Introduction

Identification of quantum phases of matter is a fundamental challenge in condensed matter physics. Conventional phases are distinguished by their order parameters, which characterize global symmetries of a system. In contrast, topological phases, characterized by non-local properties and robustness against local perturbations, defy the traditional symmetry-breaking paradigm1,2,3,4,5,6. These phases are not described by local order parameters but rather by exotic entanglement patterns, making them fundamentally distinct from conventional phases of matter.

Topological phases encompass a wide range of phenomena, including symmetry-protected topological (SPT) phases, and intrinsic topological orders. SPT phases, protected by certain global symmetries, exhibit robust edge states that are robust to local perturbations unless the symmetry is explicitly broken7,8. The concept of SPT phases has been extended to systems with gapless bulk states 9,10,11,12,13. In particular, this has led to enriched quantum criticality through additional symmetry constraints11. Intrinsic topological orders, such as those in fractional quantum Hall states and toric code, are characterized by long-range entanglement, anyonic excitations, and robust ground state degeneracies dependent on the system’s topology6. The exploration of these exotic topological phases underscores their fundamental and practical importance in the broader field of condensed matter physics. In particular, it is becoming increasingly feasible to realize and simulate a wide variety of quantum phases with the advancement of quantum technologies14,15,16,17,18. In this context, a comprehensive understanding of topological phases, the mechanisms driving phase transitions, and the distinctions between these phases and other conventional phases is becoming essential.

Nevertheless, the study of topological phases and their transitions is significantly hindered by the exponentially growing complexity of many-body quantum states, particularly in strongly interacting or higher-dimensional systems. This computational challenge limits the ability to explore the full landscape of quantum phases. In light of this, tensor network (TN) representations, such as matrix product states (MPS) and projected-entangled pair states (PEPS), have emerged as a powerful framework for exploring topological phases. Beyond their computational utility, TNs also offer conceptual insights, enabling the classification of topological phases through the interplay of local symmetries and global structures19,20,21,22,23,24,25,26,27. In particular, the gauge redundancies in the local tensors play a crucial role in describing the topological entanglement of the entire system, thereby enabling the characterization of intrinsic topological order. This gauge structure is generally represented by matrix product operators (MPOs) acting on the virtual indices of the TN, a property referred to as MPO-injectivity28,29,30,31. Many examples exhibit simpler forms of this gauge structure, the so-called \(\mathbb {G}\)-injective PEPS, where the MPO has a bond dimension of one. Furthermore, some of these examples might be referred to as 1-form symmetric PEPS32,33,34,35, in which the action on the virtual indices is directly related to the physical index. This property allows one to identify the Wilson loop operator of the system explicitly.

The zero-temperature partition function, or the norm of the ground state, \(\langle \psi | \psi \rangle\), encodes information about the long-range physics, including spontaneous symmetry breaking (SSB) and topological characteristics of the system 36. The dominant eigenvector and eigenvalue of transfer matrix (TM) \(\mathbb {H}\) can be analyzed to compute the norm, expressed as: \(\langle \psi | \psi \rangle = \underset{L \rightarrow \infty }{\lim }\ \textrm{Tr}(\mathbb {H}^L) = \underset{L \rightarrow \infty }{\lim }\ \lambda _0^L \langle 0 | 0 \rangle ,\) where L denotes the linear system size, \(\lambda _0\) is the dominant eigenvalue, and \(|0\rangle\) is the corresponding eigenvector representing the semi-infinite part of the norm. The TN representation of the wave function is highly effective for this purpose and has been extensively utilized and demonstrated to be valuable in previous studies 36,37,38,39,40,41,42,43.

Anyonic quasiparticles–emergent excitations obeying fractional exchange statistics unique to two-dimensional systems–characterize the nature of topologically ordered states. In such phases, the so-called ‘1-form’ symmetries, corresponding to the insertion of anyonic fluxes into the system, are encoded as global symmetries of the transfer matrix \(\mathbb {H}\) 36,39,40,41,42,43. By examining the spontaneous breaking of global symmetries, one can determine if the corresponding anyons are condensed or confined. In other words, the 1-form symmetry of the topologically ordered system is “shadowed” by the global symmetry of the transfer matrix, and the phase structure revealed by these global symmetries reflects the underlying topological phase of the original state.

Despite significant progress, previous studies have primarily focused on analyzing \(\mathbb {H}\) from the perspective of its global symmetries. In this work, however, we argue that analyzing the local symmetry of \(\mathbb {H}\) is crucial for characterizing phase transitions and refining the classification of topological phases. We first demonstrate that quantum phases emerging in variational wave functions, as well as the transitions between them, can be effectively identified by examining the interplay between global and local symmetries. In particular, we show that local symmetry is not merely supplementary but plays an essential role in identifying a quantum phase–specifically, a class of states whose transfer matrix is classified as belonging to the symmetry-protected topological phase–which cannot be fully characterized by global symmetries alone. Whereas SSB of TM global symmetries is well understood in terms of anyon condensation and confinement, the SPT nature of the TM and its physical implications remain largely unexplored. Through our analysis, we establish that examining the local symmetry–or equivalently, the projective representation of the global symmetries–of \(\mathbb {H}\) is essential for a comprehensive identification and classification of topological phases. Furthermore, we argue that the presence of an SPT phase in the TM serves as compelling evidence–a “smoking gun”–that the PEPS state realizes a nontrivial symmetry-enriched topological (SET) phase. We support this claim by presenting several observations that provide indirect evidence for this correspondence.

In Sec. 2.1, we discuss the global and local symmetries of \(\mathbb {H}\) and their origins. In Sec. 2.2, building on this framework, we revisit the phase diagram of the toric code under the application of a filtering operator. We examine the loop-gas ansatz state, whose transfer matrix exhibits the same global symmetries as that of the toric code. By analyzing the interplay between local and global symmetries, we demonstrate that the loop-gas ansatz TM realizes a nontrivial SPT phase with gapless edge modes, whereas the toric code TM corresponds to a trivially gapped phase without edge modes. We also investigate and classify the critical behavior at the phase boundaries between the filtered toric code and the loop-gas ansatz states. In Sec. 2.3, we extend this framework to more challenging three-dimensional models, where conventional numerical simulations face significant limitations. By analyzing only local tensors, we identify a potential nontrivial path connecting the \(3d\) toric code and X-cube states, which passes through a subdimensional critical point. Finally, we present further discussion and outlook in Sec. 3. Technical details are provided in the Supplementary Note.

Results

Global and local symmetries of transfer matrix

In this section, we investigate the global and local symmetries of the transfer matrix. In particular, we examine how the injective symmetry of the local tensor in the PEPS representation manifests as a global symmetry of the transfer matrix. We also discuss the general structure of the local symmetry of the transfer matrix.

Global symmetry

We consider a translation-invariant two-dimensional quantum many-body system defined on a lattice, where each site hosts a local spin (or qudit) degree of freedom interacting with its neighbors. The wave function of this system is represented by a uniform PEPS as follows:

(1)

where \(\textrm{tTr}_{\{\cdot \}}\) denotes the tensor trace over the virtual indices \(\{\cdot \}\), and \(s_i\) stands for the local quantum number at site i. We denote the dimension of the virtual indices as D. The PEPS formalism provides an efficient variational representation for two-dimensional quantum states with local interactions and is well suited to describe topological phases that cannot be characterized by conventional local order parameters36,37,38,39,40,41,42,43.

Figure 1
figure 1

Schematic illustrations of (a) an injective PEPS defined on a square lattice, (b) the invariance of a column of local tensors under a gauge transformation, and (c) the corresponding transfer matrix constructed from these tensors. (d) Graphical representation of the action of a 1-form symmetry operator, associated with a string operator acting on virtual bonds. (e) Illustration demonstrating the invariance of the transfer matrix under a two-site general local similarity transformation, along with (f) a special case where this transformation factorizes into a direct product of operations acting separately on each virtual index. (g) Illustration showing the relation between the anyon-pair creation operator and the local symmetry properties of the transfer matrix.

Full size image

The schematic illustration of PEPS is shown in Fig. 1(a). We further assume that the PEPS is injective, guaranteeing the existence of gauge transformations that leave the local \(T\)-tensor invariant, as illustrated in Fig. 1(a). Specifically, for a \(\mathbb {G}\)-injective PEPS, this invariance is expressed as \(g_{ll'} g_{uu'} g^{-1}_{r'r} g^{-1}_{d'd} T_{l'r'u'd'}^s = T_{lrud}^s\) where the gauge transformation g forms the group \(\mathbb {G}\). In this work, we focus on the \(\mathbb {G}\)-injective PEPS for simplicity; however, the protocol we develop can be generalized to the MPO-injective PEPS. Now, we define a TN patch consisting of a column of T-tensors:

$$\begin{aligned} \mathbb {C}_{\{l_i\}, \{r_i\}}^{\{s_i\}} \equiv \underset{\{u_i,d_i\}}{\text {tTr}} \prod _{i\in {\text {column}}} T_{l_i r_i u_i d_i}^{s_i}, \end{aligned}$$

or \(\mathbb {C}\) in short. Using the gauge invariance of the T-tensor, one can show that the \(\mathbb {C}\)-TN is invariant under transformations applied to the open virtual indices: \({\varvec{g}} \mathbb {C} {\varvec{g}}^{-1} = \mathbb {C}\) with \({\varvec{g}} \equiv g^{\otimes \textrm{column}}\) as depicted graphically in Fig.1 (b). Consequently, the transfer matrix, defined as

$$\begin{aligned} \mathbb {H}^{\{\overline{l}_i\},\{\overline{r}_i\}}_{\{l_i\},\{ r_i\}} \equiv \textrm{tTr}_{\{s_i\}} \mathbb {C}_{\{l_i\}, \{r_i\}}^{\{s_i\}} \mathbb {C}_{\{\overline{l}_i\}, \{\overline{r}_i\}}^{*\{s_i\}}, \end{aligned}$$
(2)

and graphically represented in Fig. 1 (c), remains invariant under the transformations \({\varvec{g}} \otimes \overline{{\varvec{g}}}'\) with \(g, g' \in \mathbb {G}\): \(({\varvec{g}} \otimes \overline{\varvec{g}}')\, \mathbb {H}\, ({\varvec{g}} \otimes \overline{\varvec{g}}')^{-1} = \mathbb {H}.\) Here, the operator with overline, \(\overline{o}\), acts on the virtual indices in the upper (bra) layer. This indicates that \(\mathbb {H}\) can be interpreted as a one-dimensional quantum Hamiltonian with a local Hilbert space of dimension \(D\times D\), which preserves a global \(\mathbb {G} \times \mathbb {G}\)-symmetry.

The global symmetry of TM, originating from the gauge redundancy of the local \(T\)-tensor, is directly tied to the 1-form symmetry of the underlying topologically ordered state. In the class of topological states we consider, there exists the 1-form symmetry operator acting nontrivially by changing the topological sector of the ground state,

$$\prod _{i\in C} A_i\,|\psi _a\rangle = |\psi _b\rangle ,$$

where \(C\) stands for any non-contractible closed loop, \(A_i\) is a unitary acting on site i, and \(a\) and \(b\) label distinct topological sectors connected by the 1-form symmetry. This operation is equivalent to performing a gauge transformation along the loop \(C\) as depicted in Fig. 1(d): \({\varvec{g}} \mathbb {C} = \mathbb {C} {\varvec{g}}^{-1} = \prod _i A_i \mathbb {C}\). Therefore, applying \({\varvec{g}}\) to bra (ket) layer of the TM is equivalent to applying corresponding 1-form symmetry operator to bra (ket) state. As the 1-form symmetry operator can be understood as anyonic flux insertion, the SSB of these global symmetries have been utilized to analyze the long-range properties of the state and the fate of anyonic excitations, e.g., their condensation and confinement/deconfinement 36,37,38,39,42. To detect SSB and characterize phase boundaries, one may employ the one-dimensional tensor network solvers such as the variational uniform matrix product states to compute the dominant eigenstate of the TM \(\mathbb {H}\).

Local symmetries

Local symmetry of the TM can be demonstrated by invertible transformations \({\varvec{u}}_i\) acting only on a finite patch of \(\mathbb {H}\) around site i, i.e., \({\varvec{u}}_i \mathbb {H} {\varvec{u}}_i^{-1} = \mathbb {H}\). Figure 1(e) illustrates the local transformation acting on a two-site patch of \(\mathbb {H}\). The local symmetries discussed in this study are represented by unitary operators acting on single-site or two-site patches in a tensor product form, \({\varvec{u}}_i = \prod _{j\in i} (u_j \otimes \overline{u}_j^*)\), where \(u_j\) operates solely on a single virtual index \(l_j\), as schematically illustrated in Fig. 1(f). However, our argument naturally extends to more general cases where local symmetry is realized through entangled operators acting on multi-site patches.

At the fixed point in the topologically ordered phase, the local symmetry of transfer matrix is ensured by the anyon pair creation operator on physical leg. As depicted in Fig. 1(g), the local symmetry operator applied on virtual bonds of \(\mathbb {C}\) translated into the physical operator, which creates the anyon pairs locally; \({\varvec{u}}_i \mathbb {C} {\varvec{u}_i}^{-1} = {\varvec{w}}_i \mathbb {C}\). The local symmetry implies that the anyon pair created state by \({\varvec{w}}_i\) is well defined; \(\langle {\varvec{w}}_i^\dagger {\varvec{w}}_i\rangle = 1\). Although the exact forms of these local symmetries become less transparent away from the fixed points, their existence remains evident, as they originate from the fundamental pair-creation operations of anyons. This suggests that our discussion applies broadly to tensor network representation of topologically ordered states.

Having additional knowledge of local symmetry offers significant advantages beyond what global symmetry alone can provide. First, the fate of certain global symmetries can be determined without large-scale simulations, as local symmetry may dictate them itself. More intriguingly, local symmetry enables the characterization of exotic phases beyond Landau’s paradigm, such as SPT phase. In following sections, we derive the local symmetries of \(\mathbb {H}\) and then present their invaluable utility with an exemplary PEPS ansatz extensively discussed in existing literature.

One-dimensional transfer matrices classification

In this section, we first use the toric code (TC) as an illustrative example to examine how 1-form symmetry and local anyonic pair creation manifest as global and local symmetries of the transfer matrix. Leveraging the local symmetry of the transfer matrix, we determine the phase diagram of the filtered TC state and classify the nature of criticality along its phase boundaries.

We then turn to the loop-gas (LG) ansatz state for the Kitaev honeycomb model, whose PEPS tensor exhibits the same injective symmetry as that of the TC PEPS tensor. As a result, the transfer matrices of the filtered TC and LG states are equivalent. Through an analysis of the interplay between local and global symmetries in the LG transfer matrix, we find that it is distinguished from the TC case by the presence of protected gapless edge modes.

Toric code wave function

We begin with the TC state, whose wave function can be efficiently represented as a uniform PEPS on a square lattice. The corresponding \(T\)-tensor, \(T_{lrud}^{s_1 s_2}\), carries two physical and four virtual indices, each with bond dimension \(D = 2\). See Supplementary Note for the explicit definition of the \(T\)-tensor. This tensor exhibits a gauge redundancy of the form \(Z_{ll'} Z_{uu'} Z_{r'r} Z_{d'd} T_{l'r'u'd'}^{s_1 s_2} = T_{lrud}^{s_1 s_2}\), where \(Z\) denotes the Pauli-\(Z\) matrix. As a result, the TM associated with the TC state can be regarded as an \(S=1/2\) quantum spin model that preserves a \(\mathbb {Z}_2 \times \mathbb {Z}_2\) global symmetry, generated by \({\varvec{g}}_{ZI} \equiv \prod _i (Z_i \otimes \overline{I}_i)\) and \({\varvec{g}}_{ZZ} \equiv \prod _i (Z_i \otimes \overline{Z}_i)\).

The global symmetry \({\varvec{g}}_{ZI}\) corresponds to applying the Pauli-\(Z\) operator to the virtual bonds of either the ket or bra layer. By applying the Pauli-\(Z\) operator to both the ket and bra layer, one can achieve the global symmetry \({\varvec{g}}_{ZZ}\). The operation of the Pauli-\(Z\) operator on virtual bonds can be transferred to the physical bonds as shown in

(3)

In other words, \({\varvec{g}}_{ZI}\) is equivalent to applying a 1-form symmetry operator exclusively on either the ket or bra layer, defined as

$$\begin{aligned} W_{\hat{C}}^Z = \prod _{i \in \hat{C}} Z_{i,x}, \end{aligned}$$
(4)

where \(\hat{C}\) is a non-contractible loop on the dual lattice and the subscript \(i,x\) denotes the link in the \(x\)-direction at site \(i\). This 1-form operator corresponds to the insertion of an \(m\) anyon into the system.

As a side remark, the 1-form symmetry corresponding to the insertion of an \(e\) anyon is given by

$$\begin{aligned} W_{C}^X = \prod _{i \in C} X_{i,y}, \end{aligned}$$
(5)

where \(C\) is a non-contractible loop along the direct lattice,. However, the action of \(W_{C}^X\) is canceled by the local symmetries of the tensor and therefore does not manifest as a global symmetry of the TM.

We now observe that the transfer matrix \(\mathbb {H}^\textrm{TC}\) is also invariant under two local symmetries: \({\varvec{u}}_i^z = Z_i \otimes \overline{Z}_i\) and \({\varvec{u}}_i^x = X_i \otimes \overline{X}_i\), for all sites \(i\). These operators correspond to the local creation of \(m\) and \(e\) anyons, respectively, acting simultaneously on the ket and bra layers. See Supplementary Note for a detailed derivation. This invariance implies that the norm of states with locally created anyon pairs is well-defined, indicating that both anyons are deconfined.

Note that the two local symmetry operators commute with each other. Together with Elitzur’s theorem, this implies that all physical eigenstates of \(\mathbb {H}^\textrm{TC}\) must be simultaneous eigenstates of both \({\varvec{u}}_i^z\) and \({\varvec{u}}_i^x\). Since the global symmetry \({\varvec{g}}_{ZZ}\) is given by \(\prod _i {\varvec{u}}_i^z\), it is preserved in all eigenstates. On the other hand, the global symmetry \({\varvec{g}}_{ZI}\) anticommutes with \({\varvec{u}}_i^x\), i.e., \(\{ {\varvec{g}}_{ZI}, {\varvec{u}}_i^x \} = 0\), and is therefore spontaneously broken in all eigenstates. Consequently, the eigenstates of the TM are direct products of local Bell states, indicating that the TC state corresponds to a trivial fixed-point state with zero correlation length.

Phase diagram of the filtered toric code

We now examine a deformation of the TC state through filtering operations: \(|\textrm{fTC}(h_x, h_z)\rangle \equiv \prod _i \left( 1 + h_x X_i + h_z Z_i\right) |\textrm{TC}\rangle .\) The phase diagram of the filtered TC (fTC) state has been extensively studied 43,44,45. Here, we discuss the utility of local symmetries and their interplay with global symmetries in mapping out potential phases and characterizing the nature of phase boundaries, without relying on simulations.

To begin, note that the filtering operation is an action of direct product of local operators, and therefore does not affect the injectivity of the TC state. This ensures that its TM, \(\mathbb {H}_{(h_z, h_x)}^\textrm{fTC}\), commutes with both \({\varvec{g}}_{ZI}\) and \({\varvec{g}}_{ZZ}\) for arbitrary \((h_z, h_x)\). In contrast, the \(Z\,(X)\)-filtering operation preserves only the local symmetry \({\varvec{u}}_i^{z\,(x)}\): \([{\varvec{u}}_i^z,\, \mathbb {H}^\textrm{fTC}_{(h_z, 0)}]=0\) and \([{\varvec{u}}_i^x,\, \mathbb {H}^\textrm{fTC}_{(0, h_x)}]=0\). This implies that the global \({\varvec{g}}_{ZZ}\) symmetry remains preserved in \(\mathbb {H}_{(h_z, 0)}^\textrm{fTC}\), whereas \({\varvec{g}}_{ZI}\) is always broken in \(\mathbb {H}_{(0, h_x)}^\textrm{fTC}\). Consequently, the \(Z\)-filtering operation induces at most a single phase transition, corresponding to the restoration of \({\varvec{g}}_{ZI}\). On the other hand, the breaking of \({\varvec{g}}_{ZZ}\) is the only possible transition in the \(X\)-filtering operation.

We can pinpoint the occurrence of phase transitions by analyzing the local symmetries at two points. First, at \((h_z, h_x) = (1,0)\), the TM becomes symmetric under local transformations \(Z_i \otimes \overline{I}_i\) and \(I_i \otimes \overline{Z}_i\). These local symmetries guarantee the restoration of \({\varvec{g}}_{ZI}\) in the \(Z\)-filtering operation, \(\langle {\varvec{g}}_{ZI} \rangle = 1\), indicating that the phase transition must occur at or below \(h_z = 1\), i.e., \(h_z^c \le 1\). While the exact critical point is not located, the phase diagram is consistent with the one obtained from numerical simulations 46. Similarly, two local symmetries, \(X_i \otimes \overline{I}_i\) and \(I_i \otimes \overline{X}_i\), emerge at \((h_z, h_x) = (0,1)\), ensuring the breaking of the \({\varvec{g}}_{ZZ}\) symmetry, \(\langle {\varvec{g}}_{ZZ}\rangle = 0\). This also provides the upper bound for the critical point in the \(X\)-filtering operation, \(h_x^c \le 1\). Since both transitions involve the \(\mathbb {Z}_2\) symmetry breaking, the critical points are described by the \((1+1)d\) Ising conformal field theory (CFT) with central charge \(c=1/2\).

Figure 2
figure 2

Phase diagram of the filtered TC. (a) The phase diagram is constructed exclusively based on the interplay between local and global symmetries. It highlights three phases with distinct breaking patterns of the global \({\varvec{g}}_{ZZ} \times {\varvec{g}}_{ZI}\) symmetries, along with two symmetry-enriched Ising CFT lines that separate them. (b) The phase diagram is enhanced by the inclusion of additional information regarding the global symmetry present along the dashed curve. At a point on this curve, the two distinct Ising CFTs merge into the \((\textrm{Ising})^2\) CFT with \(c=1\).

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Table 1 The unbroken symmetry \(G_\textrm{eff}\), gapped symmetry \(G_\textrm{gap}\), and the end-point of string operator \(O_i\) of the transfer matrix, \(\mathbb {H}\), for each ansatz at the critical point.
Full size table

By analyzing the effective symmetry group \(G_\textrm{eff}\) and the gapped symmetry group \(G_\textrm{gap}\), we further identify that the two critical points belong to two distinct classes among the nine different \(\mathbb {Z}_2 \times \mathbb {Z}_2\)-enriched Ising CFTs 11, as summarized in Table 1. Therefore, the two critical points cannot be smoothly connected in the parameter space \((h_z, h_x)\), preserving both \({\varvec{g}}_{ZZ}\) and \({\varvec{g}}_{ZI}\) symmetries. Instead, if they intersect, the transition must involve an additional critical point characterized by a \(c=1\) CFT 11.

The phase diagram shown in Fig. 2(a) is constructed exclusively based on the interplay between the local and global symmetries of \(\mathbb {H}_{(h_z, h_x)}^\textrm{fTC}\). In the Supplementary Note, we demonstrated the existence of a curve along which an additional accidental global symmetry emerges, causing both \({\varvec{g}}_{ZZ}\) and \({\varvec{g}}_{ZI}\) to be either simultaneously preserved or simultaneously broken along this curve. This curve is indicated by a dashed line in Fig. 2(b). Consequently, the two symmetry-enriched Ising CFTs meet at a point on this curve as shown in Fig. 2(b). This critical point is commonly referred to as \((\textrm{Ising})^2\), which corresponds to a \(c=1\) orbifold CFT 11. The critical point is known to extend along the \(h_z = h_x\) line 46, representing the direct transition between the fully symmetric and broken phases. This is the only aspect that cannot be determined through local symmetry analysis. Nonetheless, it is remarkable how long-range physics, such as SSB and the nature of phase transitions, can be readily determined from the properties of the local \(T\)-tensor or a finite patch of the TN, without relying on large-scale simulations.

It is important to emphasize that at the TC point, \((h_z,h_x) = (0,0)\), applying \({\varvec{u}}_i^z\) or \({\varvec{u}}_i^x\) creates a pair of either \(m\)- or \(e\)-anyons near the virtual index i in both ket and bra layers. In other words, these two local symmetries originate from the fact that \(m\)- and \(e\)-anyons are well-defined in the quantum state. On the other hand, at the points \((h_z, h_x) = (1,0)\) and \((h_z, h_x) = (0,1)\), the \(m\)-anyon and \(e\)-anyon, respectively, are condensed. In these cases, the pair-creation of anyons on the quantum state effectively acts as an trivial operation, which is reflected in the emergence of local symmetries in the TM: \(Z_i\otimes \overline{I}_i\) or \(I_i\otimes \overline{Z}_i\) for \(m\)-anyon condensation, and \(X_i\otimes \overline{I}_i\) or \(I_i\otimes \overline{X}_i\) for \(e\)-anyon condensation. Since these local symmetries originate from pair-creation processes of anyons–which are inherently ‘local’–such local symmetries are expected be present in the TM of any topologically ordered state. Ultimately, the phase of the fTC state at any point is determined by the interplay between the local and global symmetries of \(\mathbb {H}^\textrm{fTC}_{(h_z, h_x)}\). As a result, we believe that the argument regarding the interplay between local and global symmetries remains universally valid for any topologically ordered state.

Kitaev honeycomb loop gas state

We now demonstrate that the local symmetry of the TM serves as a powerful tool for identifying quantum phases that are indistinguishable by global symmetry considerations alone. Let us consider the so-called loop gas (LG) ansatz, \(|\textrm{LG}(\theta )\rangle\), which was introduced for the Kitaev honeycomb model (KHM) in Ref. 33. By grouping two sites within the unit cell into a single site, the LG state can be represented as a \(D=2\) PEPS on the square lattice. See the Supplementary Note for the explicit definition of the \(T\)-tensor and the variational parameter \(\theta\). One can verify that the T-tensor is also invariant under the gauge transformation: \(Z_{ll'} Z_{uu'} Z_{r'r} Z_{d'd} T_{l'r'u'd'}^{s_1 s_2} = T_{lrud}^{s_1 s_2}\). Therefore, the TM of the LG state, \(\mathbb {H}_{\theta }^\textrm{LG}\), also enjoys the global \({\varvec{g}}_{ZI} \times {\varvec{g}}_{ZZ}\) symmetry.

In \(0\le \theta \le \pi /4\), two points allow for exact solutions. At \(\theta =0\), the ansatz corresponds to the exact ground state of the KHM in the strong anisotropic limit 47, representing a \(\mathbb {Z}_2\) topologically ordered state, whose parent Hamiltonian is known as the Wen plaquette (WP) model 48,49. In contrast, at \(\theta _c = \pi /4\), it becomes a critical state characterized by \((1+1)d\) Ising CFT 33.

A naive expectation is that, since the LG state at \(\theta = 0\) corresponds to a \(\mathbb {Z}_2\) topologically ordered phase, its transfer matrix shares the same global symmetries as that of the TC. This would imply that the \(g_{ZZ}\) symmetry is preserved, while \(g_{ZI}\) is spontaneously broken. Furthermore, the TM \(\mathbb {H}_{\theta }^\textrm{LG}\) possesses the two-site local symmetry defined by

$$\begin{aligned} {\varvec{v}}_i \equiv ( v_{i-1}^* \otimes \overline{v}_{i-1})(v_i \otimes \overline{v}^*_i), \end{aligned}$$
(6)

with \(v = \frac{1}{2}(1+i)(X - Y)\), i.e., \({\varvec{v}}_i^\dagger \mathbb {H}^\textrm{LG}_{\theta } {\varvec{v}}_i = \mathbb {H}^\textrm{LG}_{\theta }\) for any \(\theta\). By multiplying these local symmetry operators together, one recovers the global symmetry operator \(g_{ZZ} = \prod _{i} {\varvec{v}}_i\), thereby ensuring the preservation of \(g_{ZZ}\) symmetry across the entire range of \(\theta\). Here, we utilized the relation \(vv^* = iZ\). Consequently, one might anticipate that \(\mathbb {H}_{(h_z,0)}^\textrm{fTC}\) and \(\mathbb {H}_{\theta }^\textrm{LG}\) are indistinguishable in terms of global symmetries, suggesting a smooth deformation between them.

However, a closer examination of the LG PEPS state at \(\theta = 0\)–which corresponds to the ground state of the WP model–reveals that the \(g_{ZI}\) symmetry is in fact preserved at this point. To be more specific, in this strong anisotropic limit, two spins in the local tensor are locked as \(s_1 = s_2\) and the effective Hilbert space is described by Pauli matrices. In terms of these effective Pauli matrices, the WP model is given by

(7)

Using the local symmetries of LG PEPS tensor, one can show that the PEPS state is the ground state of \(H_\textrm{WP}\).

The WP Hamiltonian is invariant under the following 1-form symmetry:

$$\begin{aligned} W_C^Z = \prod _{i\in C} Z_i, \end{aligned}$$
(8)

where \(C\) is a non-contractible closed loop. Utilizing the local symmetries of the tensors, one can demonstrate that the 1-form symmetry \(W_C^Z\) is equivalent to the global symmetry \(g_{ZI}\) of the transfer matrix. As detailed in the supplementary note, the PEPS state at \(\theta = 0\) is an eigenstate of \(W_C^Z\), indicating that the global symmetry \(g_{ZI}\) of the transfer matrix is preserved at this point. Consequently, the transfer matrix \(\mathbb {H}_{\theta }^\textrm{LG}\) at \(\theta = 0\) preserves both global symmetries \(g_{ZI}\) and \(g_{ZZ}\). This result contrasts with \(\mathbb {H}_{(h_z,0)}^\textrm{fTC}\) at \(h_z = 0\), where the symmetry \(g_{ZZ}\) remains intact, but \(g_{ZI}\) is spontaneously broken.

Building on the symmetry analysis at \(\theta = 0\), we now investigate the phases of \(\mathbb {H}^\textrm{LG}_{\theta }\) in the parameter range \(0 \le \theta \le \pi /4\). This can be achieved by mapping the symmetries of the model onto those of the one-dimensional cluster model, described by the Hamiltonian

$$\begin{aligned} H_{\textrm{cl}} = - \sum _{i} Z_{i-1} X_{i} Z_{i+1} - h \sum _{i \in \textrm{even}} Z_{i-1}Z_{i+1}. \end{aligned}$$
(9)

This model possesses two global symmetries:

$$\begin{aligned} g_{X}^\textrm{even} = \prod _{i\in \textrm{even}} X_i,\quad g_{X}^\textrm{odd} = \prod _{i\in \textrm{odd}} X_i. \end{aligned}$$
(10)

As the coupling parameter \(h\) increases from zero, the system undergoes a phase transition at the critical value \(h = h_c\), where the global \(\mathbb {Z}_2\) symmetry \(g_{X}^\textrm{odd}\) is spontaneously broken. This critical point is classified to the \((1+1)d\) Ising CFT with \(G_\textrm{eff} = g_{X}^\textrm{odd}\times g_{X}^\textrm{even}\) and \(G_\textrm{gap} = g_{X}^\textrm{even}\).

For non-zero \(h\), the model exhibits a local symmetry under the operation \(\hat{o}_i = Z_{i-1} X_{i} Z_{i+1}\) for even \(i\). The consecutive product of this local symmetry yields the global symmetry \(g_{X}^\textrm{even} = \prod _{i \in \textrm{even}} \hat{o}_i\), thereby ensuring the preservation of \(g_{X}^\textrm{even}\) symmetry across the entire range of \(h\). Moreover, this local symmetry gives rise to a long-range string order, characterized by the correlation

$$\lim _{|n - m|\rightarrow \infty }\left\langle \left( S_m^{X_\textrm{even}} \right) ^\dagger S_n^{X_\textrm{even}} \right\rangle = 1,$$

where the string operator \(S_n^{X_\textrm{even}}\) is explicitly defined as

$$\begin{aligned} S_n^{X_\textrm{even}} \equiv \prod _{\begin{array}{c} i\le n \\ i \in \textrm{even} \end{array}} \hat{o}_{i} = \left( \prod _{\begin{array}{c} i\le n \\ i \in \textrm{even} \end{array}} X_i \right) Z_{n+1}. \end{aligned}$$
(11)

This string operator carries a nontrivial charge under the global symmetry \(g_{X}^\textrm{odd}\), implying that the region \(h<h_c\) represents a \(\mathbb {Z}_2\times \mathbb {Z}_2\) symmetry-protected topological phase. Furthermore, this nontrivial charged string operator guarantees the presence of symmetry-protected gapless edge modes at the critical point \(h = h_c\) 11,25,50,51.

Having analyzed the cluster model, we can now interpret the phases of \(\mathbb {H}^\textrm{LG}_{\theta }\) through the following correspondence:

$$\begin{aligned} g_{X}^\textrm{odd} \rightarrow g_{ZI}, \quad g_{X}^\textrm{even} \rightarrow g_{ZZ}, \quad \hat{o}_i \rightarrow {\varvec{v}}_{i}. \end{aligned}$$
(12)

Under this mapping, the string operator \(S_n^{X_\textrm{even}}\) maps to

$$\begin{aligned} S_n^{\varvec{v}} \equiv \prod _{i \le n} {\varvec{v}}_i = \left( \prod _{i < n} Z_i \otimes \overline{Z}_i \right) (v_n \otimes \overline{v}^*_n), \end{aligned}$$
(13)

which carries a nontrivial charge under the global symmetry \(g_{ZI}\). Consequently, the gapped phase in the range \(0 \le \theta < \frac{\pi }{4}\) corresponds to a \(\mathbb {Z}_2 \times \mathbb {Z}_2\) symmetry-protected topological phase, while the critical point at \(\theta = \frac{\pi }{4}\) realizes a symmetry-enriched \((1+1)d\) Ising conformal field theory (CFT) with symmetry-protected gapless edge modes.

More specifically, the critical point \(\theta = \pi /4\) is described by a \((1+1)d\) Ising CFT characterized by the effective symmetry \(G_\textrm{eff} = {\varvec{g}}_{ZZ} \times {\varvec{g}}_{ZI}\) and the gap-preserving symmetry \(G_\textrm{gap} = {\varvec{g}}_{ZZ}\). As a result, \(G_\textrm{eff}\) and \(G_\textrm{gap}\) alone do not distinguish this critical point from that of the \(Z\)-filtered toric code (fTC) [see Table 1]. However, a crucial distinction arises from the charge of the string operator’s end-point under \({\varvec{g}}_{ZI}\). In the case of the \(Z\)-filtered TC, the string operator \(\prod _{i\le n}{\varvec{u}}_i^z = \prod _{i\le n} Z_i\otimes \overline{Z}_i\) has a trivial end-point under \({\varvec{g}}_{ZI}\). In contrast, for the LG state, the end-point operator is nontrivial, satisfying

$${\varvec{g}}_{ZI} (v_i \otimes \overline{v}_i^*) {\varvec{g}}_{ZI} = - (v_i \otimes \overline{v}_i^*).$$

This discrete distinction indicates that the two critical points correspond to distinct symmetry-enriched Ising universality classes 10. In brief, the local gauge symmetry of the KLG tensor and the local symmetry of the transfer matrix [Eq. (6)] both originate from the intrinsic 1-form symmetry of the LG state. It generates the corresponding string operators, which protect the edge excitation of the TM in direct analogy with the one-dimensional cluster model. Once the 1-form symmetry is broken, these string operators are no longer well defined, and the protection of the edge states disappears.

Figure 3
figure 3

(a) Eigenvalue spectrum of \(\mathbb {H}^\textrm{LG}_{\theta }\) under OBC. (b) Entanglement spectrum of the dominant eigenstate of \(\mathbb {H}^\textrm{LG}_{\theta }\) under PBC. The system size is fixed to \(L=10\). In both (a) and (b), a clear double degeneracy appears for entire range of \(\theta\).

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To investigate the properties of \(\mathbb {H}_{\theta }^\textrm{LG}\) numerically, we diagonalize \(\mathbb {H}^\textrm{LG}_{\theta }\) for finite system sizes as a function of \(\theta\). Under open boundary conditions (OBC), the eigenvalue spectrum consistently exhibits a twofold degeneracy across all values of \(\theta\), as illustrated in Fig. 3(a). The degenerate eigenstates are related through end-point operators, i.e., \(|1\rangle = v_1 \otimes v^*_1 |0\rangle = v^*_L \otimes v_L |0\rangle\). In contrast, under periodic boundary conditions (PBC), the dominant eigenvector becomes non-degenerate, while the bipartite entanglement spectrum retains a twofold degeneracy. These results indicate that \(\mathbb {H}_{\theta \le \theta _c}^\textrm{LG}\) belongs to an SPT phase protected by the symmetry \({\varvec{g}}_{ZI}\), rendering it topologically distinct from any phase within the fTC model. Importantly, it is not merely a global symmetry, but the system’s 1-form symmetry that fundamentally protects the topological edge states in the LG phase.

Summarizing the phases of the fTC and LG states, the overall phase diagram is presented in Fig. 4. Notably, a SSB of \({\varvec{g}}_{ZI}\) is required to deform the LG state at \(\theta = 0\) into the TC phase in the fTC model, despite both states exhibiting the same underlying topological order. As discussed above, in the strong anisotropic limit, the KHM maps onto the WP model, which–when translation symmetry is taken into account–belongs to a distinct SET phase compared to the TC model. Interestingly, this translational SET phase, to which the WP model belongs, is known to host gapless Majorana edge modes 52,53.

The fact that \(\mathbb {H}^\textrm{LG}_{\theta }\) at \(\theta = 0\) realizes an SPT phase serves as strong evidence for the presence of gapless edge modes in the translational SET phase of the WP model. Here, we argue more directly that if the transfer matrix constructed from a PEPS tensor exhibits an SPT phase, then the corresponding PEPS necessarily supports gapless edge modes.

To support this claim, we first review the interpretation of SSB in the transfer matrix and its connection to ground state degeneracy in the associated PEPS. Recall that stacking the transfer matrix \(\mathbb {H}\) leads to left and right dominant eigenstates satisfying \((\phi _L| \mathbb {H} \propto (\phi _L|\) and \(\mathbb {H} |\phi _R) \propto |\phi _R)\), and their overlap determines the norm of the corresponding PEPS: \((\phi _L | \phi _R) \propto \langle \psi | \psi \rangle\). When \(\mathbb {H}\) exhibits SSB, there exist multiple dominant eigenstates labeled by \(a\), namely \((\phi _L^a|\) and \(|\phi _R^a)\). In this case, each pair \((\phi _L^a|\), \(|\phi _R^a)\) gives rise to a distinct PEPS \(|\psi ^a\rangle\), with norm \(\langle \psi ^a | \psi ^a \rangle \propto (\phi _L^a | \phi _R^a)\), despite all being constructed from the same local tensor. These distinct PEPSs are interpreted as differing by topological insertions at the far boundaries, such as the insertion of anyons. Such operations effectively permute the dominant eigenstates of the transfer matrix (see, e.g., Ref. 36). Although the PEPSs \(|\psi ^a\rangle\) are orthogonal, they remain degenerate ground states of a common parent Hamiltonian, provided the Hamiltonian consists solely of local terms. Since all \(|\psi ^a\rangle\) share the same local tensor, they yield identical expectation values for any local observable. The operators distinguishing the different \(|\psi ^a\rangle\) involve system-size-long topological manipulations, which do not alter local measurements and hence do not lift the degeneracy.

It is also noteworthy that the global symmetry \(g_{ZI}\) is spontaneously broken in the TM of the TC, while it remains preserved in that of the WP model. In the TC, \(g_{ZI}\) corresponds to the 1-form symmetry associated with \(m\)-anyon insertion. The spontaneous breaking of \(g_{ZI}\) indicates that the PEPS ground state and the state with an inserted \(m\) anyon are orthogonal, having zero overlap. This implies that the \(m\) anyon is not condensed. In contrast, in the WP model, \(g_{ZI}\) corresponds to the 1-form symmetry associated with the insertion of the \(\epsilon = m \times e\) anyon. The preservation of \(g_{ZI}\) implies that the ground state and the state with an inserted \(\epsilon\) anyon are equivalent. However, this does not necessarily indicate anyon condensation.

Recall that the TM global symmetry \(g_{ZI}\) originates from the \(Z\)-injective symmetry of the PEPS tensor. Interestingly, we find that the WP PEPS state–equivalently, the LG ansatz at \(\theta = 0\)–exhibits two distinct injective symmetries:

(14)

Then, the \(Z\)-injective symmetry can then be decomposed into these two symmetries:

(15)

Using Eqs. (14) and (15), the global symmetry \(g_{ZI}\) can be written as a product of two emergent global symmetries:

$$g_{ZI} = g_{(vv^*)I} \times g_{(v^* v)I}.$$

Each of these global symmetries corresponds to a 1-form symmetry associated with either the \(e\) or \(m\) anyon, respectively. Notably, these 1-form symmetries anticommute with the 1-form symmetry defined in Eq. (8). As a result, the PEPS state and the state with an inserted \(e\) or \(m\) anyon are orthogonal, indicating that the TM global symmetries associated with them are spontaneously broken.

We further point out that, in the WP model, both 1-form symmetries associated with the \(e\) and \(m\) anyons are realized as global symmetries of the TM. In contrast, for the TC, only one of the two 1-form symmetries appears as a global symmetry of the TM, while the other does not. This distinction arises because the TC state is not invariant under translations that permute the \(e\) and \(m\) anyons, whereas the WP state is symmetric under such transformations. These differences reflect the fact that the TC and WP models belong to distinct classes of \(\mathbb {Z}_2\) translational SET phases.

Consequently, one may interpret the decomposition of the \(Z\)-injective symmetry in Eqs. (14) and (15) as a signature of the WP model realizing a nontrivial SET phase. Furthermore, the two injective symmetries in Eq. (14) are related to the local symmetry \({\varvec{v}}_i\) introduced in Eq. (6) (see Supplementary Note). Since the nontrivial-charge string operator \(S_n^{\varvec{v}}\) in Eq. (13) is constructed from successive applications of \({\varvec{v}}_i\), it is natural to suspect that the TM exhibiting a nontrivial SPT phase is intimately connected to the PEPS state realizing a nontrivial SET phase. A formal development of the mathematical foundation behind this connection is left for future investigation.

Figure 4
figure 4

Schematic phase diagram of the dominant eigenvalue of the transfer matrices \(\mathbb {H}_\theta ^\textrm{LG}\) and \(\mathbb {H}_{(h_z,h_x)}^\textrm{fTC}\). The colored solid lines represent different symmetry-enriched \((1+1)d\) Ising CFTs [See Table. 1], while multi-critical points are indicated by purple hexagons. The dominant eigenvector of \(\mathbb {H}_{(h_z,h_x)}^\textrm{fTC}\) in the toric code phase is a trivial direct product state breaking the \({\varvec{g}}_{ZI}\) symmetry, whereas that of \(\mathbb {H}_\theta ^\textrm{LG}\) hosts gapless edge modes protected by the \({\varvec{g}}_{ZI}\) symmetry.

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Two dimensional transfer matrices: 3d Toric code and X-cube states

Now, we discuss the local symmetry in the TM of three-dimensional quantum states and its utilization, focusing on the 3d TC and X-cube states. These states can be efficiently represented by 3d PEPS on a cubic lattice, with T-tensors having three physical indices. The TMs are then represented by two-dimensional projected entangled-pair operators (PEPO), and can be interpreted as \((2+1)d\) quantum models. See Supplementary Note for the definition of the T-tensors and detailed derivations of the global and local symmetries discussed below.

By extending the discussion in the 2d TC state, it is straightforward to show that the TMs of both states, \(\mathbb {H}^\textrm{3TC}\) and \(\mathbb {H}^\textrm{XC}\), are invariant under the global symmetries \({\varvec{g}}_{ZZ}\) and \({\varvec{g}}_{ZI}\), as well as the local symmetries \({\varvec{u}}_i^z\) and \({\varvec{u}}_i^x\). Consequently, the dominant eigenvectors are the identical product state, \(|0\rangle\), that break the \({\varvec{g}}_{ZI}\) symmetry, mirroring the 2d TC state: \(u_i^z |0\rangle = |0\rangle\) and \(u_i^x |0\rangle = |0\rangle\). In what follows, we explore two possible paths in Hilbert space that preserve both \({\varvec{g}}_{ZZ}\) and \({\varvec{g}}_{ZI}\), connecting \(\mathbb {H}^\textrm{3TC}\) and \(\mathbb {H}^\textrm{XC}\) by leveraging their local properties.

First, we consider a rather straightforward deformation: the Z-filtering operation applied to both states, i.e., \(\prod _i (1+h_z X_i) |\textrm{3TC}\rangle\) and \(\prod _i (1+h_z X_i) |\textrm{XC}\rangle\). Similar to the 2d fTC case, the Z-filtering operation preserves \({\varvec{g}}_{ZZ}\) due to \({\varvec{u}}_i^z\) but induces a phase transition involving the SSB of \({\varvec{g}}_{ZI}\), characterized by the \((2+1)d\) Ising universality class in both cases 54. The upper bound of the critical point is given by \(h_z^c \le 1\), which can be determined from the emergent local symmetries at \(h_z = 1\). This implies the existence of a path in the parameter space connecting 3d TC and XC states, which passes through two \((2+1)d\) Ising critical points.

Then, we identify a non-trivial path connecting these two states. By examining local properties of the TMs, we find that \(\mathbb {H}^\textrm{3TC}\) and \(\mathbb {H}^\textrm{XC}\) can be expressed as idempotent operators, given by:

$$\begin{aligned} \mathbb {H}^\textrm{3TC}&= \prod _{\langle ij\rangle } \left[ 1+ (X_i \otimes \overline{X}_i)(X_j \otimes \overline{X}_j) \right] , \nonumber \\ \mathbb {H}^\textrm{XC}&= \prod _{\square } \Big [1+ \prod _{i\in \square } (X_i \otimes \overline{X}_i)\Big ], \end{aligned}$$
(16)

where \(\langle ij\rangle\) denotes neighboring sites, and \(i\in \square\) represents sites on an elementary plaquette in the square lattice. The above equalities can be verified within a finite patch of the TMs, similar to how local symmetries are analyzed. See Supplementary Note for detailed derivations. Consequently, while the eigenvectors are identical, the eigenvalues are distinct: \(\mathbb {H}^\textrm{3TC} |0\rangle = Z_\textrm{Para}^{T=\infty } |0\rangle \quad \text {and} \quad \mathbb {H}^\textrm{XC} |0\rangle = Z_{\mathbb {Z}_2}^{g=\infty } |0\rangle\). Here, \(Z_\textrm{Para}^T\) denotes the classical partition functions of a trivial paramagnet, \(H_\textrm{Para} = - \sum _i Z_i\) at temperature T, while \(Z_{\mathbb {Z}_2}^{g}\) stands for the zero-temperature partition function of the \(\mathbb {Z}_2\) gauge theory on square lattice, \(H_{\mathbb {Z}_2} = - g\sum _\square \prod _{i\in \square } X_i - g^{-1} \sum _i Z_i\) at coupling constant g. As a result, the norm of two states are given by:

$$\begin{aligned} \langle 3\textrm{TC} | 3\textrm{TC} \rangle = \left[ Z_\textrm{Para}^{T=\infty } \right] ^L, \quad \langle \textrm{XC} | \textrm{XC} \rangle = \left[ Z_{\mathbb {Z}_2}^{g=\infty }\right] ^L, \end{aligned}$$
(17)

where L represents the number of layers or slices in the 3d system. Since the \(\mathbb {Z}_2\) gauge theory exhibits the \((1+1)d\) Ising critical point describing the confinement/deconfinement transition, one can envision a deformation of the XC state such that the eigenvalue, \(Z_{\mathbb {Z}_2}^g\), changes continuously from \(g=\infty\) to \(g=0\). Then, the deformation must involve a phase transition characterized by a stacking of \((1+1)d\) Ising CFTs. We do not find such a deformation with \(D=2\) PEPS, but we cannot completely rule out its existence in an enlarged PEPS Hilbert space with larger bond dimensions. Similarly, one can imagine a deformation of the 3d TC state that continuously changes the eigenvalue \(Z_\textrm{Para}^T\) to its zero-temperature limit. Along this deformation, no phase transition occurs. This implies that, since \(Z_{\mathbb {Z}_2}^{g=0} = Z_\textrm{Para}^{T=0}\), the XC and 3d TC states can be transformed into each other along this path undergoing the subdimensional criticality 55,56. We present a schematic phase diagram in Fig. 5, along with the TN representation of the eigenvalues.

Figure 5
figure 5

Schematic phase diagram illustrating two possible paths connecting the \(3d\) TC state and the X-cube state, while preserving the \(\mathbb {Z}_2\) gauge symmetry of the wavefunction. The dominant eigenvalue of the transfer matrix for the \(3d\) TC state corresponds to the classical partition function of a paramagnet in the infinite-temperature limit. Each eigenvalue can be represented by a two-dimensional TN, as depicted in the gray-shaded square. In contrast, for the X-cube state, it corresponds to the \(\mathbb {Z}_2\) gauge theory on a square lattice in the infinite coupling constant (\(g\)) limit. See text for details.

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Discussion

We have demonstrated that analyzing the local symmetry of the transfer matrix–together with its interplay with global symmetry–provides a powerful framework for distinguishing quantum phases and characterizing the nature of phase transitions, without relying on numerical simulations or classical partition function analysis. Remarkably, although long-range information is typically required to capture features such as spontaneous symmetry breaking, we have shown that these properties can be inferred from the structure of a local tensor or even a finite patch of the tensor network.

Moreover, we have established that local symmetry plays a central role in identifying topological phases, far beyond its traditional use as a y tool for phase diagram construction. In particular, by leveraging local symmetry, we have identified a SPT phase of the transfer matrix–a quantum phase that cannot be captured by global symmetries or classical partition function analysis, and whose physical interpretation in terms of topological order remains largely unexplored. This novel phase hosts edge states protected by a 1-form symmetry, rather than by the system’s global symmetry, and may serve as a signature of gapless boundary modes in a nontrivial SET phase. Motivated by this connection, we anticipate that chiral topological spin-liquid states–known to break time-reversal and parity symmetries and to host gapless chiral edge modes 57,58–may similarly exhibit SPT phases in their transfer matrices 59,60,61,62. Verifying this conjecture and developing the underlying mathematical framework remain important directions for future research.

We have also extended our approach to challenging three-dimensional models, where conventional numerical methods encounter significant limitations. By focusing on the structure of local tensors, we have proposed possible quantum phase transitions–such as sub-dimensional critical points–between the 3D toric code and X-cube phases. These transitions are difficult to detect numerically, but can be inferred from local tensor analysis. This demonstrates the broad applicability and effectiveness of our method in probing higher-dimensional systems and complex quantum phases that are otherwise inaccessible through conventional techniques.

Since the local symmetries of the transfer matrix originate from local anyon pair creation processes, our approach–based on the interplay between local and global symmetries–is expected to be broadly applicable. The general presence of local symmetries in transfer matrices has also been discussed in Ref. 63. Although their interpretation of the origin of these symmetries differs from ours, the underlying idea–that the transfer matrix generically exhibits local symmetry–is shared. A key distinction between our study and Ref. 63 lies in the object of focus. While their work investigates the interplay between local and global symmetries in the entanglement spectrum Hamiltonian (which corresponds, roughly speaking, to the ground state of the transfer matrix in our framework), they utilize this interplay to characterize so-called 1-form SPT. In contrast, our analysis centers directly on the transfer matrix itself and emphasizes its connection to SET phases. By offering a new perspective on these symmetries, our framework extends existing approaches and contributes to a deeper understanding of fundamental questions surrounding anyon condensation and confinement 36. Furthermore, it would be intriguing to examine whether our approach can be generalized to capture infinite-order transitions64 within a variational wavefunction framework.

Although our study is theoretical, the underlying topological phases and symmetry-enriched structures discussed here may be relevant to experimentally accessible systems such as Rydberg atom arrays, ultracold atoms in optical lattices, or superconducting qubit circuits, where effective spin models on two-dimensional lattices can be engineered 65,66,67,68.