Abelian and non-Abelian chiral spin liquids in a compact tensor network representation
AAbelian and Non-Abelian Chiral Spin Liquids in a Compact Tensor NetworkRepresentation
Hyun-Yong Lee, ∗ Ryui Kaneko, † Tsuyoshi Okubo, ‡ and Naoki Kawashima § Institute for Solid State Physics, University of Tokyo, Kashiwa, Chiba 277-8581, Japan Department of Physics, University of Tokyo, Tokyo 113-0033, Japan (Dated: January 27, 2020)We provide new insights into the Abelian and non-Abelian chiral Kitaev spin liquids on the starlattice using the recently proposed loop gas (LG) and string gas (SG) states [H.-Y. Lee, R. Kaneko,T. Okubo, N. Kawashima, Phys. Rev. Lett. 123, 087203 (2019)]. Those are compactly representedin the language of tensor network. By optimizing only one or two variational parameters, accurateansatze are found in the whole phase diagram of the Kitaev model on the star lattice. In particular,the variational energy of the LG state becomes exact (within machine precision) at two limits inthe model, and the criticality at one of those is analytically derived from the LG feature. It revealsthat the Abelian CSLs are well demonstrated by the short-ranged LG while the non-Abelian CSLsare adiabatically connected to the critical LG where the macroscopic loops appear. Furthermore,by constructing the minimally entangled states and exploiting their entanglement spectrum andentropy, we identify the nature of anyons and the chiral edge modes in the non-Abelian phase withthe Ising conformal field theory.
I. INTRODUCTION
Discovery of the fractional quantum Hall (FQH) effect had brought a paradigm shift in understanding of con-densed phases of matter. Exotic quantum liquid states,i.e., chiral spin liquids (CSL), were proposed as theground states of the FQH system , which cannot befeatured by Landau’s symmetry breaking theory but theso-called topological order . The topological order canbe interpreted as the pattern of long-range entangle-ment which leads to the ground state degeneracy de-pending only on the topology of system . Those CSLansatze successfully explained the nature of the FQH flu-ids such as the fractional statistics of quasiparticles (oranyons) . Furthermore, the anyons obeying the non-Abelian braiding statistics were theoretically realized inthe FQH system . Due to the robust topological degen-eracy against the local perturbations and exotic statisticsof anyons, the non-Abelian topological states have beenproposed as a promising platform for fault-tolerant quan-tum computing and thus attracted lots of attention inthe field of quantum information for the last decade .Another interesting feature of the FQH fluids and CSLs isthat the chiral gapless edge modes appear at the bound-ary of the system, and it leads to perfect heat conductionat the edge . The edge states are described by the con-formal field theories (CFT) which also characterize andhence have been employed to identify the topologicalorder . By solving the Kitaev model on the starlattice (KSM), Yao and Kivelson showed the existence ofthe CSL as an exact ground state of local Hamiltonianand found Abelian and non-Abelian phases character-ized by the topological degeneracies four and three onthe torus, respectively .The aim of this Letter is to understand the Abelianand non-Abelian Kitaev CSLs without referring to theMajorana fermion. Recently, a particular LG state and FIG. 1. (a) Schematic figures of the star lattice where the x -, y - and z -bonds defined in the model [Eq. (1)] are specifiedby red, blue and yellow colors, respectively. (b) Four localconfigurations and corresponding magnetic states of the LGstate [Eq. (3)] where | θ, γ (cid:105) is defined in Eq. (2), and x, y, z de-note each bond of the model. its extension, which is referred to as SG state, have beenproposed as ansatze for the Kitaev spin liquid (KSL) onthe honeycomb lattice in a compact tensor productstate (TPS) representation . The LG ansatz was foundto reflect most qualitative features of the KSL, and SGprovides a quantitatively accurate approximation to theKSL while keeping the qualitative features intact. Inwhat follows, we reinterpret the CSLs as the LG and SGstates and provide direct evidences identifying the topo-logical order in each phase. II. MODEL
The KSM is defined as ˆ H = − J (cid:88) (cid:104) ij (cid:105)∈ γ ˆ σ γi ˆ σ γj − J (cid:48) (cid:88) (cid:104) ij (cid:105)∈ γ (cid:48) ˆ σ γ (cid:48) i ˆ σ γ (cid:48) j , (1) a r X i v : . [ c ond - m a t . s t r- e l ] J a n where ˆ σ γi stands for the Pauli matrix with γ, γ (cid:48) = x, y, z , while (cid:104) ij (cid:105) γ and (cid:104) ij (cid:105) γ (cid:48) denote the nearest-neighborpair respectively on the intra-triangle ( γ ) and inter-triangle ( γ (cid:48) ) bonds connecting sites i and j as definedin Fig. 1 (a). Note that the Hamiltonian commuteswith two types of flux operators defined on the trian-gle plaquette ˆ V p = ˆ σ x ˆ σ y ˆ σ z and dodecagon plaquetteˆ W p = ˆ σ x ˆ σ z ˆ σ y ˆ σ x · · · ˆ σ y , where the site indices are de-fined in Fig. 1 (a). Therefore, the Hamiltonian is block-diagonalized, and each block is characterized by theset of the flux numbers or eigenvalues of flux operators { V p = ± , W p = ± } . Since the operator ˆ V p consistsof three Pauli matrices, the time-reversal transformationflips its flux number, i.e., T ˆ V p T − = − ˆ V p , and hence theTR-symmetry is spontaneously broken in eigenstates ofˆ H . It was found that the ground states do not breakany lattice symmetry (that is, the CSLs) and live in thevortex-free sector, i.e., { ˆ W p = 1 , ˆ V p = 1 } . In addition,the model exhibits a topological phase transition betweenthe non-Abelian and Abelian CSLs at J (cid:48) /J = √ . III. VARIATIONAL ANSATZEA. Loop gas states
We begin with defining a product state | Ψ( θ ) (cid:105) = ⊗ α | θ, γ α (cid:105) α with a local magnetic state given by (cid:104) θ, γ | ˆ σ γ (cid:48) | θ, γ (cid:105) = δ γγ (cid:48) cos θ + (1 − δ γγ (cid:48) ) sin θ √ , (2)and γ α being the inter-triangle bond at site α . For in-stance, at site-8 in Fig. 1 (a), we assign the state | θ, y (cid:105) .Applying the loop gas (LG) operator ˆ Q LG defined inRef. on top of | Ψ( θ ) (cid:105) results in a LG state | ψ LG ( θ ) (cid:105) ≡ ˆ Q LG | Ψ( θ ) (cid:105) , which is simply a superposition of all possibleloop configurations of magnetic states, i.e., . (3)Here, the empty site denotes the state | θ, γ α (cid:105) while theloop-occupied site stands for ˆ σ γ | θ, γ α (cid:105) depending on thedirection of loop as depicted in Fig. 1 (b). Due to the LGoperator, the desired symmetries, Z gauge redundancyand the vortex-freeness are guaranteed in the LG state .Notice that the norm of | ψ LG ( θ ) (cid:105) maps into the partitionfunction of the classical O (1) LG model with the localweights of loops being r = cos θ and q = sin θ/ √ . After simple algebra, the partitionfunction can even be mapped into that on the honeycomblattice, i.e., Z O (1) ( x ) with x being the fugacity per site: (cid:104) ψ LG ( θ ) | ψ LG ( θ ) (cid:105) = c Z O (1) (cid:18) q − r + r (cid:19) , (4) / *r*q*q c / -0.44-0.42-0.4-0.38 E exact | ψ L G | ψ S G (a) (b) dE[%] /π FIG. 2. (a) Variational energy of the optimized LG andSG ansatze (inset: dE = 1 − E/E exact of SG) (b) Optimalvariational parameter θ ∗ of | ψ LG (cid:105) as a function of φ . Here, r ∗ and q ∗ , which are determined by θ ∗ , are the optimal localweights of loop along the triangle and dodecagon plaquettes,respectively. where c is a constant(see Appendix A). Now, one canoptimize the variational parameter θ to minimize the en-ergy for a given ( J, J (cid:48) ). For simplicity, let us parameter-ize the exchange couplings as J (cid:48) /J = tan φ . However,at φ = 0, we consider J → ∞ while J (cid:48) = 1 being finiteto avoid the trivial solution at J (cid:48) = 0, and vice versa at φ = π/
2. We employ the corner transfer matrix renor-malization group (CTMRG) method to measure theenergy E = (cid:104) ψ LG | ˆ H| ψ LG (cid:105) and find θ ∗ ( φ ) minimizing theenergy at a given φ . The resulting E and θ ∗ are presentedin Fig. 2 (a) and (b), respectively, as a function of φ .Here, we present the exact energy E ex . The optimal localweights ( q ∗ , r ∗ ) = (sin θ ∗ / √ , cos θ ∗ ) are also presentedin Fig. 2 (b), which provide new insights into the natureof each phase. As one can see, the variational energyof the LG ansatz is quite accurate in 0 . π < φ ≤ . π ,where the energy deviation dE = 1 − E/E exact is lessthan 0 .
1% as shown in the inset of Fig. 2 (a). In particu-lar, the energy becomes exact (within machine precision)at φ = 0 . π at which r is maximized while q vanishes asshown in Fig. 2 (b). It indicates that the configurationswith only triangle loops and holes survive. In this sense,the Abelian CSL phase can be understood as the triangleloop gas in which longer loops are suppressed.Interestingly, the variational energy becomes also ex-act (within machine precision) as approaching the op-posite limit φ = 0 where the excitation gap closes as J (cid:48) /J . Notice that Z O (1) ( x ) in Eq. (4) becomes crit-ical at x c = 1 / √ , which leads to the critical LG stateat ( q c , r c ) = ( (cid:112) √ − , − √ q c , r c ) at φ = 0 which impliesthat the ground state is the critical LG state exhibitingmacroscopic loops. Furthermore, its low-energy physicsis described by the Ising CFT which is consistent withthe expected one . Therefore, according to thesecircumstantial evidences, we may conclude that the LGansatz at φ = 0 is the exact ground state. It tells us thatthe non-Abelian CSL around φ = 0 are well described bythe long-ranged LG states which is qualitatively distinctfrom the short-ranged feature of the Abelian CSL. Ex-cluing φ = 0, all LG ansatze map into the gapped phaseof Z O (1) ( x ), i.e., x < / √
3, ensuring its gapped nature.
B. String gas states
In the non-Abelian phase and around the phase tran-sition point, the variational energies of the LG ansatz areaway from the exact ones. Therefore, in order to lowerthe energy and find better ansatze, we apply the dimergas (DG) operator ˆ R DG as suggested in Ref. , , where the dimer on a bond (cid:104) ij (cid:105) stands for ˆ σ γi ˆ σ γj depend-ing on the bond, and different colors for the intra-bondand inter-bond. Note that one can introduce variationalparameters in the DG operator, which determine thefugacities of the dimers , and optimize them to lowerenergy. Here, we introduce two independent fugacitessuch that the one on the intra-triangle bonds ( c ) andanother one on the inter-triangle bond ( c ), i.e., ˆ R DG =ˆ R DG ( c , c ). This operator does not spoil the symme-tries and gauge structure of the ansatze . Then, weemploy the state | ψ SG ( c , c ) (cid:105) = ˆ Q LG ˆ R DG ( c , c ) | Ψ (111) (cid:105) as our ansatz which can be regarded as the SG state .Here, the state | Ψ (111) (cid:105) ≡ | Ψ(tan − √ (cid:105) is the the (111)-magnetic state where all spins point to (111)-direction.In fact, one can apply the DG operator on the generalproduct state | Ψ( θ ) (cid:105) to have an additional variationalparameter θ . Instead, for brevity, we fix the initial prod-uct state as | Ψ (111) (cid:105) and optimize c and c for a given φ .The obtained variational energy is presented in Fig. 2 (a),of which the inset is the energy deviation from the ex-act one (see Appendix B for details on the optimized pa-rameters). As one can see, the DG operator drasticallyreduces the energy and provides reasonably good ansatzeven around the transition point. Furthermore, the sec-ond derivative of the energy allows us to estimate thetransition point correctly(see Appendix B). Note that,in the non-Abelian phase, the variational energies areparticularly good around φ = 0 . π . This is because | Ψ (111) (cid:105) is used as the initial state, which is optimal at φ = 0 . π . We thus believe that one can obtain even bet-ter ansatze ( dE ∼ O (10 − )) throughout the non-Abelianphase by choosing the initial state | Ψ( θ ) (cid:105) properly. IV. MINIMALLY ENTANGLED STATES
So far we have considered the ansatz only on the in-finite system. Now we discuss the ansatz on the com-pact manifold (e.g., torus), where the topological sectorsallows us to distinguish the Abelian and non-Abelian phases. With periodic boundary conditions (PBC), oneshould also consider the so-called global flux measuredby the flux operator ˆΦ Γ = (cid:81) i ∈ Γ ˆ σ γ i i defined on a non-contractible closed path Γ . Its eigenvalues ± − y wrapping theinner tube of torus (say y -direction). One can verify thatmultiplying ˆΦ y is equivalent to the gauge twisting alonga closed path encircling the tube as illustrated below: , (5)where the five-leg tensor is composed of six onsite tensorsin the unit-cell, and . Here, the red squares denote ˆΦ y whereas the yellow onestands for the non-trivial element of the Z invariantgauge group of our ansatz, g = ˆ σ z . The detailed deriva-tion is presented in Appendix C. Note that RHS of Eq. (5)is equivalent to the procedure of creating a vortex pair,moving one of those around the loop Γ y , and then anni-hilating the pair. We define G y = ⊗ L y i =1 g as the string of g wrapping the inner tube where L y is the circumferencein units of the unit-cell, i.e., the ring of yellow tensorsin the right-hand side of Eq. (5). Therefore, acting ˆΦ y changes our ansatz | ψ (cid:105) (regardless of LG or SG) to a dif-ferent state | ψ y (cid:105) that is the G -inserted | ψ (cid:105) . However,since the square of ˆΦ y is identity, its eigenstates are sim-ply obtained by | ψ ± (cid:105) = | ψ (cid:105) ± | ψ y (cid:105) , and the subscript ± labels the global flux number, i.e., ˆΦ y | ψ ± (cid:105) = ±| ψ ± (cid:105) . Ina similar way, one can set the simultaneous eigenstates ofboth ˆΦ x and ˆΦ y , i.e., | ψ ( ± , ± ) (cid:105) living in one of four topo-logical sectors specified by ( ˆΦ x , ˆΦ y ) = ( ± , ± . Itcan be easily seen that the action of G y gives a mi-nus sign to all the configurations with odd number ofnon-contractible loops enclosing the hole of torus (say x -direction). Now, using those topologically degener-ate ansatze, we construct the so-called minimally entan-gled states (MES) , e.g., | I (cid:105) = | ψ (+ , +) (cid:105) + | ψ ( − , +) (cid:105) and / , L y =6 , L y =6 , L y =8 , L y =8 / , L y =6 , L y =6 , L y =10 , L y =10 | I | m | I | m | I | m | I | m (c) (d) = 0.02 − log − log √ | I | m (a) = 0.42 (b) | I | m L y S vN γ L y S vN − log log2 log √ | ψ L G | ψ S G | ψ L G | ψ L G **** ****
FIG. 3. The EE of | ψ LG (cid:105) on the infinitely long cylnder at(a) φ = 0 . π and (b) φ = 0 . π as a function of L y . Here, | I (cid:105) and | m (cid:105) denote two degenerate MESs (see text), and the blacksolid lines are the fitting curves. Plots of the TEE γ extractedfrom (c) the LG and (d) SG ansatze at each φ , and the greendotted line denotes the critical point ( φ c = π/ L ∗ y denotes the largest circumference for fitting the data. Forinstance, the TEE γ of L ∗ y = 6 is extracted by fitting the EEof L y = 4 and 6. | m (cid:105) = | ψ (+ , − ) (cid:105) + | ψ ( − , − ) (cid:105) characterized by each anyon ( I :trivial, m : vortex) flux threading the inner tube of torus.In this basis, one can read off the quantum dimension ( d i )of each anyon from the topological entanglement en-tropy (TEE), γ i = log( D/d i ), where i = I , m , · · · . Thetotal quantum dimension D = (cid:112)(cid:80) i d i is known tobe four for the Abelian and non-Abelian KSL phases .To this end, we employ the bulk-edge correspondence inTPS to evaluate the entanglement spectrum (ES) andentanglement entropy (EE) on the infinitely long cylin-der. Here, we impose PBC in the y -direction. Firstly,the results of TEE obtained from the LG and SG ansatzeare presented in Fig. 3. Here, (a) and (b) show the EEsin each sector obtained from | ψ LG (cid:105) at φ = 0 . π and φ = 0 . π , respectively, as a function of the circumference L y . As expected from the geometry of TPS , all EEsfollow the area law : S = αL y − γ i where α is a non-universal prefactor, and γ i is extracted by fitting the datawith linear functions (black solid lines). At φ = 0 . π , weobtained ( α, γ i ) = (0 . , . . , . I and m sector, respectively [Fig. 3 (a)]. ThoseTEE are remarkably close to the TEE in the vacuum k y φ = 0 . π, | I φ = 0 . π, | m (a) (b) k y FIG. 4. The ES of two topologically degenerate ansatze | I (cid:105) and | m (cid:105) with L y = 6 at φ = 0 . π . The level spacingsand degeneracy patterns of chiral modes in (a) and (b) areconsistent with three primary fields and their descendants inthe Ising CFT (see text for details). sector (i.e., log 2) and the σ -anyon (vortex) sector (i.e.,log √
2) of Ising anyon model . On the other hand,at φ = 0 . π [Fig. 3 (a)], both EEs almost perfectly fit to( α, γ i ) = (log 2 , log 2), which is consistent with the onefrom the toric code . Similarly, we have extracted γ i at each φ , and the results obtained from | ψ LG (cid:105) and | ψ SG (cid:105) are shown in Fig. 3 (c) and (d), respectively. Thoseof | ψ LG (cid:105) are in an excellent agreement with the ones ofIsing anyon model around φ = 0 and with the ones of thetoric code mostly in the Abelian phase [Fig. 3 (c)]. Mean-while, the SG ansatz gives almost consistent TEEs evenin the non-Abelian phase agreeing with the ones of Isinganyon model and predicts the transition point rather cor-rectly [Fig. 3 (d)].Furthermore, the identification of the topological ex-citations becomes even clearer from characteristic struc-tures in the ES . Figure 4 (a) and (b) present the ESsof | I (cid:105) and | m (cid:105) obtained by the SG ansatz at φ = 0 . π .Here, the circumference is L y = 6, and the horizontalaxis k y denotes the momentum. There are four branchesof two distinct chiral modes in the I -sector, which lin-early disperse in one direction. Those are highlightedby the red and blue solid lines in Fig. 4 (a). Assum-ing the close ESs (dashed boxes) as degenerate levels,the degeneracy pattern is consistent with the ones ofthe primary fields (blue) and ψ (red) and their descen-dants in the Ising CFT , respectively. On the otherhand, in the m -sector, we find six branches of a sin-gle chiral mode of which the degeneracy counting obeys { , , , , , , , , , · · · } , i.e., the characteristic of theprimary field σ and its descendants in the Ising CFT.In addition, the level spacings in the low-lying spectrumare in excellent agreement with the exact ones (see Ap-pendix D). From our MES setup, the state | m (cid:105) is expectedto accommodate the vortex at each boundary, and thusthe vortex is identified with the σ -anyon exhibiting non-Abelian braiding statistics . V. CONCLUSION
In this Letter, we show that the Abelian and non-Abelian CSL ground states of the KSM are well rep-resented by the LG and SG states. In particualr, atboth limits φ = 0 and π/
2, the LG states become ex-act. Further, the gap closing at φ = 0 is understood bymapping the norm of ansatz into the partition functionof the critical LG model. In addition, the fate of long-ranged loops is found to determine the Abelianess andnon-Abelianess of CSL. By constructing the MES andmeasuring its TEE, we directly show that our ansatzehost indeed the non-Abelian vortex with the quantumdimension d m = √
2. On the other hand, it becomestrivial, i.e., d m = 1, as the ansatz enters into the Abelianphase. We also identify the chiral edge modes in the non-Abelian phase with the Ising CFT, not the SU(2) Wess-Zumino-Witten theory conjectured in Ref. by exploit-ing the level spacing and their degeneracy patterns .We believe that the LG ansatze are the simplest CSLs ina compact representation, and therefore it could providea platform bridging the quantum loop models with theAbelian and non-Abelian topological states. It is alsoworth noting that our ansatze are the rigorous exampleexplicitly revealing that general TPSs can represent thechiral gapped states. In the case of the fermionic Gaus- sian TPS, there exists a no-go theorem prohibiting thechiral Gaussian TPS to be gapped. However, it was notso clear whether the theorem applies to generic TPSs ornot . We believe that our ansatze are the counterevidence against the generalization of the theorem. Intechnical aspects, two independent optimization schemes,which can be combined together, are introduced for theLG and SG ansatz. Those can be employed to studythe anisotropic Kitaev model on the honeycomb latticeand its extensions which are relevant to Kitaev materialssuch as α -RuCl . Also, the variational ansatze areof interest of the deformed topological wavefunction . ACKNOWLEDGMENTS
The computation in the present work was executedon computers at the Supercomputer Center, ISSP, Uni-versity of Tokyo, and also on K-computer (project-ID:hp190196). N.K.’s work is funded by MEXT KAKENHINo.19H01809. H.-Y.L. was supported by MEXT as “Ex-ploratory Challenge on Post-K computer” (Frontiers ofBasic Science: Challenging the Limits). T.O. was sup-ported by JSPS KAKENHI No.15K17701 and 19K03740.R.K. was supported by MEXT as ”Priority Issue on Post-K computer” (Creation of New Functional Devices andHigh-Performance Materials to Support Next-GenerationIndustries).
Appendix A: Norm of the zeroth ansatz
In this section, we show that the norm of the loop gas (LG) ansatz | ψ ( θ ) (cid:105) = ˆ Q LG | Ψ( θ ) (cid:105) can be exactly mappedinto the partition function of the O (1) loop gas model on the honeycomb lattice. To this end, we first note that theLG operator is an hermitian projector: ˆ Q † LG = ˆ Q LG and ( ˆ Q LG ) = N Γ ˆ Q LG where N Γ is the total number of loopconfigurations on the star lattice. The LG operator is efficiently represented by the tensor product operator , i.e.,ˆ Q LG = tTr (cid:81) α Q ss (cid:48) i α j α k α | s (cid:105)(cid:104) s (cid:48) | where tTr stands for the tensor trace, α labels the site index, the building block tensor Q ss (cid:48) ijk = τ ijk [(ˆ σ x ) − i (ˆ σ y ) − j (ˆ σ z ) − k ] ss (cid:48) , τ ijk = (cid:40) − i if i + j + k = 01 if i + j + k = 2 , (A1)and the indices of physical and virtual legs are s, s (cid:48) = 0 , i, j, k = 0 ,
1, respectively. Also, the virtual legs i, j and k lie on the x, y and z bonds in the model (see Fig. 1 (a) in the main text), respectively. Since the LG operatoris obtained by summing over all possible loop operators (that is, product of ˆ σ x , ˆ σ y and ˆ σ z along the loops), it isstraightforward to verify ( ˆ Q LG ) = N Γ ˆ Q LG using the manipulation rules of loop defined in Ref. . One can alsoeasily show its hermiticity using the Q -tensor in Eq. (A1). Now, let us compute the norm of the LG ansatz: (cid:104) ψ ( θ ) | ψ ( θ ) (cid:105) = (cid:104) Ψ( θ ) | ˆ Q † LG ˆ Q LG | Ψ( θ ) (cid:105) = N Γ (cid:104) Ψ( θ ) | ˆ Q LG | Ψ( θ ) (cid:105) = N Γ (cid:88) G ∈ Γ x G ( θ ) , (A2)where we used the hermiticity and idempotence (up to overall N Γ ), and G denotes each loop configuration, x G ( θ ) = (cid:104) Ψ( θ ) | ˆ Q G | Ψ( θ ) (cid:105) is the weight of a loop operator ˆ Q G . With the definition of | Ψ( θ ) (cid:105) and Eq. (2) in the main text, onecan easily verity x G ( θ ) = (sin θ ) l tG (cos θ/ √ l dG where l tG and l dG are the total lengths of partial loops along the triangleand dodecagon plaquettes, respectively, in the configuration G . Now, let us consider local loop configurations on asingle triangle plaquette . There are eight configurations on the single triangle plaquette which are depicted in theleft hand side of graphical equations below: , (A3)where r = cos θ and q = sin θ/ √ O (1) LG model on the honeycomb lattice, where theweights of hole and loop per site are 1 + r and q (1 + r ), respectively, and it is simply given by (cid:104) ψ ( θ ) | ψ ( θ ) (cid:105) = N Γ (cid:88) G (cid:48) ∈ Γ (cid:48) (1 + r ) n − n G (cid:48) [ q (1 + r )] n G (cid:48) = N Γ (1 + r ) n (cid:88) G (cid:48) ∈ Γ (cid:48) (cid:18) q (1 + r )1 + r (cid:19) n G (cid:48) . (A4)Here, Γ (cid:48) denotes all possible loop configurations on the honeycomb lattice, and n is the total number of sites onthe honeycomb lattice while n G (cid:48) stands for the total length of loops in a configuration G (cid:48) . The RHS is identicalto the partition function of the O (1) LG model on the honeycomb lattice, i.e., Z O (1) ( x ), with the loop fugacity x = q / (1 − r + r ), of which the critical point is x c = 1 / √ . Consequently, the norm of the LG ansatz maps to Z O (1) ( q / (1 − r + r )) which becomes critical at q c = (cid:113) (1 − r + r ) / √ r . Appendix B: optimal variational parameters in the first order ansatz
In this section, we present optimal variational parameters in the first order ansatz | ψ ( c , c ) (cid:105) =ˆ Q LG ˆ R DG ( c , c ) | Ψ (cid:105) , where | Ψ (cid:105) is the product state of local magnetic (111)-state. In a similar way to theLG operator, the dimer gas (DG) operator ˆ R DG ( c , c ) is also efficiently represented in the tensor network and bythe following building block tensor R ss (cid:48) ijk = ζ ijk (cid:2) (ˆ σ x ) i (ˆ σ y ) j (ˆ σ z ) k (cid:3) ss (cid:48) , ζ ijk = (cid:40) i + j + k = 0 c /c if i + j + k = 1 , (B1)where we assign c ( c ) if the non-zero element comes from the intra-triangle (inter-triangle) bond. The dimension anddirection of the virtual indices i, j, and k are the same as the ones of the Q -tensor in Eq. (A1). To be more specific, / P a r a m e t e r ( π ) / -4-2024 d E / d exact| (a) (b) FIG. 5. (a) The optimal variational parameters α and β in the first order ansatz | ψ ( α, β ) (cid:105) as a function of φ . (b) The secondderivative of the energy of the first order ansatz in terms of φ . (a) (b) FIG. 6. Schematic figure of the action of (a) the global flux operator W Γ on the tensor product states and (b) the string ofthe non-trivial element (yellow square) of the invariant gauge group of the LG operator along the y -direction. on the site 8 in Fig. 1 (a) in the main text on which the inter-triangle bond is the y -bond, we put the building blocktensor with ζ site 8 ijk = i + j + k = 0 c if j = 0 and k + i = 1 c if j = 1 and k + i = 0 . (B2)Note that the variational parameters c and c determine the fugacity of the dimer on the intra-triangle bond andinter-triangle bond, respectively, whereas the fugacity of hole is set to unity. Let us reparametrize the variationalparameters as follows: ˆ R DG ( c , c ) → ˆ R DG ( α, β ) with ζ ijk = (cid:40) cos β if i + j + k = 0sin β √ cos α/ sin β √ sin α if i + j + k = 1 . (B3)Now, one can vary two variational parameters α and β to minimize the energy expectation value of the Kitaev modelon the star lattice. Using the corner transfer matrix renormalization group method, we measured the expectationvalues and then found the optimal α and β at a given φ (see the main text) which are presented in Fig. 5 (a). Theenergy expectation values shown in Fig. 2(a) in the main text are obtained with the optimal parameters presented inFig. 5 (a). Interestingly, the optimal α , which determines the relative weight between dimers on the intra-triangle andinter-triangle bonds, increases linearly with the model parameter φ , which determines the relative strength betweenthe exchange couplings on the intra-triangle ( J ) and inter-triangle ( J (cid:48) ) bonds. Meanwhile, the parameter β , whichdetermines the relative weight between the hole and dimers, is optimized in 0 . π < β < . π . Note that one can alsointroduce complex fugacities, which lead to two more variational parameters, i.e., ( c , c ) → ( c e iη , c e iη ). However,we found that real fugacities always give the lowest energy throughout the model parameter φ . Figure 5 (b) shows thesecond derivative of the energy expectation value of | ψ (cid:105) in terms of φ . Here, the transition point expected from | ψ (cid:105) is a bit different from the exact one φ c = π/
3. However, considering the fact that the curve is obtained by numericaldifferenciation twice, its accuracy and smoothness are quite remarkable.
Appendix C: global flux
Here, we show that the LG and SG ansatze are not the eigenstates of the global flux operators, i.e., the Wilson loopoperator W Γ = (cid:81) i ∈ Γ σ α i i along a non-contractible loop Γ on a compact manifold where α = x, y and z dependingon the site i . To this end, we first note that, as shown in Ref. , the multiplication of the Pauli matrices on thephysical leg of local tensor of the LG operator is identical to the multiplication of the matrix v = (cid:18) i (cid:19) , (C1)and its conjugate on two virtual legs: σ xss (cid:48)(cid:48) Q s (cid:48)(cid:48) s (cid:48) ijk = v jj (cid:48) v ∗ kk (cid:48) Q ss (cid:48) ij (cid:48) k (cid:48) , σ yss (cid:48)(cid:48) Q s (cid:48)(cid:48) s (cid:48) ijk = v kk (cid:48) v ∗ ii (cid:48) Q ss (cid:48) i (cid:48) jk (cid:48) , σ zss (cid:48)(cid:48) Q s (cid:48)(cid:48) s (cid:48) ijk = v ii (cid:48) v ∗ jj (cid:48) Q ss (cid:48) ij (cid:48) k (cid:48) . (C2)Here, the tensor Q ss (cid:48) ijk denotes the local tensor of the LG operator, and repeated indices are implicitly summed over.Above relation can be simply described by the following graphical representation: , (C3)where the gray circle denotes the Q -tensor, black solid line stands for the physical leg (index s ) and red, blue andyellow solid lines are virtual legs (indices i, j and k ) on the x, y and z bonds of the Kitaev model. Also, the redsquare attached on the physical leg denotes the Pauli matrix. Note that a physical leg s (cid:48) is omitted in the graphicalrepresentation for simplicity. In what follows, using the above relation, we show how the LG and SG ansatze reacton the action of the global flux operator W Γ and how to construct the eigenstates of W Γ using them. Let us considerthe periodic boundary condition along the y -direction as defined in the main text and then act the operator W Γ alongthe y -direction as depicted in Fig. 6 (a). Now, using the relation in Eq. (C2), one can show that the ring of the tensornetwork, where the operator W Γ is applied, has the following equalities, , (C4)where the matrix g = σ z is the non-trivial element of the Z invariant gauge group of the LG operator . In thefirst equality, the relation in Eq. (C2) is applied, and we use relations vv † = 1 and vv T = − σ z . Finally, in the lastequality, the invariant gauge symmetry is used, i.e., g ii (cid:48) g jj (cid:48) g kk (cid:48) Q ss (cid:48) i (cid:48) j (cid:48) k (cid:48) = Q ss (cid:48) ijk . Therefore, applying the operator W Γ on our ansatze results in a different tensor network where a string of g , G = (cid:81) L y i =1 g , along the y -direction is insertedin the original state as illustrated in Fig. 6 (b). One can easily notice that such a G on the non-contractible loop cannot be eliminated by a gauge transformation, and therefore the resulting state is not identical to the original state,i.e. Fig. 6 (a) = Fig. 6 (b). That is, our ansatze are not the eigenstate of the global flux operator. However, since( W Γ ) = 1 and g = 1, it is easy to construct the eigenstate of W Γ using our ansatz, i.e., | ψ ± (cid:105) = | ψ (cid:105) ± | ψ G (cid:105) where | ψ (cid:105) denotes the LG or SG ansatz and | ψ G (cid:105) the G -inserted | ψ (cid:105) along the y -direction as shown in Fig. 6 (b). Then, thestate | ψ ± (cid:105) is the eigenstate of W Γ with the eigenvalue or global flux number ± Appendix D: entanglement spectrums and conformal towers
In the main text, the entanglement spectrums are presented as a function of the momentum k y in each topologicalsector, and their degeneracy patterns are discussed. Here, we directly compare the entanglement spectrums and theVirasoro towers of the Ising conformal field theory. In Fig. 7 (a), we compare the entanglement spectrums of thestring gas (SG) ansatz | ψ (cid:105) optimized at φ = 0 . π with the Virasoro characters of the primary operators having theconformal weights ∆ = 0 , / / . As one can see, the spectrums in the sector | I (cid:105) (left panel) can be regardedas the sum of two Virasoro towers of ∆ = 0 and 1 /
2, while the ones in the sector | m (cid:105) (right panel) match with theVirasoro tower of ∆ = 1 /
16. Furthermore, their spacings and degeneracy patterns are in excellent agreement with theexact ones up to the seventh level. In Fig. 7 (b), the entanglement spectrums of the loop gas ansatz | ψ (cid:105) optimized at i i i ( x4) ( x4) ( x4) ( x4) ( x4) ( x4) ( x4) ( x4) ( x4) ( x4) ( x4) ( x4) ( x6) ( x6) ( x6) ( x6) ( x6) ( x6) ( x6) | I | m / / | I | m (a) (b)
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