Lattice model constructions for gapless domain walls between topological phases

Domain walls between different topological phases are one of the most interesting phenomena that reveal the non-trivial bulk properties of topological phases. Very recently, gapped domain walls between different topological phases have been intensively studied. In this paper, we systematically construct a large class of lattice models for gapless domain walls between twisted and untwisted gauge theories with arbitrary finite group $G$. As simple examples, we numerically study several finite groups(including both Abelian and non-Abelian finite group such as $S_3$) in $2$D using the state-of-the-art loop optimization of tensor network renormalization algorithm. We also propose a physical mechanism for understanding the gapless nature of these particular domain wall models. Finally, by taking advantage of the classification and construction of twisted gauge theories using group cohomology theory, we generalize such constructions into arbitrary dimensions, which might provide us a systematical way to understand gapless domain walls and topological quantum phase transitions.


I. INTRODUCTION
Classification and construction topological phases of quantum matter have become an extremely important and interesting subject in modern condensed matter physics. In the past decade, great achievements have been made toward establishing a complete paradigm for understanding topological phases of quantum matter, especially for systems with strong interactions, from the concept of long range entanglement to the classification of topological phases in generic interacting bosonic and fermionic systems. [1][2][3][4][5][6][7][8][9][10][11][12][13][14]. Nevertheless, our understanding of topological phase transitions is still very limited, especially in higher dimensions. Until very recently, it has been realized that a certain class of topological phase transitions in d spatial dimensions can be realized as gapless domain walls between topological phases in d + 1 spatial dimensions [15,16]. Such a holographic principle is very attractive since the properties of gapless domain walls are closely related to the bulk properties of topological phases. It is even possible to establish a paradigm towards understanding generic gapless domain walls and topological phase transitions in future.
It has been known for a long time that domain walls between two topological states usually exhibit extremely intriguing properties. At very basic level, there are two types of fundamental domain walls: gapped domain walls and gapless domain walls. The properties of gapped domain walls can be systematically studied based on the mathematical framework of unitary modular tensor category (UMTC)theory and the corresponding tunneling matrix formulation. The physical nature of gapped domain walls is also well understood in terms of anyon condensation [17][18][19][20][21]. Nevertheless, the gapless domain walls are much more complicated and harder to understand in general. Hence a systematic way to construct and understand gapless domain walls is very desired.
In addition, it would be very useful to distinguish two types of gapless domain walls according to their thermal Hall conductance K H : those with K H = 0, and those with K H = 0. The edge modes of various fractional quantum Hall (FQH) states are natural realizations of the first kind of gapless domain walls [22], and they are well understood in terms of chiral conformal field theory(CFT) as well as (perturbative) gravitational anomaly. However, the second kind of gapless domain walls with K H = 0 are rather unexpected from simple physical considerations. Very recently, the gapless conditions for K H = 0 domain walls among different Abelian FQHs are established in terms of the mathematical concept of Lagrangian subsets [19]. The underlying physical nature of these gapless domain walls can be explained by the so-called global gravitational anomalies. Moreover, it is widely believed that CFTs will also naturally emerge for the K H = 0 gapless domain walls, however, there is still lacking of systematical understanding and concrete lattice model realization for these CFTs.
In the presence of global symmetries, gaplessness domain walls exist even in the absence of gravitational anomalies(assuming the corresponding global symmetry does not break spontanouesly or explicitly). Most notably, gapless domain walls can be constructed between different symmetry protected topological (SPT) states [4,[23][24][25][26]. For example, gapless domain walls between free fermion topological insulators/superconductors are well understood in terms of massless free Dirac/Majorana arXiv:1801.00719v2 [cond-mat.str-el] 20 Sep 2020 fermions. Unfortunately, gapless domain walls for interacting SPT states are much harder to construct and only very few special examples are well understood so far [7,[27][28][29].
Another motivation to construct and study gapless domain walls of topological phases is the novel concept of bulk-edge correspondence between topological quantum field theory (TQFT) and conformal field theory (CFT). The first concrete example is the correspondence between the 3D bulk Chern-Simons theory and the 2D boundary Wess-Zumino-Witten (WZW) model, where the space of quantum states in the bulk TQFT is identified with the space of conformal blocks of the boundary CFT [30]. Such correspondence can be viewed as an implementation of the holographic principle [31,32]. The discovery of FQH gives rise to much deeper physical understanding of the bulk-edge correspondence, where the appearance of boundary exclusion statistics is associated with the corresponding CFT [33] which has their origin from anyonic excitations [34,35] with fractional statistics. Mathematically, UMTC also provides a general framework for describing anyons and fractionalized statistics [36], and can also be used to construct TQFT [37]. Therefore, a natural setup to further extend the study of bulk-edge correspondence is the gapless domain wall between different topological phases. Apparently, domain walls are of particular interest because they are where different anyons in the two bulk topological phases meet. Understanding the dynamics of such "anyon meetings" can give us deep insights into anyon dynamics as well as topological phase transitions.
In this paper, we systematically construct lattice models of gapless domain walls with K H = 0 between twisted and untwisted gauge theories(for arbitrary finite gauge group G). Such kind of gapless domain walls are closely related to bulk topological phase transitions and can be constructed in arbitrary dimensions. As a simple example, we illustrate the major steps of constructing such a gapless domain wall between the toric code model and double semion model. Then we study the domain wall model using the state-of-the-art loop-optimization tensor network renormalization (loop-TNR) algorithm [38]. Surprisingly, we find that the low energy spectrum of the domain wall model is consistent with the su(2) 1 Wess-Zumino-Witten model even in the absence of global SU (2) symmetry. We also find a bulk picture to understand the emergence of the su(2) 1 CFT on the domain wall. We further study such kind of gapless domain walls between twisted and untwisted gauge models with several other finite group G in 2D. For Abelian group examples, we find all of them can be described by Luttinger liquid theory with a central charge c = 1. For non-Abelian group example such as S 3 , we find the gapless domain wall model can be described by a CFT with central charge c = 2. We conjecture that the su(3) 1 Wess-Zumino-Witten model could be a very good candidate for such a CFT.
On the other hand, according to the correspondence between twisted gauge theories and SPT models [27], such kind of gapless domain walls also naturally arise on the interface between the trivial and non-trivial SPT states, provided that the global symmetry on the domain wall is not broken spontaneously or explicitly. From SPT point of view, the gapless nature of the domain walls is closely related to gauge anomaly which can be systematically classified and constructed via group cohomology theory in arbitrary dimensions. [4,7]. Thus, our constructions of gapless domain walls between twisted and untwisted gauge theory models can be easily generalized into higher dimensions by using group cohomology theory. We believe that many universal properties of these gapless domain walls could also be classified by group cohomology theory. The rest of the paper is organized as follows. In Sec. II, we start with a simple example -the gapless domain walls between toric code model and double semion model. We find a conformal field theory (CFT) described by the orbifold double su(2) 1 Wess-Zumino-Witten model even in the absence of global SU (2) symmetry for such a gapless domain wall. We also study the physical mechanism for the gapless nature of domain wall models. In Sec. III, we review group cohomology and its role in the systematic classification and construction of domain walls between twisted and untwisted gauge theories. We further study examples with several other gauge group. Finally, there will be a conclusion and a discussion on how to generalize these gapless domain wall models into higher dimensions. Let us begin with the Z 2 quantum double model and twisted quantum double model, namely, the toric code model [39] and the doubled semion model [40]. They can be defined as spin-1/2 systems on a honeycomb lattice where spins live on links. The Hamiltonians are ( Fig. 1) where v, p, l denote a vertex, a plaquette, and a link respectively, and Here l∈p τ x l is the product of the τ x l around a plaquette p and l∈v τ z l is the product of the τ z l around  (1). For both models, the first term Qv is a product of τ z l on the three links connected to the vertex v, the second term is a product of τ x l on the six links in the plaquette, functions f (τ z l ) on the six links connected to the plaquette, and projector Pp.  [40].
The ground state |Ψ t.c. of H t.c. is exactly known since all the plaquette terms and vertex terms commute with each other. The string language provides us with a very intuitive way to understand the ground state wave function: we interpret the τ z l = −1 and τ z l = 1 states on a single link as the presence or absence of a string. (This string is literally an electric flux line in the Z 2 gauge theory.) The appropriate low energy Hilbert space is made up of closed string states that satisfy l∈v τ z l = 1 at every vertex. The ground state is simply a superposition of all closed string states: Such a model realizes the simplest topologically ordered state in 2D. If we put the ground state wave function Eq. (3) on a torus, there are four different topological sectors, characterized by even/odd number of large strings wrapping around a torus in both directions. Moreover, the Z 2 electric charge e can be described as the ends of a string, which are bosons and are created/annihilated in pairs. H d.s. is a less well-known model, which has the same number of ground state degeneracy on torus, but exhibits a different kind of topological order. The low energy Hilbert space of the model is again made up of closed string states. However, the ground state wavefunction of this model is very similar to the toric code wavefunction except that different closed string states are weighted by different phase factors: where n is the number of closed loops in the closed-string state X. The (−) n(X) phase factor makes the Z 2 electric charge(described as the ends of string) carry semion statistics. . (a) For H0, the first term of the Hamiltonian is a gauge flux term µ z pq µ z qr µ z rp on the three links of the triangle pqr , and the second term is a product of σ x p and a projector Op that acts on the six triangles adjacent to p. (b) For H1, the first term is the same gauge flux term, and the second term is more complex BpOp.
The above two models can be mapped to Z 2 gauge models and twisted Z 2 gauge models on dual triangular lattices [27]. Edges of the triangular lattice are perpendicular to edges of the original honeycomb lattice. Spins on the edges are mapped accordingly. Centers of hexagonal plaquettes in the honeycomb lattice correspond to vertices of the triangular lattice. We put additional spins on these vertices. For each new spin, associate a gauge transformation where p labels a vertex in the triangular lattice and pq labels the edge connecting p and q, so that dimension of the physical Hilbert space remains the same. Operator mapping compatible with the gauge transformation is then found to be The resulting Hamiltonians reads (see Fig. 2) where the product runs over six triangles adjacent to the vertex p. Apparently, µ z pq can be regarded as the Z 2 gauge connection. Both H 0 and H 1 are invariant under the Z 2 gauge transformation. We now consider a system whose upper half plane is described by the toric code model and lower half plane is described by the doubled semion model. All local terms and label the plaquettes on the domain wall in a sequential order as shown in Fig. 3a. The nontrivial Hamiltonian algebra on the domain wall can be written as where τ z m,n denotes the spin operator on the edge in between plaquettes m and n. Unfortunately, the above Hamiltonian algebra is very complicated and can not be solved in an easy way. On the other hand, this domain wall model and its operator algebra can be easily translated to the Z 2 gauge models through mappings defined in the previous section (see Fig. 3b): [σ x m , B n ] = 0, when |m − n| > 1 σ x n B n±1 = −B n±1 σ x n σ z n∓1 σ z n±2 µ z n−1,n+1 µ z n±1,n±2 (10)

C. Gauge fixing and connections to domain wall models between SPT phases
We would like to simplify the Hamiltonian algebra Eq. (10) in the low energy sector by choosing a convenient gauge and fix all spins on edges. The Z 2 flux terms −µ z pq µ z qr µ z rp commute with all other terms in both H 0 and H 1 , hence always represent an independent finite energy change. Therefore, we expect the low energy physics of a gapless domain wall to be entirely captured in the subspace where µ z pq µ z qr µ z rp = 1, i.e. no local Z 2 flux. We will show later that the domain wall is indeed gapless. The simplest configuration with no local gauge flux is µ z pq ≡ 1. Given any eigenstate of {µ z p } with no local gauge flux, the , where the product runs over six triangles pqq adjacent to site p. uniform µ z pq ≡ 1 configuration can be achieved by applying gauge transformations if and only if there is no global gauge flux going through the domain wall, or equivalently in the string-net language, when there is no global string crossing the domain wall. We will henceforth assume this is the case and use the uniform gauge.
Under this uniform gauge, the domain wall Hamiltonian algebra simplifies to Such a Hamiltonian algebra can naturally arise on the domain wall between the trivial and non-trivial Z 2 SPT phases. The corresponding bulk Hamiltonians are simply those of the gauge models H 0 , H 1 with only spins on vertices and without gauge fields on edges ( Fig. 4) where the product runs over all six triangles pqq containing p. Both systems have spin-flip Z 2 global symmetry S = p σ x p inherited from the gauge symmetry, and both have commuting local terms and unique ground states. Specifically, ground state wave functions are where {α p } is a spin configuration in σ z p eigenbasis with α p =↑ or ↓, and N dw is the number of domain walls between spin up and down regions.
As shown in Ref. 27, these two spin models realize the only two short range entangled bosonic phases with onsite Z 2 symmetry. We can, in fact, start from these two SPT models and follow the gauge coupling procedures specified in Ref. 27 to obtain the Z 2 gauge models H 0 , H 1 . On the other hand, the domain wall models between the two SPT modelsH 0 andH 1 satisfy the Hamiltonian algebra Eq. (11) automatically.

D. Effective Hamiltonian with the same operator algebra
The Hamiltonian algebra of the gauge-fixed flux free domain wall decouples from the bulk terms, so it can be realized on a purely 1+1D spin model. We can straightforwardly check that the following mapping to virtual spin operators {τ x n ,τ y n ,τ z n } preserves this algebra Eq. (11) In particular, we verify that We further unitarily transform the operators with U = τ y n , where the product runs over every other site on the domain wall. This makes allB n 's and σ n 's take the same form and enhances translational symmetry of the effective Hamiltonian, which takes the final form We will henceforth refer to this model as the Ising domain wall model. This model has a Z 2 symmetry which we will show in Sec. III is indeed the Z 2 symmetry inherited from the gauge symmetry in the bulk. This symmetry also acts as a self-dual transformation, as it transforms the two termsτ y n andτ z n−1τ x nτ z n+1 into each other. We introduce an adjustable parameter g into H dw and transform this more general model with S dw : We see that the spectrum of H dw (g) and H dw (1/g) is the same up to a factor of g, which makes g = 1 a self-dual point. The spectrum of such a self-dual model is likely to be critical. On the other hand, since the flux free Hamiltonian algebra Eq. (11) is also realized as the domain wall between a trivial and a non-trivial Z 2 SPT phases, it is natural to expect such a domain wall model to be gapless if the global Z 2 symmetry is not spontaneously broken.

E. Numerical calculations
We now perform numerical calculations on the Ising domain wall model H dw in Eq. (16). We show strong evidence that the model is indeed critical. More precisely, our numerical evidence shows that the low-energy physics is described by the su(2) 1 WZW theory, or equivalently the compactified free boson CFT at the self-dual radius.
We perform both exact diagonalization for small system size and loop-TNR calculation for larger system size. For comparison, we also perform numerical calculations on the spin-1/2 XXX Heisenberg chain which is known to be described by the su(2) 1 WZW conformal field theory at low energy. We first compute low energy spectra of both models H dw and H XXX by exact diagonalization. In Fig. 5, the lowest eigenenergies of the two models on a periodic spin chain at the size of 30 sites are plotted against corresponding lattice momenta. Both models have a typical CFT excitation tower with linear dispersion, and an identification of low energy states between the two models is clear. Starting from the ground state, degeneracies of the first few energy levels are {1, 3, 1, 6, 6, . . . } in both models. When computing lattice momenta, we use only three-site translations for the Ising domain wall model and two-site translations for the XXX model. Using a finer translational symmetry in either model would cause the unique excitation tower at k = 0 to split into three/two towers at different momenta, making comparisons difficult.
The su(2) 1 WZW theory is equivalent to a free compactified boson at the self-dual radius. A general for-mula for scaling dimensions of the compactified free boson CFT is where m, n label different primary fields and ρ is the compactification radius (in our convention, ρ = 1 is the self-dual radius). In addition to states with the above scaling dimensions, there are also states corresponding to the current operator as well as its powers and derivatives. The scaling dimensions of the latter states are all integers. More discussion on free compactified boson is given in Appendix C. At self-dual radius ρ =  Fig. 6. The system size grows exponentially with iteration steps in loop-TNR, hence the logarithmic convergence of the finite size effect is translated into a power law, which is indeed observed in Fig. 6 for both models. Even though full convergence cannot be reached before numerical errors drive the system away from the RG fixed point, we can still identify each scaling dimension's corresponding exact values based on their trend, and recover the correct degeneracies in the expected convergence limit.
Finally, we deform the Ising domain wall model without breaking its anomalous global Z 2 symmetry (17), and see how scaling dimensions change with the deformation. To find such a suitable deformation, it is easier to first make a unitary transformation The transformed Hamiltonian can then be written as a special case of a more general class of Hamiltonians all sharing one Z 2 symmetry: with In particular, H dw (g = 1) is the transformed Ising domain wall model, and H dw (g = 0) is unitarily equivalent to the XY model which is known to realize the ρ = 1 √ 2 compactified free boson CFT. We compute scaling dimensions of this deformed Ising domain wall model for 0 ≤ g ≤ 1, and find excellent agreement with compactified boson CFT with 1 √ 2 ≤ ρ ≤ 1 (Fig. 7). It is known that for a free boson CFT with ρ = 0.5, changing ρ is the only relevant direction that preserves both conformal symmetry and c = 1 [42]. We have therefore rather conclusively shown that the Ising domain wall indeed realizes the compactified free boson CFT at self-dual radius. The domain wall between the toric code and double semions can be alternatively viewed as the boundary of a stacking system of the two. It is then interesting to investigate the bulk properties of the stacking system and how they relate to the boundary domain wall.
We note that the toric code model has four types of anyons 1, e, m, f ≡ em, where e, m are bosons and the Then what is the mechanism that protects the gapless nature of the domain wall? What makes the domain wall model so special such that it is an su(2) 1 WZW CFT? For the first question, a quick answer can be achieved by ungauging the Z 2 gauge symmetry in both models and mapping the domain wall model back to a special boundary of Z 2 SPT phase. Since both toric code model and double semion model can be regarded as the deconfinement phase of Z 2 gauge theory, their domain wall is also in the deconfinement phase where the Z 2 symmetry can not be spontaneously broken. Thus, the corresponding ungauged domain wall model must be a Z 2 SPT boundary without spontaneously symmetry broken, which must be gapless. (It is well known that the boudary of 2 + 1D SPT phase must be either gapless or symmetry breaking.) A more rigorous argument can be achieved by regarding the domain wall model as a specific boundary of toric code and double semion stacking systems with eb condensation while e and b are not condensed individually. Clearly, such a condition will exclude the condensation of Lagrangian subset (1, e, b, eb) or (1, m, b, mb) and protect the gapless nature of the boundary. Very recently, a more precise mathematical language, namely, the categorical symmetry [43] is introduced to understand such a special boundary and our lattice model construction can be regarded as an explicit realization of the maximal categorical symmetry for ground state wevefunctions.
The second problem is much more subtle, and we need to analyze the anyon content for the stacking model after condensing eb. Since all the anyons with nontrivial statistics with eb are confined, the remaining anyons are consisting of one boson b (which is identified to e) and two semions ms, ms (which are identified with fs, f s). Together with the identity particle, we end up with a new double semion model with four anyons 1, b, ms, ms, whose corresponding K matrix reads: Apparently, if it is not allowed to condense b in the deconfinement phase, there is no Lagrangian subset can be condensed in the above theory and we will end up with a c = 1 CFT. It is well known that the above K-matrix describes two layers of filling fraction ν = 1/2 bosonic Laughlin states with opposite chirality. Since the edge theory of ν = 1/2 bosonic Laughlin state is described by the chiral su(2) 1 WZW model, it is quite natural that the domain wall model can be described as the stacking of two chiral su(2) 1 WZW models with opposite chirality, if we assume there is no interactions between the left and right movers. However, we stress that the emergence of SU (2) symmetry in the above gapless domain wall model is accidental and it does not apply to generic Z 2 gapless domain wall model preserving the anomalous Z 2 symmetry Eq. (24). Physically, this is because the interactions between left and right movers always exist in generic Z 2 gapless domain wall models, which makes the radius ρ deviate from 1, e.g., Hamiltonian Eq. (23) with g = 1.
We will discuss more details in next section.

III. CONSTRUCTING LATTICE MODELS OF GENERAL GAPLESS DOMAIN WALLS
Interesting properties of the Ising domain wall motivate us to construct lattice models of domain walls between more general topological orders, and find their effective field theories. In the following, we are going to utilize the duality between string-net/gauge models and SPT phases as shown in the Z 2 example, which can be explicitly generalized to arbitrary finite group G.
As with the Z 2 case, a domain wall between SPT models captures the low energy physics of a corresponding domain wall between gauge models if it is gapless and has no global flux going through. For simplicity, we will focus on this flux-free case, and directly use the lattice construction of SPT phases to study domain walls between topological phases. We will see that domain walls we construct are all gapless for G = Z N case.

A. Constructing SPT phases using group cocycles
We first briefly review the construction of a lattice model realizing a 2D SPT phase with finite on-site symmetry G [4]. We define our model on a triangular lattice. Each vertex is associated with a |G|-dim Hilbert space where local basis states |g are labeled by group elements g ∈ G.
The model is constructed with a branching structure on the lattice and a 3-cocycle in the group cohomology H 3 (G, U (1)). A branching structure is an assignment of arrows on all edges of the lattice such that there is no local oriented loop, which defines a natural ordering of vertices for each triangle. A 3-cocycle, for our purpose, is a function ν : G 4 → U (1) that satisfies two conditions ν(gg 0 , gg 1 , gg 2 , gg 3 ) = ν(g 0 , g 1 , g 2 , g 3 ) (27a) ν(g 1 , g 2 , g 3 , g 4 )ν(g 0 , g 1 , g 3 , g 4 )ν(g 0 , g 1 , g 2 , g 3 ) ν(g 0 , g 2 , g 3 , g 4 )ν(g 0 , g 1 , g 2 , g 4 ) = 1 (27b) for any g, g i ∈ G. Two 3-cocycles ν, ν are considered equivalent if they only differ by a 3-coboundary λ, i.e. ν = νλ. A 3-coboundary is a function λ : The choice of coboundary can be thought of as a gauge freedom for cocycles. Equivalent classes of 3-cocycles form the third group cohomology H 3 (G, U (1)), which itself is an Abelian group. A more detailed introduction to group cohomology can be found in Appendix A. Define a unitary transformation on the triangular lattice with a branching structure (Fig. 8) where g * is a fixed group element, the product runs over all triangles labeled by their three vertices pqr ordered according to the branching structure, and s pqr = ±1 if the triangle has anticlockwise/clockwise orientation respectively. The Hamiltonian is defined as Although U ν acts on the entire lattice, each term of the Hamiltonian acts non-trivially only on seven neighboring sites centered at p. Explicitly, with a branching structure as shown in Fig. 8, g p , g 1 g 2 g 3 g 4 g 5 g 6 |H p |g p , g 1 g 2 g 3 g 4 g 5 g 6 = ν(g 4 , g 5 , g p , g p )ν(g 5 , g p , g p , g 6 )ν(g p , g p , g 6 , g 1 ) ν(g p , g p , g 2 , g 1 )ν(g 3 , g p , g p , g 2 )ν(g 4 , g 3 , g p , g p ) . This expression has been simplified using the cocycle condition (27b). All local terms commute, so the model is exactly solvable. It has a unique ground state Both the Hamiltonian and the ground state have the G symmetry {|g p } → {|gg p }. Models realize distinct SPT phases if and only if they are defined by inequivalent 3cocycles.

B. Domain walls between general SPT phases
We now use the construction in the previous section to derive domain wall models between different SPT phases with the same symmetry G.

Consider a system on a triangular lattice with the Hamiltonian
where {•} and {•} denote vertices in the upper and lower half plane respectively (Fig. 9). Each local term H a,b p acts on seven neighboring sites centered at p. H a p and H b p are defined similarly by Eq. (30), but use two inequivalent cocycles ν a , ν b . All H a p 's commute and all H b p 's commute, leaving the domain wall as the only nontrivial part of the system.
To explicitly decouple the domain wall from the bulk, we consider the unitary transformation where A and B are regions on the lattice, as shown in Fig. 9, that mostly represent upper and lower half planes respectively. It changes local terms in the bulk into trivial one body interaction, leaving the domain wall explicitly decoupled. On the domain wall, the Hamiltonian transforms into i, j, k label a triangle whose top or bottom vertex is p, while which vertex each of them represents is assigned according to the branching structure ( Fig. 9). Due to the group structure of group cohomology H 3 (G, U (1)), we see that the domain wall is defined only by one 3cocycle ν ab rather than two. In general, different branching structures lead to different domain wall models, and H a,b p 's can be different from one another. For concreteness, we will mostly focus on the specific branching structure shown in Fig. 9. This branching structure is particularly nice, since it gives us a fully translational invariant domain wall. The corresponding Hamiltonian reads H = − n H n , g n , g n−1 g n+1 |H n |g n , g n−1 g n+1 = ν ab (g n+1 , g n , g n−1 , g * ) ν ab (g n+1 , g n , g n−1 , g * ) .
Its effective anomalous symmetry operator can also be computed Now we have obtained a general domain wall model of 2D bosonic SPT phases. To construct an explicit model, we only need to find explicit expression of group 3-cocycles.

C. ZN domain wall models
It is no coincidence that both Z N gauge theories and SPT phases in 2+1D are classified by the group cohomology H 3 (Z N , U (1)) ∼ = Z N . We note that string-net realizations of all N distinct Z N gauge theories can be constructed [44], and lattice models exist for any SPT phases with finite on-site symmetry [4].
The formula of Z N 3-cocycles is well-known [45] ν 3 (g 0 , g 1 , g 2 , g 3 ) = exp i 2πk N 2 g 10 (g 21 + g 32 − g 31 ) (38) where g i labels a group element and k labels the N different classes of 3-cocycles in H 3 (Z N , U (1)) ∼ = Z N . Using this formula in Eq. (36), we can define a Z N domain wall model that is labeled by (N, k). Hamiltonians given by (N, k) and (N, N − k) are related by complex conjugation, therefore there are effectively only N/2 distinct nontrivial domain wall models for a given N .

Re-deriving the Z2 domain wall
The simplest nontrivial domain wall model defined by (36) is given by (N, k) = (2, 1). We expect this model to be equivalent to the Ising domain wall we studied in Sec. II. The cocycle formula in this case simplifies to Substituting this formula into Eq. (36), we find which is exactly the same as Eq. (24) at g = 1, hence equivalent to the Ising domain wall model.
Entanglement entropy scaling [46] of a 48-site periodic chain as computed by density matrix renormalization group (DMRG) [47][48][49] is plotted in Fig. 10. We find very precise logarithmic scaling that is fitted with central charge very close to 1 for all models, proving criticality of these models.
We next use loop-TNR to compute lowest virtual energies of these models, normalized such that ground state energy is −1/12, compatible with a c = 1 CFT. See Fig. 11. Here, we define virtual energies via the virtualspace transfer matrix [50] of the quantum lattice model. They are characterized by a theory relating to the original lattice model by an S modular transformation. See Appendix D for details.   The loop-TNR computation does not converge for the (4, 2) model, but its converging behavior is very similar to that of the Ising domain wall model. We again see converging trends towards {0.5, 1.0, 1.5, 2.0, 2.5} and degeneracies {4, 6, 8, 17, 28} in the expected limit. We hence conjecture that it is also described by the compactified free boson CFT at self-dual radius.
The other four models' virtual energies do not match any known c = 1 CFT [42] (in the usual Euclidean space with the metric being the identity matrix). The gapless edge of an Abelian topological phase is expected to be described by a Luttinger liquid action [51] where K is an integer symmetric matrix, V is a positive definite symmetric matrix, and fields are compactified φ I = φ I + 2π. Our domain wall models can be viewed effectively as special edge models. Since c = 1 from the entanglement entropy scaling, we expect K to be a 2 × 2 matrix with eigenvalues of opposite signs. In addition, since the domain wall models are effectively 1D lattice models, we required the edge theory to be modular invariant. Hence, we are led to the conclusion that by redefining the fields φ I through a linear combination, we can simultaneously diagonalize both matrices with a congruent transformation such that This diagonalized form is equivalent to two compactified massless free chiral bosons moving in opposite directions with velocities v 1 and v 2 respectively. In the usual CFT defined with the Euclidean metric, conformal invariance requires v 1 = v 2 . In our case, v 1 = v 2 is in general not satisfied. For the general v 1 = v 2 case, we find that energies of highest weight states are (see Appendix C) and energies of descendant states are E m,n,{n 1,l },{n 2,l } = E m,n + 2 We note that an S modular transformation has the effect of mapping v r → 1/v r , which does not change the spectrum. Hence if virtual energies of a lattice model are characterized by Luttinger liquid, the lattice model itself should be described by the same theory. We proceed to fit the energies in Fig. 11 with Eq. (43) by adjusting the two free parameters ρ and v r , and find a perfect fit for all models. The fitted values are listed in Table I.
To further confirm this field theoretic description, we continuously deform the (3, 1) domain wall model without breaking the effective Z 3 symmetry defined by (37), and connect it with a compactified free boson CFT for which v 1 = v 2 . One such deformation is found to be H deform (g) = g H (3,1)  with where H (3,1) is the domain wall Hamiltonian, and H is a fine tuned Hamiltonian that realizes the ρ = √ 2/3 compactified free boson CFT. This deformed model realizes a CFT at g = 0, and recovers the (3, 1) domain wall model (up to a constant) at g = 1. Virtual energies of H deform (g) are computed with loop-TNR for 0 ≤ g ≤ 1 (Fig. 12) and fitted with the field theoretic predictions Eq. (43) to excellent agreement.

Alternative branching structure and cocycle gauges
Models we studied in the previous section all assumed a particular branching structure (Fig. 9) and cocycle gauge (38). Changing these choices amounts to a local unitary transformation in the bulk. This is inconsequential for the bulk physics, but can cause nontrivial changes on the domain wall, because the resulting local unitary transformations on the two sides may not be the same. It is important that our general conclusions remain valid for different choices of branching structures and cocycle gauges.
Using the same cocycle gauge (38), the new Z 2 domain wall Hamiltonian is H a n = This is unitarily equivalent to the Ising domain wall. Explicitly, Eq. (48) is related to Eq. (24) at g = 1 by The Z 3 domain wall is changed by this new branching structure. We again compute its virtual energies, see Fig. 13, and find that it still fits the Luttinger liquid predictions perfectly at ρ = 0.951, v r = 1.416.
Next, we keep the old branching structure in Fig. 9, but multiply each 3-cocycle in (36) with an arbitrary 3coboundary, effectively changing the cocycle gauge. Any Z N group coboundary is fully determined by N 2 independent parameters: For both Z 2 and Z 3 , we generate 16 sets of random θ mn ∈ [0, 2π), and compute virtual energies of resulting domain wall models with loop-TNR. All generated Z 2 domain walls fit a free boson CFT with compactification radius ρ ∈ [0.945, 1], and all generated Z 3 domain walls fit a Luttinger liquid theory with appropriate ρ and v 1 /v 2 . Energies of a sample of these models are shown in Fig. 15 and Fig. 16. Best fit parameters for Z 3 domain walls are listed in Table II.  Fig. 16.

D. Domain wall model of more complex Abelian group
For gapless domain wall models with product of Abelian groups such as G = Z 2 2 , we find all of them have the same central charge c = 1 and again can be described by Luttinger liquid theory with appropriate compactification radius ρ. This is not quite surprising because the gauge fluxes in these models all carry Abelian statistics, just like those simple Z N models we studied above.
Below we consider a more interesting case with G = Z 3 2 . Under a proper choice of the 3-cocycle in this case, it turns out that the gauge flux will carry non-Abelian statistics. It is well known that for G = Z 3 N , its third group cohomology H 3 (Z 3 N , U (1)) ∼ = Z 7 N has seven generators. Writing each group element as g = (g 1 , g 2 , g 3 ) with g p = 0, . . . , N − 1, the explicit expressions for the generators are [52] α pq I (g 0 , g 1 , g 2 , g 3 ) = exp i 2π N 2 g p 10 (g q 21 + g q 32 − g q 31 ) α II (g 0 , g 1 , g 2 , g 3 ) = exp i 2π N g 1 10 g 2 21 g 3 32 A 3-cocycle of Z 3 N can be written as for k pq I , k II = 0, . . . , N − 1. We will restrict ourselves to the simplest case of N = 2.
Of particular interest to us is the generator α II , which involves all three sub-Z 2 groups. We compute the scaling dimensions of the domain wall defined by the cocycle k pq I = 0, k II = 1, which corresponds to the case with non-Abelian statistics for the gauge flux. We again find such a gapless domain wall model can be described by a Luttinger liquid with c = 1. As seen in Fig. 17, the data again agrees with a compactified boson CFT, with compactification radius ρ = 0.932. Scaling dimensions of a Z 3 2 domain wall with a non-trivial αII generator. Best fit compactification radius ρ = 0.932.

E. Domain wall model of non-Abelian group
Finally, we consider an example with non-Abelian gauge group G. It is well known that the smallest nonabelian group is S 3 . We label each group element by a pair of numbers g = (g 1 , g 2 ), where g 1 = 0, 1 and g 2 = 0, 1, 2. Group multiplication rule is where n m = n mod m. The group cohomology H 3 (S 3 , U (1)) ∼ = Z 6 . An explicit formula of 3-cocycles is [52] ν 3 (g 0 , g 1 , g 2 , g 3 ) = exp i 2πk 9 a 10 (a 21 + a 32 − a 31 ) where and k = 0, . . . , 5 labels the six inequivalent 3-cocycles. We use the variational optimization algorithms for uniform matrix product states (MPS) [53,54] to obtain the ground state. Given a translationally invariant MPS, the correlation length ξ can be extracted from the spectral properties of the transfer operator [55] HereĀ s is the complex conjugate of A s . λ 1 and λ 2 are the largest and second largest eigenvalue of E. The ratio between λ 1 and λ 2 also bounds the gap of the parent Hamiltonian [56,57], The central charge c of the CFT is related to correlation length ξ by the scaling relation [58] S = c 6 lnξ (59) In MPS algorithms, both S and ξ are controlled by the numerical parameter "bond dimension" D. As D increases, we obtain a more accurate ground state with larger entanglement entropy S and larger correlation ξ.
In Fig. 19, we plot S(ξ) with respect to ξ for several D. From the data fitting we find c ∼ 2.072. In Fig. 18, we plot the gap as a function of correlation length ξ. It shows the gap is closed as ξ diverges, which confirms the gapless nature of the S3 model with k = 1.
Unfortunately, due to a very big truncation error, we can not implement the loop-TNR algorithm to compute the scaling dimension in this case. Possible candidate CFTs with c = 2 include the su(3) 1 Wess-Zumino-Witten model and two-component Luttinger liquid theory with proper compactification radii.

IV. CONCLUSIONS AND DISCUSSIONS
In conclusion, we systematically construct lattice models of gapless domain walls between twisted and untwisted gauge models with arbitrary finite group G in 2 + 1D. We then use the state-of-art loop-TNR algorithm to study several examples. For all the Abelain group cases studied here, we find all of them perfectly agree with the Luttinger liquid theory descriptions with a central charge c = 1 at low energy. For the simplest non-Abelian group case with G = S 3 , we find it is still gapless but has a larger central charge c = 2. We conjecture that the su(3) 1 Wess-Zumino-Witten model could be a very good candidate for such a CFT. We further provide a physical picture to understand the gapless nature of these domain walls based on the theory of Lagrangian subsets and anyon condensation. On the other hand, since the corresponding ungauged domain walls can be regarded as an SPT boundary with an anomalous symmetry action, they must be gapless if the anomalous symmetry is not broken spontaneously, which is true for all domain wall models constructed in this paper. We also would like to stress that for a given finite group G, our construction actually gives rise to a deformation class of gapless domain wall models parameterized by different coboundary choices, and our numerical results have shown that all these models have the same central charge but with different compactification radius and velocity ratios in general.
Finally, it is straightforward to generalize our construction of gapless domain wall models into higher dimensions using the correspondence between SPT models and twisted gauge models. All we need is a branched triangulation on a higher dimensional manifold, and an appropriate region A (Fig. 9) to define the unitary transformation U . It can be easily seen that for a higher dimensional domain wall Hamiltonian H = − i H i , each local term H i acts only on the site i and its nearest neighbors. But to write down an explicit model is rather tedious. The difficulty partly comes from the reduced translational and rotational symmetry when a branched triangulation is imposed on a higher dimensional manifold. Such reduced symmetry also increases the difficulty for numerical study of these models. Physically, we believe that these models still describe gapless phases since the anomalous symmetry cannot be broken on domain walls separating two deconfinement phases of gauge group G. From a more mathematical point of view, all these domain wall models realize the so-called categorical symmetry. Thus, our construction might even give rise to a systematical way of understanding gapless domain walls in bosonic systems, especially in 2+1D, since recent studies have shown that all topological phases in 3+1D can be realized as twisted gauge theory [59]. In future work, it will be of great interest to investigate the whether these gapless domain wall models are integrable or not. The generalization of our constructions into fermionic systems is also another important direction. in addition. More formally, we have For our purposes, it is sufficient to consider M = U (1) whose elements are simply phase factors, and G is the symmetry group of the system. G acts on U (1) in the following way: where s(g) = 1, if g ∈ G acts unitarily; −1, if g ∈ G acts anti-unitarily. (A8) To explicitly indicate this nontrivial action of antiunitary group elements, we shall from now on write M = U T (1).

Appendix B: Constructing bosonic SPT phases using group cohomology
Bosonic SPT phases can be systematically described by group cohomology theory [4]. Specifically, (d + 1)dimensional bosonic SPT phases with symmetry group G can be labeled by elements in H d+1 [G, U T (1)]. The identity element corresponds to a trivial phase (product state), and nontrivial elements correspond to nontrivial SPT phases.
Furthermore, we can construct exactly solvable lattice models of SPT phases for any finite symmetry group G and in any dimensions. Such a model is constructed on a d-dimensional simplicial complex M , which is itself the boundary of an extended (d + 1)-dimensional complex M ext with a branching structure, and we further assume that there is only one internal vertex in M ext . Each vertex in M is associated with a |G|-dimensional local Hilbert space, where basis vectors |g i are labeled by group elements g i ∈ G. The internal vertex of M ext is associated with a fixed group element g * ∈ G. The ground state wave function is given by (Fig. 20) where the sum runs over all configurations {g i } M of the vertices in M and the product runs over all simplices {ij · · · * } in M ext , and ν d+1 is a group (d + 1)-cocycle. The symmetry is on-site and acts in the following simple way: A graphical representation of the phase factor, which is a product of 3-cocycles. The "internal" vertex associated with g * was originally in the center of the cage, but can be removed using cocycle condition (either graphically or algebraically).
The ground state is trivial a product state when the cocycle is trivial. Nontrivial ground states can be obtained from this trivial state by a unitary transformation An exactly solvable Hamiltonian can be constructed for this ground state as where the sum runs over all vertices i and |φ i = gi∈G |g i . It is straightforward to check that all H i 's commute with each other, hence the solvability of this model.
Appendix C: Energy spectrum of Luttinger liquid theory In this appendix, we derive the energy spectrum (43) for the c = 1 Luttinger liquid theory whose action is given in Eq. (41). For convenience, we repeat the action here (C1) where φ I is a compact boson with φ I = φ I + 2π. As discussed in the main text, we only need to consider the c = 1 two-component theory associated with where a, b, c are real numbers satisfying the conditions a > 0 and ab > c 2 such that V is positive definite. This is a free theory and it is well studied in the literature (see e.g. Ref. 61). To be self-contained, we give a brief derivation on the relevant results used in the main text, with a focus on the case that the left and right movers have different velocities.
To find the spectrum, we perform the following change of variables:φ where φ = (φ 1 , φ 2 ) and with ρ = 4 b/a. Inserting (C3) into (C1), the action is rewritten as where v 1 = √ ab + c and v 2 = √ ab − c. The two fieldsφ 1 andφ 2 completely decouple, each of which is a standard free chiral boson. The theory (C5) can then be solved using the standard mode expansion; see e.g. Ref. 61. Below we briefly state the main results.
The energy eigenstates consist of two types. First, there is a set of highest wight states, created by the corresponding vortex operators when acting on the ground state. This energy of these states depend on the compactification radii of the fieldsφ 1 andφ 2 , which inherit from those of φ 1 and φ 2 . In the current case that φ I = φ I +2π, the general form of vortex operators are given by where m, n are integers. Acting V m,n on the ground state |0 , it creates a highest weight state |m, n ≡ V m,n |0 . These states are the primary states of the U (1) Kac-Moody algebra that one can read off from the action S edge . We rewrite V m,n = e imφ1+iñφ2 , wherẽ m = 1 √ 2 (mρ + n/ρ),ñ = 1 √ 2 (−mρ + n/ρ) (C7) The energy of the state |m, n (relative to the ground state) is given by where L is the system size and v r = v 2 /v 1 . Second, in the mode expansion of the fieldsφ 1 andφ 2 , one introduces the Fourier coefficientsã 1,l andã 2,l , where l is integer. The highest weight states are annihilated bỹ a 1,l andã 2,l with l > 0, i.e.,ã 1,l |m, n =ã 2,l |m, n = 0. On the other hand, a Fork space is spanned by acting a 1,l andã 2,l with l < 0 on each primary state |m, n : Then, the whole spectrum is generated by varying the integers m, n, and {n 1,l , n 2,l }.
Finally, we comment that the ground state energy also depends on the system size L. For periodic boundary conditions, the ground state energy is given by where c is the central charge. Since c = 1, Eq. (C11) can be used to set the energy unit π(v 1 + v 2 )/L in numerical calculations.