Intertwined order of generalized global symmetries

We thank the referee for detailed comments and a close reading of our manuscript. We provide detailed responses below to the specific points raised.

Referee:

1-This is not the first time this model, or the $\theta$-term, is discussed. I suggest the authors take a look at "Alfred Shepere, Frank Wilczek (1989), Self-dual models with theta terms, Nuclear Physics B, 320(3), 669-695." Specifically, in eq (3.5), they write down the Euclidean Villain partition function for a $d$-dim $p$-form version of the model studied in the current paper. The authors should cite this paper and clearly explain what is new in their work.

Our response: We thank the referee for bringing up this paper. Indeed, Shapere and Wilczek introduced this model in their paper, and they demonstrated that the model is self-dual, though they did not explore its global symmetries or the topological physics of its phases as we do in our paper. In Section 1, we have added a citation to their paper (and in Section 2.1, where we introduce the model) and an explanation of how we go beyond their work.

Referee:

1-On page 5, and elsewhere, the authors say that "the 1-form symmetry cannot be spontaneously broken along the 1+1d boundary." This requires a citation, or at least a clarification in the footnote. Is this related to the fact that there is no topological order in 1+1d?

Our response: Since we are dealing with a discrete one-form symmetry, yes, this fact is related to the nonexistence of topological order in (1+1)d. We have added two citations in the text where this fact is mentioned, and we also included a footnote on p. 5 where it is first stated.

Referee:

2-I am confused by (2.28). I would expect that $\mathcal{S}^2=1$, which is consistent with the first line of (2.28). However, I would also expect that $\mathcal{T}^N=1$, which doesn't seem to be consistent with the second line of (2.28). Can the authors clarify?

Our response: The condensed charges here are labeled by their embedding in a $U(1)$ theory. In a $U(1)$ theory, we do not expect $\mathcal{T}^N=1$. It is only after recalling the $\mathbb{Z}_N$ nature of the theory and analyzing the allowed probes that it is clear that $\mathcal{T}^N=1$. For example, the phase with $(0,1)$ condensed is trivial. Acting with $\mathcal{T}$, the $(-N,1)$ phase is an SPT, signaled by the fact that an open surface version of the symmetry operator for the 0-form symmetry ends on charge $-1$ electric loops. Acting on the $(0,1)$ phase with $\mathcal{T}^N$, though, gives the $(-N^2,1)$ phase, which is trivial—the open surfaces now end on charge $-N$ electric loops, which are trivial in the $\mathbb{Z}_N$ theory.

Referee:

3-In (3.9), $\Gamma$ is necessarily contractible because $\Gamma=\partial\Sigma$. However, in (3.10), $\Gamma$ doesn't have to be contractible, i.e., there is no need for $\Sigma$. The operator $e^{i\frac{N}{L}\oint_\Gamma a}$ is gauge invariant for any closed loop $\Gamma$ (contractible or not).

4-Similar to the above comment, in (3.11), there is no need of the string and the other end point. The operator $e^{i\frac{N}{L}\varphi(\mathcal{P})}$ is already gauge invariant.

Our response: We agree, which is why we referred to Eq. (3.10) as a "genuine loop operator" and Eq. (3.11) as a "genuine local operator". We have revised the discussion in this section so that it is clearer.

Referee:

5-At the end of sec 3, the authors say that whenever the two symmetries are broken to a nontrivial subgroup, they are broken to the same subgroup. Is this just an observation, or is there a proof of this fact? Also, is this a property of this particular lattice model? Because, in the general $\mathbb{Z}_N$ clock model, as well as the $\mathbb{Z}_N$ lattice gauge theory, there are $\lfloor N/2 \rfloor$ independent parameters, and one can certainly imagine a phase where the two symmetries are broken to different subgroups.

Our response: The argument is to consider the action of the most general set of $\mathcal{S}$ and $\mathcal{T}$ transformations, given in Eq. (2.39), on the phases at $\Theta=0$. The resulting phase generically has an effectively field theory that is a multi-component version of Eq. (2.37). One can explicitly check that if either the $\mathbb{Z}_N$ zero-form or $\mathbb{Z}_N$ one-form symmetries is broken to a nontrivial subgroup, then both symmetries are broken to the same subgroup. We have added a footnote sketching this proof.

Yes, other terms can be added to the action that result in phases that break the symmetries to different subgroups.

Referee:

6-I don't think writing $c_\mu=-k A_\mu/L$ mod $N$ below (4.4) is valid. One can always add a multiple of $N/L$ to $c_\mu$ while still satisfying (4.2). However, this ambiguity doesn't affect the action because $B_{\mu\nu}$ is also an integer multiple of $L$. This should be clarified.

Our response: We meant to write $c_\mu=-k A_\mu/L$ mod $N/L$ below Eq. (4.4). We have corrected this statement and added a footnote clarifying that this ambiguity does not affect the response action.

Referee:

7-In (5.1), and elsewhere in the paper, the authors refer to $\Phi$ as a background scalar field. This is incorrect. Say one starts with the action $iN/2\pi\int cdA$, where $c$ is a dynamical 1-form $U(1)$ gauge field which ensures that $A$ is a background 1-form $\mathbb{Z}_N$ gauge field. Integrating by parts, we can write the action as $iN/2\pi\int Adc \leftrightarrow i/2\pi \int (NA\rho-\phi d\rho)$, where $\phi$ is a Lagrange multiplier (hence, dynamical) that imposes $d\rho=0\implies \rho=dc$. Integrating the second term by parts gives $i/2\pi \int \rho(NA-d\phi)$ where $\phi$ is dynamical, not background. Similar comments apply to $NB=dC$.

Our response: We thank the referee for bringing this issue to our attention and have corrected it in the manuscript.

Referee:

Some minor comments:

1-Below (2.34), "integer multiples of $\theta/2\pi$" should be "integer values of $\theta/2\pi$".

2-In (2.36), $a_\mu$ should be $k_\mu$.

3-Above (4.4), the references to Eq. (4.3) should be Eq. (4.2).

Our response: We have now corrected these misprints.