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Identifying non-Abelian topological ordered state and transition by momentum polarization

Yi Zhang and Xiao-Liang QiDepartment of Physics, Stanford University, Stanford, California 94305, USA

(Dated: April 18, 2018)

Using a method called momentum polarization, we study the quasiparticle topological spin and edge-statechiral central charge of non-Abelian topological ordered states described by Gutzwiller-projected wave func-tions. Our results verify that the fractional Chern insulator state obtained by Gutzwiller projection of two partonsin bands of Chern number 2 is described byS U(2)2 Chern-Simons theory coupled to fermions, rather than thepureS U(2)2 Chern-Simons theory. In addition, by introducing an adiabatic deformation between one Chernnumber 2 band and two Chern number 1 bands, we show that the topological order in the Gutzwiller-projectedstate does not always agree with the expectation of topological field theory. Even if the parton mean-field state isadiabatically deformed, the Gutzwiller projection can introduce a topological phase transition between Abelianand non-Abelian topologically ordered states. Our approach applies to more general topologically ordered statesdescribed by Gutzwiller-projected wave functions.

I. INTRODUCTION

Topologically ordered states (TOSs) are unconventionalstates of matter with ground-state degeneracy, elementaryquasiparticle excitations with fractional statistics, and long-range quantum entanglement1. The non-Abelian TOSs are asubcategory of TOSs in which quasiparticles carry nonlocaltopological degeneracy and have received much recent atten-tion due to their potential applications in topological quan-tum computations2–4. The braiding processes of quasiparti-cles within a non-Abelian TO induce noncommuting unitarytransformations in the ground-state space instead of merelyincurring aU(1) phase factor as in the Abelian case. Candi-dates for non-Abelian TOSs include theν = 5/2 andν = 12/5fractional quantum Hall states5, which are proposed to be theMoore-Read state6 and the Read-Rezayi states7.

Unlike conventional states of matter characterized by thesymmetries preserved or those broken spontaneously, TOSsare characterized by topological properties such as ground-state degeneracy and fusion and braiding of topological quasi-particles. Except for some exactly solvable models, mostcandidate systems for TOSs can be studied only by numeri-cal methods such as the density-matrix renormalization group(DMRG)8 and the variational Monte Carlo method9. To de-termine the topological order in a numerically studied sys-tem, it is essential to develop numerical probes of topologi-cal properties. The search for more efficient and general nu-merical methods has attracted much recent attention. Vari-ous methods have been developed to characterize quasiparti-cle statistics based on direct calculation of the Berry phase1,explicit braiding of excitations10 and modular transformationof ground states with minimum entanglement entropy11. Re-cently, an additional approach has been proposed for numeri-cally extracting two topological properties of a given TOS,thetopological spins of quasiparticlesha and the edge-state chiralcentral chargec12. Physically, the topological spin determinesthe phase factorθa = ei2πha obtained by the system when aquasiparticle spins through 2π. The chiral central charge ofthe edge state determines the thermal currentIE =

c6T2 at

temperatureT13. These two quantities are essential in de-termining the TOS. The proposal is based on the concept of

momentum polarizationdefined for cylindrical systems. Fora cylindrical lattice system with periodic boundary conditionalong they direction, one can define a unitary “partial transla-tion operator”TL

y which translates the lattice sites along theydirection by one lattice constant for all sites that are in the lefthalf of the system. For a topological ground state|Φa〉 withquasiparticle typea in the cylinder, the expectation value ofTL

y is proposed to have the following asymptotic form12

λa ≡ 〈Φa|TLy |Φa〉 ≃ exp

[

2πiLy

pa − αLy

]

(1)

whereLy is the number of lattice sites in the ˆy direction,α isa nonuniversal complex constant for the leading contributionand independent of the specific topological sectora, and re-markably, the fractional part of the momentum polarizationpa

has a universal valuepa = ha − c24, which measures the com-

bination of topological spinha (modulo 1) and central chargec (modulo 24). SinceTL

y only acts only on the left half ofthe system, the momentum polarization is a quantum entan-glement property determined by the reduced density matrix ofthe left half of the system. The average valueλa has the meritof being relatively simple to evaluate in comparison with theprevious methods based on entanglement entropy11. The cal-culation of the Renyi entanglement entropy involves a swapoperator and requires a minimum of two replicas of the sys-tem, while for momentum polarization the evaluation ofTL

ydoes not need a replica so the Hilbert space for Monte Carlosampling is much smaller for the same system size. In Ref. 12,the momentum polarization was studied for two simple TOSs,the Laughlin 1/2 state in fractional Chern insulators (the def-inition of which will be given in the next paragraph) and thehoneycomb lattice Kitaev model14. The former is an Abelianstate, while the latter has a special non-Abelian state thatcanbe solved by mapping to free Majorana fermions.

In this paper, we apply the momentum polarization ap-proach to more generic non-Abelian TOSs. More specifi-cally, we study non-Abelian states described by Gutzwiller-projected wave functions15 of fractional Chern insulators(FCIs). An (integer) Chern insulator is a band insulator withnonzero quantized Hall conductance. The Hall conductanceσH = ne2

h carried by an occupied band is determined by a

2

topological invariant of the energy band, known as the ChernnumberC = n. FCIs are generalizations of Chern insula-tors to interacting systems, which have fractional Hall con-ductance and topological order. One way to understand FCIsis through the parton construction, in which the electron isconsidered as a composite particle of several “partons” car-rying fractional quantum numbers. For example, an electroncan be split into three fermionic partons, with each parton inan integer Chern insulator withC = 1. The correspondingelectron state has Hall conductance1

3e2

h and is the13 Laugh-

lin state. Gauge fields are coupled to partons to enforce theconstraint that all physical states are electron states andnoindividual parton will be observed. The parton constructioncan be expressed in ansatz ground state wave functions con-structed by the procedure of Gutzwiller projection15, which isa projection of the parton ground state into the physical elec-tron Hilbert space. Gutzwiller-projected wave functions havebeen constructed for FCI16. When two partons are glued to-gether to form a bosonic “electron”, and each parton is in astate with Chern numberC = 1, from topological effectivefield theory (which we will review later in the paper) one ex-pects to find a 1/2 bosonic Laughlin state. In contrast, if eachparton is in a state with Chern numberC = 2, the resultingelectron TOS is expected to be non-Abelian, related toS U(2)level-2 Chern-Simons (CS) theory6. The non-Abelian natureof this state has been verified by calculation of the modularSmatrix for the projected wave functions17.

In this paper, we study the momentum polarization of theGutzwiller-projected wave function for the state of two par-tons with Chern numberC = 2. In addition to confirming thenon-Abelian topological order of this state, our result containsthe following two points. First, the spin and central chargeob-tained from momentum polarization clearly distinguish tworelated but distinct topological states, theS U(2)2 CS theoryand theS U(2)2 CS theory coupled to fermions18. The particlefusion, braiding, and modularS matrix of these two theoriesare identical, but they are distinct TOSs with different edge-state chiral central chargec = 3

2 andc = 52, respectively. The

momentum polarization calculation clearly demonstrates thatthe Gutzwiller-projected parton wave function has the topo-logical order of the latter theory. Second, there is an ap-parent paradox in the statement that Gutzwiller projectionofpartonC = 2 states leads toS U(2)2 CS theory coupled tofermions. Since Chern number is the only topological invari-ant of a fermion energy band, a Chern numberC = 2 band canbe adiabatically deformed to two decoupledC = 1 bands, aslong as translation symmetry breaking is allowed. Since theGutzwiller projection of twoC = 1 partons is known to givethe Laughlin 1/2 state, it appears that one can adiabaticallydeform the non-Abelian TOS obtained from partons occupy-ing theC = 2 band to the Abelian TOS of two decoupledLaughlin 1/2 states. This is clearly in contradiction with thetopological stability of TOSs. By introducing an explicit adi-abatic deformation between aC = 2 band structure and twodecoupledC = 1 bands, we study the quasiparticle topolog-ical spin during the adiabatic interpolation. Our result showsthat there is a topological phase transition between the Abelianphase of the bilayer Laughlin state and the non-Abelian phase

of theS U(2)2 CS coupled to fermions. The topological phasetransition occurs at afinite coupling between the twoC = 1bands. In other words, the TOS obtained from Gutzwiller pro-jection ofC = 2 parton bands isnot completely determinedby the Chern number of the parton band structure, but maydepend on details of the Chern bands and the projection. Theargument based on parton “mean-field theory”,i.e., integrat-ing over partons to obtain CS gauge theory, may not predictthe correct phase. This example further emphasizes the im-portance of numerical approaches such as momentum polar-ization in identifying TOSs. Based on this numerical obser-vation, we will also discuss theoretically the effective theoryinterpretation of this topological phase transition.

The remaining of the paper is organized as follows: In Sec.II, we present our momentum polarization calculation in theGutzwiller-projected wave function of non-Abelian FCIs, af-ter reviewing the relevant background knowledge. Sec. II Apresents our projective construction and theC = 2 Chern in-sulator model; Sec. II B gives a brief field theory discussionof the corresponding TOS; Sec. II C shows our numerical re-sults from momentum polarization. We obtain the topologicalspin of the non-Abelian quasiparticlehσ = 0.321± 0.013 andthe fermion quasiparticlehψ = 0.520± 0.026 and edge centralchargec = 2.870±0.176, in agreement with theS U(2) CS the-ory coupled to fermions (hσ = 5/16 andc = 5/2). In Sec. III,we introduce the adiabatic deformation between twoC = 1bands and oneC = 2 band, and study the topological phasetransition between the two TOSs. In Sec. III A, we presentan adiabatic interpolation of the parton tight-binding Hamil-tonian. Sec. III B presents the results for the quasiparticletopological spin and ground-state degeneracy which indicatethe transition between the non-Abelian and Abelian TOS; InSec. III C, we discuss the physical interpretation of this topo-logical transition. Finally, Sec IV is devoted to a conclusionfrom our main results and discussion of open questions.

II. IDENTIFYING THE NON-ABELIAN TOS IN C = 2 FCI

A. The projective construction and C = 2 Chern insulatormodel

The projective construction is a powerful formalism foransatz wave functions of many TOS19. For our projectiveconstruction, we first introduce several species of partonsψa

as free fermions in a Chern insulator, and then constrain thepartons to recombine into physical “electrons” (which may bebosons or fermions). In the simple Gutzwiller-projected stateswe will discuss in this work, the projected wave function isdefined in first quantized language byΦ ({zi}) =

∏

aψa ({zi}).

Here{zi} with i = 1, 2, ....,N are the coordinates of all parti-cles, andψa ({zi}) is the wave function of thea-th parton.N isthe number of each parton type, which is the same as the totalelectron number of the system. The properties of the result-ing states can be numerically computed through variationalMonte Carlo calculations.

For our focused non-Abelian TOS, we start with the fol-lowing parton mean-field Hamiltonian on a two-dimensional

3

FIG. 1: An illustration of the hopping Hamiltonian in Eq. 2. Thetwo orbitals on each lattice site are shown as different layers andcolored in black and blue, respectively. The hopping is+1 (−1) alongthe solid (dashed) lines, andi/

√2 (−i/

√2) along (against) the red

arrows.

square lattice

HC=2 =∑

<i j>,I ,s

(−1)I−1c†jIsciIs +∑

<i j>,s

ei2θi j(

c†j2sci1s + c†j1sci2s

)

+1√

2

∑

<<ik>>,s

ei2θik(

c†k2sci1s − c†k1sci2s

)

+ H.C.

=∑

<i j>y,s

[(

c†j1sci1s − c†j2sci2s

)

−(

c†j2sci1s + c†j1sci2s

)]

+∑

<i j>x,s

[(

c†j1sci1s − c†j2sci2s

)

+(

c†j2sci1s + c†j1sci2s

)]

+1√

2

∑

<<ik>>,s

ei2θik(

c†k2sci1s − c†k1sci2s

)

+ H.C. (2)

where I = 1, 2 are the two orbitals on each lattice site ands =↑, ↓ labels the two flavors of partons.θi j is the azimuthalangle for the vector connectingi and j. < i j > and<< ik >>label nearest neighbor and next nearest neighbor links, while〈i j 〉x and〈i j 〉y denote nearest neighbors along the ˆx andy di-rections, respectively, as is illustrated in Fig. 1. A previousstudy17 has shown that at half filling the system is a Chern in-sulator withC = 2. The correlation lengthξ is on the orderof a lattice constant, and therefore the finite-size effects aregenerally suppressed for the system sizes we study.

In real space, the parton wave functionψa ({zi}) is a Slaterdeterminant for a completely filled valence band, wherez =(i, I ) labels both the position and orbital indices of a parton.Next we apply the Gutzwiller projection imposing the con-straintniI ↑ = niI ↓, with niIs = c†iIsciIs the number of partons ateach site and orbital. The states satisfying this constraint havetwo partons bound at each site and orbital, and are physicalelectron states with electron numberne

iI = niI ↑. This leads tothe following many-body wave function

Φ ({zi}) = ψ↑ ({zi})ψ↓ ({zi}) = ψ2↑ ({zi}) (3)

This state is the major focus of the paper. Previously, thethree topological sectors on a torus for this projective con-struction were obtained by tuning the boundary condition ofthe parton mean-field Hamiltonian in Eq. 2 and their con-nection to the corresponding threaded quasiparticle has beenestablished17. For our momentum polarization calculations,

we need to generalize the projective construction to a cylin-der. To resolve the complication from the gapless chiral edgemodes on the open edges, we start from a torus and adiabati-cally lower all hopping amplitudes across the open boundaryuntil they are much smaller than the edge modes’ finite sizegap. The residue hoppings effectively couple only the zeroenergy states atky = ±π/2 on the two edges of the cylinder,therefore the original boundary conditions of topologicalsec-tors on the torus lead to linear combinations of the zero energystates12. Since such a process involves no level crossing, wecan obtain the topological sectors on a cylinder by allowingoccupation of different parton zero-energy states on the twoedges.

B. Topological Field Theory Description

To understand the TOS described by the above projectiveconstruction, we briefly review the topological field theoryde-scription of this state. The electron operator can be expressedin partons asfiI = ciI ↑ciI ↓. This decomposition has anS U(2)gauge symmetry: for anyS U(2) matrix with α, β ∈ C and|α|2 + |β|2 = 1

(

ciI ↑ciI ↓

)

→(

α β

−β∗ α∗

) (

ciI ↑ciI ↓

)

(4)

this transformation preserves the electron operatorfiI →(

αciI ↑ + βciI ↓) (−β∗ciI ↑ + α

∗ciI ↓)

= ciI ↑ciI ↓ = fiI , and there-fore the effective theory of partons should also be gauge in-variant. The simplest possible effective theory satisfying thegauge invariant condition is obtained by a minimal couplingof the mean-field Hamiltonian (2) to anS U(2) gauge field20.A lattice S U(2) gauge field is described by gauge connectioneiai j ∈ S U(2) defined along each linki j . The Hamiltonian iswritten as

He f f =∑

<i j>,I

(−1)I−1eia jisrc†jIsciIr +

∑

<i j>

ei2θi j eia jisr(

c†j2sci1r + c†j1sci2r

)

+1√

2

∑

<<ik>>

ei2θikeiakisr(

c†k2sci1r − c†k1sci2r

)

+ H.C. (5)

wheres, r =↑, ↓ denote the two parton species, and repeatedindices are summed over.

Since the partons are gapped, it is straightforward to inte-grate them out. Due to the Chern numberC = 2 of eachparton band, integrating over the parton results in anS U(2)2non-Abelian CS theory

L = 24πǫµνρtr

[

aµ∂νaρ +23

aµaνaρ

]

(6)

However, it is not accurate to say that the topologicalfield theory describing the TOS of this parton constructionis S U(2)2 CS gauge theory, because the partons have non-trivial contribution to topological properties such as edge the-ory. The edge theory ofS U(2)2 CS theory is a chiralS U(2)2Weiss-Zumino-Witten (WZW) model21,22, while the edge the-ory of the FCI described above consists of four chiral fermions

4

S U(2)2 CS ν = 2 coupled toS U(2)2c 3/2 5/2h1 0 0hσ 3/16 5/16hψ 1/2 1/2D 3 3

TABLE I: Theoretical values of topological properties including theedge central chargec, the topological spins for quasiparticlesh1, hσ,hψ and the ground-state degeneracyD for the pureS U(2)2 CS the-ory and theν = 2 fermions coupled to anS U(2)2 gauge field (orequivalently, U(4)1

S U(2)2theory).

FIG. 2: The partial translation operatorTLy translates the left half

of the cylinder by one lattice constant along the ˆy direction. Thered arrows indicate the chiral edge modes. The topological sectora is determined by the type of quasiparticle threaded throughthecylinder, denoted by the large blue arrow.

(two from each flavor of parton) coupled to theS U(2)2 WZWmodel. Technically, the edge state of fermions coupled tothe WZW model is described by a quotient of two conformalfield theories U(4)1

S U(2)2, in whichU(4)1 describes four free chiral

fermions andS U(2)2 describes the gauge degrees of freedomwhich are removed from physical excitations.18 Although theyboth have three quasiparticles with the same fusion rule andbraiding statistics, these two theories are not topologicallyequivalent. In particular, the topological spin differs by afermionic sign for quasiparticles which correspond to an oddnumber of holes in the parton Chern insulator state. For com-parison purpose, we list the theoretical values for the quasi-particle topological spins and edge central charges for thetwotheories in Table I.

In summary, we have seen that the effective topologicalfield theory analysis suggests that the topological order intheGutzwiller-projected state isU(4)1

S U(2)2instead ofS U(2)2. How-

ever, it is essential to verify that directly for the Gutzwiller-projected wave function, as there is no guarantee that the ef-fect of Gutzwiller projection is completely equivalent to thecoupling to a gauge field in the effective field theory. This isachieved in the next section by studying the momentum polar-ization.

C. Topological spin and edge central charge from momentumpolarization calculations

Quasiparticle braiding from previous studies has deter-mined that the TOS forΦ ({zi}) is necessarily non-Abelian.

However, both theories in Table 1 are consistent with thebraiding, and therefore additional information is necessary tomake a complete identification. We numerically extract thequasiparticle topological spin and edge central charge frommomentum polarization calculations for the model in Eq. 2defined on a cylinder.

Care should be taken about the non-Abelian topologicalsector, which consists of parton states with an overall differ-ence of momentumπ on the left edge. For the expectationvalue of the partial translation operatorTL

y that translates theleft half of the cylinder by one lattice constant along the ˆydirection, see Fig. 2 for illustration, thisπ momentum dif-ference will result in contributions with opposite signs. Toovercome this difficulty, we generalizeTL

y to twist the left halfof the cylinder byl lattice constants, so that the overall phasedifference vanishes for a partial translation ofl = 2 latticeconstants. For this purpose, we takeLy

l to be integer, con-siderl sites along the ˆy direction as one unit cell, and replaceLy by Ly

l in the formula proposed in Ref. 12. Consequently,the average value ofTL

y defined byλa = 〈Φa|TLy |Φa〉 has the

following leading contributions

λa = exp

[

i2πlLy

pa − αLy

l

]

(7)

in which α is a nonuniversal complex constant independentof the specific topological sectora, while pa has a universaltopological valuepa = ha−c/24 determined by the topologicalspinha and the edge central chargec.

The quantity in Eq. 7 can be efficiently evaluated for theprojected wave functions with the variational Monte Carlomethod. For a cylinder withLx = 8, Ly = 16 andTL

y trans-lating the left half byl = 2 lattice constants for the afore-mentioned reason, numerical calculations yield arg(λ1) =−3.4449±0.0063 for the identity sector, arg(λσ) = −3.1929±0.0082 for the sector associated with the non-Abelian quasi-particle, and arg

(

λψ)

= −3.0366± 0.0257 for the fermionsector. Withh1 = 0 by definition of the identity particle, weobtain

hσ =Ly

2πl[

arg(λσ) − arg(λ1)]

= 0.321± 0.013 (8)

hψ =Ly

2πl

[

arg(

λψ)

− arg(λ1)]

= 0.520± 0.026 (9)

This is fully consistent with the theoretical value ofhthσ =

5/16 = 0.3125 for the non-Abelian quasiparticle andhthψ =

1/2 = 0.5 for the fermion quasiparticle of a theory ofν = 2fermions couple to anS U(2) gauge field .

In addition, we calculateλ1 for Lx = 8, l = 1, andvarious values ofLy. The numerical results are shown inFig. 3. To compare with Eq. 7, note that−Ly arg(λ1) =ImαL2

y − 2πp1, so the intercept of this linear fitting gives thevalue of−2πp1 = 2πc/24 = 0.7513± 0.046. The result-ing value ofc = 2.870± 0.176 is also fairly consistent withthe prediction ofcth = 5/2 according to the theory ofν = 2fermions coupled to anS U(2) gauge field. Although there isa deviation between the numerical value and the theoreticalvalue 5/2 which is probably due to the finite-size effect, the

5

FIG. 3: The value of−Ly arg(λ1) versusL2y for the identity sector

a = 1. The intercept atL2y = 0 of the linear fitting gives−2πp1 =

0.7513± 0.046. We setLx = 8 andl = 1 for all calculations.

accuracy of the result is sufficient to completely distinguishthis system from the bareS U(2)2 CS theory withhσ = 3/16andc = 3/2. This result also provides further evidence thatthe momentum polarization method for computing topologi-cal quantities is applicable to non-Abelian TOSs.

III. THE TRANSITION BETWEEN ABELIAN ANDNON-ABELIAN TOS IN PROJECTED WAVE FUNCTIONS

From the results discussed in the last section, it seems thatthe TOS of the Gutzwiller-projected wave function agreeswell with the expectation from the topological field theoryapproach. However, there is a hidden paradox in this re-sult. Since the Chern number is the only topological in-variant for a generic energy band in two dimensions, a bandwith Chern numberC = 2 is topologically equivalent to twoC = 1 bands. More explicitly, an exact mapping has beenconstructed between aC = 2 band and two decoupled Lan-dau level systems which are related by a lattice translationoperation23–25. Therefore one would naively expect that a statewith each parton in aC = 2 Chern insulator is adiabaticallyequivalent to one in which each parton occupies twoC = 1bands. However, this statement seems to contradict the factthat the Gutzwiller-projected wave function of the latter stateis Abelian. It is known that the Gutzwiller-projected wavefunction of two partons each in aC = 1 band gives a Laughlinν = 1

2 Abelian TOS11,12,16,26–28, which is also denotedS U(2)1Chern-Simons theory. Therefore one would expect that wheneach parton occupies two decoupledC = 1 bands, which canbe viewed as two decoupled layers, the Gutzwiller-projectedwave function of the whole system is simply two copies ofthe Laughlinν = 1

2 state,i.e. S U(2)1 × S U(2)1, which isan Abelian state clearly distinct from theU(4)1

S U(2)2theory we ob-

tained earlier from both effective theory and numerical results.To resolve this apparent paradox, in this section we introducean explicit interpolation between theC = 2 model used inlast section and a model with two decoupledC = 1 bands.By studying the momentum polarization of the correspondingGutzwiller-projected wave functions during this interpolation,we find a topological phase transition between the Abelian andnon-Abelian phases.

A. An adiabatic interpolation of the parent Hamiltonian

As an explicit example of the interpolation between aC = 2band and twoC = 1 bands, we consider the following partonmean-field Hamiltonian on a two-dimensional square lattice29

HΘ =√

2∑

<i j>y,s

[

cosΘ(

c†j1sci1s − c†j2sci2s

)

− sinΘ(

c†j2sci1s + c†j1sci2s

)]

+√

2∑

<i j>x,s

[

sinΘ(

c†j1sci1s − c†j2sci2s

)

+ cosΘ(

c†j2sci1s + c†j1sci2s

)]

+1√

2

∑

<<ik>>,s

ei2θik(

c†k2sci1s − c†k1sci2s

)

+ H.C. (10)

where the label definition is the same as in Eq. 2, andΘ isa continuous parameter. ForΘ = π/4, Eq. 10 returns to theHamiltonian in Eq. 2 with aC = 2 band. ForΘ = 0, theHamiltonian becomes

HΘ=0 =√

2∑

<i j>y,s

(

c†j1sci1s − c†j2sci2s

)

+√

2∑

<i j>x,s

(

c†j2sci1s + c†j1sci2s

)

+1√

2

∑

<<ik>>,s

ei2θik(

c†k2sci1s − c†k1sci2s

)

+ H.C. (11)

The hopping matrix elements are drawn in Fig. 4. Since hop-pings exist only betweenI = 1(I = 2) orbitals on thexi oddsites andI = 2(I = 1) orbitals on thexi even sites, the sys-tem can be directly decomposed into two uncoupled subsys-tems with even and odd values ofxi + I . The two subsystemsare related by a translation by one lattice constant along thex direction. Suppressing the orbital index, each of the twosubsystems has the following Hamiltonian, which is a Cherninsulator withC = 1 for each parton flavors

HC=1 =∑

〈i j 〉,sti, jc

†isc js +

∑

〈〈ik〉〉,s∆i,kc

†iscks+ H.C. (12)

where the nearest neighbor hopping amplitudeti, j is√

2 alongthe x direction and alternates between

√2 and−

√2 along the

y direction, and the next nearest neighbor is∆i,k = i/√

2 alongthe arrow and∆i,k = −i/

√2 against the arrow, see Fig. 5 for an

illustration. The unit cell contains two lattice sites. Therefore,Eq. 10 defines an interpolation between one Chern insulatorwith C = 2 and two decoupled Chern insulators each withC = 1.

It it also verified that the interpolation is adiabatic and theband gap remains finite for allΘ. Actually, the Hamiltonians

6

FIG. 4: An illustration of the hopping Hamiltonian in Eq. 11.Thetwo orbitals on each lattice site are shown in different layers andcolored in black and blue, respectively. The hoppings alongthe solid(dashed) lines are+

√2 (−√

2), and along (against) the red arrowsare i/

√2 (−i/

√2). It is straightforward to separate the system into

two uncoupled zigzag subsystems with odd and even values ofxi + I .

FIG. 5: Illustration of aC = 1 Chern insulator model on a two-dimensional square lattice. The nearest neighbor hopping amplitudesare√

2 along the square edges and−√

2 along the dashed lines. Thenext nearest neighbor hoppings are along the square diagonal withamplitude+i/

√2 along (−i/

√2 against) the arrow. The two lattice

sites in the unit cell are marked asA andB.

with differentΘ can be related by a global unitary transforma-tion on the orbital space

HΘ = U−1H0U

U = exp∑

s

[

Θ

2

(

c†i1sci2s − c†i2sci1s

)

]

(13)

The effect of the rotation on annihilation operators is

U−1

(

ci1s

ci2s

)

U =

(

cosΘ2 − sin Θ2sin Θ2 cosΘ2

) (

ci1s

ci2s

)

(14)

Consequently, the dispersion and band gap are intact with re-spect to the variation ofΘ.

Now we study the Gutzwiller-projected state correspond-ing to the parton mean-field HamiltonianHΘ. We have shownthat HΘ= π

4leads to theU(4)1

S U(2)2state. On the other hand,HΘ=0

describes two decoupled “layers”, each with two partons inC = 1 bands. The Gutzwiller projection also applies sepa-rately to the two layers, so that the resulting state is a decou-pled bilayer of the projectedC = 1 states. The projected wave

S U(2)1 × S U(2)1 CS ν = 2 coupled toS U(2)2c 2 5/2D 4 3h 0,1/4,1/4,1/2 0,5/16,1/2

TABLE II: Theoretical values of topological properties including theedge central chargec, ground-state degeneracyD, and quasiparticletopological spins for theS U(2)1 × S U(2)1 CS theory and theν = 2fermions coupled to anS U(2)2 gauge field.

functions from a Chern insulator withC = 1 have been con-firmed to be consistent with theS U(2)1 CS theory11,12,16,26–28.Correspondingly, the projected wave function of two uncou-pled Chern insulators each withC = 1 should be describableby an AbelianS U(2)1 × S U(2)1 CS theory, which has fourAbelian particles and is clearly distinct from the non-AbelianTOS established forΘ = π

4. There are major differences intheir topological properties including the torus ground-statedegeneracy, edge central charge and quasiparticle topologicalspins, as listed in Table II. Due to this topological differencebetweenΘ = 0 andΘ = π

4 , a topological phase transitionmust occur for some intermediateΘ. Since the parton groundstates before Gutzwiller projection with differentΘ are relatedby a local unitary transformation, one has to conclude that thetopological phase transition is introduced by the Gutzwillerprojection procedure. We study this topological phase transi-tion numerically in the next section.

B. The quasiparticle topological spin as a signature fortopological phase transition

First of all, we would like to determine whether there isa first-order phase transition at someΘ. Even though theinterpolation of the parton ground state before projectionisclearly adiabatic, the same is not necessarily true for the pro-jected wave function. Numerically, forHΘ defined on a sys-tem of sizeLx = Ly = 12 with periodic boundary con-ditions, we study the evolution of the projected wave func-tions with steps ofΘ as small asδΘ = π

400. VariationalMonte Carlo calculations16 indicate that for all values ofΘ ∈[0, π4 ], the overlap between neighboring steps’ wave functions

|〈Φ(Θ + δΘ)|Φ(Θ)〉| = 1 − O(

10−3)

, which clearly suggeststhat 〈δΦ(Θ)|Φ(Θ)〉 → 0 for smallδΘ → 0 and excludes thepresence of singularities. Therefore the quantum phase tran-sition must be continuous..

In particular, the open boundary conditions are equivalentfor the semion sector in the Abelian TOS and the non-Abelianquasiparticle sector in the non-Abelian TOSs, as well as forthe identity sectors in both TOS, making an adiabatic interpo-lation possible within each sector. To determine the topolog-ical phase transition point, we compute the momentum polar-ization with l = 2 for the identity and semion (non-Abelianquasiparticle) sectors of the projected wave functions on acylinder of Lx = 8 andLy = 12, 16 for each interpolation ofEq. 10. The results of topological spinh for the semion (non-

7

FIG. 6: The topological spinh for the semion (non-Abelian quasipar-ticle) sector versus various values ofΘ ∈ [0, π/4] for the projectedChern insulator in Eq. 10 from momentum polarization calculations.The red dashed line and the blue dotted line are the theoretical val-ues ofh for theS U(2)1 × S U(2)1 CS theory (hs = 1/4) andν = 2fermions coupled to anS U(2) gauge field (hσ = 5/16), respectively.

Abelian quasiparticle) sector versusΘ ∈ [0, π/4] are shownin Fig. 6. For small value ofΘ = 0.05π, the topologicalspin starts to deviate from the semionic statistics ofhs = 1/4for the Abelian TOS and evolve towardshσ = 5/16 for thenon-Abelian TOS, see Table II. Still, there is a finite regionof Θ where the value ofh represents an Abelian TOS. Forfurther verification, for a smaller value ofΘ = 0.025π, wenumerically calculated the overlaps between projected wavefunctions of various boundary conditions on anLx = Ly = 12torus16 and find that there are four linearly independent candi-date ground-state wave functions by projective construction,consistent with the AbelianS U(2)1 × S U(2)1 CS theory. Incontrast, for values such asΘ = π/4 andΘ = 3π/8 fully in theparameter region of the non-Abelian topological order, suchlinear independence is only three fold.

Our numerical results show that a topological phase transi-tion occurs at finiteΘ, which is consistent with the fact thattheΘ = 0 Abelian state is topologically stable and shouldpersist for a finite region ofΘ: the fractional Chern insulatoris an intrinsic topological ordered state protected by an exci-tation gap that is stable against small local perturbationsofarbitrary form such as weak couplings between the subsys-tems. Since the two subsystems are coupled for all nonzeroΘ, the mean-field Hamiltonian at nonzeroΘ can be viewedonly as a Chern insulator with aC = 2 band. Therefore thetopological field theory approach will predict that the TOS ofthe system is described byS U(2)2 Chern-Simons theory cou-pled toC = 2 partons, as we discussed in Sec. II. In contrast,our numerical result for smallΘ finds an Abelian TOS, whichprovides a concrete example of a case when the TOS of theGutzwiller-projected wave function is different from the pre-diction of topological field theory.

C. Theoretical interpretation of the topological phasetransition

To understand physically the topological phase transition,we first ask why the derivation of the effective field theory inSec II B does not apply toΘ = 0. For generalΘ, the con-straints on the partons induces anS U(2) gauge field alongall lattice edges in Fig. 1 that dominates the low-energy the-ory after the partons are integrated out. In theΘ = 0 limit,however, the Hamiltonian becomes Eq. 11, and all hoppingsbetween the two subsystems vanish. Therefore there are twowell-definedS U(2) gauge fields in the long wavelength limit,one for each subsystem. As is clear in Fig. 4, these twoS U(2)gauge fields exist on independent pieces and remain indepen-dent after the partons are integrated out. Integrating out theC = 1 band of the parton gives theS U(2) level 1 Chern-Simons theory, so that the topological field theory of theΘ = 0system consists of fermions coupling toS U(2)1 × S U(2)1.

At finite Θ, coupling is turned on between the two effec-tive “layers” and breaks the separateS U(2) × S U(2) gaugesymmetry into one singleS U(2). As an alternative viewof the symmetry breaking, one can carry out the unitaryrotation in Eq. 13 in reverse to transform the Hamilto-nian HΘ back toH0. In the new basis, the partons occupythe two decoupledC = 1 bands before projection. Theonly way the two independent layers are coupled is throughthe constraint. In the original basis the constraint is writ-ten asniI ↑ = niI ↓ (c†iI ↑ciI ↑ = c†iI ↓ciI ↓) in real space. Af-ter the inverse unitary transformation for a finiteΘ, the re-sulting constraints arec†i1↑ci1↑ + c†i1↑ci1↑ = c†i1↓ci1↓ + c†i1↓ci1↓

and cosΘ(

c†i1↑ci1↑ − c†i1↑ci1↑)

+ sinΘ(

c†i2↑ci1↑ + c†i1↑ci2↑)

=

cosΘ(

c†i1↓ci1↓ − c†i1↓ci1↓)

+ sinΘ(

c†i2↓ci1↓ + c†i1↓ci2↓)

. The lat-ter explicitly breaks the intra-layer charge conservationsym-metry of the parent Hamiltonian in Eq.11, defined byc†iIs →e−iφc†iIs, ciIs → eiφciIs, xi + I ∈ odd. As a consequence ofthis inter-layer coupling, the twoS U(2) gauge fields in the ef-fective theory are coupled and only a diagonalS U(2) gaugesymmetry is preserved. Physically, the holes in the twoC = 1bands are no longer distinguishable so that the two semionicquasiparticles originating from the holes in the two bands nowmerge to one particle. Consequently, the ground-state degen-eracy on a torus, effectively labeled by the quasiparticle con-tent, also decreases from four fold to three fold.

The discussion above suggests that the Abelian and non-Abelian phases are distinguished by whether the two layers(in the rotated parton basis) have separately conserved parti-cle numbers. In the Abelian (non-Abelian) phase, the separateparticle number conservation of the two layers is effective pre-served (broken). To verify this scenario, we numerically cal-culate the fluctuations of parton number in one of theC = 1layers (in the rotated parton basis):N1 =

∑

I+xi∈oddc†iI ↑ciI ↑. In

the Θ = 0 limit, the two bands are independent, thereforeN1 = N1 and the fluctuation is exactly zero. AsΘ increases,the intra-band charge conservation is broken, and thereforeone may expect an increase in theN1 fluctuation. Fig. 7 isa histogram of the number of sampled configurations in the

8

FIG. 7: A histogram of the number of sampled configurations ver-sus the parton number around its averageN1 − N1 in one of the de-composedC = 1 bands. While theN1 = N1 central peak containsmore than 98% of the configurations forΘ = 0.05π (red), the spreadfor Θ = 0.125π (black) is much wider and the percentage of theN1 = N1 configurations is only 15% suggesting thatN1 is no longera good quantum number. The results are obtained on system sizeLx = Ly = 28 with periodic boundary conditions.

projected wave function versus the parton numberN1 fluctua-tion around its average valueN1 in one of theC = 1 bands atΘ = π/20 (red) andΘ = π/8 (black). While such fluctuation isstill largely suppressed and theN1 conservation approximatelyholds atΘ = π/20 on the Abelian TOS side of the transition,it proliferates atΘ = π/8 and the intralayer charge conserva-tion no longer exists for a non-Abelian TOS. To see furtherthe connection between the parton number fluctuation and thenon-Abelian TOS, we show in Fig. 8 the mean squared devi-

ation

√

⟨

(

N1 − N1

)2⟩

/N1 versusΘ for various system sizes.

In reality, for a multiband TOS such as the topological ne-matic states24, band mixing, be it hopping or interaction, ishard to eliminate. The existence of a finiteΘc suggests thatthe Abelian TOS is stable against weak band-mixing perturba-tions. Intuitively, this is because the TOS are protected byex-citation gaps. For small band-mixing perturbations, the chargeconservation within the bands can appear as an emergent sym-metry. Nevertheless, in comparison with integer Chern insu-lators protected by the band gap, the TOS are relatively vul-nerable. A topological phase transition can occur even if theband structure remains adiabatically equivalent.

IV. CONCLUSIONS

In conclusion, we study topological properties of non-Abelian TOS using Gutzwiller-projected wave functions andthe momentum polarization approach. Our numerical resultson the topological spin and edge central charge confirm that

FIG. 8: The mean squared deviation

√

⟨

(

N1 − N1

)2⟩

/N1 versusΘ

for system sizesLx = Ly = 8,12, 16, 20, 24, 28.

projected wave functions of two partons in Chern bands withChern numberC = 2 are described by the field theory ofν = 2 fermions coupled to anS U(2) gauge field, and clearlydistinguish it from the pureS U(2)2 CS theory. In addition,we adiabatically interpolate the parent Chern insulator withC = 2 with two Chern insulators each withC = 1, and trackthe variation of topological quantities such as the topologi-cal spin and ground-state degeneracy for their correspondingTOS projected wave functions. We show that the topologicalphase transition between the non-Abelian and Abelian TOS ismarked by the breaking down of charge conservation withineach of theC = 1 Chern bands. The transition point is close tobut apart from the completely decoupled limit, in consistencywith the intuition that the corresponding Abelian TOS is pro-tected by a gap and stable against small band-mixing perturba-tions. Our result demonstrates explicitly that the topologicalorder in a Gutzwiller-projected state does not always agreewith the prediction of topological field theory, and genericallyhas to be determined by numerical calculations of topologicalproperties.

Our numerical methods based on momentum polariza-tion and the variational Monte Carlo method are generaliz-able to more complicated non-Abelian TOSs described byGutzwiller-projected wave functions. Compared to previousapproaches, momentum polarization provides an efficient wayto extract characteristic quantities given the many-body wavefunctions of a chiral topological ordered state. One open ques-tion left for future work is whether the critical behavior ofmo-mentum polarization across a topological phase transitioncanbe studied numerically and compared with any field theorydescription. Another open question is whether there is a moregeneric proof of the momentum polarization formula in Eq. 7,which has been verified numerically in several TOS, but hasnot been proved analytically except for arguments based onedge-state conformal field theory12.

We would like to thank Maissam Barkeshli, Chao-Ming

9

Jian, Ching Hua Lee and Peng Ye for insightful discussions.This work is supported by the Stanford Institute for The-

oretical Physics (YZ) and the National Science Foundationthrough the grant No. DMR-1151786 (XLQ).

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