Intrinsically/Purely Gapless-SPT from Non-Invertible Duality Transformations

We would like to thank the referee for the helpful suggestions and comments. We have add a decription of general structures of various types of gapless SPTs in the introductions. The following is our response to each comment.

1. At first, we want to clarify that the Trivial to SPT transition is not obtained by stacking an Ising^2 CFT by a Z2 x Z2 SPT. The latter corresponds to the phase transition between Z2 x Z2 SPT and Z2 x Z2 SSB phase. One lattice model realizing Z2 x Z2 SPT to trivial phase transition is

   $$H=- \sum_{j} (\sigma^z_{j-1} \tau^x_{j-\frac12} \sigma^z_{j} + \tau^z_{j-\frac12} \sigma^x_j \tau^z_{j+\frac12} + \sigma^x_j + \tau^x_{j-\frac12})$$
   . The Z2 x Z2 symmetry is generated by $U_\sigma=\prod_j \sigma^x_j$ and $U_\tau= \prod_j \tau^x_{j-\frac12}$. Under the twisted boundary condition of the first Z2, there are two ground states, carrying the opposite charge under the second Z2. Hence does not satisfy the topological feature discussed in this paper. (We require the gSPT ground state have a definite symmetry charge under twisted boundary condition.)

2. We have revised the paper to add the coefficient before SSB phases and transformations.

3. Indeed, there can be an anomalous $Z_2$ symmetry in 1+1d bosonic systems. The anomaly of Z_low in 5.19 comes from that $\int_{X_2}bA=\int{M_3}A^3$ where $X_2$ is the boundary of $M_3$ and we use the fact $\delta b=A^2$. This topological term is anomaly flow action of anomalous $Z_2$ symmetry which lives on the bounary of 2+1d $Z_2$ SPT (Levin-Gu model on the lattice).

4. The ground state degeneracy of intrinsically gapless SPT (igSPT) model Eq.(5.5) is 2 when system size L is 2 mod 4, while it is 1 in other case. Indeed, the low energy spectrum of igSPT is the same as that of boundary of Levin-Gu model, i.e., $H_{LG}=-\sum_i(X_i-Z_{i-1}X_iZ_{i+1}$), including ground state degenracy. For the boundary Levin-Gu model, we can understand its ground state degeneracy from ZY chain $H_{ZY}=-\sum(Z_iZ_{i+1}+Y_iY_{i+1})$ as they are related by the KW transformation. When size is odd, H_ZY has two ground states according to the anticommutation between symmetry operators $\prod_i X_i$ and $\prod_i Z_i$. We can choose two ground states with $\prod_i X_i=\pm 1$. When size is even, the ground state is unique with zero magentization $\sum_i X_i=0$ and the first excited states are two degenerate with $\sum_i X_i=\pm 2$. As $\prod_n X_n=i^L\exp(i\pi \sum_n X_n/2)$, the ground state has symmetry eigenvalue $i^L$ and the first excited states have symmetry eigenvalue $-(i^L)$. Under the KW transformation, only states with even eigenvalue survive. Thus the boundary Levin-Gu model has two ground state when L is 2 mod 4 and unique ground state in other cases.

5. We have edited the typos.

We thank the referees for the illuminating comments. Please find our response below.

Reply to Referre 1 We would like to thank the referee for the helpful comments. We have added an explanation of the notation of gauge degrees of freedom, and added a footnote which cites some references for discrete gauge fields in the updated version.

Reply to referee 2. We would like to thank the referee for the helpful suggestions and comments. We have add a decription of general structures of various types of gapless SPTs in the introductions. The following is our response to each comment.

1. At first, we want to clarify that the Trivial to SPT transition is not obtained by stacking an Ising^2 CFT by a Z2 x Z2 SPT. The latter corresponds to the phase transition between Z2 x Z2 SPT and Z2 x Z2 SSB phase. One lattice model realizing Z2 x Z2 SPT to trivial phase transition is

   $$H=- \sum_{j} (\sigma^z_{j-1} \tau^x_{j-\frac12} \sigma^z_{j} + \tau^z_{j-\frac12} \sigma^x_j \tau^z_{j+\frac12} + \sigma^x_j + \tau^x_{j-\frac12})$$
   . The Z2 x Z2 symmetry is generated by $U_\sigma=\prod_j \sigma^x_j$ and $U_\tau= \prod_j \tau^x_{j-\frac12}$. Under the twisted boundary condition of the first Z2, there are two ground states, carrying the opposite charge under the second Z2. Hence does not satisfy the topological feature discussed in this paper. (We require the gSPT ground state have a definite symmetry charge under twisted boundary condition.)

2. We have revised the paper to add the coefficient before SSB phases and transformations.

3. Indeed, there can be an anomalous Z2 symmetry in 1+1d bosonic systems. The anomaly of Z_low in 5.19 comes from that \int_{X_2}bA=\int{M_3}A^3 where X_2 is the boundary of M_3 and we use the fact \delta b=A^2. This topological term is anomaly flow action of anomalous Z2 symmetry which lives on the bounary of 2+1d Z2 SPT (Levin-Gu model on the lattice).

4. The ground state degeneracy of intrinsically gapless SPT (igSPT) model Eq.(5.5) is 2 when system size L is 2 mod 4, while it is 1 in other case. Indeed, the low energy spectrum of igSPT is the same as that of boundary of Levin-Gu model (H_LG=-\sum_i(X_i-Z_i-1X_iZ_i+1), including ground state degenracy. For the boundary Levin-Gu model, we can understand its ground state degeneracy from ZY chain H_ZY=-(Z_iZ_i+1+Y_iY_i+1) as they are related by the KW transformation. When size is odd, H_ZY has two ground states according to the anticommutation between symmetry operators \prod_i X_i and \prod_i Z_i. We can choose two ground states with \prod_i X_i=\pm 1. When size is even, the ground state is unique with zero magentization \sum_i X_i=0 and the first excited states are two degenerate with \sum_i X_i=\pm 2. As \prod_n X_n=i^L\exp(i\pi sum_n X_n/2), the ground state has symmetry eigenvalue i^L and the first excited states have symmetry eigenvalue -(i^L). Under the KW transformation, only states with even eigenvalue. Thus the boundary Levin-Gu model has two ground state when L is 1 mod 4 and unique ground state in other cases.

5. We have edited the typos.