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Emergent Chiral Spin Liquid: Fractional Quantum Hall Effec t in a Kagome Heisenberg Model

Shou-Shu Gong, Wei Zhu, and D. N. ShengDepartment of Physics and Astronomy, California State University, Northridge, California 91330, USA.

The fractional quantum Hall effect (FQHE) realized in two-dimensional electron systems under a magneticfield is one of the most remarkable discoveries in condensed matter physics. Interestingly, it has been proposedthat FQHE can also emerge in time-reversal invariant spin systems, known as the chiral spin liquid (CSL)characterized by the topological order and the emerging of the fractionalized quasiparticles. A CSL can naturallylead to the exotic superconductivity originating from the condense of anyonic quasiparticles. Although CSL washighly sought after for more than twenty years, it had never been found in a spin isotropic Heisenberg modelor related materials. By developing a density-matrix renormalization group based method for adiabaticallyinserting flux, we discover a FQHE in a spin-1

2isotropic kagome Heisenberg model. We identify this FQHE state

as the long-sought CSL with a uniform chiral order spontaneously breaking time reversal symmetry, which isuniquely characterized by the half-integer quantized topological Chern number protected by a robust excitationgap. The CSL is found to be at the neighbor of the previously identifiedZ2 spin liquid, which may lead to anexotic quantum phase transition between two gapped topological spin liquids.

The experimentally discovered fractional quantum Halleffect (FQHE)1–3 is the first demonstration of topologicalorder and fractional (anyonic) statistics4–8 realized in two-dimensional electronic systems under a magnetic field break-ing time-reversal symmetry (TRS). A related new state of mat-ter with fractionalized quasiparticle excitations is the topolog-ical quantum spin liquid (QSL) emerging in frustrated mag-netic systems9–18. Such spin systems, related to strongly cor-related Mott materials and holding the clue to the uncon-ventional superconductivity in doped systems, are of funda-mental importance to the condensed matter field9,10,18–20. Tounderstand the emergent physics of frustrated magnetic sys-tems, where spins escape from the conventional fate of de-veloping symmetry broken ordering, the concept of QSL withthe fractionalized quasiparticles was established8,10,14. Exper-imental candidates for such a new state of matter are iden-tified including kagome antiferromagnets21–23 and triangularorganic compounds24–26. The simplest QSL with TRS isthe gappedZ2 spin liquid, which possesses theZ2 topolog-ical order and fractionalized spinon and vison quasiparticleexcitations8,14. TheZ2 QSL is identified as an example of theresonating valence-bond liquid state, which was first proposedby Anderson10. Although explicitly demonstrated in contrivedtheoretical systems11,13,16,17, the searching of the gapped QSLin realistic Heisenberg models has always attracted much at-tention over the last twenty years. The primary example isthe recent discovered gappedZ2 QSL for kagome Heisenbergmodel (KHM) with the dominant nearest neighbor (NN) in-teractions based on the density-matrix renormalization group(DMRG) simulations27–30.

Another class of QSL with fractionalized quasiparticlesobeying fractional (anyonic) statistics is chiral spin liquid(CSL)31–35, which breaks TRS and parity symmetry while pre-serves other lattice and spin rotational symmetries. Kalmeyerand Laughlin31 first proposed that, in a time-reversal invari-ant spin system with geometry frustration, one can realize aν = 1/2 FQHE as a CSL state32 through mapping the frus-trated in-plane exchange interactions to the uniform magneticfield. A CSL is also considered to be a simple way in whichfrustrated spin systems develop topological order throughstatistics transformation to cancel out the frustration32,33. The

CSL may also lead to the exotic anyon superconductivitywith doping holes into such systems32,33. The existence ofCSL through spontaneously TRS breaking has been demon-strated in a Kitaev model on a decorated honeycomb latticewith contrived anisotropic spin interactions36 and most re-cently in a spin anisotropic kagome model37. Interestingly,based on the classical and Schwinger boson mean-field anal-yses, QSLs with different chiral spin orders have been sug-gested for extended KHM38,39. Other theoretical studies showthat one can also induce a CSL state through adding multi-spin TRS breaking chiral interactions40–43. Although CSL hasbeen explored for more than twenty years31–35,38,43,44, the ac-curate DMRG27–30 and variational Monte Carlo45 studies onvarious frustrated Heisenberg models often lead to the con-ventional ordered phases or TRS preservingZ2 and U(1)QSLs. The simple concept of realizing CSLs of the natureof FQHE through spontaneously breaking TRS and statisticstransformation32,33 remains illusive in realistic frustrated mag-netic systems.

In this article, we report a new theoretical discovery of theCSL in an extended spin-12 KHM based on the state of artDMRG simulations46,47. As illustrated in the inset of Fig.1(a),the system has the NN couplingJ = 1 as energy scale, aswell as the second and third NN couplingsJ ′ inside eachhexagon of the kagome lattice, described by the followingHamiltonian16,38:

H = J∑

〈i,j〉Si ·Sj +J ′

∑

〈〈i,j〉〉Si ·Sj +J ′

∑

〈〈〈i,j〉〉〉Si ·Sj . (1)

We perform the numerical flux insertion simulations oncylinder systems based on the newly developed adiabaticalDMRG to detect the topological Chern number, whichuniquely characterizes the chiral spin liquid. We have fullyestablished a robustν = 1/2 FQHE state for0.1 . J ′ . 0.7by observing the half-integer quantized topological Chernnumber protected by a robust excitation gap, the degenerateground states, and the uniform chiral order spontaneouslybreaking TRS.

Results

2

0 0.2 0.4 0.6 0.8 1J

0

0.05

0.1

E2

0(S) -E1

0(S)

E0

2-E0

1

ES

2-ES

1

CSL VBS

FIG. 1: Phase diagram of the spin-12J − J ′ antiferromagnetic

kagome Heisenberg model (KHM). We set the NN couplingJ = 1,and the second and third NN couplings asJ ′. In DMRG simulations,we study cylinders with lengthsLx andLy (in unit cells) along thex andy (the tilt lattice vector) directions, respectively. We also labelour system by the total number of sitesN = 3 × Lx × Ly. (a) Theenergy differences between the two lowest statesE0

2−E01 (ES

2 −ES1 )

in two different topological (vacuum and S-) sectors are shown for aN = 3 × 24 × 4 cylinder at differentJ ′, which reveal the fourdegenerating ground states in two sectors for a range of parameter0.2 . J ′ . 0.7. (b) We illustrate the uniform positive chiral order〈χi〉 = 〈Si1 · (Si2 × Si3)〉 (i1, i2, i3 ∈ △i(▽i)) on aN = 3 ×24 × 4 cylinder withJ ′ = 0.5 measured from the MES|ψ0

L〉 in thevacuum sector (identified in Fig.5), which breaks the TRS and paritysymmetry. The CSL phase is characterized by the long-range chiralcorrelations and a fractionally quantizedC = 1/2 Chern number,which identifies the state as the Laughlinν = 1/2 FQHE emergingin theJ − J ′ KHM.

Phase diagram. Our main findings are summarized in thephase diagram Fig.1(a). With the turn on of a positiveJ ′,we find a robust CSL phase in the region of0.1 . J ′ . 0.7.We design and perform the Laughlin flux insertion numericalexperiment through developing an adiabatic DMRG, whichinserts flux and obtains the ground state for each flux. Theadiabatic DMRG allows us to obtain the topological Chernnumber3,34, which characterizes the topological nature ofthe quantum phase. Our simulation experiment shows thatthe CSL is characterized by a fractionally quantized ChernnumberC = 1/2, which is a “smoking gun” evidence of theemergentν = 1/2 Laughlin FQHE state31 in the frustratedKHM. The CSL phase is also characterized by a four-fold

a

0 4 8 12 16 20 24

0.0

0.2

0.4

0.6

<Szx>

x

b

0.0 0.5 1.0 1.5 2.0

0.0

0.2

0.4

0.6

0.8

1.0

Net

spi

n tra

nsfe

r Sz | ed

ge

c

FIG. 2: Laughlin flux insertion Gedanken experiment and frac-tionalized Chern number C = 1/2 for CSL. (a) Real-space con-figuration of the spin magnetization〈Sz

x,y〉 at positionRi = (x, y)after adiabatically inserting a quantized fluxθ = 2π. The area ofthe circle is proportional to the amplitude of〈Sz

x,y〉. The red (blue)color represents the positive (negative)〈Sz

x,y〉. (b) Real-space con-figuration of the accumulated spin magnetization〈Sz

x〉 =∑

y〈Sz

x,y〉(the summation is over all the3Ly sites in each columnx) with in-creasing fluxθ. Clearly, we see a net spin-z accumulating in the rightedge of the sample, which is equivalent to the transfer of hardcorebosons (the hardcore boson numberni is related to the on-siteSz

i asni = Sz

i + 1/2) being pumped from the left edge to the right edgewithout going through the bulk. So this simulation experiment re-veals a quantum Hall system with a nonzero Hall conductance,whilethe bulk is gapped. (c) Net spin transfer∆Sz|edge to the right edge ofthe cylinder as a function ofθ. From the net spin transfer in one pe-riod of flux θ = 0 → 2π, we obtain the exact fractionally quantizedChern numberC = ∆Sz|edge = 1/2. The results are demonstratedfor a 3 × 24 × 4 cylinder atJ ′ = 0.5 using theU(1) DMRG withkeeping up to5000 states. Similar results are obtained for all thestates within the CSL phase.

degeneracy in two topological sectors. In each sector, thereis a double degeneracy representing the two sets of CSLstates with opposite chiralities. The near uniform chiralorder measured for a state spontaneously breaking TRSis illustrated in Fig.1(b). We also establish that the CSLis neighboring with theZ2 QSL previously found27–30 atJ ′ = 0, while the transition region appears to be under stronginfluence of the nonuniform Berry curvature resulting fromgauge field, which may provide new insights to many puzzlesregarding theoretical27–30 and experimental findings18,21–23

for kagome antiferromagnets.

3

Fractional quantization of topological number. To uncoverthe full topological nature of the phase at large system scale,we perform the flux inserting simulation based on the adia-batic DMRG. For conventional FQHE systems, a quantizednet charge transfer would appear as∆N = C from one edgeof the sample to the other edge after inserting one period offlux θ = 0 → 2π, corresponding to a nonzero fractionallyquantized topological invariant Chern numberC34, which isC = 1/2 for theν = 1/2 bosonic Laughlin state.

By adiabatically inserting the fluxθ in our DMRG ex-periment, we study the evolution of the local magnetization〈Sz

x,y〉, which is the spin-z average of the ground state ata local lattice siteRi = (x, y). With the increase ofθ,we measure the corresponding spin accumulations of eachground state atθ = jπ/2 (j is an integer). One example withθ = 2π is shown in Fig.2(a). We find nonzero magnetizationstarting to build up at the left and right edges of cylinder,which grows monotonically with the growing ofθ as shownin Fig. 2(b). Since our system has total spin conservation, thenet spin-z transfer∆Sz|edge (which is the total magnetizationaround the right edge of the system) is equivalent to thepumping of the hardcore bosons from the left edge to theright edge without going through the bulk. In Fig.2(c), weshow the net spin transfer∆Sz|edge as a function ofθ. A nearlinear spin pump is being realized in this chiral spin state,which is exactly quantized as∆Sz|edge = 0.5 at θ = 2π.From the fundamental correspondence between edge spintransfer and bulk Chern number48, we identify the bulk Chernnumber of the system asC = 1/2, fully characterizing thestate as the Kalmeyer-Laughlin CSL31 of ν = 1/2 FQHE.Physically, the pumping in FQHE system is achieved throughthe adiabatical rotation of the basis states of the many-bodywavefunction, which can be viewed as a non-local operationby developing a “spinon” line in the cylinder. We find theentanglement spectrum of the spinon sector obtained hereby inserting2π flux is identical to the one of the S-sectorshown below in Fig.5(b) obtained through pinning. Withfurther increasing the flux toθ = 4π, the net spin transfer∆Sz|edge = 1.0, where the system evolves back to thevacuum sector. These observations fully establish the bosonicν = 1/2 FQHE emerging in theJ − J ′ KHM. While theChern number simulations characterize the ground state asthe long-sought CSL, we will further measure the topologicaldegeneracy, chiral correlations, topological entanglemententropy, and modular matrix to demonstrate the full nature ofthe topological state in our time-reversal invariant system.

Low-energy spectrum and topological degeneracy.TheKalmeyer-Laughlin CSL has two-fold topological ground-state degeneracy, and the spontaneously TRS breaking forsuch a time-reversal invariant system must have an additionaldouble degeneracy in each topological sector. On cylindergeometry, one can control the boundary condition near thecylinder edges to target into different topological sectors27,49,which we denote as the vacuum and S-sectors, respectively.By using this technique in DMRG, we find the two lowest-energy states in each sector whose energy differencesE0

2−E01

andES2 − ES

1 drop to small values for0.2 . J ′ . 0.7. One

-2 -1 0 1 2

3

4

5

6

2

2

4 4

2

2

6

4

2

i

Sztot

2

a

-1 0 1 2

3

4

5

6b

2

2

4

66

4

2

i

Sztot

2

c

FIG. 3: Entanglement spectra characterization and MESs for theTRS broken phase. (a) and (b) are the entanglement spectra of theground states in the vacuum (|ψ0

1〉) and S- (|ψS1 〉) sectors, respec-

tively. ξi = − lnλi with λi the eigenvalues of reduced densitymatrix. The two degenerate states|ψ0

1,2〉 (|ψS1,2〉) have exactly the

same spectra. These states are real wavefunctions consistent withthe TRS of the model Hamiltonian. The lines and circles with thenear identicalξi denote the double degeneracy indicating that eachof these low-energy states is a maximum entropy state. The numbers2, 2, 4, 6 label the near degenerating pattern for the low-lying entan-glement spectra which are doubled from what one would expectfor aLaughlin FQHE state from conformal field theory. (c) Entanglemententropy profile−S of the general superposition state in the vacuumsector|ψ0〉 = c|ψ0

1〉+√1− c2eiφ|ψ0

2〉. The MESs that are pointedby red arrows and dots emerge as|ψ0

L,R〉 = 1√2(|ψ0

1〉 ± i|ψ02〉). The

MES |ψ0R(L)〉 has uniform counterclockwise (clockwise) chiral order

for each triangle as illustrated in Fig.1(b). Furthermore, if we initi-ate the DMRG state as a complex state, we automatically find sucha MES, which spontaneously breaks TRS. The results are demon-strated forJ ′ = 0.5 for aN = 3×24×6 cylinder, and near identicalresults are obtained for different parameters in the CSL phase.

example is shown in Fig.1(a) for a cylinder withLx = 24and Ly = 4. Importantly, the degenerating states|ψ0

1,2〉(|ψS

1,2〉) in each topological sector also have near identical en-tanglement spectra. The double degeneracy of entanglementspectrum for the ground states|ψ0(S)

1 〉 is explicitly shownusing two different symbols (line and circle) in Figs.5(a)and 5(b). These observations are consistent with the spon-taneously TRS breaking double degeneracy. We also find theground-state energies between the two sectors are degenerate(ES

1 /N − E01/N = 0.00001 for J ′ = 0.5 atLy = 4), which,

combined with the distinct entanglement spectra50 as shownin Figs. 5(a) and5(b) of the two sectors, establish the topo-logical degeneracy for these two sectors in the intermediatephase. By searching for other low energy excited states fromboth DMRG and exact diagonalization (ED), we exclude thatthere are other distinct topological degenerating sectorsforthe intermediate region, while a lot more lower energy statesappear nearJ ′ = 0.

4

The energy and entanglement spectra doubling are signa-tures of finding the maximally entangled states in each sec-tor, which is forced by the TRS of the system Hamiltonian(here we used a real number initial wavefunction in DMRGcalculations which forbids any spontaneous TRS breaking).To demonstrate the nature of the new quantum phase, we firstfind the minimum entangled states (MESs) in each topologicalsector28,51,52, which represent the eigenstates of the Wilson-loop (string-like) operators encircling the cylinder and are thesimplest states of the quasiparticles. In Fig.5(c), we showtwo MESs emerging (labeled by two red dots) in the vacuumsector:|ψ0

L(R)〉 = 1√2(|ψ0

1〉 ± i|ψ02〉), which are equal magni-

tude superposition of the real states with a phase difference±π/2. The MES |ψ0

L〉 breaks the TRS spontaneously anddemonstrates a uniform nonzero chirality order for each trian-gle as illustrated in Fig.1(b). The chiral order reaches a valuearound0.08 comparable to its classical value1/8. The conju-gate state|ψ0

R〉 as another MES has the opposite sign of chi-rality. The doubling of the entanglement spectra for the max-imum entropy state simply results from the superposition ofthe MESs with the same entanglement spectra. Consequently,one finds an entanglement entropy differenceln 2 comparingto the MESs as illustrated in Fig.5(c). Near identical resultsand two MESs are also found in the topological degeneratingS-sector. Furthermore, if we initiate the DMRG state with arandom complex number state, we automatically find such aMES, which spontaneously breaks TRS.

By obtaining the MES, we find the topological entan-glement entropyγ consistent with the resultln 2/2 of theν = 1/2 Laughlin state53,54. The ED calculations furtherconfirm this state on aN = 3 × 4 × 3 cluster by extractingmodular transformation matrix51,52 from the MESs of twononcontractable cuts (see Supplementary Information formore details).

Quantum phase transitions. We use both the chiral-chiralcorrelation functions and the topological Chern number ob-tained from inserting flux to identify the quantum phase dia-gram and transitions in theJ−J ′ model. In Fig.4(a), we com-pare the chiral correlations〈χiχj〉 for the states from the twotopological sectors with different system widths atJ ′ = 0.5.We find long-range correlations for the states from both topo-logical sectors, which are further enhanced with increasingsystem widthLy. To reveal the quantum phase transitions,we show the chiral correlation functions calculated from theground state of the vacuum sector for differentJ ′ in Fig. 4(b).〈χiχj〉 is positive everywhere and has the long-range order for0.1 ≤ J ′ ≤ 0.7, while transitions to other phases are detectedatJ ′ = 0.05 and0.8 by identifying the exponential decayingchiral correlations.

In the flux insertion simulations, we find that the Chernnumber remains to be quantized atC = 1/2 for the sameparameter range0.1 ≤ J ′ ≤ 0.7, thus we establish thequantum phase diagram as shown in Fig.1(a). The quantumphase transition aroundJ ′ ∼ 0.7 − 0.8 is charaterized byan excitation gap closing in the bulk of system, where wedetect a strong bulk magnetization (boson density) responseto the inserted flux. BetweenJ ′ = 0 and0.1, we detect a

2 4 6 8 10 12 14 16 18 20|i-j|

0.001

0.01

<χiχ

j>

a

2 4 6 8 10 12 14 16 18 20|i-j|

1e-05

0.0001

0.001

0.01

|<χiχ

j>|

J´=0.1J´=0.15J´=0.2J´=0.3J´=0.4J´=0.5J´=0.6J´=0.7

bLy=6, |ψ

0

1>

FIG. 4: Long-range chiral-chiral correlation function for CSL .(a) Log-linear plot of the chiral-chiral correlation function 〈χiχj〉versus the distance of triangles|i − j| along thex direction on theN = 3 × 24 × 4 and3 × 24 × 6 cylinders for the ground states inboth vacuum and S-sectors atJ ′ = 0.5. All the correlations are pos-itive and increase with the system widthLy . In the S-sector,〈χiχj〉have small edge effects due to the presence of two localized spinsfrom pinning. (b) Log-linear plot of absolute correlation|〈χiχj〉|versus|i − j| along thex direction forN = 3 × 24 × 6 cylindersat variousJ ′ obtained from the ground state in the vacuum sector.All the correlations demonstrate long-range order (and they are alsopositive) for the intermediate phase0.1 ≤ J ′ ≤ 0.7, while they de-cay exponentially atJ ′ = 0.05 and0.8, where transitions to theZ2

spin liquid and VBS phase take place. The correlations are near aconstant everywhere in the CSL phase, so we choosei = 3 and varyj to the end of the cylinder.

strong nonuniform Berry curvature resulting from the gaugefield in the inserting flux simulations, possibly indicatingtheforming of new quasiparticles and the emerging ofZ2 QSL.We also study the stability of the CSL when the second andthird neighbor couplings are different. We find the CSL phasein a region around the line withJ2 = J3. For example, whenJ2 = 0.1, the CSL is robust for0.1 . J3 . 0.3. Physically,

5

the J3 coupling suppresses the magnetic order formed inthe J1 − J2 (J2 ≃ 0.2) kagome model, thus substantiallyenlarges the non-magnetic region. Meanwhile, classicallythe J3 term will enhance a noncoplanar spin chiral order38,which may induce a CSL in the quantumJ1 − J2 − J3 modelas demonstrated here.

DiscussionIn the past twenty years, the gapped QSL in realistic mag-netic systems have attracted intensive attention. Whilethe NN or J1 − J2 KHM27–30 is the primary candidateof a possibleZ2 QSL, there are still many puzzles leftunresolved. The frustrated kagome antiferromagnetsHerbertsmithite Cu3(Zn,Mg)(OH)6Cl2 and KapellasiteCu3Zn(OH)6Cl218,21–23 are possible candidates of QSL; how-ever, they appear to be more consistent with gapless or criticalstates. At theoretical side, redundant low-energy excitationsare found for the NN KHM from ED simulations55, varia-tional studies find thatU(1) gapless QSL45,56 has relativelylow energy, and DMRG studies have not been able to identifyall the four topological sectors forZ2 QSL49. Our finding ofthe robust CSL at the neighbor of the NN KHM indicates thatthe latter is not a fully developedZ2 QSL yet, and the natureof states for the experimental relevant kagome systems maybe strongly affected by a new quantum critical point betweentwo gapped QSLs, theZ2 and the CSL. In a parallel work, aCSL has also been uncovered in an anisotropic kagome spinsystem37 with only spin-z interactions for further neighbors.We believe that our numerical findings will stimulate newtheoretical and experimental researches in this field to resolvethe nature of the quantum phases for different frustratedmagnetic systems. An exciting next step will be identifyingtheoretical models and experimental materials which canhost exotic topological superconductivity by doping differentCSLs.

MethodsDMRG is a powerful tool to study the low-lying states ofstrongly correlated electron systems46. The accuracy ofDMRG is well controlled by the number of kept statesM ,which denotes theM eigenstates of the reduced densitymatrix with the largest eigenvalues. The highly efficiencyof DMRG for one-dimensional systems or two dimensionalcylinder systems have been shown for different systems27,47.

An improvement in DMRG calculations is to implementsymmetry to reduce the Hilbert space. The spin-z or totalparticleU(1) symmetry is commonly used in DMRG, whichis preserved in many model systems. For some systemswith spin rotationalSU(2) symmetry such as the Heisenbergspin model, the more efficient choice is to apply theSU(2)symmetry57, from which we can obtain more accurate resultsfor wider systems. This algorithm has been applied to studyvarious frustrated Heisenberg systems successfully29,58,59.

Details of the SU(2) DMRG calculation. We study thefrustrated KHM without flux usingSU(2) DMRG. We studythe cylinder system with open boundaries in thex directionand periodic boundary condition in they direction. ForLy = 4 (Ly = 6) systems, we keep up to3000 (4600)SU(2) states with the DMRG truncation errorǫ ≃ 1 × 10−6

(ǫ ≃ 1 × 10−5) for most calculations. To find the groundstates in both vacuum and S- topological sectors on cylindersin the DMRG calculations, we take pinning sites in the openboundaries or insert flux to target the two different sectors49.

Adiabatic DMRG and fractionally quantized Chern num-ber. For the first time, we develop the numerical flux inser-tion experiment for cylinder systems based on the adiabaticalDMRG simulation to detect the topological Chern number34

of the bulk system, which uniquely characterizes the CSL asaν = 1/2 FQHE state emergent from theJ − J ′ Heisenbergmodel on kagome lattice. In this simulation, we impose thetwist boundary conditions along they direction by replacingtermsS+

i S−j + h.c. → eiθS+

i S−j + h.c. for all neighboring

(i, j) bonds with interactions crossing they-boundary in theHamiltonian. Starting from a smallθ ∼ 0, a state with thedefinite chirality and sign of Chern number will be randomlyselected, which remains the same through out the whole adi-abatical process ofθ = 0 → 4π. We find states with theopposite Chern numbers (C = ±1/2) in different runs of thesimulations due to spontaneously TRS breaking. A robust ex-citation gap∆ ∼ 0.24 is obtained forJ ′ = 0.5 after we createtwo spinons (atθ = 2π) at the opposite edges of the cylinder(see Fig.2(a)), which protects the CSL state. This method canbe applied to study different interacting systems and charac-terize different topological states.

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7

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Acknowledgements We thank Y. C. He for extensivediscussions. We also thank Leon Balents, Matthew P. A.Fisher, Olexei I. Motrunich and F. Duncan M. Haldane forstimulating discussions and explanations of spin liquid aswell as topological physics. This research is supported by theNational Science Foundation through grants DMR-1205734(S.S.G.), DMR-0906816 and DMR-1408560 (D.N.S.), theU.S. Department of Energy, Office of Basic Energy Sciencesunder grant No. DE-FG02-06ER46305 (W.Z.).

Author Contributions S.S.G. and W.Z. performed maincalculations based on different numerical programs theydeveloped. S.S.G., W.Z. and D.N.S. made significant con-tributions from the design of the project to the finish of themanuscript.

Additional informationCompeting financial interests:The authors declare no com-peting financial interests.

J ′ = 0.5, Ly = 4 E01/N ES

1 /N (ES1 − E0

1)/N

MSU(2) = 2000 −0.46046 −0.46038 0.00008

MSU(2) = 4000 −0.46050 −0.46048 0.00002

MSU(2) = 5000 −0.46052 −0.46051 0.00001

TABLE I: Degenerate ground-state energies in the different topo-logical sectors. The ground-state energy per site in both the vacuum(E0

1/N ) and spinon (ES1 /N ) sectors, as well as the energy difference

between the two sectors(ES1 − E0

1)/N for J ′ = 0.5 on theLy = 4cylinder. To avoid edge effects, these bulk energies are obtained bysubtracting the energies of two long cylinders with different systemlengths.MSU(2) is the keptSU(2) states.

Supplementary Information

A. TOPOLOGICAL DEGENERATE GROUND-STATEENERGY

In the gapped topological states, the ground-state energiesin different topological sectors are near degenerate on finite-size systems. With the increase of the system width, the differ-ence of the near degenerate energies vanishes exponentially.In the density-matrix renormalization group (DMRG) calcu-lations, we obtain the bulk energy per site in both the vac-uum (E0

1/N) and spinon(ES1 /N)sectors by subtracting the

energies of two long cylinders with different system lengthsin each sector.(ES

1 − E01 )/N describes the difference of the

ground-state energies in different topological sectors.Here, we show the results forJ ′ = 0.5 onLy = 4 cylinder

in Suppl. Table.I. By keeping the unconverged2000 SU(2)states, we find a small energy difference0.00008. And withincreasing kept states, the difference continues to decrease.For the well converged ground states with the DMRG trun-cation errorǫ ≃ 1 × 10−7 by keeping5000 SU(2) states,we show the energy difference is0.00001, which is consis-tent with the exact diagonalization results shown below andthe topological degeneracy in the system. This energy split-ting 0.00001 is much smaller than that in the nearest-neighbor(NN) kagome Heisenberg model0.00069.

B. TOPOLOGICAL ENTANGLEMENT ENTROPY

For the gapped quantum states with topological order, thetopological entanglement entropy (TEE)γ is proposed tocharacterize the non-local entanglement. The Renyi entropyof a subsystemA with reduced density matrixρA are definedasSn = (1 − n)−1 ln(TrρnA), where then → 1 limit givesthe Von Neuman entropy. For a topologically ordered state,Renyi entropy has the formSn = αL − γ, whereL is theboundary of the subsystem, and all other terms vanish in thelargeL limit; α is a non-universal constant, while a positiveγ is a correction to the area law of entanglement and reachesa universal value determined by total quantum dimensionDof quasiparticle excitations asγ = lnD. For theν = 1/2Laughlin state, the quantum dimension of each quasiparticle

8

is 1, leading to the total dimensionD =√2 and thus the TEE

γ = ln 2/2.By using the complex number DMRG simulations, we ob-

tain the minimal entropy state (MES) with spontaneouslybroken time-reversal symmetry and the corresponding VonNeuman entanglement entropy. With the help of theSU(2)DMRG, we could obtain the converged entropy forLy =3, 4, 5 cylinders. ForLy = 6 cylinder, we cannot get theconverged entropy because the required DMRG optimal statenumberMSU(2) is beyond our computation abilities. Thus, tofind an estimation of the entropy onLy = 6 cylinder, we studythe entropy versus1/MSU(2) as shown in Fig.5(a), and makea careful extrapolation of the data to estimate the convergedresult. ForJ ′ = 0.5, we find the entropyS = 4.49 ± 0.02.In Fig. 5(b), we make a linear fitting of the entropy datafor Ly = 4, 5, 6 cylinders atJ ′ = 0.5, and find the TEEγ = 0.34 ± 0.04, which is consistent with the TEE of theν = 1/2 Laughlin stateγ = ln 2/2.

C. SPIN-SPIN CORRELATION FUNCTION

For a gapped topological chiral spin liquid (CSL), the sys-tem is expected to have a short-range spin correlation. Wemeasure the spin-spin correlation function on the cylinderswith Ly = 4 and6 for both vacuum and spinon sectors. Wedemonstrate〈Si · Sj〉 with site i in the center of lattice andjalong the same row from the bulk to the boundary forJ ′ = 0.5in Suppl. Fig.6. The spin correlations exhibit the exponen-tial decay in both vacuum and spinon sectors. And the decaylength does not increase with growingLy from 4 to 6, whichsuggests that the spin correlation length is close to saturationwith growing system width. This observation is consistentwith a vanishing magnetic order.

D. VALENCE-BOND SOLID ORDER

To investigate the possible valence-bond solid (VBS) order,we study the dimer-dimer correlation function on cylinder sys-tems, which is defined as

D(i,j),(k,l) = 4[〈(~Si · ~Sj)(~Sk · ~Sl)〉 − 〈~Si · ~Sj〉〈~Sk · ~Sl〉], (2)

where (i, j) and (k, l) represent the nearest-neighbor (NN)bonds. First of all, we demonstrate the real space distributionsof the NN bond energies on cylinder systems. To clearly showthe fluctuations of the NN bond energies, we define the bondtexture as the bond energies subtracting a constante, whichis the average NN bond energy in the bulk of cylinder, i.e.,Bi,j = 〈Si · Sj〉 − e. As shown in Suppl. Fig.7 of the bondtextures on3 × 16 × 4 cylinders forJ ′ = 0.5 in the vacuumsector, we obtain the uniform bond textures along both thexandy directions in the bulk of cylinders. The small differencesbetween thex andy bond textures0.01 and0.004 in Suppl.Figs. 7(a) and7(b) are owing to the long cylinder geometry,which breaks the lattice rotation symmetry. The uniform bondtextures indicate the good convergence of our DMRG results.

0 0.0002 0.0004 0.0006 0.0008 0.001

1/MSU(2)

3.9

4

4.1

4.2

4.3

4.4

4.5

entropy S

J´ = 0.5, Ly= 6, DMRG

S = a + b/MSU(2)

+ c/M2

SU(2)+ d/M

3

SU(2)

a

0 2 4 6 8 10 12W

y

-1

0

1

2

3

4

5

S

J´ = 0.5, Wy = 2 Ly

γ = 0.34(4)

γ = ln2/2

b

FIG. 5: Topological entanglement entropy. (a) Entanglement en-tropy S versus DMRG optimal state number inverse1/MSU(2) forJ ′ = 0.5 on Ly = 6 cylinder. The DMRG entropy is for theMES, which is obtained from the complex number DMRG simu-lations. The data are fitted using the formulaS = a + b/MSU(2) +

c/M2SU(2) + d/M3

SU(2), from which we find the convergent entropyS = 4.49± 0.02. (b) Entanglement entropy versus system width forJ ′ = 0.5. By a linear fitting of the results forLy = 4, 5, 6, we findthe TEEγ = 0.34 ± 0.04, where the error bar is from the uncer-tainty of the entropy onLy = 6 cylinder as shown in (a). The TEEwe find is consistent with the result of theν = 1/2 Laughlin stateγ = ln 2/2.

With the uniform bond textures in the bulk, we could fur-ther study the dimer-dimer correlation functions. We set thereference bond(i, j) in the middle of cylinder. Suppl. Fig.8shows the dimer-dimer correlations on the3 × 16 × 4 cylin-ders atJ ′ = 0.5 in the vacuum sector. The black bond inthe middle denotes the reference bond(i, j), and the red andblue bonds indicate the negative and positive dimer correla-tions, respectively. We show that the dimer-dimer correlationsdecay quite fast to zero in bothx andy directions. On the3×18×6 cylinders, the dimer correlations have a similar fastdecay. The significant short-range dimer correlations strongly

9

0 2 4 6 8 10 12 14|i-j|

0.0001

0.01

1

|<SiS

j>|

Ly=4, |ψ

0

1>

Ly=6, |ψ

0

1>

a

0 2 4 6 8 10 12 14|i-j|

0.0001

0.01

1

|<SiS

j>|

Ly=4, |ψ

S

1>

Ly=6, |ψ

S

1>

b

FIG. 6: Spin-Spin correlation function. Log-linear plot of the ab-solute value of the spin-spin correlation function versus the distanceof sites|i− j| on3× 24× 4 and3× 24× 6 cylinders forJ ′ = 0.5in (a) vacuum and (b) S-sectors. The reference sitei is located in thecenter of cylinder, and sitej is chosen along the same row from thebulk to the boundary.

indicate the vanishing VBS order.

For J ′ ≥ 0.8, we find a strong VBS state with breakinglattice translational symmetry in the system. As demonstratedin Suppl. Fig.9 of the bond textures atJ ′ = 1.0 on a3×16×4cylinder, the horizontal NN bond textures are not uniform inthe bulk of cylinder but have a difference of0.01, and the tiltbonds along the vertical direction also have a difference of0.01, which are quite different from the uniform state in theCSL phase as shown in Suppl. Fig.7(b). These observationsindicate that we find the ground state with lattice translationalsymmetry breaking in both thex andy directions.

E. CHIRAL-CHIRAL CORRELATION FUNCTION

In the DMRG calculations of chiral-chiral correlation func-tion, the systems near phase boundaries require much morekept states than in deep of the CSL region to capture the long-range chiral correlations. As shown in Suppl. Fig.10 of theimprovement of chiral correlation function with the growingDMRG kept states for aN = 3×18×6 cylinder atJ ′ = 0.2 inthe vacuum sector, the system shows a fast exponential decaychiral correlation by keeping800 SU(2) states (equivalent toabout3200 U(1) states), and with increasing kept states thedecay length continues to grow. When keeping4600 SU(2)states (equivalent to about18000 U(1) states), the chiral cor-relations form a long-range correlation. Meanwhile, we onlyneed to keep about10000 U(1) states to uncover the long-range chiral correlations in deep of the CSL region such as atJ ′ = 0.4, 0.5. Therefore, the less convergent DMRG calcula-tions may find a narrower CSL phase region.

F. EXACT DIAGONALIZATION RESULTS

A. Lowest-Energy spectrum for 36-sites torus

We calculate the low-energy spectrum of theJ − J ′ modelon a36 sites kagome lattice using exact-diagonalization (ED)method. We consider a finite system with periodic boundaryconditions, as shown in Suppl. Fig.11(a). For this geometry,the two-fold topological degeneracy of theν = 1/2 FQHEare expected to live in the momentum sectorsk = (0, 0) and(0, π). Thus we obtain the low-energy spectrum in these twomomentum subspaces. As shown in Suppl. Fig.11(b) ofthe spectrum atJ ′ = 0.6, we find that two lowest states foreach momentum sector, denoted byEk=0(π)

1 , Ek=0(π)2 , are

well separated from the continuum of other excitations by agap that is about0.15. The nearly vanishing energy differencebetween two sectorsE0

1(2) − Eπ1(2) = 0.0007(0.0022) indi-

cates the emergence of the many-body magnetic translationalsymmetry. The existence of the two lowest states in each sec-tor is due to the time reversal symmetry. Therefore, our EDcalculations imply that the system has four-fold degeneracy ofground states, where two of them are from topological degen-eracy and two are from time reversal symmetry.

B. Modular matrix

The information of quantum dimension and fusion rules ofthe quasiparticles are encoded in the modularS matrix. Toextract the modularS matrix in our model, we use the methodof searching the minimal entropy states (MESs) to constructthe modular matrix. In this method, we first calculate the en-tanglement entropy through partitioning the full torus systeminto two subsystems (cylinders)A andB then tracing out thesubsystemB. Here we consider two noncontractible biparti-tions on torus geometry as shaded by light green and brownin Suppl. Fig.11(a), which is along the lattice vectors~a1,~a2,

10

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-0.002

0.002

-0.002

0.002

-0.002

-0.002

-0.002

0.002

-0.002 0.002

-0.002

-0.002

-0.002

0.002

0.002

-0.002

0.002

-0.002

-0.002

-0.002

0.002

-0.002

0.002

-0.002

0.002-0.002

-0.002

-0.002

0.002

-0.002 0.002

-0.002

-0.002

-0.002

0.002

-0.002

0.002

-0.002

0.002

-0.002

-0.002

-0.002

0.002

-0.002

0.002

-0.002

0.002

-0.001

-0.002

-0.001

0.002

-0.002 0.002

-0.001

-0.002

-0.001

0.002

-0.002

0.002

-0.002

0.002

-0.001

-0.002

-0.001

0.002

-0.002

0.002

-0.001

0.002

-0.001

-0.002

-0.001

0.002

-0.002 0.002

-0.001

-0.002

-0.001

0.002

-0.002

0.002

-0.002

0.002

-0.001

-0.002

-0.002

0.002

-0.002

0.002

-0.002

0.002

-0.002

-0.002

-0.003

0.002

-0.002 0.002

-0.003

-0.002

-0.003

0.002

-0.001

0.002

-0.001

0.002

-0.004

-0.001

-0.004

0.002

-0.001

0.002

-0.004

-0.000

-0.004

0.001

-0.004

-0.000

0.001 -0.000

-0.004

0.001

-0.004

-0.000

0.003

-0.003

0.003

-0.003

0.001

0.003

0.001

-0.003

0.003

-0.003

0.001

-0.001

0.013

-0.001

0.014

-0.001

-0.001 -0.001

0.013

-0.001

0.014

-0.002

-0.016

0.010

-0.017

0.010

0.029

-0.016

0.029

0.009

-0.016

0.010

0.029

0.009

0.016

-0.039

0.016

0.009

-0.039 0.009

0.016

-0.039

0.016

0.009

-0.028

0.056

-0.028

0.056

-0.037

-0.028

-0.037

0.056

-0.028

0.056

-0.037

-0.014

-0.014

-0.014

-0.244

-0.244

-0.244

-0.244

-0.002

b. Lx=16, Ly=4, J'=0.5, bond texture

FIG. 7: Nearest-neighbor bond textures atJ ′ = 0.5. The NN bond texturesBi,j on the3 × 16 × 4 cylinders atJ ′ = 0.5 in the vacuumsector. The numbers denote the amplitudes of bond textureBi,j = 〈Si ·Sj〉−e, wheree is the average NN bond energy in the bulk of cylinder.Here, we finde = −0.212 and−0.210 for (a) and(b), respectively. The blue (red) color represents the positive (negative) bond texture.

0.000

-0.000-0

.000

0.000

-0.000

-0.000

-0.000

-0.000

0.000

-0.000

0.000

0.000

0.000

0.000

-0.000

0.000

-0.000

-0.000

0.000

-0.000

0.000 0.

000

-0.000

0.000

0.000

-0.000

0.000

0.000

0.000

0.000

-0.000

0.000

-0.000

-0.000

0.000

-0.000

0.000

-0.000

-0.000

0.000

-0.000

0.000

-0.000-0.000

-0.000

0.0000.

000

0.000

-0.000

-0.000

-0.000

-0.000

0.000

0.000

-0.000

0.000

-0.000

-0.000

0.000

0.000

0.000

0.000

0.000

-0.000

-0.000 -0

.000

-0.001

0.001

0.000

-0.000

0.000

-0.000

-0.000

0.000

0.000

-0.000

0.000

0.000

-0.000

0.000

-0.000

0.000

0.000

0.000

-0.001

0.000

0.001 0.001

-0.001

0.002-0

.000

0.000

-0.001

-0.001

0.000

0.001

0.000

-0.000

0.000

0.000

0.000

0.000

-0.001

0.000

0.001

-0.000

0.002

-0.001

-0.001 -0

.001

-0.002

0.001

0.005

-0.004

0.001

-0.003

0.013

0.001

-0.001

0.002

-0.019

-0.001

0.014

-0.020

-0.004

-0.004

0.001

0.005

-0.002

0.001

0.002 0.001

0.002

0.001-0

.003

0.005

-0.005

-0.000

-0.027

0.000

0.038

-0.185

0.138

0.038

-0.028

0.138

0.001

0.005

0.001

-0.003

0.001

-0.005

0.002 0.

005

-0.008

-0.000

0.038

-0.022

0.004

-0.021

0.149

-0.022

-0.021

0.148

0.037

-0.008

0.004

-0.0060.000

-0.003

0.001

0.020

-0.019

-0.018

-0.028

0.007

0.069

-0.007

0.004

0.067

-0.027

0.003

-0.017

0.020

0.006

0.002

0.001

-0.019

-0.003 -0

.003

-0.004

0.002

0.005

-0.004

0.001

0.001

-0.001

0.002

-0.001

0.002

-0.001

-0.001

-0.001

-0.001

0.001

-0.004

0.002

0.005

-0.004

0.001

0.002 0.002

-0.001

0.002-0

.001

0.000

-0.000

-0.001

0.000

0.000

-0.000

0.000

-0.000

-0.000

0.000

-0.000

-0.001

0.000

0.000

-0.001

0.001

-0.000

-0.001 -0

.001

-0.000

0.000

0.000

0.000

-0.000

-0.000

0.000

-0.000

0.000

-0.000

-0.000

0.000

0.000

-0.000

-0.000

0.000

-0.000

0.000

-0.000

-0.000

0.001 0.000

-0.000

0.0000.

000

-0.000

-0.000

0.000

-0.000

0.000

0.000

-0.000

0.000

0.000

-0.000

0.000

0.000

-0.000

0.000

0.000

0.000

-0.000

-0.000 0.

000

0.000

0.000

-0.000

0.000

-0.000

-0.000

0.000

-0.000

-0.0000.000

-0.000

-0.000

0.000

-0.000

-0.000

-0.000

-0.000

0.000

-0.000

-0.000

-0.0000.000

0.000

0.000-0

.000

0.000

-0.000

-0.000

-0.000

-0.000

0.000

-0.000

0.000

0.000

0.000

0.000

-0.000

0.000

-0.000

-0.000

-0.000

-0.000

0.000 0.

000

-0.000

0.000

-0.0000.

000

-0.000

-0.000

0.000-0

.000

0.000

0.000

-0.0000.

000

0.000

-0.000

0.002

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

a. Lx=16, Ly=4, J'=0.5, dimer-dimer correlation function

-0.000

0.000

0.000

-0.000

0.000

0.000

-0.000

-0.000

-0.000

0.000

-0.000

-0.000

-0.000

0.000

-0.000

0.000

0.000 -0.000

0.000

-0.000

-0.000

0.000

0.000

0.000

0.000

-0.000

-0.000

0.000

0.000

0.000

0.000

0.000

-0.000

-0.000

-0.000

0.000

0.000

-0.000

-0.000 0.000

0.000

0.000

-0.000

-0.000

0.000

-0.000

-0.000

-0.000

-0.000

-0.000

0.000

0.000

0.000

-0.000

0.000

-0.000

-0.000

0.000

0.000

-0.000

-0.000 0.000

-0.000

0.000

0.000

-0.000

0.000

0.000

0.000

-0.000

0.000

-0.000

-0.000

0.000

0.000

-0.000

0.000

-0.000

-0.000

-0.000

0.000

0.000

0.000 -0.000

-0.000

-0.000

0.000

-0.000

0.000

-0.000

-0.000

0.000

0.000

0.000

0.000

-0.000

-0.000

0.000

-0.000

-0.000

0.000

-0.000

-0.001

0.001

0.000 -0.001

0.001

-0.000

-0.000

0.000

-0.000

-0.000

0.000

0.000

-0.003

-0.000

0.002

-0.000

0.000

0.000

-0.000

0.001

0.000

0.003

-0.003

-0.003

-0.002 0.002

-0.000

0.000

-0.000

-0.001

-0.001

0.005

0.007

-0.012

0.008

-0.002

-0.004

0.005

0.001

-0.004

0.003

0.004

-0.008

0.002

0.039

-0.022

-0.000 0.018

-0.020

0.002

0.010

-0.005

0.005

-0.029

-0.023

0.144

0.011

-0.018

-0.029

-0.007

0.016

0.004

-0.197

0.144

0.071 0.000

0.000

-0.026

-0.001

0.004

-0.029

0.000

0.071

-0.029

-0.001

0.016

-0.026

0.004

0.005

0.144

-0.023

-0.022

-0.019 0.018

0.011

0.004

-0.007

-0.005

-0.008

0.005

0.039

-0.012

0.002

-0.020

-0.001

0.005

0.010

-0.004

0.002

0.001

-0.001

0.008

0.007

-0.003

-0.004 0.002

-0.002

0.003

0.001

-0.001

0.000

-0.000

-0.003

0.000

0.003

-0.000

-0.002

-0.000

-0.000

0.000

0.000

-0.000

-0.000

-0.003

0.000

0.001

0.002 -0.001

-0.000

-0.000

0.000

0.000

0.000

-0.000

-0.001

0.000

-0.000

0.000

0.000

-0.000

-0.000

0.000

-0.000

-0.000

0.000

0.000

-0.000

0.000

0.000 -0.000

0.000

-0.000

-0.000

-0.000

-0.000

0.000

0.000

-0.000

-0.000

-0.000

0.000

0.000

0.000

-0.000

-0.000

0.000

0.000

0.000

0.000

-0.000

-0.000 0.000

-0.000

0.000

0.000

-0.000

-0.000

-0.000

0.000

0.000

0.000

-0.000

-0.000

-0.000

0.000

-0.000

-0.000

0.000

0.000

-0.000

0.000

-0.000

0.000 0.000

-0.000

0.000

0.000

-0.000

-0.000

0.000

0.000

-0.000

0.000

0.000

-0.000

0.000

-0.000

-0.000

0.000

-0.000

0.000

-0.000

-0.000

0.000

0.000 -0.000

0.000

-0.000

-0.000

0.000

-0.000

0.000

-0.000

-0.000

0.000

0.000

0.000

-0.000

-0.000

0.000

-0.000

0.000

-0.000

-0.000

0.144

-0.000

-0.000

-0.000

-0.000

b. Lx=16, Ly=4, J'=0.5, dimer-dimer correlation function

FIG. 8: Dimer-Dimer correlation function at J ′ = 0.5. Dimer-Dimer correlation forJ ′ = 0.5 on the3 × 16 × 4 cylinders in the vacuumsector. The black bond in the middle of cylinder denotes the reference bond(i, j). The blue and red bonds represent the positive and negativedimer correlations, respectively.

11

-0.013

0.005

-0.013

0.005

-0.101

-0.013

-0.101

0.005

-0.013

0.005

-0.101

-0.025

0.012

-0.026

0.012

-0.025

-0.026 -0.025

0.012

-0.026

0.012

-0.025

-0.017

0.002

-0.017

0.002

-0.022

-0.017

-0.022

0.002

-0.017

0.002-0.022

-0.011

-0.022

-0.033

-0.022

-0.011

-0.033 -0.011

-0.022

-0.033

-0.022

-0.011

-0.008

0.010

-0.008

0.010

-0.038

-0.008

-0.038

0.010

-0.008

0.010

-0.038

-0.015

-0.006

-0.032

-0.006

-0.015

-0.032 -0.015

-0.006

-0.032

-0.006

-0.015

-0.016

0.007

-0.016

0.007

-0.027

-0.016

-0.027

0.007

-0.016

0.007

-0.027

-0.009-0.019

-0.032

-0.019

-0.009

-0.032 -0.009

-0.019

-0.032

-0.019

-0.009

-0.013

0.008

-0.013

0.008

-0.032

-0.013

-0.032

0.008

-0.013

0.008

-0.032

-0.010

-0.012

-0.030

-0.012

-0.010

-0.030 -0.010

-0.012

-0.030

-0.012

-0.010

-0.017

0.006

-0.017

0.006

-0.028

-0.017

-0.028

0.006

-0.017

0.006

-0.028

-0.007

-0.019

-0.030

-0.019

-0.007

-0.030 -0.007

-0.019

-0.030

-0.019

-0.007

-0.016

0.006

-0.016

0.006

-0.030

-0.016

-0.030

0.006

-0.016

0.006

-0.030

-0.008

-0.015

-0.029

-0.015

-0.008

-0.028 -0.008

-0.015

-0.028

-0.015

-0.008

-0.017

0.005

-0.017

0.005

-0.027

-0.017

-0.027

0.005

-0.017

0.005

-0.027

-0.006

-0.018

-0.029

-0.018

-0.006

-0.029 -0.006

-0.018

-0.029

-0.018

-0.006

0.005

-0.018

0.005

-0.028

-0.018

-0.028

0.005

-0.018

0.005

-0.028

-0.007

-0.017

-0.027

-0.017

-0.007

-0.027 -0.007

-0.017

-0.027

-0.017

-0.007

-0.016

0.005

-0.016

0.005

-0.028

-0.016

-0.028

0.005

-0.016

0.005

-0.028

-0.006

-0.017

-0.029

-0.017

-0.006

-0.029 -0.006

-0.017

-0.029

-0.017

-0.006

-0.020

0.005

-0.020

0.005

-0.028

-0.020

-0.028

0.005

-0.020

0.005

-0.028

-0.008

-0.018

-0.026

-0.018

-0.008

-0.026 -0.008

-0.018

-0.026

-0.018

-0.008

-0.014

0.006

-0.014

0.006

-0.028

-0.014

-0.028

0.006

-0.014

0.006

-0.028

-0.007

-0.015

-0.031

-0.015

-0.007

-0.031 -0.007

-0.015

-0.031

-0.015

-0.007

-0.021

0.005

-0.021

0.005

-0.029

-0.021

-0.029

0.005

-0.021

0.005

-0.029

-0.012

-0.019

-0.025

-0.019

-0.012

-0.025 -0.012

-0.019

-0.025

-0.019

-0.012

-0.008

0.007

-0.008

0.007

-0.030

-0.008

-0.030

0.007

-0.008

0.007

-0.030

-0.008

-0.011

-0.036

-0.011

-0.008

-0.036 -0.008

-0.011

-0.036

-0.011

-0.008

-0.024

-0.000

-0.024

-0.000

-0.031

-0.024

-0.031

-0.000

-0.024

-0.000

-0.031

-0.022

-0.019

-0.019

-0.019

-0.022

-0.019 -0.022

-0.019

-0.019

-0.019

-0.022

0.009

0.003

0.009

0.003

-0.024

0.009

-0.024

0.002

0.009

0.002

-0.024

-0.098

-0.098

-0.098

-0.018

-0.010

-0.010

-0.010

-0.010

Lx=16, Ly=4, J'=1.0, bond texture

FIG. 9: Nearest-neighbor bond textures atJ ′ = 1.0. The NN bond texturesBi,j on the3 × 16 × 4 cylinder atJ ′ = 1.0. The numbersdenote the amplitudes of bond textureBi,j = 〈Si · Sj〉 − e, wheree is the average of the horizontal NN bond energy in the bulk of cylinder.Here, we finde = −0.0385. The blue (red) color represents the positive (negative) bond texture.

2 4 6 8 10 12 14 16|i-j|

1e-06

1e-05

0.0001

0.001

0.01

<χiχ

j>

MSU(2)

=800

MSU(2)

=1600

MSU(2)

=2600

MSU(2)

=4600

MSU(2)

=6144

FIG. 10: Chiral-Chiral correlation function . The improvement ofchiral-chiral correlation function with the growing DMRG optimalstates for a3 × 18 × 6 cylinder atJ ′ = 0.2 in the vacuum sector.MSU(2) is the keptSU(2) states for obtaining the different chiral cor-relations, which are equivalent to about3200, 6400, 10000, 18000,and24000 U(1) states.

respectively.We denote the four groundstates from ED calculation as,

|ψk=01 〉, |ψk=0

2 〉, |ψk=π1 〉, |ψk=π

2 〉. (3)

Here, each wavefunction is being chosen as a real one. All theabove four groundstates preserve the time reversal symmetryand show a vanishing chiral order. According to the discus-sions in main text, we can construct the chiral states in eachsector as

|ψk=0L,R 〉 =

1√2(|ψk=0

1 〉 ± i|ψk=02 〉) (4)

|ψk=πL,R 〉 =

1√2(|ψk=π

1 〉 ± i|ψk=π2 〉) (5)

whereL(R) represents the left (right) chirality.

a1

a2

-17.0

-16.9

-16.8

-16.7

-16.6

Ener

gy

k=(0,0) k=(0,

b

FIG. 11: (a) Geometry of the36 sites kagome lattice with latticeconstant~a1,~a2. (b) Low-energy spectrum in the momentum spacek = (0, 0), (0, π) atJ ′ = 0.6.

Then we use two chiral states with the same chirality, forexample|ψk=0

L 〉 and|ψk=πL 〉, to calculate the modular matrix.

We search for the MESs in the space of the groundstate man-ifold using the following superposition wavefunction:

|Ψ〉 = c|ψk=0L 〉+

√

1− c2eiφ|ψk=πL 〉,

wherec ∈ [0, 1] andφ ∈ [0, 2π] are the superposition param-eters. In our calculation, we find that the global MESs takeφ = 0. As shown in Suppl. Fig.12, the two orthogonalMESs alonga1-direction are respectively located atc = 1√

2

andc = − 1√2, while the MESs alonga2-direction occur at

12

-1.0 -0.5 0.0 0.5 1.03.4

3.5

3.6

3.7

3.8

3.9

4.0

4.1

4.2

Entro

py

c

a1-directiona

-1.0 -0.5 0.0 0.5 1.05.5

5.6

5.7

5.8

5.9b

Entro

py

c

a2-direction

FIG. 12: Entropy for the superposition state|Ψ〉 = c|ψk=0L 〉 +√

1− c2|ψk=πL 〉 for the partition along (a)a1-direction and (b)a2-

direction. The black arrows show parameters for the MESs.

c = 0, 1. Therefore, we have two MESs alonga1-direction

|Ξa1

1 〉 =1√2(|ψk=0

L 〉+ |ψk=πL 〉), (6)

|Ξa1

2 〉 =1√2(|ψk=0

L 〉 − |ψk=πL 〉), (7)

and the two MESs alonga2-direction,

|Ξa2

1 〉 = |ψk=0L 〉, (8)

|Ξa2

2 〉 = |ψk=πL 〉. (9)

The modularS matrix is obtained from the overlaps be-tween the MESs of the two noncontractible partition direc-tions:

S = 〈Ξa1 |Ξa2〉 = 1√2

(

1 1

1 −1

)

, (10)

which is consistent with the prediction ofSU(2)1 confor-mal field theory about theν = 1/2 bosonic Laughlin state.Through the modular matrix above, we can extract the indi-vidual quantum dimensiond11(s) = 1 for quasiparticle11(s)and the fusion rules11 × 11 = 11, 11 × s = s, s × s = 11,which also determine the characteristic semion statisticsof thes quasiparticle.