arX

iv:1

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v1 [

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2

Momentum polarization: an entanglement measure of topological spin and chiral central charge

Hong-Hao Tu1, Yi Zhang2 and Xiao-Liang Qi21 Max-Planck-Institut fur Quantenoptik, Hans-Kopfermann-Str. 1, D-85748 Garching, Germany

2 Department of Physics, Stanford University, Stanford, California 94305, USA(Dated: September 27, 2018)

Topologically ordered states are quantum states of matter with topological ground state degeneracy and quasi-particles carrying fractional quantum numbers and fractional statistics. The topological spinθa = 2πha is an im-portant property of a topological quasi-particle, which isthe Berry phase obtained in the adiabatic self-rotationof the quasi-particle by 2π. For chiral topological states with robust chiral edge states, another fundamentaltopological property is the edge state chiral central charge c. In this paper we propose a new approach to com-pute the topological spin and chiral central charge in lattice models by defining a new quantity named as themomentum polarization. Momentum polarization is defined onthe cylinder geometry as a universal subleadingterm in the average value of a “partial translation operator”. We show that the momentum polarization is a quan-tum entanglement property which can be computed from the reduced density matrix, and our analytic derivationbased on edge conformal field theory shows that the momentum polarization measures the combinationha − c

24of topological spin and central charge. Numerical results are obtained for two example systems, the non-Abelianphase of the honeycomb lattice Kitaev model, and theν = 1/2 Laughlin state of a fractional Chern insulatordescribed by a variational Monte Carlo wavefunction. The numerical results verifies the analytic formula withhigh accuracy, and further suggests that this result remains robust even when the edge states cannot be describedby a conformal field theory. Our result provides a new efficient approach to characterize and identify topologicalstates of matter from finite size numerics.

I. INTRODUCTION

Since the discovery of integer and fractional quantum Hallstates in 1980’s1,2, the research on topological states of matterhas attracted tremendous experimental and theoretical inter-est. The recent discovery of topological insulators and topo-logical superconductors have greatly expanded the knowl-edge of topological states.3–5 In general, topological states aregapped zero-temperature states of matter which cannot be de-formed to trivial states without experiencing a quantum phasetransition. Here the “trivial states” are reference stateswhichhave no quantum entanglement between any two different re-gions of space. Topological states of matter are characterizedby topological properties such as gapless edge/surface states,ground state degeneracy, bulk quasi-particle excitationswithfractional charge and fractional statistics. The fractionaliza-tion of quantum numbers and statistics can only occur in topo-logical states with ground state degeneracy, which are calledtopologically ordered states6,7.

In a topologically ordered state, a given configuration oftopological quasi-particles correspond to a finite dimensionalHilbert space. Braiding quasi-particles around each otherleads to a unitary transformation in the Hilbert space, whichis named as the fractional statistics of the particles8. Whenthe braiding of different particles leading to non-commutingtransformations, the statistics of the particle is non-Abelian,and otherwise it is Abelian. Besides the statistics, each quasi-particlea also has a fractionalized spinθa which is a phasefactor obtained by the state of the system during the self-rotation ofa by 2π. The spins of topological quasi-particlesare related to their statistics, because when two particleswindaround each other by 2π and return to their positions, fromfar away, other particles cannot determine whether there aretwo particles braiding or there is only one particle (obtainedby the “fusion” of the two particles) spinning around itself.9

The spin-statistics relation is a generalization of that for ordi-nary particles–i.e., fermions have half-odd-integer spin whilebosons have integer spin. Therefore the spin values of quasi-particles are an important set of information that distinguishesdifferent topological states. The calculation of topologicalspin is generically difficult except for algebraically definedtopological field theories9 and some ideal model states10–12.For microscopic lattice models with a topological phase, thetopological spin of quasi-particles has only been computednumerically for several special cases: 1) The honeycomb lat-tice Kitaev model studied by calculating the braiding of quasi-particles13,14; 2) The fractional Chern insulators studied bymodular transformations on a torus15–17. Such numerical re-sults are usually restricted by small system size and specificknowledge about the ground state wavefunction.

For chiral topological states with robust chiral edge states,another fundamental topological quantity is the edge statechi-ral central charge. If the edge state is described by a chiralconformal field theory (CFT), it has a chiral central chargec, which determines the heat currentIE =

π6cT2 at a given

temperature.18 The central charge also appears in the gravi-tational anomaly of the edge19 if the system is coupled to agravitational field. Since the edge states of a topological stateis only a well-defined one-dimensional state in energy belowthe bulk gap, it can at most be described by a CFT in the longwavelength limit. (As we will discuss later, there are exam-ples where the edge state is not Lorentz invariant even in thelong wavelength limit, so that it’s not a CFT at all.)

In this paper, we propose a new numerical method for thecalculation of topological spins and chiral central chargeincandidate systems of topological states, which is significantlysimpler than previous numerical methods, and can be gen-erally applied to identify topological states obtained in nu-merical approaches such as exact diagonalization and MonteCarlo. This method applies to a topological state defined oncylinder. Since a cylinder is topologically equivalent to a

2

sphere with two punctures, each topological quasiparticleacorresponds to a ground state on the cylinder, which has quasi-particlea in each puncture. This is illustrated in Fig. 1 (a). Arotation of the cylinder along the periodic directiony is equiv-alent to rotating the two particles simultaneously. By mappingto sphere we see that with reference to the direction normal tothe sphere, one particle is rotated clockwisely while the otheris rotated counter-clockwisely. Consequently, when the par-ticles have spinθa, a 2π rotation of the cylinder should leadto a Berry phase ofeiθa from one end ande−iθa from the otherend, which cancels each other as expected, since this globalrotation is a symmetry of the system. To avoid the cancela-tion, we need to define a “twist” of the cylinder where onlythe left half of the cylinder is rotated, as is illustrated inFig.1 (b). For a lattice system, we do not have a way to twist thecylinder continuously, but have to twist discretely. If we con-sider the twist Fig. 1 (b) where half of the cylinder is twistedby one lattice constant, and there areLy lattice sites in theydirection, such a twist should lead to a Berry’s phase ofeiθa/Ly

coming from the left edge of the cylinder with particle typea,without cancelation from the right edge. However, the twistlooks like a violent operation since it induces a discontinu-ous jump at the boundary between left and right halves of thecylinder. Naively this appeared to induce a non-universal cor-rection to the phase obtained during the twist, and thus makeit impossible to measure the spinθa. The key result of the cur-rent work is that the non-universal contribution is independentfrom topological sectora, which is shown in the following for-mula:

λa ≡ 〈Ga|TLy |Ga〉 = exp

[

2πiLy

pa − αLy

]

(1)

pa ≡ ha −c

24

with |Ga〉 the ground state in thea sector,c is the centralcharge of the edge CFT, andα is a non-universal complex con-stant. The key property of this formula is that the first termis universal and contains the topological informationha andc, while the second non-universal term is independent froma.We name the coefficientpa of the first term as themomentumpolarization, which has the physical meaning of the nety mo-mentum carried by the left edge in the ground state (in unit of2πLy

). We choose the unit2πLywhich is the momentum quantum

carried by a local bosonic excitation. Thereforepa ∈ Z corre-sponds to a local excitation and the fractional partpa mod 1 isthe topological contribution determined by the ground state.As we will show below, this result can be understood basedon entanglement properties of topological states.

The rest of the paper is organized as following. In Sec. IIwe provide an analytic derivation of Eq. (1) based on edgestate conformal field theory and quantum entanglement prop-erties of the topological state. In Sec. III we verify thisidea by numerical calculations in the honeycomb lattice Ki-taev model20. In the non-Abelian phase of Kitaev model, wefind that the spin of the non-Abelian anyonσ is θa =

π8, in

consistency with the expectation that this state is describedby an Ising topological quantum field theory.20 In Sec. IV

FIG. 1. (a) A global translation of the cylinder is equivalent tospinning two quasiparticles on a sphere to opposite directions. (b)The partial translationTL

y of the left part of the cylinder. In bothpanels, the red circles with arrows indicate the chiral edgestates.

this result is further verified in another topological state, thefractional Chern insulators21–24. The variational Monte Carlowavefunction15 of lattice Laughlin 1/2 state is confirmed tohave central chargec = 1 and quasiparticle chargeθa =

π2.

Finally Sec. V is the conclusion and discussions.

II. ANALYTIC RESULT

In this section we will derive Eq. (1) based on the assump-tion that the edge state of the topological state is a chiral con-formal field theory. As will be seen later in Sec. III, the nu-merical results indicate that the applicability of formula(1)goesbeyondthe edge CFT assumption, but analytically theresult can only be derived under this assumption.

To begin with, we notice that the partial translationTLy

only acts on the left half of the system, so that we can traceover the right half and define the reduced density matrixρLa = trR (|Ga〉 〈Ga|) such that

λa ≡ 〈Ga|TLy |Ga〉 = trL

(

ρLaTLy

)

(2)

The reduced density matrixρLa is generically difficult to com-pute, but for topologically ordered states with chiral edgestates, some long wavelength behavior ofρLa is known. In2008, H. Li and F. D. M. Haldane observed that the entan-glement spectrum (. i.e. the spectrum of the ”entangle-ment Hamiltonian”HE defined asHE = − logρLa ) is qual-itatively the same as that of the edge Hamiltonian25. Thisis proposed as a generic feature of the topological states,which has been verified numerically and analytically in var-ious physical systems such as general fractional quantum Hallwavefunctions26,27, free fermion topological states28,29 andKitaev model30, etc. A general analytic derivation of theentanglement-edge state correspondence has been proposedin Ref.31 for a generic topological state with chiral CFT edgestates. Here we will briefly review the result of Ref.31 whichwill be applied to the current work.

3

The basic idea behind this derivation is that a cylinder canbe considered as two cylindersA, B glued together, as is illus-trated in Fig. 2 (a). Before the gluing procedure, each cylin-der has two edge states on the two boundaries, which in thelong wavelength limit are described by a conformal field the-ory HamiltonianHA(B) = HA(B)l +HA(B)r , with l, r denoting theleft and right edge of each cylinder. A topological sector ofthecylinder labeled by quasiparticlea corresponds to conformalblocks of the edge CFT formed by a primary field with scal-ing dimensionha and its descendants. The “gluing” is doneby turning on a coupling between the two cylinders across theedge, and the coupling can be considered as a relevant cou-pling in the edge CFT. Using the property of CFT with rele-vant coupling, Ref.31 reaches the conclusion that the reduceddensity matrix of the edge stateAr is ρ(a)

Ar = Z−1ra e−βr HAr

∣

∣

∣

a,

which is a thermal density matrix of the chiral CFT restrictedto the sectora. The left edge ofA region is far from theBregion, so that the left edge CFT stays in its ground state, withno entanglement withB region. This can be described by thesame thermal density matrix with a temperatureβl → ∞. Thetotal reduced density matrix of the left half cylinder (A region)is

ρLa = ρ(a)Al ⊗ ρ

(a)Ar = Z−1

a e−βlHAl−βr HAr∣

∣

∣

a, βl = ∞, βr finite (3)

as is illustrated in Fig. 2 (b). Thereforeλa defined in Eq. (2) isthe average value of translation operatorTL

y in a CFT with leftedge at zero temperature, and right edge at finite temperature.In CFT the translation operator has the form

TLy = ei(Pl+Pr ) 2π

Ly = ei(Hl−Hr ) 2πvLy (4)

Here we have used the linear dispersion of the edge CFTHl = vPl , Hr = −vPr with v the velocity of the CFT. Nowwe introduce the charactersχa(q) of the CFT32,33 which aredefined as

χa(q) = tra(qL0) = tra(qLy2πv Hl ) = tra(q

Ly2πv Hr ) (5)

with tra the trace in the conformal blocka. From Eq. (3) and(4), λa can be expressed in the characterχa by

λa =

χa

(

e2πLy

(i−vβl))

χa

(

e2πLy

(−i−vβr ))

χa

(

e−2πLy

vβl

)

χa

(

e−2πLy

vβr

) (6)

Now we consider a large but finiteLy, such thatvβr ≪Ly ≪ vβl = ∞. The left edge is in zero temperature, so thatthe character is dominated by the ground state contribution

χa(q) = qha− c24 . Consequentlyχa

(

e2πLy

(i−vβl))

/χa

(

e−2πLy

vβl

)

=

exp[

i 2πLy

(

ha − c24

)

]

. On the other hand, the right edge is in

the high temperature limit sincevβr ≪ Ly. To compute thecharacter in that limit we can make use of the modular trans-formation property of the character

χa

(

e−2πi/τ)

=∑

b

Sabχb(ei2πτ) (7)

FIG. 2. (a) A cylinder can be considered as two separate cylin-ders glued together. Each cylinder have chiral edge states (red andblue circles) and the gluing of the two cylinders couples theedgestates at the interface. (b) After gluing the two parts of thecylinderare entangled through the interface, and the reduced density matrixof the left half cylinder is described by a thermal density matrix ofthe edge state CFT with finite “entanglement temperature”β−1

r at theright edge, and zero entanglement temperatureβ−1

l = 0 at the leftedge (see text).

with Sab the modularS matrix32. Takinge−2πi/τ = e−2πLy

vβr wehave

χa(e−2πLy

vβr ) =∑

b

Sabχb

(

e−2πLyvβr

)

(8)

Since the right-hand side is in low temperature limit, we can

approximateχb

(

e−2πLyvβr

)

≃ e−2πLyvβr (hb− c

24). In the low temper-

ature limit the trivial sectorχ1 with h1 = 0 dominates all

other sectors with positiveha, so that we haveχa(e−2πLy

vβr ) ≃Sa1e

2πLyvβr

c24 . The same modular transformation can be applied

to the numerator in Eq. (6) and leads toχa

(

e2πLy

(−i−vβr ))

≃

Sa1e2πLyi+vβr

c24 . In summary we obtain

λa = exp

[

i2πLy

(

ha −c24

)

− Ly2πc24

ivβr (i + vβr )

]

(9)

in the limit of vβr ≪ Ly, which demonstrates Eq. (1) withα = 2πc

24i

vβr (i+vβr )non-universal, but independent from topolog-

ical sectora. It should be noted that Reα = 2πc24

1vβr(1+v2β2

r )> 0,

which meansλa decays exponentially when the sizeLy in-creases.

For finite Ly/vβr , the formula above obtains finite sizecorrections coming from two origins: i) the contribution ofexcited states toχa(q) in low temperature limit. χa(q) =qha− c

24 (1+ n1aq+ ...) with n1a the number of states in the topo-logical sectora with momentum 2π/Ly. ii) The contributionof other topological sectorsb in the expansion (8), which isproportional toqhb. In the two corrections above, the mostleading contribution comes from the ground state of the topo-logical sector with minimal scaling dimension. Denoting theminimal scaling dimension byhmin, we obtain

χa

(

e−2πLy

vβr

)

≃ Sa1e2πLyvβr

c24

[

1+O(

e−2πLyvβr

hmin

)]

(10)

and similarly for the numerator of Eq. (6). Consequently thefinite size correction toλa is

λa = exp

[

i2πLy

(

ha −c24

)

− αLy

]

(

1+O(

e−Re(α)Ly))

(11)

4

with α =2πc24

ivβr (i + vβr )

We see that the finite size correction to formula (1) decaysexponentially versusLy/vβr , which suggests that this methodcan be applied well to finite size numerical calculations. Suchan exponential convergence behavior is confirmed in the nu-merical results presented in next section.

Besides the finite size correction, other corrections to theformula (1) may occur in a realistic system due to deviationsof the reduced density matrixρLa from the CFT behavior. Forexample the edge state dispersion is not strictly linear, and thestate counting may deviate from that of the pure CFT at highenergy. Somewhat surprisingly, in the two physical systemswe studied numerically, such nonuniversal corrections havenot been found, and the result (1) remains valid even if theedge state deviates strongly from a CFT.

III. NUMERICAL RESULTS ON HONEYCOMB-LATTICEKITAEV MODEL

In this section we will calculateλa numerically for thehoneycomb-lattice Kitaev model20. Kitaev model is a specialspin 1/2 model which has a non-Abelian topological phasethat we are interested in. For our purpose, we consider the fol-lowing Hamiltonian of the Kitaev model with nearest neigh-bor and second neighbor interactions:

H =∑

〈i j 〉∈x links

Jxσixσ jx +∑

〈i j 〉∈y links

Jyσiyσ jy +∑

〈i j 〉∈z links

Jzσizσ jz

+Jnn

∑

(i jk)∈△σiyσ jzσkx (12)

with x, y, z links the three types of nearest neighbor links onthe honeycomb lattice shown in Fig. 3 (a).△ in the Jnn termdenotes the upper and lower triangles shown in Fig. 3 (a), inwhich i jk runs counter-clockwisely around the triangle andjis at the upper or lower corner. For simplicity, we only intro-duced one type of second neighbor coupling instead of includ-ing all possible terms obtained by 60 degree rotations.

The unique property of the Kitaev model is that it can besolved by the following Majorana representation

σai = iγa

i ηi (13)

in which γai , a = x, y, z andηi are four Majorana fermion op-

erators defined at sitei. The Hilbert space of 4 Majoranafermions is 4-dimensional. The two physical states of the spinat sitei form a subspace of the Majorana fermion Hilbert spacewhich is determined by the condition

Di ≡ iγxi γ

yi γ

zi ηi = −iσx

i σyi σ

zi = 1 (14)

Using this representation the Hamiltonian is transformed to

H = −∑

a=x,y,z

∑

〈i j 〉∈alinks

Jaui j iηiη j − Jnn

∑

〈ik〉∈nn links

uikiηiηk

with ui j = iγai γ

aj , for 〈i j 〉 ∈ a links

uik = ui j u jk, for 〈ik〉 ∈ nn links, (i jk) ∈ △ (15)

Since[

ui j , ukl

]

= 0 for all different links〈i j 〉 , 〈kl〉, the Hamil-tonian can be viewed as a free Majorana fermion Hamiltonianof ηi with ui j = ±1 taking its eigenvalues. However the phys-ical states have to satisfy the constraint (14), which can beobtained by a projection in the following form:

|Ψ〉 =∏

i

Di + 12|ΨF({u})〉 ⊗ |{u}〉 (16)

with Di defined in Eq. (14). The link variableui j can beviewed as aZ2 gauge field coupled to the Majorana fermion,and Di acts as aZ2 gauge transformation which transformsηi → −ηi andui j → −ui j for all links connected toi. Thephysical state|Ψ〉 is a superposition of all gauge equivalentconfigurations of gauge fieldui j in direct product with thecorresponding fermion state|ΨF ({u})〉. Kitaev model has sev-eral phases including a gapless phase and Abelian and non-Abelian topologically ordered phases.20 In the following wemainly discuss the non-Abelian phase in which theηi fermionhas a band structure with a nontrivial Chern numberC = 1.There are three types of topological quasiparticles 1, σ, ψ inwhich 1 is the vacuum,ψ is the fermionic excitation ofη Ma-jorana fermion, andσ is aZ2 flux of the gauge fieldui j , witha Majorana zero mode ofη Majorana fermion trapped in it.σis the non-Abelian quasi-particle. Corresponding to the threequasi-particle sectors, there are three topological sectors forthe system on a cylinder. Depending on the flux ofui j gaugefield around the cylinder, the fermionηi has periodic or anti-periodic boundary conditions. As is shown in Fig. 3 (b), theMajorana fermion has chiral edge states due to the topologi-cal band structure. When theZ2 flux in the cylinder is−1, theedge state has anti-periodic boundary condition and momen-tum eigenvaluesk = 2π

Ly

(

n+ 12

)

. Denote theη fermion ground

state in this sector as∣

∣

∣G−F⟩

, we have|G1〉 = P∣

∣

∣G−F⟩

⊗∣

∣

∣

∣

{

u−0}⟩

with P =∏

i(Di+1)/2 the projection to gauge invariant states.u−0 is a gauge configuration with flux−1 in the cylinder. Forexample the gauge choice in Fig. 3 (b) can be made, withui j = −1 for all red links across the dashed line, andui j = 1otherwise. Theψ sector is the lowest energy excitation state∣

∣

∣Gψ

⟩

= P f†Lk f †R−k

∣

∣

∣G−F⟩

⊗∣

∣

∣

∣

{

u−0}⟩

, with f †L,Rk the creation operatorof the left and right edge states, respectively, andk = π

Ly. The

third sectorσ corresponds to the periodic boundary condition

|Gσ〉 = P∣

∣

∣G+F⟩

⊗∣

∣

∣

∣

{

u+0}⟩

. u+0 is a guage configuration with+1flux in the cylinder, which can be taken asui j = 1 for all 〈i j 〉.It should be noticed that there are two degenerate states be-fore projection due to the Majorana zero modes, but only onesurvives in the projection.

Due to the absence of quantum fluctuation of theZ2 gaugefield, the entanglement entropy and entanglement spectrum ofKitaev model can be obtained rigorously30. Using the reduceddensity matrix obtained in Ref.30, λa in Eq. (2) can be re-duced to that of the free Majorana fermion:λa = tr

(

ρFaTLFy

)

.HereρFa is the reduced density matrix of the left half cylin-der, andTLF

y is the gauge covariant translation operator act-ing on the fermionηi , which does a translation of theηi

fermions associated with a gauge transformation that trans-lates the gauge field configurationui j . This is like the mag-

5

FIG. 3. (a) Illustration of the definition of honeycomb lattice Ki-taev model in Eq. (12). Thex, y, z links are shown by the red solidlines, green dash lines and blue double lines, respectively. The sec-ond neighbor couplings are defined on the triangles shaded inorangecolor, with the labelingi, j, k of the triangle shown on the figure. (b)Illustration of the three ground states in the three topological sec-tors of the non-Abelian phase of Kitaev model. The right panelsshow the edge state energy spectrum with solid and hollow circlesstand for occupied and unoccupied quasi-particle states.|G1〉 and∣

∣

∣Gψ

⟩

correspond to the ground state and the lowest energy quasipar-ticle excitation state with fermion anti-periodic boundary conditions,while |Gσ〉 correspond to one of the degenerate ground states withfermion periodic boundary condition. The red dot in the right panelmarks the Majorana zero mode, the occupation of which does notchange the momentum polarization of|Gσ〉.

netic translation operators in a Landau level. More detailson the reduced density matrix and the calculation of momen-tum polarization are given in Appendix A. As a general re-sult for free fermions and free bosons34, ρFa can be deter-mined by the two-point function

⟨

ηiη j

⟩

(due to the Wick the-orem), and the entanglement Hamiltonian has the quadraticform HE = logZa − logρFa =

∑

i, j∈L iAi jηiη j . The constantZa

is determined by the normalization condition of the reduceddensity matrix. On the half cylinder, the entanglement Hamil-tonian can be diagonalized into the form

HE =∑

n,k

ξnk

(

f †nk fnk −12

)

(17)

with k they-direction momentum taking the values of2πLy

n and2πLy

(

n+ 12

)

(n = 0, 1, 2, ..., Ly−1) for periodic and anti-periodicboundary conditions, respectively.fnk are the quasiparticle an-nihilation operators andξnk are the eigenvalues of the entan-glement Hamiltonian. In the momentum basis, the translationoperator simply multiples a phaseeik to each quasi-particleoperatorfnk, so that

TLFy = exp

i∑

n,k

k f†nk fnk

(18)

Using Eq. (17) and Eq. (18) we obtain

λa = Z−1a tr

(

e−HETLFy

)

=∏

n,k

[

1+ eik

2+

1− eik

2tanh

ξnk

2

]

(19)

Herea = 1, σ corresponds to the anti-periodic and periodicboundary conditions, respectively. Since theψ state is an ex-cited state of the anti-periodic sector, it is not directly included

−1 0 1

−1

0

1

k/π

Ek

(a)

−1 0 1

−1

−0.5

0

0.5

1

k/π

Ek

(b)

0 50−0.4

−0.2

0

0.2

0.42π

24

−2π

48

Ly

Ly(θ

a+

αL

y)

(c)

0 50−30

−20

−10

Ly

log|δ

θ a|

(d)

FIG. 4. (a) The energy spectrum and (b) the entanglement spec-trum (ξnk in Eq. (17)) of the Kitaev model. (c) The value ofLy

(

θa(Ly) + αLy

)

versus the system sizeLy for a = 1 (blue×) andσ(red◦) sectors.θa = Im logλa. α is obtained from the slope of a lin-ear fitting ofLyθa(Ly) versusL2

y. The dashed lines mark the theoreti-

cal values given by Eq. (1) 2π(

ha − c24

)

for h1 = 0, hσ = 116. (d) The

deviation ofθa from the theoretical valueδθa = θa+αLy− 2πLy

(

ha − c24

)

versusLy plotted in a log scale. The calculations are done for a sys-tem with Lx = 40 along thex direction with zigzag edge and theparametersJx = Jy = Jz = 1, Jnn = 0.2.

in the formula above. However, it is straightforward to showthatλψ = λ1eiπ/Ly with the phase factoreiπ/Ly contributed bythe edge state fermion excitationf †Lk.

A honeycomb lattice on the cylinder can be defined withdifferent orientations. Two different simple orientations areknown as the “armchair edge” and ”zigzag edge” in thegraphene literature, as are illustrated in Fig. 3 (a). We will fo-cus on the zigzag edge, which is convenient for the reason thatwill be clear in later part of this section. The numerical resultsof λσ andλ1 are shown in Fig. 4. To compare with the formula(1), we defineθa(Ly) = Im logλa and do a linear fitting ofLyθa

versusL2y. Eq. (1) predictsLyθa(Ly) = −ImαL2

y+2πpa, so thatthe intercept of this fitting atLy = 0 gives the numerical valueof ha andc. Fig. 4 (a) shows 2πpa = Lyθa(Ly) + ImαL2

y withthe slope Imα obtained from linear fitting. The agreementwith formula (1) is apparent. As is shown in Fig. 4 (b), the

deviationδθa = θa(Ly) −[

−ImαLy +2πLy

pa

]

from the theoret-

ical value decays exponentially versusLy, in agreement withthe error estimation given by the CFT in Eq. (11).

An important question is how robust is the formula (1).Since the derivation given in Sec. (II) is based on the CFTdescription of the edge states, it is not obvious whether thedeviation of the edge state from CFT behavior will lead tocorrections to the universal 1/Ly term in Eq. (1). Interest-ingly, the numerical result above shows that the formula ap-plies to the lattice model without any correction to the 1/Ly

term. To probe the robustness of the formula (1), we can mod-ify the Hamiltonian near the cut between left and right parts

6

(a)

−1 0 1

−1

−0.5

0

0.5

1

k/π

Ek

(b)

0 20 40 60 80−0.5

0

0.52π

24

−2π

48

Ly

Ly(θ

a+

αL

y)

(c)

0 20 40 60 80−25

−20

−15

−10

−5

0

Ly

log|δ

θ a|

(d)

FIG. 5. (a) The Hamiltonian of the Kitaev model is modified byturning off the second neighbor hoppings (purple dotted lines witharrows) along the boundary between left and right parts of the cylin-der (purple dashed line). (b) The entanglement spectrum forthisgeometry, with quartic dispersing entanglement edge states. (c) Thevalue ofLy

(

θa(Ly) + αLy

)

and (d) the deviation ofθa from the theo-retical value for the modified geometry. The parameters are the sameas in Fig. 4.

of the cylinder. As is shown in Fig. 5 (a), we turn off thesecond neighbor hopping along the sites that are at the bound-ary between left and right parts. Such an edge term does notchange the topology of the system, but changes the entangle-ment spectrum. As is shown in Fig. 5 (b), the edge stateentanglement spectrum has a cubic dispersionEk ∝ k3 in thelong wavelength limit, rather than linear. Therefore even inthe long wavelength limit the entanglement HamiltonianHE

is not a chiral conformal field theory. It is reasonable to be-lieve that the physical edge (left edge) is unaffected by such achange of dispersion, since it has zero entanglement temper-ature and stays in the ground state of the topological sector.However the entangled edge (right edge) is at finite entan-glement temperature, so that its contribution to momentumshould be modified when the dispersion is changed. To oursurprise, the numerical results shown in Fig. 5 (c) (d) clearlyshows that formula (1) still applies to this case. Such a robust-ness beyond edge CFT description indicates that the quantiza-tion of momentum polarization has a deeper origin from bulktopology.

IV. PROJECTED WAVEFUNCTIONS FOR FRACTIONALCHERN INSULATORS

In this section we will calculate the topological spinand chiral central charge for aν = 1/2 lattice Laugh-lin state35–37. These lattice Laughlin states, also known asthe fractional Chern insulators, are obtainable from partonconstructions38,39, and we develop a Monte Carlo approachfor calculating their momentum polarization. Based on Eq.(1), our numerical results for topological spin and chiral cen-

tral charge are obtained for theν = 1/2 lattice Laughlin state,which agree very well with the theoretical predictions fromCFT. This suggests that momentum polarization is a veryuseful quantity for identifying chiral topological order fromground-state wave functions of interacting systems.

The fractional Chern insulators are lattice analogies of frac-tional quantum Hall states21–24. When bosons or fermionsfractionally fill a nearly flat band with nonzero Chern number,incompressible quantum liquid states with fractional quan-tized Hall conductance can appear as stable ground states ofa repulsive interaction Hamiltonian. In this work, we focuson a particular example of fractional Chern insulator, i.e.aν = 1/2 lattice Laughlin state of hardcore bosons. Let usconsider aN × N square lattice with single parton sites. Toconstruct aν = 1/2 lattice Laughlin state, we use the par-ton construction in Ref.15,40 and split the hardcore boson ateach site into two fermionic partons,b†i = c†i↑c

†i↓. The physi-

cal Hilbert space of hardcore bosons requires either two par-tons or no parton at each site, corresponding to the pres-ence or absence of hardcore boson, respectively. To builda hardcore boson wave function from partons, a projectorPG is needed to remove those unphysical configurations withsingle-parton sites. Then, a hardcore boson state can be ob-tained by acting the projectorPG on the parton wave function|Φa〉 = PG |φa〉↑ |φa〉↓.

Now we assume that the fermionic partons are in the groundstate of the following Chern insulator (See Fig. 6)

HC =∑

〈i j 〉,σti j c†iσc jσ + i

∑

〈〈ik〉〉,σ∆ikc†iσckσ (20)

Here the nearest-neighbor hopping integralsti j are equal tot along x direction and take the valuet (−t) in odd (even)columns alongy direction. The next-nearest-neighbor hop-ping integrals∆i j are∆ (−∆) if the hopping direction is along(against) the arrow in Fig. 6. In this work we focus ont = 1and∆ = 1/2. For periodic boundary conditions along bothdirections, the Hamiltonian (20) has two bands with ChernnumberC = ±1. At half filling, the ground state of par-tons has a completely filled lower band. After projectingthis parton band insulator state onto hardcore bosons, it hasbeen shown40 that the resulting projected wave function is aν = 1/2 lattice Laughlin state with two-fold degenerate on atorus, and its topological sectors are related with the bound-ary conditions of partons: the identity sector (semion sec-tor) is given by|Φ1 >= |0, 0 > +|π, 0 >= |0, π > +|π, π >

(|Φs >= |0, 0 > −|π, 0 >= |0, π > −|π, π >), where|Ψx,Ψy >

is the projected wavefunction with periodic (Ψx = 0) or anti-period (Ψx = π) parton boundary conditions alongx and sim-ilarly Ψy = 0, π alongy.

The parton construction of theν = 1/2 lattice Laughlinground states on a cylinder has further complication due to itsgapless chiral edge modes. To shed some light on the two-fold degenerate ground-state projected wavefunctions, imag-ine that we start on a torus and adiabatically lower all hop-ping amplitudes across thex boundary until they are muchsmaller than the chiral edge modes’ finite size gap propor-tional to L−1. As such a process involves no gap closing orlevel crossing, the wavefunction’s topological properties, es-

7

A B A B

t

t

-t

x

y

FIG. 6. Illustration of the Chern insulator Hamiltonian (20). Thetwo sublattices in the unit cell are labeled by A and B. The nearest-neighbor hopping integrals aret alongx direction and odd columnsalongy direction, and−t along even columns (dashed lines) alongydirection. The next-nearest-neighbor hopping integrals are∆ (−∆) ifthe hopping direction is along (against) the arrow.

pecially the eigenvalue of Wilson loops alongy identifyingthe topological sectors should not change. Consider the spe-cial case when the chiral edge modes on the two edges ofthe cylinder have one exactly zero energy state each:cσL andcσR. On comparison with the openx boundary condition case,the small inter-edge hopping amplitudes will not modify thechiral edge modes, except for the coupling of the two zeroenergy statescL andcR in the form ofδ

(

∑

σ c†σLcσR + h.c.

)

.Hereδ > 0 ( δ < 0) is remnant from periodic (anti-periodic)boundary condition alongx, respectively. In this way, the de-generacy betweencL andcR is lifted, and the correspondingprojected wavefunctions by filling all negative energy par-ton states are

∣

∣

∣0,Ψy

⟩

= PG

(

c†L↑ − c†R↑) (

c†L↓ − c†R↓)

|Φ′〉 and∣

∣

∣π,Ψy

⟩

= PG

(

c†L↑ + c†R↑) (

c†L↓ + c†R↓)

|Φ′〉, respectively, where

|Φ′〉 =∏

ǫα<0 c†↑(α)c†↓(α)|0〉 is obtained with all other partonstates deeper in the valence bands filled. As previous knowl-edge about the torus geometry case would imply, the two topo-logical sectors on a cylinder correspond the symmetrizationand anti-symmetrization of theδ > 0 andδ < 0 cases, whichgive:

|Φ1 > = PG

(

c†L↑c†L↓ + c†R↑c

†R↓

) ∣

∣

∣Φ′⟩

= PGc†L↑c†L↓

∣

∣

∣Φ′⟩

(21)

|Φs > = PG

(

c†R↑c†L↓ + c†L↑c

†R↓

) ∣

∣

∣Φ′⟩

= PGc†L↑c†R↓

∣

∣

∣Φ′⟩

(22)

In the last step of both equations we have used the particlehole symmetry of the Hamiltonian (20).|Φa〉 , a = 1, scan bewritten in the equivalent form of|Φa〉 = PG |φa〉↑ |φa〉↓, withthe two parton wavefunctions|φa〉↑,↓ defined by

|φ1〉σ = c†Lσ∏

ǫα<0

c†σ(α)|0〉 (23)

|φs〉σ =c†Lσ + ic†Rσ√

2

∏

ǫα<0

c†σ(α)|0〉 (24)

with σ =↑, ↓.

FIG. 7. Upper: TheLyθa data of variousL2y values for the identity

sector (blue crosses) and the quasiparticle sector (red dots) of a frac-tional Chern insulator. The intercept of the linear fits givethe valueof 2π(c/24 − ha). For the identity sector, the data is well consis-tent with linear fit (blue solid line) and suggestsc = 1.078± 0.091;lower: the difference inLyθa corresponds to the measured values ofhs = (Lyθ1 − Lyθs)/2π = 0.2521± 0.0063, the dotted line is the theo-retical valuehs = 1/4.

The identification of topological sectors above is furtherconfirmed by using a Monte Carlo algorithm16 to calculatethe overlap between a series of projected wavefunctions of theform

∣

∣

∣Φ(p,q)

⟩

= PG

∣

∣

∣φ(p,q)

⟩

↑

∣

∣

∣φ(p,q)

⟩

↓, with

∣

∣

∣φ(p,q)

⟩

σ=

(

pc†Lσ + qc†Rσ)∏

ǫα<0

c†σ(α)|0〉 (25)

and p, q complex coefficients. The over-lap of the states with trial values (p, q) =

(0, 1), (1, 0), (1/√

2,±1/√

2), (1/√

2, i/√

2) confirmsthat the only two linearly independent basis states arethe two topological ground states (p, q) = (1, 0) and(p, q) = (1/

√2, i/√

2), corresponding to Eq. (23) and (24).To develop a Monte Carlo method to compute the momen-

tum polarization, we note that the expectation value of the

8

twist operatorTLy for |Φa〉 can be express as

〈Φa|TLy |Φa〉 =

∑

α

|〈α|Φa〉|2〈Φa|TL

y α〉〈Φa|α〉

(26)

where |α〉 corresponds to a lattice configuration withN2/2hardcore bosons, and|TL

y α〉 a configuration with the posi-tions of hardcore bosons at the left half of the cylinder be-ing translated by one lattice spacing iny. Based on Eq.(26), 〈Φa|TL

y |Φa〉 can be straightforwardly evaluated by treat-ing |〈α|Φa〉|2 as the probability for Monte Carlo updates and〈Φa|TL

y α〉/〈Φa|α〉 as the Monte Carlo measurable. For thepresentν = 1/2 Laughlin state wavefunctions, these quanti-ties can be expressed in terms of determinant squares and up-dated efficiently with the inverse-matrix update techniques41.We emphasize that this Monte Carlo approach is also appli-cable for other chiral topological states from fermionic partonconstructions42, whose wavefunctions take the form of deter-minants and/or Pfaffians.

The results from this Monte Carlo method are shown in Fig.7. According to Eq. 1, the intercept of the linear fit ofLyθa

versusL2y for the identity sector corresponds to the measured

central charge of the systemc = 1.078± 0.091, whereas thedifference inLyθa between the two sectors corresponds to 2πh,so the measured topological spin ishs = 0.252±0.006. Theseare well consistent with the known theoretical values for themodelc = 1 andh = 1/4, and suggest Eq. (1) is also applica-ble to the parton constructed systems.

V. DISCUSSION AND CONCLUSION

In summary, we have proposed a new efficient method forcomputing the quasi-particle topological spin and chiral cen-tral charge for chiral topologically ordered states. By defin-ing the partial translation operatorTL

y which translates the leftpart of the cylinder by one lattice constant, the combinationof topological spinha and chiral central chargec can be ex-tracted from a universal term2πLy

(

ha − c24

)

in the phase of the

ground state average values ofTLy in each topological sector.

ha − c24 is the momentum polarization of the ground state in

the topological sectora. We have derived the formula of cen-tral charge and topological spin (Eq. (1)) from the edge CFTform of the reduced density matrix, and also numerically ver-ified the formula in two distinct topological states, the honey-comb lattice Kitaev model and theν = 1/2 bosonic Laugh-lin state in fractional Chern insulators. Compared to previousnumerical approaches which computes braid matrix of quasi-particles13,14 and modularS,T matrices15–17, the method wepropose is more efficient.

It is interesting to note that the momentum polarization de-fined on cylinder may be related to the characterization ofone-dimensional symmetry protected topological states. If weview the cylinder as a one-dimensional (1D) chain, with thecylinder direction as an “extra-dimension”, the system canbe viewed as a one-dimensional system with aZN discretesymmetry. HereZN is the translation alongy direction with

N = Ly. In 1D there is no intrinsic topological state but thereare symmetry-protected topological states, which have non-trivial edge states at each end of the 1D chain protected byglobal symmetries of the system. The type of topological statefor a given symmetry groupG is classified by the inequivalentprojective representations ofG, or equivalently by the secondcohomology of the groupH2(G,U(1))43–45. When the edgestate carries a nontrivial projective representation of the groupG, there will be a nontrivial series of group operationsgi suchthatgNgN−1...g1 = 1 but the sequential action ofgi on the edgestate leads to a nontrivial phase factoreiθ. This phase factorwill be canceled the other edge so that the whole system is stillinvariant inG. Compared with the momentum polarizationwe see that the idea is very similar. A phaseθ = 2π

(

h− c24

)

is obtained by the left edge upon a 2π rotation of the cylin-der, which is canceled by the right edge. However, the groupcohomology ofZN (and its continuum limitU(1)) is trivial.46

Therefore if only the translationZN symmetry is concerned,the projective representation labeled by the fractional momen-tum h − c

24 can be linearized, and no nontrivial classificationis obtained. More information is required besides translationsymmetry to give a fundamental reason for the accurate quan-tization ofha, c in lattice models that is observed in the currentwork. Physically, the missing information is probably relatedto the locality of the model in 2D, which distinguishes a 2Dmodel on the cylinder from a more generic 1D model withZN

symmetry.

In the two example systems studied in this paper, the groundstates in each topological sector is obtained from the knowl-edge to these states. However, it should be noted that in gen-eral our method does not require knowledge on topologicalsectors. If|Gα〉 , α = 1, 2, ..., ndenote then orthogonal groundstates on the cylinder in an unknown basis,|Gα〉may be super-positions of different topological sectors|Ga〉. An importantobservation is that the partial translationTL

y does not changethe topological sector, so that〈Ga|TL

y |Gb〉 ∝ δab is diagonal.Physically, the reason is that topological sectora cannot bemeasured by local probes but can only be measured by cre-ating a pair of quasi-particles, taking one of them around thecylinder and then reannihilate them. We can choose the pathof the quasi-particle so that it’s far from the boundaries ofthe cylinder and the partition in the middle. As long as thisis satisfied, such a quasi-particle motion is unaffected by theoperation of partial translationTL

y . Therefore the topologicalsector has to remain the same after partial translation. Con-sequently, in a generic basis one just needs to calculate thematrixΛαβ = 〈Gα|TL

y

∣

∣

∣Gβ

⟩

and diagonalize it. The eigenval-ues areλa with the form of Eq. (1).

Besides the examples given in the current work, the mo-mentum polarization can be applied a wide range of othertopological states. The Monte Carlo method of computingthe momentum polarization discussed in Sec. IV generallyapplies to other projected wavefunctions obtained by vari-ous parton constructions. In particular, this approach canbe used to detect topological spin and chiral central chargein non-Abelian chiral topological states, which will be help-ful in identifying the topological order in more complicated

9

many-body wave functions. For instance, natural candidatesfor testing the momentum polarization scheme include SU(n)k

states16 and SO(2n + 1)1 states47, whose trial wavefunctionsand chiral edge CFTs are known. The momentum polarizationresults of these non-Abelian states will be reported in futureworks.

Our approach may also have interesting applications intopological states described by density matrix renormaliza-tion group (DMRG)48. DMRG applies to all gapped 1D stateswell, which is equivalent to finding a variational ground statewith the form of matrix product states (MPS)49. MPS refersto states of the form|Ψ〉 =

∑

{νi }[

Tν11 Tν2

2 ...TνNN

]

αβ|{νi}〉, with νi

labels the physical states at each site, andT iνi

a matrix with“internal” indicesα, β. The DMRG/MPS approach has beenapplied to 2D topological states on a cylinder50–53, which canbe viewed as a 1D gapped state with a large number of statesat each site, or one with a finite but long range interaction.Recently, the infinite DMRG, or infinite MPS, description hasbeen studied for FQH states54 and FCI states17, which sim-plifies the variational state by assuming a translation invari-ant ansatz and taking the infinite length limit. Besides theMPS states obtained numerically, the model wavefunctionsof FQH states obtained from CFT correlation functions havealso been shown to have the infinite MPS form when definedon a cylinder55,56 (with infinite internal dimension). The infi-

nite MPS is particularly suitable for the study of the momen-tum polarization we propose, since it provides a description ofthe reduced density matrix using the internal auxiliary space(called the Schmidt states). In the long cylinder limit, themo-mentum polarization can be directly computed from the ma-tricesTµ at each site. An interesting open question is whetherit is possible to obtain a generic proof of the universal quanti-zation of the momentum polarizationha− c

24 from the infiniteMPS approach, which, if exists, will be more rigorous thanthe CFT approach taken in the current work. Another inter-esting direction is to generalize the infinite MPS approach ofRef. 54 to cylinder FQH to FCI states by using the Wannierstate representation57–59.

We would like to note that Ref. 17 has studied the modularS,T matrices by a similar approach as Ref. 15, and Ref. 54has independently studied the topological spin in an approachsimilar to ours, but with a continuous instead of discrete twistoperator which is specially defined for cylinder FQH states.

Acknowledgement. We would like to acknowledge thehelpful discussions with Meng Cheng, J. Ignacio Cirac, Tay-lor L. Hughes, Anne E. B. Nielsen, Frank Pollmann, GermanSierra, Zhenghan Wang and Edward Witten. This work is sup-ported in parts by the EU project AQUTE (HHT), the Stan-ford Institute for Theoretical Physics (YZ) and the NationalScience Foundation through the grant No. DMR-1151786(XLQ).

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Appendix A: Reduced density matrix and momentum polarization of Kitaev model

We first review the structure of the reduced density matrix ofKitaev model obtained in Ref.30. Without losing generality, in thefollowing we assumeLy to be even. The first step is to separate the degrees of freedomto the left and right region. In the enlargedHilbert space, the Majorana fermionsηi and most link variablesui j are already separated into the two regionsL andR, and theonly degrees of freedom that need special treating is the link variablesui j on the links crossing the boundary. By pairing up theneighbor links across the boundary, we can redefine the link variables as is shown in Fig. 8 In the 4 boundary sites 1, 2, 3, 4, twolink variables are defined asu12 = iγz

1γz2, u34 = iγz

3γz4. Now we define the new link variablesw13 = iγz

1γz3, w42 = iγz

4γz2. It is easy

to see thatwi j ’s commute with each other and they both anticommute withui j ’s. Also w13w42 = u12u34. In the 4-dimensionalHilbert space of the 4 Majorana fermions (before projection), choosing the eigenstates ofu12, u34 we can find a basis in whichu12 = σ

z ⊗ 1, u34 = 1⊗ σz, w13 = σx ⊗ σx, w42 = −σy ⊗ σy. Therefore the eigenstates ofw13,w42 are

|+−〉 = 1√

2(|↑↓〉 + |↓↑〉) , |++〉 = 1

√2

(|↑↑〉 + |↓↓〉)

|−−〉 = 1√

2(|↑↑〉 − |↓↓〉) , |−+〉 = 1

√2

(|↑↓〉 − |↓↑〉) (A1)

|↑↑〉 = 1√

2(|++〉 + |−−〉) (A2)

with the two± labeling the eigenvalues ofw13,w24 respectively, and↑, ↓ labels those ofu12, u34. A state∣

∣

∣

∣

{

ui j

}⟩

with fixed valuesof ui j on all links can be written in the new basis. For simplicity wecan always make a gauge choice so thatui j = 1 for all linkscrossing the boundary. For such

{

ui j

}

, we have

∣

∣

∣

∣

{

ui j

}⟩

= 2−Ly/4∑

wi j=±1

∣

∣

∣

∣

{

uLi j ,wi j

}⟩

L⊗

∣

∣

∣

∣

{

uRi j ,wi j

}⟩

R(A3)

A physical ground state has the form

|G〉 = P(∣

∣

∣

∣

GF

({

ui j

})⟩

⊗∣

∣

∣

∣

{

ui j

}⟩

)

(A4)

with P =∏

iDi+1

2 . We write theηi fermion ground state∣

∣

∣

∣

GF

({

ui j

})⟩

in the Schmit decomposition form

∣

∣

∣

∣

GF

({

ui j

})⟩

=∑

N

αN

∣

∣

∣

∣

ΨLN

({

uLi j

})⟩

∣

∣

∣

∣

ΨRN

({

uRi j

})⟩

(A5)

with∣

∣

∣ΨLN

⟩

and∣

∣

∣ΨRN

⟩

a set of orthogonal basis of left and right sideη fermions respectively, andαN the Schmit eigenvalues

satisfying∑

N |αN |2 = 1. Such a Schmit decomposition for quadratic fermion problem can be reduced to a simple single-particle

11

problem34, the detail of which will be discussed later in this appendix. Thus the ground state is written as

|G〉 = 2−Ly/4∑

wi j=±1

∑

N

αNPL

[∣

∣

∣

∣ΨL

N

({

uLi j

})⟩

∣

∣

∣

∣

{

uLi j ,wi j

}⟩

L

]

⊗ PR

[∣

∣

∣

∣ΨR

N

({

uRi j

})⟩

∣

∣

∣

∣

{

uRi j ,wi j

}⟩

R

]

(A6)

in which we have writtenP = PLPR since the projection is local and separable to two subsystems. If we expandPR =∏

i∈RDi+1

2 ,there are two terms which preserve the gauge field configurationsuR

i j ,wi j , which are 1 and∏

i∈R Di . The second term does agauge transformation to all sitesηi → −ηi but preserves all the link variables. Therefore we can write

PR

[∣

∣

∣

∣

ΨRN

({

uRi j

})⟩

∣

∣

∣

∣

{

uRi j ,wi j

}⟩

R

]

=∑

{

u′Ri j ,w′i j

}

≃{

uRi j ,wi j

}

1+∏

i∈R Di

2

∣

∣

∣

∣

ΨRN

({

u′Ri j})⟩

∣

∣

∣

∣

{

u′Ri j ,w′i j

}⟩

R(A7)

and similar forPL. Here≃ denotes gauge equivalence. For two statesPR

[∣

∣

∣

∣ΨR

N

({

uRi j

})⟩

∣

∣

∣

∣

{

uRi j ,wi j

}⟩

R

]

with different boundary

configurationswi j , after the projection they are still orthogonal since it is not possible to do a gauge transformation towi j

without affectingui j on neighboring links. Also the number of orthogonal states in Eq. (A7) is the same for all configurations.Therefore the reduced density matrix is

ρL = Const. ·∑

wi j=±1

∑

N

|αN |2 PL

[∣

∣

∣

∣

ΨLN

({

uLi j

})⟩

∣

∣

∣

∣

{

uLi j ,wi j

}⟩

L

] [

⟨

ΨLN

({

uLi j

})

∣

∣

∣

∣

⟨{

uLi j ,wi j

}

∣

∣

∣

∣

L

]

PL

= Const. ·∑

{

u′Li j ,w′i j

}

,{

u′′Li j ,w′′i j

}

≃{

uLi j ,wi j

}

∑

N

|αN |21+

∏

i∈L Di

2

∣

∣

∣

∣

ΨLN

({

u′Li j})⟩

∣

∣

∣

∣

{

u′Li j ,w′i j

}⟩

L

⟨

ΨLN

({

u′′Li j})

∣

∣

∣

∣

⟨{

u′Li j ,w′′i j

}

∣

∣

∣

∣

L

1+∏

i∈L Di

2(A8)

The translation operatorTLy translates bothηi andui j ,wi j . The global projection1+

∏

i∈R Di

2 only cares about the total fermionnumber parity, and commutes withTL

y . Thus

λ ≡ Tr(

TLy ρL

)

= Const. ·∑

wi j=±1

∑

{

u′Li j ,w′i j

}

≃{

uLi j ,wi j

}

∑

N

|αN |2⟨

ΨLN

({

Tu′Li j})

∣

∣

∣

∣

1+∏

i∈L Di

2TLη

y

∣

∣

∣

∣

ΨLN

({

u′Li j})⟩

(A9)

HereTLηy denotes the action ofTL

y to η subspace, andTu′Li j is configuration obtained from the translation ofu′Li j by one lattice

constant. For all gauge equivalent configurations{

u′Li j}

, the matrix element in Eq. (A9) has the same value, since the gauge trans-formations tou′ andTu′ cancels each other after being acted byTL

y . Therefore we can just take the representative configurationui j and write

λ ≡ Tr(

TLy ρL

)

= Const. ·∑

wi j=±1

∑

N

|αN |2⟨

ΨLN

({

TuLi j

})

∣

∣

∣

∣

1+∏

i∈L Di

2TLη

y

∣

∣

∣

∣

ΨLN

({

uLi j

})⟩

(A10)

Now we do some more analysis toDi . We can write∏

i∈LDi = Fη

∏

〈i j 〉∈Lui j

∏

i j

wi j (A11)

with Fη the fermion number parity ofη fermions. To write down this formula some suitable orientation convention should bemade to the links sinceui j = −u ji , but the detail of this convention is not important here. Theprojector 1+

∏

i∈L Di

2 imposed theconstraint

∏

i∈L Di = 1 which requiresFη =∏

〈i j 〉∈L ui j∏

i j wi j . Therefore in the sum over 2Ly/2 possible configurations ofwi j ,

half of them has the fermion projected toFη = 1 and the other half projected toFη = −1. Since the Schmit state∣

∣

∣

∣

ΨLN

({

uLi j

})⟩

isindependent from the boundary variablewi j , the sum over the two projections simply give

λ =∑

N

|αN |2⟨

ΨLN

({

TuLi j

})

∣

∣

∣

∣TLη

y

∣

∣

∣

∣ΨL

N

({

uLi j

})⟩

(A12)

Defining the gauge transformationV({

uLi j

})

by

V({

TuLi j

}

,{

uLi j

})

∣

∣

∣

∣

ΨLN

({

uLi j

})⟩

=

∣

∣

∣

∣

ΨLN

({

TuLi j

})⟩

(A13)

and the covariant translation operator

TLFy = V†

({

TuLi j

}

,{

uLi j

})

TLηy (A14)

12

we have

λ = Tr(

TLFy ρLF

)

(A15)

with ρLF =∑

N

|αN |2∣

∣

∣

∣

ΨLN

({

uLi j

})⟩ ⟨

ΨLN

({

uLi j

})

∣

∣

∣

∣

(A16)

To see the effect ofTLFy more explicitly, we discuss the periodic and antiperiodic sectors separately. For the periodic sector

with∏

〈i j 〉 ui j = 1 around the cylinder, we can simply take the gauge choiceui j = 1 for all links. Since this gauge is alreadytranslation invariant,TLF

y is the ordinary translation operator. The momentum eigenstatesfnk has the wavefunction

fnk =∑

i

φnk(ix)e−ikyiηi (A17)

with k = 2πLy

n, n ∈ Z, and (ix, iy) the two-dimensional coordinate ofi site. The translation operator acts as

TLFy−1ηiT

LFy = ηi+y, TLF

y−1

fnkTLFy = eik fnk (A18)

The anti-periodic sector is a bit more complicated. We can choose a gauge withui j = −1 on all links across a horizontal line,as is shown by the black solid line in Fig. 8 (b), andui j = 1 on all other links. For convenience we label the horizontalchainsparallel to the branchcut line byn = 1, 2, ..., Ly as is illustrated in Fig. 8 (b) by the blue solid line. We choose the labeling suchthat the branchcut line is at the boundary, betweenn = 1 andn = Ly. In this gauge, the gauge transformationV

({

TuLi j

}

,{

uLi j

})

needs to translate the branchcut line by one lattice constant, which is defined by

VηiV† = −ηi , ∀ i ∈ chain 1 (A19)

Now if we define

fnk =∑

i

φnk(ix)e−ikyiηi (A20)

with yi ∈ [1, Ly] andk = 2πLy

(

n− 12

)

, n = 1, 2, ..., Ly, the functione−ikyi has a jump betweenyi = 1 andyi = Ly. Therefore

TLηy−1

fnkTLηy =

∑

ix

Ly∑

iy=2

φnk(ix)e−ik(yi−1)ηi + φnk(ix)e−ikLyη(ix,1)

= eik∑

ix

Ly∑

iy=2

φnk(ix)e−ikyiηi − φnk(ix)e−ikη(ix,1)

(A21)

Thus we see that the gauge transformationV correctly removes the additional minus sign and obtain

TLFy−1

fnkTLFy = eik fnk (A22)

Due to Eq. (A18) and (A22) we see thatTLFy in the particular gauge choice can be written in the form of Eq. (18)

TLFy = exp

i∑

n,k

k f†nk fnk

(A23)

for both periodic and anti-periodic flux sectors. Using the fact thatf †nk fnk has eigenvalues 0, 1 we can write

TLFy =

∏

nk

exp[

ik f †nk fnk

]

=∏

nk

[

f †nk fnkeik +

(

1− f †nk fnk

)]

(A24)

Using this equation and the entanglement Hamiltonian (17),it is straightforward to obtain Eq. (19).As the last part of this appendix we briefly review the free fermion reduced density matrixρLF calculated in Ref.34. All

multi-point functions〈G|∏2N

i=1 ηi |G〉 of the free fermion ground state satisfies the Wick theorem, so that all correlation functionsare determined by the two-point function

Ci j = 〈G| ηiη j |G〉 − δi j (A25)

13

FIG. 8. (a) The definition of boundary variableswi j on four sites 1,2, 3,4. The boundary is drawn as the purple vertical dashed line. (b)Illustration of the gauge choice for anti-periodic boundary condition, with allui j = 1 except those crossing the branchcut line (thick black solidline). The sites on the lattice are organized into chains parallel to the branchcut line, shown by the blue thick solid lines. The chain labeling isgiven on the left.

In particular,Ci j for i, j ∈ L in the left half cylinder determines all multi-point correlation functions in the left half cylinder.Therefore the reduced density matrix must i) also satisfy Wick theorem, and ii) reproduces the two-point functions. Thefirstcondition requiresρLF to have the quadratic (thermal) form

ρLF = e−HE = exp

−12

∑

i, j∈Lhi jηiη j

(A26)

If we diagonalizeHE into the form given in Eq. (17)

HE =∑

n,k

ξnk

(

f †nk fnk −12

)

(A27)

the correlation function will be

Tr(

ρLF f †nk fnk

)

=1

eξnk + 1(A28)

and corresponding Tr(

ρLF fnk f †nk

)

, which should agree with the eigenvalues ofCi j with the same eigenvectors. Therefore in thematrix level we have

C = tanhh ⇒ h =12

[

log(I +C) − log(I −C)]

(A29)

This equation determinesh and its spectrumξnk from correlation functionC.