproperties on system symmetries to be observed

There aremanyways to apply the Trotter-Suzukimappingto the partition function for the class of quantum spin chains(1) resulting in different classical partition functions Thosethat we obtain are of the form

119885119860= sum

all states119890minus120573clHcl(119904119894119895)119891 (119904

119894119895) (8a)

119885119861= sum

restricted states119890minus120573clHcl(119904119894119895) (8b)

119885119862= sum

all configurationsprod

119894

120596119894 (8c)

119885119863= sum

all states119890minus120573clHcl(120590119894119895120591119894119895119904119894119895) (8d)

where Hcl is the effective classical Hamiltonian In (8a) and(8b)Hcl is a real function of the classical spin variables 119904

119894119895=

plusmn1 and in (8d) it is a complex function of the classical spinvariables 120590

119894119895 120591

119894119895 119904

119894119895= plusmn1 which represent the eigenvalues

of the Pauli matrices 120590119909119894 120590119910

119894 and 120590119911

119894 respectively The second

index in the classical variables 120590119894119895 120591

119894119895 119904

119894119895is due to the extra

dimension appearing when applying the Trotter formula (7)The function 119891(119904

119894119895) is also a real function of the classical

spin variables 119904119894119895= plusmn1 and we find that if 119891(119904

119894119895) =

1 then (8a) has the familiar form of a classical partitionfunction with Hcl representing the Hamiltonian describingthe effective classical system The same is true for (8b) and(8d) but (8b) has additional constraints on the spin states and(8d) involves imaginary interaction coefficients The form in(8c) is that of a vertex model with vertex weights given by 120596

119894

Examples of equivalent partition functions with each of theseforms will be given in the following sections

We begin to present our results by first restricting toquantum systems with nearest neighbour interactions onlyThe extensions to longer range interactions are detailed inAppendix B

21 Nearest Neighbour Interactions Restricting (1) to near-est neighbour interactions gives the well known one-dimensional quantum119883119884model4

H119883119884

= minus

119872

sum

119895=1

(119869119909

119895120590119909

119895120590119909

119895+1+ 119869

119910

119895120590119910

119895120590119910

119895+1+ ℎ120590

119911

119895) (9)

where 119869119909119895= minus(12)(119860

119895119895+1+ 120574119861

119895119895+1) 119869119910

119895= minus(12)(119860

119895119895+1minus

120574119861119895119895+1) This mapping is achieved by using Jordan-Wigner

transformations

119887dagger

119895=1

2(119898

2119895+1+ 119894119898

2119895) =

1

2(120590

119909

119895+ 119894120590

119910

119895)

119895minus1

prod

119897=1

(minus120590119911

)

119887119895=1

2(119898

2119895+1minus 119894119898

2119895) =

1

2(120590

119909

119895minus 119894120590

119910

119895)

119895minus1

prod

119897=1

(minus120590119911

119897)

(10)

where

1198982119895+1

= 120590119909

119895

119895minus1

prod

119897=0

(minus120590119911

119895)

1198982119895= 120590

119910

119895

119895minus1

prod

119897=0

(minus120590119911

119895)

(11)

or inversely as

120590119911

119895= 119894119898

2119895119898

2119895+1

120590119909

119895= 119898

2119895+1

119895minus1

prod

119897=0

(minus1198941198982119897119898

2119897+1)

120590119910

119895= 119898

2119895

119895minus1

prod

119897=0

(minus1198941198982119897119898

2119897+1)

(12)

The 119898119895s are thus Hermitian and obey the anticommutation

relations 119898119895 119898

119896 = 2120575

119895119896

211 A Class of Classical Ising Type Models with NearestNeighbour Interactions When we restrict to 120574 = 1 and119861119895119895+1

= 119860119895119895+1

= 119869119894 (9) becomes a class of quantum Ising type

models in a transverse magnetic field with site-dependentcoupling parameters Suzuki showed [3] that the partitionfunction for such a system can be mapped5 to that for a classof two-dimensional classical Ising models with HamiltonianHcl given by

Hcl = minus119899

sum

119901=1

119872

sum

119895=1

(119869ℎ

119895119904119895119901119904119895+1119901

+ 119869V119904119895119901119904119895119901+1

) (13)

with parameter relations

120573cl119869V=1

2log coth

120573quℎ

119899

120573cl119869ℎ

119895=120573qu

119899119869119895

(14)

where 120573qu (cl) is the inverse temperature of the quantum(classical) system

Thus we have an equivalence between our class of quan-tum spin chains under these restrictions and a class of two-dimensional classical Ising models also with site-dependentcoupling parameters in one direction and a constant coupling


parameter in the other From (14) we see that the magneticfield ℎ driving the phase transition in the ground state of thequantum system plays the role of temperature 120573cl driving thefinite temperature phase transition of the classical system

This mapping holds in the limit 119899 rarr infin which wouldresult in anisotropic couplings for the class of classical Isingmodels unless we also take 120573qu rarr infin This thereforeprovides us with a connection between the ground stateproperties of the class of quantum systems and the finitetemperature properties of the classical systems

In this case we can also use this mapping to write theexpectation value of any function 119891(120590119911) with respect to theground state of the class of quantum systems as

⟨119891 (120590119911

)⟩qu = ⟨119891 (119904)⟩cl (15)

where ⟨119891(119904)⟩cl is the finite temperature expectation ofthe corresponding function of classical spin variables withrespect to the class of classical systems (13)

Some examples of this are the spin correlation functionsbetween two or more spins in the ground state of the class ofquantum systems in the 119911 direction which can be interpretedas the equivalent correlator between classical spins in thesame row of the corresponding class of classical systems (13)

⟨120590119911

119895120590119911

119895+119903⟩qu= ⟨119904

119895119901119904119895+119903119901⟩cl

⟨

119903

prod

119895=1

120590119911

119895⟩

qu

= ⟨

119903

prod

119895=1

119904119895119901⟩

cl

(16)

212 A Class of Classical Ising Type Models with AdditionalConstraints on the Spin States Similarly the Trotter-Suzukimapping can be applied to the partition function for the 119883119884model (9) in full generality In this case we first order theterms in the partition function in the following way

119885 = lim119899rarrinfin

Tr [V119886V

119887]119899

V120572= sum

119895isin120572

119890(120573qu119899)H

119911

119895 119890(120573qu119899)H

119909

119895 119890(120573qu119899)H

119910

119895 119890(120573qu119899)H

119911

119895

(17)

where H120583

119895= 119869

120583

119895120590120583

119895120590120583

119895+1for 120583 isin 119909 119910 H119911

119895= (ℎ4)(120590

119911

119895+ 120590

119911

119895+1)

and 120572 denotes either 119886 or 119887 which are the sets of odd and evenintegers respectively

We then insert 2119899 copies of the identity operator in the120590119911 basis I

119904119901= sum

119904119901

| 119904119901⟩⟨ 119904

119901| where | 119904

119901⟩ = |119904

1119901 119904

2119901 119904

119872119901⟩

between each of the 2119899 terms in (17)


Tr I 1199041

V119886I 1199042

V119887sdot sdot sdot I

1199042119899minus1V

119886I 1199042119899

V119887


sum

119904119895119901

2119899

prod

119901isin119886

⟨ 119904119901

10038161003816100381610038161003816V

119886

10038161003816100381610038161003816119904119901+1⟩ ⟨ 119904

119901+1

10038161003816100381610038161003816V

119887

10038161003816100381610038161003816119904119901+2⟩

(18)

The remaining matrix elements in (18) are given by

⟨ 119904119901

10038161003816100381610038161003816V

120572

10038161003816100381610038161003816119904119901+1⟩ =

119872

prod

119895isin120572

⟨119904119895119901 119904

119895+1119901

10038161003816100381610038161003816M10038161003816100381610038161003816119904119895119901+1

119904119895+1119901+1

⟩ (19)

where

M =

(((((((

(

119890120573quℎ119899 cosh(

2120573qu120574

119899119861119895) 0 0 sinh(

2120573qu120574

119899119861119895)

0 cosh(2120573qu

119899119860

119895) sinh(

2120573qu

119899119860

119895) 0

0 sinh(2120573qu

119899119860

119895) cosh(

2120573qu

119899119860

119895) 0

sinh(2120573qu120574

119899119861119895) 0 0 119890

minusℎ119899 cosh(2120573qu120574

119899119861119895)

)))))))

)

(20)

It is then possible to write the terms (19) in exponentialform as

⟨ 119904119901

10038161003816100381610038161003816V

120572

10038161003816100381610038161003816119904119901+1⟩ =

119872

prod

119895isin120572

119890minus120573clH119895119901 (21)

whereH119895119901

can be written as

H119895119901= minus1

4(119869

V119895119904119895119901119904119895119901+1

+ 119869ℎ

119895119904119895119901119904119895+1119901

+ 119869119889

119895119904119895+1119901

119904119895119901+1

+ 119867(119904119895119901+ 119904

119895+1119901) + 119862

119895)

(22)

or more symmetrically as

H119895119901= minus1

4(119869

ℎ

119895(119904

119895119901119904119895+1119901

+ 119904119895119901+1

119904119895+1119901+1

)

+ 119869V119895(119904

119895119901119904119895119901+1

+ 119904119895+1119901

119904119895+1119901+1

)

+ 119869119889

119895(119904

119895119901119904119895+1119901+1

+ 119904119895119901+1

119904119895+1119901

)

+ 119867(119904119895119901+ 119904

119895+1119901+ 119904

119895119901+1+ 119904

119895+1119901+1) + 119862

119895)

(23)

where

120573cl119869ℎ

119895= log

sinh (4120573qu119899) 120574119861119895sinh (4120573qu119899)119860119895


120573cl119869119889

119895= log

tanh (2120573qu119899)119860119895

tanh (2120574120573qu119899) 119861119895

120573cl119869V119895= log coth

2120574120573qu

119899119861119895coth

2120573qu

119899119860

119895

120573cl119867 =120573quℎ

119899

120573cl119862119895= log sinh

2120573qu

119899119860

119895sinh

2120574120573qu

119899119861119895

(24)

as long aswe have the additional restriction that the four spinsbordering each shaded square in Figure 1 obey

119904119895119901119904119895+1119901

119904119895119901+1

119904119895+1119901+1

= 1 (25)

This guarantees that each factor in the partition function isdifferent from zero

Thus we obtain a partition function equivalent to that fora class of two-dimensional classical Ising type models on a119872times 2119899 lattice with classical HamiltonianHcl given by

Hcl =2119899

sum

119901isin119886

119872

sum

119895isin119886

H119895119901+

2119899

sum

119901isin119887

119872

sum

119895isin119887

H119895119901 (26)

whereH119895119901

can have the form (22) or (23) with the additionalconstraint (25)

In this case we see that the classical spin variables ateach site of the two-dimensional lattice only interact withother spins bordering the same shaded square representedschematically in Figure 1 with an even number of these fourinteracting spins being spun up and down (from condition(25))

This mapping holds in the limit 119899 rarr infin which wouldresult in coupling parameters 119869ℎ

119895 119869

119889

119895 119867 rarr 0 and 119869V

119895rarr infin

unless we also take 120573qu rarr infin Therefore this again gives us aconnection between the ground state properties of this classof quantum systems and the finite temperature properties ofthe classical systems

Again we have the same relationship between expectationvalues (15) and (16)

22 A Class of Classical Ising Type Models with ImaginaryInteraction Coefficients Alternatively lifting the restriction(25) we instead can obtain a class of classical systemsdescribed by aHamiltonian containing imaginary interactioncoefficients

Hcl = minus119899

sum

119901=1

119872

sum

119895=1

(119869120590

119895120590119895119901120590119895+1119901

+ 119869120591

119895120591119895119901120591119895+1119901

+ 119894119869120591119895119901(120590

119895119901minus 120590

119895119901+1))

(27)

Trotter

direction

p darr

Lattice direction jrarr

Figure 1 Lattice representation of a class of classical systemsequivalent to the general class of quantum systems (9) Spins onlyinteract with other spins bordering the same shaded square

with parameter relations given by

120573cl119869120590

119895=120573qu

119899119869119909

119895

120573cl119869120591

119895=120573qu

119899119869119910

119895

120573cl119869 =1

2arctan 1

sinh (120573qu119899) ℎ

(28)

To achieve this we first apply the Trotter-Suzukimappingto the quantum partition function divided in the followingway


Tr [U1U

2]119899

U1= 119890

(120573qu2119899)H119909119890(120573qu2119899)H119911119890

(120573qu2119899)H119910

U2= 119890

(120573qu2119899)H119910119890(120573qu2119899)H119911119890

(120573qu2119899)H119909

(29)

where this time H120583

= sum119872

119895=1119869120583

119895120590120583

119895120590120583

119895+1for 120583 isin 119909 119910 and H119911

=

sum119872

119895=1120590119911

119895

Next insert 119899 of each of the identity operators I119901

=

sum119901

|119901⟩⟨

119901| and I

120591119901= sum

120591119901

| 120591119901⟩⟨ 120591

119901| which are in the 120590119909 and

120590119910 basis respectively into (29) obtaining


Tr I1

U1I 1205911

U2I2

U1I 1205912sdot sdot sdot I

1205912119899U

2


sum

120590119895119901 120591119895119901

119899

prod

119901=1

⟨119901

10038161003816100381610038161003816U

1

10038161003816100381610038161003816120591119901⟩ ⟨ 120591

119901

10038161003816100381610038161003816U

2

10038161003816100381610038161003816119901+1⟩

(30)


It is then possible to rewrite the remaining matrix ele-ments in (30) as complex exponentials

⟨119901

10038161003816100381610038161003816U

1

10038161003816100381610038161003816120591119901⟩ ⟨ 120591

119901

10038161003816100381610038161003816U

2

10038161003816100381610038161003816119901+1⟩

= 119890(120573qu119899)((12)(H

119909

119901+H119909119901+1

)+H119910

119901) ⟨119901

10038161003816100381610038161003816119890(120573qu2119899)H

119911 10038161003816100381610038161003816120591119901⟩ ⟨ 120591

119901

10038161003816100381610038161003816

sdot 119890(120573qu2119899)H

119911 10038161003816100381610038161003816119901+1⟩

= 1198622119872

119890(120573qu119899)((12)(H

119909

119901+H119909119901+1

)+H119910

119901)+(1198942)119863sum119872

119895=1120591119895119901(120590119895119901minus120590119895119901+1)

(31)

where H119909

119901= sum

119872

119895=1119869119909

120590119895119901120590119895+1119901

H119910

119901= sum

119872

119895=1119869119910

120591119895119901120591119895+1119901

119863 =(12) arctan(1 sinh(120573qu119899)ℎ) and 119862 = (12) cosh((120573qu119899)ℎ)and we have used

⟨120590119895119901

10038161003816100381610038161003816119890119886120590119911

11989510038161003816100381610038161003816120591119895119901⟩

=1

2cosh (2119886) 119890119894(12) arctan(1 sinh(2119886))120590119895119901120591119895119901

(32)

The classical system with Hamiltonian given by (27) canbe depicted as in Figure 2 where the two types of classical spinvariables 120590

119895119901and 120591

119895119901can be visualised as each representing

two-dimensional lattices on two separate planes as shown inthe top diagram in Figure 2 One can imagine ldquounfoldingrdquothe three-dimensional interaction surface shown in the topdiagram in Figure 2 into the two-dimensional plane shown inthe bottom diagram with new classical spin variables labelledby

119895119901

As in previous cases this mapping holds in the limit 119899 rarrinfin which would result in coupling parameters 119869120590

119895 119869120591

119895rarr infin

and 119869 rarr 1205874120573cl unless we also take 120573qu rarr 0 Thereforeit gives us a connection between the ground state propertiesof the class of quantum systems and the finite temperatureproperties of the classical ones

We can use this mapping to write the expectation value ofany function 119891(120590119909) or 119891(120590119910) with respect to the groundstate of the class of quantum systems (9) as

⟨119891 (120590119909

)⟩qu = ⟨119891 (120590)⟩cl

⟨119891 (120590119910

)⟩qu = ⟨119891 (120591)⟩cl (33)

where ⟨119891(120590)⟩cl and ⟨119891(120591)⟩cl are the finite temperatureexpectation values of the equivalent function of classical spinvariables with respect to the class of classical systems (27)6

An example of this is the two-spin correlation functionbetween spins in the ground state of the class of quantumsystems (9) in the 119909 and 119910 direction which can be interpretedas the two-spin correlation function between spins in thesame odd and even rows of the corresponding class ofclassical systems (13) respectively

⟨120590119909

119895120590119909

119895+119903⟩qu= ⟨120590

119895119901120590119895+119903119901⟩cl

⟨120590119910

119895120590119910

119895+119903⟩qu= ⟨120591

119895119901120591119895+119903119901⟩cl

(34)

23 A Class of Classical Vertex Models Another interpre-tation of the partition function obtained using the Trotter-Suzuki mapping following a similar method to that of [17] isthat corresponding to a vertex model

This can be seen by applying the Trotter-Suzuki mappingto the quantum partition function ordered as in (17) andinserting 2119899 identity operators as in (18) with remainingmatrix elements given once more by (19) This time insteadof writing them in exponential form as in (21) we interpreteach matrix element as a weight corresponding to a differentvertex configuration at every point (119895 119901) of the lattice

⟨ 119904119901

10038161003816100381610038161003816119890(120573qu119899)V120572 10038161003816100381610038161003816

119904119901+1⟩

=

119872

prod

119895isin120572

120596119895

(119904119895119901 119904

119895+1119901 119904

119895119901+1 119904

119895+1119901+1)

(35)

As such the partition function can be thought of as corre-sponding to a class of two-dimensional classical vertex mod-els on a (1198722+119899)times(1198722+119899) lattice as shown in Figure 4 with119872119899 vertices each with a weight 120596119895

(119904119895119901119904119895+1119901

119904119895119901+1

119904119895+1119901+1

)

given by one of the following

120596119895

1(+1 +1 +1 +1) = 119890

ℎ120573qu119899 cosh(2120573qu120574

119899119861119895)

120596119895

2(minus1 minus1 minus1 minus1) = 119890

minus120573quℎ119899 cosh(2120574120573qu

119899119861119895)

120596119895

3(minus1 +1 +1 minus1) = 120596

119895

4(+1 minus1 minus1 +1)

= sinh(2120573qu

119899119860

119895)

120596119895

5(+1 minus1 +1 minus1) = 120596

119895

6(minus1 +1 minus1 +1)

= cosh(2120573qu

119899119860

119895)

120596119895

7(minus1 minus1 +1 +1) = 120596

119895

8(+1 +1 minus1 minus1)

= sinh(2120573qu120574

119899119861119895)

(36)

thus leading to a class of 8-vertex models with the usual8 possible respective vertex configurations as shown inFigure 3

The values of these weights depend upon the column119895 = 1 119872 of the original lattice thus each column hasits own separate set of 8 weights as represented by thedifferent colours of the circles at the vertices in each columnin Figure 4

Once again this mapping holds in the limit 119899 rarr infinwhich would result in weights 120596119894

3 120596

119894

4 120596

119894

7 120596

119894

8rarr 0 and

weights 120596119894

1 120596

119894

2 120596

119894

5 120596

119894

6rarr 1 unless we also take 120573qu rarr infin It

thus gives us a connection between the ground state proper-ties of the class of quantum systems and the finite temperatureproperties of the corresponding classical systems

24 Algebraic Form for the Classical Partition FunctionFinally one last form for the partition function can beobtained using the same method as in Section 212 such that


12059041

12059031

12059021

12059011

12059042

12059032

12059022

12059012

12059043

12059033

12059023

12059013

12059141

12059131

12059121

12059111

12059142

12059132

12059122

12059112

12059143

12059133

12059123

12059113

1 2 3 4 5 6

120590j1 = j1

120591j1 = j2

120590j2 = j3

120591j2 = j4

120590j3 = j5

120591j3 = j6

Trotter

direction

p darr


Figure 2 Lattice representation of a class of classical systems equivalent to the class of quantum systems (9) The blue (thick solid) linesrepresent interactions with coefficients dictated by 119869120590

119895and the red (thick dashed) lines by 119869120591

119895 and the 119869

119895coupling constants correspond to the

green (thin solid) lines which connect these two lattice interaction planes

1205961 1205962 1205963 1205964 1205965 1205966 1205967 1205968

Figure 3 The 8 allowed vertex configurations

the quantum partition function is mapped to one involvingentries from matrices given by (20) This time howeverinstead of applying the extra constraint (25) we can write thepartition function as


sum

120590119895119901=plusmn1

1

4(

119899

prod

119901isin119886

119872

prod

119895isin119886

+

119899

prod

119901isin119887

119872

prod

119895isin119887

)

sdot [(1 minus 119904119895119901119904119895+1119901

) (1 + 119904119895119901119904119895119901+1

) cosh2120573qu

119899119860

119895119895+1

+ (1 minus 119904119895119901119904119895+1119901

) (1 minus 119904119895119901119904119895119901+1

) sinh2120573qu

119899119860

119895119895+1

+ (1 + 119904119895119901119904119895+1119901

) (1 minus 119904119895119901119904119895119901+1

) sinh2120573qu120574

119899119861119895119895+1

+ (1 + 119904119895119901119904119895119901+1

) (1 + 119904119895119901119904119895119901+1

) 119890(120573qu119899)ℎ119904119895119901

sdot cosh2120573qu120574

119899119861119895119895+1]

(37)

25 Longer Range Interactions The Trotter-Suzuki mappingcan similarly be applied to the class of quantum systems (1)with longer range interactions to obtain partition functions


Trotter

direction

p darr


Figure 4 Lattice representation demonstrating how configurationsof spins on the dotted vertices (represented by arrows uarrdarr) give riseto arrow configurations about the solid vertices

equivalent to classical systems with rather cumbersomedescriptions examples of which can be found in Appendix B

3 Method of Coherent States

An alternative method to map the partition function for theclass of quantum spin chains (1) as studied in [6] onto thatcorresponding to a class of classical systems with equivalentcritical properties is to use the method of coherent states [18]

To use such a method for spin operators 119878119894 = (ℏ2)120590119894we first apply the Jordan-Wigner transformations (10) oncemore to map the Hamiltonian (1) onto one involving Paulioperators 120590119894 119894 isin 119909 119910 119911

Hqu =1

2sum

1le119895le119896le119872

((119860119895119896+ 120574119861

119895119896) 120590

119909

119895120590119909

119896

+ (119860119895119896minus 120574119861

119895119896) 120590

119910

119895120590119910

119896)(

119896minus1

prod

119897=119895+1

minus 120590119911

119897) minus ℎ

119872

sum

119895=1

120590119911

119895

(38)

We then construct a path integral expression for thequantum partition function for (38) First we divide thequantum partition function into 119899 pieces

119885 = Tr 119890minus120573Hqu = Tr [119890minusΔ120591Hqu119890minusΔ120591

Hqu sdot sdot sdot 119890minusΔ120591

Hqu]

= TrV119899

(39)

where Δ120591 = 120573119899 and V = 119890minusΔ120591Hqu

Next we insert resolutions of the identity in the infiniteset of spin coherent states |N⟩ between each of the 119899 factorsin (39) such that we obtain

119885 = int sdot sdot sdot int

119872

prod

119894=1

119889N (120591119894) ⟨N (120591

119872)1003816100381610038161003816 119890

minusΔ120591H 1003816100381610038161003816N (120591119872minus1)⟩

sdot ⟨N (120591119872minus1)1003816100381610038161003816 119890

minusΔ120591H 1003816100381610038161003816N (120591119872minus2)⟩ sdot sdot sdot ⟨N (120591

1)1003816100381610038161003816

sdot 119890minusΔ120591H 1003816100381610038161003816N (120591119872)⟩

(40)

Taking the limit119872 rarr infin such that

⟨N (120591)| 119890minusΔ120591Hqu(S) |N (120591 minus Δ120591)⟩ = ⟨N (120591)|

sdot (1 minus Δ120591Hqu (S)) (|N (120591)⟩ minus Δ120591119889

119889120591|N (120591)⟩)

= ⟨N (120591) | N (120591)⟩ minus Δ120591 ⟨N (120591)| 119889119889120591|N (120591)⟩

minus Δ120591 ⟨N (120591)| Hqu (S) |N (120591)⟩ + 119874 ((Δ120591)2

)

= 119890minusΔ120591(⟨N(120591)|(119889119889120591)|N(120591)⟩+H(N(120591)))

Δ120591

119872

sum

119894=1

997888rarr int

120573

0

119889120591

119872

prod

119894=1

119889N (120591119894) 997888rarr DN (120591)

(41)

we can rewrite (40) as

119885 = int

N(120573)

N(0)

DN (120591) 119890minusint

120573

0119889120591H(N(120591))minusS119861 (42)

where H(N(120591)) now has the form of a Hamiltonian corre-sponding to a two-dimensional classical system and

S119861= int

120573

0

119889120591 ⟨N (120591)| 119889119889120591|N (120591)⟩ (43)

appears through the overlap between the coherent statesat two infinitesimally separated steps Δ120591 and is purelyimaginary This is the appearance of the Berry phase in theaction [18 19] Despite being imaginary this term gives thecorrect equation of motion for spin systems [19]

The coherent states for spin operators labeled by thecontinuous vector N in three dimensions can be visualisedas a classical spin (unit vector) pointing in direction N suchthat they have the property

⟨N| S |N⟩ = N (44)

They are constructed by applying a rotation operator to aninitial state to obtain all the other states as described in [18]such that we end up with

⟨N| 119878119894 |N⟩ = minus119878119873119894

(45)


with119873119894s given by

N = (119873119909

119873119910

119873119911

) = (sin 120579 cos120601 sin 120579 sin120601 cos 120579)

0 le 120579 le 120587 0 le 120601 le 2120587

(46)

Thus when our quantum Hamiltonian Hqu is given by(38) H(N(120591)) in (42) now has the form of a Hamiltoniancorresponding to a two-dimensional classical system given by

H (N (120591)) = ⟨N (120591)| Hqu |N (120591)⟩

= sum

1le119895le119896le119872

((119860119895119896+ 120574119861

119895119896)119873

119909

119895(120591)119873

119909

119896(120591)

+ (119860119895119896minus 120574119861

119895119896)119873

119910

119895(120591)119873

119910

119896(120591))

119896minus1

prod

119897=119895+1

(minus119873119911

119897(120591))

minus ℎ

119872

sum

119895=1

119873119911

119895(120591) = sum

1le119895le119896le119872

(119860119895119896

cos (120601119895(120591) minus 120601

119896(120591))

+ 119861119895119896120574 cos (120601

119895(120591) + 120601

119896(120591))) sin (120579

119895(120591))

sdot sin (120579119896(120591))

119896minus1

prod

119897=119895+1

(minus cos (120579119897(120591))) minus ℎ

119872

sum

119895=1

cos (120579119895(120591))

(47)

4 Simultaneous Diagonalisation ofthe Quantum Hamiltonian andthe Transfer Matrix

This section presents a particular type of equivalence betweenone-dimensional quantum and two-dimensional classicalmodels established by commuting the quantumHamiltonianwith the transfer matrix of the classical system under certainparameter relations between the corresponding systemsSuzuki [2] used this method to prove an equivalence betweenthe one-dimensional generalised quantum 119883119884 model andthe two-dimensional Ising and dimer models under specificparameter restrictions between the two systems In particularhe proved that this equivalence holds when the quantumsystem is restricted to nearest neighbour or nearest and nextnearest neighbour interactions

Here we extend the work of Suzuki [2] establishing thistype of equivalence between the class of quantum spin chains(1) for all interaction lengths when the system is restricted topossessing symmetries corresponding to that of the unitarygroup only7 and the two-dimensional Ising and dimermodelsunder certain restrictions amongst coupling parameters Forthe Ising model we use both transfer matrices forming twoseparate sets of parameter relations under which the systemsare equivalentWhere possible we connect critical propertiesof the corresponding systems providing a pathway withwhich to show that the critical properties of these classicalsystems are also influenced by symmetry

All discussions regarding the general class of quantumsystems (1) in this section refer to the family correspondingto 119880(119873) symmetry only in which case we find that

[HquVcl] = 0 (48)

under appropriate relationships amongst parameters of thequantum and classical systems when Vcl is the transfermatrix for either the two-dimensional Ising model withHamiltonian given by

H = minus

119873

sum

119894

119872

sum

119895

(1198691119904119894119895119904119894+1119895

+ 1198692119904119894119895119904119894119895+1) (49)

or the dimer modelA dimer is a rigid rod covering exactly two neighbouring

vertices either vertically or horizontally The model we referto is one consisting of a square planar lattice with119873 rows and119872 columns with an allowed configuration being when eachof the119873119872 vertices is covered exactly once such that

2ℎ + 2V = 119873119872 (50)

where ℎ and V are the number of horizontal and verticaldimers respectively The partition function is given by

119885 = sum

allowed configs119909ℎ

119910V= 119910

1198721198732

sum


(51)

where 119909 and 119910 are the appropriate ldquoactivitiesrdquo and 120572 = 119909119910The transform used to diagonalise both of these classical

systems as well as the class of quantum spin chains (1) can bewritten as

120578dagger

119902=119890minus1198941205874

radic119872sum

119895

119890minus(2120587119894119872)119902119895

(119887dagger

119895119906119902+ 119894119887

119895V119902)

120578119902=1198901198941205874

radic119872sum

119895

119890(2120587119894119872)119902119895

(119887119895119906119902minus 119894119887

dagger

119895V119902)

(52)

where the 120578119902s are the Fermi operators in which the systems

are left in diagonal form This diagonal form is given by (3)for the quantum system and for the transfer matrix for theIsing model by8 [20]

V+(minus)

= (2 sinh 21198701)1198732

119890minussum119902120598119902(120578dagger

119902120578119902minus12) (53)

where119870119894= 120573119869

119894and 120598

119902is the positive root of9

cosh 120598119902= cosh 2119870lowast

1cosh 2119870

2

minus sinh 2119870lowast

1sinh 2119870

2cos 119902

(54)

The dimer model on a two-dimensional lattice was firstsolved byKasteleyn [21] via a combinatorialmethod reducingthe problem to the evaluation of a Pfaffian Lieb [22] laterformulated the dimer-monomer problem in terms of transfermatrices such thatVcl = V2

119863is left in the diagonal form given

by

V2

119863

= prod

0le119902le120587

(120582119902(120578

dagger

119902120578119902+ 120578

dagger

minus119902120578minus119902minus 1) + (1 + 2120572

2sin2

119902)) (55)


with

120582119902= 2120572 sin 119902 (1 + 1205722sin2

119902)12

(56)

For the class of quantum spin chains (1) as well as eachof these classical models we have that the ratio of terms intransform (52) is given by

2V119902119906119902

1199062119902minus V2

119902

=

119886119902

119887119902

for Hqu

sin 119902cosh 2119870lowast

1cos 119902 minus sinh 2119870lowast

1coth 2119870

2

for V

sin 119902 (1 minus tanh 2119870lowast

1tanh119870

2cos 119902)

cos 119902 minus tanh1198702tanh 2119870lowast

1cos2119902 minus tanh 2119870lowast

1 sinh 2119870

2

for V1015840

minus1

120572 sin 119902for V2

119863

(57)

which as we show in the following sections will provide uswith relationships between parameters under which theseclassical systems are equivalent to the quantum systems

41The IsingModel with TransferMatrixV We see from (57)that the Hamiltonian (1) commutes with the transfer matrixV if we require that

119886119902

119887119902

=sin 119902

cosh 2119870lowast


1coth 2119870

2

(58)

This provides us with the following relations betweenparameters under which this equivalence holds10

sinh 2119870lowast

1coth 2119870

2= minus119886 (119871 minus 1)

119887 (119871)

tanh2119870lowast

1=119886 (119871) minus 119887 (119871)

119886 (119871) + 119887 (119871)

119886 (119871 minus 1)

119886 (119871) + 119887 (119871)= minus coth 2119870

2tanh119870lowast

1

(59)

or inversely as

cosh 2119870lowast

1=119886 (119871)

119887 (119871)

tanh 21198702= minus

1

119886 (119871 minus 1)

radic(119886 (119871))2

minus (119887 (119871))2

(60)

where

119886 (119871) = 119886 (119871)

[1198712]

sum

119897=0

(119871

2119897)

119887 (119871) = 119887 (119871)

[(119871minus1)2]

sum

119897=0

(119871

2119897 + 1)

119886 (0) = Γ

(61)

From (60) we see that this equivalence holds when119886 (119871)

119887 (119871)ge 1

1198862

(119871) le 1198862

(119871 minus 1) + 1198872

(119871)

(62)

For 119871 gt 1 we also have the added restrictions on theparameters that

119871

sum

119896=1

119887 (119896)

[(119871minus1)2]

sum

119897=0

(119896

2119897 + 1)

119897

sum

119894=1

(minus1)119894 cos119896minus2119894119902

+

119871minus1

sum

119896=1

119887 (119896) cos119896119902 = 0

(63)

Γ +

119871minus2

sum

119896=1

119886 (119896) cos119896119902

+

119871

sum

119896=1

119886 (119896)

[1198962]

sum

119897=0

(119896

2119897)

119897

sum

119894=1

(119897

119894) (minus1)

119894 cos119871minus2119894119902 = 0

(64)

which implies that all coefficients of cos119894119902 for 0 le 119894 lt 119871 in(63) and of cos119894119902 for 0 le 119894 lt 119871 minus 1 in (64) are zero11

When only nearest neighbour interactions are present in(1) (119871 = 1) with 119886(119896) = 119887(119896) = 0 for 119896 = 1 we recover Suzukirsquosresult [2]

The critical properties of the class of quantum systems canbe analysed from the dispersion relation (4) which under theabove parameter restrictions is given by

10038161003816100381610038161003816Λ

119902

10038161003816100381610038161003816= 2

119901+110038161003816100381610038161003816cos(119871minus1)11990210038161003816100381610038161003816

sdot ((119886 (119871) cos 119902 + 119886 (119871 minus 1))2 + 1198872

(119871) sin2

119902)

12

(65)

which is gapless for 119871 gt 1 for all parameter valuesThe critical temperature for the Ising model [20] is given

by

119870lowast

1= 119870

2 (66)


which using (59) and (60) gives

119886 (119871) = plusmn119886 (119871 minus 1) (67)

This means that (65) becomes

10038161003816100381610038161003816Λ

119902

10038161003816100381610038161003816= 2

119901+110038161003816100381610038161003816119886 (119871) cos(119871minus1)11990210038161003816100381610038161003816

sdot ((cos 119902 plusmn 1)2 + (119887 (119871)119886 (119871)

)

2

sin2

119902)

12

(68)

which is now gapless for all 119871 gt 1 and for 119871 = 1 (67)is the well known critical value for the external field for thequantum119883119884model

The correlation function between two spins in the samerow in the classical Ising model at finite temperature canalso be written in terms of those in the ground state of thequantum model

⟨120590119909

119895119896120590119909

119895+119903119896⟩Is= ⟨Ψ

0

1003816100381610038161003816Vminus12

1120590119909

119895120590119909

119895+119903V12

1

1003816100381610038161003816Ψ0⟩

= ⟨Φ0

1003816100381610038161003816Vminus12

1120590119909

119895120590119909

119895+119903V12

1

1003816100381610038161003816Φ0⟩

= ⟨(Vminus12

1120590119909

119895V12

1) (Vminus12

1120590119909

119895+119903V12

1)⟩

qu

= cosh2119870lowast

1⟨120590

119909

119895120590119909

119895+119903⟩qu

minus sinh2119870lowast

1⟨120590

119910

119895120590119910

119895+119903⟩qu

(69)

using the fact that ⟨120590119909119895120590119910

119895+119903⟩qu = ⟨120590

119910

119895120590119909

119895+119903⟩qu = 0 for 119903 = 0 and

Ψ0= Φ

0 (70)

from (3) (48) and (53) where Ψ0is the eigenvector corre-

sponding to the maximum eigenvalue of V and Φ0is the

ground state eigenvector for the general class of quantumsystems (1) (restricted to 119880(119873) symmetry)

This implies that the correspondence between criticalproperties (ie correlation functions) is not limited to quan-tum systems with short range interactions (as Suzuki [2]found) but also holds for a more general class of quantumsystems for a fixed relationship between the magnetic fieldand coupling parameters as dictated by (64) and (63) whichwe see from (65) results in a gapless system

42 The Ising Model with Transfer Matrix V1015840 From (57) theHamiltonian for the quantum spin chains (1) commutes withtransfer matrix V1015840 if we set119886119902

119887119902

=sin 119902 (1 minus tanh 2119870lowast

1tanh119870

2cos 119902)



1 sinh 2119870

2

(71)

This provides us with the following relations betweenparameters under which this equivalence holds when the

class of quantum spin chains (1) has an interaction length119871 gt 1

tanh 2119870lowast

1tanh119870

2= minus

119887 (119871)

119887 (119871 minus 1)= minus

119886 (119871)

119887 (119871 minus 1)

119886 (119871 minus 1)

119887 (119871 minus 1)= 1

tanh 2119870lowast

1

sinh 21198702

= minus119886lowast

(119871)

119887 (119871 minus 1)

(72)

or inversely as

sinh21198702=

119886 (119871)

2 (119886lowast

(119871))

tanh 2119870lowast

1= minus

1

119886 (119871 minus 1)radic119886 (119871) (2119886

lowast

(119871) + 119886 (119871))

(73)

where

119886lowast

(119871) = 119886 (119871 minus 2) minus 119886 (119871)

[1198712]

sum

119897=0

(119871

2119897) 119897 (74)

From (73) we see that this equivalence holds when

119886 (119871) (2119886lowast

(119871) + 119886 (119871)) le 1198862

(119871 minus 1) (75)

When 119871 gt 2 we have further restrictions upon theparameters of the class of quantum systems (1) namely

119871minus2

sum

119896=1

119887 (119896) cos119896119902

+

119871

sum

119896=1

119887 (119896)

[(119896minus1)2]

sum

119897=0

(119896

2119897 + 1)

119897

sum

119894=1

(119897

119894) (minus1)

119894 cos119896minus2119894119902

= 0

(76)

Γ +

119871minus3

sum

119896=1

119896cos119896119902 minus119871minus1

sum

119896=1

119886 (119896)

[1198962]

sum

119897=0

(119896

2119897) 119897cos119896minus2119902

+

119871

sum

119896=1

119886 (119896)

[1198962]

sum

119897=0

(119896

2119897)

119897

sum

119894=2

(119897

119894) (minus1)

119894 cos119896minus2119894119902 = 0

(77)

This implies that coefficients of cos119894119902 for 0 le 119894 lt 119871 minus 1 in (76)and of cos119894119902 for 0 le 119894 lt 119871 minus 2 in (77) are zero

Under these parameter restrictions the dispersion rela-tion is given by

10038161003816100381610038161003816Λ

119902

10038161003816100381610038161003816= 2

119901+110038161003816100381610038161003816cos119871minus211990210038161003816100381610038161003816

sdot ((cos 119902 (119886 (119871) cos 119902 + 119886 (119871 minus 1)) + 119886lowast (119871))2

+ sin2

119902 (119887 (119871) cos 119902 + 119887 (119871 minus 1)))12

(78)

which is gapless for 119871 gt 2 for all parameter values


The critical temperature for the Isingmodel (66) becomes

minus119886 (119871 minus 1) = 119886lowast

(119871) + 119886 (119871) (79)

using (72) and (73)Substituting (79) into (78) we obtain

10038161003816100381610038161003816Λ

119902

10038161003816100381610038161003816= 2

119901+110038161003816100381610038161003816cos119871minus211990210038161003816100381610038161003816

sdot ((119886 (119871) cos 119902 minus 119886lowast (119871))2 (cos 119902 minus 1)2

+ sin2

119902 (119887 (119871) cos 119902 + 119887 (119871 minus 1)))12

(80)

which we see is now gapless for all 119871 ge 2 (for 119871 = 2 this clearlycorresponds to a critical value of Γ causing the energy gap toclose)

In this case we can once again write the correlationfunction for spins in the same row of the classical Isingmodelat finite temperature in terms of those in the ground state ofthe quantum model as

⟨120590119909

119895119896120590119909

119895+119903119896⟩Is= ⟨120590

119909

119895120590119909

119895+119903⟩qu (81)

where Ψ1015840

0is the eigenvector corresponding to the maximum

eigenvalue of V1015840 and

Ψ1015840

0= Φ

0 (82)

Once more this implies that the correspondence betweencritical properties such as correlation functions is not limitedto quantum systems with short range interactions it alsoholds for longer range interactions for a fixed relationshipbetween the magnetic field and coupling parameters whichcauses the systems to be gapless

43 The Dimer Model with Transfer Matrix V2

119863 In this case

when the class of quantum spin chains (1) has a maximuminteraction length 119871 gt 1 it is possible to find relationshipsbetween parameters for which an equivalence is obtainedbetween it and the two-dimensional dimer model For detailsand examples see Appendix C When 119886(119896) = 119887(119896) = 0 for119896 gt 2 we recover Suzukirsquos result [2]

Table 1The structure of functions 119886(119895) and 119887(119895) dictating the entriesof matrices A = A minus 2ℎI and B = 120574B which reflect the respectivesymmetry groups The 119892

119897s are the Fourier coefficients of the symbol

119892M(120579) ofM

119872 Note that for all symmetry classes other than 119880(119873)

120574 = 0 and thus B = 0

Classicalcompact group

Structure of matrices Matrix entries119860

119895119896(119861

119895119896) (M

119899)119895119896

119880(119873) 119886(119895 minus 119896) (119887(119895 minus 119896)) 119892119895minus119896

119895 119896 ge 0119874

+

(2119873) 119886(119895 minus 119896) + 119886(119895 + 119896) 1198920if 119895 = 119896 = 0radic2119892

119897if

either 119895 = 0 119896 = 119897or 119895 = 119897 119896 = 0

119892119895minus119896+ 119892

119895+119896 119895 119896 gt 0

Sp(2119873) 119886(119895 minus 119896) minus 119886(119895 + 119896 + 2) 119892119895minus119896minus 119892

119895+119896+2 119895 119896 ge 0

119874plusmn

(2119873 + 1) 119886(119895 minus 119896) ∓ 119886(119895 + 119896 + 1) 119892119895minus119896∓ 119892

119895+119896+1 119895 119896 ge 0

119874minus

(2119873 + 2) 119886(119895 minus 119896) minus 119886(119895 + 119896 + 2) 119892119895minus119896minus 119892

119895+119896+2 119895 119896 ge 0

Appendices

A Symmetry Classes

See Table 1

B Longer Range Interactions

B1 Nearest and Next Nearest Neighbour Interactions Theclass of quantum systems (1) with nearest and next nearestneighbour interactions can be mapped12 onto

Hqu = minus119872

sum

119895=1

(119869119909

119895120590119909

119895120590119909

119895+1+ 119869

119910

119895120590119910

119895120590119910

119895+1

minus (1198691015840119909

119895120590119909

119895120590119909

119895+2+ 119869

1015840119910

119895120590119910

119895120590119910

119895+2) 120590

119911

119895+1+ ℎ120590

119911

119895)

(B1)

where 1198691015840119909119895= (12)(119860

119895119895+2+ 120574119861

119895119895+2) and 1198691015840119910

119895= (12)(119860

119895119895+2minus

120574119861119895119895+2) using the Jordan Wigner transformations (10)

We apply the Trotter-Suzuki mapping to the partitionfunction for (B1) with operators in the Hamiltonian orderedas


Tr [119890(120573qu119899)H119909

119886 119890(120573qu2119899)H

119911

119890(120573qu119899)H

119910

119887 119890(120573qu119899)H

119910

119886 119890(120573qu2119899)H

119911

119890(120573qu119899)H

119909

119887 ]119899

(B2)

where again 119886 and 119887 are the set of odd and even integersrespectively and H120583

120572= sum

119872

119895isin120572((12)119869

120583

119895(120590

120583

119895120590120583

119895+1+ 120590

120583

119895+1120590120583

119895+2) minus

1198691015840120583

119895120590120583

119895120590119911

119895+1120590120583

119895+2) and H119911

= ℎsum119872

119895=1120590119911

119895 for 120583 isin 119909 119910 and once

more 120572 denotes either 119886 or 119887

For thismodel we need to insert 4119899 identity operators into(B2) We use 119899 in each of the 120590119909 and 120590119910 bases and 2119899 in the120590119911 basis in the following way


Tr [I1205901119890(120573qu119899)H

119909

119886 I1199041119890(120573qu2119899)H

119911

119890(120573qu119899)H

119910

119887 I1205911119890(120573qu119899)H

119910

119886 I1199041119890(120573qu2119899)H

119911

119890(120573qu119899)H

119909

119887 ]119899


sum

120590119895119901 120591119895119901119904119895119901

119899

prod

119901=1

[⟨119901

10038161003816100381610038161003816119890(120573qu119899)H

119909

119886100381610038161003816100381610038161199042119901minus1⟩ ⟨ 119904

2119901minus1

10038161003816100381610038161003816119890(120573qu2119899)H

119911

119890(120573qu119899)H

119910

119887

10038161003816100381610038161003816120591119901⟩ ⟨ 120591

119901

10038161003816100381610038161003816119890(120573qu119899)H

119910

119886100381610038161003816100381610038161199042119901⟩ ⟨ 119904

2119901

10038161003816100381610038161003816119890(120573qu2119899)H

119911

119890(120573qu119899)H

119909

11988710038161003816100381610038161003816119901+1⟩]

(B3)


For this system it is then possible to rewrite the remainingmatrix elements in (B3) in complex scalar exponential formby first writing

⟨119901

10038161003816100381610038161003816119890(120573119899)

H119909119886100381610038161003816100381610038161199042119901minus1⟩ ⟨ 119904

2119901minus1

10038161003816100381610038161003816119890(120573qu2119899)H

119911

119890(120573qu119899)H

119910

119887

10038161003816100381610038161003816120591119901⟩ ⟨ 120591

2119901minus1

10038161003816100381610038161003816119890(120573qu119899)H

119910

119886100381610038161003816100381610038161199042119901⟩ ⟨ 119904

2119901

10038161003816100381610038161003816119890(120573qu2119899)H

119911

119890(120573qu119899)H

119909

119887100381610038161003816100381610038162119901⟩

= 119890(120573qu119899)H

119909

119886(119901)

119890(120573qu2119899)H

119911(2119901minus1)

119890(120573qu119899)H

119910

119887(119901)

119890(120573qu119899)H

119910

119886(119901)

119890(120573qu2119899)H

119911(2119901)

119890(120573qu119899)H

119909

119887(119901)

⟨119901| 119904

2119901minus1⟩ ⟨ 119904

2119901minus1| 120591

119901⟩ ⟨ 120591

119901| 119904

2119901⟩

sdot ⟨ 1199042119901|

119901+1⟩

(B4)

where H119909

120572(119901) = sum

119872

119895isin120572((12)119869

119909

119895(120590

119895119901120590119895+1119901

+ 120590119895+1119901

120590119895+2119901

) +

1198691015840119909

119895+1120590119895119901119904119895+1119901

120590119895+2119901

) H119910

120572(119901) = sum

119872

119895isin120572((12)119869

119910

119895(120591

119895119901120591119895+1119901

+

120591119895+1119901

120591119895+2119901

) + 1198691015840119910

119895+1120591119895119901119904119895+1119901

120591119895+2119901

) andH119911

(119901) = sum119872

119895=1119904119895119901 We

can then evaluate the remaining matrix elements as

⟨119901| 119904

2119901minus1⟩ ⟨ 119904

2119901minus1| 120591

119901⟩ ⟨ 120591

2119901minus1| 119904

2119901⟩ ⟨ 119904

2119901|

119901+1⟩

=1

24119872

sdot

119872

prod

119895=1

119890(1198941205874)(minus1199041198952119901minus1+1199041198952119901+1205901198951199011199041198952119901minus1minus120590119895119901+11199042119901+120591119895119901(1199041198952119901minus1199041198952119901minus1))

(B5)

Thus we obtain a partition function with the same formas that corresponding to a class of two-dimensional classicalIsing type systems on119872times4119899 latticewith classicalHamiltonianHcl given by

minus 120573clHcl =120573qu

119899

119899

sum

119901=1

(sum

119895isin119886

(119869119909

119895

2(120590

119895119901120590119895+1119901

+ 120590119895+1119901

120590119895+2119901

) minus 1198691015840119909

119895+1120590119895119901119904119895+1119901

120590119895+2119901

)

+sum

119895isin119887

(

119869119910

119895

2(120591

119895119901120591119895+1119901

+ 120591119895+1119901

120591119895+2119901

) minus 1198691015840119910

119895+1120591119895119901119904119895+12119901minus1

120591119895+2119901

)

+ sum

119895isin119886

(

119869119910

119895

2(120591

119895119901120591119895+1119901

+ 120591119895+1119901

120591119895+2119901

) minus 1198691015840119910

119895+1120591119895119901119904119895+12119901

120591119895+2119901

)

+sum

119895isin119887

(119869119909

119895

2(120590

119895119901+1120590119895+1119901+1

+ 120590119895+1119901+1

120590119895+2119901+1

) minus 1198691015840119909

119895+1120590119895119901+1

119904119895+12119901

120590119895+2119901+1

))

+

119899

sum

119901=1

(

119872

sum

119895=1

((120573quℎ

2119899minus119894120587

4) 119904

1198952119901+ (120573quℎ

2119899+119894120587

4) 119904

1198952119901) +

119872

sum

119895=1

119894120587

4(120590

1198951199011199041198952119901minus1

minus 120590119895119901+1

1199042119901+ 120591

119895119901(119904

1198952119901minus 119904

1198952119901minus1)))

+ 4119899119872 ln 2

(B6)

A schematic representation of this model on a two-dimensional lattice is given in Figure 5 with a yellowborder representing a unit cell which can be repeated ineither direction The horizontal and diagonal blue and redlines represent interaction coefficients 119869119909 1198691015840119909 and 119869119910 1198691015840119910respectively and the imaginary interaction coefficients arerepresented by the dotted green linesThere is also a complexmagnetic field term ((120573qu2119899)ℎ plusmn 1198941205874) applied to each site inevery second row as represented by the black circles

This mapping holds in the limit 119899 rarr infin whichwould result in coupling parameters (120573qu119899)119869

119909 (120573qu119899)119869119910

(120573qu119899)1198691015840119909 (120573qu119899)119869

1015840119910 and (120573qu119899)ℎ rarr 0 unless we also take120573qu rarr infin Therefore this gives us a connection between theground state properties of the class of quantum systems andthe finite temperature properties of the classical systems

Similarly to the nearest neighbour case the partitionfunction for this extended class of quantum systems can alsobe mapped to a class of classical vertex models (as we saw forthe nearest neighbour case in Section 21) or a class of classicalmodels with up to 6 spin interactions around a plaquette withsome extra constraints applied to the model (as we saw forthe nearest neighbour case in Section 21) We will not give


S1

S2

S3

S4

1205901

1205911

1205902

1205912

1 2 3 4 5 6 7 8


Trotter

direction

p darr

Figure 5 Lattice representation of a class of classical systemsequivalent to the class of quantum systems (1) restricted to nearestand next nearest neighbours

the derivation of these as they are quite cumbersome andfollow the same steps as outlined previously for the nearestneighbour cases and instead we include only the schematicrepresentations of possible equivalent classical lattices Theinterested reader can find the explicit computations in [23]

Firstly in Figure 6 we present a schematic representationof the latter of these two interpretations a two-dimensionallattice of spins which interact with up to 6 other spins aroundthe plaquettes shaded in grey

To imagine what the corresponding vertex models wouldlook like picture a line protruding from the lattice pointsbordering the shaded region and meeting in the middle ofit A schematic representation of two possible options for thisis shown in Figure 7

B2 Long-Range Interactions For completeness we includethe description of a classical system obtained by apply-ing the Trotter-Suzuki mapping to the partition functionfor the general class of quantum systems (1) without anyrestrictions

We can now apply the Trotter expansion (7) to the quan-tum partition function with operators in the Hamiltonian(38) ordered as


Tr[

[

119872minus1

prod

119895=1

(119890(120573qu119899)H

119909

119895119895+1119890(120573qu119899)H

119909

119895119895+2 sdot sdot sdot 119890(120573qu119899)H

119909

119895119872119890(120573qu2119899(119872minus1))

H119911119890(120573qu119899)H

119910

119895119872 sdot sdot sdot 119890(120573qu119899)H

119910

119895119895+2119890(120573qu119899)H

119910

119895119895+1)]

]

119899


Tr[

[

119872

prod

119895=1

((

119872minus119895

prod

119896=1

119890(120573qu119899)H

119909

119895119895+119896)119890(120573qu2119899(119872minus1))

H119911(

119872minus119895minus1

prod

119896=0

119890(120573qu119899)H

119910

119895119872minus119896))]

]

119899

(B7)

where H120583

119895119896= 119869

120583

119895119896120590120583

119895120590120583

119896prod

119896minus1

119897=1(minus120590

119911

119897) for 120583 isin 119909 119910 and H119911

=

ℎsum119872

119895=1120590119911

119895

For this model we need to insert 3119872119899 identity operators119899119872 in each of the 120590119909 120590119910 and 120590119911 bases into (B7) in thefollowing way


Tr[

[

119872minus1

prod

119895=1

(I120590119895(

119872minus119895

prod

119896=1

119890(120573qu119899)H

119909

119895119895+119896)119890(120573qu(119872minus1)119899)

H119911I119904119895(


prod

119896=0

119890(120573qu119899)H

119910

119895119872minus119896) I120591119895)]

]

119899


sum

120590119895119901 120591119895119901

119899minus1

prod

119901=0

119872minus1

prod

119895=1

(⟨119895+119895119901

10038161003816100381610038161003816(

119872minus119895

prod

119896=1

119890(120573qu119899)H

119909

119895119895+119896)119890(120573qu119899(119872minus1))

H119911 10038161003816100381610038161003816119904119895+119895119901⟩ ⟨ 119904

119895+119895119901

10038161003816100381610038161003816(


prod

119896=0

119890(120573qu119899)H

119910

119895119872minus119896)10038161003816100381610038161003816120591119895+119895119901⟩ ⟨ 120591

119895+119895119901|

119895+119895119901+1⟩)

(B8)


⟨119895+119895119901

10038161003816100381610038161003816(

119872minus119895

prod

119896=1

119890(120573qu119899)H

119909

119895119895+119896)119890(120573qu119899(119872minus1))

H119911 10038161003816100381610038161003816119904119895+119895119901⟩

sdot ⟨ 119904119895+119895119901

10038161003816100381610038161003816(


prod

119896=0

119890(120573qu119899)H

119910

119895119872minus119896)10038161003816100381610038161003816120591119895+119895119901⟩ ⟨ 120591

119895+119895119901|

119895+119895119901+1

⟩

= 119890(120573qu119899)sum

119872minus119895

119896=1(H119909119895119895+119896

(119901)+H119910

119895119895+119896(119901)+(1119899(119872minus1))H119911)

⟨119895+119895119901

|

119904119895+119895119901⟩ ⟨ 119904

119895+119895119901| 120591

119895+119895119901⟩ ⟨ 120591

119895+119895119901|

119895+119895119901+1⟩

(B9)


S1

S2

S3

S4

S5

S6

S7

S8

S9

S10

1 2 3 4 5 6 7 8 9


Trotter

direction

p darr

Figure 6 Lattice representation of a class of classical systems equivalent to the class of quantum systems (1) restricted to nearest and nextnearest neighbour interactions The shaded areas indicate which particles interact together

Figure 7 Possible vertex representations

where H119909

119895119896(119901) = sum

119872

119896=119895+1119869119909

119895119896120590119895119901120590119896119901prod

119896minus1

119897=119895+1(minus119904

119897119901) H119910

119895119896(119901) =

sum119872

119896=119895+1119869119910

119895119896120591119895119901120591119896119901prod

119896minus1

119897=119895+1(minus119904

119897119901) andH119911

119901= ℎsum

119872

119895=1120590119911

119895119901 Finally

evaluate the remaining terms as

⟨119901| 119904

119901⟩ ⟨ 119904

119901| 120591

119901⟩ ⟨ 120591

119901|

119901+1⟩ = (

1

2radic2)

119872

sdot

119872

prod

119895=1

119890(1198941205874)((1minus120590119895119901)(1minus119904119895119901)+120591119895119901(1minus119904119895119901)minus120590119895119901+1120591119895119901)

(B10)

The partition function now has the same form as that of aclass of two-dimensional classical Isingmodels on a119872times3119872119899lattice with classical HamiltonianHcl given by

minus 120573clHcl =119899minus1

sum

119901=1

119872

sum

119895=1

(120573qu

119899

119872

sum

119896=119895+1

(119869119909

119895119896120590119895119895+119895119901

120590119896119895+119895119901

+ 119869119910

119895119896120591119895119895+119895119901

120591119896119895+119895119901

)

119896minus1

prod

119897=119895+1

(minus119904119897119901) + (

120573qu

119899 (119872 minus 1)ℎ minus119894120587

4) 119904

119895119895+119895119901

+119894120587

4(1 minus 120590

119895119895+119895119901+ 120591

119895119895+119895119901+ 120590

119895119895+119895119901119904119895119895+119895119901

minus 120591119895119895+119895119901

119904119895119895+119895119901

minus 120590119895119895+119895119901+1

120591119895119895+119895119901

)) + 1198991198722 ln 1

2radic2

(B11)

A schematic representation of this class of classical sys-tems on a two-dimensional lattice is given in Figure 8 wherethe blue and red lines represent interaction coefficients 119869119909

119895119896

and 119869119910119895119896 respectively the black lines are where they are both

present and the imaginary interaction coefficients are givenby the dotted green lines The black circles also represent

a complex field ((120573qu119899(119872 minus 1))ℎ minus 1198941205874) acting on eachindividual particle in every second row

This mapping holds in the limit 119899 rarr infin which wouldresult in coupling parameters (120573qu119899)119869

119909

119895119896 (120573qu119899)119869

119910

119895119896 and

(120573qu119899)ℎ rarr 0 unless we also take 120573qu rarr infin Therefore thisgives us a connection between the ground state properties of


1205901

S1

1205911

S2

1205902

S3

1205912

S4

1205903

S5

1 2 3 4 5 6 7 8 9 10

Trotter

direction

p darr


Figure 8 Lattice representation of a classical system equivalent tothe general class of quantum systems

the quantum system and the finite temperature properties ofthe classical system

C Systems Equivalent to the Dimer Model

We give here some explicit examples of relationships betweenparameters under which our general class of quantum spinchains (1) is equivalent to the two-dimensional classical dimermodel using transfer matrix V2

119863(55)

(i) When 119871 = 1 from (57) we have

minus1

120572 sin 119902=119887 (1) sin 119902

Γ + 119886 (1) cos 119902 (C1)

therefore it is not possible to establish an equivalencein this case

(ii) When 119871 = 2 from (57) we have

minus1

120572 sin 119902=

119887 (1)

minus2119886 (2) sin 119902

if Γ = minus119886 (2) 119886 (1) = 119887 (2) = 0

(C2)

Thus the systems are equivalent under the parameterrelations

120572 =2119886 (2)

119887 (1) Γ = minus119886 (2) 119886 (1) = 119887 (2) = 0 (C3)

(iii) When 119871 = 3 from (57) we have

minus1

120572 sin 119902= minus

119887 (1) minus 119887 (3) + 119887 (2) cos 1199022 sin 119902 (119886 (2) + 119886 (3) cos 119902)

if Γ = minus119886 (2) 119886 (1) = minus119886 (3) 119887 (3) = 0

(C4)


120572 =2119886 (3)

119887 (2)

119886 (2)

119886 (3)=119887 (1) minus 119887 (3)

119887 (2)

Γ = minus119886 (2) 119886 (1) = minus119886 (3) 119887 (3) = 0

(C5)

Therefore we find that in general when 119871 gt 1 we can use(57) to prove that we have an equivalence if

minus1

120572 sin 119902

=sin 119902sum119898

119896=1119887 (119896)sum

[(119896minus1)2]

119897=0( 119896

2119897+1)sum

119897

119894=0( 119897119894) (minus1)

minus119894 cos119896minus2119894minus1119902Γ + 119886 (1) cos 119902 + sum119898

119896=2119886 (119896)sum

[1198962]

119897=0(minus1)

119897

( 119896

2119897) sin2119897

119902cos119896minus2119897119902

(C6)

We can write the sum in the denominator of (C6) as

[1198982]

sum

119895=1

119886 (2119895) + cos 119902[1198982]

sum

119895=1

119886 (2119895 + 1) + sin2

119902

sdot (

[1198982]

sum

119895=1

119886 (2119895)

119895

sum

119894=1

(119895

119894) (minus1)

119894 sin2(119894minus1)

119902

+ cos 119902[(119898minus1)2]

sum

119895=1

119886 (2119895 + 1)

119895

sum

119894=1

(119895

119894) (minus1)

119894 sin2(119894minus1)

119902

+

119898

sum

119896=2

119886 (119896)

[1198962]

sum

119897=1

(minus1)119897

(119896

2119897) sin2(119897minus1)

119902cos119896minus2119897119902)

(C7)

This gives us the following conditions

Γ = minus

[1198982]

sum

119895=1

119886 (2119895)

119886 (1) = minus

[(119898+1)2]

sum

119895=1

119886 (2119895 + 1) = 0

(C8)


We can then rewrite the remaining terms in the denomi-nator (C7) as

sin2

119902(

[1198982]

sum

119895=1

119886 (2119895)

119895

sum

119894=1

(119895

119894)

119894minus1

sum

119901=0

(minus1)119894+119901 cos2119901119902

+

[(119898minus1)2]

sum

119895=1

119886 (2119895 + 1)

119895

sum

119894=1

(119895

119894)

119894minus1

sum

119901=0

(minus1)119894+119901 cos2119901+1119902 +

[(119898minus1)2]

sum

119895=1

119886 (2119895 + 1)

119895

sum

119897=1

(2119895 + 1

2119897)

119897minus1

sum

119901=0

(minus1)119901+119897 cos2(119895minus119901minus1)+1119902

+

[1198982]

sum

119895=1

119886 (2119895)

119895

sum

119897=1

(2119895

2119897)

119897minus1

sum

119901=0

(minus1)119901+119897 cos2(119895minus119901minus1)119902)

(C9)

Finally we equate coefficients of matching powers ofcos 119902 in the numerator in (C6) and denominator (C9) Forexample this demands that 119887(119898) = 0

Disclosure

No empirical or experimental data were created during thisstudy

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors are grateful to Professor Shmuel Fishman forhelpful discussions and to Professor Ingo Peschel for bringingsome references to their attention J Hutchinson is pleased tothank Nick Jones for several insightful remarks the EPSRCfor support during her PhD and the Leverhulme Trustfor further support F Mezzadri was partially supported byEPSRC research Grant EPL0103051

Endnotes

1 The thickness 119870 of a band matrix is defined by thecondition 119860

119895119896= 0 if |119895 minus 119896| gt 119870 where 119870 is a positive

integer

2 For the other symmetry classes see [8]

3 This is observed through the structure of matrices 119860119895119896

and 119861119895119896

summarised in Table 1 inherited by the classicalsystems

4 We can ignore boundary term effects since we areinterested in the thermodynamic limit only

5 Up to an overall constant

6 Recall from the picture on the right in Figure 2 that the120590 and 120591 represent alternate rows of the lattice

7 Thus matrices 119860119895119896

and 119861119895119896

have Toeplitz structure asgiven by Table 1

8 The superscripts +(minus) represent anticyclic and cyclicboundary conditions respectively

9 This is for the symmetrisation V = V12

1V

2V12

1of

the transfer matrix the other possibility is with V1015840

=

V12

2V

1V12

2 whereV

1= (2 sinh 2119870

1)1198722

119890minus119870lowast

1sum119872

119894120590119909

119894 V2=

1198901198702 sum119872

119894=1120590119911

119894120590119911

119894+1 and tanh119870lowast

119894= 119890

minus2119870119894 10 Here we have used De Moivrersquos Theorem and the

binomial formula to rewrite the summations in 119886119902and

119887119902(5) as

119886119902= Γ +

119871

sum

119896=1

119886 (119896)

[1198962]

sum

119897=0

(119896

2119897)

119897

sum

119894=0

(119897

119894) (minus1)

minus119894 cos119896minus2119894119902

119887119902= tan 119902

sdot

119871

sum

119896=1

119887 (119896)

[(119896minus1)2]

sum

119897=0

(119896

2119897 + 1)

119897

sum

119894=0

(119897

119894) (minus1)

minus119894 cos119896minus2119894119902

(lowast)

11 For example setting the coefficient of (cos 119902)0 to zeroimplies that Γ = minussum[(119871minus1)2]

119895=1(minus1)

119895

119886(2119895)

12 Once again we ignore boundary term effects due to ourinterest in phenomena in the thermodynamic limit only

References

[1] R J Baxter ldquoOne-dimensional anisotropic Heisenberg chainrdquoAnnals of Physics vol 70 pp 323ndash337 1972

[2] M Suzuki ldquoRelationship among exactly soluble models ofcritical phenomena Irdquo Progress of Theoretical Physics vol 46no 5 pp 1337ndash1359 1971

[3] M Suzuki ldquoRelationship between d-dimensional quantal spinsystems and (119889 + 1)-dimensional Ising systemsrdquo Progress ofTheoretical Physics vol 56 pp 1454ndash1469 1976

[4] D P Landau and K BinderAGuide toMonte Carlo Simulationsin Statistical Physics Cambridge University Press 2014


[5] M SuzukiQuantumMonte CarloMethods inCondensedMatterPhysics World Scientific 1993

[6] J Hutchinson J P Keating and F Mezzadri ldquoRandom matrixtheory and critical phenomena in quantum spin chainsrdquo Physi-cal Review E vol 92 no 3 Article ID 032106 2015

[7] E Lieb T Schultz and D Mattis ldquoTwo soluble models of anantiferromagnetic chainrdquo Annals of Physics vol 16 no 3 pp407ndash466 1961

[8] J P Keating and F Mezzadri ldquoRandom matrix theory andentanglement in quantum spin chainsrdquo Communications inMathematical Physics vol 252 no 1ndash3 pp 543ndash579 2004

[9] J P Keating and F Mezzadri ldquoEntanglement in quantum spinchains symmetry classes of random matrices and conformalfield theoryrdquo Physical Review Letters vol 94 no 5 Article ID050501 2005

[10] A Altland and M R Zirnbauer ldquoRandom matrix theory of achaotic Andreev quantum dotrdquo Physical Review Letters vol 76no 18 pp 3420ndash3423 1996

[11] A Altland and M R Zirnbauer ldquoNonstandard symmetryclasses in mesoscopic normal-superconducting hybrid struc-turesrdquo Physical Review B vol 55 no 2 pp 1142ndash1161 1997

[12] M R Zirnbauer ldquoRiemannian symmetric superspaces andtheir origin in random-matrix theoryrdquo Journal of MathematicalPhysics vol 37 no 10 pp 4986ndash5018 1996

[13] S Krinsky ldquoEquivalence of the free fermion model to theground state of the linear XY modelrdquo Physics Letters A vol 39no 3 pp 169ndash170 1972

[14] I Peschel ldquoOn the correlation functions of fully frustrated two-dimensional Ising systemsrdquo Zeitschrift fur Physik B CondensedMatter vol 45 no 4 pp 339ndash344 1982

[15] K Minami ldquoEquivalence between the two-dimensional Isingmodel and the quantum XY chain with randomness and withopen boundaryrdquo EPL vol 108 no 3 Article ID 30001 2014

[16] F Igloi and P Lajko ldquoSurface magnetization and surfacecorrelations in aperiodic Isingmodelsrdquo Journal of Physics A vol29 no 16 pp 4803ndash4814 1996

[17] M Barma and B S Shastry ldquoClassical equivalents of one-dimensional quantum-mechanical systemsrdquo Physical Review Bvol 18 no 7 article 3351 1978

[18] S Sachdev Quantum Phase Transitions Wiley Online Library2007

[19] X G Wen Quantum Field Theory of Many-Body Systems fromthe Origin of Sound to an Origin of Light and Electrons OxfordUniversity Press New York NY USA 2004

[20] T D Schultz D C Mattis and E H Lieb ldquoTwo-dimensionalIsing model as a soluble problem of many fermionsrdquo Reviews ofModern Physics vol 36 pp 856ndash871 1964

[21] PW Kasteleyn ldquoDimer statistics and phase transitionsrdquo Journalof Mathematical Physics vol 4 no 2 pp 287ndash293 1963

[22] EH Lieb ldquoSolution of the dimer problemby the transfermatrixmethodrdquo Journal of Mathematical Physics vol 8 no 12 pp2339ndash2341 1967

[23] J Hutchinson Random matrix theory and critical phenomena[PhD thesis] University of Bristol 2014

Submit your manuscripts athttpwwwhindawicom

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Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


containing only real entries without loss of generality andperiodic boundary conditions 119887

119872+119895= 119887

119895are assumed We

can think of119860119895119896

and 119861119895119896

as band matrices whose thickness1determines the length of the spin-spin interaction

This model can be diagonalised [7] so that

H = sum

119902

10038161003816100381610038161003816Λ

119902

10038161003816100381610038161003816120578dagger

119902120578119902+ 119862 (3)

with the dispersion relation |Λ119902| determined bymatrices119860

119895119896

and 119861119895119896 the 120578

119902s are Fermi operators and 119862 is a constant

In [8 9] Keating andMezzadri restricted theHamiltonian(1) to possess symmetries corresponding to the Haarmeasureof each of the classical compact groups 119880(119873) 119874+

(2119873)Sp(2119873) 119874+

(2119873 + 1) 119874minus

(2119873 + 1) and 119874minus

(2119873 + 2) enablingthe calculation of |Λ

119902| using techniques from randommatrix

theory This corresponds to a symmetry classification of spinchains similar to that introduced for disordered systems byAltland and Zirnbauer [10ndash12] These symmetry propertieswere encoded into the structure of the matrices119860

119895119896and 119861

119895119896

as summarised in Appendix A For example when restrictedto 119880(119873) symmetry2 [8 9]

Λ119902= 4(Γ +

119871

sum

119896=1

(119886 (119896) cos 119896119902 + 119894119887 (119896) sin 119896119902))

= 4 (119886119902+ 119894119887

119902)

(4)

where 119871 = [(119872 minus 1)2] with [ ] denoting the integer part ofits argument The real and imaginary parts of (4) are

119886119902= Γ +

119871

sum

119896=1

119886 (119896) cos 119896119902

119887119902=

119871

sum

119896=1

119887 (119896) sin 119896119902

(5)

Furthermore

Γ =1

2

119886 (0) if 119872 is odd

119886 (0) + (minus1)119897

119886 (119872

2) if 119872 is even

119902 =2120587119897

119872 119897 = 0 119872 minus 1

(6)

In general the symmetry constraints were achieved using realfunctions 119886(119895) and 119887(119895) even and odd functions of Z119872Zrespectively to dictate the entries of matrices119860

119895119896and 119861

119895119896 as

reported in Appendix AExploiting the formalism developed in [8 9] enabled us

to compute the critical properties of this class of spin chains[6] demonstrating a dependence of the critical exponents onsystem symmetries and establishing universality for this classof quantum systems

Having established universality for the above class ofquantum spin chains in [6] we now make use of QCmappings to obtain a class of classical systemswith equivalent

critical properties establishing universality for this class ofclassical systems as well by extension This is our main goal

There is no systematic technique to construct a classi-cal 119889 + 1-dimensional lattice system from a quantum 119889-dimensional one there is no alternative but to develop an adhoc approach for each case This is usually a major challengeFurthermore such a mapping is not unique However overthe years severalmodels have been proved equivalent Suzuki[3] introduced a powerfulmethod based onTrotterrsquos formulaAnother technique exploits the fact that if the quantumHamiltonian commutes with the transfer matrix of a classicalsystem then they are equivalent This idea was used bySuzuki [2] to prove the equivalence between the generalisedquantum 119883119884 model and the two-dimensional Ising anddimer models Krinsky [13] showed that the eight-vertex freefermion model with an electric field is equivalent to theground state of the 119883119884 model in the presence of a magneticfield Peschel [14] demonstrated that the quantum119883119884modelcan be mapped to Ising type models with three differentfrustrated lattice structures Lifting the translational invari-ance Minami [15] proved the equivalence of the 119883119884 modelto a class of two-dimensional Ising models with nonuniforminteraction coefficients Igloi and Lajko [16] showed that thequantum Ising model with site-dependent coupling param-eters in a transverse magnetic field is equivalent to an Isingmodel on a square lattice with a diagonally layered structureSome of these systems are not translation invariant but theyall have nearest neighbour spin-spin interactions to ourknowledge there is no system with long-range interactionsfor which classical-quantum equivalence has previously beenproved

A quantum and a classical system are equivalent if theirpartition functions are the same such a correspondencehowever is not unique as different classical systems canbe equivalent to the same quantum system We will hereadopt the following different approaches to map the partitionfunctions of the quantum spin chains (1) onto those of ageneral class of two-dimensional classical systems

(i) The Trotter-Suzuki formula (Section 2)(ii) The method of coherent states (Section 3)(iii) The simultaneous diagonalisation of the quantum

Hamiltonian and the transfer matrix for the classicalsystem (Section 4)

2 Trotter-Suzuki Mapping

This approach was developed by Suzuki [3] who applied theTrotter product formula

119890119860+119861


(119890119860119899

119890119861119899

)119899

[119860 119861] = 0 (7)

to map the partition function for a 119889-dimensional quantumsystem to that for a (119889 + 1)-dimensional classical one Inparticular he applied it to the partition function of a 119889-dimensional quantum Ising model in a transverse magneticfield mapping it to that of a (119889 + 1)-dimensional classicalIsingmodel [3] He then proved the equivalence of the critical





119885119860= sum


119894119895) (8a)

119885119861= sum


119885119862= sum


119894

120596119894 (8c)

119885119863= sum



119894119895=


119894119895 120591

119894119895 119904



119894 and 120590119911



119894119895 119904





119894119895) =


119894




H119883119884

= minus

119872

sum

119895=1

(119869119909

119895120590119909

119895120590119909

119895+1+ 119869

119910

119895120590119910

119895120590119910

119895+1+ ℎ120590

119911

119895) (9)

where 119869119909119895= minus(12)(119860

119895119895+1+ 120574119861

119895119895+1) 119869119910

119895= minus(12)(119860

119895119895+1minus


transformations

119887dagger

119895=1

2(119898

2119895+1+ 119894119898

2119895) =

1

2(120590

119909

119895+ 119894120590

119910

119895)

119895minus1

prod

119897=1

(minus120590119911

)

119887119895=1

2(119898

2119895+1minus 119894119898

2119895) =

1

2(120590

119909

119895minus 119894120590

119910

119895)

119895minus1

prod

119897=1

(minus120590119911

119897)

(10)

where

1198982119895+1

= 120590119909

119895

119895minus1

prod

119897=0

(minus120590119911

119895)

1198982119895= 120590

119910

119895

119895minus1

prod

119897=0

(minus120590119911

119895)

(11)

or inversely as

120590119911

119895= 119894119898

2119895119898

2119895+1

120590119909

119895= 119898

2119895+1

119895minus1

prod

119897=0

(minus1198941198982119897119898

2119897+1)

120590119910

119895= 119898

2119895

119895minus1

prod

119897=0

(minus1198941198982119897119898

2119897+1)

(12)


relations 119898119895 119898

119896 = 2120575

119895119896


= 119860119895119895+1



Hcl = minus119899

sum

119901=1

119872

sum

119895=1

(119869ℎ

119895119904119895119901119904119895+1119901

+ 119869V119904119895119901119904119895119901+1

) (13)


120573cl119869V=1

2log coth

120573quℎ

119899

120573cl119869ℎ

119895=120573qu

119899119869119895

(14)







⟨119891 (120590119911

)⟩qu = ⟨119891 (119904)⟩cl (15)



⟨120590119911

119895120590119911

119895+119903⟩qu= ⟨119904

119895119901119904119895+119903119901⟩cl

⟨

119903

prod

119895=1

120590119911

119895⟩

qu

= ⟨

119903

prod

119895=1

119904119895119901⟩

cl

(16)



Tr [V119886V

119887]119899

V120572= sum

119895isin120572

119890(120573qu119899)H

119911

119895 119890(120573qu119899)H

119909

119895 119890(120573qu119899)H

119910

119895 119890(120573qu119899)H

119911

119895

(17)

where H120583

119895= 119869

120583

119895120590120583

119895120590120583

119895+1for 120583 isin 119909 119910 H119911

119895= (ℎ4)(120590

119911

119895+ 120590

119911

119895+1)



119904119901= sum

119904119901

| 119904119901⟩⟨ 119904

119901| where | 119904

119901⟩ = |119904

1119901 119904

2119901 119904

119872119901⟩



Tr I 1199041

V119886I 1199042


1199042119899minus1V

119886I 1199042119899

V119887


sum

119904119895119901

2119899

prod

119901isin119886

⟨ 119904119901

10038161003816100381610038161003816V

119886

10038161003816100381610038161003816119904119901+1⟩ ⟨ 119904

119901+1

10038161003816100381610038161003816V

119887

10038161003816100381610038161003816119904119901+2⟩

(18)


⟨ 119904119901

10038161003816100381610038161003816V

120572

10038161003816100381610038161003816119904119901+1⟩ =

119872

prod

119895isin120572

⟨119904119895119901 119904

119895+1119901

10038161003816100381610038161003816M10038161003816100381610038161003816119904119895119901+1

119904119895+1119901+1

⟩ (19)

where

M =

(((((((

(

119890120573quℎ119899 cosh(

2120573qu120574

119899119861119895) 0 0 sinh(

2120573qu120574

119899119861119895)

0 cosh(2120573qu

119899119860

119895) sinh(

2120573qu

119899119860

119895) 0

0 sinh(2120573qu

119899119860

119895) cosh(

2120573qu

119899119860

119895) 0

sinh(2120573qu120574

119899119861119895) 0 0 119890

minusℎ119899 cosh(2120573qu120574

119899119861119895)

)))))))

)

(20)


⟨ 119904119901

10038161003816100381610038161003816V

120572

10038161003816100381610038161003816119904119901+1⟩ =

119872

prod

119895isin120572

119890minus120573clH119895119901 (21)

whereH119895119901

can be written as

H119895119901= minus1

4(119869

V119895119904119895119901119904119895119901+1

+ 119869ℎ

119895119904119895119901119904119895+1119901

+ 119869119889

119895119904119895+1119901

119904119895119901+1

+ 119867(119904119895119901+ 119904

119895+1119901) + 119862

119895)

(22)


H119895119901= minus1

4(119869

ℎ

119895(119904

119895119901119904119895+1119901

+ 119904119895119901+1

119904119895+1119901+1

)

+ 119869V119895(119904

119895119901119904119895119901+1

+ 119904119895+1119901

119904119895+1119901+1

)

+ 119869119889

119895(119904

119895119901119904119895+1119901+1

+ 119904119895119901+1

119904119895+1119901

)

+ 119867(119904119895119901+ 119904

119895+1119901+ 119904

119895119901+1+ 119904

119895+1119901+1) + 119862

119895)

(23)

where

120573cl119869ℎ

119895= log

sinh (4120573qu119899) 120574119861119895sinh (4120573qu119899)119860119895


120573cl119869119889

119895= log

tanh (2120573qu119899)119860119895

tanh (2120574120573qu119899) 119861119895

120573cl119869V119895= log coth

2120574120573qu

119899119861119895coth

2120573qu

119899119860

119895

120573cl119867 =120573quℎ

119899

120573cl119862119895= log sinh

2120573qu

119899119860

119895sinh

2120574120573qu

119899119861119895

(24)


119904119895119901119904119895+1119901

119904119895119901+1

119904119895+1119901+1

= 1 (25)



Hcl =2119899

sum

119901isin119886

119872

sum

119895isin119886

H119895119901+

2119899

sum

119901isin119887

119872

sum

119895isin119887

H119895119901 (26)

whereH119895119901




119895 119869

119889

119895 119867 rarr 0 and 119869V

119895rarr infin




Hcl = minus119899

sum

119901=1

119872

sum

119895=1

(119869120590

119895120590119895119901120590119895+1119901

+ 119869120591

119895120591119895119901120591119895+1119901

+ 119894119869120591119895119901(120590

119895119901minus 120590

119895119901+1))

(27)

Trotter

direction

p darr




120573cl119869120590

119895=120573qu

119899119869119909

119895

120573cl119869120591

119895=120573qu

119899119869119910

119895

120573cl119869 =1

2arctan 1

sinh (120573qu119899) ℎ

(28)



Tr [U1U

2]119899

U1= 119890

(120573qu2119899)H119909119890(120573qu2119899)H119911119890

(120573qu2119899)H119910

U2= 119890

(120573qu2119899)H119910119890(120573qu2119899)H119911119890

(120573qu2119899)H119909

(29)


= sum119872

119895=1119869120583

119895120590120583

119895120590120583

119895+1for 120583 isin 119909 119910 and H119911

=

sum119872

119895=1120590119911

119895


=

sum119901

|119901⟩⟨

119901| and I

120591119901= sum

120591119901

| 120591119901⟩⟨ 120591




Tr I1

U1I 1205911

U2I2


1205912119899U

2


sum

120590119895119901 120591119895119901

119899

prod

119901=1

⟨119901

10038161003816100381610038161003816U

1

10038161003816100381610038161003816120591119901⟩ ⟨ 120591

119901

10038161003816100381610038161003816U

2

10038161003816100381610038161003816119901+1⟩

(30)



⟨119901

10038161003816100381610038161003816U

1

10038161003816100381610038161003816120591119901⟩ ⟨ 120591

119901

10038161003816100381610038161003816U

2

10038161003816100381610038161003816119901+1⟩

= 119890(120573qu119899)((12)(H

119909

119901+H119909119901+1

)+H119910

119901) ⟨119901

10038161003816100381610038161003816119890(120573qu2119899)H

119911 10038161003816100381610038161003816120591119901⟩ ⟨ 120591

119901

10038161003816100381610038161003816

sdot 119890(120573qu2119899)H

119911 10038161003816100381610038161003816119901+1⟩

= 1198622119872

119890(120573qu119899)((12)(H

119909

119901+H119909119901+1

)+H119910

119901)+(1198942)119863sum119872

119895=1120591119895119901(120590119895119901minus120590119895119901+1)

(31)

where H119909

119901= sum

119872

119895=1119869119909

120590119895119901120590119895+1119901

H119910

119901= sum

119872

119895=1119869119910

120591119895119901120591119895+1119901


⟨120590119895119901

10038161003816100381610038161003816119890119886120590119911

11989510038161003816100381610038161003816120591119895119901⟩

=1

2cosh (2119886) 119890119894(12) arctan(1 sinh(2119886))120590119895119901120591119895119901

(32)


119895119901and 120591



119895119901


119895 119869120591

119895rarr infin



⟨119891 (120590119909

)⟩qu = ⟨119891 (120590)⟩cl

⟨119891 (120590119910

)⟩qu = ⟨119891 (120591)⟩cl (33)



⟨120590119909

119895120590119909

119895+119903⟩qu= ⟨120590

119895119901120590119895+119903119901⟩cl

⟨120590119910

119895120590119910

119895+119903⟩qu= ⟨120591

119895119901120591119895+119903119901⟩cl

(34)



⟨ 119904119901

10038161003816100381610038161003816119890(120573qu119899)V120572 10038161003816100381610038161003816

119904119901+1⟩

=

119872

prod

119895isin120572

120596119895

(119904119895119901 119904

119895+1119901 119904

119895119901+1 119904

119895+1119901+1)

(35)


(119904119895119901119904119895+1119901

119904119895119901+1

119904119895+1119901+1

)


120596119895

1(+1 +1 +1 +1) = 119890

ℎ120573qu119899 cosh(2120573qu120574

119899119861119895)

120596119895


minus120573quℎ119899 cosh(2120574120573qu

119899119861119895)

120596119895

3(minus1 +1 +1 minus1) = 120596

119895


= sinh(2120573qu

119899119860

119895)

120596119895

5(+1 minus1 +1 minus1) = 120596

119895


= cosh(2120573qu

119899119860

119895)

120596119895

7(minus1 minus1 +1 +1) = 120596

119895


= sinh(2120573qu120574

119899119861119895)

(36)




3 120596

119894

4 120596

119894

7 120596

119894

8rarr 0 and

weights 120596119894

1 120596

119894

2 120596

119894

5 120596

119894





12059041

12059031

12059021

12059011

12059042

12059032

12059022

12059012

12059043

12059033

12059023

12059013

12059141

12059131

12059121

12059111

12059142

12059132

12059122

12059112

12059143

12059133

12059123

12059113

1 2 3 4 5 6

120590j1 = j1

120591j1 = j2

120590j2 = j3

120591j2 = j4

120590j3 = j5

120591j3 = j6

Trotter

direction

p darr




119895 and the 119869



1205961 1205962 1205963 1205964 1205965 1205966 1205967 1205968




sum

120590119895119901=plusmn1

1

4(

119899

prod

119901isin119886

119872

prod

119895isin119886

+

119899

prod

119901isin119887

119872

prod

119895isin119887

)

sdot [(1 minus 119904119895119901119904119895+1119901

) (1 + 119904119895119901119904119895119901+1

) cosh2120573qu

119899119860

119895119895+1

+ (1 minus 119904119895119901119904119895+1119901

) (1 minus 119904119895119901119904119895119901+1

) sinh2120573qu

119899119860

119895119895+1

+ (1 + 119904119895119901119904119895+1119901

) (1 minus 119904119895119901119904119895119901+1

) sinh2120573qu120574

119899119861119895119895+1

+ (1 + 119904119895119901119904119895119901+1

) (1 + 119904119895119901119904119895119901+1

) 119890(120573qu119899)ℎ119904119895119901


119899119861119895119895+1]

(37)



Trotter

direction

p darr







Hqu =1

2sum

1le119895le119896le119872

((119860119895119896+ 120574119861

119895119896) 120590

119909

119895120590119909

119896

+ (119860119895119896minus 120574119861

119895119896) 120590

119910

119895120590119910

119896)(

119896minus1

prod

119897=119895+1

minus 120590119911

119897) minus ℎ

119872

sum

119895=1

120590119911

119895

(38)




Hqu]

= TrV119899

(39)




119872

prod

119894=1

119889N (120591119894) ⟨N (120591

119872)1003816100381610038161003816 119890

minusΔ120591H 1003816100381610038161003816N (120591119872minus1)⟩

sdot ⟨N (120591119872minus1)1003816100381610038161003816 119890


1)1003816100381610038161003816

sdot 119890minusΔ120591H 1003816100381610038161003816N (120591119872)⟩

(40)




119889120591|N (120591)⟩)

= ⟨N (120591) | N (120591)⟩ minus Δ120591 ⟨N (120591)| 119889119889120591|N (120591)⟩

minus Δ120591 ⟨N (120591)| Hqu (S) |N (120591)⟩ + 119874 ((Δ120591)2

)

= 119890minusΔ120591(⟨N(120591)|(119889119889120591)|N(120591)⟩+H(N(120591)))

Δ120591

119872

sum

119894=1

997888rarr int

120573

0

119889120591

119872

prod

119894=1

119889N (120591119894) 997888rarr DN (120591)

(41)


119885 = int

N(120573)

N(0)

DN (120591) 119890minusint

120573

0119889120591H(N(120591))minusS119861 (42)


S119861= int

120573

0

119889120591 ⟨N (120591)| 119889119889120591|N (120591)⟩ (43)



⟨N| S |N⟩ = N (44)


⟨N| 119878119894 |N⟩ = minus119878119873119894

(45)



N = (119873119909

119873119910

119873119911


0 le 120579 le 120587 0 le 120601 le 2120587

(46)


H (N (120591)) = ⟨N (120591)| Hqu |N (120591)⟩

= sum

1le119895le119896le119872

((119860119895119896+ 120574119861

119895119896)119873

119909

119895(120591)119873

119909

119896(120591)

+ (119860119895119896minus 120574119861

119895119896)119873

119910

119895(120591)119873

119910

119896(120591))

119896minus1

prod

119897=119895+1

(minus119873119911

119897(120591))

minus ℎ

119872

sum

119895=1

119873119911

119895(120591) = sum

1le119895le119896le119872

(119860119895119896

cos (120601119895(120591) minus 120601

119896(120591))

+ 119861119895119896120574 cos (120601

119895(120591) + 120601

119896(120591))) sin (120579

119895(120591))

sdot sin (120579119896(120591))

119896minus1

prod

119897=119895+1

(minus cos (120579119897(120591))) minus ℎ

119872

sum

119895=1

cos (120579119895(120591))

(47)





[HquVcl] = 0 (48)


H = minus

119873

sum

119894

119872

sum

119895

(1198691119904119894119895119904119894+1119895

+ 1198692119904119894119895119904119894119895+1) (49)



2ℎ + 2V = 119873119872 (50)


119885 = sum


119910V= 119910

1198721198732

sum


(51)



120578dagger

119902=119890minus1198941205874

radic119872sum

119895

119890minus(2120587119894119872)119902119895

(119887dagger

119895119906119902+ 119894119887

119895V119902)

120578119902=1198901198941205874

radic119872sum

119895

119890(2120587119894119872)119902119895

(119887119895119906119902minus 119894119887

dagger

119895V119902)

(52)



V+(minus)

= (2 sinh 21198701)1198732

119890minussum119902120598119902(120578dagger

119902120578119902minus12) (53)

where119870119894= 120573119869

119894and 120598


cosh 120598119902= cosh 2119870lowast

1cosh 2119870

2


1sinh 2119870

2cos 119902

(54)



by

V2

119863

= prod

0le119902le120587

(120582119902(120578

dagger

119902120578119902+ 120578

dagger


2sin2

119902)) (55)


with

120582119902= 2120572 sin 119902 (1 + 1205722sin2

119902)12

(56)


2V119902119906119902

1199062119902minus V2

119902

=

119886119902

119887119902

for Hqu



1coth 2119870

2

for V


1tanh119870

2cos 119902)



1 sinh 2119870

2

for V1015840

minus1

120572 sin 119902for V2

119863

(57)



119886119902

119887119902

=sin 119902

cosh 2119870lowast


1coth 2119870

2

(58)


sinh 2119870lowast

1coth 2119870

2= minus119886 (119871 minus 1)

119887 (119871)

tanh2119870lowast

1=119886 (119871) minus 119887 (119871)

119886 (119871) + 119887 (119871)

119886 (119871 minus 1)

119886 (119871) + 119887 (119871)= minus coth 2119870

2tanh119870lowast

1

(59)

or inversely as

cosh 2119870lowast

1=119886 (119871)

119887 (119871)


1

119886 (119871 minus 1)

radic(119886 (119871))2

minus (119887 (119871))2

(60)

where

119886 (119871) = 119886 (119871)

[1198712]

sum

119897=0

(119871

2119897)

119887 (119871) = 119887 (119871)

[(119871minus1)2]

sum

119897=0

(119871

2119897 + 1)

119886 (0) = Γ

(61)


119887 (119871)ge 1

1198862

(119871) le 1198862

(119871 minus 1) + 1198872

(119871)

(62)


119871

sum

119896=1

119887 (119896)

[(119871minus1)2]

sum

119897=0

(119896

2119897 + 1)

119897

sum

119894=1

(minus1)119894 cos119896minus2119894119902

+

119871minus1

sum

119896=1

119887 (119896) cos119896119902 = 0

(63)

Γ +

119871minus2

sum

119896=1

119886 (119896) cos119896119902

+

119871

sum

119896=1

119886 (119896)

[1198962]

sum

119897=0

(119896

2119897)

119897

sum

119894=1

(119897

119894) (minus1)

119894 cos119871minus2119894119902 = 0

(64)




10038161003816100381610038161003816Λ

119902

10038161003816100381610038161003816= 2

119901+110038161003816100381610038161003816cos(119871minus1)11990210038161003816100381610038161003816

sdot ((119886 (119871) cos 119902 + 119886 (119871 minus 1))2 + 1198872

(119871) sin2

119902)

12

(65)


by

119870lowast

1= 119870

2 (66)



119886 (119871) = plusmn119886 (119871 minus 1) (67)


10038161003816100381610038161003816Λ

119902

10038161003816100381610038161003816= 2

119901+110038161003816100381610038161003816119886 (119871) cos(119871minus1)11990210038161003816100381610038161003816

sdot ((cos 119902 plusmn 1)2 + (119887 (119871)119886 (119871)

)

2

sin2

119902)

12

(68)



⟨120590119909

119895119896120590119909

119895+119903119896⟩Is= ⟨Ψ

0

1003816100381610038161003816Vminus12

1120590119909

119895120590119909

119895+119903V12

1

1003816100381610038161003816Ψ0⟩

= ⟨Φ0

1003816100381610038161003816Vminus12

1120590119909

119895120590119909

119895+119903V12

1

1003816100381610038161003816Φ0⟩

= ⟨(Vminus12

1120590119909

119895V12

1) (Vminus12

1120590119909

119895+119903V12

1)⟩

qu

= cosh2119870lowast

1⟨120590

119909

119895120590119909

119895+119903⟩qu


1⟨120590

119910

119895120590119910

119895+119903⟩qu

(69)


119895+119903⟩qu = ⟨120590

119910

119895120590119909

119895+119903⟩qu = 0 for 119903 = 0 and

Ψ0= Φ

0 (70)






119887119902


1tanh119870

2cos 119902)



1 sinh 2119870

2

(71)



tanh 2119870lowast

1tanh119870

2= minus

119887 (119871)

119887 (119871 minus 1)= minus

119886 (119871)

119887 (119871 minus 1)

119886 (119871 minus 1)

119887 (119871 minus 1)= 1

tanh 2119870lowast

1

sinh 21198702

= minus119886lowast

(119871)

119887 (119871 minus 1)

(72)

or inversely as

sinh21198702=

119886 (119871)

2 (119886lowast

(119871))

tanh 2119870lowast

1= minus

1

119886 (119871 minus 1)radic119886 (119871) (2119886

lowast

(119871) + 119886 (119871))

(73)

where

119886lowast

(119871) = 119886 (119871 minus 2) minus 119886 (119871)

[1198712]

sum

119897=0

(119871

2119897) 119897 (74)


119886 (119871) (2119886lowast

(119871) + 119886 (119871)) le 1198862

(119871 minus 1) (75)


119871minus2

sum

119896=1

119887 (119896) cos119896119902

+

119871

sum

119896=1

119887 (119896)

[(119896minus1)2]

sum

119897=0

(119896

2119897 + 1)

119897

sum

119894=1

(119897

119894) (minus1)

119894 cos119896minus2119894119902

= 0

(76)

Γ +

119871minus3

sum

119896=1

119896cos119896119902 minus119871minus1

sum

119896=1

119886 (119896)

[1198962]

sum

119897=0

(119896

2119897) 119897cos119896minus2119902

+

119871

sum

119896=1

119886 (119896)

[1198962]

sum

119897=0

(119896

2119897)

119897

sum

119894=2

(119897

119894) (minus1)

119894 cos119896minus2119894119902 = 0

(77)



10038161003816100381610038161003816Λ

119902

10038161003816100381610038161003816= 2

119901+110038161003816100381610038161003816cos119871minus211990210038161003816100381610038161003816


+ sin2

119902 (119887 (119871) cos 119902 + 119887 (119871 minus 1)))12

(78)





(119871) + 119886 (119871) (79)


10038161003816100381610038161003816Λ

119902

10038161003816100381610038161003816= 2

119901+110038161003816100381610038161003816cos119871minus211990210038161003816100381610038161003816


+ sin2

119902 (119887 (119871) cos 119902 + 119887 (119871 minus 1)))12

(80)



⟨120590119909

119895119896120590119909

119895+119903119896⟩Is= ⟨120590

119909

119895120590119909

119895+119903⟩qu (81)

where Ψ1015840



Ψ1015840

0= Φ

0 (82)



119863 In this case




119892M(120579) ofM


120574 = 0 and thus B = 0



119895119896(119861

119895119896) (M

119899)119895119896


119895 119896 ge 0119874

+

(2119873) 119886(119895 minus 119896) + 119886(119895 + 119896) 1198920if 119895 = 119896 = 0radic2119892

119897if

either 119895 = 0 119896 = 119897or 119895 = 119897 119896 = 0

119892119895minus119896+ 119892

119895+119896 119895 119896 gt 0


119895+119896+2 119895 119896 ge 0

119874plusmn

(2119873 + 1) 119886(119895 minus 119896) ∓ 119886(119895 + 119896 + 1) 119892119895minus119896∓ 119892

119895+119896+1 119895 119896 ge 0

119874minus


119895+119896+2 119895 119896 ge 0

Appendices

A Symmetry Classes

See Table 1



Hqu = minus119872

sum

119895=1

(119869119909

119895120590119909

119895120590119909

119895+1+ 119869

119910

119895120590119910

119895120590119910

119895+1

minus (1198691015840119909

119895120590119909

119895120590119909

119895+2+ 119869

1015840119910

119895120590119910

119895120590119910

119895+2) 120590

119911

119895+1+ ℎ120590

119911

119895)

(B1)

where 1198691015840119909119895= (12)(119860

119895119895+2+ 120574119861

119895119895+2) and 1198691015840119910

119895= (12)(119860

119895119895+2minus




Tr [119890(120573qu119899)H119909

119886 119890(120573qu2119899)H

119911

119890(120573qu119899)H

119910

119887 119890(120573qu119899)H

119910

119886 119890(120573qu2119899)H

119911

119890(120573qu119899)H

119909

119887 ]119899

(B2)


120572= sum

119872

119895isin120572((12)119869

120583

119895(120590

120583

119895120590120583

119895+1+ 120590

120583

119895+1120590120583

119895+2) minus

1198691015840120583

119895120590120583

119895120590119911

119895+1120590120583

119895+2) and H119911

= ℎsum119872

119895=1120590119911

119895 for 120583 isin 119909 119910 and once




Tr [I1205901119890(120573qu119899)H

119909

119886 I1199041119890(120573qu2119899)H

119911

119890(120573qu119899)H

119910

119887 I1205911119890(120573qu119899)H

119910

119886 I1199041119890(120573qu2119899)H

119911

119890(120573qu119899)H

119909

119887 ]119899


sum

120590119895119901 120591119895119901119904119895119901

119899

prod

119901=1

[⟨119901

10038161003816100381610038161003816119890(120573qu119899)H

119909

119886100381610038161003816100381610038161199042119901minus1⟩ ⟨ 119904

2119901minus1

10038161003816100381610038161003816119890(120573qu2119899)H

119911

119890(120573qu119899)H

119910

119887

10038161003816100381610038161003816120591119901⟩ ⟨ 120591

119901

10038161003816100381610038161003816119890(120573qu119899)H

119910

119886100381610038161003816100381610038161199042119901⟩ ⟨ 119904

2119901

10038161003816100381610038161003816119890(120573qu2119899)H

119911

119890(120573qu119899)H

119909

11988710038161003816100381610038161003816119901+1⟩]

(B3)



⟨119901

10038161003816100381610038161003816119890(120573119899)

H119909119886100381610038161003816100381610038161199042119901minus1⟩ ⟨ 119904

2119901minus1

10038161003816100381610038161003816119890(120573qu2119899)H

119911

119890(120573qu119899)H

119910

119887

10038161003816100381610038161003816120591119901⟩ ⟨ 120591

2119901minus1

10038161003816100381610038161003816119890(120573qu119899)H

119910

119886100381610038161003816100381610038161199042119901⟩ ⟨ 119904

2119901

10038161003816100381610038161003816119890(120573qu2119899)H

119911

119890(120573qu119899)H

119909

119887100381610038161003816100381610038162119901⟩

= 119890(120573qu119899)H

119909

119886(119901)

119890(120573qu2119899)H

119911(2119901minus1)

119890(120573qu119899)H

119910

119887(119901)

119890(120573qu119899)H

119910

119886(119901)

119890(120573qu2119899)H

119911(2119901)

119890(120573qu119899)H

119909

119887(119901)

⟨119901| 119904

2119901minus1⟩ ⟨ 119904

2119901minus1| 120591

119901⟩ ⟨ 120591

119901| 119904

2119901⟩

sdot ⟨ 1199042119901|

119901+1⟩

(B4)

where H119909

120572(119901) = sum

119872

119895isin120572((12)119869

119909

119895(120590

119895119901120590119895+1119901

+ 120590119895+1119901

120590119895+2119901

) +

1198691015840119909

119895+1120590119895119901119904119895+1119901

120590119895+2119901

) H119910

120572(119901) = sum

119872

119895isin120572((12)119869

119910

119895(120591

119895119901120591119895+1119901

+

120591119895+1119901

120591119895+2119901

) + 1198691015840119910

119895+1120591119895119901119904119895+1119901

120591119895+2119901

) andH119911

(119901) = sum119872

119895=1119904119895119901 We


⟨119901| 119904

2119901minus1⟩ ⟨ 119904

2119901minus1| 120591

119901⟩ ⟨ 120591

2119901minus1| 119904

2119901⟩ ⟨ 119904

2119901|

119901+1⟩

=1

24119872

sdot

119872

prod

119895=1


(B5)



119899

119899

sum

119901=1

(sum

119895isin119886

(119869119909

119895

2(120590

119895119901120590119895+1119901

+ 120590119895+1119901

120590119895+2119901

) minus 1198691015840119909

119895+1120590119895119901119904119895+1119901

120590119895+2119901

)

+sum

119895isin119887

(

119869119910

119895

2(120591

119895119901120591119895+1119901

+ 120591119895+1119901

120591119895+2119901

) minus 1198691015840119910

119895+1120591119895119901119904119895+12119901minus1

120591119895+2119901

)

+ sum

119895isin119886

(

119869119910

119895

2(120591

119895119901120591119895+1119901

+ 120591119895+1119901

120591119895+2119901

) minus 1198691015840119910

119895+1120591119895119901119904119895+12119901

120591119895+2119901

)

+sum

119895isin119887

(119869119909

119895

2(120590

119895119901+1120590119895+1119901+1

+ 120590119895+1119901+1

120590119895+2119901+1

) minus 1198691015840119909

119895+1120590119895119901+1

119904119895+12119901

120590119895+2119901+1

))

+

119899

sum

119901=1

(

119872

sum

119895=1

((120573quℎ

2119899minus119894120587

4) 119904

1198952119901+ (120573quℎ

2119899+119894120587

4) 119904

1198952119901) +

119872

sum

119895=1

119894120587

4(120590

1198951199011199041198952119901minus1

minus 120590119895119901+1

1199042119901+ 120591

119895119901(119904

1198952119901minus 119904

1198952119901minus1)))

+ 4119899119872 ln 2

(B6)



119909 (120573qu119899)119869119910

(120573qu119899)1198691015840119909 (120573qu119899)119869




S1

S2

S3

S4

1205901

1205911

1205902

1205912

1 2 3 4 5 6 7 8


Trotter

direction

p darr








Tr[

[

119872minus1

prod

119895=1

(119890(120573qu119899)H

119909

119895119895+1119890(120573qu119899)H

119909


119909

119895119872119890(120573qu2119899(119872minus1))

H119911119890(120573qu119899)H

119910


119910

119895119895+2119890(120573qu119899)H

119910

119895119895+1)]

]

119899


Tr[

[

119872

prod

119895=1

((

119872minus119895

prod

119896=1

119890(120573qu119899)H

119909

119895119895+119896)119890(120573qu2119899(119872minus1))

H119911(


prod

119896=0

119890(120573qu119899)H

119910

119895119872minus119896))]

]

119899

(B7)

where H120583

119895119896= 119869

120583

119895119896120590120583

119895120590120583

119896prod

119896minus1

119897=1(minus120590

119911

119897) for 120583 isin 119909 119910 and H119911

=

ℎsum119872

119895=1120590119911

119895



Tr[

[

119872minus1

prod

119895=1

(I120590119895(

119872minus119895

prod

119896=1

119890(120573qu119899)H

119909

119895119895+119896)119890(120573qu(119872minus1)119899)

H119911I119904119895(


prod

119896=0

119890(120573qu119899)H

119910

119895119872minus119896) I120591119895)]

]

119899


sum

120590119895119901 120591119895119901

119899minus1

prod

119901=0

119872minus1

prod

119895=1

(⟨119895+119895119901

10038161003816100381610038161003816(

119872minus119895

prod

119896=1

119890(120573qu119899)H

119909

119895119895+119896)119890(120573qu119899(119872minus1))

H119911 10038161003816100381610038161003816119904119895+119895119901⟩ ⟨ 119904

119895+119895119901

10038161003816100381610038161003816(


prod

119896=0

119890(120573qu119899)H

119910

119895119872minus119896)10038161003816100381610038161003816120591119895+119895119901⟩ ⟨ 120591

119895+119895119901|

119895+119895119901+1⟩)

(B8)


⟨119895+119895119901

10038161003816100381610038161003816(

119872minus119895

prod

119896=1

119890(120573qu119899)H

119909

119895119895+119896)119890(120573qu119899(119872minus1))

H119911 10038161003816100381610038161003816119904119895+119895119901⟩

sdot ⟨ 119904119895+119895119901

10038161003816100381610038161003816(


prod

119896=0

119890(120573qu119899)H

119910

119895119872minus119896)10038161003816100381610038161003816120591119895+119895119901⟩ ⟨ 120591

119895+119895119901|

119895+119895119901+1

⟩

= 119890(120573qu119899)sum

119872minus119895

119896=1(H119909119895119895+119896

(119901)+H119910

119895119895+119896(119901)+(1119899(119872minus1))H119911)

⟨119895+119895119901

|

119904119895+119895119901⟩ ⟨ 119904

119895+119895119901| 120591

119895+119895119901⟩ ⟨ 120591

119895+119895119901|

119895+119895119901+1⟩

(B9)


S1

S2

S3

S4

S5

S6

S7

S8

S9

S10

1 2 3 4 5 6 7 8 9


Trotter

direction

p darr



where H119909

119895119896(119901) = sum

119872

119896=119895+1119869119909

119895119896120590119895119901120590119896119901prod

119896minus1

119897=119895+1(minus119904

119897119901) H119910

119895119896(119901) =

sum119872

119896=119895+1119869119910

119895119896120591119895119901120591119896119901prod

119896minus1

119897=119895+1(minus119904

119897119901) andH119911

119901= ℎsum

119872

119895=1120590119911

119895119901 Finally


⟨119901| 119904

119901⟩ ⟨ 119904

119901| 120591

119901⟩ ⟨ 120591

119901|

119901+1⟩ = (

1

2radic2)

119872

sdot

119872

prod

119895=1


(B10)



sum

119901=1

119872

sum

119895=1

(120573qu

119899

119872

sum

119896=119895+1

(119869119909

119895119896120590119895119895+119895119901

120590119896119895+119895119901

+ 119869119910

119895119896120591119895119895+119895119901

120591119896119895+119895119901

)

119896minus1

prod

119897=119895+1

(minus119904119897119901) + (

120573qu

119899 (119872 minus 1)ℎ minus119894120587

4) 119904

119895119895+119895119901

+119894120587

4(1 minus 120590

119895119895+119895119901+ 120591

119895119895+119895119901+ 120590

119895119895+119895119901119904119895119895+119895119901

minus 120591119895119895+119895119901

119904119895119895+119895119901

minus 120590119895119895+119895119901+1

120591119895119895+119895119901

)) + 1198991198722 ln 1

2radic2

(B11)


119895119896





119909

119895119896 (120573qu119899)119869

119910

119895119896 and



1205901

S1

1205911

S2

1205902

S3

1205912

S4

1205903

S5

1 2 3 4 5 6 7 8 9 10

Trotter

direction

p darr






119863(55)


minus1

120572 sin 119902=119887 (1) sin 119902

Γ + 119886 (1) cos 119902 (C1)



minus1

120572 sin 119902=

119887 (1)

minus2119886 (2) sin 119902

if Γ = minus119886 (2) 119886 (1) = 119887 (2) = 0

(C2)


120572 =2119886 (2)

119887 (1) Γ = minus119886 (2) 119886 (1) = 119887 (2) = 0 (C3)


minus1

120572 sin 119902= minus

119887 (1) minus 119887 (3) + 119887 (2) cos 1199022 sin 119902 (119886 (2) + 119886 (3) cos 119902)

if Γ = minus119886 (2) 119886 (1) = minus119886 (3) 119887 (3) = 0

(C4)


120572 =2119886 (3)

119887 (2)

119886 (2)

119886 (3)=119887 (1) minus 119887 (3)

119887 (2)

Γ = minus119886 (2) 119886 (1) = minus119886 (3) 119887 (3) = 0

(C5)


minus1

120572 sin 119902

=sin 119902sum119898

119896=1119887 (119896)sum

[(119896minus1)2]

119897=0( 119896

2119897+1)sum

119897

119894=0( 119897119894) (minus1)


119896=2119886 (119896)sum

[1198962]

119897=0(minus1)

119897

( 119896

2119897) sin2119897

119902cos119896minus2119897119902

(C6)


[1198982]

sum

119895=1

119886 (2119895) + cos 119902[1198982]

sum

119895=1

119886 (2119895 + 1) + sin2

119902

sdot (

[1198982]

sum

119895=1

119886 (2119895)

119895

sum

119894=1

(119895

119894) (minus1)

119894 sin2(119894minus1)

119902

+ cos 119902[(119898minus1)2]

sum

119895=1

119886 (2119895 + 1)

119895

sum

119894=1

(119895

119894) (minus1)

119894 sin2(119894minus1)

119902

+

119898

sum

119896=2

119886 (119896)

[1198962]

sum

119897=1

(minus1)119897

(119896

2119897) sin2(119897minus1)

119902cos119896minus2119897119902)

(C7)


Γ = minus

[1198982]

sum

119895=1

119886 (2119895)

119886 (1) = minus

[(119898+1)2]

sum

119895=1

119886 (2119895 + 1) = 0

(C8)



sin2

119902(

[1198982]

sum

119895=1

119886 (2119895)

119895

sum

119894=1

(119895

119894)

119894minus1

sum

119901=0

(minus1)119894+119901 cos2119901119902

+

[(119898minus1)2]

sum

119895=1

119886 (2119895 + 1)

119895

sum

119894=1

(119895

119894)

119894minus1

sum

119901=0

(minus1)119894+119901 cos2119901+1119902 +

[(119898minus1)2]

sum

119895=1

119886 (2119895 + 1)

119895

sum

119897=1

(2119895 + 1

2119897)

119897minus1

sum

119901=0


+

[1198982]

sum

119895=1

119886 (2119895)

119895

sum

119897=1

(2119895

2119897)

119897minus1

sum

119901=0


(C9)


Disclosure




Acknowledgments


Endnotes



integer



and 119861119895119896





7 Thus matrices 119860119895119896

and 119861119895119896




1V

2V12

1of


=

V12

2V

1V12

2 whereV

1= (2 sinh 2119870

1)1198722


1sum119872

119894120590119909

119894 V2=

1198901198702 sum119872

119894=1120590119911

119894120590119911


119894= 119890



119887119902(5) as

119886119902= Γ +

119871

sum

119896=1

119886 (119896)

[1198962]

sum

119897=0

(119896

2119897)

119897

sum

119894=0

(119897

119894) (minus1)

minus119894 cos119896minus2119894119902

119887119902= tan 119902

sdot

119871

sum

119896=1

119887 (119896)

[(119896minus1)2]

sum

119897=0

(119896

2119897 + 1)

119897

sum

119894=0

(119897

119894) (minus1)

minus119894 cos119896minus2119894119902

(lowast)


119895=1(minus1)

119895

119886(2119895)


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













119885119860= sum


119894119895) (8a)

119885119861= sum


119885119862= sum


119894

120596119894 (8c)

119885119863= sum



119894119895=


119894119895 120591

119894119895 119904



119894 and 120590119911



119894119895 119904





119894119895) =


119894




H119883119884

= minus

119872

sum

119895=1

(119869119909

119895120590119909

119895120590119909

119895+1+ 119869

119910

119895120590119910

119895120590119910

119895+1+ ℎ120590

119911

119895) (9)

where 119869119909119895= minus(12)(119860

119895119895+1+ 120574119861

119895119895+1) 119869119910

119895= minus(12)(119860

119895119895+1minus


transformations

119887dagger

119895=1

2(119898

2119895+1+ 119894119898

2119895) =

1

2(120590

119909

119895+ 119894120590

119910

119895)

119895minus1

prod

119897=1

(minus120590119911

)

119887119895=1

2(119898

2119895+1minus 119894119898

2119895) =

1

2(120590

119909

119895minus 119894120590

119910

119895)

119895minus1

prod

119897=1

(minus120590119911

119897)

(10)

where

1198982119895+1

= 120590119909

119895

119895minus1

prod

119897=0

(minus120590119911

119895)

1198982119895= 120590

119910

119895

119895minus1

prod

119897=0

(minus120590119911

119895)

(11)

or inversely as

120590119911

119895= 119894119898

2119895119898

2119895+1

120590119909

119895= 119898

2119895+1

119895minus1

prod

119897=0

(minus1198941198982119897119898

2119897+1)

120590119910

119895= 119898

2119895

119895minus1

prod

119897=0

(minus1198941198982119897119898

2119897+1)

(12)


relations 119898119895 119898

119896 = 2120575

119895119896


= 119860119895119895+1



Hcl = minus119899

sum

119901=1

119872

sum

119895=1

(119869ℎ

119895119904119895119901119904119895+1119901

+ 119869V119904119895119901119904119895119901+1

) (13)


120573cl119869V=1

2log coth

120573quℎ

119899

120573cl119869ℎ

119895=120573qu

119899119869119895

(14)







⟨119891 (120590119911

)⟩qu = ⟨119891 (119904)⟩cl (15)



⟨120590119911

119895120590119911

119895+119903⟩qu= ⟨119904

119895119901119904119895+119903119901⟩cl

⟨

119903

prod

119895=1

120590119911

119895⟩

qu

= ⟨

119903

prod

119895=1

119904119895119901⟩

cl

(16)



Tr [V119886V

119887]119899

V120572= sum

119895isin120572

119890(120573qu119899)H

119911

119895 119890(120573qu119899)H

119909

119895 119890(120573qu119899)H

119910

119895 119890(120573qu119899)H

119911

119895

(17)

where H120583

119895= 119869

120583

119895120590120583

119895120590120583

119895+1for 120583 isin 119909 119910 H119911

119895= (ℎ4)(120590

119911

119895+ 120590

119911

119895+1)



119904119901= sum

119904119901

| 119904119901⟩⟨ 119904

119901| where | 119904

119901⟩ = |119904

1119901 119904

2119901 119904

119872119901⟩



Tr I 1199041

V119886I 1199042


1199042119899minus1V

119886I 1199042119899

V119887


sum

119904119895119901

2119899

prod

119901isin119886

⟨ 119904119901

10038161003816100381610038161003816V

119886

10038161003816100381610038161003816119904119901+1⟩ ⟨ 119904

119901+1

10038161003816100381610038161003816V

119887

10038161003816100381610038161003816119904119901+2⟩

(18)


⟨ 119904119901

10038161003816100381610038161003816V

120572

10038161003816100381610038161003816119904119901+1⟩ =

119872

prod

119895isin120572

⟨119904119895119901 119904

119895+1119901

10038161003816100381610038161003816M10038161003816100381610038161003816119904119895119901+1

119904119895+1119901+1

⟩ (19)

where

M =

(((((((

(

119890120573quℎ119899 cosh(

2120573qu120574

119899119861119895) 0 0 sinh(

2120573qu120574

119899119861119895)

0 cosh(2120573qu

119899119860

119895) sinh(

2120573qu

119899119860

119895) 0

0 sinh(2120573qu

119899119860

119895) cosh(

2120573qu

119899119860

119895) 0

sinh(2120573qu120574

119899119861119895) 0 0 119890

minusℎ119899 cosh(2120573qu120574

119899119861119895)

)))))))

)

(20)


⟨ 119904119901

10038161003816100381610038161003816V

120572

10038161003816100381610038161003816119904119901+1⟩ =

119872

prod

119895isin120572

119890minus120573clH119895119901 (21)

whereH119895119901

can be written as

H119895119901= minus1

4(119869

V119895119904119895119901119904119895119901+1

+ 119869ℎ

119895119904119895119901119904119895+1119901

+ 119869119889

119895119904119895+1119901

119904119895119901+1

+ 119867(119904119895119901+ 119904

119895+1119901) + 119862

119895)

(22)


H119895119901= minus1

4(119869

ℎ

119895(119904

119895119901119904119895+1119901

+ 119904119895119901+1

119904119895+1119901+1

)

+ 119869V119895(119904

119895119901119904119895119901+1

+ 119904119895+1119901

119904119895+1119901+1

)

+ 119869119889

119895(119904

119895119901119904119895+1119901+1

+ 119904119895119901+1

119904119895+1119901

)

+ 119867(119904119895119901+ 119904

119895+1119901+ 119904

119895119901+1+ 119904

119895+1119901+1) + 119862

119895)

(23)

where

120573cl119869ℎ

119895= log

sinh (4120573qu119899) 120574119861119895sinh (4120573qu119899)119860119895


120573cl119869119889

119895= log

tanh (2120573qu119899)119860119895

tanh (2120574120573qu119899) 119861119895

120573cl119869V119895= log coth

2120574120573qu

119899119861119895coth

2120573qu

119899119860

119895

120573cl119867 =120573quℎ

119899

120573cl119862119895= log sinh

2120573qu

119899119860

119895sinh

2120574120573qu

119899119861119895

(24)


119904119895119901119904119895+1119901

119904119895119901+1

119904119895+1119901+1

= 1 (25)



Hcl =2119899

sum

119901isin119886

119872

sum

119895isin119886

H119895119901+

2119899

sum

119901isin119887

119872

sum

119895isin119887

H119895119901 (26)

whereH119895119901




119895 119869

119889

119895 119867 rarr 0 and 119869V

119895rarr infin




Hcl = minus119899

sum

119901=1

119872

sum

119895=1

(119869120590

119895120590119895119901120590119895+1119901

+ 119869120591

119895120591119895119901120591119895+1119901

+ 119894119869120591119895119901(120590

119895119901minus 120590

119895119901+1))

(27)

Trotter

direction

p darr




120573cl119869120590

119895=120573qu

119899119869119909

119895

120573cl119869120591

119895=120573qu

119899119869119910

119895

120573cl119869 =1

2arctan 1

sinh (120573qu119899) ℎ

(28)



Tr [U1U

2]119899

U1= 119890

(120573qu2119899)H119909119890(120573qu2119899)H119911119890

(120573qu2119899)H119910

U2= 119890

(120573qu2119899)H119910119890(120573qu2119899)H119911119890

(120573qu2119899)H119909

(29)


= sum119872

119895=1119869120583

119895120590120583

119895120590120583

119895+1for 120583 isin 119909 119910 and H119911

=

sum119872

119895=1120590119911

119895


=

sum119901

|119901⟩⟨

119901| and I

120591119901= sum

120591119901

| 120591119901⟩⟨ 120591




Tr I1

U1I 1205911

U2I2


1205912119899U

2


sum

120590119895119901 120591119895119901

119899

prod

119901=1

⟨119901

10038161003816100381610038161003816U

1

10038161003816100381610038161003816120591119901⟩ ⟨ 120591

119901

10038161003816100381610038161003816U

2

10038161003816100381610038161003816119901+1⟩

(30)



⟨119901

10038161003816100381610038161003816U

1

10038161003816100381610038161003816120591119901⟩ ⟨ 120591

119901

10038161003816100381610038161003816U

2

10038161003816100381610038161003816119901+1⟩

= 119890(120573qu119899)((12)(H

119909

119901+H119909119901+1

)+H119910

119901) ⟨119901

10038161003816100381610038161003816119890(120573qu2119899)H

119911 10038161003816100381610038161003816120591119901⟩ ⟨ 120591

119901

10038161003816100381610038161003816

sdot 119890(120573qu2119899)H

119911 10038161003816100381610038161003816119901+1⟩

= 1198622119872

119890(120573qu119899)((12)(H

119909

119901+H119909119901+1

)+H119910

119901)+(1198942)119863sum119872

119895=1120591119895119901(120590119895119901minus120590119895119901+1)

(31)

where H119909

119901= sum

119872

119895=1119869119909

120590119895119901120590119895+1119901

H119910

119901= sum

119872

119895=1119869119910

120591119895119901120591119895+1119901


⟨120590119895119901

10038161003816100381610038161003816119890119886120590119911

11989510038161003816100381610038161003816120591119895119901⟩

=1

2cosh (2119886) 119890119894(12) arctan(1 sinh(2119886))120590119895119901120591119895119901

(32)


119895119901and 120591



119895119901


119895 119869120591

119895rarr infin



⟨119891 (120590119909

)⟩qu = ⟨119891 (120590)⟩cl

⟨119891 (120590119910

)⟩qu = ⟨119891 (120591)⟩cl (33)



⟨120590119909

119895120590119909

119895+119903⟩qu= ⟨120590

119895119901120590119895+119903119901⟩cl

⟨120590119910

119895120590119910

119895+119903⟩qu= ⟨120591

119895119901120591119895+119903119901⟩cl

(34)



⟨ 119904119901

10038161003816100381610038161003816119890(120573qu119899)V120572 10038161003816100381610038161003816

119904119901+1⟩

=

119872

prod

119895isin120572

120596119895

(119904119895119901 119904

119895+1119901 119904

119895119901+1 119904

119895+1119901+1)

(35)


(119904119895119901119904119895+1119901

119904119895119901+1

119904119895+1119901+1

)


120596119895

1(+1 +1 +1 +1) = 119890

ℎ120573qu119899 cosh(2120573qu120574

119899119861119895)

120596119895


minus120573quℎ119899 cosh(2120574120573qu

119899119861119895)

120596119895

3(minus1 +1 +1 minus1) = 120596

119895


= sinh(2120573qu

119899119860

119895)

120596119895

5(+1 minus1 +1 minus1) = 120596

119895


= cosh(2120573qu

119899119860

119895)

120596119895

7(minus1 minus1 +1 +1) = 120596

119895


= sinh(2120573qu120574

119899119861119895)

(36)




3 120596

119894

4 120596

119894

7 120596

119894

8rarr 0 and

weights 120596119894

1 120596

119894

2 120596

119894

5 120596

119894





12059041

12059031

12059021

12059011

12059042

12059032

12059022

12059012

12059043

12059033

12059023

12059013

12059141

12059131

12059121

12059111

12059142

12059132

12059122

12059112

12059143

12059133

12059123

12059113

1 2 3 4 5 6

120590j1 = j1

120591j1 = j2

120590j2 = j3

120591j2 = j4

120590j3 = j5

120591j3 = j6

Trotter

direction

p darr




119895 and the 119869



1205961 1205962 1205963 1205964 1205965 1205966 1205967 1205968




sum

120590119895119901=plusmn1

1

4(

119899

prod

119901isin119886

119872

prod

119895isin119886

+

119899

prod

119901isin119887

119872

prod

119895isin119887

)

sdot [(1 minus 119904119895119901119904119895+1119901

) (1 + 119904119895119901119904119895119901+1

) cosh2120573qu

119899119860

119895119895+1

+ (1 minus 119904119895119901119904119895+1119901

) (1 minus 119904119895119901119904119895119901+1

) sinh2120573qu

119899119860

119895119895+1

+ (1 + 119904119895119901119904119895+1119901

) (1 minus 119904119895119901119904119895119901+1

) sinh2120573qu120574

119899119861119895119895+1

+ (1 + 119904119895119901119904119895119901+1

) (1 + 119904119895119901119904119895119901+1

) 119890(120573qu119899)ℎ119904119895119901


119899119861119895119895+1]

(37)



Trotter

direction

p darr







Hqu =1

2sum

1le119895le119896le119872

((119860119895119896+ 120574119861

119895119896) 120590

119909

119895120590119909

119896

+ (119860119895119896minus 120574119861

119895119896) 120590

119910

119895120590119910

119896)(

119896minus1

prod

119897=119895+1

minus 120590119911

119897) minus ℎ

119872

sum

119895=1

120590119911

119895

(38)




Hqu]

= TrV119899

(39)




119872

prod

119894=1

119889N (120591119894) ⟨N (120591

119872)1003816100381610038161003816 119890

minusΔ120591H 1003816100381610038161003816N (120591119872minus1)⟩

sdot ⟨N (120591119872minus1)1003816100381610038161003816 119890


1)1003816100381610038161003816

sdot 119890minusΔ120591H 1003816100381610038161003816N (120591119872)⟩

(40)




119889120591|N (120591)⟩)

= ⟨N (120591) | N (120591)⟩ minus Δ120591 ⟨N (120591)| 119889119889120591|N (120591)⟩

minus Δ120591 ⟨N (120591)| Hqu (S) |N (120591)⟩ + 119874 ((Δ120591)2

)

= 119890minusΔ120591(⟨N(120591)|(119889119889120591)|N(120591)⟩+H(N(120591)))

Δ120591

119872

sum

119894=1

997888rarr int

120573

0

119889120591

119872

prod

119894=1

119889N (120591119894) 997888rarr DN (120591)

(41)


119885 = int

N(120573)

N(0)

DN (120591) 119890minusint

120573

0119889120591H(N(120591))minusS119861 (42)


S119861= int

120573

0

119889120591 ⟨N (120591)| 119889119889120591|N (120591)⟩ (43)



⟨N| S |N⟩ = N (44)


⟨N| 119878119894 |N⟩ = minus119878119873119894

(45)



N = (119873119909

119873119910

119873119911


0 le 120579 le 120587 0 le 120601 le 2120587

(46)


H (N (120591)) = ⟨N (120591)| Hqu |N (120591)⟩

= sum

1le119895le119896le119872

((119860119895119896+ 120574119861

119895119896)119873

119909

119895(120591)119873

119909

119896(120591)

+ (119860119895119896minus 120574119861

119895119896)119873

119910

119895(120591)119873

119910

119896(120591))

119896minus1

prod

119897=119895+1

(minus119873119911

119897(120591))

minus ℎ

119872

sum

119895=1

119873119911

119895(120591) = sum

1le119895le119896le119872

(119860119895119896

cos (120601119895(120591) minus 120601

119896(120591))

+ 119861119895119896120574 cos (120601

119895(120591) + 120601

119896(120591))) sin (120579

119895(120591))

sdot sin (120579119896(120591))

119896minus1

prod

119897=119895+1

(minus cos (120579119897(120591))) minus ℎ

119872

sum

119895=1

cos (120579119895(120591))

(47)





[HquVcl] = 0 (48)


H = minus

119873

sum

119894

119872

sum

119895

(1198691119904119894119895119904119894+1119895

+ 1198692119904119894119895119904119894119895+1) (49)



2ℎ + 2V = 119873119872 (50)


119885 = sum


119910V= 119910

1198721198732

sum


(51)



120578dagger

119902=119890minus1198941205874

radic119872sum

119895

119890minus(2120587119894119872)119902119895

(119887dagger

119895119906119902+ 119894119887

119895V119902)

120578119902=1198901198941205874

radic119872sum

119895

119890(2120587119894119872)119902119895

(119887119895119906119902minus 119894119887

dagger

119895V119902)

(52)



V+(minus)

= (2 sinh 21198701)1198732

119890minussum119902120598119902(120578dagger

119902120578119902minus12) (53)

where119870119894= 120573119869

119894and 120598


cosh 120598119902= cosh 2119870lowast

1cosh 2119870

2


1sinh 2119870

2cos 119902

(54)



by

V2

119863

= prod

0le119902le120587

(120582119902(120578

dagger

119902120578119902+ 120578

dagger


2sin2

119902)) (55)


with

120582119902= 2120572 sin 119902 (1 + 1205722sin2

119902)12

(56)


2V119902119906119902

1199062119902minus V2

119902

=

119886119902

119887119902

for Hqu



1coth 2119870

2

for V


1tanh119870

2cos 119902)



1 sinh 2119870

2

for V1015840

minus1

120572 sin 119902for V2

119863

(57)



119886119902

119887119902

=sin 119902

cosh 2119870lowast


1coth 2119870

2

(58)


sinh 2119870lowast

1coth 2119870

2= minus119886 (119871 minus 1)

119887 (119871)

tanh2119870lowast

1=119886 (119871) minus 119887 (119871)

119886 (119871) + 119887 (119871)

119886 (119871 minus 1)

119886 (119871) + 119887 (119871)= minus coth 2119870

2tanh119870lowast

1

(59)

or inversely as

cosh 2119870lowast

1=119886 (119871)

119887 (119871)


1

119886 (119871 minus 1)

radic(119886 (119871))2

minus (119887 (119871))2

(60)

where

119886 (119871) = 119886 (119871)

[1198712]

sum

119897=0

(119871

2119897)

119887 (119871) = 119887 (119871)

[(119871minus1)2]

sum

119897=0

(119871

2119897 + 1)

119886 (0) = Γ

(61)


119887 (119871)ge 1

1198862

(119871) le 1198862

(119871 minus 1) + 1198872

(119871)

(62)


119871

sum

119896=1

119887 (119896)

[(119871minus1)2]

sum

119897=0

(119896

2119897 + 1)

119897

sum

119894=1

(minus1)119894 cos119896minus2119894119902

+

119871minus1

sum

119896=1

119887 (119896) cos119896119902 = 0

(63)

Γ +

119871minus2

sum

119896=1

119886 (119896) cos119896119902

+

119871

sum

119896=1

119886 (119896)

[1198962]

sum

119897=0

(119896

2119897)

119897

sum

119894=1

(119897

119894) (minus1)

119894 cos119871minus2119894119902 = 0

(64)




10038161003816100381610038161003816Λ

119902

10038161003816100381610038161003816= 2

119901+110038161003816100381610038161003816cos(119871minus1)11990210038161003816100381610038161003816

sdot ((119886 (119871) cos 119902 + 119886 (119871 minus 1))2 + 1198872

(119871) sin2

119902)

12

(65)


by

119870lowast

1= 119870

2 (66)



119886 (119871) = plusmn119886 (119871 minus 1) (67)


10038161003816100381610038161003816Λ

119902

10038161003816100381610038161003816= 2

119901+110038161003816100381610038161003816119886 (119871) cos(119871minus1)11990210038161003816100381610038161003816

sdot ((cos 119902 plusmn 1)2 + (119887 (119871)119886 (119871)

)

2

sin2

119902)

12

(68)



⟨120590119909

119895119896120590119909

119895+119903119896⟩Is= ⟨Ψ

0

1003816100381610038161003816Vminus12

1120590119909

119895120590119909

119895+119903V12

1

1003816100381610038161003816Ψ0⟩

= ⟨Φ0

1003816100381610038161003816Vminus12

1120590119909

119895120590119909

119895+119903V12

1

1003816100381610038161003816Φ0⟩

= ⟨(Vminus12

1120590119909

119895V12

1) (Vminus12

1120590119909

119895+119903V12

1)⟩

qu

= cosh2119870lowast

1⟨120590

119909

119895120590119909

119895+119903⟩qu


1⟨120590

119910

119895120590119910

119895+119903⟩qu

(69)


119895+119903⟩qu = ⟨120590

119910

119895120590119909

119895+119903⟩qu = 0 for 119903 = 0 and

Ψ0= Φ

0 (70)






119887119902


1tanh119870

2cos 119902)



1 sinh 2119870

2

(71)



tanh 2119870lowast

1tanh119870

2= minus

119887 (119871)

119887 (119871 minus 1)= minus

119886 (119871)

119887 (119871 minus 1)

119886 (119871 minus 1)

119887 (119871 minus 1)= 1

tanh 2119870lowast

1

sinh 21198702

= minus119886lowast

(119871)

119887 (119871 minus 1)

(72)

or inversely as

sinh21198702=

119886 (119871)

2 (119886lowast

(119871))

tanh 2119870lowast

1= minus

1

119886 (119871 minus 1)radic119886 (119871) (2119886

lowast

(119871) + 119886 (119871))

(73)

where

119886lowast

(119871) = 119886 (119871 minus 2) minus 119886 (119871)

[1198712]

sum

119897=0

(119871

2119897) 119897 (74)


119886 (119871) (2119886lowast

(119871) + 119886 (119871)) le 1198862

(119871 minus 1) (75)


119871minus2

sum

119896=1

119887 (119896) cos119896119902

+

119871

sum

119896=1

119887 (119896)

[(119896minus1)2]

sum

119897=0

(119896

2119897 + 1)

119897

sum

119894=1

(119897

119894) (minus1)

119894 cos119896minus2119894119902

= 0

(76)

Γ +

119871minus3

sum

119896=1

119896cos119896119902 minus119871minus1

sum

119896=1

119886 (119896)

[1198962]

sum

119897=0

(119896

2119897) 119897cos119896minus2119902

+

119871

sum

119896=1

119886 (119896)

[1198962]

sum

119897=0

(119896

2119897)

119897

sum

119894=2

(119897

119894) (minus1)

119894 cos119896minus2119894119902 = 0

(77)



10038161003816100381610038161003816Λ

119902

10038161003816100381610038161003816= 2

119901+110038161003816100381610038161003816cos119871minus211990210038161003816100381610038161003816


+ sin2

119902 (119887 (119871) cos 119902 + 119887 (119871 minus 1)))12

(78)





(119871) + 119886 (119871) (79)


10038161003816100381610038161003816Λ

119902

10038161003816100381610038161003816= 2

119901+110038161003816100381610038161003816cos119871minus211990210038161003816100381610038161003816


+ sin2

119902 (119887 (119871) cos 119902 + 119887 (119871 minus 1)))12

(80)



⟨120590119909

119895119896120590119909

119895+119903119896⟩Is= ⟨120590

119909

119895120590119909

119895+119903⟩qu (81)

where Ψ1015840



Ψ1015840

0= Φ

0 (82)



119863 In this case




119892M(120579) ofM


120574 = 0 and thus B = 0



119895119896(119861

119895119896) (M

119899)119895119896


119895 119896 ge 0119874

+

(2119873) 119886(119895 minus 119896) + 119886(119895 + 119896) 1198920if 119895 = 119896 = 0radic2119892

119897if

either 119895 = 0 119896 = 119897or 119895 = 119897 119896 = 0

119892119895minus119896+ 119892

119895+119896 119895 119896 gt 0


119895+119896+2 119895 119896 ge 0

119874plusmn

(2119873 + 1) 119886(119895 minus 119896) ∓ 119886(119895 + 119896 + 1) 119892119895minus119896∓ 119892

119895+119896+1 119895 119896 ge 0

119874minus


119895+119896+2 119895 119896 ge 0

Appendices

A Symmetry Classes

See Table 1



Hqu = minus119872

sum

119895=1

(119869119909

119895120590119909

119895120590119909

119895+1+ 119869

119910

119895120590119910

119895120590119910

119895+1

minus (1198691015840119909

119895120590119909

119895120590119909

119895+2+ 119869

1015840119910

119895120590119910

119895120590119910

119895+2) 120590

119911

119895+1+ ℎ120590

119911

119895)

(B1)

where 1198691015840119909119895= (12)(119860

119895119895+2+ 120574119861

119895119895+2) and 1198691015840119910

119895= (12)(119860

119895119895+2minus




Tr [119890(120573qu119899)H119909

119886 119890(120573qu2119899)H

119911

119890(120573qu119899)H

119910

119887 119890(120573qu119899)H

119910

119886 119890(120573qu2119899)H

119911

119890(120573qu119899)H

119909

119887 ]119899

(B2)


120572= sum

119872

119895isin120572((12)119869

120583

119895(120590

120583

119895120590120583

119895+1+ 120590

120583

119895+1120590120583

119895+2) minus

1198691015840120583

119895120590120583

119895120590119911

119895+1120590120583

119895+2) and H119911

= ℎsum119872

119895=1120590119911

119895 for 120583 isin 119909 119910 and once




Tr [I1205901119890(120573qu119899)H

119909

119886 I1199041119890(120573qu2119899)H

119911

119890(120573qu119899)H

119910

119887 I1205911119890(120573qu119899)H

119910

119886 I1199041119890(120573qu2119899)H

119911

119890(120573qu119899)H

119909

119887 ]119899


sum

120590119895119901 120591119895119901119904119895119901

119899

prod

119901=1

[⟨119901

10038161003816100381610038161003816119890(120573qu119899)H

119909

119886100381610038161003816100381610038161199042119901minus1⟩ ⟨ 119904

2119901minus1

10038161003816100381610038161003816119890(120573qu2119899)H

119911

119890(120573qu119899)H

119910

119887

10038161003816100381610038161003816120591119901⟩ ⟨ 120591

119901

10038161003816100381610038161003816119890(120573qu119899)H

119910

119886100381610038161003816100381610038161199042119901⟩ ⟨ 119904

2119901

10038161003816100381610038161003816119890(120573qu2119899)H

119911

119890(120573qu119899)H

119909

11988710038161003816100381610038161003816119901+1⟩]

(B3)



⟨119901

10038161003816100381610038161003816119890(120573119899)

H119909119886100381610038161003816100381610038161199042119901minus1⟩ ⟨ 119904

2119901minus1

10038161003816100381610038161003816119890(120573qu2119899)H

119911

119890(120573qu119899)H

119910

119887

10038161003816100381610038161003816120591119901⟩ ⟨ 120591

2119901minus1

10038161003816100381610038161003816119890(120573qu119899)H

119910

119886100381610038161003816100381610038161199042119901⟩ ⟨ 119904

2119901

10038161003816100381610038161003816119890(120573qu2119899)H

119911

119890(120573qu119899)H

119909

119887100381610038161003816100381610038162119901⟩

= 119890(120573qu119899)H

119909

119886(119901)

119890(120573qu2119899)H

119911(2119901minus1)

119890(120573qu119899)H

119910

119887(119901)

119890(120573qu119899)H

119910

119886(119901)

119890(120573qu2119899)H

119911(2119901)

119890(120573qu119899)H

119909

119887(119901)

⟨119901| 119904

2119901minus1⟩ ⟨ 119904

2119901minus1| 120591

119901⟩ ⟨ 120591

119901| 119904

2119901⟩

sdot ⟨ 1199042119901|

119901+1⟩

(B4)

where H119909

120572(119901) = sum

119872

119895isin120572((12)119869

119909

119895(120590

119895119901120590119895+1119901

+ 120590119895+1119901

120590119895+2119901

) +

1198691015840119909

119895+1120590119895119901119904119895+1119901

120590119895+2119901

) H119910

120572(119901) = sum

119872

119895isin120572((12)119869

119910

119895(120591

119895119901120591119895+1119901

+

120591119895+1119901

120591119895+2119901

) + 1198691015840119910

119895+1120591119895119901119904119895+1119901

120591119895+2119901

) andH119911

(119901) = sum119872

119895=1119904119895119901 We


⟨119901| 119904

2119901minus1⟩ ⟨ 119904

2119901minus1| 120591

119901⟩ ⟨ 120591

2119901minus1| 119904

2119901⟩ ⟨ 119904

2119901|

119901+1⟩

=1

24119872

sdot

119872

prod

119895=1


(B5)



119899

119899

sum

119901=1

(sum

119895isin119886

(119869119909

119895

2(120590

119895119901120590119895+1119901

+ 120590119895+1119901

120590119895+2119901

) minus 1198691015840119909

119895+1120590119895119901119904119895+1119901

120590119895+2119901

)

+sum

119895isin119887

(

119869119910

119895

2(120591

119895119901120591119895+1119901

+ 120591119895+1119901

120591119895+2119901

) minus 1198691015840119910

119895+1120591119895119901119904119895+12119901minus1

120591119895+2119901

)

+ sum

119895isin119886

(

119869119910

119895

2(120591

119895119901120591119895+1119901

+ 120591119895+1119901

120591119895+2119901

) minus 1198691015840119910

119895+1120591119895119901119904119895+12119901

120591119895+2119901

)

+sum

119895isin119887

(119869119909

119895

2(120590

119895119901+1120590119895+1119901+1

+ 120590119895+1119901+1

120590119895+2119901+1

) minus 1198691015840119909

119895+1120590119895119901+1

119904119895+12119901

120590119895+2119901+1

))

+

119899

sum

119901=1

(

119872

sum

119895=1

((120573quℎ

2119899minus119894120587

4) 119904

1198952119901+ (120573quℎ

2119899+119894120587

4) 119904

1198952119901) +

119872

sum

119895=1

119894120587

4(120590

1198951199011199041198952119901minus1

minus 120590119895119901+1

1199042119901+ 120591

119895119901(119904

1198952119901minus 119904

1198952119901minus1)))

+ 4119899119872 ln 2

(B6)



119909 (120573qu119899)119869119910

(120573qu119899)1198691015840119909 (120573qu119899)119869




S1

S2

S3

S4

1205901

1205911

1205902

1205912

1 2 3 4 5 6 7 8


Trotter

direction

p darr








Tr[

[

119872minus1

prod

119895=1

(119890(120573qu119899)H

119909

119895119895+1119890(120573qu119899)H

119909


119909

119895119872119890(120573qu2119899(119872minus1))

H119911119890(120573qu119899)H

119910


119910

119895119895+2119890(120573qu119899)H

119910

119895119895+1)]

]

119899


Tr[

[

119872

prod

119895=1

((

119872minus119895

prod

119896=1

119890(120573qu119899)H

119909

119895119895+119896)119890(120573qu2119899(119872minus1))

H119911(


prod

119896=0

119890(120573qu119899)H

119910

119895119872minus119896))]

]

119899

(B7)

where H120583

119895119896= 119869

120583

119895119896120590120583

119895120590120583

119896prod

119896minus1

119897=1(minus120590

119911

119897) for 120583 isin 119909 119910 and H119911

=

ℎsum119872

119895=1120590119911

119895



Tr[

[

119872minus1

prod

119895=1

(I120590119895(

119872minus119895

prod

119896=1

119890(120573qu119899)H

119909

119895119895+119896)119890(120573qu(119872minus1)119899)

H119911I119904119895(


prod

119896=0

119890(120573qu119899)H

119910

119895119872minus119896) I120591119895)]

]

119899


sum

120590119895119901 120591119895119901

119899minus1

prod

119901=0

119872minus1

prod

119895=1

(⟨119895+119895119901

10038161003816100381610038161003816(

119872minus119895

prod

119896=1

119890(120573qu119899)H

119909

119895119895+119896)119890(120573qu119899(119872minus1))

H119911 10038161003816100381610038161003816119904119895+119895119901⟩ ⟨ 119904

119895+119895119901

10038161003816100381610038161003816(


prod

119896=0

119890(120573qu119899)H

119910

119895119872minus119896)10038161003816100381610038161003816120591119895+119895119901⟩ ⟨ 120591

119895+119895119901|

119895+119895119901+1⟩)

(B8)


⟨119895+119895119901

10038161003816100381610038161003816(

119872minus119895

prod

119896=1

119890(120573qu119899)H

119909

119895119895+119896)119890(120573qu119899(119872minus1))

H119911 10038161003816100381610038161003816119904119895+119895119901⟩

sdot ⟨ 119904119895+119895119901

10038161003816100381610038161003816(


prod

119896=0

119890(120573qu119899)H

119910

119895119872minus119896)10038161003816100381610038161003816120591119895+119895119901⟩ ⟨ 120591

119895+119895119901|

119895+119895119901+1

⟩

= 119890(120573qu119899)sum

119872minus119895

119896=1(H119909119895119895+119896

(119901)+H119910

119895119895+119896(119901)+(1119899(119872minus1))H119911)

⟨119895+119895119901

|

119904119895+119895119901⟩ ⟨ 119904

119895+119895119901| 120591

119895+119895119901⟩ ⟨ 120591

119895+119895119901|

119895+119895119901+1⟩

(B9)


S1

S2

S3

S4

S5

S6

S7

S8

S9

S10

1 2 3 4 5 6 7 8 9


Trotter

direction

p darr



where H119909

119895119896(119901) = sum

119872

119896=119895+1119869119909

119895119896120590119895119901120590119896119901prod

119896minus1

119897=119895+1(minus119904

119897119901) H119910

119895119896(119901) =

sum119872

119896=119895+1119869119910

119895119896120591119895119901120591119896119901prod

119896minus1

119897=119895+1(minus119904

119897119901) andH119911

119901= ℎsum

119872

119895=1120590119911

119895119901 Finally


⟨119901| 119904

119901⟩ ⟨ 119904

119901| 120591

119901⟩ ⟨ 120591

119901|

119901+1⟩ = (

1

2radic2)

119872

sdot

119872

prod

119895=1


(B10)



sum

119901=1

119872

sum

119895=1

(120573qu

119899

119872

sum

119896=119895+1

(119869119909

119895119896120590119895119895+119895119901

120590119896119895+119895119901

+ 119869119910

119895119896120591119895119895+119895119901

120591119896119895+119895119901

)

119896minus1

prod

119897=119895+1

(minus119904119897119901) + (

120573qu

119899 (119872 minus 1)ℎ minus119894120587

4) 119904

119895119895+119895119901

+119894120587

4(1 minus 120590

119895119895+119895119901+ 120591

119895119895+119895119901+ 120590

119895119895+119895119901119904119895119895+119895119901

minus 120591119895119895+119895119901

119904119895119895+119895119901

minus 120590119895119895+119895119901+1

120591119895119895+119895119901

)) + 1198991198722 ln 1

2radic2

(B11)


119895119896





119909

119895119896 (120573qu119899)119869

119910

119895119896 and



1205901

S1

1205911

S2

1205902

S3

1205912

S4

1205903

S5

1 2 3 4 5 6 7 8 9 10

Trotter

direction

p darr






119863(55)


minus1

120572 sin 119902=119887 (1) sin 119902

Γ + 119886 (1) cos 119902 (C1)



minus1

120572 sin 119902=

119887 (1)

minus2119886 (2) sin 119902

if Γ = minus119886 (2) 119886 (1) = 119887 (2) = 0

(C2)


120572 =2119886 (2)

119887 (1) Γ = minus119886 (2) 119886 (1) = 119887 (2) = 0 (C3)


minus1

120572 sin 119902= minus

119887 (1) minus 119887 (3) + 119887 (2) cos 1199022 sin 119902 (119886 (2) + 119886 (3) cos 119902)

if Γ = minus119886 (2) 119886 (1) = minus119886 (3) 119887 (3) = 0

(C4)


120572 =2119886 (3)

119887 (2)

119886 (2)

119886 (3)=119887 (1) minus 119887 (3)

119887 (2)

Γ = minus119886 (2) 119886 (1) = minus119886 (3) 119887 (3) = 0

(C5)


minus1

120572 sin 119902

=sin 119902sum119898

119896=1119887 (119896)sum

[(119896minus1)2]

119897=0( 119896

2119897+1)sum

119897

119894=0( 119897119894) (minus1)


119896=2119886 (119896)sum

[1198962]

119897=0(minus1)

119897

( 119896

2119897) sin2119897

119902cos119896minus2119897119902

(C6)


[1198982]

sum

119895=1

119886 (2119895) + cos 119902[1198982]

sum

119895=1

119886 (2119895 + 1) + sin2

119902

sdot (

[1198982]

sum

119895=1

119886 (2119895)

119895

sum

119894=1

(119895

119894) (minus1)

119894 sin2(119894minus1)

119902

+ cos 119902[(119898minus1)2]

sum

119895=1

119886 (2119895 + 1)

119895

sum

119894=1

(119895

119894) (minus1)

119894 sin2(119894minus1)

119902

+

119898

sum

119896=2

119886 (119896)

[1198962]

sum

119897=1

(minus1)119897

(119896

2119897) sin2(119897minus1)

119902cos119896minus2119897119902)

(C7)


Γ = minus

[1198982]

sum

119895=1

119886 (2119895)

119886 (1) = minus

[(119898+1)2]

sum

119895=1

119886 (2119895 + 1) = 0

(C8)



sin2

119902(

[1198982]

sum

119895=1

119886 (2119895)

119895

sum

119894=1

(119895

119894)

119894minus1

sum

119901=0

(minus1)119894+119901 cos2119901119902

+

[(119898minus1)2]

sum

119895=1

119886 (2119895 + 1)

119895

sum

119894=1

(119895

119894)

119894minus1

sum

119901=0

(minus1)119894+119901 cos2119901+1119902 +

[(119898minus1)2]

sum

119895=1

119886 (2119895 + 1)

119895

sum

119897=1

(2119895 + 1

2119897)

119897minus1

sum

119901=0


+

[1198982]

sum

119895=1

119886 (2119895)

119895

sum

119897=1

(2119895

2119897)

119897minus1

sum

119901=0


(C9)


Disclosure




Acknowledgments


Endnotes



integer



and 119861119895119896





7 Thus matrices 119860119895119896

and 119861119895119896




1V

2V12

1of


=

V12

2V

1V12

2 whereV

1= (2 sinh 2119870

1)1198722


1sum119872

119894120590119909

119894 V2=

1198901198702 sum119872

119894=1120590119911

119894120590119911


119894= 119890



119887119902(5) as

119886119902= Γ +

119871

sum

119896=1

119886 (119896)

[1198962]

sum

119897=0

(119896

2119897)

119897

sum

119894=0

(119897

119894) (minus1)

minus119894 cos119896minus2119894119902

119887119902= tan 119902

sdot

119871

sum

119896=1

119887 (119896)

[(119896minus1)2]

sum

119897=0

(119896

2119897 + 1)

119897

sum

119894=0

(119897

119894) (minus1)

minus119894 cos119896minus2119894119902

(lowast)


119895=1(minus1)

119895

119886(2119895)


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













⟨119891 (120590119911

)⟩qu = ⟨119891 (119904)⟩cl (15)



⟨120590119911

119895120590119911

119895+119903⟩qu= ⟨119904

119895119901119904119895+119903119901⟩cl

⟨

119903

prod

119895=1

120590119911

119895⟩

qu

= ⟨

119903

prod

119895=1

119904119895119901⟩

cl

(16)



Tr [V119886V

119887]119899

V120572= sum

119895isin120572

119890(120573qu119899)H

119911

119895 119890(120573qu119899)H

119909

119895 119890(120573qu119899)H

119910

119895 119890(120573qu119899)H

119911

119895

(17)

where H120583

119895= 119869

120583

119895120590120583

119895120590120583

119895+1for 120583 isin 119909 119910 H119911

119895= (ℎ4)(120590

119911

119895+ 120590

119911

119895+1)



119904119901= sum

119904119901

| 119904119901⟩⟨ 119904

119901| where | 119904

119901⟩ = |119904

1119901 119904

2119901 119904

119872119901⟩



Tr I 1199041

V119886I 1199042


1199042119899minus1V

119886I 1199042119899

V119887


sum

119904119895119901

2119899

prod

119901isin119886

⟨ 119904119901

10038161003816100381610038161003816V

119886

10038161003816100381610038161003816119904119901+1⟩ ⟨ 119904

119901+1

10038161003816100381610038161003816V

119887

10038161003816100381610038161003816119904119901+2⟩

(18)


⟨ 119904119901

10038161003816100381610038161003816V

120572

10038161003816100381610038161003816119904119901+1⟩ =

119872

prod

119895isin120572

⟨119904119895119901 119904

119895+1119901

10038161003816100381610038161003816M10038161003816100381610038161003816119904119895119901+1

119904119895+1119901+1

⟩ (19)

where

M =

(((((((

(

119890120573quℎ119899 cosh(

2120573qu120574

119899119861119895) 0 0 sinh(

2120573qu120574

119899119861119895)

0 cosh(2120573qu

119899119860

119895) sinh(

2120573qu

119899119860

119895) 0

0 sinh(2120573qu

119899119860

119895) cosh(

2120573qu

119899119860

119895) 0

sinh(2120573qu120574

119899119861119895) 0 0 119890

minusℎ119899 cosh(2120573qu120574

119899119861119895)

)))))))

)

(20)


⟨ 119904119901

10038161003816100381610038161003816V

120572

10038161003816100381610038161003816119904119901+1⟩ =

119872

prod

119895isin120572

119890minus120573clH119895119901 (21)

whereH119895119901

can be written as

H119895119901= minus1

4(119869

V119895119904119895119901119904119895119901+1

+ 119869ℎ

119895119904119895119901119904119895+1119901

+ 119869119889

119895119904119895+1119901

119904119895119901+1

+ 119867(119904119895119901+ 119904

119895+1119901) + 119862

119895)

(22)


H119895119901= minus1

4(119869

ℎ

119895(119904

119895119901119904119895+1119901

+ 119904119895119901+1

119904119895+1119901+1

)

+ 119869V119895(119904

119895119901119904119895119901+1

+ 119904119895+1119901

119904119895+1119901+1

)

+ 119869119889

119895(119904

119895119901119904119895+1119901+1

+ 119904119895119901+1

119904119895+1119901

)

+ 119867(119904119895119901+ 119904

119895+1119901+ 119904

119895119901+1+ 119904

119895+1119901+1) + 119862

119895)

(23)

where

120573cl119869ℎ

119895= log

sinh (4120573qu119899) 120574119861119895sinh (4120573qu119899)119860119895


120573cl119869119889

119895= log

tanh (2120573qu119899)119860119895

tanh (2120574120573qu119899) 119861119895

120573cl119869V119895= log coth

2120574120573qu

119899119861119895coth

2120573qu

119899119860

119895

120573cl119867 =120573quℎ

119899

120573cl119862119895= log sinh

2120573qu

119899119860

119895sinh

2120574120573qu

119899119861119895

(24)


119904119895119901119904119895+1119901

119904119895119901+1

119904119895+1119901+1

= 1 (25)



Hcl =2119899

sum

119901isin119886

119872

sum

119895isin119886

H119895119901+

2119899

sum

119901isin119887

119872

sum

119895isin119887

H119895119901 (26)

whereH119895119901




119895 119869

119889

119895 119867 rarr 0 and 119869V

119895rarr infin




Hcl = minus119899

sum

119901=1

119872

sum

119895=1

(119869120590

119895120590119895119901120590119895+1119901

+ 119869120591

119895120591119895119901120591119895+1119901

+ 119894119869120591119895119901(120590

119895119901minus 120590

119895119901+1))

(27)

Trotter

direction

p darr




120573cl119869120590

119895=120573qu

119899119869119909

119895

120573cl119869120591

119895=120573qu

119899119869119910

119895

120573cl119869 =1

2arctan 1

sinh (120573qu119899) ℎ

(28)



Tr [U1U

2]119899

U1= 119890

(120573qu2119899)H119909119890(120573qu2119899)H119911119890

(120573qu2119899)H119910

U2= 119890

(120573qu2119899)H119910119890(120573qu2119899)H119911119890

(120573qu2119899)H119909

(29)


= sum119872

119895=1119869120583

119895120590120583

119895120590120583

119895+1for 120583 isin 119909 119910 and H119911

=

sum119872

119895=1120590119911

119895


=

sum119901

|119901⟩⟨

119901| and I

120591119901= sum

120591119901

| 120591119901⟩⟨ 120591




Tr I1

U1I 1205911

U2I2


1205912119899U

2


sum

120590119895119901 120591119895119901

119899

prod

119901=1

⟨119901

10038161003816100381610038161003816U

1

10038161003816100381610038161003816120591119901⟩ ⟨ 120591

119901

10038161003816100381610038161003816U

2

10038161003816100381610038161003816119901+1⟩

(30)



⟨119901

10038161003816100381610038161003816U

1

10038161003816100381610038161003816120591119901⟩ ⟨ 120591

119901

10038161003816100381610038161003816U

2

10038161003816100381610038161003816119901+1⟩

= 119890(120573qu119899)((12)(H

119909

119901+H119909119901+1

)+H119910

119901) ⟨119901

10038161003816100381610038161003816119890(120573qu2119899)H

119911 10038161003816100381610038161003816120591119901⟩ ⟨ 120591

119901

10038161003816100381610038161003816

sdot 119890(120573qu2119899)H

119911 10038161003816100381610038161003816119901+1⟩

= 1198622119872

119890(120573qu119899)((12)(H

119909

119901+H119909119901+1

)+H119910

119901)+(1198942)119863sum119872

119895=1120591119895119901(120590119895119901minus120590119895119901+1)

(31)

where H119909

119901= sum

119872

119895=1119869119909

120590119895119901120590119895+1119901

H119910

119901= sum

119872

119895=1119869119910

120591119895119901120591119895+1119901


⟨120590119895119901

10038161003816100381610038161003816119890119886120590119911

11989510038161003816100381610038161003816120591119895119901⟩

=1

2cosh (2119886) 119890119894(12) arctan(1 sinh(2119886))120590119895119901120591119895119901

(32)


119895119901and 120591



119895119901


119895 119869120591

119895rarr infin



⟨119891 (120590119909

)⟩qu = ⟨119891 (120590)⟩cl

⟨119891 (120590119910

)⟩qu = ⟨119891 (120591)⟩cl (33)



⟨120590119909

119895120590119909

119895+119903⟩qu= ⟨120590

119895119901120590119895+119903119901⟩cl

⟨120590119910

119895120590119910

119895+119903⟩qu= ⟨120591

119895119901120591119895+119903119901⟩cl

(34)



⟨ 119904119901

10038161003816100381610038161003816119890(120573qu119899)V120572 10038161003816100381610038161003816

119904119901+1⟩

=

119872

prod

119895isin120572

120596119895

(119904119895119901 119904

119895+1119901 119904

119895119901+1 119904

119895+1119901+1)

(35)


(119904119895119901119904119895+1119901

119904119895119901+1

119904119895+1119901+1

)


120596119895

1(+1 +1 +1 +1) = 119890

ℎ120573qu119899 cosh(2120573qu120574

119899119861119895)

120596119895


minus120573quℎ119899 cosh(2120574120573qu

119899119861119895)

120596119895

3(minus1 +1 +1 minus1) = 120596

119895


= sinh(2120573qu

119899119860

119895)

120596119895

5(+1 minus1 +1 minus1) = 120596

119895


= cosh(2120573qu

119899119860

119895)

120596119895

7(minus1 minus1 +1 +1) = 120596

119895


= sinh(2120573qu120574

119899119861119895)

(36)




3 120596

119894

4 120596

119894

7 120596

119894

8rarr 0 and

weights 120596119894

1 120596

119894

2 120596

119894

5 120596

119894





12059041

12059031

12059021

12059011

12059042

12059032

12059022

12059012

12059043

12059033

12059023

12059013

12059141

12059131

12059121

12059111

12059142

12059132

12059122

12059112

12059143

12059133

12059123

12059113

1 2 3 4 5 6

120590j1 = j1

120591j1 = j2

120590j2 = j3

120591j2 = j4

120590j3 = j5

120591j3 = j6

Trotter

direction

p darr




119895 and the 119869



1205961 1205962 1205963 1205964 1205965 1205966 1205967 1205968




sum

120590119895119901=plusmn1

1

4(

119899

prod

119901isin119886

119872

prod

119895isin119886

+

119899

prod

119901isin119887

119872

prod

119895isin119887

)

sdot [(1 minus 119904119895119901119904119895+1119901

) (1 + 119904119895119901119904119895119901+1

) cosh2120573qu

119899119860

119895119895+1

+ (1 minus 119904119895119901119904119895+1119901

) (1 minus 119904119895119901119904119895119901+1

) sinh2120573qu

119899119860

119895119895+1

+ (1 + 119904119895119901119904119895+1119901

) (1 minus 119904119895119901119904119895119901+1

) sinh2120573qu120574

119899119861119895119895+1

+ (1 + 119904119895119901119904119895119901+1

) (1 + 119904119895119901119904119895119901+1

) 119890(120573qu119899)ℎ119904119895119901


119899119861119895119895+1]

(37)



Trotter

direction

p darr







Hqu =1

2sum

1le119895le119896le119872

((119860119895119896+ 120574119861

119895119896) 120590

119909

119895120590119909

119896

+ (119860119895119896minus 120574119861

119895119896) 120590

119910

119895120590119910

119896)(

119896minus1

prod

119897=119895+1

minus 120590119911

119897) minus ℎ

119872

sum

119895=1

120590119911

119895

(38)




Hqu]

= TrV119899

(39)




119872

prod

119894=1

119889N (120591119894) ⟨N (120591

119872)1003816100381610038161003816 119890

minusΔ120591H 1003816100381610038161003816N (120591119872minus1)⟩

sdot ⟨N (120591119872minus1)1003816100381610038161003816 119890


1)1003816100381610038161003816

sdot 119890minusΔ120591H 1003816100381610038161003816N (120591119872)⟩

(40)




119889120591|N (120591)⟩)

= ⟨N (120591) | N (120591)⟩ minus Δ120591 ⟨N (120591)| 119889119889120591|N (120591)⟩

minus Δ120591 ⟨N (120591)| Hqu (S) |N (120591)⟩ + 119874 ((Δ120591)2

)

= 119890minusΔ120591(⟨N(120591)|(119889119889120591)|N(120591)⟩+H(N(120591)))

Δ120591

119872

sum

119894=1

997888rarr int

120573

0

119889120591

119872

prod

119894=1

119889N (120591119894) 997888rarr DN (120591)

(41)


119885 = int

N(120573)

N(0)

DN (120591) 119890minusint

120573

0119889120591H(N(120591))minusS119861 (42)


S119861= int

120573

0

119889120591 ⟨N (120591)| 119889119889120591|N (120591)⟩ (43)



⟨N| S |N⟩ = N (44)


⟨N| 119878119894 |N⟩ = minus119878119873119894

(45)



N = (119873119909

119873119910

119873119911


0 le 120579 le 120587 0 le 120601 le 2120587

(46)


H (N (120591)) = ⟨N (120591)| Hqu |N (120591)⟩

= sum

1le119895le119896le119872

((119860119895119896+ 120574119861

119895119896)119873

119909

119895(120591)119873

119909

119896(120591)

+ (119860119895119896minus 120574119861

119895119896)119873

119910

119895(120591)119873

119910

119896(120591))

119896minus1

prod

119897=119895+1

(minus119873119911

119897(120591))

minus ℎ

119872

sum

119895=1

119873119911

119895(120591) = sum

1le119895le119896le119872

(119860119895119896

cos (120601119895(120591) minus 120601

119896(120591))

+ 119861119895119896120574 cos (120601

119895(120591) + 120601

119896(120591))) sin (120579

119895(120591))

sdot sin (120579119896(120591))

119896minus1

prod

119897=119895+1

(minus cos (120579119897(120591))) minus ℎ

119872

sum

119895=1

cos (120579119895(120591))

(47)





[HquVcl] = 0 (48)


H = minus

119873

sum

119894

119872

sum

119895

(1198691119904119894119895119904119894+1119895

+ 1198692119904119894119895119904119894119895+1) (49)



2ℎ + 2V = 119873119872 (50)


119885 = sum


119910V= 119910

1198721198732

sum


(51)



120578dagger

119902=119890minus1198941205874

radic119872sum

119895

119890minus(2120587119894119872)119902119895

(119887dagger

119895119906119902+ 119894119887

119895V119902)

120578119902=1198901198941205874

radic119872sum

119895

119890(2120587119894119872)119902119895

(119887119895119906119902minus 119894119887

dagger

119895V119902)

(52)



V+(minus)

= (2 sinh 21198701)1198732

119890minussum119902120598119902(120578dagger

119902120578119902minus12) (53)

where119870119894= 120573119869

119894and 120598


cosh 120598119902= cosh 2119870lowast

1cosh 2119870

2


1sinh 2119870

2cos 119902

(54)



by

V2

119863

= prod

0le119902le120587

(120582119902(120578

dagger

119902120578119902+ 120578

dagger


2sin2

119902)) (55)


with

120582119902= 2120572 sin 119902 (1 + 1205722sin2

119902)12

(56)


2V119902119906119902

1199062119902minus V2

119902

=

119886119902

119887119902

for Hqu



1coth 2119870

2

for V


1tanh119870

2cos 119902)



1 sinh 2119870

2

for V1015840

minus1

120572 sin 119902for V2

119863

(57)



119886119902

119887119902

=sin 119902

cosh 2119870lowast


1coth 2119870

2

(58)


sinh 2119870lowast

1coth 2119870

2= minus119886 (119871 minus 1)

119887 (119871)

tanh2119870lowast

1=119886 (119871) minus 119887 (119871)

119886 (119871) + 119887 (119871)

119886 (119871 minus 1)

119886 (119871) + 119887 (119871)= minus coth 2119870

2tanh119870lowast

1

(59)

or inversely as

cosh 2119870lowast

1=119886 (119871)

119887 (119871)


1

119886 (119871 minus 1)

radic(119886 (119871))2

minus (119887 (119871))2

(60)

where

119886 (119871) = 119886 (119871)

[1198712]

sum

119897=0

(119871

2119897)

119887 (119871) = 119887 (119871)

[(119871minus1)2]

sum

119897=0

(119871

2119897 + 1)

119886 (0) = Γ

(61)


119887 (119871)ge 1

1198862

(119871) le 1198862

(119871 minus 1) + 1198872

(119871)

(62)


119871

sum

119896=1

119887 (119896)

[(119871minus1)2]

sum

119897=0

(119896

2119897 + 1)

119897

sum

119894=1

(minus1)119894 cos119896minus2119894119902

+

119871minus1

sum

119896=1

119887 (119896) cos119896119902 = 0

(63)

Γ +

119871minus2

sum

119896=1

119886 (119896) cos119896119902

+

119871

sum

119896=1

119886 (119896)

[1198962]

sum

119897=0

(119896

2119897)

119897

sum

119894=1

(119897

119894) (minus1)

119894 cos119871minus2119894119902 = 0

(64)




10038161003816100381610038161003816Λ

119902

10038161003816100381610038161003816= 2

119901+110038161003816100381610038161003816cos(119871minus1)11990210038161003816100381610038161003816

sdot ((119886 (119871) cos 119902 + 119886 (119871 minus 1))2 + 1198872

(119871) sin2

119902)

12

(65)


by

119870lowast

1= 119870

2 (66)



119886 (119871) = plusmn119886 (119871 minus 1) (67)


10038161003816100381610038161003816Λ

119902

10038161003816100381610038161003816= 2

119901+110038161003816100381610038161003816119886 (119871) cos(119871minus1)11990210038161003816100381610038161003816

sdot ((cos 119902 plusmn 1)2 + (119887 (119871)119886 (119871)

)

2

sin2

119902)

12

(68)



⟨120590119909

119895119896120590119909

119895+119903119896⟩Is= ⟨Ψ

0

1003816100381610038161003816Vminus12

1120590119909

119895120590119909

119895+119903V12

1

1003816100381610038161003816Ψ0⟩

= ⟨Φ0

1003816100381610038161003816Vminus12

1120590119909

119895120590119909

119895+119903V12

1

1003816100381610038161003816Φ0⟩

= ⟨(Vminus12

1120590119909

119895V12

1) (Vminus12

1120590119909

119895+119903V12

1)⟩

qu

= cosh2119870lowast

1⟨120590

119909

119895120590119909

119895+119903⟩qu


1⟨120590

119910

119895120590119910

119895+119903⟩qu

(69)


119895+119903⟩qu = ⟨120590

119910

119895120590119909

119895+119903⟩qu = 0 for 119903 = 0 and

Ψ0= Φ

0 (70)






119887119902


1tanh119870

2cos 119902)



1 sinh 2119870

2

(71)



tanh 2119870lowast

1tanh119870

2= minus

119887 (119871)

119887 (119871 minus 1)= minus

119886 (119871)

119887 (119871 minus 1)

119886 (119871 minus 1)

119887 (119871 minus 1)= 1

tanh 2119870lowast

1

sinh 21198702

= minus119886lowast

(119871)

119887 (119871 minus 1)

(72)

or inversely as

sinh21198702=

119886 (119871)

2 (119886lowast

(119871))

tanh 2119870lowast

1= minus

1

119886 (119871 minus 1)radic119886 (119871) (2119886

lowast

(119871) + 119886 (119871))

(73)

where

119886lowast

(119871) = 119886 (119871 minus 2) minus 119886 (119871)

[1198712]

sum

119897=0

(119871

2119897) 119897 (74)


119886 (119871) (2119886lowast

(119871) + 119886 (119871)) le 1198862

(119871 minus 1) (75)


119871minus2

sum

119896=1

119887 (119896) cos119896119902

+

119871

sum

119896=1

119887 (119896)

[(119896minus1)2]

sum

119897=0

(119896

2119897 + 1)

119897

sum

119894=1

(119897

119894) (minus1)

119894 cos119896minus2119894119902

= 0

(76)

Γ +

119871minus3

sum

119896=1

119896cos119896119902 minus119871minus1

sum

119896=1

119886 (119896)

[1198962]

sum

119897=0

(119896

2119897) 119897cos119896minus2119902

+

119871

sum

119896=1

119886 (119896)

[1198962]

sum

119897=0

(119896

2119897)

119897

sum

119894=2

(119897

119894) (minus1)

119894 cos119896minus2119894119902 = 0

(77)



10038161003816100381610038161003816Λ

119902

10038161003816100381610038161003816= 2

119901+110038161003816100381610038161003816cos119871minus211990210038161003816100381610038161003816


+ sin2

119902 (119887 (119871) cos 119902 + 119887 (119871 minus 1)))12

(78)





(119871) + 119886 (119871) (79)


10038161003816100381610038161003816Λ

119902

10038161003816100381610038161003816= 2

119901+110038161003816100381610038161003816cos119871minus211990210038161003816100381610038161003816


+ sin2

119902 (119887 (119871) cos 119902 + 119887 (119871 minus 1)))12

(80)



⟨120590119909

119895119896120590119909

119895+119903119896⟩Is= ⟨120590

119909

119895120590119909

119895+119903⟩qu (81)

where Ψ1015840



Ψ1015840

0= Φ

0 (82)



119863 In this case




119892M(120579) ofM


120574 = 0 and thus B = 0



119895119896(119861

119895119896) (M

119899)119895119896


119895 119896 ge 0119874

+

(2119873) 119886(119895 minus 119896) + 119886(119895 + 119896) 1198920if 119895 = 119896 = 0radic2119892

119897if

either 119895 = 0 119896 = 119897or 119895 = 119897 119896 = 0

119892119895minus119896+ 119892

119895+119896 119895 119896 gt 0


119895+119896+2 119895 119896 ge 0

119874plusmn

(2119873 + 1) 119886(119895 minus 119896) ∓ 119886(119895 + 119896 + 1) 119892119895minus119896∓ 119892

119895+119896+1 119895 119896 ge 0

119874minus


119895+119896+2 119895 119896 ge 0

Appendices

A Symmetry Classes

See Table 1



Hqu = minus119872

sum

119895=1

(119869119909

119895120590119909

119895120590119909

119895+1+ 119869

119910

119895120590119910

119895120590119910

119895+1

minus (1198691015840119909

119895120590119909

119895120590119909

119895+2+ 119869

1015840119910

119895120590119910

119895120590119910

119895+2) 120590

119911

119895+1+ ℎ120590

119911

119895)

(B1)

where 1198691015840119909119895= (12)(119860

119895119895+2+ 120574119861

119895119895+2) and 1198691015840119910

119895= (12)(119860

119895119895+2minus




Tr [119890(120573qu119899)H119909

119886 119890(120573qu2119899)H

119911

119890(120573qu119899)H

119910

119887 119890(120573qu119899)H

119910

119886 119890(120573qu2119899)H

119911

119890(120573qu119899)H

119909

119887 ]119899

(B2)


120572= sum

119872

119895isin120572((12)119869

120583

119895(120590

120583

119895120590120583

119895+1+ 120590

120583

119895+1120590120583

119895+2) minus

1198691015840120583

119895120590120583

119895120590119911

119895+1120590120583

119895+2) and H119911

= ℎsum119872

119895=1120590119911

119895 for 120583 isin 119909 119910 and once




Tr [I1205901119890(120573qu119899)H

119909

119886 I1199041119890(120573qu2119899)H

119911

119890(120573qu119899)H

119910

119887 I1205911119890(120573qu119899)H

119910

119886 I1199041119890(120573qu2119899)H

119911

119890(120573qu119899)H

119909

119887 ]119899


sum

120590119895119901 120591119895119901119904119895119901

119899

prod

119901=1

[⟨119901

10038161003816100381610038161003816119890(120573qu119899)H

119909

119886100381610038161003816100381610038161199042119901minus1⟩ ⟨ 119904

2119901minus1

10038161003816100381610038161003816119890(120573qu2119899)H

119911

119890(120573qu119899)H

119910

119887

10038161003816100381610038161003816120591119901⟩ ⟨ 120591

119901

10038161003816100381610038161003816119890(120573qu119899)H

119910

119886100381610038161003816100381610038161199042119901⟩ ⟨ 119904

2119901

10038161003816100381610038161003816119890(120573qu2119899)H

119911

119890(120573qu119899)H

119909

11988710038161003816100381610038161003816119901+1⟩]

(B3)



⟨119901

10038161003816100381610038161003816119890(120573119899)

H119909119886100381610038161003816100381610038161199042119901minus1⟩ ⟨ 119904

2119901minus1

10038161003816100381610038161003816119890(120573qu2119899)H

119911

119890(120573qu119899)H

119910

119887

10038161003816100381610038161003816120591119901⟩ ⟨ 120591

2119901minus1

10038161003816100381610038161003816119890(120573qu119899)H

119910

119886100381610038161003816100381610038161199042119901⟩ ⟨ 119904

2119901

10038161003816100381610038161003816119890(120573qu2119899)H

119911

119890(120573qu119899)H

119909

119887100381610038161003816100381610038162119901⟩

= 119890(120573qu119899)H

119909

119886(119901)

119890(120573qu2119899)H

119911(2119901minus1)

119890(120573qu119899)H

119910

119887(119901)

119890(120573qu119899)H

119910

119886(119901)

119890(120573qu2119899)H

119911(2119901)

119890(120573qu119899)H

119909

119887(119901)

⟨119901| 119904

2119901minus1⟩ ⟨ 119904

2119901minus1| 120591

119901⟩ ⟨ 120591

119901| 119904

2119901⟩

sdot ⟨ 1199042119901|

119901+1⟩

(B4)

where H119909

120572(119901) = sum

119872

119895isin120572((12)119869

119909

119895(120590

119895119901120590119895+1119901

+ 120590119895+1119901

120590119895+2119901

) +

1198691015840119909

119895+1120590119895119901119904119895+1119901

120590119895+2119901

) H119910

120572(119901) = sum

119872

119895isin120572((12)119869

119910

119895(120591

119895119901120591119895+1119901

+

120591119895+1119901

120591119895+2119901

) + 1198691015840119910

119895+1120591119895119901119904119895+1119901

120591119895+2119901

) andH119911

(119901) = sum119872

119895=1119904119895119901 We


⟨119901| 119904

2119901minus1⟩ ⟨ 119904

2119901minus1| 120591

119901⟩ ⟨ 120591

2119901minus1| 119904

2119901⟩ ⟨ 119904

2119901|

119901+1⟩

=1

24119872

sdot

119872

prod

119895=1


(B5)



119899

119899

sum

119901=1

(sum

119895isin119886

(119869119909

119895

2(120590

119895119901120590119895+1119901

+ 120590119895+1119901

120590119895+2119901

) minus 1198691015840119909

119895+1120590119895119901119904119895+1119901

120590119895+2119901

)

+sum

119895isin119887

(

119869119910

119895

2(120591

119895119901120591119895+1119901

+ 120591119895+1119901

120591119895+2119901

) minus 1198691015840119910

119895+1120591119895119901119904119895+12119901minus1

120591119895+2119901

)

+ sum

119895isin119886

(

119869119910

119895

2(120591

119895119901120591119895+1119901

+ 120591119895+1119901

120591119895+2119901

) minus 1198691015840119910

119895+1120591119895119901119904119895+12119901

120591119895+2119901

)

+sum

119895isin119887

(119869119909

119895

2(120590

119895119901+1120590119895+1119901+1

+ 120590119895+1119901+1

120590119895+2119901+1

) minus 1198691015840119909

119895+1120590119895119901+1

119904119895+12119901

120590119895+2119901+1

))

+

119899

sum

119901=1

(

119872

sum

119895=1

((120573quℎ

2119899minus119894120587

4) 119904

1198952119901+ (120573quℎ

2119899+119894120587

4) 119904

1198952119901) +

119872

sum

119895=1

119894120587

4(120590

1198951199011199041198952119901minus1

minus 120590119895119901+1

1199042119901+ 120591

119895119901(119904

1198952119901minus 119904

1198952119901minus1)))

+ 4119899119872 ln 2

(B6)



119909 (120573qu119899)119869119910

(120573qu119899)1198691015840119909 (120573qu119899)119869




S1

S2

S3

S4

1205901

1205911

1205902

1205912

1 2 3 4 5 6 7 8


Trotter

direction

p darr








Tr[

[

119872minus1

prod

119895=1

(119890(120573qu119899)H

119909

119895119895+1119890(120573qu119899)H

119909


119909

119895119872119890(120573qu2119899(119872minus1))

H119911119890(120573qu119899)H

119910


119910

119895119895+2119890(120573qu119899)H

119910

119895119895+1)]

]

119899


Tr[

[

119872

prod

119895=1

((

119872minus119895

prod

119896=1

119890(120573qu119899)H

119909

119895119895+119896)119890(120573qu2119899(119872minus1))

H119911(


prod

119896=0

119890(120573qu119899)H

119910

119895119872minus119896))]

]

119899

(B7)

where H120583

119895119896= 119869

120583

119895119896120590120583

119895120590120583

119896prod

119896minus1

119897=1(minus120590

119911

119897) for 120583 isin 119909 119910 and H119911

=

ℎsum119872

119895=1120590119911

119895



Tr[

[

119872minus1

prod

119895=1

(I120590119895(

119872minus119895

prod

119896=1

119890(120573qu119899)H

119909

119895119895+119896)119890(120573qu(119872minus1)119899)

H119911I119904119895(


prod

119896=0

119890(120573qu119899)H

119910

119895119872minus119896) I120591119895)]

]

119899


sum

120590119895119901 120591119895119901

119899minus1

prod

119901=0

119872minus1

prod

119895=1

(⟨119895+119895119901

10038161003816100381610038161003816(

119872minus119895

prod

119896=1

119890(120573qu119899)H

119909

119895119895+119896)119890(120573qu119899(119872minus1))

H119911 10038161003816100381610038161003816119904119895+119895119901⟩ ⟨ 119904

119895+119895119901

10038161003816100381610038161003816(


prod

119896=0

119890(120573qu119899)H

119910

119895119872minus119896)10038161003816100381610038161003816120591119895+119895119901⟩ ⟨ 120591

119895+119895119901|

119895+119895119901+1⟩)

(B8)


⟨119895+119895119901

10038161003816100381610038161003816(

119872minus119895

prod

119896=1

119890(120573qu119899)H

119909

119895119895+119896)119890(120573qu119899(119872minus1))

H119911 10038161003816100381610038161003816119904119895+119895119901⟩

sdot ⟨ 119904119895+119895119901

10038161003816100381610038161003816(


prod

119896=0

119890(120573qu119899)H

119910

119895119872minus119896)10038161003816100381610038161003816120591119895+119895119901⟩ ⟨ 120591

119895+119895119901|

119895+119895119901+1

⟩

= 119890(120573qu119899)sum

119872minus119895

119896=1(H119909119895119895+119896

(119901)+H119910

119895119895+119896(119901)+(1119899(119872minus1))H119911)

⟨119895+119895119901

|

119904119895+119895119901⟩ ⟨ 119904

119895+119895119901| 120591

119895+119895119901⟩ ⟨ 120591

119895+119895119901|

119895+119895119901+1⟩

(B9)


S1

S2

S3

S4

S5

S6

S7

S8

S9

S10

1 2 3 4 5 6 7 8 9


Trotter

direction

p darr



where H119909

119895119896(119901) = sum

119872

119896=119895+1119869119909

119895119896120590119895119901120590119896119901prod

119896minus1

119897=119895+1(minus119904

119897119901) H119910

119895119896(119901) =

sum119872

119896=119895+1119869119910

119895119896120591119895119901120591119896119901prod

119896minus1

119897=119895+1(minus119904

119897119901) andH119911

119901= ℎsum

119872

119895=1120590119911

119895119901 Finally


⟨119901| 119904

119901⟩ ⟨ 119904

119901| 120591

119901⟩ ⟨ 120591

119901|

119901+1⟩ = (

1

2radic2)

119872

sdot

119872

prod

119895=1


(B10)



sum

119901=1

119872

sum

119895=1

(120573qu

119899

119872

sum

119896=119895+1

(119869119909

119895119896120590119895119895+119895119901

120590119896119895+119895119901

+ 119869119910

119895119896120591119895119895+119895119901

120591119896119895+119895119901

)

119896minus1

prod

119897=119895+1

(minus119904119897119901) + (

120573qu

119899 (119872 minus 1)ℎ minus119894120587

4) 119904

119895119895+119895119901

+119894120587

4(1 minus 120590

119895119895+119895119901+ 120591

119895119895+119895119901+ 120590

119895119895+119895119901119904119895119895+119895119901

minus 120591119895119895+119895119901

119904119895119895+119895119901

minus 120590119895119895+119895119901+1

120591119895119895+119895119901

)) + 1198991198722 ln 1

2radic2

(B11)


119895119896





119909

119895119896 (120573qu119899)119869

119910

119895119896 and



1205901

S1

1205911

S2

1205902

S3

1205912

S4

1205903

S5

1 2 3 4 5 6 7 8 9 10

Trotter

direction

p darr






119863(55)


minus1

120572 sin 119902=119887 (1) sin 119902

Γ + 119886 (1) cos 119902 (C1)



minus1

120572 sin 119902=

119887 (1)

minus2119886 (2) sin 119902

if Γ = minus119886 (2) 119886 (1) = 119887 (2) = 0

(C2)


120572 =2119886 (2)

119887 (1) Γ = minus119886 (2) 119886 (1) = 119887 (2) = 0 (C3)


minus1

120572 sin 119902= minus

119887 (1) minus 119887 (3) + 119887 (2) cos 1199022 sin 119902 (119886 (2) + 119886 (3) cos 119902)

if Γ = minus119886 (2) 119886 (1) = minus119886 (3) 119887 (3) = 0

(C4)


120572 =2119886 (3)

119887 (2)

119886 (2)

119886 (3)=119887 (1) minus 119887 (3)

119887 (2)

Γ = minus119886 (2) 119886 (1) = minus119886 (3) 119887 (3) = 0

(C5)


minus1

120572 sin 119902

=sin 119902sum119898

119896=1119887 (119896)sum

[(119896minus1)2]

119897=0( 119896

2119897+1)sum

119897

119894=0( 119897119894) (minus1)


119896=2119886 (119896)sum

[1198962]

119897=0(minus1)

119897

( 119896

2119897) sin2119897

119902cos119896minus2119897119902

(C6)


[1198982]

sum

119895=1

119886 (2119895) + cos 119902[1198982]

sum

119895=1

119886 (2119895 + 1) + sin2

119902

sdot (

[1198982]

sum

119895=1

119886 (2119895)

119895

sum

119894=1

(119895

119894) (minus1)

119894 sin2(119894minus1)

119902

+ cos 119902[(119898minus1)2]

sum

119895=1

119886 (2119895 + 1)

119895

sum

119894=1

(119895

119894) (minus1)

119894 sin2(119894minus1)

119902

+

119898

sum

119896=2

119886 (119896)

[1198962]

sum

119897=1

(minus1)119897

(119896

2119897) sin2(119897minus1)

119902cos119896minus2119897119902)

(C7)


Γ = minus

[1198982]

sum

119895=1

119886 (2119895)

119886 (1) = minus

[(119898+1)2]

sum

119895=1

119886 (2119895 + 1) = 0

(C8)



sin2

119902(

[1198982]

sum

119895=1

119886 (2119895)

119895

sum

119894=1

(119895

119894)

119894minus1

sum

119901=0

(minus1)119894+119901 cos2119901119902

+

[(119898minus1)2]

sum

119895=1

119886 (2119895 + 1)

119895

sum

119894=1

(119895

119894)

119894minus1

sum

119901=0

(minus1)119894+119901 cos2119901+1119902 +

[(119898minus1)2]

sum

119895=1

119886 (2119895 + 1)

119895

sum

119897=1

(2119895 + 1

2119897)

119897minus1

sum

119901=0


+

[1198982]

sum

119895=1

119886 (2119895)

119895

sum

119897=1

(2119895

2119897)

119897minus1

sum

119901=0


(C9)


Disclosure




Acknowledgments


Endnotes



integer



and 119861119895119896





7 Thus matrices 119860119895119896

and 119861119895119896




1V

2V12

1of


=

V12

2V

1V12

2 whereV

1= (2 sinh 2119870

1)1198722


1sum119872

119894120590119909

119894 V2=

1198901198702 sum119872

119894=1120590119911

119894120590119911


119894= 119890



119887119902(5) as

119886119902= Γ +

119871

sum

119896=1

119886 (119896)

[1198962]

sum

119897=0

(119896

2119897)

119897

sum

119894=0

(119897

119894) (minus1)

minus119894 cos119896minus2119894119902

119887119902= tan 119902

sdot

119871

sum

119896=1

119887 (119896)

[(119896minus1)2]

sum

119897=0

(119896

2119897 + 1)

119897

sum

119894=0

(119897

119894) (minus1)

minus119894 cos119896minus2119894119902

(lowast)


119895=1(minus1)

119895

119886(2119895)


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










120573cl119869119889

119895= log

tanh (2120573qu119899)119860119895

tanh (2120574120573qu119899) 119861119895

120573cl119869V119895= log coth

2120574120573qu

119899119861119895coth

2120573qu

119899119860

119895

120573cl119867 =120573quℎ

119899

120573cl119862119895= log sinh

2120573qu

119899119860

119895sinh

2120574120573qu

119899119861119895

(24)


119904119895119901119904119895+1119901

119904119895119901+1

119904119895+1119901+1

= 1 (25)



Hcl =2119899

sum

119901isin119886

119872

sum

119895isin119886

H119895119901+

2119899

sum

119901isin119887

119872

sum

119895isin119887

H119895119901 (26)

whereH119895119901




119895 119869

119889

119895 119867 rarr 0 and 119869V

119895rarr infin




Hcl = minus119899

sum

119901=1

119872

sum

119895=1

(119869120590

119895120590119895119901120590119895+1119901

+ 119869120591

119895120591119895119901120591119895+1119901

+ 119894119869120591119895119901(120590

119895119901minus 120590

119895119901+1))

(27)

Trotter

direction

p darr




120573cl119869120590

119895=120573qu

119899119869119909

119895

120573cl119869120591

119895=120573qu

119899119869119910

119895

120573cl119869 =1

2arctan 1

sinh (120573qu119899) ℎ

(28)



Tr [U1U

2]119899

U1= 119890

(120573qu2119899)H119909119890(120573qu2119899)H119911119890

(120573qu2119899)H119910

U2= 119890

(120573qu2119899)H119910119890(120573qu2119899)H119911119890

(120573qu2119899)H119909

(29)


= sum119872

119895=1119869120583

119895120590120583

119895120590120583

119895+1for 120583 isin 119909 119910 and H119911

=

sum119872

119895=1120590119911

119895


=

sum119901

|119901⟩⟨

119901| and I

120591119901= sum

120591119901

| 120591119901⟩⟨ 120591




Tr I1

U1I 1205911

U2I2


1205912119899U

2


sum

120590119895119901 120591119895119901

119899

prod

119901=1

⟨119901

10038161003816100381610038161003816U

1

10038161003816100381610038161003816120591119901⟩ ⟨ 120591

119901

10038161003816100381610038161003816U

2

10038161003816100381610038161003816119901+1⟩

(30)



⟨119901

10038161003816100381610038161003816U

1

10038161003816100381610038161003816120591119901⟩ ⟨ 120591

119901

10038161003816100381610038161003816U

2

10038161003816100381610038161003816119901+1⟩

= 119890(120573qu119899)((12)(H

119909

119901+H119909119901+1

)+H119910

119901) ⟨119901

10038161003816100381610038161003816119890(120573qu2119899)H

119911 10038161003816100381610038161003816120591119901⟩ ⟨ 120591

119901

10038161003816100381610038161003816

sdot 119890(120573qu2119899)H

119911 10038161003816100381610038161003816119901+1⟩

= 1198622119872

119890(120573qu119899)((12)(H

119909

119901+H119909119901+1

)+H119910

119901)+(1198942)119863sum119872

119895=1120591119895119901(120590119895119901minus120590119895119901+1)

(31)

where H119909

119901= sum

119872

119895=1119869119909

120590119895119901120590119895+1119901

H119910

119901= sum

119872

119895=1119869119910

120591119895119901120591119895+1119901


⟨120590119895119901

10038161003816100381610038161003816119890119886120590119911

11989510038161003816100381610038161003816120591119895119901⟩

=1

2cosh (2119886) 119890119894(12) arctan(1 sinh(2119886))120590119895119901120591119895119901

(32)


119895119901and 120591



119895119901


119895 119869120591

119895rarr infin



⟨119891 (120590119909

)⟩qu = ⟨119891 (120590)⟩cl

⟨119891 (120590119910

)⟩qu = ⟨119891 (120591)⟩cl (33)



⟨120590119909

119895120590119909

119895+119903⟩qu= ⟨120590

119895119901120590119895+119903119901⟩cl

⟨120590119910

119895120590119910

119895+119903⟩qu= ⟨120591

119895119901120591119895+119903119901⟩cl

(34)



⟨ 119904119901

10038161003816100381610038161003816119890(120573qu119899)V120572 10038161003816100381610038161003816

119904119901+1⟩

=

119872

prod

119895isin120572

120596119895

(119904119895119901 119904

119895+1119901 119904

119895119901+1 119904

119895+1119901+1)

(35)


(119904119895119901119904119895+1119901

119904119895119901+1

119904119895+1119901+1

)


120596119895

1(+1 +1 +1 +1) = 119890

ℎ120573qu119899 cosh(2120573qu120574

119899119861119895)

120596119895


minus120573quℎ119899 cosh(2120574120573qu

119899119861119895)

120596119895

3(minus1 +1 +1 minus1) = 120596

119895


= sinh(2120573qu

119899119860

119895)

120596119895

5(+1 minus1 +1 minus1) = 120596

119895


= cosh(2120573qu

119899119860

119895)

120596119895

7(minus1 minus1 +1 +1) = 120596

119895


= sinh(2120573qu120574

119899119861119895)

(36)




3 120596

119894

4 120596

119894

7 120596

119894

8rarr 0 and

weights 120596119894

1 120596

119894

2 120596

119894

5 120596

119894





12059041

12059031

12059021

12059011

12059042

12059032

12059022

12059012

12059043

12059033

12059023

12059013

12059141

12059131

12059121

12059111

12059142

12059132

12059122

12059112

12059143

12059133

12059123

12059113

1 2 3 4 5 6

120590j1 = j1

120591j1 = j2

120590j2 = j3

120591j2 = j4

120590j3 = j5

120591j3 = j6

Trotter

direction

p darr




119895 and the 119869



1205961 1205962 1205963 1205964 1205965 1205966 1205967 1205968




sum

120590119895119901=plusmn1

1

4(

119899

prod

119901isin119886

119872

prod

119895isin119886

+

119899

prod

119901isin119887

119872

prod

119895isin119887

)

sdot [(1 minus 119904119895119901119904119895+1119901

) (1 + 119904119895119901119904119895119901+1

) cosh2120573qu

119899119860

119895119895+1

+ (1 minus 119904119895119901119904119895+1119901

) (1 minus 119904119895119901119904119895119901+1

) sinh2120573qu

119899119860

119895119895+1

+ (1 + 119904119895119901119904119895+1119901

) (1 minus 119904119895119901119904119895119901+1

) sinh2120573qu120574

119899119861119895119895+1

+ (1 + 119904119895119901119904119895119901+1

) (1 + 119904119895119901119904119895119901+1

) 119890(120573qu119899)ℎ119904119895119901


119899119861119895119895+1]

(37)



Trotter

direction

p darr







Hqu =1

2sum

1le119895le119896le119872

((119860119895119896+ 120574119861

119895119896) 120590

119909

119895120590119909

119896

+ (119860119895119896minus 120574119861

119895119896) 120590

119910

119895120590119910

119896)(

119896minus1

prod

119897=119895+1

minus 120590119911

119897) minus ℎ

119872

sum

119895=1

120590119911

119895

(38)




Hqu]

= TrV119899

(39)




119872

prod

119894=1

119889N (120591119894) ⟨N (120591

119872)1003816100381610038161003816 119890

minusΔ120591H 1003816100381610038161003816N (120591119872minus1)⟩

sdot ⟨N (120591119872minus1)1003816100381610038161003816 119890


1)1003816100381610038161003816

sdot 119890minusΔ120591H 1003816100381610038161003816N (120591119872)⟩

(40)




119889120591|N (120591)⟩)

= ⟨N (120591) | N (120591)⟩ minus Δ120591 ⟨N (120591)| 119889119889120591|N (120591)⟩

minus Δ120591 ⟨N (120591)| Hqu (S) |N (120591)⟩ + 119874 ((Δ120591)2

)

= 119890minusΔ120591(⟨N(120591)|(119889119889120591)|N(120591)⟩+H(N(120591)))

Δ120591

119872

sum

119894=1

997888rarr int

120573

0

119889120591

119872

prod

119894=1

119889N (120591119894) 997888rarr DN (120591)

(41)


119885 = int

N(120573)

N(0)

DN (120591) 119890minusint

120573

0119889120591H(N(120591))minusS119861 (42)


S119861= int

120573

0

119889120591 ⟨N (120591)| 119889119889120591|N (120591)⟩ (43)



⟨N| S |N⟩ = N (44)


⟨N| 119878119894 |N⟩ = minus119878119873119894

(45)



N = (119873119909

119873119910

119873119911


0 le 120579 le 120587 0 le 120601 le 2120587

(46)


H (N (120591)) = ⟨N (120591)| Hqu |N (120591)⟩

= sum

1le119895le119896le119872

((119860119895119896+ 120574119861

119895119896)119873

119909

119895(120591)119873

119909

119896(120591)

+ (119860119895119896minus 120574119861

119895119896)119873

119910

119895(120591)119873

119910

119896(120591))

119896minus1

prod

119897=119895+1

(minus119873119911

119897(120591))

minus ℎ

119872

sum

119895=1

119873119911

119895(120591) = sum

1le119895le119896le119872

(119860119895119896

cos (120601119895(120591) minus 120601

119896(120591))

+ 119861119895119896120574 cos (120601

119895(120591) + 120601

119896(120591))) sin (120579

119895(120591))

sdot sin (120579119896(120591))

119896minus1

prod

119897=119895+1

(minus cos (120579119897(120591))) minus ℎ

119872

sum

119895=1

cos (120579119895(120591))

(47)





[HquVcl] = 0 (48)


H = minus

119873

sum

119894

119872

sum

119895

(1198691119904119894119895119904119894+1119895

+ 1198692119904119894119895119904119894119895+1) (49)



2ℎ + 2V = 119873119872 (50)


119885 = sum


119910V= 119910

1198721198732

sum


(51)



120578dagger

119902=119890minus1198941205874

radic119872sum

119895

119890minus(2120587119894119872)119902119895

(119887dagger

119895119906119902+ 119894119887

119895V119902)

120578119902=1198901198941205874

radic119872sum

119895

119890(2120587119894119872)119902119895

(119887119895119906119902minus 119894119887

dagger

119895V119902)

(52)



V+(minus)

= (2 sinh 21198701)1198732

119890minussum119902120598119902(120578dagger

119902120578119902minus12) (53)

where119870119894= 120573119869

119894and 120598


cosh 120598119902= cosh 2119870lowast

1cosh 2119870

2


1sinh 2119870

2cos 119902

(54)



by

V2

119863

= prod

0le119902le120587

(120582119902(120578

dagger

119902120578119902+ 120578

dagger


2sin2

119902)) (55)


with

120582119902= 2120572 sin 119902 (1 + 1205722sin2

119902)12

(56)


2V119902119906119902

1199062119902minus V2

119902

=

119886119902

119887119902

for Hqu



1coth 2119870

2

for V


1tanh119870

2cos 119902)



1 sinh 2119870

2

for V1015840

minus1

120572 sin 119902for V2

119863

(57)



119886119902

119887119902

=sin 119902

cosh 2119870lowast


1coth 2119870

2

(58)


sinh 2119870lowast

1coth 2119870

2= minus119886 (119871 minus 1)

119887 (119871)

tanh2119870lowast

1=119886 (119871) minus 119887 (119871)

119886 (119871) + 119887 (119871)

119886 (119871 minus 1)

119886 (119871) + 119887 (119871)= minus coth 2119870

2tanh119870lowast

1

(59)

or inversely as

cosh 2119870lowast

1=119886 (119871)

119887 (119871)


1

119886 (119871 minus 1)

radic(119886 (119871))2

minus (119887 (119871))2

(60)

where

119886 (119871) = 119886 (119871)

[1198712]

sum

119897=0

(119871

2119897)

119887 (119871) = 119887 (119871)

[(119871minus1)2]

sum

119897=0

(119871

2119897 + 1)

119886 (0) = Γ

(61)


119887 (119871)ge 1

1198862

(119871) le 1198862

(119871 minus 1) + 1198872

(119871)

(62)


119871

sum

119896=1

119887 (119896)

[(119871minus1)2]

sum

119897=0

(119896

2119897 + 1)

119897

sum

119894=1

(minus1)119894 cos119896minus2119894119902

+

119871minus1

sum

119896=1

119887 (119896) cos119896119902 = 0

(63)

Γ +

119871minus2

sum

119896=1

119886 (119896) cos119896119902

+

119871

sum

119896=1

119886 (119896)

[1198962]

sum

119897=0

(119896

2119897)

119897

sum

119894=1

(119897

119894) (minus1)

119894 cos119871minus2119894119902 = 0

(64)




10038161003816100381610038161003816Λ

119902

10038161003816100381610038161003816= 2

119901+110038161003816100381610038161003816cos(119871minus1)11990210038161003816100381610038161003816

sdot ((119886 (119871) cos 119902 + 119886 (119871 minus 1))2 + 1198872

(119871) sin2

119902)

12

(65)


by

119870lowast

1= 119870

2 (66)



119886 (119871) = plusmn119886 (119871 minus 1) (67)


10038161003816100381610038161003816Λ

119902

10038161003816100381610038161003816= 2

119901+110038161003816100381610038161003816119886 (119871) cos(119871minus1)11990210038161003816100381610038161003816

sdot ((cos 119902 plusmn 1)2 + (119887 (119871)119886 (119871)

)

2

sin2

119902)

12

(68)



⟨120590119909

119895119896120590119909

119895+119903119896⟩Is= ⟨Ψ

0

1003816100381610038161003816Vminus12

1120590119909

119895120590119909

119895+119903V12

1

1003816100381610038161003816Ψ0⟩

= ⟨Φ0

1003816100381610038161003816Vminus12

1120590119909

119895120590119909

119895+119903V12

1

1003816100381610038161003816Φ0⟩

= ⟨(Vminus12

1120590119909

119895V12

1) (Vminus12

1120590119909

119895+119903V12

1)⟩

qu

= cosh2119870lowast

1⟨120590

119909

119895120590119909

119895+119903⟩qu


1⟨120590

119910

119895120590119910

119895+119903⟩qu

(69)


119895+119903⟩qu = ⟨120590

119910

119895120590119909

119895+119903⟩qu = 0 for 119903 = 0 and

Ψ0= Φ

0 (70)






119887119902


1tanh119870

2cos 119902)



1 sinh 2119870

2

(71)



tanh 2119870lowast

1tanh119870

2= minus

119887 (119871)

119887 (119871 minus 1)= minus

119886 (119871)

119887 (119871 minus 1)

119886 (119871 minus 1)

119887 (119871 minus 1)= 1

tanh 2119870lowast

1

sinh 21198702

= minus119886lowast

(119871)

119887 (119871 minus 1)

(72)

or inversely as

sinh21198702=

119886 (119871)

2 (119886lowast

(119871))

tanh 2119870lowast

1= minus

1

119886 (119871 minus 1)radic119886 (119871) (2119886

lowast

(119871) + 119886 (119871))

(73)

where

119886lowast

(119871) = 119886 (119871 minus 2) minus 119886 (119871)

[1198712]

sum

119897=0

(119871

2119897) 119897 (74)


119886 (119871) (2119886lowast

(119871) + 119886 (119871)) le 1198862

(119871 minus 1) (75)


119871minus2

sum

119896=1

119887 (119896) cos119896119902

+

119871

sum

119896=1

119887 (119896)

[(119896minus1)2]

sum

119897=0

(119896

2119897 + 1)

119897

sum

119894=1

(119897

119894) (minus1)

119894 cos119896minus2119894119902

= 0

(76)

Γ +

119871minus3

sum

119896=1

119896cos119896119902 minus119871minus1

sum

119896=1

119886 (119896)

[1198962]

sum

119897=0

(119896

2119897) 119897cos119896minus2119902

+

119871

sum

119896=1

119886 (119896)

[1198962]

sum

119897=0

(119896

2119897)

119897

sum

119894=2

(119897

119894) (minus1)

119894 cos119896minus2119894119902 = 0

(77)



10038161003816100381610038161003816Λ

119902

10038161003816100381610038161003816= 2

119901+110038161003816100381610038161003816cos119871minus211990210038161003816100381610038161003816


+ sin2

119902 (119887 (119871) cos 119902 + 119887 (119871 minus 1)))12

(78)





(119871) + 119886 (119871) (79)


10038161003816100381610038161003816Λ

119902

10038161003816100381610038161003816= 2

119901+110038161003816100381610038161003816cos119871minus211990210038161003816100381610038161003816


+ sin2

119902 (119887 (119871) cos 119902 + 119887 (119871 minus 1)))12

(80)



⟨120590119909

119895119896120590119909

119895+119903119896⟩Is= ⟨120590

119909

119895120590119909

119895+119903⟩qu (81)

where Ψ1015840



Ψ1015840

0= Φ

0 (82)



119863 In this case




119892M(120579) ofM


120574 = 0 and thus B = 0



119895119896(119861

119895119896) (M

119899)119895119896


119895 119896 ge 0119874

+

(2119873) 119886(119895 minus 119896) + 119886(119895 + 119896) 1198920if 119895 = 119896 = 0radic2119892

119897if

either 119895 = 0 119896 = 119897or 119895 = 119897 119896 = 0

119892119895minus119896+ 119892

119895+119896 119895 119896 gt 0


119895+119896+2 119895 119896 ge 0

119874plusmn

(2119873 + 1) 119886(119895 minus 119896) ∓ 119886(119895 + 119896 + 1) 119892119895minus119896∓ 119892

119895+119896+1 119895 119896 ge 0

119874minus


119895+119896+2 119895 119896 ge 0

Appendices

A Symmetry Classes

See Table 1



Hqu = minus119872

sum

119895=1

(119869119909

119895120590119909

119895120590119909

119895+1+ 119869

119910

119895120590119910

119895120590119910

119895+1

minus (1198691015840119909

119895120590119909

119895120590119909

119895+2+ 119869

1015840119910

119895120590119910

119895120590119910

119895+2) 120590

119911

119895+1+ ℎ120590

119911

119895)

(B1)

where 1198691015840119909119895= (12)(119860

119895119895+2+ 120574119861

119895119895+2) and 1198691015840119910

119895= (12)(119860

119895119895+2minus




Tr [119890(120573qu119899)H119909

119886 119890(120573qu2119899)H

119911

119890(120573qu119899)H

119910

119887 119890(120573qu119899)H

119910

119886 119890(120573qu2119899)H

119911

119890(120573qu119899)H

119909

119887 ]119899

(B2)


120572= sum

119872

119895isin120572((12)119869

120583

119895(120590

120583

119895120590120583

119895+1+ 120590

120583

119895+1120590120583

119895+2) minus

1198691015840120583

119895120590120583

119895120590119911

119895+1120590120583

119895+2) and H119911

= ℎsum119872

119895=1120590119911

119895 for 120583 isin 119909 119910 and once




Tr [I1205901119890(120573qu119899)H

119909

119886 I1199041119890(120573qu2119899)H

119911

119890(120573qu119899)H

119910

119887 I1205911119890(120573qu119899)H

119910

119886 I1199041119890(120573qu2119899)H

119911

119890(120573qu119899)H

119909

119887 ]119899


sum

120590119895119901 120591119895119901119904119895119901

119899

prod

119901=1

[⟨119901

10038161003816100381610038161003816119890(120573qu119899)H

119909

119886100381610038161003816100381610038161199042119901minus1⟩ ⟨ 119904

2119901minus1

10038161003816100381610038161003816119890(120573qu2119899)H

119911

119890(120573qu119899)H

119910

119887

10038161003816100381610038161003816120591119901⟩ ⟨ 120591

119901

10038161003816100381610038161003816119890(120573qu119899)H

119910

119886100381610038161003816100381610038161199042119901⟩ ⟨ 119904

2119901

10038161003816100381610038161003816119890(120573qu2119899)H

119911

119890(120573qu119899)H

119909

11988710038161003816100381610038161003816119901+1⟩]

(B3)



⟨119901

10038161003816100381610038161003816119890(120573119899)

H119909119886100381610038161003816100381610038161199042119901minus1⟩ ⟨ 119904

2119901minus1

10038161003816100381610038161003816119890(120573qu2119899)H

119911

119890(120573qu119899)H

119910

119887

10038161003816100381610038161003816120591119901⟩ ⟨ 120591

2119901minus1

10038161003816100381610038161003816119890(120573qu119899)H

119910

119886100381610038161003816100381610038161199042119901⟩ ⟨ 119904

2119901

10038161003816100381610038161003816119890(120573qu2119899)H

119911

119890(120573qu119899)H

119909

119887100381610038161003816100381610038162119901⟩

= 119890(120573qu119899)H

119909

119886(119901)

119890(120573qu2119899)H

119911(2119901minus1)

119890(120573qu119899)H

119910

119887(119901)

119890(120573qu119899)H

119910

119886(119901)

119890(120573qu2119899)H

119911(2119901)

119890(120573qu119899)H

119909

119887(119901)

⟨119901| 119904

2119901minus1⟩ ⟨ 119904

2119901minus1| 120591

119901⟩ ⟨ 120591

119901| 119904

2119901⟩

sdot ⟨ 1199042119901|

119901+1⟩

(B4)

where H119909

120572(119901) = sum

119872

119895isin120572((12)119869

119909

119895(120590

119895119901120590119895+1119901

+ 120590119895+1119901

120590119895+2119901

) +

1198691015840119909

119895+1120590119895119901119904119895+1119901

120590119895+2119901

) H119910

120572(119901) = sum

119872

119895isin120572((12)119869

119910

119895(120591

119895119901120591119895+1119901

+

120591119895+1119901

120591119895+2119901

) + 1198691015840119910

119895+1120591119895119901119904119895+1119901

120591119895+2119901

) andH119911

(119901) = sum119872

119895=1119904119895119901 We


⟨119901| 119904

2119901minus1⟩ ⟨ 119904

2119901minus1| 120591

119901⟩ ⟨ 120591

2119901minus1| 119904

2119901⟩ ⟨ 119904

2119901|

119901+1⟩

=1

24119872

sdot

119872

prod

119895=1


(B5)



119899

119899

sum

119901=1

(sum

119895isin119886

(119869119909

119895

2(120590

119895119901120590119895+1119901

+ 120590119895+1119901

120590119895+2119901

) minus 1198691015840119909

119895+1120590119895119901119904119895+1119901

120590119895+2119901

)

+sum

119895isin119887

(

119869119910

119895

2(120591

119895119901120591119895+1119901

+ 120591119895+1119901

120591119895+2119901

) minus 1198691015840119910

119895+1120591119895119901119904119895+12119901minus1

120591119895+2119901

)

+ sum

119895isin119886

(

119869119910

119895

2(120591

119895119901120591119895+1119901

+ 120591119895+1119901

120591119895+2119901

) minus 1198691015840119910

119895+1120591119895119901119904119895+12119901

120591119895+2119901

)

+sum

119895isin119887

(119869119909

119895

2(120590

119895119901+1120590119895+1119901+1

+ 120590119895+1119901+1

120590119895+2119901+1

) minus 1198691015840119909

119895+1120590119895119901+1

119904119895+12119901

120590119895+2119901+1

))

+

119899

sum

119901=1

(

119872

sum

119895=1

((120573quℎ

2119899minus119894120587

4) 119904

1198952119901+ (120573quℎ

2119899+119894120587

4) 119904

1198952119901) +

119872

sum

119895=1

119894120587

4(120590

1198951199011199041198952119901minus1

minus 120590119895119901+1

1199042119901+ 120591

119895119901(119904

1198952119901minus 119904

1198952119901minus1)))

+ 4119899119872 ln 2

(B6)



119909 (120573qu119899)119869119910

(120573qu119899)1198691015840119909 (120573qu119899)119869




S1

S2

S3

S4

1205901

1205911

1205902

1205912

1 2 3 4 5 6 7 8


Trotter

direction

p darr








Tr[

[

119872minus1

prod

119895=1

(119890(120573qu119899)H

119909

119895119895+1119890(120573qu119899)H

119909


119909

119895119872119890(120573qu2119899(119872minus1))

H119911119890(120573qu119899)H

119910


119910

119895119895+2119890(120573qu119899)H

119910

119895119895+1)]

]

119899


Tr[

[

119872

prod

119895=1

((

119872minus119895

prod

119896=1

119890(120573qu119899)H

119909

119895119895+119896)119890(120573qu2119899(119872minus1))

H119911(


prod

119896=0

119890(120573qu119899)H

119910

119895119872minus119896))]

]

119899

(B7)

where H120583

119895119896= 119869

120583

119895119896120590120583

119895120590120583

119896prod

119896minus1

119897=1(minus120590

119911

119897) for 120583 isin 119909 119910 and H119911

=

ℎsum119872

119895=1120590119911

119895



Tr[

[

119872minus1

prod

119895=1

(I120590119895(

119872minus119895

prod

119896=1

119890(120573qu119899)H

119909

119895119895+119896)119890(120573qu(119872minus1)119899)

H119911I119904119895(


prod

119896=0

119890(120573qu119899)H

119910

119895119872minus119896) I120591119895)]

]

119899


sum

120590119895119901 120591119895119901

119899minus1

prod

119901=0

119872minus1

prod

119895=1

(⟨119895+119895119901

10038161003816100381610038161003816(

119872minus119895

prod

119896=1

119890(120573qu119899)H

119909

119895119895+119896)119890(120573qu119899(119872minus1))

H119911 10038161003816100381610038161003816119904119895+119895119901⟩ ⟨ 119904

119895+119895119901

10038161003816100381610038161003816(


prod

119896=0

119890(120573qu119899)H

119910

119895119872minus119896)10038161003816100381610038161003816120591119895+119895119901⟩ ⟨ 120591

119895+119895119901|

119895+119895119901+1⟩)

(B8)


⟨119895+119895119901

10038161003816100381610038161003816(

119872minus119895

prod

119896=1

119890(120573qu119899)H

119909

119895119895+119896)119890(120573qu119899(119872minus1))

H119911 10038161003816100381610038161003816119904119895+119895119901⟩

sdot ⟨ 119904119895+119895119901

10038161003816100381610038161003816(


prod

119896=0

119890(120573qu119899)H

119910

119895119872minus119896)10038161003816100381610038161003816120591119895+119895119901⟩ ⟨ 120591

119895+119895119901|

119895+119895119901+1

⟩

= 119890(120573qu119899)sum

119872minus119895

119896=1(H119909119895119895+119896

(119901)+H119910

119895119895+119896(119901)+(1119899(119872minus1))H119911)

⟨119895+119895119901

|

119904119895+119895119901⟩ ⟨ 119904

119895+119895119901| 120591

119895+119895119901⟩ ⟨ 120591

119895+119895119901|

119895+119895119901+1⟩

(B9)


S1

S2

S3

S4

S5

S6

S7

S8

S9

S10

1 2 3 4 5 6 7 8 9


Trotter

direction

p darr



where H119909

119895119896(119901) = sum

119872

119896=119895+1119869119909

119895119896120590119895119901120590119896119901prod

119896minus1

119897=119895+1(minus119904

119897119901) H119910

119895119896(119901) =

sum119872

119896=119895+1119869119910

119895119896120591119895119901120591119896119901prod

119896minus1

119897=119895+1(minus119904

119897119901) andH119911

119901= ℎsum

119872

119895=1120590119911

119895119901 Finally


⟨119901| 119904

119901⟩ ⟨ 119904

119901| 120591

119901⟩ ⟨ 120591

119901|

119901+1⟩ = (

1

2radic2)

119872

sdot

119872

prod

119895=1


(B10)



sum

119901=1

119872

sum

119895=1

(120573qu

119899

119872

sum

119896=119895+1

(119869119909

119895119896120590119895119895+119895119901

120590119896119895+119895119901

+ 119869119910

119895119896120591119895119895+119895119901

120591119896119895+119895119901

)

119896minus1

prod

119897=119895+1

(minus119904119897119901) + (

120573qu

119899 (119872 minus 1)ℎ minus119894120587

4) 119904

119895119895+119895119901

+119894120587

4(1 minus 120590

119895119895+119895119901+ 120591

119895119895+119895119901+ 120590

119895119895+119895119901119904119895119895+119895119901

minus 120591119895119895+119895119901

119904119895119895+119895119901

minus 120590119895119895+119895119901+1

120591119895119895+119895119901

)) + 1198991198722 ln 1

2radic2

(B11)


119895119896





119909

119895119896 (120573qu119899)119869

119910

119895119896 and



1205901

S1

1205911

S2

1205902

S3

1205912

S4

1205903

S5

1 2 3 4 5 6 7 8 9 10

Trotter

direction

p darr






119863(55)


minus1

120572 sin 119902=119887 (1) sin 119902

Γ + 119886 (1) cos 119902 (C1)



minus1

120572 sin 119902=

119887 (1)

minus2119886 (2) sin 119902

if Γ = minus119886 (2) 119886 (1) = 119887 (2) = 0

(C2)


120572 =2119886 (2)

119887 (1) Γ = minus119886 (2) 119886 (1) = 119887 (2) = 0 (C3)


minus1

120572 sin 119902= minus

119887 (1) minus 119887 (3) + 119887 (2) cos 1199022 sin 119902 (119886 (2) + 119886 (3) cos 119902)

if Γ = minus119886 (2) 119886 (1) = minus119886 (3) 119887 (3) = 0

(C4)


120572 =2119886 (3)

119887 (2)

119886 (2)

119886 (3)=119887 (1) minus 119887 (3)

119887 (2)

Γ = minus119886 (2) 119886 (1) = minus119886 (3) 119887 (3) = 0

(C5)


minus1

120572 sin 119902

=sin 119902sum119898

119896=1119887 (119896)sum

[(119896minus1)2]

119897=0( 119896

2119897+1)sum

119897

119894=0( 119897119894) (minus1)


119896=2119886 (119896)sum

[1198962]

119897=0(minus1)

119897

( 119896

2119897) sin2119897

119902cos119896minus2119897119902

(C6)


[1198982]

sum

119895=1

119886 (2119895) + cos 119902[1198982]

sum

119895=1

119886 (2119895 + 1) + sin2

119902

sdot (

[1198982]

sum

119895=1

119886 (2119895)

119895

sum

119894=1

(119895

119894) (minus1)

119894 sin2(119894minus1)

119902

+ cos 119902[(119898minus1)2]

sum

119895=1

119886 (2119895 + 1)

119895

sum

119894=1

(119895

119894) (minus1)

119894 sin2(119894minus1)

119902

+

119898

sum

119896=2

119886 (119896)

[1198962]

sum

119897=1

(minus1)119897

(119896

2119897) sin2(119897minus1)

119902cos119896minus2119897119902)

(C7)


Γ = minus

[1198982]

sum

119895=1

119886 (2119895)

119886 (1) = minus

[(119898+1)2]

sum

119895=1

119886 (2119895 + 1) = 0

(C8)



sin2

119902(

[1198982]

sum

119895=1

119886 (2119895)

119895

sum

119894=1

(119895

119894)

119894minus1

sum

119901=0

(minus1)119894+119901 cos2119901119902

+

[(119898minus1)2]

sum

119895=1

119886 (2119895 + 1)

119895

sum

119894=1

(119895

119894)

119894minus1

sum

119901=0

(minus1)119894+119901 cos2119901+1119902 +

[(119898minus1)2]

sum

119895=1

119886 (2119895 + 1)

119895

sum

119897=1

(2119895 + 1

2119897)

119897minus1

sum

119901=0


+

[1198982]

sum

119895=1

119886 (2119895)

119895

sum

119897=1

(2119895

2119897)

119897minus1

sum

119901=0


(C9)


Disclosure




Acknowledgments


Endnotes



integer



and 119861119895119896





7 Thus matrices 119860119895119896

and 119861119895119896




1V

2V12

1of


=

V12

2V

1V12

2 whereV

1= (2 sinh 2119870

1)1198722


1sum119872

119894120590119909

119894 V2=

1198901198702 sum119872

119894=1120590119911

119894120590119911


119894= 119890



119887119902(5) as

119886119902= Γ +

119871

sum

119896=1

119886 (119896)

[1198962]

sum

119897=0

(119896

2119897)

119897

sum

119894=0

(119897

119894) (minus1)

minus119894 cos119896minus2119894119902

119887119902= tan 119902

sdot

119871

sum

119896=1

119887 (119896)

[(119896minus1)2]

sum

119897=0

(119896

2119897 + 1)

119897

sum

119894=0

(119897

119894) (minus1)

minus119894 cos119896minus2119894119902

(lowast)


119895=1(minus1)

119895

119886(2119895)


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











⟨119901

10038161003816100381610038161003816U

1

10038161003816100381610038161003816120591119901⟩ ⟨ 120591

119901

10038161003816100381610038161003816U

2

10038161003816100381610038161003816119901+1⟩

= 119890(120573qu119899)((12)(H

119909

119901+H119909119901+1

)+H119910

119901) ⟨119901

10038161003816100381610038161003816119890(120573qu2119899)H

119911 10038161003816100381610038161003816120591119901⟩ ⟨ 120591

119901

10038161003816100381610038161003816

sdot 119890(120573qu2119899)H

119911 10038161003816100381610038161003816119901+1⟩

= 1198622119872

119890(120573qu119899)((12)(H

119909

119901+H119909119901+1

)+H119910

119901)+(1198942)119863sum119872

119895=1120591119895119901(120590119895119901minus120590119895119901+1)

(31)

where H119909

119901= sum

119872

119895=1119869119909

120590119895119901120590119895+1119901

H119910

119901= sum

119872

119895=1119869119910

120591119895119901120591119895+1119901


⟨120590119895119901

10038161003816100381610038161003816119890119886120590119911

11989510038161003816100381610038161003816120591119895119901⟩

=1

2cosh (2119886) 119890119894(12) arctan(1 sinh(2119886))120590119895119901120591119895119901

(32)


119895119901and 120591



119895119901


119895 119869120591

119895rarr infin



⟨119891 (120590119909

)⟩qu = ⟨119891 (120590)⟩cl

⟨119891 (120590119910

)⟩qu = ⟨119891 (120591)⟩cl (33)



⟨120590119909

119895120590119909

119895+119903⟩qu= ⟨120590

119895119901120590119895+119903119901⟩cl

⟨120590119910

119895120590119910

119895+119903⟩qu= ⟨120591

119895119901120591119895+119903119901⟩cl

(34)



⟨ 119904119901

10038161003816100381610038161003816119890(120573qu119899)V120572 10038161003816100381610038161003816

119904119901+1⟩

=

119872

prod

119895isin120572

120596119895

(119904119895119901 119904

119895+1119901 119904

119895119901+1 119904

119895+1119901+1)

(35)


(119904119895119901119904119895+1119901

119904119895119901+1

119904119895+1119901+1

)


120596119895

1(+1 +1 +1 +1) = 119890

ℎ120573qu119899 cosh(2120573qu120574

119899119861119895)

120596119895


minus120573quℎ119899 cosh(2120574120573qu

119899119861119895)

120596119895

3(minus1 +1 +1 minus1) = 120596

119895


= sinh(2120573qu

119899119860

119895)

120596119895

5(+1 minus1 +1 minus1) = 120596

119895


= cosh(2120573qu

119899119860

119895)

120596119895

7(minus1 minus1 +1 +1) = 120596

119895


= sinh(2120573qu120574

119899119861119895)

(36)




3 120596

119894

4 120596

119894

7 120596

119894

8rarr 0 and

weights 120596119894

1 120596

119894

2 120596

119894

5 120596

119894





12059041

12059031

12059021

12059011

12059042

12059032

12059022

12059012

12059043

12059033

12059023

12059013

12059141

12059131

12059121

12059111

12059142

12059132

12059122

12059112

12059143

12059133

12059123

12059113

1 2 3 4 5 6

120590j1 = j1

120591j1 = j2

120590j2 = j3

120591j2 = j4

120590j3 = j5

120591j3 = j6

Trotter

direction

p darr




119895 and the 119869



1205961 1205962 1205963 1205964 1205965 1205966 1205967 1205968




sum

120590119895119901=plusmn1

1

4(

119899

prod

119901isin119886

119872

prod

119895isin119886

+

119899

prod

119901isin119887

119872

prod

119895isin119887

)

sdot [(1 minus 119904119895119901119904119895+1119901

) (1 + 119904119895119901119904119895119901+1

) cosh2120573qu

119899119860

119895119895+1

+ (1 minus 119904119895119901119904119895+1119901

) (1 minus 119904119895119901119904119895119901+1

) sinh2120573qu

119899119860

119895119895+1

+ (1 + 119904119895119901119904119895+1119901

) (1 minus 119904119895119901119904119895119901+1

) sinh2120573qu120574

119899119861119895119895+1

+ (1 + 119904119895119901119904119895119901+1

) (1 + 119904119895119901119904119895119901+1

) 119890(120573qu119899)ℎ119904119895119901


119899119861119895119895+1]

(37)



Trotter

direction

p darr







Hqu =1

2sum

1le119895le119896le119872

((119860119895119896+ 120574119861

119895119896) 120590

119909

119895120590119909

119896

+ (119860119895119896minus 120574119861

119895119896) 120590

119910

119895120590119910

119896)(

119896minus1

prod

119897=119895+1

minus 120590119911

119897) minus ℎ

119872

sum

119895=1

120590119911

119895

(38)




Hqu]

= TrV119899

(39)




119872

prod

119894=1

119889N (120591119894) ⟨N (120591

119872)1003816100381610038161003816 119890

minusΔ120591H 1003816100381610038161003816N (120591119872minus1)⟩

sdot ⟨N (120591119872minus1)1003816100381610038161003816 119890


1)1003816100381610038161003816

sdot 119890minusΔ120591H 1003816100381610038161003816N (120591119872)⟩

(40)




119889120591|N (120591)⟩)

= ⟨N (120591) | N (120591)⟩ minus Δ120591 ⟨N (120591)| 119889119889120591|N (120591)⟩

minus Δ120591 ⟨N (120591)| Hqu (S) |N (120591)⟩ + 119874 ((Δ120591)2

)

= 119890minusΔ120591(⟨N(120591)|(119889119889120591)|N(120591)⟩+H(N(120591)))

Δ120591

119872

sum

119894=1

997888rarr int

120573

0

119889120591

119872

prod

119894=1

119889N (120591119894) 997888rarr DN (120591)

(41)


119885 = int

N(120573)

N(0)

DN (120591) 119890minusint

120573

0119889120591H(N(120591))minusS119861 (42)


S119861= int

120573

0

119889120591 ⟨N (120591)| 119889119889120591|N (120591)⟩ (43)



⟨N| S |N⟩ = N (44)


⟨N| 119878119894 |N⟩ = minus119878119873119894

(45)



N = (119873119909

119873119910

119873119911


0 le 120579 le 120587 0 le 120601 le 2120587

(46)


H (N (120591)) = ⟨N (120591)| Hqu |N (120591)⟩

= sum

1le119895le119896le119872

((119860119895119896+ 120574119861

119895119896)119873

119909

119895(120591)119873

119909

119896(120591)

+ (119860119895119896minus 120574119861

119895119896)119873

119910

119895(120591)119873

119910

119896(120591))

119896minus1

prod

119897=119895+1

(minus119873119911

119897(120591))

minus ℎ

119872

sum

119895=1

119873119911

119895(120591) = sum

1le119895le119896le119872

(119860119895119896

cos (120601119895(120591) minus 120601

119896(120591))

+ 119861119895119896120574 cos (120601

119895(120591) + 120601

119896(120591))) sin (120579

119895(120591))

sdot sin (120579119896(120591))

119896minus1

prod

119897=119895+1

(minus cos (120579119897(120591))) minus ℎ

119872

sum

119895=1

cos (120579119895(120591))

(47)





[HquVcl] = 0 (48)


H = minus

119873

sum

119894

119872

sum

119895

(1198691119904119894119895119904119894+1119895

+ 1198692119904119894119895119904119894119895+1) (49)



2ℎ + 2V = 119873119872 (50)


119885 = sum


119910V= 119910

1198721198732

sum


(51)



120578dagger

119902=119890minus1198941205874

radic119872sum

119895

119890minus(2120587119894119872)119902119895

(119887dagger

119895119906119902+ 119894119887

119895V119902)

120578119902=1198901198941205874

radic119872sum

119895

119890(2120587119894119872)119902119895

(119887119895119906119902minus 119894119887

dagger

119895V119902)

(52)



V+(minus)

= (2 sinh 21198701)1198732

119890minussum119902120598119902(120578dagger

119902120578119902minus12) (53)

where119870119894= 120573119869

119894and 120598


cosh 120598119902= cosh 2119870lowast

1cosh 2119870

2


1sinh 2119870

2cos 119902

(54)



by

V2

119863

= prod

0le119902le120587

(120582119902(120578

dagger

119902120578119902+ 120578

dagger


2sin2

119902)) (55)


with

120582119902= 2120572 sin 119902 (1 + 1205722sin2

119902)12

(56)


2V119902119906119902

1199062119902minus V2

119902

=

119886119902

119887119902

for Hqu



1coth 2119870

2

for V


1tanh119870

2cos 119902)



1 sinh 2119870

2

for V1015840

minus1

120572 sin 119902for V2

119863

(57)



119886119902

119887119902

=sin 119902

cosh 2119870lowast


1coth 2119870

2

(58)


sinh 2119870lowast

1coth 2119870

2= minus119886 (119871 minus 1)

119887 (119871)

tanh2119870lowast

1=119886 (119871) minus 119887 (119871)

119886 (119871) + 119887 (119871)

119886 (119871 minus 1)

119886 (119871) + 119887 (119871)= minus coth 2119870

2tanh119870lowast

1

(59)

or inversely as

cosh 2119870lowast

1=119886 (119871)

119887 (119871)


1

119886 (119871 minus 1)

radic(119886 (119871))2

minus (119887 (119871))2

(60)

where

119886 (119871) = 119886 (119871)

[1198712]

sum

119897=0

(119871

2119897)

119887 (119871) = 119887 (119871)

[(119871minus1)2]

sum

119897=0

(119871

2119897 + 1)

119886 (0) = Γ

(61)


119887 (119871)ge 1

1198862

(119871) le 1198862

(119871 minus 1) + 1198872

(119871)

(62)


119871

sum

119896=1

119887 (119896)

[(119871minus1)2]

sum

119897=0

(119896

2119897 + 1)

119897

sum

119894=1

(minus1)119894 cos119896minus2119894119902

+

119871minus1

sum

119896=1

119887 (119896) cos119896119902 = 0

(63)

Γ +

119871minus2

sum

119896=1

119886 (119896) cos119896119902

+

119871

sum

119896=1

119886 (119896)

[1198962]

sum

119897=0

(119896

2119897)

119897

sum

119894=1

(119897

119894) (minus1)

119894 cos119871minus2119894119902 = 0

(64)




10038161003816100381610038161003816Λ

119902

10038161003816100381610038161003816= 2

119901+110038161003816100381610038161003816cos(119871minus1)11990210038161003816100381610038161003816

sdot ((119886 (119871) cos 119902 + 119886 (119871 minus 1))2 + 1198872

(119871) sin2

119902)

12

(65)


by

119870lowast

1= 119870

2 (66)



119886 (119871) = plusmn119886 (119871 minus 1) (67)


10038161003816100381610038161003816Λ

119902

10038161003816100381610038161003816= 2

119901+110038161003816100381610038161003816119886 (119871) cos(119871minus1)11990210038161003816100381610038161003816

sdot ((cos 119902 plusmn 1)2 + (119887 (119871)119886 (119871)

)

2

sin2

119902)

12

(68)



⟨120590119909

119895119896120590119909

119895+119903119896⟩Is= ⟨Ψ

0

1003816100381610038161003816Vminus12

1120590119909

119895120590119909

119895+119903V12

1

1003816100381610038161003816Ψ0⟩

= ⟨Φ0

1003816100381610038161003816Vminus12

1120590119909

119895120590119909

119895+119903V12

1

1003816100381610038161003816Φ0⟩

= ⟨(Vminus12

1120590119909

119895V12

1) (Vminus12

1120590119909

119895+119903V12

1)⟩

qu

= cosh2119870lowast

1⟨120590

119909

119895120590119909

119895+119903⟩qu


1⟨120590

119910

119895120590119910

119895+119903⟩qu

(69)


119895+119903⟩qu = ⟨120590

119910

119895120590119909

119895+119903⟩qu = 0 for 119903 = 0 and

Ψ0= Φ

0 (70)






119887119902


1tanh119870

2cos 119902)



1 sinh 2119870

2

(71)



tanh 2119870lowast

1tanh119870

2= minus

119887 (119871)

119887 (119871 minus 1)= minus

119886 (119871)

119887 (119871 minus 1)

119886 (119871 minus 1)

119887 (119871 minus 1)= 1

tanh 2119870lowast

1

sinh 21198702

= minus119886lowast

(119871)

119887 (119871 minus 1)

(72)

or inversely as

sinh21198702=

119886 (119871)

2 (119886lowast

(119871))

tanh 2119870lowast

1= minus

1

119886 (119871 minus 1)radic119886 (119871) (2119886

lowast

(119871) + 119886 (119871))

(73)

where

119886lowast

(119871) = 119886 (119871 minus 2) minus 119886 (119871)

[1198712]

sum

119897=0

(119871

2119897) 119897 (74)


119886 (119871) (2119886lowast

(119871) + 119886 (119871)) le 1198862

(119871 minus 1) (75)


119871minus2

sum

119896=1

119887 (119896) cos119896119902

+

119871

sum

119896=1

119887 (119896)

[(119896minus1)2]

sum

119897=0

(119896

2119897 + 1)

119897

sum

119894=1

(119897

119894) (minus1)

119894 cos119896minus2119894119902

= 0

(76)

Γ +

119871minus3

sum

119896=1

119896cos119896119902 minus119871minus1

sum

119896=1

119886 (119896)

[1198962]

sum

119897=0

(119896

2119897) 119897cos119896minus2119902

+

119871

sum

119896=1

119886 (119896)

[1198962]

sum

119897=0

(119896

2119897)

119897

sum

119894=2

(119897

119894) (minus1)

119894 cos119896minus2119894119902 = 0

(77)



10038161003816100381610038161003816Λ

119902

10038161003816100381610038161003816= 2

119901+110038161003816100381610038161003816cos119871minus211990210038161003816100381610038161003816


+ sin2

119902 (119887 (119871) cos 119902 + 119887 (119871 minus 1)))12

(78)





(119871) + 119886 (119871) (79)


10038161003816100381610038161003816Λ

119902

10038161003816100381610038161003816= 2

119901+110038161003816100381610038161003816cos119871minus211990210038161003816100381610038161003816


+ sin2

119902 (119887 (119871) cos 119902 + 119887 (119871 minus 1)))12

(80)



⟨120590119909

119895119896120590119909

119895+119903119896⟩Is= ⟨120590

119909

119895120590119909

119895+119903⟩qu (81)

where Ψ1015840



Ψ1015840

0= Φ

0 (82)



119863 In this case




119892M(120579) ofM


120574 = 0 and thus B = 0



119895119896(119861

119895119896) (M

119899)119895119896


119895 119896 ge 0119874

+

(2119873) 119886(119895 minus 119896) + 119886(119895 + 119896) 1198920if 119895 = 119896 = 0radic2119892

119897if

either 119895 = 0 119896 = 119897or 119895 = 119897 119896 = 0

119892119895minus119896+ 119892

119895+119896 119895 119896 gt 0


119895+119896+2 119895 119896 ge 0

119874plusmn

(2119873 + 1) 119886(119895 minus 119896) ∓ 119886(119895 + 119896 + 1) 119892119895minus119896∓ 119892

119895+119896+1 119895 119896 ge 0

119874minus


119895+119896+2 119895 119896 ge 0

Appendices

A Symmetry Classes

See Table 1



Hqu = minus119872

sum

119895=1

(119869119909

119895120590119909

119895120590119909

119895+1+ 119869

119910

119895120590119910

119895120590119910

119895+1

minus (1198691015840119909

119895120590119909

119895120590119909

119895+2+ 119869

1015840119910

119895120590119910

119895120590119910

119895+2) 120590

119911

119895+1+ ℎ120590

119911

119895)

(B1)

where 1198691015840119909119895= (12)(119860

119895119895+2+ 120574119861

119895119895+2) and 1198691015840119910

119895= (12)(119860

119895119895+2minus




Tr [119890(120573qu119899)H119909

119886 119890(120573qu2119899)H

119911

119890(120573qu119899)H

119910

119887 119890(120573qu119899)H

119910

119886 119890(120573qu2119899)H

119911

119890(120573qu119899)H

119909

119887 ]119899

(B2)


120572= sum

119872

119895isin120572((12)119869

120583

119895(120590

120583

119895120590120583

119895+1+ 120590

120583

119895+1120590120583

119895+2) minus

1198691015840120583

119895120590120583

119895120590119911

119895+1120590120583

119895+2) and H119911

= ℎsum119872

119895=1120590119911

119895 for 120583 isin 119909 119910 and once




Tr [I1205901119890(120573qu119899)H

119909

119886 I1199041119890(120573qu2119899)H

119911

119890(120573qu119899)H

119910

119887 I1205911119890(120573qu119899)H

119910

119886 I1199041119890(120573qu2119899)H

119911

119890(120573qu119899)H

119909

119887 ]119899


sum

120590119895119901 120591119895119901119904119895119901

119899

prod

119901=1

[⟨119901

10038161003816100381610038161003816119890(120573qu119899)H

119909

119886100381610038161003816100381610038161199042119901minus1⟩ ⟨ 119904

2119901minus1

10038161003816100381610038161003816119890(120573qu2119899)H

119911

119890(120573qu119899)H

119910

119887

10038161003816100381610038161003816120591119901⟩ ⟨ 120591

119901

10038161003816100381610038161003816119890(120573qu119899)H

119910

119886100381610038161003816100381610038161199042119901⟩ ⟨ 119904

2119901

10038161003816100381610038161003816119890(120573qu2119899)H

119911

119890(120573qu119899)H

119909

11988710038161003816100381610038161003816119901+1⟩]

(B3)



⟨119901

10038161003816100381610038161003816119890(120573119899)

H119909119886100381610038161003816100381610038161199042119901minus1⟩ ⟨ 119904

2119901minus1

10038161003816100381610038161003816119890(120573qu2119899)H

119911

119890(120573qu119899)H

119910

119887

10038161003816100381610038161003816120591119901⟩ ⟨ 120591

2119901minus1

10038161003816100381610038161003816119890(120573qu119899)H

119910

119886100381610038161003816100381610038161199042119901⟩ ⟨ 119904

2119901

10038161003816100381610038161003816119890(120573qu2119899)H

119911

119890(120573qu119899)H

119909

119887100381610038161003816100381610038162119901⟩

= 119890(120573qu119899)H

119909

119886(119901)

119890(120573qu2119899)H

119911(2119901minus1)

119890(120573qu119899)H

119910

119887(119901)

119890(120573qu119899)H

119910

119886(119901)

119890(120573qu2119899)H

119911(2119901)

119890(120573qu119899)H

119909

119887(119901)

⟨119901| 119904

2119901minus1⟩ ⟨ 119904

2119901minus1| 120591

119901⟩ ⟨ 120591

119901| 119904

2119901⟩

sdot ⟨ 1199042119901|

119901+1⟩

(B4)

where H119909

120572(119901) = sum

119872

119895isin120572((12)119869

119909

119895(120590

119895119901120590119895+1119901

+ 120590119895+1119901

120590119895+2119901

) +

1198691015840119909

119895+1120590119895119901119904119895+1119901

120590119895+2119901

) H119910

120572(119901) = sum

119872

119895isin120572((12)119869

119910

119895(120591

119895119901120591119895+1119901

+

120591119895+1119901

120591119895+2119901

) + 1198691015840119910

119895+1120591119895119901119904119895+1119901

120591119895+2119901

) andH119911

(119901) = sum119872

119895=1119904119895119901 We


⟨119901| 119904

2119901minus1⟩ ⟨ 119904

2119901minus1| 120591

119901⟩ ⟨ 120591

2119901minus1| 119904

2119901⟩ ⟨ 119904

2119901|

119901+1⟩

=1

24119872

sdot

119872

prod

119895=1


(B5)



119899

119899

sum

119901=1

(sum

119895isin119886

(119869119909

119895

2(120590

119895119901120590119895+1119901

+ 120590119895+1119901

120590119895+2119901

) minus 1198691015840119909

119895+1120590119895119901119904119895+1119901

120590119895+2119901

)

+sum

119895isin119887

(

119869119910

119895

2(120591

119895119901120591119895+1119901

+ 120591119895+1119901

120591119895+2119901

) minus 1198691015840119910

119895+1120591119895119901119904119895+12119901minus1

120591119895+2119901

)

+ sum

119895isin119886

(

119869119910

119895

2(120591

119895119901120591119895+1119901

+ 120591119895+1119901

120591119895+2119901

) minus 1198691015840119910

119895+1120591119895119901119904119895+12119901

120591119895+2119901

)

+sum

119895isin119887

(119869119909

119895

2(120590

119895119901+1120590119895+1119901+1

+ 120590119895+1119901+1

120590119895+2119901+1

) minus 1198691015840119909

119895+1120590119895119901+1

119904119895+12119901

120590119895+2119901+1

))

+

119899

sum

119901=1

(

119872

sum

119895=1

((120573quℎ

2119899minus119894120587

4) 119904

1198952119901+ (120573quℎ

2119899+119894120587

4) 119904

1198952119901) +

119872

sum

119895=1

119894120587

4(120590

1198951199011199041198952119901minus1

minus 120590119895119901+1

1199042119901+ 120591

119895119901(119904

1198952119901minus 119904

1198952119901minus1)))

+ 4119899119872 ln 2

(B6)



119909 (120573qu119899)119869119910

(120573qu119899)1198691015840119909 (120573qu119899)119869




S1

S2

S3

S4

1205901

1205911

1205902

1205912

1 2 3 4 5 6 7 8


Trotter

direction

p darr








Tr[

[

119872minus1

prod

119895=1

(119890(120573qu119899)H

119909

119895119895+1119890(120573qu119899)H

119909


119909

119895119872119890(120573qu2119899(119872minus1))

H119911119890(120573qu119899)H

119910


119910

119895119895+2119890(120573qu119899)H

119910

119895119895+1)]

]

119899


Tr[

[

119872

prod

119895=1

((

119872minus119895

prod

119896=1

119890(120573qu119899)H

119909

119895119895+119896)119890(120573qu2119899(119872minus1))

H119911(


prod

119896=0

119890(120573qu119899)H

119910

119895119872minus119896))]

]

119899

(B7)

where H120583

119895119896= 119869

120583

119895119896120590120583

119895120590120583

119896prod

119896minus1

119897=1(minus120590

119911

119897) for 120583 isin 119909 119910 and H119911

=

ℎsum119872

119895=1120590119911

119895



Tr[

[

119872minus1

prod

119895=1

(I120590119895(

119872minus119895

prod

119896=1

119890(120573qu119899)H

119909

119895119895+119896)119890(120573qu(119872minus1)119899)

H119911I119904119895(


prod

119896=0

119890(120573qu119899)H

119910

119895119872minus119896) I120591119895)]

]

119899


sum

120590119895119901 120591119895119901

119899minus1

prod

119901=0

119872minus1

prod

119895=1

(⟨119895+119895119901

10038161003816100381610038161003816(

119872minus119895

prod

119896=1

119890(120573qu119899)H

119909

119895119895+119896)119890(120573qu119899(119872minus1))

H119911 10038161003816100381610038161003816119904119895+119895119901⟩ ⟨ 119904

119895+119895119901

10038161003816100381610038161003816(


prod

119896=0

119890(120573qu119899)H

119910

119895119872minus119896)10038161003816100381610038161003816120591119895+119895119901⟩ ⟨ 120591

119895+119895119901|

119895+119895119901+1⟩)

(B8)


⟨119895+119895119901

10038161003816100381610038161003816(

119872minus119895

prod

119896=1

119890(120573qu119899)H

119909

119895119895+119896)119890(120573qu119899(119872minus1))

H119911 10038161003816100381610038161003816119904119895+119895119901⟩

sdot ⟨ 119904119895+119895119901

10038161003816100381610038161003816(


prod

119896=0

119890(120573qu119899)H

119910

119895119872minus119896)10038161003816100381610038161003816120591119895+119895119901⟩ ⟨ 120591

119895+119895119901|

119895+119895119901+1

⟩

= 119890(120573qu119899)sum

119872minus119895

119896=1(H119909119895119895+119896

(119901)+H119910

119895119895+119896(119901)+(1119899(119872minus1))H119911)

⟨119895+119895119901

|

119904119895+119895119901⟩ ⟨ 119904

119895+119895119901| 120591

119895+119895119901⟩ ⟨ 120591

119895+119895119901|

119895+119895119901+1⟩

(B9)


S1

S2

S3

S4

S5

S6

S7

S8

S9

S10

1 2 3 4 5 6 7 8 9


Trotter

direction

p darr



where H119909

119895119896(119901) = sum

119872

119896=119895+1119869119909

119895119896120590119895119901120590119896119901prod

119896minus1

119897=119895+1(minus119904

119897119901) H119910

119895119896(119901) =

sum119872

119896=119895+1119869119910

119895119896120591119895119901120591119896119901prod

119896minus1

119897=119895+1(minus119904

119897119901) andH119911

119901= ℎsum

119872

119895=1120590119911

119895119901 Finally


⟨119901| 119904

119901⟩ ⟨ 119904

119901| 120591

119901⟩ ⟨ 120591

119901|

119901+1⟩ = (

1

2radic2)

119872

sdot

119872

prod

119895=1


(B10)



sum

119901=1

119872

sum

119895=1

(120573qu

119899

119872

sum

119896=119895+1

(119869119909

119895119896120590119895119895+119895119901

120590119896119895+119895119901

+ 119869119910

119895119896120591119895119895+119895119901

120591119896119895+119895119901

)

119896minus1

prod

119897=119895+1

(minus119904119897119901) + (

120573qu

119899 (119872 minus 1)ℎ minus119894120587

4) 119904

119895119895+119895119901

+119894120587

4(1 minus 120590

119895119895+119895119901+ 120591

119895119895+119895119901+ 120590

119895119895+119895119901119904119895119895+119895119901

minus 120591119895119895+119895119901

119904119895119895+119895119901

minus 120590119895119895+119895119901+1

120591119895119895+119895119901

)) + 1198991198722 ln 1

2radic2

(B11)


119895119896





119909

119895119896 (120573qu119899)119869

119910

119895119896 and



1205901

S1

1205911

S2

1205902

S3

1205912

S4

1205903

S5

1 2 3 4 5 6 7 8 9 10

Trotter

direction

p darr






119863(55)


minus1

120572 sin 119902=119887 (1) sin 119902

Γ + 119886 (1) cos 119902 (C1)



minus1

120572 sin 119902=

119887 (1)

minus2119886 (2) sin 119902

if Γ = minus119886 (2) 119886 (1) = 119887 (2) = 0

(C2)


120572 =2119886 (2)

119887 (1) Γ = minus119886 (2) 119886 (1) = 119887 (2) = 0 (C3)


minus1

120572 sin 119902= minus

119887 (1) minus 119887 (3) + 119887 (2) cos 1199022 sin 119902 (119886 (2) + 119886 (3) cos 119902)

if Γ = minus119886 (2) 119886 (1) = minus119886 (3) 119887 (3) = 0

(C4)


120572 =2119886 (3)

119887 (2)

119886 (2)

119886 (3)=119887 (1) minus 119887 (3)

119887 (2)

Γ = minus119886 (2) 119886 (1) = minus119886 (3) 119887 (3) = 0

(C5)


minus1

120572 sin 119902

=sin 119902sum119898

119896=1119887 (119896)sum

[(119896minus1)2]

119897=0( 119896

2119897+1)sum

119897

119894=0( 119897119894) (minus1)


119896=2119886 (119896)sum

[1198962]

119897=0(minus1)

119897

( 119896

2119897) sin2119897

119902cos119896minus2119897119902

(C6)


[1198982]

sum

119895=1

119886 (2119895) + cos 119902[1198982]

sum

119895=1

119886 (2119895 + 1) + sin2

119902

sdot (

[1198982]

sum

119895=1

119886 (2119895)

119895

sum

119894=1

(119895

119894) (minus1)

119894 sin2(119894minus1)

119902

+ cos 119902[(119898minus1)2]

sum

119895=1

119886 (2119895 + 1)

119895

sum

119894=1

(119895

119894) (minus1)

119894 sin2(119894minus1)

119902

+

119898

sum

119896=2

119886 (119896)

[1198962]

sum

119897=1

(minus1)119897

(119896

2119897) sin2(119897minus1)

119902cos119896minus2119897119902)

(C7)


Γ = minus

[1198982]

sum

119895=1

119886 (2119895)

119886 (1) = minus

[(119898+1)2]

sum

119895=1

119886 (2119895 + 1) = 0

(C8)



sin2

119902(

[1198982]

sum

119895=1

119886 (2119895)

119895

sum

119894=1

(119895

119894)

119894minus1

sum

119901=0

(minus1)119894+119901 cos2119901119902

+

[(119898minus1)2]

sum

119895=1

119886 (2119895 + 1)

119895

sum

119894=1

(119895

119894)

119894minus1

sum

119901=0

(minus1)119894+119901 cos2119901+1119902 +

[(119898minus1)2]

sum

119895=1

119886 (2119895 + 1)

119895

sum

119897=1

(2119895 + 1

2119897)

119897minus1

sum

119901=0


+

[1198982]

sum

119895=1

119886 (2119895)

119895

sum

119897=1

(2119895

2119897)

119897minus1

sum

119901=0


(C9)


Disclosure




Acknowledgments


Endnotes



integer



and 119861119895119896





7 Thus matrices 119860119895119896

and 119861119895119896




1V

2V12

1of


=

V12

2V

1V12

2 whereV

1= (2 sinh 2119870

1)1198722


1sum119872

119894120590119909

119894 V2=

1198901198702 sum119872

119894=1120590119911

119894120590119911


119894= 119890



119887119902(5) as

119886119902= Γ +

119871

sum

119896=1

119886 (119896)

[1198962]

sum

119897=0

(119896

2119897)

119897

sum

119894=0

(119897

119894) (minus1)

minus119894 cos119896minus2119894119902

119887119902= tan 119902

sdot

119871

sum

119896=1

119887 (119896)

[(119896minus1)2]

sum

119897=0

(119896

2119897 + 1)

119897

sum

119894=0

(119897

119894) (minus1)

minus119894 cos119896minus2119894119902

(lowast)


119895=1(minus1)

119895

119886(2119895)


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










12059041

12059031

12059021

12059011

12059042

12059032

12059022

12059012

12059043

12059033

12059023

12059013

12059141

12059131

12059121

12059111

12059142

12059132

12059122

12059112

12059143

12059133

12059123

12059113

1 2 3 4 5 6

120590j1 = j1

120591j1 = j2

120590j2 = j3

120591j2 = j4

120590j3 = j5

120591j3 = j6

Trotter

direction

p darr




119895 and the 119869



1205961 1205962 1205963 1205964 1205965 1205966 1205967 1205968




sum

120590119895119901=plusmn1

1

4(

119899

prod

119901isin119886

119872

prod

119895isin119886

+

119899

prod

119901isin119887

119872

prod

119895isin119887

)

sdot [(1 minus 119904119895119901119904119895+1119901

) (1 + 119904119895119901119904119895119901+1

) cosh2120573qu

119899119860

119895119895+1

+ (1 minus 119904119895119901119904119895+1119901

) (1 minus 119904119895119901119904119895119901+1

) sinh2120573qu

119899119860

119895119895+1

+ (1 + 119904119895119901119904119895+1119901

) (1 minus 119904119895119901119904119895119901+1

) sinh2120573qu120574

119899119861119895119895+1

+ (1 + 119904119895119901119904119895119901+1

) (1 + 119904119895119901119904119895119901+1

) 119890(120573qu119899)ℎ119904119895119901


119899119861119895119895+1]

(37)



Trotter

direction

p darr







Hqu =1

2sum

1le119895le119896le119872

((119860119895119896+ 120574119861

119895119896) 120590

119909

119895120590119909

119896

+ (119860119895119896minus 120574119861

119895119896) 120590

119910

119895120590119910

119896)(

119896minus1

prod

119897=119895+1

minus 120590119911

119897) minus ℎ

119872

sum

119895=1

120590119911

119895

(38)




Hqu]

= TrV119899

(39)




119872

prod

119894=1

119889N (120591119894) ⟨N (120591

119872)1003816100381610038161003816 119890

minusΔ120591H 1003816100381610038161003816N (120591119872minus1)⟩

sdot ⟨N (120591119872minus1)1003816100381610038161003816 119890


1)1003816100381610038161003816

sdot 119890minusΔ120591H 1003816100381610038161003816N (120591119872)⟩

(40)




119889120591|N (120591)⟩)

= ⟨N (120591) | N (120591)⟩ minus Δ120591 ⟨N (120591)| 119889119889120591|N (120591)⟩

minus Δ120591 ⟨N (120591)| Hqu (S) |N (120591)⟩ + 119874 ((Δ120591)2

)

= 119890minusΔ120591(⟨N(120591)|(119889119889120591)|N(120591)⟩+H(N(120591)))

Δ120591

119872

sum

119894=1

997888rarr int

120573

0

119889120591

119872

prod

119894=1

119889N (120591119894) 997888rarr DN (120591)

(41)


119885 = int

N(120573)

N(0)

DN (120591) 119890minusint

120573

0119889120591H(N(120591))minusS119861 (42)


S119861= int

120573

0

119889120591 ⟨N (120591)| 119889119889120591|N (120591)⟩ (43)



⟨N| S |N⟩ = N (44)


⟨N| 119878119894 |N⟩ = minus119878119873119894

(45)



N = (119873119909

119873119910

119873119911


0 le 120579 le 120587 0 le 120601 le 2120587

(46)


H (N (120591)) = ⟨N (120591)| Hqu |N (120591)⟩

= sum

1le119895le119896le119872

((119860119895119896+ 120574119861

119895119896)119873

119909

119895(120591)119873

119909

119896(120591)

+ (119860119895119896minus 120574119861

119895119896)119873

119910

119895(120591)119873

119910

119896(120591))

119896minus1

prod

119897=119895+1

(minus119873119911

119897(120591))

minus ℎ

119872

sum

119895=1

119873119911

119895(120591) = sum

1le119895le119896le119872

(119860119895119896

cos (120601119895(120591) minus 120601

119896(120591))

+ 119861119895119896120574 cos (120601

119895(120591) + 120601

119896(120591))) sin (120579

119895(120591))

sdot sin (120579119896(120591))

119896minus1

prod

119897=119895+1

(minus cos (120579119897(120591))) minus ℎ

119872

sum

119895=1

cos (120579119895(120591))

(47)





[HquVcl] = 0 (48)


H = minus

119873

sum

119894

119872

sum

119895

(1198691119904119894119895119904119894+1119895

+ 1198692119904119894119895119904119894119895+1) (49)



2ℎ + 2V = 119873119872 (50)


119885 = sum


119910V= 119910

1198721198732

sum


(51)



120578dagger

119902=119890minus1198941205874

radic119872sum

119895

119890minus(2120587119894119872)119902119895

(119887dagger

119895119906119902+ 119894119887

119895V119902)

120578119902=1198901198941205874

radic119872sum

119895

119890(2120587119894119872)119902119895

(119887119895119906119902minus 119894119887

dagger

119895V119902)

(52)



V+(minus)

= (2 sinh 21198701)1198732

119890minussum119902120598119902(120578dagger

119902120578119902minus12) (53)

where119870119894= 120573119869

119894and 120598


cosh 120598119902= cosh 2119870lowast

1cosh 2119870

2


1sinh 2119870

2cos 119902

(54)



by

V2

119863

= prod

0le119902le120587

(120582119902(120578

dagger

119902120578119902+ 120578

dagger


2sin2

119902)) (55)


with

120582119902= 2120572 sin 119902 (1 + 1205722sin2

119902)12

(56)


2V119902119906119902

1199062119902minus V2

119902

=

119886119902

119887119902

for Hqu



1coth 2119870

2

for V


1tanh119870

2cos 119902)



1 sinh 2119870

2

for V1015840

minus1

120572 sin 119902for V2

119863

(57)



119886119902

119887119902

=sin 119902

cosh 2119870lowast


1coth 2119870

2

(58)


sinh 2119870lowast

1coth 2119870

2= minus119886 (119871 minus 1)

119887 (119871)

tanh2119870lowast

1=119886 (119871) minus 119887 (119871)

119886 (119871) + 119887 (119871)

119886 (119871 minus 1)

119886 (119871) + 119887 (119871)= minus coth 2119870

2tanh119870lowast

1

(59)

or inversely as

cosh 2119870lowast

1=119886 (119871)

119887 (119871)


1

119886 (119871 minus 1)

radic(119886 (119871))2

minus (119887 (119871))2

(60)

where

119886 (119871) = 119886 (119871)

[1198712]

sum

119897=0

(119871

2119897)

119887 (119871) = 119887 (119871)

[(119871minus1)2]

sum

119897=0

(119871

2119897 + 1)

119886 (0) = Γ

(61)


119887 (119871)ge 1

1198862

(119871) le 1198862

(119871 minus 1) + 1198872

(119871)

(62)


119871

sum

119896=1

119887 (119896)

[(119871minus1)2]

sum

119897=0

(119896

2119897 + 1)

119897

sum

119894=1

(minus1)119894 cos119896minus2119894119902

+

119871minus1

sum

119896=1

119887 (119896) cos119896119902 = 0

(63)

Γ +

119871minus2

sum

119896=1

119886 (119896) cos119896119902

+

119871

sum

119896=1

119886 (119896)

[1198962]

sum

119897=0

(119896

2119897)

119897

sum

119894=1

(119897

119894) (minus1)

119894 cos119871minus2119894119902 = 0

(64)




10038161003816100381610038161003816Λ

119902

10038161003816100381610038161003816= 2

119901+110038161003816100381610038161003816cos(119871minus1)11990210038161003816100381610038161003816

sdot ((119886 (119871) cos 119902 + 119886 (119871 minus 1))2 + 1198872

(119871) sin2

119902)

12

(65)


by

119870lowast

1= 119870

2 (66)



119886 (119871) = plusmn119886 (119871 minus 1) (67)


10038161003816100381610038161003816Λ

119902

10038161003816100381610038161003816= 2

119901+110038161003816100381610038161003816119886 (119871) cos(119871minus1)11990210038161003816100381610038161003816

sdot ((cos 119902 plusmn 1)2 + (119887 (119871)119886 (119871)

)

2

sin2

119902)

12

(68)



⟨120590119909

119895119896120590119909

119895+119903119896⟩Is= ⟨Ψ

0

1003816100381610038161003816Vminus12

1120590119909

119895120590119909

119895+119903V12

1

1003816100381610038161003816Ψ0⟩

= ⟨Φ0

1003816100381610038161003816Vminus12

1120590119909

119895120590119909

119895+119903V12

1

1003816100381610038161003816Φ0⟩

= ⟨(Vminus12

1120590119909

119895V12

1) (Vminus12

1120590119909

119895+119903V12

1)⟩

qu

= cosh2119870lowast

1⟨120590

119909

119895120590119909

119895+119903⟩qu


1⟨120590

119910

119895120590119910

119895+119903⟩qu

(69)


119895+119903⟩qu = ⟨120590

119910

119895120590119909

119895+119903⟩qu = 0 for 119903 = 0 and

Ψ0= Φ

0 (70)






119887119902


1tanh119870

2cos 119902)



1 sinh 2119870

2

(71)



tanh 2119870lowast

1tanh119870

2= minus

119887 (119871)

119887 (119871 minus 1)= minus

119886 (119871)

119887 (119871 minus 1)

119886 (119871 minus 1)

119887 (119871 minus 1)= 1

tanh 2119870lowast

1

sinh 21198702

= minus119886lowast

(119871)

119887 (119871 minus 1)

(72)

or inversely as

sinh21198702=

119886 (119871)

2 (119886lowast

(119871))

tanh 2119870lowast

1= minus

1

119886 (119871 minus 1)radic119886 (119871) (2119886

lowast

(119871) + 119886 (119871))

(73)

where

119886lowast

(119871) = 119886 (119871 minus 2) minus 119886 (119871)

[1198712]

sum

119897=0

(119871

2119897) 119897 (74)


119886 (119871) (2119886lowast

(119871) + 119886 (119871)) le 1198862

(119871 minus 1) (75)


119871minus2

sum

119896=1

119887 (119896) cos119896119902

+

119871

sum

119896=1

119887 (119896)

[(119896minus1)2]

sum

119897=0

(119896

2119897 + 1)

119897

sum

119894=1

(119897

119894) (minus1)

119894 cos119896minus2119894119902

= 0

(76)

Γ +

119871minus3

sum

119896=1

119896cos119896119902 minus119871minus1

sum

119896=1

119886 (119896)

[1198962]

sum

119897=0

(119896

2119897) 119897cos119896minus2119902

+

119871

sum

119896=1

119886 (119896)

[1198962]

sum

119897=0

(119896

2119897)

119897

sum

119894=2

(119897

119894) (minus1)

119894 cos119896minus2119894119902 = 0

(77)



10038161003816100381610038161003816Λ

119902

10038161003816100381610038161003816= 2

119901+110038161003816100381610038161003816cos119871minus211990210038161003816100381610038161003816


+ sin2

119902 (119887 (119871) cos 119902 + 119887 (119871 minus 1)))12

(78)





(119871) + 119886 (119871) (79)


10038161003816100381610038161003816Λ

119902

10038161003816100381610038161003816= 2

119901+110038161003816100381610038161003816cos119871minus211990210038161003816100381610038161003816


+ sin2

119902 (119887 (119871) cos 119902 + 119887 (119871 minus 1)))12

(80)



⟨120590119909

119895119896120590119909

119895+119903119896⟩Is= ⟨120590

119909

119895120590119909

119895+119903⟩qu (81)

where Ψ1015840



Ψ1015840

0= Φ

0 (82)



119863 In this case




119892M(120579) ofM


120574 = 0 and thus B = 0



119895119896(119861

119895119896) (M

119899)119895119896


119895 119896 ge 0119874

+

(2119873) 119886(119895 minus 119896) + 119886(119895 + 119896) 1198920if 119895 = 119896 = 0radic2119892

119897if

either 119895 = 0 119896 = 119897or 119895 = 119897 119896 = 0

119892119895minus119896+ 119892

119895+119896 119895 119896 gt 0


119895+119896+2 119895 119896 ge 0

119874plusmn

(2119873 + 1) 119886(119895 minus 119896) ∓ 119886(119895 + 119896 + 1) 119892119895minus119896∓ 119892

119895+119896+1 119895 119896 ge 0

119874minus


119895+119896+2 119895 119896 ge 0

Appendices

A Symmetry Classes

See Table 1



Hqu = minus119872

sum

119895=1

(119869119909

119895120590119909

119895120590119909

119895+1+ 119869

119910

119895120590119910

119895120590119910

119895+1

minus (1198691015840119909

119895120590119909

119895120590119909

119895+2+ 119869

1015840119910

119895120590119910

119895120590119910

119895+2) 120590

119911

119895+1+ ℎ120590

119911

119895)

(B1)

where 1198691015840119909119895= (12)(119860

119895119895+2+ 120574119861

119895119895+2) and 1198691015840119910

119895= (12)(119860

119895119895+2minus




Tr [119890(120573qu119899)H119909

119886 119890(120573qu2119899)H

119911

119890(120573qu119899)H

119910

119887 119890(120573qu119899)H

119910

119886 119890(120573qu2119899)H

119911

119890(120573qu119899)H

119909

119887 ]119899

(B2)


120572= sum

119872

119895isin120572((12)119869

120583

119895(120590

120583

119895120590120583

119895+1+ 120590

120583

119895+1120590120583

119895+2) minus

1198691015840120583

119895120590120583

119895120590119911

119895+1120590120583

119895+2) and H119911

= ℎsum119872

119895=1120590119911

119895 for 120583 isin 119909 119910 and once




Tr [I1205901119890(120573qu119899)H

119909

119886 I1199041119890(120573qu2119899)H

119911

119890(120573qu119899)H

119910

119887 I1205911119890(120573qu119899)H

119910

119886 I1199041119890(120573qu2119899)H

119911

119890(120573qu119899)H

119909

119887 ]119899


sum

120590119895119901 120591119895119901119904119895119901

119899

prod

119901=1

[⟨119901

10038161003816100381610038161003816119890(120573qu119899)H

119909

119886100381610038161003816100381610038161199042119901minus1⟩ ⟨ 119904

2119901minus1

10038161003816100381610038161003816119890(120573qu2119899)H

119911

119890(120573qu119899)H

119910

119887

10038161003816100381610038161003816120591119901⟩ ⟨ 120591

119901

10038161003816100381610038161003816119890(120573qu119899)H

119910

119886100381610038161003816100381610038161199042119901⟩ ⟨ 119904

2119901

10038161003816100381610038161003816119890(120573qu2119899)H

119911

119890(120573qu119899)H

119909

11988710038161003816100381610038161003816119901+1⟩]

(B3)



⟨119901

10038161003816100381610038161003816119890(120573119899)

H119909119886100381610038161003816100381610038161199042119901minus1⟩ ⟨ 119904

2119901minus1

10038161003816100381610038161003816119890(120573qu2119899)H

119911

119890(120573qu119899)H

119910

119887

10038161003816100381610038161003816120591119901⟩ ⟨ 120591

2119901minus1

10038161003816100381610038161003816119890(120573qu119899)H

119910

119886100381610038161003816100381610038161199042119901⟩ ⟨ 119904

2119901

10038161003816100381610038161003816119890(120573qu2119899)H

119911

119890(120573qu119899)H

119909

119887100381610038161003816100381610038162119901⟩

= 119890(120573qu119899)H

119909

119886(119901)

119890(120573qu2119899)H

119911(2119901minus1)

119890(120573qu119899)H

119910

119887(119901)

119890(120573qu119899)H

119910

119886(119901)

119890(120573qu2119899)H

119911(2119901)

119890(120573qu119899)H

119909

119887(119901)

⟨119901| 119904

2119901minus1⟩ ⟨ 119904

2119901minus1| 120591

119901⟩ ⟨ 120591

119901| 119904

2119901⟩

sdot ⟨ 1199042119901|

119901+1⟩

(B4)

where H119909

120572(119901) = sum

119872

119895isin120572((12)119869

119909

119895(120590

119895119901120590119895+1119901

+ 120590119895+1119901

120590119895+2119901

) +

1198691015840119909

119895+1120590119895119901119904119895+1119901

120590119895+2119901

) H119910

120572(119901) = sum

119872

119895isin120572((12)119869

119910

119895(120591

119895119901120591119895+1119901

+

120591119895+1119901

120591119895+2119901

) + 1198691015840119910

119895+1120591119895119901119904119895+1119901

120591119895+2119901

) andH119911

(119901) = sum119872

119895=1119904119895119901 We


⟨119901| 119904

2119901minus1⟩ ⟨ 119904

2119901minus1| 120591

119901⟩ ⟨ 120591

2119901minus1| 119904

2119901⟩ ⟨ 119904

2119901|

119901+1⟩

=1

24119872

sdot

119872

prod

119895=1


(B5)



119899

119899

sum

119901=1

(sum

119895isin119886

(119869119909

119895

2(120590

119895119901120590119895+1119901

+ 120590119895+1119901

120590119895+2119901

) minus 1198691015840119909

119895+1120590119895119901119904119895+1119901

120590119895+2119901

)

+sum

119895isin119887

(

119869119910

119895

2(120591

119895119901120591119895+1119901

+ 120591119895+1119901

120591119895+2119901

) minus 1198691015840119910

119895+1120591119895119901119904119895+12119901minus1

120591119895+2119901

)

+ sum

119895isin119886

(

119869119910

119895

2(120591

119895119901120591119895+1119901

+ 120591119895+1119901

120591119895+2119901

) minus 1198691015840119910

119895+1120591119895119901119904119895+12119901

120591119895+2119901

)

+sum

119895isin119887

(119869119909

119895

2(120590

119895119901+1120590119895+1119901+1

+ 120590119895+1119901+1

120590119895+2119901+1

) minus 1198691015840119909

119895+1120590119895119901+1

119904119895+12119901

120590119895+2119901+1

))

+

119899

sum

119901=1

(

119872

sum

119895=1

((120573quℎ

2119899minus119894120587

4) 119904

1198952119901+ (120573quℎ

2119899+119894120587

4) 119904

1198952119901) +

119872

sum

119895=1

119894120587

4(120590

1198951199011199041198952119901minus1

minus 120590119895119901+1

1199042119901+ 120591

119895119901(119904

1198952119901minus 119904

1198952119901minus1)))

+ 4119899119872 ln 2

(B6)



119909 (120573qu119899)119869119910

(120573qu119899)1198691015840119909 (120573qu119899)119869




S1

S2

S3

S4

1205901

1205911

1205902

1205912

1 2 3 4 5 6 7 8


Trotter

direction

p darr








Tr[

[

119872minus1

prod

119895=1

(119890(120573qu119899)H

119909

119895119895+1119890(120573qu119899)H

119909


119909

119895119872119890(120573qu2119899(119872minus1))

H119911119890(120573qu119899)H

119910


119910

119895119895+2119890(120573qu119899)H

119910

119895119895+1)]

]

119899


Tr[

[

119872

prod

119895=1

((

119872minus119895

prod

119896=1

119890(120573qu119899)H

119909

119895119895+119896)119890(120573qu2119899(119872minus1))

H119911(


prod

119896=0

119890(120573qu119899)H

119910

119895119872minus119896))]

]

119899

(B7)

where H120583

119895119896= 119869

120583

119895119896120590120583

119895120590120583

119896prod

119896minus1

119897=1(minus120590

119911

119897) for 120583 isin 119909 119910 and H119911

=

ℎsum119872

119895=1120590119911

119895



Tr[

[

119872minus1

prod

119895=1

(I120590119895(

119872minus119895

prod

119896=1

119890(120573qu119899)H

119909

119895119895+119896)119890(120573qu(119872minus1)119899)

H119911I119904119895(


prod

119896=0

119890(120573qu119899)H

119910

119895119872minus119896) I120591119895)]

]

119899


sum

120590119895119901 120591119895119901

119899minus1

prod

119901=0

119872minus1

prod

119895=1

(⟨119895+119895119901

10038161003816100381610038161003816(

119872minus119895

prod

119896=1

119890(120573qu119899)H

119909

119895119895+119896)119890(120573qu119899(119872minus1))

H119911 10038161003816100381610038161003816119904119895+119895119901⟩ ⟨ 119904

119895+119895119901

10038161003816100381610038161003816(


prod

119896=0

119890(120573qu119899)H

119910

119895119872minus119896)10038161003816100381610038161003816120591119895+119895119901⟩ ⟨ 120591

119895+119895119901|

119895+119895119901+1⟩)

(B8)


⟨119895+119895119901

10038161003816100381610038161003816(

119872minus119895

prod

119896=1

119890(120573qu119899)H

119909

119895119895+119896)119890(120573qu119899(119872minus1))

H119911 10038161003816100381610038161003816119904119895+119895119901⟩

sdot ⟨ 119904119895+119895119901

10038161003816100381610038161003816(


prod

119896=0

119890(120573qu119899)H

119910

119895119872minus119896)10038161003816100381610038161003816120591119895+119895119901⟩ ⟨ 120591

119895+119895119901|

119895+119895119901+1

⟩

= 119890(120573qu119899)sum

119872minus119895

119896=1(H119909119895119895+119896

(119901)+H119910

119895119895+119896(119901)+(1119899(119872minus1))H119911)

⟨119895+119895119901

|

119904119895+119895119901⟩ ⟨ 119904

119895+119895119901| 120591

119895+119895119901⟩ ⟨ 120591

119895+119895119901|

119895+119895119901+1⟩

(B9)


S1

S2

S3

S4

S5

S6

S7

S8

S9

S10

1 2 3 4 5 6 7 8 9


Trotter

direction

p darr



where H119909

119895119896(119901) = sum

119872

119896=119895+1119869119909

119895119896120590119895119901120590119896119901prod

119896minus1

119897=119895+1(minus119904

119897119901) H119910

119895119896(119901) =

sum119872

119896=119895+1119869119910

119895119896120591119895119901120591119896119901prod

119896minus1

119897=119895+1(minus119904

119897119901) andH119911

119901= ℎsum

119872

119895=1120590119911

119895119901 Finally


⟨119901| 119904

119901⟩ ⟨ 119904

119901| 120591

119901⟩ ⟨ 120591

119901|

119901+1⟩ = (

1

2radic2)

119872

sdot

119872

prod

119895=1


(B10)



sum

119901=1

119872

sum

119895=1

(120573qu

119899

119872

sum

119896=119895+1

(119869119909

119895119896120590119895119895+119895119901

120590119896119895+119895119901

+ 119869119910

119895119896120591119895119895+119895119901

120591119896119895+119895119901

)

119896minus1

prod

119897=119895+1

(minus119904119897119901) + (

120573qu

119899 (119872 minus 1)ℎ minus119894120587

4) 119904

119895119895+119895119901

+119894120587

4(1 minus 120590

119895119895+119895119901+ 120591

119895119895+119895119901+ 120590

119895119895+119895119901119904119895119895+119895119901

minus 120591119895119895+119895119901

119904119895119895+119895119901

minus 120590119895119895+119895119901+1

120591119895119895+119895119901

)) + 1198991198722 ln 1

2radic2

(B11)


119895119896





119909

119895119896 (120573qu119899)119869

119910

119895119896 and



1205901

S1

1205911

S2

1205902

S3

1205912

S4

1205903

S5

1 2 3 4 5 6 7 8 9 10

Trotter

direction

p darr






119863(55)


minus1

120572 sin 119902=119887 (1) sin 119902

Γ + 119886 (1) cos 119902 (C1)



minus1

120572 sin 119902=

119887 (1)

minus2119886 (2) sin 119902

if Γ = minus119886 (2) 119886 (1) = 119887 (2) = 0

(C2)


120572 =2119886 (2)

119887 (1) Γ = minus119886 (2) 119886 (1) = 119887 (2) = 0 (C3)


minus1

120572 sin 119902= minus

119887 (1) minus 119887 (3) + 119887 (2) cos 1199022 sin 119902 (119886 (2) + 119886 (3) cos 119902)

if Γ = minus119886 (2) 119886 (1) = minus119886 (3) 119887 (3) = 0

(C4)


120572 =2119886 (3)

119887 (2)

119886 (2)

119886 (3)=119887 (1) minus 119887 (3)

119887 (2)

Γ = minus119886 (2) 119886 (1) = minus119886 (3) 119887 (3) = 0

(C5)


minus1

120572 sin 119902

=sin 119902sum119898

119896=1119887 (119896)sum

[(119896minus1)2]

119897=0( 119896

2119897+1)sum

119897

119894=0( 119897119894) (minus1)


119896=2119886 (119896)sum

[1198962]

119897=0(minus1)

119897

( 119896

2119897) sin2119897

119902cos119896minus2119897119902

(C6)


[1198982]

sum

119895=1

119886 (2119895) + cos 119902[1198982]

sum

119895=1

119886 (2119895 + 1) + sin2

119902

sdot (

[1198982]

sum

119895=1

119886 (2119895)

119895

sum

119894=1

(119895

119894) (minus1)

119894 sin2(119894minus1)

119902

+ cos 119902[(119898minus1)2]

sum

119895=1

119886 (2119895 + 1)

119895

sum

119894=1

(119895

119894) (minus1)

119894 sin2(119894minus1)

119902

+

119898

sum

119896=2

119886 (119896)

[1198962]

sum

119897=1

(minus1)119897

(119896

2119897) sin2(119897minus1)

119902cos119896minus2119897119902)

(C7)


Γ = minus

[1198982]

sum

119895=1

119886 (2119895)

119886 (1) = minus

[(119898+1)2]

sum

119895=1

119886 (2119895 + 1) = 0

(C8)



sin2

119902(

[1198982]

sum

119895=1

119886 (2119895)

119895

sum

119894=1

(119895

119894)

119894minus1

sum

119901=0

(minus1)119894+119901 cos2119901119902

+

[(119898minus1)2]

sum

119895=1

119886 (2119895 + 1)

119895

sum

119894=1

(119895

119894)

119894minus1

sum

119901=0

(minus1)119894+119901 cos2119901+1119902 +

[(119898minus1)2]

sum

119895=1

119886 (2119895 + 1)

119895

sum

119897=1

(2119895 + 1

2119897)

119897minus1

sum

119901=0


+

[1198982]

sum

119895=1

119886 (2119895)

119895

sum

119897=1

(2119895

2119897)

119897minus1

sum

119901=0


(C9)


Disclosure




Acknowledgments


Endnotes



integer



and 119861119895119896





7 Thus matrices 119860119895119896

and 119861119895119896




1V

2V12

1of


=

V12

2V

1V12

2 whereV

1= (2 sinh 2119870

1)1198722


1sum119872

119894120590119909

119894 V2=

1198901198702 sum119872

119894=1120590119911

119894120590119911


119894= 119890



119887119902(5) as

119886119902= Γ +

119871

sum

119896=1

119886 (119896)

[1198962]

sum

119897=0

(119896

2119897)

119897

sum

119894=0

(119897

119894) (minus1)

minus119894 cos119896minus2119894119902

119887119902= tan 119902

sdot

119871

sum

119896=1

119887 (119896)

[(119896minus1)2]

sum

119897=0

(119896

2119897 + 1)

119897

sum

119894=0

(119897

119894) (minus1)

minus119894 cos119896minus2119894119902

(lowast)


119895=1(minus1)

119895

119886(2119895)


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Trotter

direction

p darr







Hqu =1

2sum

1le119895le119896le119872

((119860119895119896+ 120574119861

119895119896) 120590

119909

119895120590119909

119896

+ (119860119895119896minus 120574119861

119895119896) 120590

119910

119895120590119910

119896)(

119896minus1

prod

119897=119895+1

minus 120590119911

119897) minus ℎ

119872

sum

119895=1

120590119911

119895

(38)




Hqu]

= TrV119899

(39)




119872

prod

119894=1

119889N (120591119894) ⟨N (120591

119872)1003816100381610038161003816 119890

minusΔ120591H 1003816100381610038161003816N (120591119872minus1)⟩

sdot ⟨N (120591119872minus1)1003816100381610038161003816 119890


1)1003816100381610038161003816

sdot 119890minusΔ120591H 1003816100381610038161003816N (120591119872)⟩

(40)




119889120591|N (120591)⟩)

= ⟨N (120591) | N (120591)⟩ minus Δ120591 ⟨N (120591)| 119889119889120591|N (120591)⟩

minus Δ120591 ⟨N (120591)| Hqu (S) |N (120591)⟩ + 119874 ((Δ120591)2

)

= 119890minusΔ120591(⟨N(120591)|(119889119889120591)|N(120591)⟩+H(N(120591)))

Δ120591

119872

sum

119894=1

997888rarr int

120573

0

119889120591

119872

prod

119894=1

119889N (120591119894) 997888rarr DN (120591)

(41)


119885 = int

N(120573)

N(0)

DN (120591) 119890minusint

120573

0119889120591H(N(120591))minusS119861 (42)


S119861= int

120573

0

119889120591 ⟨N (120591)| 119889119889120591|N (120591)⟩ (43)



⟨N| S |N⟩ = N (44)


⟨N| 119878119894 |N⟩ = minus119878119873119894

(45)



N = (119873119909

119873119910

119873119911


0 le 120579 le 120587 0 le 120601 le 2120587

(46)


H (N (120591)) = ⟨N (120591)| Hqu |N (120591)⟩

= sum

1le119895le119896le119872

((119860119895119896+ 120574119861

119895119896)119873

119909

119895(120591)119873

119909

119896(120591)

+ (119860119895119896minus 120574119861

119895119896)119873

119910

119895(120591)119873

119910

119896(120591))

119896minus1

prod

119897=119895+1

(minus119873119911

119897(120591))

minus ℎ

119872

sum

119895=1

119873119911

119895(120591) = sum

1le119895le119896le119872

(119860119895119896

cos (120601119895(120591) minus 120601

119896(120591))

+ 119861119895119896120574 cos (120601

119895(120591) + 120601

119896(120591))) sin (120579

119895(120591))

sdot sin (120579119896(120591))

119896minus1

prod

119897=119895+1

(minus cos (120579119897(120591))) minus ℎ

119872

sum

119895=1

cos (120579119895(120591))

(47)





[HquVcl] = 0 (48)


H = minus

119873

sum

119894

119872

sum

119895

(1198691119904119894119895119904119894+1119895

+ 1198692119904119894119895119904119894119895+1) (49)



2ℎ + 2V = 119873119872 (50)


119885 = sum


119910V= 119910

1198721198732

sum


(51)



120578dagger

119902=119890minus1198941205874

radic119872sum

119895

119890minus(2120587119894119872)119902119895

(119887dagger

119895119906119902+ 119894119887

119895V119902)

120578119902=1198901198941205874

radic119872sum

119895

119890(2120587119894119872)119902119895

(119887119895119906119902minus 119894119887

dagger

119895V119902)

(52)



V+(minus)

= (2 sinh 21198701)1198732

119890minussum119902120598119902(120578dagger

119902120578119902minus12) (53)

where119870119894= 120573119869

119894and 120598


cosh 120598119902= cosh 2119870lowast

1cosh 2119870

2


1sinh 2119870

2cos 119902

(54)



by

V2

119863

= prod

0le119902le120587

(120582119902(120578

dagger

119902120578119902+ 120578

dagger


2sin2

119902)) (55)


with

120582119902= 2120572 sin 119902 (1 + 1205722sin2

119902)12

(56)


2V119902119906119902

1199062119902minus V2

119902

=

119886119902

119887119902

for Hqu



1coth 2119870

2

for V


1tanh119870

2cos 119902)



1 sinh 2119870

2

for V1015840

minus1

120572 sin 119902for V2

119863

(57)



119886119902

119887119902

=sin 119902

cosh 2119870lowast


1coth 2119870

2

(58)


sinh 2119870lowast

1coth 2119870

2= minus119886 (119871 minus 1)

119887 (119871)

tanh2119870lowast

1=119886 (119871) minus 119887 (119871)

119886 (119871) + 119887 (119871)

119886 (119871 minus 1)

119886 (119871) + 119887 (119871)= minus coth 2119870

2tanh119870lowast

1

(59)

or inversely as

cosh 2119870lowast

1=119886 (119871)

119887 (119871)


1

119886 (119871 minus 1)

radic(119886 (119871))2

minus (119887 (119871))2

(60)

where

119886 (119871) = 119886 (119871)

[1198712]

sum

119897=0

(119871

2119897)

119887 (119871) = 119887 (119871)

[(119871minus1)2]

sum

119897=0

(119871

2119897 + 1)

119886 (0) = Γ

(61)


119887 (119871)ge 1

1198862

(119871) le 1198862

(119871 minus 1) + 1198872

(119871)

(62)


119871

sum

119896=1

119887 (119896)

[(119871minus1)2]

sum

119897=0

(119896

2119897 + 1)

119897

sum

119894=1

(minus1)119894 cos119896minus2119894119902

+

119871minus1

sum

119896=1

119887 (119896) cos119896119902 = 0

(63)

Γ +

119871minus2

sum

119896=1

119886 (119896) cos119896119902

+

119871

sum

119896=1

119886 (119896)

[1198962]

sum

119897=0

(119896

2119897)

119897

sum

119894=1

(119897

119894) (minus1)

119894 cos119871minus2119894119902 = 0

(64)




10038161003816100381610038161003816Λ

119902

10038161003816100381610038161003816= 2

119901+110038161003816100381610038161003816cos(119871minus1)11990210038161003816100381610038161003816

sdot ((119886 (119871) cos 119902 + 119886 (119871 minus 1))2 + 1198872

(119871) sin2

119902)

12

(65)


by

119870lowast

1= 119870

2 (66)



119886 (119871) = plusmn119886 (119871 minus 1) (67)


10038161003816100381610038161003816Λ

119902

10038161003816100381610038161003816= 2

119901+110038161003816100381610038161003816119886 (119871) cos(119871minus1)11990210038161003816100381610038161003816

sdot ((cos 119902 plusmn 1)2 + (119887 (119871)119886 (119871)

)

2

sin2

119902)

12

(68)



⟨120590119909

119895119896120590119909

119895+119903119896⟩Is= ⟨Ψ

0

1003816100381610038161003816Vminus12

1120590119909

119895120590119909

119895+119903V12

1

1003816100381610038161003816Ψ0⟩

= ⟨Φ0

1003816100381610038161003816Vminus12

1120590119909

119895120590119909

119895+119903V12

1

1003816100381610038161003816Φ0⟩

= ⟨(Vminus12

1120590119909

119895V12

1) (Vminus12

1120590119909

119895+119903V12

1)⟩

qu

= cosh2119870lowast

1⟨120590

119909

119895120590119909

119895+119903⟩qu


1⟨120590

119910

119895120590119910

119895+119903⟩qu

(69)


119895+119903⟩qu = ⟨120590

119910

119895120590119909

119895+119903⟩qu = 0 for 119903 = 0 and

Ψ0= Φ

0 (70)






119887119902


1tanh119870

2cos 119902)



1 sinh 2119870

2

(71)



tanh 2119870lowast

1tanh119870

2= minus

119887 (119871)

119887 (119871 minus 1)= minus

119886 (119871)

119887 (119871 minus 1)

119886 (119871 minus 1)

119887 (119871 minus 1)= 1

tanh 2119870lowast

1

sinh 21198702

= minus119886lowast

(119871)

119887 (119871 minus 1)

(72)

or inversely as

sinh21198702=

119886 (119871)

2 (119886lowast

(119871))

tanh 2119870lowast

1= minus

1

119886 (119871 minus 1)radic119886 (119871) (2119886

lowast

(119871) + 119886 (119871))

(73)

where

119886lowast

(119871) = 119886 (119871 minus 2) minus 119886 (119871)

[1198712]

sum

119897=0

(119871

2119897) 119897 (74)


119886 (119871) (2119886lowast

(119871) + 119886 (119871)) le 1198862

(119871 minus 1) (75)


119871minus2

sum

119896=1

119887 (119896) cos119896119902

+

119871

sum

119896=1

119887 (119896)

[(119896minus1)2]

sum

119897=0

(119896

2119897 + 1)

119897

sum

119894=1

(119897

119894) (minus1)

119894 cos119896minus2119894119902

= 0

(76)

Γ +

119871minus3

sum

119896=1

119896cos119896119902 minus119871minus1

sum

119896=1

119886 (119896)

[1198962]

sum

119897=0

(119896

2119897) 119897cos119896minus2119902

+

119871

sum

119896=1

119886 (119896)

[1198962]

sum

119897=0

(119896

2119897)

119897

sum

119894=2

(119897

119894) (minus1)

119894 cos119896minus2119894119902 = 0

(77)



10038161003816100381610038161003816Λ

119902

10038161003816100381610038161003816= 2

119901+110038161003816100381610038161003816cos119871minus211990210038161003816100381610038161003816


+ sin2

119902 (119887 (119871) cos 119902 + 119887 (119871 minus 1)))12

(78)





(119871) + 119886 (119871) (79)


10038161003816100381610038161003816Λ

119902

10038161003816100381610038161003816= 2

119901+110038161003816100381610038161003816cos119871minus211990210038161003816100381610038161003816


+ sin2

119902 (119887 (119871) cos 119902 + 119887 (119871 minus 1)))12

(80)



⟨120590119909

119895119896120590119909

119895+119903119896⟩Is= ⟨120590

119909

119895120590119909

119895+119903⟩qu (81)

where Ψ1015840



Ψ1015840

0= Φ

0 (82)



119863 In this case




119892M(120579) ofM


120574 = 0 and thus B = 0



119895119896(119861

119895119896) (M

119899)119895119896


119895 119896 ge 0119874

+

(2119873) 119886(119895 minus 119896) + 119886(119895 + 119896) 1198920if 119895 = 119896 = 0radic2119892

119897if

either 119895 = 0 119896 = 119897or 119895 = 119897 119896 = 0

119892119895minus119896+ 119892

119895+119896 119895 119896 gt 0


119895+119896+2 119895 119896 ge 0

119874plusmn

(2119873 + 1) 119886(119895 minus 119896) ∓ 119886(119895 + 119896 + 1) 119892119895minus119896∓ 119892

119895+119896+1 119895 119896 ge 0

119874minus


119895+119896+2 119895 119896 ge 0

Appendices

A Symmetry Classes

See Table 1



Hqu = minus119872

sum

119895=1

(119869119909

119895120590119909

119895120590119909

119895+1+ 119869

119910

119895120590119910

119895120590119910

119895+1

minus (1198691015840119909

119895120590119909

119895120590119909

119895+2+ 119869

1015840119910

119895120590119910

119895120590119910

119895+2) 120590

119911

119895+1+ ℎ120590

119911

119895)

(B1)

where 1198691015840119909119895= (12)(119860

119895119895+2+ 120574119861

119895119895+2) and 1198691015840119910

119895= (12)(119860

119895119895+2minus




Tr [119890(120573qu119899)H119909

119886 119890(120573qu2119899)H

119911

119890(120573qu119899)H

119910

119887 119890(120573qu119899)H

119910

119886 119890(120573qu2119899)H

119911

119890(120573qu119899)H

119909

119887 ]119899

(B2)


120572= sum

119872

119895isin120572((12)119869

120583

119895(120590

120583

119895120590120583

119895+1+ 120590

120583

119895+1120590120583

119895+2) minus

1198691015840120583

119895120590120583

119895120590119911

119895+1120590120583

119895+2) and H119911

= ℎsum119872

119895=1120590119911

119895 for 120583 isin 119909 119910 and once




Tr [I1205901119890(120573qu119899)H

119909

119886 I1199041119890(120573qu2119899)H

119911

119890(120573qu119899)H

119910

119887 I1205911119890(120573qu119899)H

119910

119886 I1199041119890(120573qu2119899)H

119911

119890(120573qu119899)H

119909

119887 ]119899


sum

120590119895119901 120591119895119901119904119895119901

119899

prod

119901=1

[⟨119901

10038161003816100381610038161003816119890(120573qu119899)H

119909

119886100381610038161003816100381610038161199042119901minus1⟩ ⟨ 119904

2119901minus1

10038161003816100381610038161003816119890(120573qu2119899)H

119911

119890(120573qu119899)H

119910

119887

10038161003816100381610038161003816120591119901⟩ ⟨ 120591

119901

10038161003816100381610038161003816119890(120573qu119899)H

119910

119886100381610038161003816100381610038161199042119901⟩ ⟨ 119904

2119901

10038161003816100381610038161003816119890(120573qu2119899)H

119911

119890(120573qu119899)H

119909

11988710038161003816100381610038161003816119901+1⟩]

(B3)



⟨119901

10038161003816100381610038161003816119890(120573119899)

H119909119886100381610038161003816100381610038161199042119901minus1⟩ ⟨ 119904

2119901minus1

10038161003816100381610038161003816119890(120573qu2119899)H

119911

119890(120573qu119899)H

119910

119887

10038161003816100381610038161003816120591119901⟩ ⟨ 120591

2119901minus1

10038161003816100381610038161003816119890(120573qu119899)H

119910

119886100381610038161003816100381610038161199042119901⟩ ⟨ 119904

2119901

10038161003816100381610038161003816119890(120573qu2119899)H

119911

119890(120573qu119899)H

119909

119887100381610038161003816100381610038162119901⟩

= 119890(120573qu119899)H

119909

119886(119901)

119890(120573qu2119899)H

119911(2119901minus1)

119890(120573qu119899)H

119910

119887(119901)

119890(120573qu119899)H

119910

119886(119901)

119890(120573qu2119899)H

119911(2119901)

119890(120573qu119899)H

119909

119887(119901)

⟨119901| 119904

2119901minus1⟩ ⟨ 119904

2119901minus1| 120591

119901⟩ ⟨ 120591

119901| 119904

2119901⟩

sdot ⟨ 1199042119901|

119901+1⟩

(B4)

where H119909

120572(119901) = sum

119872

119895isin120572((12)119869

119909

119895(120590

119895119901120590119895+1119901

+ 120590119895+1119901

120590119895+2119901

) +

1198691015840119909

119895+1120590119895119901119904119895+1119901

120590119895+2119901

) H119910

120572(119901) = sum

119872

119895isin120572((12)119869

119910

119895(120591

119895119901120591119895+1119901

+

120591119895+1119901

120591119895+2119901

) + 1198691015840119910

119895+1120591119895119901119904119895+1119901

120591119895+2119901

) andH119911

(119901) = sum119872

119895=1119904119895119901 We


⟨119901| 119904

2119901minus1⟩ ⟨ 119904

2119901minus1| 120591

119901⟩ ⟨ 120591

2119901minus1| 119904

2119901⟩ ⟨ 119904

2119901|

119901+1⟩

=1

24119872

sdot

119872

prod

119895=1


(B5)



119899

119899

sum

119901=1

(sum

119895isin119886

(119869119909

119895

2(120590

119895119901120590119895+1119901

+ 120590119895+1119901

120590119895+2119901

) minus 1198691015840119909

119895+1120590119895119901119904119895+1119901

120590119895+2119901

)

+sum

119895isin119887

(

119869119910

119895

2(120591

119895119901120591119895+1119901

+ 120591119895+1119901

120591119895+2119901

) minus 1198691015840119910

119895+1120591119895119901119904119895+12119901minus1

120591119895+2119901

)

+ sum

119895isin119886

(

119869119910

119895

2(120591

119895119901120591119895+1119901

+ 120591119895+1119901

120591119895+2119901

) minus 1198691015840119910

119895+1120591119895119901119904119895+12119901

120591119895+2119901

)

+sum

119895isin119887

(119869119909

119895

2(120590

119895119901+1120590119895+1119901+1

+ 120590119895+1119901+1

120590119895+2119901+1

) minus 1198691015840119909

119895+1120590119895119901+1

119904119895+12119901

120590119895+2119901+1

))

+

119899

sum

119901=1

(

119872

sum

119895=1

((120573quℎ

2119899minus119894120587

4) 119904

1198952119901+ (120573quℎ

2119899+119894120587

4) 119904

1198952119901) +

119872

sum

119895=1

119894120587

4(120590

1198951199011199041198952119901minus1

minus 120590119895119901+1

1199042119901+ 120591

119895119901(119904

1198952119901minus 119904

1198952119901minus1)))

+ 4119899119872 ln 2

(B6)



119909 (120573qu119899)119869119910

(120573qu119899)1198691015840119909 (120573qu119899)119869




S1

S2

S3

S4

1205901

1205911

1205902

1205912

1 2 3 4 5 6 7 8


Trotter

direction

p darr








Tr[

[

119872minus1

prod

119895=1

(119890(120573qu119899)H

119909

119895119895+1119890(120573qu119899)H

119909


119909

119895119872119890(120573qu2119899(119872minus1))

H119911119890(120573qu119899)H

119910


119910

119895119895+2119890(120573qu119899)H

119910

119895119895+1)]

]

119899


Tr[

[

119872

prod

119895=1

((

119872minus119895

prod

119896=1

119890(120573qu119899)H

119909

119895119895+119896)119890(120573qu2119899(119872minus1))

H119911(


prod

119896=0

119890(120573qu119899)H

119910

119895119872minus119896))]

]

119899

(B7)

where H120583

119895119896= 119869

120583

119895119896120590120583

119895120590120583

119896prod

119896minus1

119897=1(minus120590

119911

119897) for 120583 isin 119909 119910 and H119911

=

ℎsum119872

119895=1120590119911

119895



Tr[

[

119872minus1

prod

119895=1

(I120590119895(

119872minus119895

prod

119896=1

119890(120573qu119899)H

119909

119895119895+119896)119890(120573qu(119872minus1)119899)

H119911I119904119895(


prod

119896=0

119890(120573qu119899)H

119910

119895119872minus119896) I120591119895)]

]

119899


sum

120590119895119901 120591119895119901

119899minus1

prod

119901=0

119872minus1

prod

119895=1

(⟨119895+119895119901

10038161003816100381610038161003816(

119872minus119895

prod

119896=1

119890(120573qu119899)H

119909

119895119895+119896)119890(120573qu119899(119872minus1))

H119911 10038161003816100381610038161003816119904119895+119895119901⟩ ⟨ 119904

119895+119895119901

10038161003816100381610038161003816(


prod

119896=0

119890(120573qu119899)H

119910

119895119872minus119896)10038161003816100381610038161003816120591119895+119895119901⟩ ⟨ 120591

119895+119895119901|

119895+119895119901+1⟩)

(B8)


⟨119895+119895119901

10038161003816100381610038161003816(

119872minus119895

prod

119896=1

119890(120573qu119899)H

119909

119895119895+119896)119890(120573qu119899(119872minus1))

H119911 10038161003816100381610038161003816119904119895+119895119901⟩

sdot ⟨ 119904119895+119895119901

10038161003816100381610038161003816(


prod

119896=0

119890(120573qu119899)H

119910

119895119872minus119896)10038161003816100381610038161003816120591119895+119895119901⟩ ⟨ 120591

119895+119895119901|

119895+119895119901+1

⟩

= 119890(120573qu119899)sum

119872minus119895

119896=1(H119909119895119895+119896

(119901)+H119910

119895119895+119896(119901)+(1119899(119872minus1))H119911)

⟨119895+119895119901

|

119904119895+119895119901⟩ ⟨ 119904

119895+119895119901| 120591

119895+119895119901⟩ ⟨ 120591

119895+119895119901|

119895+119895119901+1⟩

(B9)


S1

S2

S3

S4

S5

S6

S7

S8

S9

S10

1 2 3 4 5 6 7 8 9


Trotter

direction

p darr



where H119909

119895119896(119901) = sum

119872

119896=119895+1119869119909

119895119896120590119895119901120590119896119901prod

119896minus1

119897=119895+1(minus119904

119897119901) H119910

119895119896(119901) =

sum119872

119896=119895+1119869119910

119895119896120591119895119901120591119896119901prod

119896minus1

119897=119895+1(minus119904

119897119901) andH119911

119901= ℎsum

119872

119895=1120590119911

119895119901 Finally


⟨119901| 119904

119901⟩ ⟨ 119904

119901| 120591

119901⟩ ⟨ 120591

119901|

119901+1⟩ = (

1

2radic2)

119872

sdot

119872

prod

119895=1


(B10)



sum

119901=1

119872

sum

119895=1

(120573qu

119899

119872

sum

119896=119895+1

(119869119909

119895119896120590119895119895+119895119901

120590119896119895+119895119901

+ 119869119910

119895119896120591119895119895+119895119901

120591119896119895+119895119901

)

119896minus1

prod

119897=119895+1

(minus119904119897119901) + (

120573qu

119899 (119872 minus 1)ℎ minus119894120587

4) 119904

119895119895+119895119901

+119894120587

4(1 minus 120590

119895119895+119895119901+ 120591

119895119895+119895119901+ 120590

119895119895+119895119901119904119895119895+119895119901

minus 120591119895119895+119895119901

119904119895119895+119895119901

minus 120590119895119895+119895119901+1

120591119895119895+119895119901

)) + 1198991198722 ln 1

2radic2

(B11)


119895119896





119909

119895119896 (120573qu119899)119869

119910

119895119896 and



1205901

S1

1205911

S2

1205902

S3

1205912

S4

1205903

S5

1 2 3 4 5 6 7 8 9 10

Trotter

direction

p darr






119863(55)


minus1

120572 sin 119902=119887 (1) sin 119902

Γ + 119886 (1) cos 119902 (C1)



minus1

120572 sin 119902=

119887 (1)

minus2119886 (2) sin 119902

if Γ = minus119886 (2) 119886 (1) = 119887 (2) = 0

(C2)


120572 =2119886 (2)

119887 (1) Γ = minus119886 (2) 119886 (1) = 119887 (2) = 0 (C3)


minus1

120572 sin 119902= minus

119887 (1) minus 119887 (3) + 119887 (2) cos 1199022 sin 119902 (119886 (2) + 119886 (3) cos 119902)

if Γ = minus119886 (2) 119886 (1) = minus119886 (3) 119887 (3) = 0

(C4)


120572 =2119886 (3)

119887 (2)

119886 (2)

119886 (3)=119887 (1) minus 119887 (3)

119887 (2)

Γ = minus119886 (2) 119886 (1) = minus119886 (3) 119887 (3) = 0

(C5)


minus1

120572 sin 119902

=sin 119902sum119898

119896=1119887 (119896)sum

[(119896minus1)2]

119897=0( 119896

2119897+1)sum

119897

119894=0( 119897119894) (minus1)


119896=2119886 (119896)sum

[1198962]

119897=0(minus1)

119897

( 119896

2119897) sin2119897

119902cos119896minus2119897119902

(C6)


[1198982]

sum

119895=1

119886 (2119895) + cos 119902[1198982]

sum

119895=1

119886 (2119895 + 1) + sin2

119902

sdot (

[1198982]

sum

119895=1

119886 (2119895)

119895

sum

119894=1

(119895

119894) (minus1)

119894 sin2(119894minus1)

119902

+ cos 119902[(119898minus1)2]

sum

119895=1

119886 (2119895 + 1)

119895

sum

119894=1

(119895

119894) (minus1)

119894 sin2(119894minus1)

119902

+

119898

sum

119896=2

119886 (119896)

[1198962]

sum

119897=1

(minus1)119897

(119896

2119897) sin2(119897minus1)

119902cos119896minus2119897119902)

(C7)


Γ = minus

[1198982]

sum

119895=1

119886 (2119895)

119886 (1) = minus

[(119898+1)2]

sum

119895=1

119886 (2119895 + 1) = 0

(C8)



sin2

119902(

[1198982]

sum

119895=1

119886 (2119895)

119895

sum

119894=1

(119895

119894)

119894minus1

sum

119901=0

(minus1)119894+119901 cos2119901119902

+

[(119898minus1)2]

sum

119895=1

119886 (2119895 + 1)

119895

sum

119894=1

(119895

119894)

119894minus1

sum

119901=0

(minus1)119894+119901 cos2119901+1119902 +

[(119898minus1)2]

sum

119895=1

119886 (2119895 + 1)

119895

sum

119897=1

(2119895 + 1

2119897)

119897minus1

sum

119901=0


+

[1198982]

sum

119895=1

119886 (2119895)

119895

sum

119897=1

(2119895

2119897)

119897minus1

sum

119901=0


(C9)


Disclosure




Acknowledgments


Endnotes



integer



and 119861119895119896





7 Thus matrices 119860119895119896

and 119861119895119896




1V

2V12

1of


=

V12

2V

1V12

2 whereV

1= (2 sinh 2119870

1)1198722


1sum119872

119894120590119909

119894 V2=

1198901198702 sum119872

119894=1120590119911

119894120590119911


119894= 119890



119887119902(5) as

119886119902= Γ +

119871

sum

119896=1

119886 (119896)

[1198962]

sum

119897=0

(119896

2119897)

119897

sum

119894=0

(119897

119894) (minus1)

minus119894 cos119896minus2119894119902

119887119902= tan 119902

sdot

119871

sum

119896=1

119887 (119896)

[(119896minus1)2]

sum

119897=0

(119896

2119897 + 1)

119897

sum

119894=0

(119897

119894) (minus1)

minus119894 cos119896minus2119894119902

(lowast)


119895=1(minus1)

119895

119886(2119895)


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











N = (119873119909

119873119910

119873119911


0 le 120579 le 120587 0 le 120601 le 2120587

(46)


H (N (120591)) = ⟨N (120591)| Hqu |N (120591)⟩

= sum

1le119895le119896le119872

((119860119895119896+ 120574119861

119895119896)119873

119909

119895(120591)119873

119909

119896(120591)

+ (119860119895119896minus 120574119861

119895119896)119873

119910

119895(120591)119873

119910

119896(120591))

119896minus1

prod

119897=119895+1

(minus119873119911

119897(120591))

minus ℎ

119872

sum

119895=1

119873119911

119895(120591) = sum

1le119895le119896le119872

(119860119895119896

cos (120601119895(120591) minus 120601

119896(120591))

+ 119861119895119896120574 cos (120601

119895(120591) + 120601

119896(120591))) sin (120579

119895(120591))

sdot sin (120579119896(120591))

119896minus1

prod

119897=119895+1

(minus cos (120579119897(120591))) minus ℎ

119872

sum

119895=1

cos (120579119895(120591))

(47)





[HquVcl] = 0 (48)


H = minus

119873

sum

119894

119872

sum

119895

(1198691119904119894119895119904119894+1119895

+ 1198692119904119894119895119904119894119895+1) (49)



2ℎ + 2V = 119873119872 (50)


119885 = sum


119910V= 119910

1198721198732

sum


(51)



120578dagger

119902=119890minus1198941205874

radic119872sum

119895

119890minus(2120587119894119872)119902119895

(119887dagger

119895119906119902+ 119894119887

119895V119902)

120578119902=1198901198941205874

radic119872sum

119895

119890(2120587119894119872)119902119895

(119887119895119906119902minus 119894119887

dagger

119895V119902)

(52)



V+(minus)

= (2 sinh 21198701)1198732

119890minussum119902120598119902(120578dagger

119902120578119902minus12) (53)

where119870119894= 120573119869

119894and 120598


cosh 120598119902= cosh 2119870lowast

1cosh 2119870

2


1sinh 2119870

2cos 119902

(54)



by

V2

119863

= prod

0le119902le120587

(120582119902(120578

dagger

119902120578119902+ 120578

dagger


2sin2

119902)) (55)


with

120582119902= 2120572 sin 119902 (1 + 1205722sin2

119902)12

(56)


2V119902119906119902

1199062119902minus V2

119902

=

119886119902

119887119902

for Hqu



1coth 2119870

2

for V


1tanh119870

2cos 119902)



1 sinh 2119870

2

for V1015840

minus1

120572 sin 119902for V2

119863

(57)



119886119902

119887119902

=sin 119902

cosh 2119870lowast


1coth 2119870

2

(58)


sinh 2119870lowast

1coth 2119870

2= minus119886 (119871 minus 1)

119887 (119871)

tanh2119870lowast

1=119886 (119871) minus 119887 (119871)

119886 (119871) + 119887 (119871)

119886 (119871 minus 1)

119886 (119871) + 119887 (119871)= minus coth 2119870

2tanh119870lowast

1

(59)

or inversely as

cosh 2119870lowast

1=119886 (119871)

119887 (119871)


1

119886 (119871 minus 1)

radic(119886 (119871))2

minus (119887 (119871))2

(60)

where

119886 (119871) = 119886 (119871)

[1198712]

sum

119897=0

(119871

2119897)

119887 (119871) = 119887 (119871)

[(119871minus1)2]

sum

119897=0

(119871

2119897 + 1)

119886 (0) = Γ

(61)


119887 (119871)ge 1

1198862

(119871) le 1198862

(119871 minus 1) + 1198872

(119871)

(62)


119871

sum

119896=1

119887 (119896)

[(119871minus1)2]

sum

119897=0

(119896

2119897 + 1)

119897

sum

119894=1

(minus1)119894 cos119896minus2119894119902

+

119871minus1

sum

119896=1

119887 (119896) cos119896119902 = 0

(63)

Γ +

119871minus2

sum

119896=1

119886 (119896) cos119896119902

+

119871

sum

119896=1

119886 (119896)

[1198962]

sum

119897=0

(119896

2119897)

119897

sum

119894=1

(119897

119894) (minus1)

119894 cos119871minus2119894119902 = 0

(64)




10038161003816100381610038161003816Λ

119902

10038161003816100381610038161003816= 2

119901+110038161003816100381610038161003816cos(119871minus1)11990210038161003816100381610038161003816

sdot ((119886 (119871) cos 119902 + 119886 (119871 minus 1))2 + 1198872

(119871) sin2

119902)

12

(65)


by

119870lowast

1= 119870

2 (66)



119886 (119871) = plusmn119886 (119871 minus 1) (67)


10038161003816100381610038161003816Λ

119902

10038161003816100381610038161003816= 2

119901+110038161003816100381610038161003816119886 (119871) cos(119871minus1)11990210038161003816100381610038161003816

sdot ((cos 119902 plusmn 1)2 + (119887 (119871)119886 (119871)

)

2

sin2

119902)

12

(68)



⟨120590119909

119895119896120590119909

119895+119903119896⟩Is= ⟨Ψ

0

1003816100381610038161003816Vminus12

1120590119909

119895120590119909

119895+119903V12

1

1003816100381610038161003816Ψ0⟩

= ⟨Φ0

1003816100381610038161003816Vminus12

1120590119909

119895120590119909

119895+119903V12

1

1003816100381610038161003816Φ0⟩

= ⟨(Vminus12

1120590119909

119895V12

1) (Vminus12

1120590119909

119895+119903V12

1)⟩

qu

= cosh2119870lowast

1⟨120590

119909

119895120590119909

119895+119903⟩qu


1⟨120590

119910

119895120590119910

119895+119903⟩qu

(69)


119895+119903⟩qu = ⟨120590

119910

119895120590119909

119895+119903⟩qu = 0 for 119903 = 0 and

Ψ0= Φ

0 (70)






119887119902


1tanh119870

2cos 119902)



1 sinh 2119870

2

(71)



tanh 2119870lowast

1tanh119870

2= minus

119887 (119871)

119887 (119871 minus 1)= minus

119886 (119871)

119887 (119871 minus 1)

119886 (119871 minus 1)

119887 (119871 minus 1)= 1

tanh 2119870lowast

1

sinh 21198702

= minus119886lowast

(119871)

119887 (119871 minus 1)

(72)

or inversely as

sinh21198702=

119886 (119871)

2 (119886lowast

(119871))

tanh 2119870lowast

1= minus

1

119886 (119871 minus 1)radic119886 (119871) (2119886

lowast

(119871) + 119886 (119871))

(73)

where

119886lowast

(119871) = 119886 (119871 minus 2) minus 119886 (119871)

[1198712]

sum

119897=0

(119871

2119897) 119897 (74)


119886 (119871) (2119886lowast

(119871) + 119886 (119871)) le 1198862

(119871 minus 1) (75)


119871minus2

sum

119896=1

119887 (119896) cos119896119902

+

119871

sum

119896=1

119887 (119896)

[(119896minus1)2]

sum

119897=0

(119896

2119897 + 1)

119897

sum

119894=1

(119897

119894) (minus1)

119894 cos119896minus2119894119902

= 0

(76)

Γ +

119871minus3

sum

119896=1

119896cos119896119902 minus119871minus1

sum

119896=1

119886 (119896)

[1198962]

sum

119897=0

(119896

2119897) 119897cos119896minus2119902

+

119871

sum

119896=1

119886 (119896)

[1198962]

sum

119897=0

(119896

2119897)

119897

sum

119894=2

(119897

119894) (minus1)

119894 cos119896minus2119894119902 = 0

(77)



10038161003816100381610038161003816Λ

119902

10038161003816100381610038161003816= 2

119901+110038161003816100381610038161003816cos119871minus211990210038161003816100381610038161003816


+ sin2

119902 (119887 (119871) cos 119902 + 119887 (119871 minus 1)))12

(78)





(119871) + 119886 (119871) (79)


10038161003816100381610038161003816Λ

119902

10038161003816100381610038161003816= 2

119901+110038161003816100381610038161003816cos119871minus211990210038161003816100381610038161003816


+ sin2

119902 (119887 (119871) cos 119902 + 119887 (119871 minus 1)))12

(80)



⟨120590119909

119895119896120590119909

119895+119903119896⟩Is= ⟨120590

119909

119895120590119909

119895+119903⟩qu (81)

where Ψ1015840



Ψ1015840

0= Φ

0 (82)



119863 In this case




119892M(120579) ofM


120574 = 0 and thus B = 0



119895119896(119861

119895119896) (M

119899)119895119896


119895 119896 ge 0119874

+

(2119873) 119886(119895 minus 119896) + 119886(119895 + 119896) 1198920if 119895 = 119896 = 0radic2119892

119897if

either 119895 = 0 119896 = 119897or 119895 = 119897 119896 = 0

119892119895minus119896+ 119892

119895+119896 119895 119896 gt 0


119895+119896+2 119895 119896 ge 0

119874plusmn

(2119873 + 1) 119886(119895 minus 119896) ∓ 119886(119895 + 119896 + 1) 119892119895minus119896∓ 119892

119895+119896+1 119895 119896 ge 0

119874minus


119895+119896+2 119895 119896 ge 0

Appendices

A Symmetry Classes

See Table 1



Hqu = minus119872

sum

119895=1

(119869119909

119895120590119909

119895120590119909

119895+1+ 119869

119910

119895120590119910

119895120590119910

119895+1

minus (1198691015840119909

119895120590119909

119895120590119909

119895+2+ 119869

1015840119910

119895120590119910

119895120590119910

119895+2) 120590

119911

119895+1+ ℎ120590

119911

119895)

(B1)

where 1198691015840119909119895= (12)(119860

119895119895+2+ 120574119861

119895119895+2) and 1198691015840119910

119895= (12)(119860

119895119895+2minus




Tr [119890(120573qu119899)H119909

119886 119890(120573qu2119899)H

119911

119890(120573qu119899)H

119910

119887 119890(120573qu119899)H

119910

119886 119890(120573qu2119899)H

119911

119890(120573qu119899)H

119909

119887 ]119899

(B2)


120572= sum

119872

119895isin120572((12)119869

120583

119895(120590

120583

119895120590120583

119895+1+ 120590

120583

119895+1120590120583

119895+2) minus

1198691015840120583

119895120590120583

119895120590119911

119895+1120590120583

119895+2) and H119911

= ℎsum119872

119895=1120590119911

119895 for 120583 isin 119909 119910 and once




Tr [I1205901119890(120573qu119899)H

119909

119886 I1199041119890(120573qu2119899)H

119911

119890(120573qu119899)H

119910

119887 I1205911119890(120573qu119899)H

119910

119886 I1199041119890(120573qu2119899)H

119911

119890(120573qu119899)H

119909

119887 ]119899


sum

120590119895119901 120591119895119901119904119895119901

119899

prod

119901=1

[⟨119901

10038161003816100381610038161003816119890(120573qu119899)H

119909

119886100381610038161003816100381610038161199042119901minus1⟩ ⟨ 119904

2119901minus1

10038161003816100381610038161003816119890(120573qu2119899)H

119911

119890(120573qu119899)H

119910

119887

10038161003816100381610038161003816120591119901⟩ ⟨ 120591

119901

10038161003816100381610038161003816119890(120573qu119899)H

119910

119886100381610038161003816100381610038161199042119901⟩ ⟨ 119904

2119901

10038161003816100381610038161003816119890(120573qu2119899)H

119911

119890(120573qu119899)H

119909

11988710038161003816100381610038161003816119901+1⟩]

(B3)



⟨119901

10038161003816100381610038161003816119890(120573119899)

H119909119886100381610038161003816100381610038161199042119901minus1⟩ ⟨ 119904

2119901minus1

10038161003816100381610038161003816119890(120573qu2119899)H

119911

119890(120573qu119899)H

119910

119887

10038161003816100381610038161003816120591119901⟩ ⟨ 120591

2119901minus1

10038161003816100381610038161003816119890(120573qu119899)H

119910

119886100381610038161003816100381610038161199042119901⟩ ⟨ 119904

2119901

10038161003816100381610038161003816119890(120573qu2119899)H

119911

119890(120573qu119899)H

119909

119887100381610038161003816100381610038162119901⟩

= 119890(120573qu119899)H

119909

119886(119901)

119890(120573qu2119899)H

119911(2119901minus1)

119890(120573qu119899)H

119910

119887(119901)

119890(120573qu119899)H

119910

119886(119901)

119890(120573qu2119899)H

119911(2119901)

119890(120573qu119899)H

119909

119887(119901)

⟨119901| 119904

2119901minus1⟩ ⟨ 119904

2119901minus1| 120591

119901⟩ ⟨ 120591

119901| 119904

2119901⟩

sdot ⟨ 1199042119901|

119901+1⟩

(B4)

where H119909

120572(119901) = sum

119872

119895isin120572((12)119869

119909

119895(120590

119895119901120590119895+1119901

+ 120590119895+1119901

120590119895+2119901

) +

1198691015840119909

119895+1120590119895119901119904119895+1119901

120590119895+2119901

) H119910

120572(119901) = sum

119872

119895isin120572((12)119869

119910

119895(120591

119895119901120591119895+1119901

+

120591119895+1119901

120591119895+2119901

) + 1198691015840119910

119895+1120591119895119901119904119895+1119901

120591119895+2119901

) andH119911

(119901) = sum119872

119895=1119904119895119901 We


⟨119901| 119904

2119901minus1⟩ ⟨ 119904

2119901minus1| 120591

119901⟩ ⟨ 120591

2119901minus1| 119904

2119901⟩ ⟨ 119904

2119901|

119901+1⟩

=1

24119872

sdot

119872

prod

119895=1


(B5)



119899

119899

sum

119901=1

(sum

119895isin119886

(119869119909

119895

2(120590

119895119901120590119895+1119901

+ 120590119895+1119901

120590119895+2119901

) minus 1198691015840119909

119895+1120590119895119901119904119895+1119901

120590119895+2119901

)

+sum

119895isin119887

(

119869119910

119895

2(120591

119895119901120591119895+1119901

+ 120591119895+1119901

120591119895+2119901

) minus 1198691015840119910

119895+1120591119895119901119904119895+12119901minus1

120591119895+2119901

)

+ sum

119895isin119886

(

119869119910

119895

2(120591

119895119901120591119895+1119901

+ 120591119895+1119901

120591119895+2119901

) minus 1198691015840119910

119895+1120591119895119901119904119895+12119901

120591119895+2119901

)

+sum

119895isin119887

(119869119909

119895

2(120590

119895119901+1120590119895+1119901+1

+ 120590119895+1119901+1

120590119895+2119901+1

) minus 1198691015840119909

119895+1120590119895119901+1

119904119895+12119901

120590119895+2119901+1

))

+

119899

sum

119901=1

(

119872

sum

119895=1

((120573quℎ

2119899minus119894120587

4) 119904

1198952119901+ (120573quℎ

2119899+119894120587

4) 119904

1198952119901) +

119872

sum

119895=1

119894120587

4(120590

1198951199011199041198952119901minus1

minus 120590119895119901+1

1199042119901+ 120591

119895119901(119904

1198952119901minus 119904

1198952119901minus1)))

+ 4119899119872 ln 2

(B6)



119909 (120573qu119899)119869119910

(120573qu119899)1198691015840119909 (120573qu119899)119869




S1

S2

S3

S4

1205901

1205911

1205902

1205912

1 2 3 4 5 6 7 8


Trotter

direction

p darr








Tr[

[

119872minus1

prod

119895=1

(119890(120573qu119899)H

119909

119895119895+1119890(120573qu119899)H

119909


119909

119895119872119890(120573qu2119899(119872minus1))

H119911119890(120573qu119899)H

119910


119910

119895119895+2119890(120573qu119899)H

119910

119895119895+1)]

]

119899


Tr[

[

119872

prod

119895=1

((

119872minus119895

prod

119896=1

119890(120573qu119899)H

119909

119895119895+119896)119890(120573qu2119899(119872minus1))

H119911(


prod

119896=0

119890(120573qu119899)H

119910

119895119872minus119896))]

]

119899

(B7)

where H120583

119895119896= 119869

120583

119895119896120590120583

119895120590120583

119896prod

119896minus1

119897=1(minus120590

119911

119897) for 120583 isin 119909 119910 and H119911

=

ℎsum119872

119895=1120590119911

119895



Tr[

[

119872minus1

prod

119895=1

(I120590119895(

119872minus119895

prod

119896=1

119890(120573qu119899)H

119909

119895119895+119896)119890(120573qu(119872minus1)119899)

H119911I119904119895(


prod

119896=0

119890(120573qu119899)H

119910

119895119872minus119896) I120591119895)]

]

119899


sum

120590119895119901 120591119895119901

119899minus1

prod

119901=0

119872minus1

prod

119895=1

(⟨119895+119895119901

10038161003816100381610038161003816(

119872minus119895

prod

119896=1

119890(120573qu119899)H

119909

119895119895+119896)119890(120573qu119899(119872minus1))

H119911 10038161003816100381610038161003816119904119895+119895119901⟩ ⟨ 119904

119895+119895119901

10038161003816100381610038161003816(


prod

119896=0

119890(120573qu119899)H

119910

119895119872minus119896)10038161003816100381610038161003816120591119895+119895119901⟩ ⟨ 120591

119895+119895119901|

119895+119895119901+1⟩)

(B8)


⟨119895+119895119901

10038161003816100381610038161003816(

119872minus119895

prod

119896=1

119890(120573qu119899)H

119909

119895119895+119896)119890(120573qu119899(119872minus1))

H119911 10038161003816100381610038161003816119904119895+119895119901⟩

sdot ⟨ 119904119895+119895119901

10038161003816100381610038161003816(


prod

119896=0

119890(120573qu119899)H

119910

119895119872minus119896)10038161003816100381610038161003816120591119895+119895119901⟩ ⟨ 120591

119895+119895119901|

119895+119895119901+1

⟩

= 119890(120573qu119899)sum

119872minus119895

119896=1(H119909119895119895+119896

(119901)+H119910

119895119895+119896(119901)+(1119899(119872minus1))H119911)

⟨119895+119895119901

|

119904119895+119895119901⟩ ⟨ 119904

119895+119895119901| 120591

119895+119895119901⟩ ⟨ 120591

119895+119895119901|

119895+119895119901+1⟩

(B9)


S1

S2

S3

S4

S5

S6

S7

S8

S9

S10

1 2 3 4 5 6 7 8 9


Trotter

direction

p darr



where H119909

119895119896(119901) = sum

119872

119896=119895+1119869119909

119895119896120590119895119901120590119896119901prod

119896minus1

119897=119895+1(minus119904

119897119901) H119910

119895119896(119901) =

sum119872

119896=119895+1119869119910

119895119896120591119895119901120591119896119901prod

119896minus1

119897=119895+1(minus119904

119897119901) andH119911

119901= ℎsum

119872

119895=1120590119911

119895119901 Finally


⟨119901| 119904

119901⟩ ⟨ 119904

119901| 120591

119901⟩ ⟨ 120591

119901|

119901+1⟩ = (

1

2radic2)

119872

sdot

119872

prod

119895=1


(B10)



sum

119901=1

119872

sum

119895=1

(120573qu

119899

119872

sum

119896=119895+1

(119869119909

119895119896120590119895119895+119895119901

120590119896119895+119895119901

+ 119869119910

119895119896120591119895119895+119895119901

120591119896119895+119895119901

)

119896minus1

prod

119897=119895+1

(minus119904119897119901) + (

120573qu

119899 (119872 minus 1)ℎ minus119894120587

4) 119904

119895119895+119895119901

+119894120587

4(1 minus 120590

119895119895+119895119901+ 120591

119895119895+119895119901+ 120590

119895119895+119895119901119904119895119895+119895119901

minus 120591119895119895+119895119901

119904119895119895+119895119901

minus 120590119895119895+119895119901+1

120591119895119895+119895119901

)) + 1198991198722 ln 1

2radic2

(B11)


119895119896





119909

119895119896 (120573qu119899)119869

119910

119895119896 and



1205901

S1

1205911

S2

1205902

S3

1205912

S4

1205903

S5

1 2 3 4 5 6 7 8 9 10

Trotter

direction

p darr






119863(55)


minus1

120572 sin 119902=119887 (1) sin 119902

Γ + 119886 (1) cos 119902 (C1)



minus1

120572 sin 119902=

119887 (1)

minus2119886 (2) sin 119902

if Γ = minus119886 (2) 119886 (1) = 119887 (2) = 0

(C2)


120572 =2119886 (2)

119887 (1) Γ = minus119886 (2) 119886 (1) = 119887 (2) = 0 (C3)


minus1

120572 sin 119902= minus

119887 (1) minus 119887 (3) + 119887 (2) cos 1199022 sin 119902 (119886 (2) + 119886 (3) cos 119902)

if Γ = minus119886 (2) 119886 (1) = minus119886 (3) 119887 (3) = 0

(C4)


120572 =2119886 (3)

119887 (2)

119886 (2)

119886 (3)=119887 (1) minus 119887 (3)

119887 (2)

Γ = minus119886 (2) 119886 (1) = minus119886 (3) 119887 (3) = 0

(C5)


minus1

120572 sin 119902

=sin 119902sum119898

119896=1119887 (119896)sum

[(119896minus1)2]

119897=0( 119896

2119897+1)sum

119897

119894=0( 119897119894) (minus1)


119896=2119886 (119896)sum

[1198962]

119897=0(minus1)

119897

( 119896

2119897) sin2119897

119902cos119896minus2119897119902

(C6)


[1198982]

sum

119895=1

119886 (2119895) + cos 119902[1198982]

sum

119895=1

119886 (2119895 + 1) + sin2

119902

sdot (

[1198982]

sum

119895=1

119886 (2119895)

119895

sum

119894=1

(119895

119894) (minus1)

119894 sin2(119894minus1)

119902

+ cos 119902[(119898minus1)2]

sum

119895=1

119886 (2119895 + 1)

119895

sum

119894=1

(119895

119894) (minus1)

119894 sin2(119894minus1)

119902

+

119898

sum

119896=2

119886 (119896)

[1198962]

sum

119897=1

(minus1)119897

(119896

2119897) sin2(119897minus1)

119902cos119896minus2119897119902)

(C7)


Γ = minus

[1198982]

sum

119895=1

119886 (2119895)

119886 (1) = minus

[(119898+1)2]

sum

119895=1

119886 (2119895 + 1) = 0

(C8)



sin2

119902(

[1198982]

sum

119895=1

119886 (2119895)

119895

sum

119894=1

(119895

119894)

119894minus1

sum

119901=0

(minus1)119894+119901 cos2119901119902

+

[(119898minus1)2]

sum

119895=1

119886 (2119895 + 1)

119895

sum

119894=1

(119895

119894)

119894minus1

sum

119901=0

(minus1)119894+119901 cos2119901+1119902 +

[(119898minus1)2]

sum

119895=1

119886 (2119895 + 1)

119895

sum

119897=1

(2119895 + 1

2119897)

119897minus1

sum

119901=0


+

[1198982]

sum

119895=1

119886 (2119895)

119895

sum

119897=1

(2119895

2119897)

119897minus1

sum

119901=0


(C9)


Disclosure




Acknowledgments


Endnotes



integer



and 119861119895119896





7 Thus matrices 119860119895119896

and 119861119895119896




1V

2V12

1of


=

V12

2V

1V12

2 whereV

1= (2 sinh 2119870

1)1198722


1sum119872

119894120590119909

119894 V2=

1198901198702 sum119872

119894=1120590119911

119894120590119911


119894= 119890



119887119902(5) as

119886119902= Γ +

119871

sum

119896=1

119886 (119896)

[1198962]

sum

119897=0

(119896

2119897)

119897

sum

119894=0

(119897

119894) (minus1)

minus119894 cos119896minus2119894119902

119887119902= tan 119902

sdot

119871

sum

119896=1

119887 (119896)

[(119896minus1)2]

sum

119897=0

(119896

2119897 + 1)

119897

sum

119894=0

(119897

119894) (minus1)

minus119894 cos119896minus2119894119902

(lowast)


119895=1(minus1)

119895

119886(2119895)


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










with

120582119902= 2120572 sin 119902 (1 + 1205722sin2

119902)12

(56)


2V119902119906119902

1199062119902minus V2

119902

=

119886119902

119887119902

for Hqu



1coth 2119870

2

for V


1tanh119870

2cos 119902)



1 sinh 2119870

2

for V1015840

minus1

120572 sin 119902for V2

119863

(57)



119886119902

119887119902

=sin 119902

cosh 2119870lowast


1coth 2119870

2

(58)


sinh 2119870lowast

1coth 2119870

2= minus119886 (119871 minus 1)

119887 (119871)

tanh2119870lowast

1=119886 (119871) minus 119887 (119871)

119886 (119871) + 119887 (119871)

119886 (119871 minus 1)

119886 (119871) + 119887 (119871)= minus coth 2119870

2tanh119870lowast

1

(59)

or inversely as

cosh 2119870lowast

1=119886 (119871)

119887 (119871)


1

119886 (119871 minus 1)

radic(119886 (119871))2

minus (119887 (119871))2

(60)

where

119886 (119871) = 119886 (119871)

[1198712]

sum

119897=0

(119871

2119897)

119887 (119871) = 119887 (119871)

[(119871minus1)2]

sum

119897=0

(119871

2119897 + 1)

119886 (0) = Γ

(61)


119887 (119871)ge 1

1198862

(119871) le 1198862

(119871 minus 1) + 1198872

(119871)

(62)


119871

sum

119896=1

119887 (119896)

[(119871minus1)2]

sum

119897=0

(119896

2119897 + 1)

119897

sum

119894=1

(minus1)119894 cos119896minus2119894119902

+

119871minus1

sum

119896=1

119887 (119896) cos119896119902 = 0

(63)

Γ +

119871minus2

sum

119896=1

119886 (119896) cos119896119902

+

119871

sum

119896=1

119886 (119896)

[1198962]

sum

119897=0

(119896

2119897)

119897

sum

119894=1

(119897

119894) (minus1)

119894 cos119871minus2119894119902 = 0

(64)




10038161003816100381610038161003816Λ

119902

10038161003816100381610038161003816= 2

119901+110038161003816100381610038161003816cos(119871minus1)11990210038161003816100381610038161003816

sdot ((119886 (119871) cos 119902 + 119886 (119871 minus 1))2 + 1198872

(119871) sin2

119902)

12

(65)


by

119870lowast

1= 119870

2 (66)



119886 (119871) = plusmn119886 (119871 minus 1) (67)


10038161003816100381610038161003816Λ

119902

10038161003816100381610038161003816= 2

119901+110038161003816100381610038161003816119886 (119871) cos(119871minus1)11990210038161003816100381610038161003816

sdot ((cos 119902 plusmn 1)2 + (119887 (119871)119886 (119871)

)

2

sin2

119902)

12

(68)



⟨120590119909

119895119896120590119909

119895+119903119896⟩Is= ⟨Ψ

0

1003816100381610038161003816Vminus12

1120590119909

119895120590119909

119895+119903V12

1

1003816100381610038161003816Ψ0⟩

= ⟨Φ0

1003816100381610038161003816Vminus12

1120590119909

119895120590119909

119895+119903V12

1

1003816100381610038161003816Φ0⟩

= ⟨(Vminus12

1120590119909

119895V12

1) (Vminus12

1120590119909

119895+119903V12

1)⟩

qu

= cosh2119870lowast

1⟨120590

119909

119895120590119909

119895+119903⟩qu


1⟨120590

119910

119895120590119910

119895+119903⟩qu

(69)


119895+119903⟩qu = ⟨120590

119910

119895120590119909

119895+119903⟩qu = 0 for 119903 = 0 and

Ψ0= Φ

0 (70)






119887119902


1tanh119870

2cos 119902)



1 sinh 2119870

2

(71)



tanh 2119870lowast

1tanh119870

2= minus

119887 (119871)

119887 (119871 minus 1)= minus

119886 (119871)

119887 (119871 minus 1)

119886 (119871 minus 1)

119887 (119871 minus 1)= 1

tanh 2119870lowast

1

sinh 21198702

= minus119886lowast

(119871)

119887 (119871 minus 1)

(72)

or inversely as

sinh21198702=

119886 (119871)

2 (119886lowast

(119871))

tanh 2119870lowast

1= minus

1

119886 (119871 minus 1)radic119886 (119871) (2119886

lowast

(119871) + 119886 (119871))

(73)

where

119886lowast

(119871) = 119886 (119871 minus 2) minus 119886 (119871)

[1198712]

sum

119897=0

(119871

2119897) 119897 (74)


119886 (119871) (2119886lowast

(119871) + 119886 (119871)) le 1198862

(119871 minus 1) (75)


119871minus2

sum

119896=1

119887 (119896) cos119896119902

+

119871

sum

119896=1

119887 (119896)

[(119896minus1)2]

sum

119897=0

(119896

2119897 + 1)

119897

sum

119894=1

(119897

119894) (minus1)

119894 cos119896minus2119894119902

= 0

(76)

Γ +

119871minus3

sum

119896=1

119896cos119896119902 minus119871minus1

sum

119896=1

119886 (119896)

[1198962]

sum

119897=0

(119896

2119897) 119897cos119896minus2119902

+

119871

sum

119896=1

119886 (119896)

[1198962]

sum

119897=0

(119896

2119897)

119897

sum

119894=2

(119897

119894) (minus1)

119894 cos119896minus2119894119902 = 0

(77)



10038161003816100381610038161003816Λ

119902

10038161003816100381610038161003816= 2

119901+110038161003816100381610038161003816cos119871minus211990210038161003816100381610038161003816


+ sin2

119902 (119887 (119871) cos 119902 + 119887 (119871 minus 1)))12

(78)





(119871) + 119886 (119871) (79)


10038161003816100381610038161003816Λ

119902

10038161003816100381610038161003816= 2

119901+110038161003816100381610038161003816cos119871minus211990210038161003816100381610038161003816


+ sin2

119902 (119887 (119871) cos 119902 + 119887 (119871 minus 1)))12

(80)



⟨120590119909

119895119896120590119909

119895+119903119896⟩Is= ⟨120590

119909

119895120590119909

119895+119903⟩qu (81)

where Ψ1015840



Ψ1015840

0= Φ

0 (82)



119863 In this case




119892M(120579) ofM


120574 = 0 and thus B = 0



119895119896(119861

119895119896) (M

119899)119895119896


119895 119896 ge 0119874

+

(2119873) 119886(119895 minus 119896) + 119886(119895 + 119896) 1198920if 119895 = 119896 = 0radic2119892

119897if

either 119895 = 0 119896 = 119897or 119895 = 119897 119896 = 0

119892119895minus119896+ 119892

119895+119896 119895 119896 gt 0


119895+119896+2 119895 119896 ge 0

119874plusmn

(2119873 + 1) 119886(119895 minus 119896) ∓ 119886(119895 + 119896 + 1) 119892119895minus119896∓ 119892

119895+119896+1 119895 119896 ge 0

119874minus


119895+119896+2 119895 119896 ge 0

Appendices

A Symmetry Classes

See Table 1



Hqu = minus119872

sum

119895=1

(119869119909

119895120590119909

119895120590119909

119895+1+ 119869

119910

119895120590119910

119895120590119910

119895+1

minus (1198691015840119909

119895120590119909

119895120590119909

119895+2+ 119869

1015840119910

119895120590119910

119895120590119910

119895+2) 120590

119911

119895+1+ ℎ120590

119911

119895)

(B1)

where 1198691015840119909119895= (12)(119860

119895119895+2+ 120574119861

119895119895+2) and 1198691015840119910

119895= (12)(119860

119895119895+2minus




Tr [119890(120573qu119899)H119909

119886 119890(120573qu2119899)H

119911

119890(120573qu119899)H

119910

119887 119890(120573qu119899)H

119910

119886 119890(120573qu2119899)H

119911

119890(120573qu119899)H

119909

119887 ]119899

(B2)


120572= sum

119872

119895isin120572((12)119869

120583

119895(120590

120583

119895120590120583

119895+1+ 120590

120583

119895+1120590120583

119895+2) minus

1198691015840120583

119895120590120583

119895120590119911

119895+1120590120583

119895+2) and H119911

= ℎsum119872

119895=1120590119911

119895 for 120583 isin 119909 119910 and once




Tr [I1205901119890(120573qu119899)H

119909

119886 I1199041119890(120573qu2119899)H

119911

119890(120573qu119899)H

119910

119887 I1205911119890(120573qu119899)H

119910

119886 I1199041119890(120573qu2119899)H

119911

119890(120573qu119899)H

119909

119887 ]119899


sum

120590119895119901 120591119895119901119904119895119901

119899

prod

119901=1

[⟨119901

10038161003816100381610038161003816119890(120573qu119899)H

119909

119886100381610038161003816100381610038161199042119901minus1⟩ ⟨ 119904

2119901minus1

10038161003816100381610038161003816119890(120573qu2119899)H

119911

119890(120573qu119899)H

119910

119887

10038161003816100381610038161003816120591119901⟩ ⟨ 120591

119901

10038161003816100381610038161003816119890(120573qu119899)H

119910

119886100381610038161003816100381610038161199042119901⟩ ⟨ 119904

2119901

10038161003816100381610038161003816119890(120573qu2119899)H

119911

119890(120573qu119899)H

119909

11988710038161003816100381610038161003816119901+1⟩]

(B3)



⟨119901

10038161003816100381610038161003816119890(120573119899)

H119909119886100381610038161003816100381610038161199042119901minus1⟩ ⟨ 119904

2119901minus1

10038161003816100381610038161003816119890(120573qu2119899)H

119911

119890(120573qu119899)H

119910

119887

10038161003816100381610038161003816120591119901⟩ ⟨ 120591

2119901minus1

10038161003816100381610038161003816119890(120573qu119899)H

119910

119886100381610038161003816100381610038161199042119901⟩ ⟨ 119904

2119901

10038161003816100381610038161003816119890(120573qu2119899)H

119911

119890(120573qu119899)H

119909

119887100381610038161003816100381610038162119901⟩

= 119890(120573qu119899)H

119909

119886(119901)

119890(120573qu2119899)H

119911(2119901minus1)

119890(120573qu119899)H

119910

119887(119901)

119890(120573qu119899)H

119910

119886(119901)

119890(120573qu2119899)H

119911(2119901)

119890(120573qu119899)H

119909

119887(119901)

⟨119901| 119904

2119901minus1⟩ ⟨ 119904

2119901minus1| 120591

119901⟩ ⟨ 120591

119901| 119904

2119901⟩

sdot ⟨ 1199042119901|

119901+1⟩

(B4)

where H119909

120572(119901) = sum

119872

119895isin120572((12)119869

119909

119895(120590

119895119901120590119895+1119901

+ 120590119895+1119901

120590119895+2119901

) +

1198691015840119909

119895+1120590119895119901119904119895+1119901

120590119895+2119901

) H119910

120572(119901) = sum

119872

119895isin120572((12)119869

119910

119895(120591

119895119901120591119895+1119901

+

120591119895+1119901

120591119895+2119901

) + 1198691015840119910

119895+1120591119895119901119904119895+1119901

120591119895+2119901

) andH119911

(119901) = sum119872

119895=1119904119895119901 We


⟨119901| 119904

2119901minus1⟩ ⟨ 119904

2119901minus1| 120591

119901⟩ ⟨ 120591

2119901minus1| 119904

2119901⟩ ⟨ 119904

2119901|

119901+1⟩

=1

24119872

sdot

119872

prod

119895=1


(B5)



119899

119899

sum

119901=1

(sum

119895isin119886

(119869119909

119895

2(120590

119895119901120590119895+1119901

+ 120590119895+1119901

120590119895+2119901

) minus 1198691015840119909

119895+1120590119895119901119904119895+1119901

120590119895+2119901

)

+sum

119895isin119887

(

119869119910

119895

2(120591

119895119901120591119895+1119901

+ 120591119895+1119901

120591119895+2119901

) minus 1198691015840119910

119895+1120591119895119901119904119895+12119901minus1

120591119895+2119901

)

+ sum

119895isin119886

(

119869119910

119895

2(120591

119895119901120591119895+1119901

+ 120591119895+1119901

120591119895+2119901

) minus 1198691015840119910

119895+1120591119895119901119904119895+12119901

120591119895+2119901

)

+sum

119895isin119887

(119869119909

119895

2(120590

119895119901+1120590119895+1119901+1

+ 120590119895+1119901+1

120590119895+2119901+1

) minus 1198691015840119909

119895+1120590119895119901+1

119904119895+12119901

120590119895+2119901+1

))

+

119899

sum

119901=1

(

119872

sum

119895=1

((120573quℎ

2119899minus119894120587

4) 119904

1198952119901+ (120573quℎ

2119899+119894120587

4) 119904

1198952119901) +

119872

sum

119895=1

119894120587

4(120590

1198951199011199041198952119901minus1

minus 120590119895119901+1

1199042119901+ 120591

119895119901(119904

1198952119901minus 119904

1198952119901minus1)))

+ 4119899119872 ln 2

(B6)



119909 (120573qu119899)119869119910

(120573qu119899)1198691015840119909 (120573qu119899)119869




S1

S2

S3

S4

1205901

1205911

1205902

1205912

1 2 3 4 5 6 7 8


Trotter

direction

p darr








Tr[

[

119872minus1

prod

119895=1

(119890(120573qu119899)H

119909

119895119895+1119890(120573qu119899)H

119909


119909

119895119872119890(120573qu2119899(119872minus1))

H119911119890(120573qu119899)H

119910


119910

119895119895+2119890(120573qu119899)H

119910

119895119895+1)]

]

119899


Tr[

[

119872

prod

119895=1

((

119872minus119895

prod

119896=1

119890(120573qu119899)H

119909

119895119895+119896)119890(120573qu2119899(119872minus1))

H119911(


prod

119896=0

119890(120573qu119899)H

119910

119895119872minus119896))]

]

119899

(B7)

where H120583

119895119896= 119869

120583

119895119896120590120583

119895120590120583

119896prod

119896minus1

119897=1(minus120590

119911

119897) for 120583 isin 119909 119910 and H119911

=

ℎsum119872

119895=1120590119911

119895



Tr[

[

119872minus1

prod

119895=1

(I120590119895(

119872minus119895

prod

119896=1

119890(120573qu119899)H

119909

119895119895+119896)119890(120573qu(119872minus1)119899)

H119911I119904119895(


prod

119896=0

119890(120573qu119899)H

119910

119895119872minus119896) I120591119895)]

]

119899


sum

120590119895119901 120591119895119901

119899minus1

prod

119901=0

119872minus1

prod

119895=1

(⟨119895+119895119901

10038161003816100381610038161003816(

119872minus119895

prod

119896=1

119890(120573qu119899)H

119909

119895119895+119896)119890(120573qu119899(119872minus1))

H119911 10038161003816100381610038161003816119904119895+119895119901⟩ ⟨ 119904

119895+119895119901

10038161003816100381610038161003816(


prod

119896=0

119890(120573qu119899)H

119910

119895119872minus119896)10038161003816100381610038161003816120591119895+119895119901⟩ ⟨ 120591

119895+119895119901|

119895+119895119901+1⟩)

(B8)


⟨119895+119895119901

10038161003816100381610038161003816(

119872minus119895

prod

119896=1

119890(120573qu119899)H

119909

119895119895+119896)119890(120573qu119899(119872minus1))

H119911 10038161003816100381610038161003816119904119895+119895119901⟩

sdot ⟨ 119904119895+119895119901

10038161003816100381610038161003816(


prod

119896=0

119890(120573qu119899)H

119910

119895119872minus119896)10038161003816100381610038161003816120591119895+119895119901⟩ ⟨ 120591

119895+119895119901|

119895+119895119901+1

⟩

= 119890(120573qu119899)sum

119872minus119895

119896=1(H119909119895119895+119896

(119901)+H119910

119895119895+119896(119901)+(1119899(119872minus1))H119911)

⟨119895+119895119901

|

119904119895+119895119901⟩ ⟨ 119904

119895+119895119901| 120591

119895+119895119901⟩ ⟨ 120591

119895+119895119901|

119895+119895119901+1⟩

(B9)


S1

S2

S3

S4

S5

S6

S7

S8

S9

S10

1 2 3 4 5 6 7 8 9


Trotter

direction

p darr



where H119909

119895119896(119901) = sum

119872

119896=119895+1119869119909

119895119896120590119895119901120590119896119901prod

119896minus1

119897=119895+1(minus119904

119897119901) H119910

119895119896(119901) =

sum119872

119896=119895+1119869119910

119895119896120591119895119901120591119896119901prod

119896minus1

119897=119895+1(minus119904

119897119901) andH119911

119901= ℎsum

119872

119895=1120590119911

119895119901 Finally


⟨119901| 119904

119901⟩ ⟨ 119904

119901| 120591

119901⟩ ⟨ 120591

119901|

119901+1⟩ = (

1

2radic2)

119872

sdot

119872

prod

119895=1


(B10)



sum

119901=1

119872

sum

119895=1

(120573qu

119899

119872

sum

119896=119895+1

(119869119909

119895119896120590119895119895+119895119901

120590119896119895+119895119901

+ 119869119910

119895119896120591119895119895+119895119901

120591119896119895+119895119901

)

119896minus1

prod

119897=119895+1

(minus119904119897119901) + (

120573qu

119899 (119872 minus 1)ℎ minus119894120587

4) 119904

119895119895+119895119901

+119894120587

4(1 minus 120590

119895119895+119895119901+ 120591

119895119895+119895119901+ 120590

119895119895+119895119901119904119895119895+119895119901

minus 120591119895119895+119895119901

119904119895119895+119895119901

minus 120590119895119895+119895119901+1

120591119895119895+119895119901

)) + 1198991198722 ln 1

2radic2

(B11)


119895119896





119909

119895119896 (120573qu119899)119869

119910

119895119896 and



1205901

S1

1205911

S2

1205902

S3

1205912

S4

1205903

S5

1 2 3 4 5 6 7 8 9 10

Trotter

direction

p darr






119863(55)


minus1

120572 sin 119902=119887 (1) sin 119902

Γ + 119886 (1) cos 119902 (C1)



minus1

120572 sin 119902=

119887 (1)

minus2119886 (2) sin 119902

if Γ = minus119886 (2) 119886 (1) = 119887 (2) = 0

(C2)


120572 =2119886 (2)

119887 (1) Γ = minus119886 (2) 119886 (1) = 119887 (2) = 0 (C3)


minus1

120572 sin 119902= minus

119887 (1) minus 119887 (3) + 119887 (2) cos 1199022 sin 119902 (119886 (2) + 119886 (3) cos 119902)

if Γ = minus119886 (2) 119886 (1) = minus119886 (3) 119887 (3) = 0

(C4)


120572 =2119886 (3)

119887 (2)

119886 (2)

119886 (3)=119887 (1) minus 119887 (3)

119887 (2)

Γ = minus119886 (2) 119886 (1) = minus119886 (3) 119887 (3) = 0

(C5)


minus1

120572 sin 119902

=sin 119902sum119898

119896=1119887 (119896)sum

[(119896minus1)2]

119897=0( 119896

2119897+1)sum

119897

119894=0( 119897119894) (minus1)


119896=2119886 (119896)sum

[1198962]

119897=0(minus1)

119897

( 119896

2119897) sin2119897

119902cos119896minus2119897119902

(C6)


[1198982]

sum

119895=1

119886 (2119895) + cos 119902[1198982]

sum

119895=1

119886 (2119895 + 1) + sin2

119902

sdot (

[1198982]

sum

119895=1

119886 (2119895)

119895

sum

119894=1

(119895

119894) (minus1)

119894 sin2(119894minus1)

119902

+ cos 119902[(119898minus1)2]

sum

119895=1

119886 (2119895 + 1)

119895

sum

119894=1

(119895

119894) (minus1)

119894 sin2(119894minus1)

119902

+

119898

sum

119896=2

119886 (119896)

[1198962]

sum

119897=1

(minus1)119897

(119896

2119897) sin2(119897minus1)

119902cos119896minus2119897119902)

(C7)


Γ = minus

[1198982]

sum

119895=1

119886 (2119895)

119886 (1) = minus

[(119898+1)2]

sum

119895=1

119886 (2119895 + 1) = 0

(C8)



sin2

119902(

[1198982]

sum

119895=1

119886 (2119895)

119895

sum

119894=1

(119895

119894)

119894minus1

sum

119901=0

(minus1)119894+119901 cos2119901119902

+

[(119898minus1)2]

sum

119895=1

119886 (2119895 + 1)

119895

sum

119894=1

(119895

119894)

119894minus1

sum

119901=0

(minus1)119894+119901 cos2119901+1119902 +

[(119898minus1)2]

sum

119895=1

119886 (2119895 + 1)

119895

sum

119897=1

(2119895 + 1

2119897)

119897minus1

sum

119901=0


+

[1198982]

sum

119895=1

119886 (2119895)

119895

sum

119897=1

(2119895

2119897)

119897minus1

sum

119901=0


(C9)


Disclosure




Acknowledgments


Endnotes



integer



and 119861119895119896





7 Thus matrices 119860119895119896

and 119861119895119896




1V

2V12

1of


=

V12

2V

1V12

2 whereV

1= (2 sinh 2119870

1)1198722


1sum119872

119894120590119909

119894 V2=

1198901198702 sum119872

119894=1120590119911

119894120590119911


119894= 119890



119887119902(5) as

119886119902= Γ +

119871

sum

119896=1

119886 (119896)

[1198962]

sum

119897=0

(119896

2119897)

119897

sum

119894=0

(119897

119894) (minus1)

minus119894 cos119896minus2119894119902

119887119902= tan 119902

sdot

119871

sum

119896=1

119887 (119896)

[(119896minus1)2]

sum

119897=0

(119896

2119897 + 1)

119897

sum

119894=0

(119897

119894) (minus1)

minus119894 cos119896minus2119894119902

(lowast)


119895=1(minus1)

119895

119886(2119895)


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119886 (119871) = plusmn119886 (119871 minus 1) (67)


10038161003816100381610038161003816Λ

119902

10038161003816100381610038161003816= 2

119901+110038161003816100381610038161003816119886 (119871) cos(119871minus1)11990210038161003816100381610038161003816

sdot ((cos 119902 plusmn 1)2 + (119887 (119871)119886 (119871)

)

2

sin2

119902)

12

(68)



⟨120590119909

119895119896120590119909

119895+119903119896⟩Is= ⟨Ψ

0

1003816100381610038161003816Vminus12

1120590119909

119895120590119909

119895+119903V12

1

1003816100381610038161003816Ψ0⟩

= ⟨Φ0

1003816100381610038161003816Vminus12

1120590119909

119895120590119909

119895+119903V12

1

1003816100381610038161003816Φ0⟩

= ⟨(Vminus12

1120590119909

119895V12

1) (Vminus12

1120590119909

119895+119903V12

1)⟩

qu

= cosh2119870lowast

1⟨120590

119909

119895120590119909

119895+119903⟩qu


1⟨120590

119910

119895120590119910

119895+119903⟩qu

(69)


119895+119903⟩qu = ⟨120590

119910

119895120590119909

119895+119903⟩qu = 0 for 119903 = 0 and

Ψ0= Φ

0 (70)






119887119902


1tanh119870

2cos 119902)



1 sinh 2119870

2

(71)



tanh 2119870lowast

1tanh119870

2= minus

119887 (119871)

119887 (119871 minus 1)= minus

119886 (119871)

119887 (119871 minus 1)

119886 (119871 minus 1)

119887 (119871 minus 1)= 1

tanh 2119870lowast

1

sinh 21198702

= minus119886lowast

(119871)

119887 (119871 minus 1)

(72)

or inversely as

sinh21198702=

119886 (119871)

2 (119886lowast

(119871))

tanh 2119870lowast

1= minus

1

119886 (119871 minus 1)radic119886 (119871) (2119886

lowast

(119871) + 119886 (119871))

(73)

where

119886lowast

(119871) = 119886 (119871 minus 2) minus 119886 (119871)

[1198712]

sum

119897=0

(119871

2119897) 119897 (74)


119886 (119871) (2119886lowast

(119871) + 119886 (119871)) le 1198862

(119871 minus 1) (75)


119871minus2

sum

119896=1

119887 (119896) cos119896119902

+

119871

sum

119896=1

119887 (119896)

[(119896minus1)2]

sum

119897=0

(119896

2119897 + 1)

119897

sum

119894=1

(119897

119894) (minus1)

119894 cos119896minus2119894119902

= 0

(76)

Γ +

119871minus3

sum

119896=1

119896cos119896119902 minus119871minus1

sum

119896=1

119886 (119896)

[1198962]

sum

119897=0

(119896

2119897) 119897cos119896minus2119902

+

119871

sum

119896=1

119886 (119896)

[1198962]

sum

119897=0

(119896

2119897)

119897

sum

119894=2

(119897

119894) (minus1)

119894 cos119896minus2119894119902 = 0

(77)



10038161003816100381610038161003816Λ

119902

10038161003816100381610038161003816= 2

119901+110038161003816100381610038161003816cos119871minus211990210038161003816100381610038161003816


+ sin2

119902 (119887 (119871) cos 119902 + 119887 (119871 minus 1)))12

(78)





(119871) + 119886 (119871) (79)


10038161003816100381610038161003816Λ

119902

10038161003816100381610038161003816= 2

119901+110038161003816100381610038161003816cos119871minus211990210038161003816100381610038161003816


+ sin2

119902 (119887 (119871) cos 119902 + 119887 (119871 minus 1)))12

(80)



⟨120590119909

119895119896120590119909

119895+119903119896⟩Is= ⟨120590

119909

119895120590119909

119895+119903⟩qu (81)

where Ψ1015840



Ψ1015840

0= Φ

0 (82)



119863 In this case




119892M(120579) ofM


120574 = 0 and thus B = 0



119895119896(119861

119895119896) (M

119899)119895119896


119895 119896 ge 0119874

+

(2119873) 119886(119895 minus 119896) + 119886(119895 + 119896) 1198920if 119895 = 119896 = 0radic2119892

119897if

either 119895 = 0 119896 = 119897or 119895 = 119897 119896 = 0

119892119895minus119896+ 119892

119895+119896 119895 119896 gt 0


119895+119896+2 119895 119896 ge 0

119874plusmn

(2119873 + 1) 119886(119895 minus 119896) ∓ 119886(119895 + 119896 + 1) 119892119895minus119896∓ 119892

119895+119896+1 119895 119896 ge 0

119874minus


119895+119896+2 119895 119896 ge 0

Appendices

A Symmetry Classes

See Table 1



Hqu = minus119872

sum

119895=1

(119869119909

119895120590119909

119895120590119909

119895+1+ 119869

119910

119895120590119910

119895120590119910

119895+1

minus (1198691015840119909

119895120590119909

119895120590119909

119895+2+ 119869

1015840119910

119895120590119910

119895120590119910

119895+2) 120590

119911

119895+1+ ℎ120590

119911

119895)

(B1)

where 1198691015840119909119895= (12)(119860

119895119895+2+ 120574119861

119895119895+2) and 1198691015840119910

119895= (12)(119860

119895119895+2minus




Tr [119890(120573qu119899)H119909

119886 119890(120573qu2119899)H

119911

119890(120573qu119899)H

119910

119887 119890(120573qu119899)H

119910

119886 119890(120573qu2119899)H

119911

119890(120573qu119899)H

119909

119887 ]119899

(B2)


120572= sum

119872

119895isin120572((12)119869

120583

119895(120590

120583

119895120590120583

119895+1+ 120590

120583

119895+1120590120583

119895+2) minus

1198691015840120583

119895120590120583

119895120590119911

119895+1120590120583

119895+2) and H119911

= ℎsum119872

119895=1120590119911

119895 for 120583 isin 119909 119910 and once




Tr [I1205901119890(120573qu119899)H

119909

119886 I1199041119890(120573qu2119899)H

119911

119890(120573qu119899)H

119910

119887 I1205911119890(120573qu119899)H

119910

119886 I1199041119890(120573qu2119899)H

119911

119890(120573qu119899)H

119909

119887 ]119899


sum

120590119895119901 120591119895119901119904119895119901

119899

prod

119901=1

[⟨119901

10038161003816100381610038161003816119890(120573qu119899)H

119909

119886100381610038161003816100381610038161199042119901minus1⟩ ⟨ 119904

2119901minus1

10038161003816100381610038161003816119890(120573qu2119899)H

119911

119890(120573qu119899)H

119910

119887

10038161003816100381610038161003816120591119901⟩ ⟨ 120591

119901

10038161003816100381610038161003816119890(120573qu119899)H

119910

119886100381610038161003816100381610038161199042119901⟩ ⟨ 119904

2119901

10038161003816100381610038161003816119890(120573qu2119899)H

119911

119890(120573qu119899)H

119909

11988710038161003816100381610038161003816119901+1⟩]

(B3)



⟨119901

10038161003816100381610038161003816119890(120573119899)

H119909119886100381610038161003816100381610038161199042119901minus1⟩ ⟨ 119904

2119901minus1

10038161003816100381610038161003816119890(120573qu2119899)H

119911

119890(120573qu119899)H

119910

119887

10038161003816100381610038161003816120591119901⟩ ⟨ 120591

2119901minus1

10038161003816100381610038161003816119890(120573qu119899)H

119910

119886100381610038161003816100381610038161199042119901⟩ ⟨ 119904

2119901

10038161003816100381610038161003816119890(120573qu2119899)H

119911

119890(120573qu119899)H

119909

119887100381610038161003816100381610038162119901⟩

= 119890(120573qu119899)H

119909

119886(119901)

119890(120573qu2119899)H

119911(2119901minus1)

119890(120573qu119899)H

119910

119887(119901)

119890(120573qu119899)H

119910

119886(119901)

119890(120573qu2119899)H

119911(2119901)

119890(120573qu119899)H

119909

119887(119901)

⟨119901| 119904

2119901minus1⟩ ⟨ 119904

2119901minus1| 120591

119901⟩ ⟨ 120591

119901| 119904

2119901⟩

sdot ⟨ 1199042119901|

119901+1⟩

(B4)

where H119909

120572(119901) = sum

119872

119895isin120572((12)119869

119909

119895(120590

119895119901120590119895+1119901

+ 120590119895+1119901

120590119895+2119901

) +

1198691015840119909

119895+1120590119895119901119904119895+1119901

120590119895+2119901

) H119910

120572(119901) = sum

119872

119895isin120572((12)119869

119910

119895(120591

119895119901120591119895+1119901

+

120591119895+1119901

120591119895+2119901

) + 1198691015840119910

119895+1120591119895119901119904119895+1119901

120591119895+2119901

) andH119911

(119901) = sum119872

119895=1119904119895119901 We


⟨119901| 119904

2119901minus1⟩ ⟨ 119904

2119901minus1| 120591

119901⟩ ⟨ 120591

2119901minus1| 119904

2119901⟩ ⟨ 119904

2119901|

119901+1⟩

=1

24119872

sdot

119872

prod

119895=1


(B5)



119899

119899

sum

119901=1

(sum

119895isin119886

(119869119909

119895

2(120590

119895119901120590119895+1119901

+ 120590119895+1119901

120590119895+2119901

) minus 1198691015840119909

119895+1120590119895119901119904119895+1119901

120590119895+2119901

)

+sum

119895isin119887

(

119869119910

119895

2(120591

119895119901120591119895+1119901

+ 120591119895+1119901

120591119895+2119901

) minus 1198691015840119910

119895+1120591119895119901119904119895+12119901minus1

120591119895+2119901

)

+ sum

119895isin119886

(

119869119910

119895

2(120591

119895119901120591119895+1119901

+ 120591119895+1119901

120591119895+2119901

) minus 1198691015840119910

119895+1120591119895119901119904119895+12119901

120591119895+2119901

)

+sum

119895isin119887

(119869119909

119895

2(120590

119895119901+1120590119895+1119901+1

+ 120590119895+1119901+1

120590119895+2119901+1

) minus 1198691015840119909

119895+1120590119895119901+1

119904119895+12119901

120590119895+2119901+1

))

+

119899

sum

119901=1

(

119872

sum

119895=1

((120573quℎ

2119899minus119894120587

4) 119904

1198952119901+ (120573quℎ

2119899+119894120587

4) 119904

1198952119901) +

119872

sum

119895=1

119894120587

4(120590

1198951199011199041198952119901minus1

minus 120590119895119901+1

1199042119901+ 120591

119895119901(119904

1198952119901minus 119904

1198952119901minus1)))

+ 4119899119872 ln 2

(B6)



119909 (120573qu119899)119869119910

(120573qu119899)1198691015840119909 (120573qu119899)119869




S1

S2

S3

S4

1205901

1205911

1205902

1205912

1 2 3 4 5 6 7 8


Trotter

direction

p darr








Tr[

[

119872minus1

prod

119895=1

(119890(120573qu119899)H

119909

119895119895+1119890(120573qu119899)H

119909


119909

119895119872119890(120573qu2119899(119872minus1))

H119911119890(120573qu119899)H

119910


119910

119895119895+2119890(120573qu119899)H

119910

119895119895+1)]

]

119899


Tr[

[

119872

prod

119895=1

((

119872minus119895

prod

119896=1

119890(120573qu119899)H

119909

119895119895+119896)119890(120573qu2119899(119872minus1))

H119911(


prod

119896=0

119890(120573qu119899)H

119910

119895119872minus119896))]

]

119899

(B7)

where H120583

119895119896= 119869

120583

119895119896120590120583

119895120590120583

119896prod

119896minus1

119897=1(minus120590

119911

119897) for 120583 isin 119909 119910 and H119911

=

ℎsum119872

119895=1120590119911

119895



Tr[

[

119872minus1

prod

119895=1

(I120590119895(

119872minus119895

prod

119896=1

119890(120573qu119899)H

119909

119895119895+119896)119890(120573qu(119872minus1)119899)

H119911I119904119895(


prod

119896=0

119890(120573qu119899)H

119910

119895119872minus119896) I120591119895)]

]

119899


sum

120590119895119901 120591119895119901

119899minus1

prod

119901=0

119872minus1

prod

119895=1

(⟨119895+119895119901

10038161003816100381610038161003816(

119872minus119895

prod

119896=1

119890(120573qu119899)H

119909

119895119895+119896)119890(120573qu119899(119872minus1))

H119911 10038161003816100381610038161003816119904119895+119895119901⟩ ⟨ 119904

119895+119895119901

10038161003816100381610038161003816(


prod

119896=0

119890(120573qu119899)H

119910

119895119872minus119896)10038161003816100381610038161003816120591119895+119895119901⟩ ⟨ 120591

119895+119895119901|

119895+119895119901+1⟩)

(B8)


⟨119895+119895119901

10038161003816100381610038161003816(

119872minus119895

prod

119896=1

119890(120573qu119899)H

119909

119895119895+119896)119890(120573qu119899(119872minus1))

H119911 10038161003816100381610038161003816119904119895+119895119901⟩

sdot ⟨ 119904119895+119895119901

10038161003816100381610038161003816(


prod

119896=0

119890(120573qu119899)H

119910

119895119872minus119896)10038161003816100381610038161003816120591119895+119895119901⟩ ⟨ 120591

119895+119895119901|

119895+119895119901+1

⟩

= 119890(120573qu119899)sum

119872minus119895

119896=1(H119909119895119895+119896

(119901)+H119910

119895119895+119896(119901)+(1119899(119872minus1))H119911)

⟨119895+119895119901

|

119904119895+119895119901⟩ ⟨ 119904

119895+119895119901| 120591

119895+119895119901⟩ ⟨ 120591

119895+119895119901|

119895+119895119901+1⟩

(B9)


S1

S2

S3

S4

S5

S6

S7

S8

S9

S10

1 2 3 4 5 6 7 8 9


Trotter

direction

p darr



where H119909

119895119896(119901) = sum

119872

119896=119895+1119869119909

119895119896120590119895119901120590119896119901prod

119896minus1

119897=119895+1(minus119904

119897119901) H119910

119895119896(119901) =

sum119872

119896=119895+1119869119910

119895119896120591119895119901120591119896119901prod

119896minus1

119897=119895+1(minus119904

119897119901) andH119911

119901= ℎsum

119872

119895=1120590119911

119895119901 Finally


⟨119901| 119904

119901⟩ ⟨ 119904

119901| 120591

119901⟩ ⟨ 120591

119901|

119901+1⟩ = (

1

2radic2)

119872

sdot

119872

prod

119895=1


(B10)



sum

119901=1

119872

sum

119895=1

(120573qu

119899

119872

sum

119896=119895+1

(119869119909

119895119896120590119895119895+119895119901

120590119896119895+119895119901

+ 119869119910

119895119896120591119895119895+119895119901

120591119896119895+119895119901

)

119896minus1

prod

119897=119895+1

(minus119904119897119901) + (

120573qu

119899 (119872 minus 1)ℎ minus119894120587

4) 119904

119895119895+119895119901

+119894120587

4(1 minus 120590

119895119895+119895119901+ 120591

119895119895+119895119901+ 120590

119895119895+119895119901119904119895119895+119895119901

minus 120591119895119895+119895119901

119904119895119895+119895119901

minus 120590119895119895+119895119901+1

120591119895119895+119895119901

)) + 1198991198722 ln 1

2radic2

(B11)


119895119896





119909

119895119896 (120573qu119899)119869

119910

119895119896 and



1205901

S1

1205911

S2

1205902

S3

1205912

S4

1205903

S5

1 2 3 4 5 6 7 8 9 10

Trotter

direction

p darr






119863(55)


minus1

120572 sin 119902=119887 (1) sin 119902

Γ + 119886 (1) cos 119902 (C1)



minus1

120572 sin 119902=

119887 (1)

minus2119886 (2) sin 119902

if Γ = minus119886 (2) 119886 (1) = 119887 (2) = 0

(C2)


120572 =2119886 (2)

119887 (1) Γ = minus119886 (2) 119886 (1) = 119887 (2) = 0 (C3)


minus1

120572 sin 119902= minus

119887 (1) minus 119887 (3) + 119887 (2) cos 1199022 sin 119902 (119886 (2) + 119886 (3) cos 119902)

if Γ = minus119886 (2) 119886 (1) = minus119886 (3) 119887 (3) = 0

(C4)


120572 =2119886 (3)

119887 (2)

119886 (2)

119886 (3)=119887 (1) minus 119887 (3)

119887 (2)

Γ = minus119886 (2) 119886 (1) = minus119886 (3) 119887 (3) = 0

(C5)


minus1

120572 sin 119902

=sin 119902sum119898

119896=1119887 (119896)sum

[(119896minus1)2]

119897=0( 119896

2119897+1)sum

119897

119894=0( 119897119894) (minus1)


119896=2119886 (119896)sum

[1198962]

119897=0(minus1)

119897

( 119896

2119897) sin2119897

119902cos119896minus2119897119902

(C6)


[1198982]

sum

119895=1

119886 (2119895) + cos 119902[1198982]

sum

119895=1

119886 (2119895 + 1) + sin2

119902

sdot (

[1198982]

sum

119895=1

119886 (2119895)

119895

sum

119894=1

(119895

119894) (minus1)

119894 sin2(119894minus1)

119902

+ cos 119902[(119898minus1)2]

sum

119895=1

119886 (2119895 + 1)

119895

sum

119894=1

(119895

119894) (minus1)

119894 sin2(119894minus1)

119902

+

119898

sum

119896=2

119886 (119896)

[1198962]

sum

119897=1

(minus1)119897

(119896

2119897) sin2(119897minus1)

119902cos119896minus2119897119902)

(C7)


Γ = minus

[1198982]

sum

119895=1

119886 (2119895)

119886 (1) = minus

[(119898+1)2]

sum

119895=1

119886 (2119895 + 1) = 0

(C8)



sin2

119902(

[1198982]

sum

119895=1

119886 (2119895)

119895

sum

119894=1

(119895

119894)

119894minus1

sum

119901=0

(minus1)119894+119901 cos2119901119902

+

[(119898minus1)2]

sum

119895=1

119886 (2119895 + 1)

119895

sum

119894=1

(119895

119894)

119894minus1

sum

119901=0

(minus1)119894+119901 cos2119901+1119902 +

[(119898minus1)2]

sum

119895=1

119886 (2119895 + 1)

119895

sum

119897=1

(2119895 + 1

2119897)

119897minus1

sum

119901=0


+

[1198982]

sum

119895=1

119886 (2119895)

119895

sum

119897=1

(2119895

2119897)

119897minus1

sum

119901=0


(C9)


Disclosure




Acknowledgments


Endnotes



integer



and 119861119895119896





7 Thus matrices 119860119895119896

and 119861119895119896




1V

2V12

1of


=

V12

2V

1V12

2 whereV

1= (2 sinh 2119870

1)1198722


1sum119872

119894120590119909

119894 V2=

1198901198702 sum119872

119894=1120590119911

119894120590119911


119894= 119890



119887119902(5) as

119886119902= Γ +

119871

sum

119896=1

119886 (119896)

[1198962]

sum

119897=0

(119896

2119897)

119897

sum

119894=0

(119897

119894) (minus1)

minus119894 cos119896minus2119894119902

119887119902= tan 119902

sdot

119871

sum

119896=1

119887 (119896)

[(119896minus1)2]

sum

119897=0

(119896

2119897 + 1)

119897

sum

119894=0

(119897

119894) (minus1)

minus119894 cos119896minus2119894119902

(lowast)


119895=1(minus1)

119895

119886(2119895)


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












(119871) + 119886 (119871) (79)


10038161003816100381610038161003816Λ

119902

10038161003816100381610038161003816= 2

119901+110038161003816100381610038161003816cos119871minus211990210038161003816100381610038161003816


+ sin2

119902 (119887 (119871) cos 119902 + 119887 (119871 minus 1)))12

(80)



⟨120590119909

119895119896120590119909

119895+119903119896⟩Is= ⟨120590

119909

119895120590119909

119895+119903⟩qu (81)

where Ψ1015840



Ψ1015840

0= Φ

0 (82)



119863 In this case




119892M(120579) ofM


120574 = 0 and thus B = 0



119895119896(119861

119895119896) (M

119899)119895119896


119895 119896 ge 0119874

+

(2119873) 119886(119895 minus 119896) + 119886(119895 + 119896) 1198920if 119895 = 119896 = 0radic2119892

119897if

either 119895 = 0 119896 = 119897or 119895 = 119897 119896 = 0

119892119895minus119896+ 119892

119895+119896 119895 119896 gt 0


119895+119896+2 119895 119896 ge 0

119874plusmn

(2119873 + 1) 119886(119895 minus 119896) ∓ 119886(119895 + 119896 + 1) 119892119895minus119896∓ 119892

119895+119896+1 119895 119896 ge 0

119874minus


119895+119896+2 119895 119896 ge 0

Appendices

A Symmetry Classes

See Table 1



Hqu = minus119872

sum

119895=1

(119869119909

119895120590119909

119895120590119909

119895+1+ 119869

119910

119895120590119910

119895120590119910

119895+1

minus (1198691015840119909

119895120590119909

119895120590119909

119895+2+ 119869

1015840119910

119895120590119910

119895120590119910

119895+2) 120590

119911

119895+1+ ℎ120590

119911

119895)

(B1)

where 1198691015840119909119895= (12)(119860

119895119895+2+ 120574119861

119895119895+2) and 1198691015840119910

119895= (12)(119860

119895119895+2minus




Tr [119890(120573qu119899)H119909

119886 119890(120573qu2119899)H

119911

119890(120573qu119899)H

119910

119887 119890(120573qu119899)H

119910

119886 119890(120573qu2119899)H

119911

119890(120573qu119899)H

119909

119887 ]119899

(B2)


120572= sum

119872

119895isin120572((12)119869

120583

119895(120590

120583

119895120590120583

119895+1+ 120590

120583

119895+1120590120583

119895+2) minus

1198691015840120583

119895120590120583

119895120590119911

119895+1120590120583

119895+2) and H119911

= ℎsum119872

119895=1120590119911

119895 for 120583 isin 119909 119910 and once




Tr [I1205901119890(120573qu119899)H

119909

119886 I1199041119890(120573qu2119899)H

119911

119890(120573qu119899)H

119910

119887 I1205911119890(120573qu119899)H

119910

119886 I1199041119890(120573qu2119899)H

119911

119890(120573qu119899)H

119909

119887 ]119899


sum

120590119895119901 120591119895119901119904119895119901

119899

prod

119901=1

[⟨119901

10038161003816100381610038161003816119890(120573qu119899)H

119909

119886100381610038161003816100381610038161199042119901minus1⟩ ⟨ 119904

2119901minus1

10038161003816100381610038161003816119890(120573qu2119899)H

119911

119890(120573qu119899)H

119910

119887

10038161003816100381610038161003816120591119901⟩ ⟨ 120591

119901

10038161003816100381610038161003816119890(120573qu119899)H

119910

119886100381610038161003816100381610038161199042119901⟩ ⟨ 119904

2119901

10038161003816100381610038161003816119890(120573qu2119899)H

119911

119890(120573qu119899)H

119909

11988710038161003816100381610038161003816119901+1⟩]

(B3)



⟨119901

10038161003816100381610038161003816119890(120573119899)

H119909119886100381610038161003816100381610038161199042119901minus1⟩ ⟨ 119904

2119901minus1

10038161003816100381610038161003816119890(120573qu2119899)H

119911

119890(120573qu119899)H

119910

119887

10038161003816100381610038161003816120591119901⟩ ⟨ 120591

2119901minus1

10038161003816100381610038161003816119890(120573qu119899)H

119910

119886100381610038161003816100381610038161199042119901⟩ ⟨ 119904

2119901

10038161003816100381610038161003816119890(120573qu2119899)H

119911

119890(120573qu119899)H

119909

119887100381610038161003816100381610038162119901⟩

= 119890(120573qu119899)H

119909

119886(119901)

119890(120573qu2119899)H

119911(2119901minus1)

119890(120573qu119899)H

119910

119887(119901)

119890(120573qu119899)H

119910

119886(119901)

119890(120573qu2119899)H

119911(2119901)

119890(120573qu119899)H

119909

119887(119901)

⟨119901| 119904

2119901minus1⟩ ⟨ 119904

2119901minus1| 120591

119901⟩ ⟨ 120591

119901| 119904

2119901⟩

sdot ⟨ 1199042119901|

119901+1⟩

(B4)

where H119909

120572(119901) = sum

119872

119895isin120572((12)119869

119909

119895(120590

119895119901120590119895+1119901

+ 120590119895+1119901

120590119895+2119901

) +

1198691015840119909

119895+1120590119895119901119904119895+1119901

120590119895+2119901

) H119910

120572(119901) = sum

119872

119895isin120572((12)119869

119910

119895(120591

119895119901120591119895+1119901

+

120591119895+1119901

120591119895+2119901

) + 1198691015840119910

119895+1120591119895119901119904119895+1119901

120591119895+2119901

) andH119911

(119901) = sum119872

119895=1119904119895119901 We


⟨119901| 119904

2119901minus1⟩ ⟨ 119904

2119901minus1| 120591

119901⟩ ⟨ 120591

2119901minus1| 119904

2119901⟩ ⟨ 119904

2119901|

119901+1⟩

=1

24119872

sdot

119872

prod

119895=1


(B5)



119899

119899

sum

119901=1

(sum

119895isin119886

(119869119909

119895

2(120590

119895119901120590119895+1119901

+ 120590119895+1119901

120590119895+2119901

) minus 1198691015840119909

119895+1120590119895119901119904119895+1119901

120590119895+2119901

)

+sum

119895isin119887

(

119869119910

119895

2(120591

119895119901120591119895+1119901

+ 120591119895+1119901

120591119895+2119901

) minus 1198691015840119910

119895+1120591119895119901119904119895+12119901minus1

120591119895+2119901

)

+ sum

119895isin119886

(

119869119910

119895

2(120591

119895119901120591119895+1119901

+ 120591119895+1119901

120591119895+2119901

) minus 1198691015840119910

119895+1120591119895119901119904119895+12119901

120591119895+2119901

)

+sum

119895isin119887

(119869119909

119895

2(120590

119895119901+1120590119895+1119901+1

+ 120590119895+1119901+1

120590119895+2119901+1

) minus 1198691015840119909

119895+1120590119895119901+1

119904119895+12119901

120590119895+2119901+1

))

+

119899

sum

119901=1

(

119872

sum

119895=1

((120573quℎ

2119899minus119894120587

4) 119904

1198952119901+ (120573quℎ

2119899+119894120587

4) 119904

1198952119901) +

119872

sum

119895=1

119894120587

4(120590

1198951199011199041198952119901minus1

minus 120590119895119901+1

1199042119901+ 120591

119895119901(119904

1198952119901minus 119904

1198952119901minus1)))

+ 4119899119872 ln 2

(B6)



119909 (120573qu119899)119869119910

(120573qu119899)1198691015840119909 (120573qu119899)119869




S1

S2

S3

S4

1205901

1205911

1205902

1205912

1 2 3 4 5 6 7 8


Trotter

direction

p darr








Tr[

[

119872minus1

prod

119895=1

(119890(120573qu119899)H

119909

119895119895+1119890(120573qu119899)H

119909


119909

119895119872119890(120573qu2119899(119872minus1))

H119911119890(120573qu119899)H

119910


119910

119895119895+2119890(120573qu119899)H

119910

119895119895+1)]

]

119899


Tr[

[

119872

prod

119895=1

((

119872minus119895

prod

119896=1

119890(120573qu119899)H

119909

119895119895+119896)119890(120573qu2119899(119872minus1))

H119911(


prod

119896=0

119890(120573qu119899)H

119910

119895119872minus119896))]

]

119899

(B7)

where H120583

119895119896= 119869

120583

119895119896120590120583

119895120590120583

119896prod

119896minus1

119897=1(minus120590

119911

119897) for 120583 isin 119909 119910 and H119911

=

ℎsum119872

119895=1120590119911

119895



Tr[

[

119872minus1

prod

119895=1

(I120590119895(

119872minus119895

prod

119896=1

119890(120573qu119899)H

119909

119895119895+119896)119890(120573qu(119872minus1)119899)

H119911I119904119895(


prod

119896=0

119890(120573qu119899)H

119910

119895119872minus119896) I120591119895)]

]

119899


sum

120590119895119901 120591119895119901

119899minus1

prod

119901=0

119872minus1

prod

119895=1

(⟨119895+119895119901

10038161003816100381610038161003816(

119872minus119895

prod

119896=1

119890(120573qu119899)H

119909

119895119895+119896)119890(120573qu119899(119872minus1))

H119911 10038161003816100381610038161003816119904119895+119895119901⟩ ⟨ 119904

119895+119895119901

10038161003816100381610038161003816(


prod

119896=0

119890(120573qu119899)H

119910

119895119872minus119896)10038161003816100381610038161003816120591119895+119895119901⟩ ⟨ 120591

119895+119895119901|

119895+119895119901+1⟩)

(B8)


⟨119895+119895119901

10038161003816100381610038161003816(

119872minus119895

prod

119896=1

119890(120573qu119899)H

119909

119895119895+119896)119890(120573qu119899(119872minus1))

H119911 10038161003816100381610038161003816119904119895+119895119901⟩

sdot ⟨ 119904119895+119895119901

10038161003816100381610038161003816(


prod

119896=0

119890(120573qu119899)H

119910

119895119872minus119896)10038161003816100381610038161003816120591119895+119895119901⟩ ⟨ 120591

119895+119895119901|

119895+119895119901+1

⟩

= 119890(120573qu119899)sum

119872minus119895

119896=1(H119909119895119895+119896

(119901)+H119910

119895119895+119896(119901)+(1119899(119872minus1))H119911)

⟨119895+119895119901

|

119904119895+119895119901⟩ ⟨ 119904

119895+119895119901| 120591

119895+119895119901⟩ ⟨ 120591

119895+119895119901|

119895+119895119901+1⟩

(B9)


S1

S2

S3

S4

S5

S6

S7

S8

S9

S10

1 2 3 4 5 6 7 8 9


Trotter

direction

p darr



where H119909

119895119896(119901) = sum

119872

119896=119895+1119869119909

119895119896120590119895119901120590119896119901prod

119896minus1

119897=119895+1(minus119904

119897119901) H119910

119895119896(119901) =

sum119872

119896=119895+1119869119910

119895119896120591119895119901120591119896119901prod

119896minus1

119897=119895+1(minus119904

119897119901) andH119911

119901= ℎsum

119872

119895=1120590119911

119895119901 Finally


⟨119901| 119904

119901⟩ ⟨ 119904

119901| 120591

119901⟩ ⟨ 120591

119901|

119901+1⟩ = (

1

2radic2)

119872

sdot

119872

prod

119895=1


(B10)



sum

119901=1

119872

sum

119895=1

(120573qu

119899

119872

sum

119896=119895+1

(119869119909

119895119896120590119895119895+119895119901

120590119896119895+119895119901

+ 119869119910

119895119896120591119895119895+119895119901

120591119896119895+119895119901

)

119896minus1

prod

119897=119895+1

(minus119904119897119901) + (

120573qu

119899 (119872 minus 1)ℎ minus119894120587

4) 119904

119895119895+119895119901

+119894120587

4(1 minus 120590

119895119895+119895119901+ 120591

119895119895+119895119901+ 120590

119895119895+119895119901119904119895119895+119895119901

minus 120591119895119895+119895119901

119904119895119895+119895119901

minus 120590119895119895+119895119901+1

120591119895119895+119895119901

)) + 1198991198722 ln 1

2radic2

(B11)


119895119896





119909

119895119896 (120573qu119899)119869

119910

119895119896 and



1205901

S1

1205911

S2

1205902

S3

1205912

S4

1205903

S5

1 2 3 4 5 6 7 8 9 10

Trotter

direction

p darr






119863(55)


minus1

120572 sin 119902=119887 (1) sin 119902

Γ + 119886 (1) cos 119902 (C1)



minus1

120572 sin 119902=

119887 (1)

minus2119886 (2) sin 119902

if Γ = minus119886 (2) 119886 (1) = 119887 (2) = 0

(C2)


120572 =2119886 (2)

119887 (1) Γ = minus119886 (2) 119886 (1) = 119887 (2) = 0 (C3)


minus1

120572 sin 119902= minus

119887 (1) minus 119887 (3) + 119887 (2) cos 1199022 sin 119902 (119886 (2) + 119886 (3) cos 119902)

if Γ = minus119886 (2) 119886 (1) = minus119886 (3) 119887 (3) = 0

(C4)


120572 =2119886 (3)

119887 (2)

119886 (2)

119886 (3)=119887 (1) minus 119887 (3)

119887 (2)

Γ = minus119886 (2) 119886 (1) = minus119886 (3) 119887 (3) = 0

(C5)


minus1

120572 sin 119902

=sin 119902sum119898

119896=1119887 (119896)sum

[(119896minus1)2]

119897=0( 119896

2119897+1)sum

119897

119894=0( 119897119894) (minus1)


119896=2119886 (119896)sum

[1198962]

119897=0(minus1)

119897

( 119896

2119897) sin2119897

119902cos119896minus2119897119902

(C6)


[1198982]

sum

119895=1

119886 (2119895) + cos 119902[1198982]

sum

119895=1

119886 (2119895 + 1) + sin2

119902

sdot (

[1198982]

sum

119895=1

119886 (2119895)

119895

sum

119894=1

(119895

119894) (minus1)

119894 sin2(119894minus1)

119902

+ cos 119902[(119898minus1)2]

sum

119895=1

119886 (2119895 + 1)

119895

sum

119894=1

(119895

119894) (minus1)

119894 sin2(119894minus1)

119902

+

119898

sum

119896=2

119886 (119896)

[1198962]

sum

119897=1

(minus1)119897

(119896

2119897) sin2(119897minus1)

119902cos119896minus2119897119902)

(C7)


Γ = minus

[1198982]

sum

119895=1

119886 (2119895)

119886 (1) = minus

[(119898+1)2]

sum

119895=1

119886 (2119895 + 1) = 0

(C8)



sin2

119902(

[1198982]

sum

119895=1

119886 (2119895)

119895

sum

119894=1

(119895

119894)

119894minus1

sum

119901=0

(minus1)119894+119901 cos2119901119902

+

[(119898minus1)2]

sum

119895=1

119886 (2119895 + 1)

119895

sum

119894=1

(119895

119894)

119894minus1

sum

119901=0

(minus1)119894+119901 cos2119901+1119902 +

[(119898minus1)2]

sum

119895=1

119886 (2119895 + 1)

119895

sum

119897=1

(2119895 + 1

2119897)

119897minus1

sum

119901=0


+

[1198982]

sum

119895=1

119886 (2119895)

119895

sum

119897=1

(2119895

2119897)

119897minus1

sum

119901=0


(C9)


Disclosure




Acknowledgments


Endnotes



integer



and 119861119895119896





7 Thus matrices 119860119895119896

and 119861119895119896




1V

2V12

1of


=

V12

2V

1V12

2 whereV

1= (2 sinh 2119870

1)1198722


1sum119872

119894120590119909

119894 V2=

1198901198702 sum119872

119894=1120590119911

119894120590119911


119894= 119890



119887119902(5) as

119886119902= Γ +

119871

sum

119896=1

119886 (119896)

[1198962]

sum

119897=0

(119896

2119897)

119897

sum

119894=0

(119897

119894) (minus1)

minus119894 cos119896minus2119894119902

119887119902= tan 119902

sdot

119871

sum

119896=1

119887 (119896)

[(119896minus1)2]

sum

119897=0

(119896

2119897 + 1)

119897

sum

119894=0

(119897

119894) (minus1)

minus119894 cos119896minus2119894119902

(lowast)


119895=1(minus1)

119895

119886(2119895)


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











⟨119901

10038161003816100381610038161003816119890(120573119899)

H119909119886100381610038161003816100381610038161199042119901minus1⟩ ⟨ 119904

2119901minus1

10038161003816100381610038161003816119890(120573qu2119899)H

119911

119890(120573qu119899)H

119910

119887

10038161003816100381610038161003816120591119901⟩ ⟨ 120591

2119901minus1

10038161003816100381610038161003816119890(120573qu119899)H

119910

119886100381610038161003816100381610038161199042119901⟩ ⟨ 119904

2119901

10038161003816100381610038161003816119890(120573qu2119899)H

119911

119890(120573qu119899)H

119909

119887100381610038161003816100381610038162119901⟩

= 119890(120573qu119899)H

119909

119886(119901)

119890(120573qu2119899)H

119911(2119901minus1)

119890(120573qu119899)H

119910

119887(119901)

119890(120573qu119899)H

119910

119886(119901)

119890(120573qu2119899)H

119911(2119901)

119890(120573qu119899)H

119909

119887(119901)

⟨119901| 119904

2119901minus1⟩ ⟨ 119904

2119901minus1| 120591

119901⟩ ⟨ 120591

119901| 119904

2119901⟩

sdot ⟨ 1199042119901|

119901+1⟩

(B4)

where H119909

120572(119901) = sum

119872

119895isin120572((12)119869

119909

119895(120590

119895119901120590119895+1119901

+ 120590119895+1119901

120590119895+2119901

) +

1198691015840119909

119895+1120590119895119901119904119895+1119901

120590119895+2119901

) H119910

120572(119901) = sum

119872

119895isin120572((12)119869

119910

119895(120591

119895119901120591119895+1119901

+

120591119895+1119901

120591119895+2119901

) + 1198691015840119910

119895+1120591119895119901119904119895+1119901

120591119895+2119901

) andH119911

(119901) = sum119872

119895=1119904119895119901 We


⟨119901| 119904

2119901minus1⟩ ⟨ 119904

2119901minus1| 120591

119901⟩ ⟨ 120591

2119901minus1| 119904

2119901⟩ ⟨ 119904

2119901|

119901+1⟩

=1

24119872

sdot

119872

prod

119895=1


(B5)



119899

119899

sum

119901=1

(sum

119895isin119886

(119869119909

119895

2(120590

119895119901120590119895+1119901

+ 120590119895+1119901

120590119895+2119901

) minus 1198691015840119909

119895+1120590119895119901119904119895+1119901

120590119895+2119901

)

+sum

119895isin119887

(

119869119910

119895

2(120591

119895119901120591119895+1119901

+ 120591119895+1119901

120591119895+2119901

) minus 1198691015840119910

119895+1120591119895119901119904119895+12119901minus1

120591119895+2119901

)

+ sum

119895isin119886

(

119869119910

119895

2(120591

119895119901120591119895+1119901

+ 120591119895+1119901

120591119895+2119901

) minus 1198691015840119910

119895+1120591119895119901119904119895+12119901

120591119895+2119901

)

+sum

119895isin119887

(119869119909

119895

2(120590

119895119901+1120590119895+1119901+1

+ 120590119895+1119901+1

120590119895+2119901+1

) minus 1198691015840119909

119895+1120590119895119901+1

119904119895+12119901

120590119895+2119901+1

))

+

119899

sum

119901=1

(

119872

sum

119895=1

((120573quℎ

2119899minus119894120587

4) 119904

1198952119901+ (120573quℎ

2119899+119894120587

4) 119904

1198952119901) +

119872

sum

119895=1

119894120587

4(120590

1198951199011199041198952119901minus1

minus 120590119895119901+1

1199042119901+ 120591

119895119901(119904

1198952119901minus 119904

1198952119901minus1)))

+ 4119899119872 ln 2

(B6)



119909 (120573qu119899)119869119910

(120573qu119899)1198691015840119909 (120573qu119899)119869




S1

S2

S3

S4

1205901

1205911

1205902

1205912

1 2 3 4 5 6 7 8


Trotter

direction

p darr








Tr[

[

119872minus1

prod

119895=1

(119890(120573qu119899)H

119909

119895119895+1119890(120573qu119899)H

119909


119909

119895119872119890(120573qu2119899(119872minus1))

H119911119890(120573qu119899)H

119910


119910

119895119895+2119890(120573qu119899)H

119910

119895119895+1)]

]

119899


Tr[

[

119872

prod

119895=1

((

119872minus119895

prod

119896=1

119890(120573qu119899)H

119909

119895119895+119896)119890(120573qu2119899(119872minus1))

H119911(


prod

119896=0

119890(120573qu119899)H

119910

119895119872minus119896))]

]

119899

(B7)

where H120583

119895119896= 119869

120583

119895119896120590120583

119895120590120583

119896prod

119896minus1

119897=1(minus120590

119911

119897) for 120583 isin 119909 119910 and H119911

=

ℎsum119872

119895=1120590119911

119895



Tr[

[

119872minus1

prod

119895=1

(I120590119895(

119872minus119895

prod

119896=1

119890(120573qu119899)H

119909

119895119895+119896)119890(120573qu(119872minus1)119899)

H119911I119904119895(


prod

119896=0

119890(120573qu119899)H

119910

119895119872minus119896) I120591119895)]

]

119899


sum

120590119895119901 120591119895119901

119899minus1

prod

119901=0

119872minus1

prod

119895=1

(⟨119895+119895119901

10038161003816100381610038161003816(

119872minus119895

prod

119896=1

119890(120573qu119899)H

119909

119895119895+119896)119890(120573qu119899(119872minus1))

H119911 10038161003816100381610038161003816119904119895+119895119901⟩ ⟨ 119904

119895+119895119901

10038161003816100381610038161003816(


prod

119896=0

119890(120573qu119899)H

119910

119895119872minus119896)10038161003816100381610038161003816120591119895+119895119901⟩ ⟨ 120591

119895+119895119901|

119895+119895119901+1⟩)

(B8)


⟨119895+119895119901

10038161003816100381610038161003816(

119872minus119895

prod

119896=1

119890(120573qu119899)H

119909

119895119895+119896)119890(120573qu119899(119872minus1))

H119911 10038161003816100381610038161003816119904119895+119895119901⟩

sdot ⟨ 119904119895+119895119901

10038161003816100381610038161003816(


prod

119896=0

119890(120573qu119899)H

119910

119895119872minus119896)10038161003816100381610038161003816120591119895+119895119901⟩ ⟨ 120591

119895+119895119901|

119895+119895119901+1

⟩

= 119890(120573qu119899)sum

119872minus119895

119896=1(H119909119895119895+119896

(119901)+H119910

119895119895+119896(119901)+(1119899(119872minus1))H119911)

⟨119895+119895119901

|

119904119895+119895119901⟩ ⟨ 119904

119895+119895119901| 120591

119895+119895119901⟩ ⟨ 120591

119895+119895119901|

119895+119895119901+1⟩

(B9)


S1

S2

S3

S4

S5

S6

S7

S8

S9

S10

1 2 3 4 5 6 7 8 9


Trotter

direction

p darr



where H119909

119895119896(119901) = sum

119872

119896=119895+1119869119909

119895119896120590119895119901120590119896119901prod

119896minus1

119897=119895+1(minus119904

119897119901) H119910

119895119896(119901) =

sum119872

119896=119895+1119869119910

119895119896120591119895119901120591119896119901prod

119896minus1

119897=119895+1(minus119904

119897119901) andH119911

119901= ℎsum

119872

119895=1120590119911

119895119901 Finally


⟨119901| 119904

119901⟩ ⟨ 119904

119901| 120591

119901⟩ ⟨ 120591

119901|

119901+1⟩ = (

1

2radic2)

119872

sdot

119872

prod

119895=1


(B10)



sum

119901=1

119872

sum

119895=1

(120573qu

119899

119872

sum

119896=119895+1

(119869119909

119895119896120590119895119895+119895119901

120590119896119895+119895119901

+ 119869119910

119895119896120591119895119895+119895119901

120591119896119895+119895119901

)

119896minus1

prod

119897=119895+1

(minus119904119897119901) + (

120573qu

119899 (119872 minus 1)ℎ minus119894120587

4) 119904

119895119895+119895119901

+119894120587

4(1 minus 120590

119895119895+119895119901+ 120591

119895119895+119895119901+ 120590

119895119895+119895119901119904119895119895+119895119901

minus 120591119895119895+119895119901

119904119895119895+119895119901

minus 120590119895119895+119895119901+1

120591119895119895+119895119901

)) + 1198991198722 ln 1

2radic2

(B11)


119895119896





119909

119895119896 (120573qu119899)119869

119910

119895119896 and



1205901

S1

1205911

S2

1205902

S3

1205912

S4

1205903

S5

1 2 3 4 5 6 7 8 9 10

Trotter

direction

p darr






119863(55)


minus1

120572 sin 119902=119887 (1) sin 119902

Γ + 119886 (1) cos 119902 (C1)



minus1

120572 sin 119902=

119887 (1)

minus2119886 (2) sin 119902

if Γ = minus119886 (2) 119886 (1) = 119887 (2) = 0

(C2)


120572 =2119886 (2)

119887 (1) Γ = minus119886 (2) 119886 (1) = 119887 (2) = 0 (C3)


minus1

120572 sin 119902= minus

119887 (1) minus 119887 (3) + 119887 (2) cos 1199022 sin 119902 (119886 (2) + 119886 (3) cos 119902)

if Γ = minus119886 (2) 119886 (1) = minus119886 (3) 119887 (3) = 0

(C4)


120572 =2119886 (3)

119887 (2)

119886 (2)

119886 (3)=119887 (1) minus 119887 (3)

119887 (2)

Γ = minus119886 (2) 119886 (1) = minus119886 (3) 119887 (3) = 0

(C5)


minus1

120572 sin 119902

=sin 119902sum119898

119896=1119887 (119896)sum

[(119896minus1)2]

119897=0( 119896

2119897+1)sum

119897

119894=0( 119897119894) (minus1)


119896=2119886 (119896)sum

[1198962]

119897=0(minus1)

119897

( 119896

2119897) sin2119897

119902cos119896minus2119897119902

(C6)


[1198982]

sum

119895=1

119886 (2119895) + cos 119902[1198982]

sum

119895=1

119886 (2119895 + 1) + sin2

119902

sdot (

[1198982]

sum

119895=1

119886 (2119895)

119895

sum

119894=1

(119895

119894) (minus1)

119894 sin2(119894minus1)

119902

+ cos 119902[(119898minus1)2]

sum

119895=1

119886 (2119895 + 1)

119895

sum

119894=1

(119895

119894) (minus1)

119894 sin2(119894minus1)

119902

+

119898

sum

119896=2

119886 (119896)

[1198962]

sum

119897=1

(minus1)119897

(119896

2119897) sin2(119897minus1)

119902cos119896minus2119897119902)

(C7)


Γ = minus

[1198982]

sum

119895=1

119886 (2119895)

119886 (1) = minus

[(119898+1)2]

sum

119895=1

119886 (2119895 + 1) = 0

(C8)



sin2

119902(

[1198982]

sum

119895=1

119886 (2119895)

119895

sum

119894=1

(119895

119894)

119894minus1

sum

119901=0

(minus1)119894+119901 cos2119901119902

+

[(119898minus1)2]

sum

119895=1

119886 (2119895 + 1)

119895

sum

119894=1

(119895

119894)

119894minus1

sum

119901=0

(minus1)119894+119901 cos2119901+1119902 +

[(119898minus1)2]

sum

119895=1

119886 (2119895 + 1)

119895

sum

119897=1

(2119895 + 1

2119897)

119897minus1

sum

119901=0


+

[1198982]

sum

119895=1

119886 (2119895)

119895

sum

119897=1

(2119895

2119897)

119897minus1

sum

119901=0


(C9)


Disclosure




Acknowledgments


Endnotes



integer



and 119861119895119896





7 Thus matrices 119860119895119896

and 119861119895119896




1V

2V12

1of


=

V12

2V

1V12

2 whereV

1= (2 sinh 2119870

1)1198722


1sum119872

119894120590119909

119894 V2=

1198901198702 sum119872

119894=1120590119911

119894120590119911


119894= 119890



119887119902(5) as

119886119902= Γ +

119871

sum

119896=1

119886 (119896)

[1198962]

sum

119897=0

(119896

2119897)

119897

sum

119894=0

(119897

119894) (minus1)

minus119894 cos119896minus2119894119902

119887119902= tan 119902

sdot

119871

sum

119896=1

119887 (119896)

[(119896minus1)2]

sum

119897=0

(119896

2119897 + 1)

119897

sum

119894=0

(119897

119894) (minus1)

minus119894 cos119896minus2119894119902

(lowast)


119895=1(minus1)

119895

119886(2119895)


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










S1

S2

S3

S4

1205901

1205911

1205902

1205912

1 2 3 4 5 6 7 8


Trotter

direction

p darr








Tr[

[

119872minus1

prod

119895=1

(119890(120573qu119899)H

119909

119895119895+1119890(120573qu119899)H

119909


119909

119895119872119890(120573qu2119899(119872minus1))

H119911119890(120573qu119899)H

119910


119910

119895119895+2119890(120573qu119899)H

119910

119895119895+1)]

]

119899


Tr[

[

119872

prod

119895=1

((

119872minus119895

prod

119896=1

119890(120573qu119899)H

119909

119895119895+119896)119890(120573qu2119899(119872minus1))

H119911(


prod

119896=0

119890(120573qu119899)H

119910

119895119872minus119896))]

]

119899

(B7)

where H120583

119895119896= 119869

120583

119895119896120590120583

119895120590120583

119896prod

119896minus1

119897=1(minus120590

119911

119897) for 120583 isin 119909 119910 and H119911

=

ℎsum119872

119895=1120590119911

119895



Tr[

[

119872minus1

prod

119895=1

(I120590119895(

119872minus119895

prod

119896=1

119890(120573qu119899)H

119909

119895119895+119896)119890(120573qu(119872minus1)119899)

H119911I119904119895(


prod

119896=0

119890(120573qu119899)H

119910

119895119872minus119896) I120591119895)]

]

119899


sum

120590119895119901 120591119895119901

119899minus1

prod

119901=0

119872minus1

prod

119895=1

(⟨119895+119895119901

10038161003816100381610038161003816(

119872minus119895

prod

119896=1

119890(120573qu119899)H

119909

119895119895+119896)119890(120573qu119899(119872minus1))

H119911 10038161003816100381610038161003816119904119895+119895119901⟩ ⟨ 119904

119895+119895119901

10038161003816100381610038161003816(


prod

119896=0

119890(120573qu119899)H

119910

119895119872minus119896)10038161003816100381610038161003816120591119895+119895119901⟩ ⟨ 120591

119895+119895119901|

119895+119895119901+1⟩)

(B8)


⟨119895+119895119901

10038161003816100381610038161003816(

119872minus119895

prod

119896=1

119890(120573qu119899)H

119909

119895119895+119896)119890(120573qu119899(119872minus1))

H119911 10038161003816100381610038161003816119904119895+119895119901⟩

sdot ⟨ 119904119895+119895119901

10038161003816100381610038161003816(


prod

119896=0

119890(120573qu119899)H

119910

119895119872minus119896)10038161003816100381610038161003816120591119895+119895119901⟩ ⟨ 120591

119895+119895119901|

119895+119895119901+1

⟩

= 119890(120573qu119899)sum

119872minus119895

119896=1(H119909119895119895+119896

(119901)+H119910

119895119895+119896(119901)+(1119899(119872minus1))H119911)

⟨119895+119895119901

|

119904119895+119895119901⟩ ⟨ 119904

119895+119895119901| 120591

119895+119895119901⟩ ⟨ 120591

119895+119895119901|

119895+119895119901+1⟩

(B9)


S1

S2

S3

S4

S5

S6

S7

S8

S9

S10

1 2 3 4 5 6 7 8 9


Trotter

direction

p darr



where H119909

119895119896(119901) = sum

119872

119896=119895+1119869119909

119895119896120590119895119901120590119896119901prod

119896minus1

119897=119895+1(minus119904

119897119901) H119910

119895119896(119901) =

sum119872

119896=119895+1119869119910

119895119896120591119895119901120591119896119901prod

119896minus1

119897=119895+1(minus119904

119897119901) andH119911

119901= ℎsum

119872

119895=1120590119911

119895119901 Finally


⟨119901| 119904

119901⟩ ⟨ 119904

119901| 120591

119901⟩ ⟨ 120591

119901|

119901+1⟩ = (

1

2radic2)

119872

sdot

119872

prod

119895=1


(B10)



sum

119901=1

119872

sum

119895=1

(120573qu

119899

119872

sum

119896=119895+1

(119869119909

119895119896120590119895119895+119895119901

120590119896119895+119895119901

+ 119869119910

119895119896120591119895119895+119895119901

120591119896119895+119895119901

)

119896minus1

prod

119897=119895+1

(minus119904119897119901) + (

120573qu

119899 (119872 minus 1)ℎ minus119894120587

4) 119904

119895119895+119895119901

+119894120587

4(1 minus 120590

119895119895+119895119901+ 120591

119895119895+119895119901+ 120590

119895119895+119895119901119904119895119895+119895119901

minus 120591119895119895+119895119901

119904119895119895+119895119901

minus 120590119895119895+119895119901+1

120591119895119895+119895119901

)) + 1198991198722 ln 1

2radic2

(B11)


119895119896





119909

119895119896 (120573qu119899)119869

119910

119895119896 and



1205901

S1

1205911

S2

1205902

S3

1205912

S4

1205903

S5

1 2 3 4 5 6 7 8 9 10

Trotter

direction

p darr






119863(55)


minus1

120572 sin 119902=119887 (1) sin 119902

Γ + 119886 (1) cos 119902 (C1)



minus1

120572 sin 119902=

119887 (1)

minus2119886 (2) sin 119902

if Γ = minus119886 (2) 119886 (1) = 119887 (2) = 0

(C2)


120572 =2119886 (2)

119887 (1) Γ = minus119886 (2) 119886 (1) = 119887 (2) = 0 (C3)


minus1

120572 sin 119902= minus

119887 (1) minus 119887 (3) + 119887 (2) cos 1199022 sin 119902 (119886 (2) + 119886 (3) cos 119902)

if Γ = minus119886 (2) 119886 (1) = minus119886 (3) 119887 (3) = 0

(C4)


120572 =2119886 (3)

119887 (2)

119886 (2)

119886 (3)=119887 (1) minus 119887 (3)

119887 (2)

Γ = minus119886 (2) 119886 (1) = minus119886 (3) 119887 (3) = 0

(C5)


minus1

120572 sin 119902

=sin 119902sum119898

119896=1119887 (119896)sum

[(119896minus1)2]

119897=0( 119896

2119897+1)sum

119897

119894=0( 119897119894) (minus1)


119896=2119886 (119896)sum

[1198962]

119897=0(minus1)

119897

( 119896

2119897) sin2119897

119902cos119896minus2119897119902

(C6)


[1198982]

sum

119895=1

119886 (2119895) + cos 119902[1198982]

sum

119895=1

119886 (2119895 + 1) + sin2

119902

sdot (

[1198982]

sum

119895=1

119886 (2119895)

119895

sum

119894=1

(119895

119894) (minus1)

119894 sin2(119894minus1)

119902

+ cos 119902[(119898minus1)2]

sum

119895=1

119886 (2119895 + 1)

119895

sum

119894=1

(119895

119894) (minus1)

119894 sin2(119894minus1)

119902

+

119898

sum

119896=2

119886 (119896)

[1198962]

sum

119897=1

(minus1)119897

(119896

2119897) sin2(119897minus1)

119902cos119896minus2119897119902)

(C7)


Γ = minus

[1198982]

sum

119895=1

119886 (2119895)

119886 (1) = minus

[(119898+1)2]

sum

119895=1

119886 (2119895 + 1) = 0

(C8)



sin2

119902(

[1198982]

sum

119895=1

119886 (2119895)

119895

sum

119894=1

(119895

119894)

119894minus1

sum

119901=0

(minus1)119894+119901 cos2119901119902

+

[(119898minus1)2]

sum

119895=1

119886 (2119895 + 1)

119895

sum

119894=1

(119895

119894)

119894minus1

sum

119901=0

(minus1)119894+119901 cos2119901+1119902 +

[(119898minus1)2]

sum

119895=1

119886 (2119895 + 1)

119895

sum

119897=1

(2119895 + 1

2119897)

119897minus1

sum

119901=0


+

[1198982]

sum

119895=1

119886 (2119895)

119895

sum

119897=1

(2119895

2119897)

119897minus1

sum

119901=0


(C9)


Disclosure




Acknowledgments


Endnotes



integer



and 119861119895119896





7 Thus matrices 119860119895119896

and 119861119895119896




1V

2V12

1of


=

V12

2V

1V12

2 whereV

1= (2 sinh 2119870

1)1198722


1sum119872

119894120590119909

119894 V2=

1198901198702 sum119872

119894=1120590119911

119894120590119911


119894= 119890



119887119902(5) as

119886119902= Γ +

119871

sum

119896=1

119886 (119896)

[1198962]

sum

119897=0

(119896

2119897)

119897

sum

119894=0

(119897

119894) (minus1)

minus119894 cos119896minus2119894119902

119887119902= tan 119902

sdot

119871

sum

119896=1

119887 (119896)

[(119896minus1)2]

sum

119897=0

(119896

2119897 + 1)

119897

sum

119894=0

(119897

119894) (minus1)

minus119894 cos119896minus2119894119902

(lowast)


119895=1(minus1)

119895

119886(2119895)


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










S1

S2

S3

S4

S5

S6

S7

S8

S9

S10

1 2 3 4 5 6 7 8 9


Trotter

direction

p darr



where H119909

119895119896(119901) = sum

119872

119896=119895+1119869119909

119895119896120590119895119901120590119896119901prod

119896minus1

119897=119895+1(minus119904

119897119901) H119910

119895119896(119901) =

sum119872

119896=119895+1119869119910

119895119896120591119895119901120591119896119901prod

119896minus1

119897=119895+1(minus119904

119897119901) andH119911

119901= ℎsum

119872

119895=1120590119911

119895119901 Finally


⟨119901| 119904

119901⟩ ⟨ 119904

119901| 120591

119901⟩ ⟨ 120591

119901|

119901+1⟩ = (

1

2radic2)

119872

sdot

119872

prod

119895=1


(B10)



sum

119901=1

119872

sum

119895=1

(120573qu

119899

119872

sum

119896=119895+1

(119869119909

119895119896120590119895119895+119895119901

120590119896119895+119895119901

+ 119869119910

119895119896120591119895119895+119895119901

120591119896119895+119895119901

)

119896minus1

prod

119897=119895+1

(minus119904119897119901) + (

120573qu

119899 (119872 minus 1)ℎ minus119894120587

4) 119904

119895119895+119895119901

+119894120587

4(1 minus 120590

119895119895+119895119901+ 120591

119895119895+119895119901+ 120590

119895119895+119895119901119904119895119895+119895119901

minus 120591119895119895+119895119901

119904119895119895+119895119901

minus 120590119895119895+119895119901+1

120591119895119895+119895119901

)) + 1198991198722 ln 1

2radic2

(B11)


119895119896





119909

119895119896 (120573qu119899)119869

119910

119895119896 and



1205901

S1

1205911

S2

1205902

S3

1205912

S4

1205903

S5

1 2 3 4 5 6 7 8 9 10

Trotter

direction

p darr






119863(55)


minus1

120572 sin 119902=119887 (1) sin 119902

Γ + 119886 (1) cos 119902 (C1)



minus1

120572 sin 119902=

119887 (1)

minus2119886 (2) sin 119902

if Γ = minus119886 (2) 119886 (1) = 119887 (2) = 0

(C2)


120572 =2119886 (2)

119887 (1) Γ = minus119886 (2) 119886 (1) = 119887 (2) = 0 (C3)


minus1

120572 sin 119902= minus

119887 (1) minus 119887 (3) + 119887 (2) cos 1199022 sin 119902 (119886 (2) + 119886 (3) cos 119902)

if Γ = minus119886 (2) 119886 (1) = minus119886 (3) 119887 (3) = 0

(C4)


120572 =2119886 (3)

119887 (2)

119886 (2)

119886 (3)=119887 (1) minus 119887 (3)

119887 (2)

Γ = minus119886 (2) 119886 (1) = minus119886 (3) 119887 (3) = 0

(C5)


minus1

120572 sin 119902

=sin 119902sum119898

119896=1119887 (119896)sum

[(119896minus1)2]

119897=0( 119896

2119897+1)sum

119897

119894=0( 119897119894) (minus1)


119896=2119886 (119896)sum

[1198962]

119897=0(minus1)

119897

( 119896

2119897) sin2119897

119902cos119896minus2119897119902

(C6)


[1198982]

sum

119895=1

119886 (2119895) + cos 119902[1198982]

sum

119895=1

119886 (2119895 + 1) + sin2

119902

sdot (

[1198982]

sum

119895=1

119886 (2119895)

119895

sum

119894=1

(119895

119894) (minus1)

119894 sin2(119894minus1)

119902

+ cos 119902[(119898minus1)2]

sum

119895=1

119886 (2119895 + 1)

119895

sum

119894=1

(119895

119894) (minus1)

119894 sin2(119894minus1)

119902

+

119898

sum

119896=2

119886 (119896)

[1198962]

sum

119897=1

(minus1)119897

(119896

2119897) sin2(119897minus1)

119902cos119896minus2119897119902)

(C7)


Γ = minus

[1198982]

sum

119895=1

119886 (2119895)

119886 (1) = minus

[(119898+1)2]

sum

119895=1

119886 (2119895 + 1) = 0

(C8)



sin2

119902(

[1198982]

sum

119895=1

119886 (2119895)

119895

sum

119894=1

(119895

119894)

119894minus1

sum

119901=0

(minus1)119894+119901 cos2119901119902

+

[(119898minus1)2]

sum

119895=1

119886 (2119895 + 1)

119895

sum

119894=1

(119895

119894)

119894minus1

sum

119901=0

(minus1)119894+119901 cos2119901+1119902 +

[(119898minus1)2]

sum

119895=1

119886 (2119895 + 1)

119895

sum

119897=1

(2119895 + 1

2119897)

119897minus1

sum

119901=0


+

[1198982]

sum

119895=1

119886 (2119895)

119895

sum

119897=1

(2119895

2119897)

119897minus1

sum

119901=0


(C9)


Disclosure




Acknowledgments


Endnotes



integer



and 119861119895119896





7 Thus matrices 119860119895119896

and 119861119895119896




1V

2V12

1of


=

V12

2V

1V12

2 whereV

1= (2 sinh 2119870

1)1198722


1sum119872

119894120590119909

119894 V2=

1198901198702 sum119872

119894=1120590119911

119894120590119911


119894= 119890



119887119902(5) as

119886119902= Γ +

119871

sum

119896=1

119886 (119896)

[1198962]

sum

119897=0

(119896

2119897)

119897

sum

119894=0

(119897

119894) (minus1)

minus119894 cos119896minus2119894119902

119887119902= tan 119902

sdot

119871

sum

119896=1

119887 (119896)

[(119896minus1)2]

sum

119897=0

(119896

2119897 + 1)

119897

sum

119894=0

(119897

119894) (minus1)

minus119894 cos119896minus2119894119902

(lowast)


119895=1(minus1)

119895

119886(2119895)


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










1205901

S1

1205911

S2

1205902

S3

1205912

S4

1205903

S5

1 2 3 4 5 6 7 8 9 10

Trotter

direction

p darr






119863(55)


minus1

120572 sin 119902=119887 (1) sin 119902

Γ + 119886 (1) cos 119902 (C1)



minus1

120572 sin 119902=

119887 (1)

minus2119886 (2) sin 119902

if Γ = minus119886 (2) 119886 (1) = 119887 (2) = 0

(C2)


120572 =2119886 (2)

119887 (1) Γ = minus119886 (2) 119886 (1) = 119887 (2) = 0 (C3)


minus1

120572 sin 119902= minus

119887 (1) minus 119887 (3) + 119887 (2) cos 1199022 sin 119902 (119886 (2) + 119886 (3) cos 119902)

if Γ = minus119886 (2) 119886 (1) = minus119886 (3) 119887 (3) = 0

(C4)


120572 =2119886 (3)

119887 (2)

119886 (2)

119886 (3)=119887 (1) minus 119887 (3)

119887 (2)

Γ = minus119886 (2) 119886 (1) = minus119886 (3) 119887 (3) = 0

(C5)


minus1

120572 sin 119902

=sin 119902sum119898

119896=1119887 (119896)sum

[(119896minus1)2]

119897=0( 119896

2119897+1)sum

119897

119894=0( 119897119894) (minus1)


119896=2119886 (119896)sum

[1198962]

119897=0(minus1)

119897

( 119896

2119897) sin2119897

119902cos119896minus2119897119902

(C6)


[1198982]

sum

119895=1

119886 (2119895) + cos 119902[1198982]

sum

119895=1

119886 (2119895 + 1) + sin2

119902

sdot (

[1198982]

sum

119895=1

119886 (2119895)

119895

sum

119894=1

(119895

119894) (minus1)

119894 sin2(119894minus1)

119902

+ cos 119902[(119898minus1)2]

sum

119895=1

119886 (2119895 + 1)

119895

sum

119894=1

(119895

119894) (minus1)

119894 sin2(119894minus1)

119902

+

119898

sum

119896=2

119886 (119896)

[1198962]

sum

119897=1

(minus1)119897

(119896

2119897) sin2(119897minus1)

119902cos119896minus2119897119902)

(C7)


Γ = minus

[1198982]

sum

119895=1

119886 (2119895)

119886 (1) = minus

[(119898+1)2]

sum

119895=1

119886 (2119895 + 1) = 0

(C8)



sin2

119902(

[1198982]

sum

119895=1

119886 (2119895)

119895

sum

119894=1

(119895

119894)

119894minus1

sum

119901=0

(minus1)119894+119901 cos2119901119902

+

[(119898minus1)2]

sum

119895=1

119886 (2119895 + 1)

119895

sum

119894=1

(119895

119894)

119894minus1

sum

119901=0

(minus1)119894+119901 cos2119901+1119902 +

[(119898minus1)2]

sum

119895=1

119886 (2119895 + 1)

119895

sum

119897=1

(2119895 + 1

2119897)

119897minus1

sum

119901=0


+

[1198982]

sum

119895=1

119886 (2119895)

119895

sum

119897=1

(2119895

2119897)

119897minus1

sum

119901=0


(C9)


Disclosure




Acknowledgments


Endnotes



integer



and 119861119895119896





7 Thus matrices 119860119895119896

and 119861119895119896




1V

2V12

1of


=

V12

2V

1V12

2 whereV

1= (2 sinh 2119870

1)1198722


1sum119872

119894120590119909

119894 V2=

1198901198702 sum119872

119894=1120590119911

119894120590119911


119894= 119890



119887119902(5) as

119886119902= Γ +

119871

sum

119896=1

119886 (119896)

[1198962]

sum

119897=0

(119896

2119897)

119897

sum

119894=0

(119897

119894) (minus1)

minus119894 cos119896minus2119894119902

119887119902= tan 119902

sdot

119871

sum

119896=1

119887 (119896)

[(119896minus1)2]

sum

119897=0

(119896

2119897 + 1)

119897

sum

119894=0

(119897

119894) (minus1)

minus119894 cos119896minus2119894119902

(lowast)


119895=1(minus1)

119895

119886(2119895)


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











sin2

119902(

[1198982]

sum

119895=1

119886 (2119895)

119895

sum

119894=1

(119895

119894)

119894minus1

sum

119901=0

(minus1)119894+119901 cos2119901119902

+

[(119898minus1)2]

sum

119895=1

119886 (2119895 + 1)

119895

sum

119894=1

(119895

119894)

119894minus1

sum

119901=0

(minus1)119894+119901 cos2119901+1119902 +

[(119898minus1)2]

sum

119895=1

119886 (2119895 + 1)

119895

sum

119897=1

(2119895 + 1

2119897)

119897minus1

sum

119901=0


+

[1198982]

sum

119895=1

119886 (2119895)

119895

sum

119897=1

(2119895

2119897)

119897minus1

sum

119901=0


(C9)


Disclosure




Acknowledgments


Endnotes



integer



and 119861119895119896





7 Thus matrices 119860119895119896

and 119861119895119896




1V

2V12

1of


=

V12

2V

1V12

2 whereV

1= (2 sinh 2119870

1)1198722


1sum119872

119894120590119909

119894 V2=

1198901198702 sum119872

119894=1120590119911

119894120590119911


119894= 119890



119887119902(5) as

119886119902= Γ +

119871

sum

119896=1

119886 (119896)

[1198962]

sum

119897=0

(119896

2119897)

119897

sum

119894=0

(119897

119894) (minus1)

minus119894 cos119896minus2119894119902

119887119902= tan 119902

sdot

119871

sum

119896=1

119887 (119896)

[(119896minus1)2]

sum

119897=0

(119896

2119897 + 1)

119897

sum

119894=0

(119897

119894) (minus1)

minus119894 cos119896minus2119894119902

(lowast)


119895=1(minus1)

119895

119886(2119895)


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra




































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Documents

Research Article On Relations between One-Dimensional ...downloads.hindawi.com/journals/amp/2015/652026.pdf · Research Article On Relations between One-Dimensional Quantum and Two-Dimensional