Block Entropy of the XYZ Model:Block Entropy of the XYZ Model:Exact Results and Numerical StudyExact Results and Numerical Study

Cristian Degli Esposti BoschiCNISM and Dipartimento di Fisica, Università di Bologna

Stefano EvangelistiTesi di laurea in Fisica, Università di Bologna, 

Elisa Ercolessi, Francesco Ravanini, Fabio Ortolani, Dipartimento di Fisica and Sezione INFN, Bologna

Italian Quantum Information Science Conference,5­8 nov 2009, Scuola Normale Superiore, Pisa

Given a bipartitebipartite quantum system (A+B), thevon Neumannvon Neumann entropy

quantifies the entanglemententanglement of A with B if the whole system is in a pure statepure state

in particular the ground state of the Hamiltonian on (A+B)

S=−tr AA logA

A=trB =∣ ⟩ ⟨∣

0=∣0 ⟩ ⟨0∣

H AB∣0 ⟩=E 0∣0 ⟩

● Critical points of 1D quantum systems are typically described by (1+1) conformal field theories.

● Calabrese and Cardy [J. Stat. Mech. P06002 (2004)] computed exactly the block entropy in the conformal case. (Note added on the arXiv: corrections are necessary with block composed of disjoint blocks: Bari group, and others...).

● As far as the noncriticalnoncritical (or massive or gapped) case is concerned, only two problems are addressed: Free bosonic theory and integrable ntegrable models with a corner transfer matrixmodels with a corner transfer matrix. General prediction for a single interval (one internal end)

S=c /6 ln /a

Spin-1/2 particles on a chain of L sites

Particular cases:

● Only one nonvanishing coupling, sayClassical-like 1D Ising modelClassical-like 1D Ising model

● One vanishing coupling, say XY modelXY model. Critical when Solved by means of Jordan-Wigner + Bogolioubov transformations and asymptotics of Toeplitz determinants. Review in the following paper...

The model studied hereThe model studied here

H XYZ=−∑ jJ x j

x j1

x J y jy j1

y J z jz j1

z

J x

J z

J x=J y

Its & Korepin, “The Fisher-Hartwig formula and entanglement entropy”, J. Stat. Phys., Online first (29/9/2009)

Three nonvanishing couplingsThree nonvanishing couplings:

At least two of them have the same sign, sayand . This common sign can always be taken positive or negative by means of a -rotation of the (conventionally) odd spins about z-axis.

The physical behaviour is dictated by the sign of the other coupling :

● Ferromagnetism

● Antiferromagnetism (favourable for entanglement due to the tendency to form singlets)

J xJ y

J z

J z0

J z0

Integrability and connection with 2D Integrability and connection with 2D models in classical statistical mechanicsmodels in classical statistical mechanics

● Back to the XXZ model, it is exactly solved by means of the Bethe Bethe ansatzansatz.

● It is also integrableintegrable: essentially this means that it admits an infinite number of (independent) conserved quantities. There is only a small number of “evident” symmetries:

Arbitrary rotation about z-axis [U(1)], -rotation about x- or y-axis [Z

2], left-right reflection,

possibly translational.

● In the fully anisotropic XYZ model, the internal spin symmetry is reduced to Z

2 X Z

2. Nonetheless

it is still integrablestill integrable.

● It is now convenient to exploit the dimensional crossover rule, that establishes a correspondence between D-dimensional quantum systems and classical systems in D+ spatial dimensions. Frequently one has =1, the additional dimension being represented by time.

● For some 1D quantum lattice problems at zero temperature, the Hamiltonian is related to the trasfer matrix T of an equivalent 2D classical trasfer matrix T of an equivalent 2D classical model in the canonical ensemblemodel in the canonical ensemble, and the two commute.

XYZ corresponds to a 8-vertexXYZ corresponds to a 8-vertex model and XXZ reduces to 6-vertex model; both are integrableintegrable.

The parameters in the quantum model are related to the Boltzmann weights of classical variables configurations (“arrows”) at the sites (“vertex”) of a suitable 2D lattice.

Remarkable properties [Peschel, Kaulke and Legeza, Ann. Physik (Leipzig) 8, 133 (1999)]:

(1)(1) The block density matrix (the same used in the DMRG) is formally written as the partition function of the classical model with a cutcut

With a clever choice of the lattice orientation and of the boundary conditions A=B=C=D

Row-to-row transfer matrix T

block=A B C D

Corner transfer matricesCorner transfer matrices

(2)(2) Formally the block density matrix is written as a thermal state of a harmonic oscillator

and K has integer eigenvaluesinteger eigenvalues.

Hence, every quantity related to can be computed exactly; the only thing to know is the parameter parameter . In particular

block∝exp−K

block

S=∑ j=1

∞ j1exp j

ln [1exp − j]

Strictly valid in the thermodynamic limit in noncritical cases. At criticality ~1/ ln L

∣J y∣≤J x≤−J z

Baxter's parametrisation of the XYZ model with Jacobian functions

J x : J y : J z=1 : :

=1k sn2

i1−k sn2

i=

−cnidni1−k sn2

i

0k1 0 I k ' k '=1−k2Principal regime

Complete elliptic integral of the first kind

Once a common energy scale is fixed by, say, the other two couplings can be varied by tuning kand . However the entropy depends only onthe entropy depends only on

=/ I k

J x

Details and reproduction of known limiting cases in Ercolessi, Evangelisti and Ravanini, “Exact entanglement entropy of the XYZ model and its sine-Gordon limit”, arXiv: 0905.400

0.61

2

46

Numerical checks Numerical checks (DMRG)(DMRG)

Early attempsEarly attemps (i. e. forgetting about symmetriessymmetries) with XXZ model J x=J y=1

Saturation ??????Expected at 1.195

Even-oddeffects at not sufficientlylarge sizes

Analogous to even-odd effect due toopen boundary conditionsopen boundary conditionsin critical systems (vanishes algebraically)

Laflorencie et al.,PRL 96, 100603 (2006)

Symmetry breaking in the gapped phaseSymmetry breaking in the gapped phase

● XXZXXZ: When the ground state is twofold degenerate in the thermodyamic limit and the Z

2

up-down symmetry is spontaneously broken. There are two ground states with opposite value of the staggered (antiferro) magnetisation

The excitations above each of these two states are gapped with

≤−1

M s=∑ j

−1j⟨ j

z ⟩L

G∝exp− 2

22−−1 (BKTtransition)

an orderparameter

At finite size the ground state is unique but the energy difference between the two states and vanishes exponentially

∣ ⟩∣− ⟩

E∝exp −L / L0

L pwith L0∝G−1

● When L increases the two states are numericallydegenerate and the DMRG will pick up a linear combination ∣ ⟩L=∣ ⟩L∣− ⟩LThis state (with real coefficients) has an average staggered magnetisation

from which we can read off , once Ms is known

by a symmetry breaking construction.

M s ,DMRG=2∣∣2−1M s

Entropy of the linear combinationEntropy of the linear combination

block=trenv∣ ⟩ ⟨∣=∣∣2∣∣

2−2

±=trenv∣± ⟩ ⟨±∣ =trenv

∣ ⟩ ⟨−∣∣− ⟩ ⟨∣2

are connected by the up-down symmetry and

give the same entropy. Moreover the trace over the environment (subsystem B) is expected to have a major contribution from the states obtained by restricting to B. So ifif∣± ⟩ ≃0

S ≃S []Sdeg

Sdeg=−∣∣2log∣∣

2−1−∣∣

2 log 1−∣∣

2

Symmetry breaking fieldsSymmetry breaking fields

−J z sleft1z −J zL

z srightPinning fixed spins

Staggered magnetic field along z: −hs∑ j−1

j j

z

● XYZXYZ: The fully anisotropic case is critical only when the two largest couplings have equal magnitude and disordered when . We can adopt the same symmetry breaking mechanisms to select only one of the two ground states.

● This was an implicit choice in the corner transferm matrix construction!

For the XXZ case a small staggered field (to set to zero at the end of calculations) lifts the quasi-degeneracy at finite size as soon as that is approximately

M s hs LE

∣∣1

L−ln M s hs

G

Characteristicsize dependslogarithmicallyon h

s

Moving towards the criticalpoint the saturation from above (stag. fields) occursat a larger value and on longer length scales

Fully anisotropicFully anisotropiccase (XYZ)case (XYZ)

● Different choices of parametersbut same same

● DMRG caveats

DMRG caveatsDMRG caveats

The influence of pinning fixed spins may be so strong that not only the ground-state degeneracy is lifted, but also the choice of block states retained during the DMRG is strongly biased: only a few states (in fact too few) that minimise the interaction with the boundary are retained and the representation in the truncated Hilbert space in the next steps is poor.

Practical solution: Even if the model is not strictly Z

2 (up-down) invariant we consider at

each step the symmetrised density matrix

blockPblock P P j= jx

Successful tests Successful tests (pinning fixed spins)

Still even-odd effects beforea characteristic length scale

XYZXYZ:

Accurate quantitative check: Scaling with L

Further developmentsFurther developments:

● The XYZ model with impurities is still integrable.Is it still related to the 8-vertex classical 2D model, so that one can borrow these results?

● Ambitious goal: Reproduce numerically the scaling limit that leads to the quantum sine-quantum sine-Gordon field theory Gordon field theory [Luther, PRB 14, 2153 (1976)] so to check quantitatively the exactexact formula for the block entropy of the sine-Gordon model (Ercolessi, Evangelisti, Ravanini, 2009).

ASG=∫d2 x {12∂

2g cos[x ]}S=

c6

lng ,

aU

cristian.degliesposti@unibo.it

www.df.unibo.it/fismat/theory

Thanks for your attentionThanks for your attention

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Supported by MIUR, PRIN n. 2007JHLPEZ

Definitions of Jacobian elliptic functionsDefinitions of Jacobian elliptic functions

snu=sin

u=∫0

d x 1−k2 sin x −1/2

cnu=cos dnu=1−k2 sin

Elliptic integral of the first kind

Delta amplitude