The phase transitions between $Z_n\times Z_n$ bosonic topological phases in 1+1 D, and a constraint on the central charge for the critical points between bosonic symmetry protected topological phases

The study of continuous phase transitions triggered by spontaneous symmetry breaking has brought revolutionary ideas to physics. Recently, through the discovery of symmetry protected topological phases, it is realized that continuous quantum phase transition can also occur between states with the same symmetry but different topology. Here we study a specific class of such phase transitions in 1+1 dimensions -- the phase transition between bosonic topological phases protected by $Z_n\times Z_n$. We find in all cases the critical point possesses two gap opening relevant operators: one leads to a Landau-forbidden symmetry breaking phase transition and the other to the topological phase transition. We also obtained a constraint on the central charge for general phase transitions between symmetry protected bosonic topological phases in 1+1D.


Introduction and the outline
Last five years witnessed a fast progress in the understanding of a new type of quantum disordered states -symmetry protected topological states (SPTs) [1][2][3]. These states exhibit a full energy gap in closed (i.e., boundary-free) geometry and exhibit the full symmetry of the Hamiltonian. However, these states are grouped into different "topological classes" such that it is not possible to cross from one topological class to another without closing the energy gap while preserving the symmetry. Our goal is to understand the difference (if any) between the traditional Landau type and this new kind of "topological" phase transitions.
Because the Landau-type phase transitions are triggered by the fluctuations of bosonic order parameters over space-time, to minimize the obvious difference we focus on the phase transitions between bosonic SPT phases [3]. Hence we do not address the phase transition between fermionic topological insulators or superconductors [1,2]. Moreover, to make everything as concrete as possible we shall focus on one space dimension and to topological phase transitions which have dynamical exponent equal to one (hence can be described by conformal field theories (CFTs)). We spend most of the space studying a specific class of such phase transitions -the phase transition between bosonic SPTs protected by Z n × Z n .
We spend most of the space describing the study of a specific class of such phase transitions -the phase transition between bosonic SPTs protected by Z n × Z n . Here we combine a blend of analytic and numerical methods to arrive at a rather complete picture for such critical points. From studying these phase transitions we observe an interesting fact, namely whenever the transition is direct (i.e., when there are no intervening phases) and continuous the central charge (c) of the CFT is always greater or equal to one. Near the end of the paper, we obtain a constraint on the central charge for CFTs describing bosonic SPT phase transitions: namely, c ≥ 1. Therefore, none of the best known "minimal models [4]" can be the CFT for bosonic SPT phase transitions! According to the group cohomology classification [3], in one space dimension, the group Z n × Z n protects n different topological classes of SPTs. If we "stack" a pair of SPTs (which can belong to either the same or different topological class) on top of each other and turn on all symmetry allowed interactions, a new SPT will emerge to describe the combined system. An abelian (cohomology) group H 2 (Z n × Z n , U(1)) = Z n (here the superscript "2" refers to the space-time dimension) classifies the SPT phases and describes the stacking operation. Here each topological class is represented by an element (i.e., 0, ..., n − 1) of H 2 (Z n × Z n , U(1)) = Z n and the "stacking" operation is isomorphic to the mod(n) addition of these elements.
To understand the phase transitions between different classes of SPTs it is sufficient to focus on the transition between the trivial state (which corresponds to the "0" of Z n ) and the non-trivial SPT corresponding to the "1" of Z n . The transition between phases correspond to other adjacent elements of Z n , e.g., (m, m + 1), will be in the same universality class as that between (0, 1). Transitions between "non-adjacent" topological classes will generically spit into successive transitions between adjacent classes.
There are 11 sections in the main text. In these sections we restrain from heavy mathematics, i.e., we simply state the main results and provide simple arguments. There are 6 appendices where mathematical details can be found. The outline of this paper is the follows. In section 2 we present the exactly solvable fixed point hamiltonians for the trivial and non-trivial Z n × Z n protected SPT phases. In section 3 we present a hamiltonian that interpolates between the fixed point hamiltonians in section 2. A single parameter tunes this hamiltonian through the SPT phase transition. Section 4 introduces a non-local transformation that maps the hamiltonian in section 3 to that of two n-state clock models with spatially twisted boundary condition and Hilbert space constraint. In particular, at criticality, we show that the partition function of the transformed hamiltonian corresponds to an "orbifolded" Z n × Z n clock model. In section 5 we discuss the effects of orbifolding on the phases of the clock model and show the results are consistent with what one expect for the SPT phases. Section 6 gives the phase diagram of the hamiltonian given in section 3. In section 7 we show that from the point of view of the orbifolded clock model the SPT transition corresponds to a Landau forbidden transition. In section 8 we present the conformal field theories for the SPT phase transitions discussed up to that point. Section 9 presents our numerical density matrix renormalization group results. We compare these results with the prediction of section 8. Section 10 presents the argument that the central charge of the CFTs that describe SPT phase transitions must be greater or equal to one. Finally, section 11 is the conclusion.
In Appendix A, we provide a brief review of the key ingredients of the 1 + 1D group cohomology, namely, the notions of cocycles and projective representations. After that, we show how to use cocycles to construct solvable fixed point SPT hamiltonians. Appendix B summarizes the non-local transformation that maps the hamiltonian in section 3 of the main text to that of two n-state clock models with spatially twisted boundary condition and Hilbert space constraint. In Appendix C we show that the partition function associated with the hamiltonian in Appendix B (and section 3 of the main text) corresponds to that of "orbifolded" Z n × Z n clock model. Appendices D, E, F present the modular invariant partition functions of the orbifold Z 2 × Z 2 , Z 3 × Z 3 and Z 4 × Z 4 clock models, respectively. In these appendices, we examine the primary scaling operator content of the modular invariant conformal field theory. In addition, we study the symmetry transformation properties of various Verma modules and the scaling dimension of primary scaling operators, particularly that of the gap opening operator. Appendix G summarizes the details of the density matrix renormalization group calculation. Finally, in Appendix H we briefly review the symmetry of the minimal model conformal field theories.

Exactly solvable "fixed point" Hamiltonians for the SPTs
Each SPT phase is characterized by an exactly solvable "fixed point" Hamiltonian. In Appendix A we briefly review the construction of these Hamiltonians using the "cocycles" associated with the cohomology group [5,6]. For the case relevant to our discussion the following lattice Hamiltonians can be derived [7] so that its ground state belong to the "0" and "1" topological classes of H 2 (Z n × Z n , U(1)) = Z n These Hamiltonians are defined on 1D rings consisting of N sites. For each site labeled by i the local Hilbert space is spanned by |g 2i−1 , g 2i := |g 2i−1 ⊗ |g 2i where (g 2i−1 , g 2i ) ∈ Z n × Z n with g 2i−1 , g 2i = 0, 1, ..., n − 1. The total Hilbert space is the tensor product of the local Hilbert space for each site. For the convenience of future discussions from now on we shall refer to (2i − 1, 2i) as defining a "cell", and call |g 2i−1 and |g 2i as basis states defined for "site" 2i − 1 and 2i. The operators M j and R j in Equation (1) are defined by M j |g j := |g j + 1 mod n, and R j |g j := η g j n |g j where η n = e i2π/n .
From Equation (2) we deduce the following commutation relation between M and R: Due to this commutation relation, it can be checked that the n × n matrices associated with M j and R j form a projective representation of the Z n × Z n group multiplication law (see appendix A.2 for the definition of projective representations). Finally periodic boundary condition is imposed on Equation (1) which requires Under these definitions Equation (1) is invariant under the global Z n × Z n group generated by The form of Hamiltonians given in Equation (1) is quite asymmetric between M and R. We can make it more symmetric by performing the following unitary transformation on the local cell basis as follow This results in the following transformations of the operators in Equation (1) It is straightforward to show that after these transformations the new operators obey the same commutation relation as Equation (3). Moreover, it can also be shown that R obeys the same boundary condition, namely, R 2N +1 = R 1 and R 2N +2 = R 2 . In addition, it is also straightforward to show that under Equation (6) the generators of the Z n × Z n group become Thus alternating "site" carries the projective and anti-projective representation of Z n × Z n . Under Equation (6) the Hamiltonian H 0 and H 1 become These Hamiltonians are pictorially depicted in Fig. 1(a,b). Note that while H 0 ( Fig. 1(a)) couples sites within the same cell, H 1 couples sites belong to adjacent cells ( Fig. 1(b)). Because both H 0 and H 1 consist of decoupled pairs of sites (the coupling terms associated with different pairs commute with one another) they can be exactly diagonalized. The result shows a unique ground state with a fully gapped spectrum for both H 0 and H 1 . Using Equation (7) it is simple to show that the ground states are invariant under Z n × Z n . The fact that H 0 and H 1 describe inequivalent SPTs can be inferred by forming an interface of H 0 and H 1 as shown in Fig. 1(c). A decoupled site (red) emerges. Localizing on this site there are degenerate gapless excitations carrying a projective representation of the Z n × Z n [23]. The fact that gapless excitations must exist at the interface between the ground states of H 0 and H 1 attests to that fact that these states belong to inequivalent topological classes of H 2 (Z n × Z n , U(1)) = Z n .

An interpolating Hamiltonian describing the phase transition between Z n × Z n SPTs
To study the phase transition between the ground state of H 0 and the ground state of H 1 we construct the following Hamiltonian which interpolates between H 0 and H 1 as follows With both H 0 and H 1 present the Hamiltonian given in Equation (9) is no longer easily solvable. However, in the following, we present analytic results showing (1) for 2 ≤ n ≤ 4 the phase transition occurs at λ = 1/2, (2) the central charge, the conformal field theory and its associated primary scaling operators at the phase transitions. For n ≥ 5 there is a gapless phase centered around λ = 1/2 hence the phase transition is not direct. Moreover for the interesting case of n = 3 we will present the numerical density matrix renormalization group results which confirm our analytic solution.

Mapping to "orbifold" Z n × Z n clock chains
In Appendix B we show that Equation (9) can be mapped onto a Z n × Z n clock model with spatially twisted boundary condition and a Hilbert space constraint. In Appendix C we further show that these amount to "orbifolding".
The mapping is reminiscent of the duality transformation in a single Z n clock model. The mapping is achieved via the following transformations: After the mapping, the Hamiltonian in Equation (9) is transformed to Here M and R obey the same commutation relations as M and R in Equation (3). Equation (11) is the quantum Hamiltonian for two Z n clock models [8], one defined on the even and one on the odd sites, respectively. However, generated by the mapping, Equation (11) is supplemented with a twisted spatial boundary condition and a constraint: Constraint: Here B is an operator that commutes with all the Rs and Ms. The eigenvalues of B are b = 1, η n , ..., η n−1 n (recall that η n = e i2π/n ). In terms of the transformed variables, the generators of the original Z n × Z n group are given by The spatially twisted boundary condition Equation (12) and the constraint Equation (13) (which turns into a time direction boundary condition twist in the path integral representation of the partition function) execute the "orbifolding" (see later). By swapping the even and odd chains Equation (11) exhibit the duality. This implies the self-dual point at λ = 1/2 is special. In particular, if there is a single critical point as a function of λ, it must occur at λ = 1/2. Incidentally, if we put aside Equation (12) and Equation (13), λ = 1/2 is where each of the clock chains in Equation (11) becomes critical.
As we will show later the effects of Equation (12) and Equation (13) (i.e., orbifold) is to change the primary scaling operator content of the critical CFT from that of the direct product of two Z n clock models. However they do not jeopardize the criticality, nor do they change the central charge. We shall return to these more technical points later. At the meantime let's first study the effects of Equation (12) and Equation (13) on the phases.

The effect of orbifold on the phases
Knowing the behavior of the single Z n clock chain, Equation (11) suggests for λ < 1/2 the odd-site chain will spontaneously break the Z n symmetry while the even chain remains disordered. The ground state will lie in the b (the eigenvalue of B ) = 1 sector on account of the twisted boundary condition. For λ > 1/2 the behaviors of the even and odd chains exchange, and the ground state remains in the b = 1 sector. On the surface, such symmetry breaking should lead to ground state degeneracy which is inconsistent with the fact that both SPTs (for λ < 1/2 and λ > 1/2) should have unique groundstate.
This paradox is resolved if we take into account of the constraint in Equation (13). For simplicity let's look at the limiting cases. For λ = 0 the ground state of Equation (11) is |g, g, ..., g odd ⊗ |p, p, ..., p even ⊗ |b = 1 (15) where g = 0, ..., n − 1. Here the "paramagnet state" |p for each site is defined as As expected, such ground state is n-fold degenerate and it does not satisfy the constraint of Equation (13). However, if we form the symmetric superposition of the odd-site symmetry breaking states ⎛ ⎝ 1 √ n n−1 g=0 |g, g, ..., g odd ⎞ ⎠ ⊗ |p, p, ..., p even ⊗ |b = 1 (17) the constraint is satisfied and the state is non-degenerate. Obviously, Equation (17) is invariant under the Z n × Z n generated by Equation (14). Although Equation (17) is non-degenerate, the two-point correlation function R 2j +1 R † 2k+1 still shows long-range order. Almost exactly the same arguments, with odd and even switched, apply to the λ = 1 limit. The only difference is instead of observing |p, p, ..., p even being invariant under the action of N j =1 M 2j we need to observe that 1 √ n n−1 g=0 |g, g, ..., g even is invariant. As λ deviates from the limiting values, so long as it does not cross any phase transition the above argument should remain qualitatively unchanged. In this way we understand the effects of Equation (12) and Equation (13) on the phases.

The phase diagram
Since upon orbifolding the phases of the decoupled Z n × Z n clock models seamlessly evolve into the SPT phases we shall construct that phase diagram using what's know about the phase structure of the clock model. It is known that a single Z n clock chain shows an order-disorder phase transition at a single critical point for n ≤ 4, while there is an intermediate gapless phase for n ≥ 5 we conclude the phase diagram is shown in Fig. 2(a,b). Since our goal is to study the continuous phase transition between SPTs we focus on n ≤ 4.

SPT transitions as "Landau-forbidden" phase transitions
According to Landau's rule, transitions between phases whose symmetry groups do not have subgroup relationship should generically be first order. Continuous phase transitions between such phases are regarded as "Landau forbidden" in the literature.
As discussed earlier, in terms of the orbifolded Z n × Z n clock chains, the two phases on either side of the SPT phase transition correspond to the breaking of the Z n symmetry in one of the clock chain but not the other. In the following, we elaborate on this statement.
For λ < 1/2 although the ground state in Equation (17) is non-degenerate, the two-point correlation function R 2j +1 R † 2k+1 shows long-range order. When the odd and even chains are switched the same argument applies to the λ > 1/2 limit. If we define it is easy to show that equations (11), (12) and (13) commute with them, hence the Z n × Z n group they generate are also the symmetry of the problem. However it is important not to confuse Z n × Z n with the original Z n × Z n group (which is generated by Equation (14)). With respect to the Z n × Z n symmetry the two phases (realized for λ < 1/2 and λ > 1/2) breaks two different Z n factors, hence the symmetry groups of the two phases have no subgroup relationship, thus if a continuous phase transition between them exists it is a Landau forbidden transition. In fact, it is the original Z n × Z n symmetry that "fine tunes" the system to realize such non-generic continuous phase transition.

The CFT at the SPT phase transition for n = 2, 3, 4
It is known that the central charge of the CFT describing the criticality of a single Z n clock chain is c = 1/2, 4/5, 1 for n = 2, 3, 4. Thus the central charge of the CFT describing the simultaneous criticality of two decoupled Z n clock chains should be c = 1, 8/5, 2 for Z 2 × Z 2 , Z 3 × Z 3 and Z 4 × Z 4 . This is summarized in Table 1.
Of course, we do not have two decoupled clock chains. The spatial boundary condition twist (Equation (12)) and the constraint (Equation (13)), namely the orbifolding, couples the two Table 1 The central charges associated with the critical point of the Z n × Z n SPT phase transitions for n = 2, 3, 4.

Symmetry group
Central charge chains together. The purpose of this section is to address the effects of orbifolding on the criticality of the two decoupled chains. Let's start with the conformal field theory of a single Z n clock chain. The partition function of such CFT on a torus is given by Here the indices a, b labels the Verma modules. Each Verma module is spanned by states associated with a primary scaling operator and its descendants through the operator-state correspondence. Each Verma module carries an irreducible representation of the conformal group. The parameter q in Equation (19) is equal to e 2πiτ , where τ is the modular parameter of the spacetime torus (see Fig. D.8). χ a (q) and χ b (q) are, respectively, the partition function associated with the holomorphic Verma module a and antiholomorphic Verma module b. The matrix M ab has non-negative integer entries. The partition function of the two decoupled Z n clock chains that are simultaneously critical is given by It turns out that the effect of orbifold is to change In particular, N (1,1);(1,1) = 1, i.e., the tensor product of the ground state of the two clock chains is also the ground state of the orbifold model. Moreover, for those N (a,c);(b,d) > 0 the scaling dimension of the holomorphic primary operator (a, c) is h (a,c) = h a + h c and that of the antiholomorphic primary operator The fact that the ground state of the orbifold model remain the same as the tensor product of the ground states of the decoupled clock chains implies The latter identity can be seen from the fact that the central charge can be computed from the entanglement entropy, which is a pure ground state property. Thus, after the orbifold, the system is still conformal invariant (i.e. quantum critical) and the central charge is unaffected by the orbifold. This argument allows us to conclude that the central charge of the Z n × Z n (n = 2, 3, 4) SPT phase transition is indeed given in Table 1.
In Appendices D, E, and F we go through the details of obtaining the modular invariant partition function for the orbifold Z n × Z n (n = 2, 3, 4) clock chains. We examine the primary scaling operator content of the modular invariant conformal field theory. In addition, we study the symmetry transformation properties of various Verma modules and the scaling dimension of primary scaling operators, in particular, that of the gap opening operator. In Table 2 we list the first few most relevant scaling operators and their scaling dimension for n = 2, 3, 4. Entries in blue are invariant under Z n × Z n .

Numerical DMRG study of the Z 3 × Z 3 SPT phase transition
In this section, we report the results of numerical density matrix renormalization group calculation for the Z 3 × Z 3 transition. The purpose is to check our analytic predictions in the last section. The details of the numerical calculations are presented in Appendix G. First, we demonstrate that λ = 1/2 in Equation (9) is indeed a critical point. Let's look at the second derivative of the ground state energy with respect to λ for both open and periodic boundary conditions with different system sizes (Fig. 3). The results clearly suggest a second-order phase transition at λ c = 1/2 where the second order energy derivative diverges.
Next, we compute the central charge at λ = 1/2. This is done by computing the entanglement entropy, which is calculated from the reduced density matrix by tracing out the degrees of freedom associated with N − l sites in a system with total N sites. In Fig. 4 we plot the von Neumann entanglement entropy S against x = N π sin(πl/N ) where l is the number of sites that are not traced out. CFT predicts S = c 6 ln(x) + const for the open boundary condition and S = c 3 ln(x) + const for periodic boundary condition [9]. From the numerics we find c = 1.599 (9). This result is in nearly perfect agreement with our analytic prediction c = 8/5.
In addition to the above results, we have also calculated the gap as a function of λ. In fitting the result to we estimate the gap exponent to be α = 0.855 (1) for open boundary condition (Fig. 5) and α = 0.847(1) for periodic boundary condition (Fig. 6). These results are in good agreement with the analytic prediction α = 5/6 (see Appendix E.4).

The constraint on the central charge
After an examination of Table 1 it is easy to notice that c ≥ 1 for all Z n × Z n SPT phase transitions. Moreover, for all the cases we know, including SPTs protected by continuous groups, all 1D (z = 1) bosonic SPT phase transitions are described by CFT with c ≥ 1. In the following present an argument that the CFT of all 1D bosonic SPT phase transition must have c ≥ 1.
We proceed by showing that the c < 1 CFTs cannot be the critical theory for bosonic SPT transitions. The 1D CFTs that are unitary and have c < 1 are the so-called minimal models. In Appendix H we summarize the argument in Ref. [10] where it is shown that the maximum  on-site internal symmetry ("on-site" symmetries are the ones consisting of product over local transformations that act on the local, e.g. site or group of sites, Hilbert space) that these CFTs can possess are either Z 2 or S 3 . Since the critical point of the bosonic SPT phase transitions must possess the same on-site symmetry as the phases on either side, and neither Z 2 nor S 3 can protect non-trivial bosonic SPTs in 1D (i.e., H 2 (Z 2 , U(1)) = H 2 (S 3 , U(1)) = Z 1 ), we conclude that the CFTs corresponding to the minimal model cannot possibly be the critical theory for bosonic SPT phase transitions. This leaves the c ≥ 1 CFTs the only possible candidates as the critical theory for bosonic SPT phase transitions.

Conclusions
In this paper, we present an analytic theory for the phase transition between symmetry protected topological states protected by the Z n × Z n symmetry group. We have shown that for 2 ≤ n ≤ 4 a direct, continuous, topological phase transition exists. In contrast for n ≥ 5 the transition from the topological trivial to non-trivial SPTs is intervened by an intermediate gapless phase. Our theory predicts that for n = 2, 3, 4 the central charge of the CFT describing the SPT phase transitions are c = 1, 8/5 and 2, respectively. We perform explicit numerical density matrix renormalization group calculations for the interesting case of n = 3 to confirm our analytic predictions.
We expect treatment analogous to what's outlined in this paper can be generalized to the phase transitions between SPTs protected by symmetry group Z n 1 × Z n 2 × .... In addition, we provide the proof for a conjectured put forward in a previous unpublished preprint [11] that the central charge of the CFTs describing bosonic SPT transitions must be greater or equal to 1. Thus all c < 1 CFTs cannot be the critical theory for bosonic phase transitions. However, we have not yet answered the question "are all CFTs with c ≥ 1 capable of describing topological phase transitions." Of course upon non-local transformation the c < 1 minimal models can be viewed as the critical theory for parafermion SPT transitions. Indeed, the c = 1/2 Ising conformal field theory describes the critical Majorana chain. The c = 4/5 three-state Potts model CFT describes the critical point of Z 3 parafermion chain. We suspect that the parafermion models escape the classification of either the K theory or the cohomology group because of its non-local commutation relation.
In space dimension greater than one, we do not know a model which definitively exhibits a continuous phase transition between bosonic SPTs. This is due partly to the likelihood of spontaneous breaking of the discrete protection symmetry when d ≥ 2. In addition, even if the continuous phase transition exists, it is more difficult to study these phase transitions, even numerically. However a "holographic theory" was developed for phase transitions between SPT phases which satisfy the "no double-stacking constraint" [6]. That theory predicts the criti-cal point should exhibit "delocalized boundary excitations" of the non-trivial SPT, which are extended "string" or "membrane" like objects with gapless excitation residing on them. We expect this kind of critical point to be fundamentally different from the Landau-like critical point. Clearly many future studies are warranted for the understanding of these interesting phase transitions. Z n × Z n SPT Hamiltonian in 1D We briefly review the definition of cocycles in the group cohomology, and describe a procedure [3,5] to construct fixed point SPT hamiltonians (1) that are relevant to this paper.

A.1. Cocycle
In 1D a cocycle associated with group G is an U(1) valued function ν(g 0 , g 1 , g 2 ) where the argument g i ∈ G which satisfies ν(gg 0 , gg 1 , gg 2 ) = ν(g 0 , g 1 , g 2 ). Here we only consider the group realized by unitary representation. Moreover, ν satisfies the following cocycle condition for certain c(g 0 , g 1 ) satisfying c(gg 0 , gg 1 ) = c(g 0 , g 1 ) we say it is a coboundary. It may be checked that a coboundary automatically satisfies the cocycle condition Equation (A.1). Two cocycles related by the multiplication of a coboundary are viewed as equivalent.
The equivalence classes of cocycles form H 2 (G, U(1)) -the 2 nd cohomology group of G with U(1) coefficient. Bosonic G-symmetric SPTs in 1 space dimensions are "classified" by H 2 (G, U(1)), i.e., each equivalent class of SPTs is in one to one correspondence with an element of the abelian group H 2 (G, U (1)). The binary operation of the abelian group corresponds to the "stacking" operation, i.e., laying two SPTs on top of each other and turning on all symmetry allowed interactions.

A.2. Projective representation
In quantum mechanics, symmetry operators are usually realized as matrices R(g) acting on Hilbert space. Usually these matrices form a linear representation of the symmetry group, namely, However, two quantum states differ by an U(1) phase are regarded as the same quantum mechanically. Thus, one should relax Equation (A.4) by allowing a phase ambiguity ω, namely, When Equation (A.5) is satisfied we say that R(g) form a projective representation of the original symmetry group. Obviously, linear representation where ω(g 1 , g 2 ) = 1 is a special case of projective representation. In the literature linear representations are usually viewed as "trivial" projective representations. Associativity under group multiplication, namely, In addition the phase ambiguity of quantum states obviously allows one to multiply all R(g) by an U(1) phase φ(g), namely, This phase transformation results in Consequently ωs related by Equation (A.8) should be regarded as equivalent. It turns out that in 1D, cocycles of group cohomology can be interpreted as projective representations. The easiest way to see it is by defining ω(g 1 , g 2 ) and φ(g 1 ) in terms of the cocycle ν and the coboundary c defined in the last subsection, namely, where e is the identity group element of G. In terms of ω the cocycle condition becomes (∂ω)(g 1 , g 2 , g 3 ) := (∂ν)(e, g 1 , g 1 g 2 , g 1 g 2 g 3 ) (∂φ)(g 1 , g 2 ) := (∂c)(e, g 1 , which is exactly the factor appearing in Equation (A.8).

A.3. Construction of Hamiltonian
Here we describe how to construct solvable Hamiltonians, one for each equivalence class of the SPTs. Consider a 1D ring consists of N lattice sites. The Hilbert space for each site i is spanned by {|g i } where g i ∈ G, and the total Hilbert space is spanned by the tensor product of the site basis, i.e., |{g i } = i |g i . For each class of the SPTs (or for each element of H 2 (G, U(1))) picks a representing cocycle ν(g 0 , g 1 , g 2 ). The "fixed point" ground state, which is a particular representative of a whole equivalent class of SPTs, associated with the cocycle ν is equal to (Ref. [3] Section IX) Here e represents the identity element of G. It is attached to "0" site at the center of the ring as shown in Fig. A.7. σ (i, i + 1) = ±1 depending on the orientation of the triangle 0, i, i + 1.
The orientation of each link in the triangle is represented by an arrow pointing from the site labeled by a smaller site index to the site labeled by the bigger index. From the link orientation we determine the triangle orientation by following the majority of the link orientation and the right-hand rule). Finally periodic boundary condition requires g N+1 = g 1 . The Hamiltonian whose exact ground state is Equation (A.12) is where J > 0. The operator B i only changes the basis states on site i, and (A.14) For G = Z n × Z n , there are n inequivalent SPT classes and H 2 (Z n × Z n , U(1)) = Z n . Each equivalent class of H 2 (Z n × Z n , U(1)) is represented by a cocycle ν((e, e), (g 1 , g 2 ), (g 3 , g 4 )) = η kg 2 g 3 n , where η n = e i2π/n In the above (g 2i−1 , g 2i ) ∈ Z n × Z n are the Z n elements associated with site i, and k ∈ {0, 1, . . . , n − 1} each correspond to a different element of Z n (H 2 (Z n × Z n , U(1))). In the main text, we refer to |g 2i−1 , g 2i as the "cell basis" which is the tensor product of the "site basis" |g 2i−1 and |g 2i . The fixed point Hamiltonian constructed using the procedure discussed above is where B 2i−1 changes the state |g 2i−1 and B 2i changes the state |g 2i . Explicitly calculating the matrix element (Equation (A.14)) for the cases, i.e., k = 0 and k = 1, relevant to our consideration (recall that we are interested in the quantum phase transition between SPTs correspond to the "0" and "1" elements of Z n ) it can be shown that where M j and R j are defined by Equation (2) of the main text.

Appendix B. The mapping to Z n × Z n clock models with spatially twisted boundary condition and a Hilbert space constraint
In this section, we show that Equation (8) and Equation (9) of the main text can be mapped onto an "orbifold" Z n × Z n clock models. The mapping is similar to the "duality transformation" of the Z n clock model. The mapping is given by Here the tilde operators obey the same commutation relation as the un-tilde ones. Due to the periodic boundary condition on R, namely, R 2N +1 = R 1 and R 2N+2 = R 2 the line of Equation Moreover, if we also impose the periodic boundary condition on R j a similar constraint on M i , namely, is obtained. Since there is no such constraint on M i in the original problem we need to impose a "twisted" boundary condition on R j : where B commutes with all R j and M j . Moreover B has eigenvalues b = 1, η n , ..., η n−1 n , i.e., B|b =b|b .
Substituting Equation (B.1) into Equation (8) and Equation (9) of the main text we obtain the following expression of the transformed Hamiltonian It is important to note that Equation On the surface Equation (B.5) describes two decoupled Z n clock models living on even and odd sites, respectively. However the notion of "decoupled chains" is deceptive because the constraint in Equation (B.2) couples them together.

Appendix C. The notion of "orbifold"
A useful way to implement the constraint Equation (B.2) is to apply the projection operator where Z n-clock q s ,q τ represents clock model partition function under the space and time twisted boundary condition characterized by q s and q τ . In Equation (C.5) Z n-clock q s ,q τ appears twice on right-hand side because without orbifold (i.e., summing over space and time twisted boundary conditions) we have two independent n-state clock models. Averaging over the partition function under space and time boundary condition twists is the "orbifolding" [12]. Note that here the spatial boundary condition twist is generated by one of the Z n generator, namely B , in Equation (B.7). However, the time twist is generated by Q = 2N j =1 M j , which is a symmetry of the Z n × Z n clock Hamiltonian, Equation (B.5), but it is not the generator for the other Z n in Equation (B.7).

D.1. Review of modular invariant partition function for the Ising model
The Ising model shows an order-disorder phase transition. At the critical point, the Hamiltonian is given by where M i , R i are Pauli matrices σ x and σ z respectively (we use M, R rather than σ x , σ z for the consistency of notation). The central charge of a single critical Ising chain is c = 1 2 . Its conformal field theory is the M(4, 3) minimal model. The primary scaling operators are labeled by two pairs of indices (r, s) and (r , s ) each label the "holomorphic" and the "anti-holomorphic" part of the operator. The ranges of these indices are given by 1 ≤ s ≤ r ≤ 2 and 1 ≤ s ≤ r ≤ 2. The scaling dimensions of the holomorphic and anti-holomorphic parts of these operators are given by  Through the operator-state correspondence, each of these primary fields and their associated "descendants" form the basis of a Hilbert space (the "Verma module") which carries an irreducible representation of the conformal group. Now consider the partition function of the CFT on a spacetime torus (see Fig. D.8). The prototype torus is obtained from identifying opposite edges of the parallelogram having (0, 1, 1 + τ, τ ) as the complex coordinates of its four vertices (τ ∈ upper half complex plane). On such a torus, the partition function is given by with q = e i2πτ and q = e −i2πτ . Here the trace Tr (r,s) and Tr (r s,s ) are taken within the Verma module labeled by (r, s) and (r , s ). In the literature χ r,s and χ r ,s are referred to as "characters". For the CFT to be consistent, its partition function must be "modular invariant" [13]. The modular group consists of discrete coordinate transformations that leave the lattice whose fundamental domain is given by Fig. D.8 invariant. This group is generated by the T (τ → τ + 1) and the S (τ → −1/τ ) transformations. When acted upon by these transformations the characters χ r,s (with a similar expression for χ r ,s ) transform according to Here S, T are known matrices and the transformation matrices for the anti-holomorphic χ are the complex conjugate of those of the holomorphic ones.
The requirement of modular invariance, namely,  δ (r,s),(r ,s ) . The corresponding partition function is given by: The explicit form of χ (r,s) is given by equation (8.15) of Ref. [14]. The conformal dimensions of primary fields and their eigenvalues under the action of the Z 2 generator are summarized in Table D.3 [14].

D.2. Constructing the orbifold partition function for the Z 2 × Z 2 critical theory
With the brief review of the modular invariant partition function of the critical Ising model we are ready to construct the partition function for the orbifolded Z 2 × Z 2 model defined by of Equation (C.5): 1) is given in Ref. [14]. It is also shown in the same reference that Z  Table D. 4 The quantum numbers of the first few primary operators of the orbifold where the τ dependence is suppressed. When expanded in terms of χ r,sχr ,s the first term yields 4 terms (henceforth referred to as group I terms). The second term yields 2 terms (group II terms). The third term yields 4 terms (group III terms). Due to the prefactor 2 in the second term on the right-hand side of Equation (D.8), terms in group II appear with multiplicity 2. It turns out that this partition function is the same as the XY model. The first few energy levels with h +h < 2 and their quantum numbers are listed in Table D.4.

D.3. Transformation properties under the action of Z 2 × Z 2
To see how the contributing Verma modules of Equation (D.8) transform under the action of Z 2 × Z 2 , we construct operators that project the Hilbert space into subspaces carrying various irreducible representations of Z 2 × Z 2 . Let G A = B and G B = i∈even M i be the generators of Z 2 × Z 2 . The operator that projects into subspace with eigenvalues (η a 2 , η b 2 ) (here η 2 = −1 and a, b = 0, 1) under the action of G A and G B is given by To filter out the Verma modules that transform according to this particular irreducible representation, we compute Tr P ab Q q τ e −β(H even +H odd ) For example, which means only group I transform as the identity representation of Z 2 × Z 2 . For other P ab the results are summarized in Table D.5  Table D.5 Transformation properties of the contributing Verma modules in Equation (D.8) under the action of G A and G B . For group II, the doublet records the transformation properties of the multiplicity two Verma modules in Equation (D.8). Group

D.4. Scaling dimension for the operator driving the Z 2 × Z 2 SPT transition
The operator that drives the SPT phase transition must be (1) relevant, (2) translational invariant and (3) invariant under Z 2 × Z 2 . In Equation (D.8) the only term that contains operators (there are two such operators due to the multiplicity 2) satisfy these conditions is 2|χ I χ | 2 . The scaling dimension of (I )(Ī ) is h +h = 1 < 2 hence it is relevant. The momentum of this operator is h −h = 0 hence is translation invariant. Moreover according to Table D.5 there operators are invariant under Z 2 × Z 2 . It turns out that one of these two relevant operators drives a symmetry breaking transition while the other drives the SPT transition (see Fig. D.10). From the scaling dimension h +h = 1 we predict the gap exponent to be 1 2−1 = 1.

E.1. Review of modular invariant partition function for the 3 states Potts model
The construction of the orbifold partition function for the Z 3 × Z 3 case closely mirrors the Z 2 × Z 2 case. But instead of two critical Ising chains, we now have two critical Potts chains. We first review the known results for the modular invariant Z 3 clock model (equivalent to the 3-state Potts model). The 3-state Potts model shows an order-disorder phase transition. At the critical point the Hamiltonian is given by where R j = 1, η 3 , η 2 3 (η 3 = e i2π/3 ) and R j M k = η δ jk 3 M k R j . The conformal field theory for the critical 3-state Potts model belong to the well known "minimal" model M (6,5) [14,15]. The central charge is and the primary scaling operators are labeled by two pairs of indices (r, s) and (r , s ) each label the "holomorphic" and the "anti-holomorphic" part of the operator. The range of these indices are given by 1 ≤ s ≤ r ≤ 4 and 1 ≤ s ≤ r ≤ 4. The scaling dimensions of the holomorphic and anti-holomorphic parts of these operators are given by It is easy to check that h r,s = h 5−r,6−s and h r ,s =h 5−r ,6−s hence there are 10 distinct primary fields in the holomorphic and anti-holomorphic sector each. Requiring modular invariance (D.4) for c = 4/5 yields two possible such M's: one with M (r,s);(r ,s ) = δ (r,s),(r ,s ) describing the "tetra-critical Ising model", and the other corresponds to the 3-state Potts model described by the following partition function [14]: where χ I := χ 1,1 + χ 4,1 , χ := χ 2,1 + χ 3,1 , χ ψ := χ 4,3 , χ σ := χ 3,3 (E.4) Note that out of the 10 possible primary operators in each holomorphic/anti-holomorphic sector only six of them contribute to the partition function. In addition, the diagonal combination of the (3,3) and (4, 3) operators from each sector appear twice. The explicit form of χ (r,s) is given by equation (8.15) of Ref. [14]. The conformal dimensions of primary fields and their eigenvalues under the action of the Z 3 generator are summarized in Table E.6 [14].

E.2. Constructing the orbifold partition function for the Z 3 × Z 3 critical theory
With the brief review of the modular invariant partition function of the critical 3-state Potts model we are ready to construct the partition function for the orbifolded Z 3 × Z 3 model defined by of Equation (C.5): and Z  are given in Ref. [14]. Using Z 3-Potts q s ,q τ (τ ) = Z 3-Potts q s ,q τ q s (τ + 1) = Z 3-Potts Using the S, T matrices of the 3-states Potts model we can compute all these terms. Substituting the results into Equation (E.5) we obtain the orbifolded Z 3 × Z 3 partition function: where the τ dependence is suppressed. When expanded in terms of χ r,sχr ,s the first term yields 64 terms (henceforth referred as group I terms Following the same procedure in D.3 we construct operators that project the Hilbert space into subspaces carrying various irreducible representation of Z 3 × Z 3 which is generated by G A = B and G B = i∈even M i . The projector into subspace with eigenvalues (η a 3 , η b 3 ) (here η 3 = e i2π/3 and a, b = 0, 1, 2) under the action of G A and G B is given by  .7).
Analogous to Equation (D.10) we filter out the Verma modules that transform according to this particular irreducible representation by computing For example, which means only group I transform as the identity representation of Z 3 × Z 3 . For other P ab the results are summarized in Table E.8.

E.4. Scaling dimension for the operator driving the Z 3 × Z 3 SPT transition
From Table E.7 and Equation (E.10) it is seen that the translation-invariant (i.e. h −h = 0), relevant (i.e. h +h < 2), Z 3 × Z 3 invariant operators either have scaling dimensions 4/5 or 8/5. Through a comparison with the numerical result for the gap exponent in section 9 of the main text, we identify one of the operators with scaling dimension 4/5 as responsible for the opening of the energy gap in the SPT phase transition. The predicted gap exponent is 1 2−4/5 = 5/6 which agrees reasonably well with the numerical gap exponent. Moreover similar to the Z 2 × Z 2 case there are two operators with the same scaling dimension (4/5). Again one of these operators drives a symmetry breaking transition while the other drives the SPT transition, hence the phase diagram is similar to Fig. D.10. follows. For the Z 4 clock model the Hilbert space for each site j is 4-dimensional. In the following, we shall regard this 4-dimensional Hilbert space as the tensor product of two 2-dimensional Hilbert spaces associated with site 2j − 1 and 2j . We then view each of the 2-dimensional space as the Hilbert space of an Ising spin. In this way the Z 4 clock model with N sites can be viewed as an Ising model with 2N sites.
More explicitly, under the unitary transformation U = i U i , where where X i and Z i denote the 2 × 2 Pauli matrices σ x i and σ z i . Thus the partition function of the Z 4 clock model under periodic boundary condition is given by The fact that Ising model has central charge c = 1/2 implies the central charge of the critical Z 4 clock model to be 1/2 + 1/2 = 1.
CFT with c = 1 has infinitely many Verma modules [17]. The scaling dimension of the primary fields, which can take any non-negative values, is parametrized by h = x 2 /4 where x is a non-negative real number. The characters associated with these Verma modules are given [18] by Because later on we shall perform orbifolding it is necessary to consider the Z 4 clock model under twisted spatial boundary condition. With the spatial boundary condition twisted by the Z 4 generator, i.e., R N+1 = η 4 R 1 , the last two terms, namely Z 2N Z 2 + Z 2N−1 Z 1 in Equation (F.2), are replaced by In the language of Ising model, the above replacement creates an overpass connecting the even chain to the odd chain and a sign change of one bond (the red bond in Fig. F.11(b)). Thus we arrive at an Ising chain twice as long and with the spatial boundary condition twisted by the Z 2 generator. As a result The reason the modular parameter of the Ising partition function is half that of the Z 4 clock partition function is because the Ising chain has twice the length in the spatial direction. The same argument applies if the boundary is twisted by the inverse of the Z 4 generator (R N+1 = η 3 4 R 1 ) instead, i.e., Similarly, when the spatial direction is R N+1 = η 2 4 R 1 , the Hamiltonian of the Ising model becomes that of two decoupled Ising chain each having a sign-flipped bond equivalent to the Z 2 twisted boundary condition (see Fig. F.11(c)). The resulting partition function is given by Using the known S and T matrices for the Ising model, other Z 4-clock (q s ,q t ) (τ ) can be determined where The χ h in the above equations are given by Equation (F.3). The scaling dimensions of the highest weight states associated with the Verma modules that generate these χ h are summarized in Table F.9. Let's refer to the six terms in Equation (F.8) as Groups I, II, III, IV, V, and VI respectively. Due to the prefactor of 2, Group III elements appear in doublets. Due to the prefactor of 4, Groups IV, V and VI elements appear with multiplicity 2. In Table F.10 we list the first few primary fields with scaling dimension h +h < 2 and their quantum numbers. The results are summarized in Table F.11. The term 2|χ I χ | 2 in Equation (F.8) yields two primary fields with scaling dimension h +h = 1 (hence are relevant) and are invariant under Z 4 × Z 4 and translation. Hence they are qualified as the gap generating operator. The gap exponent is 1 2−1 = 1. Similar to the Z 2 × Z 2 and Z 3 × Z 3 cases there are two operators with the same scaling dimension (1). As the above two cases one of these operators drives a symmetry breaking transition while the other drives the SPT transition, hence the phase diagram is similar to Fig. D.10. (Color online) The space-time torus with spatial and temporal boundary condition twisted by group elements g s and g τ . The path in red picks up the group element g τ g s , while the path in blue picks up the group element g s g τ . Since the path in red can be deformed into the path in blue, g s and g τ need to commute so that the boundary condition is self-consistent. modular invariant. Therefore to detect whether an on-site symmetry group contains Z m as an abelian subgroup we just need to see whether it possible to assign Z m irreducible representations to the Verma modules so that after orbifolding the partition function is modular invariant. For discrete groups after knowing all abelian subgroups we can reconstruct the total group G. This is essentially the strategy followed by Ref. [10].
More explicitly, let the Hilbert space be consistent with a spatial boundary condition involving a twist generated by ρ g s (ρ is the generator of certain abelian subgroup Z m and g s = 0, ..., m − 1) where V i (V i ) is the ith Verma module in the holomorphic (anti-holomorphic) sector and M

(g s ) ij
is a non-negative integer labeling the multiplicity of the V i ⊗V j modules. Moreover, for the CFT to have a unique ground state, we require the vacuum module (i = 1) only shows up once in the periodic sector, i.e., M (g s ) 11 = δ 0,g s . Next we assign irreducible representation to the Verma modules: where g τ = 0, ..., m − 1, η m = e i2π m and Q(g τ ; g s , i, j, k) ∈ 0, ..., m − 1 is called "symmetry charge" in Ref. [10]. Combine Equation (H.1) and Equation (H.2) we obtain the following spacetime boundary twisted partition function on a torus with modular parameter τ Z g s ,g τ (τ ) = T r H (gs ) (q L 0 −c/24qL 0 −c/24 q τ ) =

H.1. The consistency conditions
So far the abelian subgroup Z m as well as M (g s ) ij and Q(g τ ; g s , i, j, k) are unknown. They need to be determined subjected to the following consistency conditions. (1) When there is no spatial boundary condition twist the Hilbert space in Equation (H.1) must return to that of the periodic boundary condition. Moreover in the case where there is also no time boundary condition twist the partition function must agree with the modular invariant partition function Z 0,0 (τ ).
(1)-(3) pose strong constraints on the possible abelian subgroup Z m and the allowed assignment of the irreducible representations (i.e. Q(g τ ; g s , i, j, k)) to each Verma module.

H.2. The on-site symmetry of minimal models
Under constants (1)-(3) in the previous subsection Ref. [10] solved the possible abelian subgroups and their symmetry representations for the all minimal models. By patching these abelian subgroups together the author reached the following conclusion: the on-site symmetries of the unitary minimal models are exactly the same as those predicted by the lattice RSOS models [20]. Hence there is no emergent symmetry! Thus, for most of the unitary minimal models the symmetry is Z 2 . The only exceptions are 3-states Potts and tri-critical 3-state Potts models where the symmetry is S 3 . Finally for the minimal model labeled by E 7 , E 8 , where there is no symmetry.