Detecting transition between Abelian and non-Abelian topological orders through symmetric tensor networks
DDetecting transition between Abelian and non-Abelian topological ordersthrough symmetric tensor networks
Yu-Hsueh Chen, Ching-Yu Huang, ∗ and Ying-Jer Kao † Department of Physics and Center for Theoretical Physics, National Taiwan University, Taipei 10607, Taiwan Department of Applied Physics, Tunghai University, Taichung 40704, Taiwan
We propose a unified scheme to identify phase transitions out of the Z Abelian topological order, includingthe transition to a non-Abelian chiral spin liquid. Using loop gas and and string gas states [H.-Y. Lee, R. Kaneko,T. Okubo, N. Kawashima, Phys. Rev. Lett. 123, 087203 (2019)] on the star lattice Kitaev model as an example,we compute the overlap of minimally entangled states through transfer matrices. We demonstrate that, similarto the anyon condensation, continuous deformation of a Z -injective projected entangled-pair state (PEPS) alsoallows us to study the transition between Abelian and non-Abelian topological orders. We show that the chargeand flux anyons defined in the Abelian phase transmute into the σ anyon in the non-Abelian topological order.Furthermore, we show that both the LG and SG states have infinite correlation length in the non-Abelian regime,consistent with the recent claim that a chiral PEPS has a gapless parent Hamiltonian. I. INTRODUCTION
In the past decades, significant efforts have been devotedto understand topologically ordered phases and topologicalphase transitions. Topological phases [1, 2] cannot be char-acterized by a local order parameter and can be characterizedby properties such as the ground-state degeneracy, and non-trivial quasiparticle statistics [3–7]. Recently, it is realizedthat these states can be understood through the quantum en-tanglement, using notions such as topological entanglemententropy [8–10], and entanglement spectrum [11–13]. Thelatter reveals the edge physics of the topological state, andcan be easily computed for the projected entangled-pair states(PEPS) [14], a tensor network that has been successfully rep-resented the ground state wavefunction for systems with bothconventional and topological orders. For a symmetry group G , the G -injective PEPS [15] forms a natural framework torepresent topological ordered systems. It encodes topologicalproperties in the local symmetries on the virtual dimensions.The parent Hamiltonian of a G -injective PEPS automaticallyencodes the degenerate ground state subspace and supportsanyonic excitations, allowing us to study the topological prop-erties through its entanglement degrees of freedom. However,it is shown that a G -injective PEPS does not guarantee a topo-logically ordered phase since the system can be driven intoa topologically trivial phase by a physical deformation of thelocal tensor [16–22]. For example, the phases and phase tran-sitions of the two dimensional (2D) toric code (TC) modelwith finite string tension, whose ground state is representedby a Z -injective PEPS [18], can be fully understood withinthis framework. Similar idea of detecting topological phasetransitions has also been generalized to non-Abelian cases re-cently [23–25].A new class of Z -injective ansatz called loop gas (LG) andstring gas (SG) is constructed to represent the ground state ofthe Kitaev models [5] on the honeycomb lattice [26–28]. Sur-prisingly, when the same ansatz is applied to the Kitaev model ∗ [email protected] † [email protected] on the star lattice, the entanglement entropy and spectrum sug-gest that flux anyon in the Z -topological order become the σ anyon in the non-Abelian chiral spin liquid (CSL) [29]. How-ever, exact results show that the ground state subspace shouldbe three dimensional in the non-Abelian regime [30], incon-sistent with the four-fold degenerate ground state structure forthe Z -injective PEPS.In this paper, we propose to use the overlap of MESs [31] asa unified framework to understand the phase transitions out ofa Z topological order. Using the gauge-symmetry preservedhigher-order tensor renormalization group (GSP-HOTRG) al-gorithm, the overlap can be obtained from the eigenvalue ofthe corresponding transfer matrix . Our results show that sim-ilar to the anyon condensation, the transition from an Abelianto a non-Abelian topologically ordered phase can be under-stood as both the charge and flux transmute into the σ anyon,resolving the mismatch between the dimension of the groundstate subspace for a Z -injective PEPS and degeneracy of theCSL ground state of the star lattice Kitaev model. We alsoshow that the correlation lengths of LG and SG states divergein the CSL regime, consistent with the recent claim that theparent Hamiltonian of a chiral PEPS is gapless [32–34].This paper is organized as follows, In Sec. II, we brieflyreview the properties of Z -injective PEPS. In Sec. III, weshow the relation between the overlap of MESs and the trans-fer matrix. In Sec. IV, we revisit the toric code with string ten-sion and demonstrate how to detect anyon condensation tran-sitions using the MES overlap picture. In Sec. V, we apply themethod to the Kitaev model on the star lattice and show howto detect the Abelian to non-Abelian topological order tran-sition using MES overlap. We show that the flux and chargeanyons transmute into the σ anyon from the full transfer ma-trix spectrum. We conclude in Sec. VI. II. SYMMETRIC PEPS AND ANYONS
A translational invariant PEPS wave function can be writtenin terms of a local tensor A iαβγδ with the physical index i and a r X i v : . [ c ond - m a t . s t r- e l ] F e b FIG. 1. (a) A Z -injective PEPS is invariant under A ( u g ⊗ u g ⊗ u † g ⊗ u † g ) = A , where g ∈ { I, Z } . On the other hand, wecan find an operator R α such that it transforms non-trivially un-der the group action. (b) For any Z -injective PEPS A , we canconstruct a parent Hamiltonian such that its ground state subspaceon the torus is spanned by | ψ A ( g, h ) (cid:105) , ∀ g, h ∈ { I, Z } (c) Theanyon excitation can be constructed by either attaching a string of u g (flux) or applying R α (charge) on the virtual dimension. (d) TheMES basis relating to anyon excitations can be constructed through | ψ A ( g, α ) (cid:105) = (cid:80) h ∈ Z χ α ( h ) | ψ A ( g, h ) (cid:105) . virtual indices α, β, γ, δ as | ψ A (cid:105) = (cid:88) i ,...,i N tTr (cid:0) A i A i . . . A i N (cid:1) | i , i , . . . , i N (cid:105) , (1)where the tensorial trace is over the virtual indices. A Z -invariant PEPS [Fig. 1(a)] is represented by a local ten-sor A that is invariant under the global Z symmetry, i.e., A ( u g ⊗ u g ⊗ u † g ⊗ u † g ) = A , where u g is a representationof the group Z with g ∈ { I, Z } . If Z is the only sym-metry of A , we say that A is Z -injective [15]. For a Z -injective PEPS, the ground state subspace of the parent Hamil-tonian is spanned by the two non-contractible loop operators, ( u ⊗ L x g , u ⊗ L y h ) , ∀ g, h ∈ { I, Z } acting on the states, which wedenote it as | ψ A ( g, h ) (cid:105) [Fig. 1(b)]. This arises from the factthat the parent Hamiltonian cannot detect these loop operatorslocally, and we can always use the pull-through condition todeform the non-contractible loop operators. The Z -injectivetensor naturally supports anyonic excitations that cannot becreated locally on the systems. For example, a flux excitationcan be created by attaching a string of u g , g ∈ { I, Z } on thevirtual bond. A charge excitation can be created by acting anoperator R α on the virtual bond which transform non-triviallyunder the group action R α u g = χ α ( g ) u g R α , where χ is thecharacter and α designates the irreducible representation of Z [Fig. 1(c)] [15, 18].Note that far away from the renormalization group fixedpoint, the excitation is dispersive and local action of u g and R α may not correspond to the eigenstates of the parent Hamil-tonian. Instead, the excited states should be created by thesuperposition of local excitations. However, as shown inRef. [17], these local actions remain crucial to extract anyonicinformation. III. MINIMAL ENTANGLED STATES AND TRANSFERMATRICES
Ground states subspace and the anyonic excitation areclosely related, and we can construct a special ground statebasis, the minimally entangled states (MESs), to reflect theanyonic excitation of the topological phases. Basically, theMES basis can be obtained by creating a pair of anyonson a torus, wrapping them around a closed non-contractableloop, and finally annihilating them. To be specific, a MESis the eigenstate of the Wilson loop operator with a def-inite type of anyon excitation; therefore we can constructthe MESs in the ground state subspace by | ψ A ( g, α ) (cid:105) = (cid:80) h ∈ Z χ α ( h ) | ψ A ( g, h ) (cid:105) , with g ∈ { I, Z } [Fig. 1(d)]. Wethen follow the same notation in Ref. [16] to denote g = I ( Z ) as 0( π )-flux and the parity α , as even (trivial) andodd (non-trivial). The four MESs | I (cid:105) , | e (cid:105) , | m (cid:105) , | (cid:15) (cid:105) then cor-respond to | ψ A (0 , e ) (cid:105) , | ψ A (0 , o ) (cid:105) , | ψ A ( π, e ) (cid:105) , | ψ A ( π, o ) (cid:105) , re-spectively. The overlap of MESs provides the informationabout the identities of the anyonic excitations and can be ob-tained from a transfer matrix (TM) [Fig. 2 (a) ]. Starting froma local tensor A representing a Z topological order, we form adouble tensor E [Fig. 2 (b)] by contracting the physical indicesof A and its adjoint A ∗ , E ≡ (cid:80) s ( A si,j,k,l ) × ( A si (cid:48) ,j (cid:48) ,k (cid:48) ,l (cid:48) ) ∗ . Thecorresponding transfer matrix is given by T ≡ tTr( E E · · · E L ) . (2)The minimally entangled topological sectors corresponding tothe quasi-particles can be obtained by inserting the string op-erator Z g = u Zg , ( g = 0 , along the cylinder direction andchoosing the projector P α ( α = even, odd) in either the bra orket. This gives us 16 blocks of transfer matrices. Since eachtopological sector corresponds to an MES, we can label eachblock by the overlap of MESs (cid:104) a | b (cid:105) , with a, b = I, e, m, (cid:15) .The transfer matrices (Fig. 2 (c)) are defined as T (cid:104) a | b (cid:105) = T g (cid:48) ,α (cid:48) g,α ≡ P α (cid:48) α (cid:2) tTr( E E · · · E L Z g (cid:48) g ) (cid:3) P α (cid:48) α . (3)Here a = ( g, α ) and b = ( g (cid:48) , α (cid:48) ) . g ( α ) and g (cid:48) ( α (cid:48) ) label thestring (parity ) operator in the ket and bra layer. The largesteigenvalue λ (cid:104) a | b (cid:105) of T (cid:104) a | b (cid:105) gives the overlap of the two MESsin the thermodynamic limit.In the following, we will directly refer the transfer matrixfor a given block as (cid:104) a | b (cid:105) . The transfer matrices can be di-vided into four types: (a) α = α (cid:48) and g = g (cid:48) corresponds tothe regular TM computing the norm of the MES, (cid:104) a | a (cid:105) , with a = I, e, m, (cid:15) ( red color in Fig. 2(c)), (b) α (cid:54) = α (cid:48) and g = g (cid:48) corresponds to the TM measuring the charge difference be-tween the bra and ket (blue), (c) α = α (cid:48) and g (cid:54) = g (cid:48) corre-sponds to the TM measuring the flux difference between thebra and ket (yellow), (d) α (cid:54) = α (cid:48) and g (cid:54) = g (cid:48) corresponds to theTMs measuring the both charge and flux (fermion) differencebetween the bra and ket (green).Far away from the renormalization group fixed point, thesubleading eigenvalues of the transfer matrix can be inter-preted as the excitation energy [17, 35], and the dominanteigenvalues of the transfer matrices measuring the charge(flux) difference can be interpreted as the charge (flux) ex-citation energy [17]. Similarly, the green blocks in Fig. 2(c) ( FIG. 2. (a) Overlap of two-dimensional infinite PEPS states can beregarded as a one-dimensional transfer matrix. (b) A double tensoris formed by contracting tensor A on a lattice site a with its adjointtensor A ∗ over the physical index. (c) Sixteen blocks of the transfermatrices. α (cid:54) = α (cid:48) and g (cid:54) = g (cid:48) ) correspond to the fermionic excitation.This remarkable correspondence makes the relation betweenMES and anyon more apparent, allowing us to use these 16transfer matrices to study anyon condensation. As we willdiscuss in the following section, by tuning the wave functionwithout spoiling the Z -injectivity, it is possible to drive thesystem from a topological ordered phase to a topological triv-ial phase through the condensation of anyons.It is also possible that an anyon can transmute into anothertype of anyon. For example, when the D ( Z ) quantum doublemodel is continuously deformed to the toric code or the doublesemion model, some of the anyons distinct in the D ( Z ) phasecan be identified as the same [18, 19]. Similarly, in the caseof the Z -injective PEPS, we can ask which anyons we canidentify as the same.Since the TM is periodic around the cylinder, we can la-bel the states with the momentum quantum number. Inter-estingly, it has been shown in Ref. [17] that the momentumquantum number of | (cid:15) (cid:105) will be shifted by half a spacing, i.e., k = 2 π ( n + ) /L , where n = 0 , . . . , L − and L is the cir-cumference of the cylinder. This momentum polarization [36]makes | (cid:15) (cid:105) impossible to become other MESs. This makessense in the anyon condensation picture that fermion can nevercondense (corresponds to | (cid:15) (cid:105) (cid:54) = | I (cid:105) ), which has been provenin the more general case [18]. Discarding the possibility that | (cid:15) (cid:105) becomes | e (cid:105) or | m (cid:105) , we are left with the only choice toidentify | e (cid:105) and | m (cid:105) as the same state. Later we will see thatthe LG and SG states satisfy this condition.In order to compute the dominant eigenvalue of each TMblock in the thermodynamic limit, we employ the higher-ordertensor renormalization group method to merge the sites. Tokeep the block structure during coarse graining, it is importantto carefully preserve the symmetry. Here we develop a GSP-HOTRG method which extends the idea in Ref. [37] to com-pute the spectrum of the transfer matrices by coarse-grainingalong the cylinder direction (see App. A for details). FIG. 3. (a) Dominant eigenvalues of the transfer matrices for TCwith L = 1 (dashed lines), L = 8 (dotted lines), and L = 256 (solidlines). (b) Dominant eigenvalues of the transfer matrix for β x = 0 and β z = β cz with D cut = 80 . IV. TORIC CODE WITH FINITE STRING TENSION
Here we revisit the toric code (TC) model with finitestring tension, which is the simplest example with the phasetransition from a topological order to a topologically trivialphase [16, 17]. We add the string tension by applying the op-erator Q e ( β x , β z ) = exp (cid:16) β x σ xe + β z σ ze (cid:17) to the TC, | Ψ( β x , β z ) (cid:105) = (cid:89) e Q e ( β x , β z ) × (cid:89) v (cid:32) (cid:89) e (cid:51) v σ xe (cid:33) (cid:89) p (cid:89) e ∈ ∂p σ ze | Ω (cid:105) , (4)where e/p labels the the vertex/plaquette, and | Ω (cid:105) indicatesthe fully polarized spin state | Ω (cid:105) = ⊗ e | ↑(cid:105) e . For β x = 0 and β z → ∞ , the system is driven to the charge condensed (CC)phase. On the other hand, as β x → ∞ and β z = 0, the systemis driven to the flux condensed (FC) phase. Therefore, it is ex-pected that by tuning the parameters β x , β z , phase transitionswill occur.At the Z topological order (TO) fixed point, the eigenval-ues λ (cid:104) a | a (cid:105) = 1 , ( a = I, e, m, (cid:15) ) , and zero otherwise, indicat-ing these four MESs are orthonormal. At the fixed point of theCC phase, λ (cid:104) I | I (cid:105) = λ (cid:104) I | e (cid:105) = λ (cid:104) e | I (cid:105) = λ (cid:104) e | e (cid:105) = 1 , and zerootherwise, suggesting that sector | e (cid:105) is identical to | I (cid:105) while | m (cid:105) and | (cid:15) (cid:105) are confined. Similarly, at the fixed point of theFC phase, we have λ (cid:104) I | I (cid:105) = λ (cid:104) I | m (cid:105) = λ (cid:104) m | I (cid:105) = λ (cid:104) m | m (cid:105) = 1 .Along the β x = 0 axis, there exists a phase transitionfrom the TO to the CC phase as shown in Fig. 3. As we in-crease β z , only the regular and charge difference TMs [redand blue blocks in Fig. 2(c)] are non-zero, and there are onlyfour distinct eigenvalues: λ (cid:104) I | I (cid:105) = λ (cid:104) e | e (cid:105) , λ (cid:104) m | m (cid:105) = λ (cid:104) (cid:15) | (cid:15) (cid:105) , λ (cid:104) I | e (cid:105) = λ (cid:104) e | I (cid:105) , λ (cid:104) m | (cid:15) (cid:105) = λ (cid:104) (cid:15) | m (cid:105) . We choose one in eachas a representative. The system exhibits phase transition at β z = β cz ≈ . [ Fig. 3(a)].We find λ (cid:104) a | b (cid:105) does not have significant change for L > ; in the following, we will use L = 256 data to representthe thermodynamic limit. For β z < β cz , λ (cid:104) m | m (cid:105) = λ (cid:104) I | I (cid:105) = 1 and λ (cid:104) I | e (cid:105) = λ (cid:104) m | (cid:15) (cid:105) < , which is consistent with Fig. 2(c)that the blocks in red and blue blocks should be regarded asthe same, respectively. For β z > β cz , λ (cid:104) I | e (cid:105) = λ (cid:104) I | I (cid:105) = 1 and λ (cid:104) m | m (cid:105) = λ (cid:104) m | (cid:15) (cid:105) < . This indicates that the classificationin Fig. 2(c) no longer applies; instead, we should identify theblocks in the first (fourth) column as the same. At L = 1 and , for β z < β cz , while λ (cid:104) I | e (cid:105) = λ (cid:104) m | (cid:15) (cid:105) in the thermodynamiclimit, λ (cid:104) I | e (cid:105) is always larger than λ (cid:104) m | (cid:15) (cid:105) . This is in fact due tothe π/L shift of momentum for (cid:15) as mentioned in Sec. III. Inparticular, at β z = β cz , λ (cid:104) m | m (cid:105) and λ (cid:104) I | e (cid:105) are always the sameregardless of the system size, as shown in Fig. 3(b). There-fore, we can accurately identify the critical point by using thecrossing of λ (cid:104) m | m (cid:105) and λ (cid:104) I | e (cid:105) merely from a single tensor.Note that this is only true for β x = 0 . For β x (cid:54) = 0 , if we con-tinuously deform β z , the crossing point will shift as we keepincreasing L . This arises from the fact that at in this scenario,not only regular and charge difference blocks but the flux andfermion difference blocks are also non-zero. However, evenif we fix β x to other values than , after L > , the crossingpoint is almost fixed, meaning that we can still identify thecritical point using very small system sizes.At L = 256 , λ (cid:104) I | e (cid:105) = λ (cid:104) m | m (cid:105) → . We can interpret λ (cid:104) I | e (cid:105) → as | e (cid:105) begins to condense and λ (cid:104) m | m (cid:105) → as | m (cid:105) begins to confine. On the other hand, as mentioned inSec. III since no other MES can become exactly the same as | (cid:15) (cid:105) , we know that λ (cid:104) m | (cid:15) (cid:105) → indicates a gapless excitation.Similarly, for the TO to FC transition, only the regular andflux difference TMs (red and yellow blocks in Fig. 2(b)) arenon-zero. In fact, all the above observations and argumentscan be directly adopted to this case once we switch the chargedifference to flux difference.This idea of using the non-vanishing TMs to distinguishphases, surprisingly, can also be extended to the Abelian tonon-Abelian TO transition, as we will demonstrate in the nextsection. V. KITAEV MODEL ON THE STAR LATTICE
We extend the ideas developed in the Sec. IV to study thephase transition from an Abelian to a non-Abelian spin TO.We will study the quantum phase transition of the Kitaevmodel on the star lattice by studying TMs built from the Z -injective LG and SG states [26–28]. The Hamiltonian is de-fined as [30] H = − J (cid:88) (cid:104) i,j (cid:105) γ S γi S γj − J (cid:48) (cid:88) (cid:104) ij (cid:105)∈ γ (cid:48) S γ (cid:48) i S γ (cid:48) j (5)where (cid:104) i, j (cid:105) γ and (cid:104) i, j (cid:105) (cid:48) γ are the pairs on the intratriangle( γ = x, y, z ) and the intertriangle ( γ (cid:48) = x (cid:48) , y (cid:48) , z (cid:48) ) linksconnecting site i ans j as shown in Fig.4(a), respectively.The Hamiltonian can be block diagonalized by the eigen-values of two types of flux operators defined on the trian-gle plaquette ˆ V p = ˆ σ z ˆ σ x ˆ σ y and the dodecagon plaquete ˆ W p = ˆ σ x ˆ σ z ˆ σ y ... ˆ σ y , where ˆ σ i , i = x, y, z is the Pauli ma-trix. To gain insights into the model, we consider two ex-treme limits: the isolated-dimer limit ( J = 0 , J (cid:48) = 1 ) and the FIG. 4. (a) Star lattice with Kitaev-like interactions. (b) Theisolated-dimer and isolated-triangle limits of the Hamiltonian. (c)The initial product state of the LG state. isolated-triangle limit ( J = 1 , J (cid:48) = 0 ) [Fig.4(b)]. The pertur-bative study in Ref. [38] shows that in the isolated-dimer limit,while the Hamiltonian does not map exactly onto the standardtoric code, the ground state is the same as the toric code on thehoneycomb lattice. On the other hand, the isolated-trianglelimit can be mapped onto the Kitaev honeycomb model atthe isotropic point which exhibits a non-Abelian topologicalorder. This suggests that there should exist a phase transi-tion between the two phases. Exact results shows that themodel has two distinct gapped phases: Z topological orderwhen J (cid:48) /J > √ and non-Abelian CSL with Ising anyon,when J (cid:48) /J < √ . The latter can be distinguished from theformer by its three-fold topological degeneracies which canbe labeled using MES basis in Ising anyon: | I (cid:105) , | σ (cid:105) , | (cid:15) (cid:105) . Inboth regime, the ground states live in the vortex-free sector { W p = 1 , V p = 1 } . A. Loop gas and string gas states
Let us first consider an LG operator: 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) | with the LG tensor defined as Q = I , Q = − iU x , Q = − iU y , Q = − iU z . (6)where U γ = e iπS γ , γ = x, y, z is the π -rotation operator fora given spin (Fig. 5(a)) [28]. One can easily verify that theLG tensor is invariant under the global Z symmetry on thevirtual dimension: Q ( u g ⊗ u g ⊗ u g ) = Q with u g = I , σ z .Therefore, applying ˆ Q LG on any injective PEPS yields a Z -injective PEPS. ˆ Q LG is a projector to the vortex-free spacesuch that ˆ W p ˆ Q LG = ˆ Q LG ˆ W p = ˆ Q LG , ˆ V p ˆ Q LG = ˆ Q LG ˆ V p =ˆ Q LG . Another interesting property of ˆ Q LG is that the creationof flux anyon pair discussed in Sec.II now corresponds to twovortices W p = − at the endpoint of the string u ⊗ Lg [26].The LG state can then be obtained by applying ˆ Q LG onan initial product state | Ψ( θ ) (cid:105) = ⊗ α | ψ α ( θ ) (cid:105) where α is thesites index for a given triangular plaquette and | ψ α ( θ ) (cid:105) = | θ, x α (cid:105)| θ, y α (cid:105)| θ, z α (cid:105) (Fig. 4(c)). The magnetic state | θ, γ α (cid:105) satisfies (cid:104) θ, γ | σ γ (cid:48) | θ, γ (cid:105) = δ γ (cid:48) γ cos θ + (1 − δ γ (cid:48) γ ) sin θ √ , (7)where θ is a tunable parameter and γ, γ (cid:48) = x, y, z . Tosimplify the notation, we follow the convention in Ref. [29]to parametrize the Hamiltonian H = H ( φ ) with J (cid:48) =sin( φ ) , J = cos( φ ) . The ground state for a given Hamil-tonian H ( φ ) can then be obtained by variationally optimizingthe parameter θ to find the lowest energy.To gain more insights, we first consider two limits wherethe LG state is the exact ground state. In the isolated-dimerlimit, H ( φ = π/ , the ground state degeneracy is expo-nentially large with the system size, and one of the groundstate basis is a product state | Ψ (cid:105) = ⊗ α | ψ α (cid:105) with | ψ (cid:105) = (cid:0) | x, + (cid:105)| y, + (cid:105)| z, + (cid:105) (cid:1) , where | γ, ±(cid:105) is the eigenvector of σ γ with ± eigenvalues for γ = x, y, z . This basis is the initialproduct state for the LG states with θ = 0 since (cid:104) γ | σ γ (cid:48) | γ (cid:105) = δ γ (cid:48) γ , and thus one can identify | γ, + (cid:105) = | θ = 0 , γ (cid:105) . If weslightly deviate from φ = π/ , the state is no longer theground state. However, we expect the ground state of themodel to live in the vortex-free sector, and we can apply ˆ Q LG to project it back to the vortex-free space, again giving the LGstate at θ = 0 .Using the fact that Q , Q , Q are the π -rotation op-erator (up to a phase factor) around the x, y, z -axes, one canderive the resulting state of the LG operator on | θ = 0 , γ (cid:105) (i.e., | γ, + (cid:105) ) as in Fig. 5(b). Now we can combine three dif-ferent initial states together to form a triangular product state: | x, + (cid:105)| y, + (cid:105)| z, + (cid:105) (Fig. 5 (c)). For a given set of virtual in-dices, the triangular LG state can be written exactly as the sumof two terms due to the Z -invariance. Using the relation inFig. 5(b), one can find that the two terms are exactly the same,as shown in Fig. 5(c). Furthermore, the physical state withdifferent virtual indices, e.g., | y, + (cid:105) , | y, −(cid:105) and | z, + (cid:105) , | z, −(cid:105) in Fig. 5(c), are orthogonal. This property, combining withthe fact the the tensor is Z -invariant, guarantees that the LGstate with θ = 0 is Z -isometric [15]. Interestingly, the Z -isometry suggest that LG state at θ = 0 is the RG fixed pointof Z topological order. This is consistent with the fact thatin the isolated-dimer limit, the ground state is the same as thethat for the TC on the honeycomb lattice [38]. Note that sinceSG states is an extension for LG states, it is also Z -isometricat φ = π/ . The property of Z -isometry allows us to viewthe toric code with string tension (discussed in Sec. IV) andLG, SG with different parameters on the same footing.On the other hand, in the isolated-triangle limit, H ( φ = 0) ,the LG state is the exact ground state [29, 38]. However, for < φ < π , the energy of the optimized LG state is higherthan the exact value [29, 30]. Therefore, instead of using theoptimized LG state for a specific Hamiltonian H ( φ ) , in thefollowing we tune the parameter θ in the LG state to study itsproperty.Similarly, for the SG state, we introduce the dimer gas (DG)operator ˆ R DG ( α, β ) = tTr (cid:81) γ R ss (cid:48) i γ j γ k γ ( α, β ) | s (cid:105)(cid:104) s (cid:48) | with aDG tensor R ss (cid:48) ijk ( α, β ) = ζ ijk ( α, β )[( σ x ) i ( σ y ) j ( σ z ) k ] ss (cid:48) . (8) FIG. 5. (a) Non-zero elements of the LG tensor. Here we denote thethe virtual index 0(1) as black(red) leg. (b) The resulting states (upto a phase factor) of LG operators acting on | x, + (cid:105) for a given setof virtual indices. Similar expression can be derived for the initialstates | y ( z ) , + (cid:105) . (c) The resulting states (up to a phase factor) ofLG operators acting on | ψ (cid:105) = | x, + (cid:105)| y, + (cid:105)| z, + (cid:105) for a given set ofvirtual indices. Here we use the thick lines to denote that the virtuallegs are contracted. where ζ ijk ( α, β ) = (cid:40) cos β if i + j + k = 0sin α if i + j + k = 1 . (9)The SG state can be constructed as | ψ SG ( α, β ) (cid:105) =ˆ Q LG ˆ R DG ( α, β ) | ψ ( θ = tan − √ (cid:105) [29]. Since the SG stateyields quite accurate ground state energy for the star-latticeKitaev model, instead of considering the SG state as a two-parameter family of Z -injective tensor, we label them usingthe Hamiltonian parameter φ instead. B. Overlap of minimally entangled states
In the following, we study the topological properties of theLG and SG states by computing the overlap of MESs, whichcorresponds to the dominant eigenvalues of the TM blocks, λ (cid:104) a | b (cid:105) .As we continuously change the parameter for both the LGand SG states, we find that only the regular and the TMsmeasuring the fermion difference (red and green blocks inFig. 2(c)) are non-zero, and there are only four distinct eigen-values, λ (cid:104) I | I (cid:105) = λ (cid:104) (cid:15) | (cid:15) (cid:105) , λ (cid:104) m | m (cid:105) = λ (cid:104) e | e (cid:105) , λ (cid:104) m | e (cid:105) = λ (cid:104) e | m (cid:105) ,and λ (cid:104) I | (cid:15) (cid:105) = λ (cid:104) (cid:15) | I (cid:105) , while others are zero. Therefore, in eachcategory we choose one as the representative. Note that since λ (cid:104) I | I (cid:105) is always the largest, we normalize it to 1.Figure 6 shows the overlaps of the LG states for L = 1 , ,and . For L = 1 , , we can see that as θ increase from to θ c = cos − (2 − √ , λ (cid:104) m | m (cid:105) gradually decrease and λ (cid:104) m | e (cid:105) gradually increase. At θ = θ c , these two eigenvalues are ex-actly the same, meaning that we have reached the transitionpoint.However, if we keep increasing θ , the λ (cid:104) m | m (cid:105) and λ (cid:104) m | e (cid:105) do not cross. Unlike the level crossing for the charge or FIG. 6. (a) Dominant eigenvalues of the transfer matrix for LG stateswith L = 1 (dashed lines), L = 8 (dotted lines), and L = 256 (solidlines). (b) Dominant eigenvalues of the transfer matrix for θ = θ c with D cut = 80 . flux condensation transition in the toric code with string ten-sion (see Sec. IV), here the dominant eigenvalues for the TMblocks only touch, indicating that θ < θ c and θ > θ c arein the same phase. At system size L = 256 , we find that λ (cid:104) m | e (cid:105) = λ (cid:104) I | (cid:15) (cid:105) < for θ (cid:54) = θ c . This means that (cid:104) m | e (cid:105) and (cid:104) I | (cid:15) (cid:105) should be regarded as the same excitation, and we groupthem with the same (green) color in Fig. 2(c). Therefore, allthe LG states live in the topologically ordered phase, exceptat θ = θ c .At the transition point θ = θ c , all λ ’s approach 1 as the sizeincreases. Similar to the transition points studied in Sec. IV, λ (cid:104) I | (cid:15) (cid:105) → indicates a gapless excitation; however, we caninterpret λ (cid:104) m | e (cid:105) → as | e (cid:105) and | m (cid:105) become the same state..In addition, the spectrum of (cid:104) m | (cid:15) (cid:105) shows no matter how largethe system size is, it will always have two degenerate fixedpoint; on the other hand, (cid:104) m | m (cid:105) and (cid:104) (cid:15) | (cid:15) (cid:105) have a unique fixedpoint. This means that | m (cid:105) can never be identified with | (cid:15) (cid:105) .We find that λ (cid:104) m | e (cid:105) is always equal to λ (cid:104) m | m (cid:105) = λ (cid:104) e | e (cid:105) re-gardless of the system size (see Fig. 6(b)). This argumentcan also be supported by the calculation of topological en-tropy γ . We find that both | e (cid:105) and | m (cid:105) can yield exactly thesame γ = 1 / × ln(2) at θ = θ c , while | I (cid:105) and | e (cid:105) give γ = ln 2 . Also, by identifying | e (cid:105) with | m (cid:105) , we can explainthe three-fold degeneracy of the ground state with the correcttotal quantum dimension D = (cid:113) + 1 + ( √ = 2 .To obtain the SG state for each φ , we optimize the free pa-rameters α and β in Eq. (8) to obtain the variational groundstate of Eq. (5). Figure 7 shows the dominant eigenvalues ofthe transfer matrices constructed from the SG state. Recallthat the ground state of the star lattice Kitaev is an Abelianspin liquid at π/ < φ < π , and non-Abelian at < φ < π/ .For φ = π , the SG state is Z -isometric just like the LGstate at θ = 0 . For L = 1 , as φ decreases from φ = π to φ ≈ . π , λ (cid:104) m | m (cid:105) gradually decreases and λ (cid:104) m | e (cid:105) grad-ually increases. At φ ≈ . π , these two eigenvalues be-come identical. Different from the transition in toric codecharge condensation, if we keep decreasing φ , both λ (cid:104) m | m (cid:105) and λ (cid:104) m | e (cid:105) increase together. In fact, λ (cid:104) m | e (cid:105) should never be-come lager than λ (cid:104) m | m (cid:105) since the dominant eigenvalues of theregular transfer matrix will always be larger than other blocks / i | j / I | I I | Im | mm | eI | FIG. 7. The dominant eigenvalues of the transfer matrices for the SGstates with L = 1 (dashed lines), L = 8 (dotted lines), and L = 256(solid lines). for the norm of the states to stay positive. However, the trendfor φ < . π is also different from the LG case. The in-crease of both λ (cid:104) m | m (cid:105) and λ (cid:104) m | e (cid:105) strongly suggest that | e (cid:105) becomes | m (cid:105) in that regime. This is also consistent with thetopological entanglement entropy of | m (cid:105) becomes / at φ < . π [29]. As φ → , both λ (cid:104) m | m (cid:105) and λ (cid:104) m | e (cid:105) beginto decrease to the same point as φ ≈ . π , suggesting thatwe have two transition points at φ = 0 and φ = 0 . π .As we further increase the circumference to L = 256 , all λ → for φ < . π . The diverging correlation length sug-gests that the parent Hamiltonian of the SG states is gapless inthis regime. However, as shown in Ref. [33], there might ex-ist other non-frustration-free gapped Hamiltonian, in our casethe Kitaev star lattice Hamiltonian, which can also be well ap-proximated by the SG states. This result is also compatiblewith the no-go theorem [32] that the parent Hamiltonian of achiral PEPS is gapless. C. Transfer Matrix Spectrum
The full spectrum of the transfer matrices labeled by themomentum quantum numbers can be used to support our pic-ture that | e (cid:105) and | m (cid:105) are identical [17]. In Fig. 8, we ob-serve that not only the dominant eigenvalues match λ (cid:104) e | e (cid:105) = λ (cid:104) m | m (cid:105) = λ (cid:104) m | e (cid:105) , but their full spectra also match. Thismeans that the two MES | m (cid:105) and | e (cid:105) are exactly the samestate. In contrast, while in the thermodynamic limit λ (cid:104) I | I (cid:105) = λ (cid:104) I | (cid:15) (cid:105) , their spectra are always different. This strongly sug-gests that λ (cid:104) e | e (cid:105) = λ (cid:104) m | m (cid:105) = λ (cid:104) m | e (cid:105) is due to the degeneracyof the state while λ (cid:104) I | I (cid:105) = λ (cid:104) I | (cid:15) (cid:105) is due to the mode softening.Recall the discussion in Sec. IV, once we drive the Z -injective wave functions from the TO phase to the CC phase,the original MES basis is no longer the appropriate basis. The | e (cid:105) becomes exactly the same as | I (cid:105) , and the | m (cid:105) is not a phys-ical normalizable state. Similarly, for the LG and SG states inthe non-Abelian regime, there exist no charge and flux anyonsanymore. Combining with the calculation of entanglement k / l o g ( || ) I | Ie | eI | e | mm | m | FIG. 8. Transfer matrix spectrum of LG at θ = θ c with L = 6 . entropy in Ref. [29], we can regard the charge and flux trans-mute into σ anyon for φ < . π . For the honeycomb Kitaevmodel, the ground state is Abelian when | J x | ≥ | J y | + | J z | ,and the extreme limit can be directly mapped to the TC [8].There, the charge and flux lives in alternating rows of pla-quettes. On the other hand, in the non-Abelian phase, allthe plaquettes should be regarded equal and the vortex exci-tation is the σ − anyon. Since the Abelian and non-Abelianlimit can be respectively mapped to the TC and the isotropichoneycomb Kitaev model, there exists a critical point where | e (cid:105) and | m (cid:105) transmute into | σ (cid:105) . However, the string of u g and the charge operator R α encode the fusion and braidingrules for the Z topological order, which can not describe thenon-Abelian case. This is the limitation of the Z classifica-tion built in the Z -injective PEPS. As we show in Sec. V B,the parent Hamiltonian is gapless for φ < . π , which doesnot support gapped Z anyons such as | e (cid:105) and | m (cid:105) . Withinthe constraint of the Z -injective PEPS, the best approximatewave function for a non-Abelian CSL is to make | e (cid:105) and | m (cid:105) identical. On the other hand, the star lattice Kitaev model is not the parent Hamiltonian of the SG states and excitationscan be gapped for φ < . π . This means that the anyonicexcitation described by a Z -injective PEPS may not be thetrue excitation of the model. Nevertheless, one can create anexcitation by the string action u ⊗ Lg = (ˆ σ z ) ⊗ L , which will cre-ate a vortex pair with W p = − at the endpoints of the string. VI. DISCUSSION AND OUTLOOK
Now we have a unified picture to describe the transitionsfrom TO to CC, FC, and non-Abelian phases in terms of the16 blocks of the TM. The TO to CC transition can be detectedwhen the blue blocks in Fig. 2(c) become distinct. To be morespecific, as | I (cid:105) and | e (cid:105) become the same state, | m (cid:105) and | (cid:15) (cid:105) areconfined, and thus (cid:104) I | e (cid:105) and (cid:104) m | (cid:15) (cid:105) are different. Similarly, theemergence of the FC(non-Abelian) phases can be observed asthe yellow(green) blocks become distinct. Different from theCC and FC case, the non-Abelian case is not accompanied with the confinement of other particles. Since the parity evensector of a Z -invariant tensor is always non-zero, the vac-uum state | I (cid:105) is always normalizable. In addition, the fact thatno other MES can become | (cid:15) (cid:105) , we conclude that Z -injectivetensors can only detect three types of anyon transitions fromidentifying the MESs: | e (cid:105) = | I (cid:105) , | m (cid:105) = | I (cid:105) , or | e (cid:105) = | m (cid:105) .However, there exist other types of topological phase tran-sitions beyond this scheme. For instance, in Ref. [22], it isshown that the self-dual phase transition point of the TC wavefunction corresponds to the Kramners-Wannier duality of theAshkin-Teller model, where none of the MESs become iden-tical.In the current work, we use the Z -injective PEPS as anexample to identify and classify topological phase transitionsout the the Z TO. However, this scheme can be easily gen-eralized to G -injective and MPO-injective PEPS [39]. TheGSP-HOTRG method developed here is a powerful tool to de-termine whether the system undergoes a phase transition whenthe PEPS tensor acquires virtual symmetry. The low compu-tation cost of the GSP-HOTRG, same as HOTRG, makes itsuitable to perform finite-size scaling analysis, which can beused to extract scaling dimensions at the critical point. Furtherstudies along these directions are worth pursuing. ACKNOWLEDGMENTS
This work is partially supported by the Ministry of Scienceand Technology (MOST) of Taiwan under grants No. 108-2112-M-002-020-MY3, No. 107-2112-M-002-016-MY3, andNo. 108-2112-M-029-006-MY3. We thank J. Genzor for col-laboration on related work.
Appendix A: Gauge-Symmetry Preserved HOTRG
We apply the HOTRG coarse-graining scheme to mergetensors along the cylinder direction. Two sites are mergedinto a single site, generating a rank-6 tensor E (cid:48) = (cid:80) y E x ,y ,x ,y E x (cid:48) ,y ,x (cid:48) ,y (cid:48) , where the indices of E start onthe right and go around clockwise to the top. This can be re-garded as a rank-4 tensor by formally grouping the two indices( x , x (cid:48) ) on the right to one index, and similarly the two onthe left to another. The bond dimension of tensor E (cid:48) along thecylinder direction is the square of the original bond dimensionof tensor E . Applying an appropriate isometry U truncates thesize of these squared bond dimensions to a fixed number, say, D cut , and a truncated tensor ˜ E can be obtained (Fig. 9).For Z topologically ordered phases, we have the Z gaugesymmetry. In our calculation, we only preserve Z symmetryof the double tensor E , i.e., we identify (+ , +) , ( − , − ) as theeven sector and (+ , − ) , ( − , +) as the odd sector. Merging tworank-4 E tensors along the y -axis, we obtain a new rank-6 ten-sor E (cid:48) which retains the Z structure. In order to determine the Z symmetric isometries, U g ( g = 0 , , we perform eigen-value decomposition in each block of the Z block-diagonal FIG. 9. (a) Start from a one-site double tensor. (b) Merge two doubletensors to form a new rank-6 tensor. (c) Apply appropriate isometries U which trunctates the bond dimension tensor, M x x (cid:48) ,x x (cid:48) = (cid:88) x ,x (cid:48) ,y ,y E (cid:48) x ,x (cid:48) ,y ,x ,x (cid:48) ,y E (cid:48)∗ x ,x (cid:48) ,y ,x ,x (cid:48) ,y . (A1) Applying the isometry U g in each block generate a truncatedtensor ˜ E which preserve the Z symmetry. After p steps ofGSP-HOTRG, the final tensor represents a chain of p tensorsthat preserves the gauge symmetry. 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