Characterization of Topological Insulators:

Chern Numbers for Ground State Multiplet

Yasuhiro HATSUGAI�

Department of Applied Physics, University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656

(Received February 18, 2005; accepted February 21, 2005)

We propose the use of generic Chern numbers for the characterization of topological insulators. It issuitable for the numerical characterization of low-dimensional quantum liquids, in which strong quantumfluctuations prevent the development of conventional orders. Using the twisting parameters of boundaryconditions, the non-Abelian Chern numbers are defined for a few low-lying states near the ground state ina finite system, which is a ground state multiplet with a possible (topological) degeneracy. We define thesystem as a topological insulator when energies of the multiplet are well separated from the above.Translational invariant twists up to a unitary equivalence are crucial to picking up only bulk propertieswithout edge states. As a simple example, the setup is applied to a two-dimensional XXZ-spin systemwith an Ising anisotropy, in which the ground state multiplet is composed of doubly almost degeneratestates. It gives a vanishing Chern number due to symmetry. Also Chern numbers for the genericfractional quantum Hall states are discussed briefly.

KEYWORDS: Chern numbers, degeneracy, topological orders, quantum Liquids, insulatorsDOI: 10.1143/JPSJ.74.1374

A crucial role of phases is one of the intrinsic features ofquantum mechanics and has long been investigated. Amongthe phases, those which have intrinsic geometric origins arenow understood as geometrical phases.1) Aharonov–Bohmeffects and Dirac monopoles are typical and classicexamples whose geometrical phases are fundamental. Thegeometrical features of gauge theories are another proto-type.2) Also, studies of Berry’s phases revealed that thegeometric phases and the gauge structure are closely relatedand derived by restricting a physical Hilbert space.3,4)

On the other hand, using order parameters, symmetrybreaking is one of the most fundamental concepts in modernphysics. This standard setup can characterize most of theordered states and describe phase transitions and criticalphenomena. However, in low-dimensional quantum systems,such as electrons with a strong correlation and spins,quantum fluctuations prevent the development of conven-tional orders even at a zero temperature. In these systems,the quantum phases of many-particle ground states varywildly in space and time, which destroy the standard orders.

The wild quantum phases are not random but obey somehidden restriction rules and reflect the features of thequantum mechanical wave function. Some of them are wellknown today such as Marshall sign rules in spin systems5)

and the fractional statistics of quasi-particle (hole) wave-functions in the fractional quantum Hall effect (FQH).6)

String order parameters in the Haldane spin chains7–9) andthe quasi off-diagonal order in the FQH10) are also discussedon the basis of the feature. Generically, quantum states withthe characteristic geometric phases are considered to possessnontrivial topological orders.11)

Recently, a wide variety of interesting and physicallyimportant phenomena have been clarified on the basis of aconcept of the topological order. Some of them includequantum Hall effects,12,13) solitons in polyacetylens,14)

adiabatic transports of charge and spins,15) itinerant magnet-ism and spintronics,16) anomalous Hall conductances,17–19)

polarizations in insulators,20) two-dimensional carbonsheets,21,22) anisotropic superconductors,22–24) and string-netcondensations.25) They are currently extensively studied.

A local phase of the many-particle wave function isarbitrary but there is some correlation with the phases of itsneighbors, which generates some gauge structures. Fromthese viewpoints, the templates of such systems are quantumHall states, particularly integer quantum Hall (IQH) states.There are apparently different QH states with differentquantized Hall conductances. However, any symmetries arenot broken among the states but they are clearly differentphysical systems. These states are characterized by thequantized Hall conductances which have an intrinsictopological origin.12) The topological origin of the Hallconductance is clear as determined by the Chern numberexpression.13,26,27) On the basis of the observation, wepropose to characterize the topological orders by usinggeneric Chern numbers.28) The Chern numbers in topolog-ical ordered systems are order parameters in conventionalordered phases. In the same manner as the conventionalphase transition is characterized by a sudden change in orderparameters, topological phase transitions are characterizedby a discontinuous change in the Chern numbers.

To have a well-defined Chern number, we need thepresence of a generic gap.28) Topological insulators aredefined as physical systems with this generic energy gap.Then the Chern numbers are always integers and thetopological phase transition is characterized by a discontin-uous change in the Chern numbers which are alwaysintegers. This integral property of the Chern number impliesa stability of the characterization against a small perturba-tion. However, as in the case of the edge states of thequantum Hall effect29) and Kennedy’s triplet states in theHaldane spin chains,30) the topological ordered state arequite sensitive to a geometrical change of the physicalsystem such as the presence of edges and boundaries. Itcontrasts with the conventional order in which boundaryconditions are always negligible in the thermodynamic limit.Therefore, a translational invariance is fundamentally

LETTERS

�E-mail: hatsugai@pothos.t.u-tokyo.ac.jp

Journal of the Physical Society of Japan

Vol. 74, No. 5, May, 2005, pp. 1374–1377

#2005 The Physical Society of Japan

1374

important for describing topological ordered states. On theother hand, in many cases, as far as physical observables areconcerned, a topological order is hidden in a bulk and onlyreveals its physical significance near the boundaries of thesystem.

Generically, to define the Chern number C for a physical(many-particle) wave function, j ðxÞi, we need to requirethe wave function to depend on multiple parameters,namely, x 2 V, where dimV � 2. The most natural param-eters that do not disturb bulk properties are multipleAharonov–Bohm fluxes on a genus g Riemann surface.When the topological order is nontrivial, there can beinevitable topological degeneracies,11) such as a qg-folddegeneracy of the FQH state with filling factor 1=q on atorus.31–33) The degeneracy of a generic ground state will bediscussed later. Here we just point out that one has toconsider non-Abelian gauge structures arising from it.4,28)

This is crucial for a numerical concrete characterization ofthe topological insulators. In particular, an explicit gaugefixing for the degenerate multiplet is required to performcalculations.28)

Quantum Spin Systems as Topological Insulators: Todescribe the characterization of the topological order, let usconsider a generic translational invariant spin-1/2 Hamil-tonian on a d-dimension orthogonal lattice

HP ¼ Hðh‘Þ ¼Xm

Tmh‘ððSðr1Þ;Sðr2Þ; . . .ÞTmy;

where Tm ¼ Tm1

1 ; . . . ;Tmd

d , m ¼ ðm1; . . . ;mdÞ and SðrÞ ¼tðSxðrÞ; SyðrÞ; SzðrÞÞ is a spin-1/2 operator at a lattice site rand h‘ is a local Hamiltonian which depends on several spinsat r1; r2; . . .. It generically breaks several symmetriesexplicitly such as a parity, a chiral symmetry, and a timereversal symmetry. The operator T� is a translation in the �-direction, T�SðrÞTy

� ¼ Sðrþ a�Þ (a� is a unit translation inthe � direction). We use the periodic boundary conditionTL�� ¼ 1 (m� ¼ 0; . . . ; d) to avoid the disturbance of bulk

properties by possible edge states. We propose the use oftwisted boundary conditions for the spin model and take thetwists as the parameters x as discussed below.

Local Gauge Transformation and Twists: Let usconsider a local gauge transformation of the string type,that is, local spin rotations at a unit cell label m as

S0�ðr

m� Þ ¼ Qð�ÞSðrm� Þ

with 3� 3 matrix Qð�Þ ¼ e�X, � ¼ m � � where

X�� ¼1

2in� Tr ������;

� ¼ ð�1; . . . ; �dÞ and n ¼ ðnx; ny; nzÞ ðjnj ¼ 1Þ is a fixedrotation axis. Also, � is a label that distinguishes intra-unitcell spins. The simplest example is given by taking n ¼ð0; 0; 1Þ as

Qzð�Þ ¼cos � � sin � 0

sin � cos � 0

0 0 1

0B@

1CA with � ¼ m � �:

We further assume the local Hamiltonian h‘ is of the gaugeinteraction type as

h‘ðSðr1Þ;Sðr2Þ; . . .Þ ¼ hGð� ¼ 0; fri � rjg;Sðr1Þ;Sðr2Þ; . . .Þ

¼ hGð�; fri � rjg;S0�ðr1Þ;S

0�ðr2Þ; . . .Þ

� h�‘ðS0�ðr1Þ;S

0�ðr2Þ; . . .Þ

with some function hG. That is, the twisting parametersonly affect the Hamiltonian through the relative positions oflocal spins. Examples of such interactions for the aboverotation around the z-axis are h

pair‘ ¼ tSðr1ÞJSðr2Þ with

J ¼ J diagð1; 1; �Þ and hsb‘ ¼ JcSðr1Þ � ðSðr2Þ � Sðr3ÞÞ ¼Jci jkSiðr1ÞSjðr2ÞSkðr3Þ.34) They transform respectively ashpair ¼ J

2ðe�ið�1��2ÞSþ1�1

S�2�2þ h.c.Þ þ �Sz1�1S

z2�2, hsb ¼ Jc

2Sz1�1

ðie�ið�2��3ÞSþ2�2S�3�3

þ h.c.Þ þ ðcyclic perm.Þ, �i ¼ mi � �, i ¼1, 2, 3 where mi is a unit cell labeling of the spin SðriÞ.

The Hamiltonian HP is periodic in the original S-representation but is not periodic in the twisted S0

�-represen-tation as S0ðTL�rÞ ¼ T

0L�� S0ðrÞ with T

0L�� ¼ expð�n̂n���L�XÞ.

Now let us define a translational invariant twistedHamiltonians HT by a representation by the twisted S0

� as

HT ð�Þ ¼Xm

T 0mh�‘ðS0�ðr1Þ;S

0�ðr2Þ; . . .ÞT

0my

with a periodic boundary condition T0L�� ¼ 1. In the original

spin operators S, HT is given by HP with the twistedboundary condition T

L�� ¼ expðþ��L�XÞ.

The two Hamiltonians, HP and HT , are both translation-ally invariant in S- and S0

�-representations, respectively. Onemay expect a macroscopic OðVÞ energy difference betweentheir ground state energies. However, as discussed, thecontribution should be at the most Oðj@�V jÞ due to the gaugeinvariance where j@�Vj is a (hyper) area of the systemperpendicular to the r�-axis where V ¼ L1 � � � Ld. That is, thedifference in energy should be a finite size effect. Thus, thedifference between HP and HT is negligible in thethermodynamic limit V ! 1 when we generally discussthe bulk properties.

Another important point for the present construction isthat the twisted Hamiltonian HT is translational invariant inthe S0

�-representation. Then edge states never appear in anyrepresentation, which is particularly important for picking uponly bulk properties through probes under twisted boundaryconditions.

Degeneracies and Ground State Multiplet: A topolog-ical order on a non-zero genus Riemann surface is one of thereasons for the ground state degeneracy, which is thetopological degeneracy.11) The simplest example is a many-particle state with two-dimensional periodic boundaryconditions.31,32) Also, if the system has a standard symmetrybreaking, such as Ising orders, a finite system has almostdegenerate ground states corresponding to the linear combi-nations of the symmetry-broken states.38) For a finite system,the degeneracy can be lifted and the splitting is estimated as�e�CV for symmetry-broken states. Moreover, if the groundstate has a finite spin moment which may not be macro-scopic as in a ferromagnet, spin degeneracy occurs. Some ofthese degeneracies can be approximated for a finite systemand may be sensitive to the boundary condition and twistingparameters, such as ��’s. In these cases, the lowest energygap is not a physical one and may vanish in the thermody-namic limit. The physical energy gap for the bulk is anenergy gap above these almost degenerate states. We definea ground state multiplet by a collection of these almostdegenerate states near the ground state and define a Chern

J. Phys. Soc. Jpn., Vol. 74, No. 5, May, 2005 LETTERS Y. HATSUGAI 1375

number for this ground state multiplet.35) (See Fig. 1.) Sincethe two Hamiltonians HP and HT ð�Þ differ only in boundary,the bulk properties of the two should be the same. Then, fortopological insulators, the energy gap above the ground statemultiplet is stable against perturbations. If the ground statemultiplet is well separated from the above in a finite system,we do not need to take the thermodynamic limit.

The twist we are proposing is a boundary perturbation in aparticular representation. However, it also preserves thetranslational symmetry up to a unitary equivalence. Then, onthe basis of the discussion of edge state, we expect an energyseparation of topological degeneracy, that is, a band width ofthe multiplet by the twist as e�L= where L is the lineardimension of the system and is a typical length scale of theground state multiplet. It can be different from the conven-tional broken symmetric cases.

Chern Numbers for Spins: Let us define a totalparameter space by V ¼ fð�1; . . . ; �dÞj�� 2 ½0; 2�=L��g.Since expð2�XÞ ¼ I3, we have HT ð� þ ð. . . ; 0; �� þ 2�=L�;0; . . .Þ ¼ HT ð�Þ in the S0 representation. Then the twistedHamiltonian HT ð�Þ is well defined on V without boundariesas V ¼ Td.36) Any two-dimensional integration surface SðVÞ without boundaries is used to define the Chern numbers.S ¼ T2

i j ¼ fð�i; �jÞg is the simplest example. Now, we definea ground state multiplet �ðxÞ; x 2 V. It is a N � q matrix as�ðxÞ ¼ ðj 1ðxÞi; . . . ; j qðxÞiÞ with HT ðxÞj jðxÞi ¼ jj jðxÞi;j ¼ 1; 2; . . ., i j; ði < jÞ, where j jðxÞi is a column vectorin a many-spin Hilbert space with a dimension N, and q is adimension of the ground state multiplet. The generic energygap condition for the multiplet is given as qðxÞ < qþ1ðxÞ,8x 2 S. This is a definition of the topological insulators.

We define a non-Abelian connection one-form A, whichis a q� q matrix, as A ¼ �yd� and a field strength two-form F ¼ dAþA2. The first Chern number2) is then definedby

CS ¼1

2�i

ZS

TrF ¼1

2�i

ZS

Tr dA:

It is a topological integer which is stable against perturbationunless the generic gap collapses. We use these integersdepending on the choice of S to characterize the topologicalorders. Changing a basis within the multiplet space, �0ðxÞ ¼�ðxÞ!ðxÞ, (!!y ¼ Iq) gives a gauge transformation A0 ¼�0d�0 ¼ !�1A!þ !�1d! and F 0 ¼ !�1F!.4,28) The Chernnumber is gauge-invariant but we need to fix the gauge toevaluate this expression.28) We take a generic arbitrarymultiplet, �, and define an overlap matrix as O� ¼ �yP�where P ¼ ��y is a projection into the ground statemultiplet, which is gauge-invariant. Then we define regions

S�R , R ¼ 1; 2; . . ., as (infinitesimally) small neighborhoods of

zeros of detO�ðxÞ and S�0 as the remaining of S. Then the

first Chern number is written as

CS ¼ �NT�ðSÞ ¼ �

XR�1

nR�ðS�R Þ; nR�ðS

�R Þ ¼

1

2�

I@S�

R

d0�:

The field � is defined as �ð ~��;�Þ ¼ Arg det ~��yP� ¼Arg det �� Arg det ~�� where ~�� is also another genericarbitrary multiplet, � ¼ �y� and ~�� ¼ �y ~��. The matrices� and ~�� depend on the choice of the multiplet � but thedifference in the arguments is gauge-invariant. The field �

depends on the choice of � and ~��, but the total vorticityNT�ðSÞ is gauge-invariant and independent of the choice. The

field � reflects a phase sensitivity of the multiplet to the twistwhen one fixes � and ~��. It is illustrative to show � and itsupplies information of the ground state multiplet. Alsowhen the integration surface S is contractible to a point thatmaintains a generic energy gap, the Chern number vanishesdue to a topological stability.

Ex.1: Two-Dimensional Spin Model: The present for-mulation can be effective for the characterization oftopological ordered phases in any dimensions, such as chiralspin states.34) To have a finite Chern number for the presenttwists in the example, one may need to break time reversalsymmetry as for the quantum Hall states.37) The simplestexample of h‘ can be the sum of local pair-spin interactionshpair‘ with a symmetry breaking term hsb‘ discussed above as

h‘ ¼P

pair hpair‘ þ hsb‘ . Here, let us show an example with a

degeneracy to demonstrate the present general procedure.Considering only a nearest-neighbor exchange interactionand assume n ¼ ð0; 0; 1Þ, the local Hamiltonian is given ash�‘ ¼

P�¼x;y Jð12 ðe

�i��S��þS�� þ ei�

�S���S�þÞ þ �S

��zS�z Þ where

S�� ¼ S�ðT�rÞ and S� ¼ S�ðrÞ. In this case, the twistedboundary condition for HT ð�Þ in S is given by the followingmatrix T

L�� ¼ Qzð�Þ with � ¼ �xLx þ �yLy. Then we can take

a two-dimensional torus T2 ¼ fð�x; �yÞj0 �� 2�=L�g forthe integration surface S. This is a nearest neighbor XXZ

model on a square lattice with twists. With an Isinganisotropy, � > 1, the ground state of an infinite systemhas a long-range order and has a finite energy gap. We shownumerical results for a system with J ¼ 1 and � ¼ 1:3. Theground state of a finite size system is given by a bondingstate between two symmetry-broken states with antiferro-magnetic (Ising) order. The next lowest state is the anti-bonding state and the energy separation between is expectedto be / e�LxLy=2 where is a typical length scale. Aphysical Ising gap for flipping one ordered spin is given theone above, that is, the second lowest gap. Therefore, theground state multiplet is composed of the two low-lyingstates including the finite size ground state. [See Fig. 2(a).]The field � of the two-dimensional ground state multiplet isshown in Fig. 2(b). As discussed in the reference,28) theChern number is the sum of vorticity at the zeros ofdetO� ¼ j det �j2. In the present example, it is 0 as expectedfor a chiral symmetric system.

Ex.2: Many-particle States in the First QuantizedForm: The same procedure is also applied to a many-particle state in the first quantized form, such as the genericFQH states �kðx; r1; �1; . . . ; rN ; �NÞ where k denotes a labelof the (topological) degeneracy of the ground state multipletand x ¼ ð�x; �yÞ is a set of parameters specifying twisted

x

Ene

rgy

x i xf

Fig. 1. Schematic spectral flow with parameters x.

1376 J. Phys. Soc. Jpn., Vol. 74, No. 5, May, 2005 LETTERS Y. HATSUGAI

boundary conditions on a torus.32) By taking a referencemultiplet as �ðr1; �1; . . . ; rN ; �NÞ ¼ ��1�1

�ðr1 � r1Þ � � � ��N�N�ðrN � rNÞ, ¼ 1; . . . ; q. The Chern number for the degen-erate multiplet is given by the field �ðxÞ ¼ Arg det ~��y� withf�gk ¼ �kðx; r1; �

1; . . . ; r

N ; �

NÞ and f~��g0k ¼ �kðx; r

0

1 ; �0

1 ;

. . . ; r0

N ; �0

NÞ, where k, and 0 vary over f1; . . . ; qg. TheChern number is evaluated as a total vorticity of � at thezeros of j det �j2.28)

We thank X.-G. Wen for fruitful discussion. Part of thepresent work was supported by a Grant-in-Aid from theJapanese Ministry of Education, Culture, Sport, Science andTechnology, and the JFE 21st Century Foundation. Also partof the computation in this work has been done using thefacilities of the Supercomputer Center, ISSP, University ofTokyo.

Note Added: After finishing this work, we have noticedtwo related papers on Abelian Chern numbers in interactingsystems.39,40) In the present letter, we have proposed ageneral framework for the characterization of topologicalinsulator by the non-Abelian Chern numbers preserving thetranslational invariance. The (topological) degeneracy re-quires inevitably non-Abelian extension.

1) Geometric Phases in Physics, ed. A. Shapere and F. Wilczek (World

Scientific, Singapore, 1989).

2) T. Eguchi, P. B. Gilkey and A. J. Hanson: Phys. Rep. 66 (1980) 213.

3) M. V. Berry: Proc. R. Soc. London, Ser. A 392 (1984) 45.

4) F. Wilczek and A. Zee: Phys. Rev. Lett. 52 (1984) 2111.

5) W. Marshall: Proc. R. Soc. London, Ser. A 232 (1955) 48.

6) R. B. Laughlin: Phys. Rev. Lett. 50 (1983) 1395.

7) M. den Nijs and K. Rommelse: Phys. Rev. B 40 (1989) 4709.

8) S. Girvin and D. Arovas: Phys. Scr. T27 (1989) 156.

9) Y. Hatsugai: J. Phys. Soc. Jpn. 61 (1992) 3856.

10) S. Girvin and A. H. MacDonald: Phys. Rev. Lett. 58 (1987) 1252.

11) X. G. Wen: Phys. Rev. B 40 (1989) 7387.

12) R. B. Laughlin: Phys. Rev. B 23 (1981) 5632.

13) D. J. Thouless, M. Kohmoto, P. Nightingale and M. den Nijs: Phys.

Rev. Lett. 49 (1982) 405.

14) W. P. Su, J. R. Schrieffer and A. J. Heeger: Phys. Rev. Lett. 42 (1979)

1698.

15) D. J. Thouless: Phys. Rev. B 27 (1983) 6083.

16) Y. Taguchi, Y. Oohara, H. Yoshizawa, N. Nagaosa and Y. Tokura:

Science 291 (2001) 2573.

17) T. Jungwirth, Q. Niu and A. H. MacDonald: Phys. Rev. Lett. 88 (2002)

207208.

18) S. Murakami, N. Nagaosa and S.-C. Zhang: Science 5 (2003) 1348.

19) F. D. Haldane: Phys. Rev. Lett. 93 (2004) 206602.

20) R. D. King-Smith and D. Vanderblit: Phys. Rev. B 47 (1993) 1651.

21) F. D. M. Haldane: Phys. Rev. Lett. 61 (1988) 2015.

22) S. Ryu and Y. Hatsugai: Phys. Rev. Lett. 89 (2002) 077002.

23) T. Senthil, J. B. Marston and M. Fisher: Phys. Rev. B 60 (1999) 4245.

24) Y. Hatsugai, S. Ryu and M. Kohmoto: Phys. Rev. B 70 (2004) 054502.

25) X. G. Wen: Phys. Rev. Lett. 90 (2003) 016803.

26) M. Kohmoto: Ann. Phys. (N.Y.) 160 355 (1985).

27) Q. Niu, D. J. Thouless and Y. Wu: Phys. Rev. B 31 (1985) 3372.

28) Y. Hatsugai: J. Phys. Soc. Jpn. 73 (2004) 2604.

29) B. I. Halperin: Phys. Rev. B 25 (1982) 2185.

30) T. Kennedy: J. Phys.: Condens. Matter 2 (1990) 5737.

31) D. Yoshioka, B. I. Halperin and P. A. Lee: Phys. Rev. Lett. 50 (1983)

1219.

32) F. D. M. Haldane and E. H. Rezayi: Phys. Rev. B 31 (1985) 2529.

33) F. D. M. Haldane: Phys. Rev. Lett. 55 (1985) 2095.

34) X. G. Wen, F. Wilczek and A. Zee: Phys. Rev. B 39 (1989) 11413.

35) Since the Hamiltonian HT ð�Þ is translationally invariant, ½HT ; T 0�� ¼ 0.

That is T 0� ¼ eik� with k� ¼ 2�m�=L�, m� ¼ 0, �1, �2, . . ., the

ground state multiplet may compose of states with different k�. We

may define the Chern numbers using these conserved quantum

numbers. However it is natural to discuss (almost) degenerate states as

a ground state multiplet.28)

36) When we use a representation by S�, HT ð� þ ð. . . ; 0; �� þ 2�=L�;

0; . . .ÞÞ 6¼ HT ð�Þ and it is not well defined on the surface S. It is just

well defined up to an unitary equivalence.

37) Nontrivial examples with finite Chern numbers for the present spin

twists are under studies in collaboration with X.-G. Wen (unpub-

lished).

38) Y. Hatsugai: Phys. Rev. B 56 (1997) 12183.

39) F. D. M. Haldane and D. Arovas: Phys. Rev. B 52 (1995) 4223.

40) D. N. Sheng, X. Wan, E. H. Rezayi, K. Yang and F. D. M. Haldane:

Phys. Rev. Lett. 90 (2003) 256802.

0

−12

−11

−10

2π/ L

θ x

x

0

2π/ yL

yθ

y

2π/ xLxθ0

2π/L

yθ

0

(a) (b)

Ene

rgy

Fig. 2. (a) Three lowest energies of XXZ model on 4� 4 square lattice

with twists in total Sz ¼ 0 sector are shown. (� ¼ 1:3) (b) Field � of

ground state multiplet composed of two lowest eigen states for some

choice of �, ~��.

J. Phys. Soc. Jpn., Vol. 74, No. 5, May, 2005 LETTERS Y. HATSUGAI 1377