Modulated symmetries from generalized Lieb-Schultz-Mattis anomalies

Reply to the third referee:

1) "It appears that the manuscript is mostly centered around gauging symmetries by minimally coupling particular models with these symmetries. However, the gauging induced dualities should apply to all Hamiltonians with the appropriate symmetries subjected to projection onto the appropriate sub-Hilbert space. Authors should add more explanation from this point of view, i.e., gauging-induced dualities as isomorphisms between algebras of symmetric local operators."

--We thank referee for raising this point. In the revised paper (footnote. 6), we make comments explaining that gauging induces a mapping between algebras whereas in our work, we implement gauging by minimally couple model with a symmetry to investigate modulated symmetry emerge.

2) "In relation to the above point, it can be beneficial to discuss what kind of phases can be realized in systems with the LSM-like anomalies the authors considered and what these phases are mapped to under the gauging map the authors considered. For example, when an LSM-like anomaly is present, there cannot be non-degenerate gapped ground state that is simultaneously symmetric."

--Aside from Ref. [45], which discusses phase diagram of a dipole Ising chain by mapping this chain to the one with LSM involving 0-form and translation symmetries, studying phase diagram of a system with modulated symmetries has largely remained unexplored. In the amended paper, we add comments on this issue in the discussion section.

3)" In the prelude of Sec. 3, the authors make the claim that the phase factor depending on the area of the system somewhat is beyond what is conventionally called as LSM anomaly. This is not correct as the generalization of LSM anomalies with 0-form symmetries to higher than 1D predates generalization to those with higher-form symmetries. For example, an important paper authors should cite is Yao, Oshikawa, PhysRevLett.126.217201. I find this paragraph very misleading."

-- Before delving into the details, in footnote. 3 of the revised manuscript, we clarify our usage of terminology: by LSM anomalies we mean anomalies involving 0-form and translation symmetries whereas by generalized LSM anomalies we refer to anomalies which involve higher-form and translation symmetries.

4) "The equation on page 3 is not precise as the projective phase between p- and q-form symmetries should also depend on some linking number in general."

-- We thank referee for asking this point. Indeed, in our work, s=d-p-q is always greater than 0, hence p-and q-form symmetry operators have always overlap whose area is proportional to L^s. In footnote. 4, in the revised paper, we make comments on it.

5) "Authors use operators with hat and without hat to denote different things. I found this notation confusing."

--Variables with hat (which is now tilde in the amended paper) is introduced to emphasize that they are used in the intermediate steps of gauging, which are written as the ones without tilde [See, e.g., (9)] after completion of gauging. We add an explanation on it in footnote. 7 in revised version.

6) "In the label of Fig. 2, it should read "...defined in (14), that respect...". I also do not understand the sublabel (b) which I think should be rewritten."

--We rectify the grammar and add more explanations in the caption.

7) "Above Eq. (32) it should read "...0-form modulated symmetry..." -- We correct the spelling. Thank you.

8)"I do not understand what authors mean by "...to make the gauged theory dynamically trivial." -- We mean it by making the theory so that it does not admit excess flux. We correct this sentence in an amended manuscript.

Reply to the second referee:

1-, In page 16, the first paragraph, the last sentence, QI[Σ−p] should be QI[Σd−p]?
2-, In page 20, the next to last paragraph, the last sentence, "a" should be "an".
3-, In page 24, formula 89, the operators in the second term should be Z's not X's.
4-, In page 32, around formula 116, N should already be abelian when the authors state that formula 116 is a central extension that is characterized by H2(K,N).

-- We thank the referee for carefully reading our paper. In the revised version of our paper, we correct all the typos you raised (points 1-3). Also, regarding point 4, we remove the sentence stating that N is Abelian, since, as the referee points out, N is already Abelian when we say the central extension is characterized by H^2(K,N).

Reply to the first referee:

We thank the referee for carefully reading our paper.

1. “In the abstract, “In one-dimensional a spin chain” -> “In a one-dimensional spin chain”

-- We have corrected this wording.

1. “In (21), it was stated that TyξZx,yT−1y=ξZ†2ξZx,y. This equation holds only in the ground state subspace, where Bly=1 and is not true on the full tensor-factorized Hilbert space. The authors should clarify this point.”

-- We thank the referee for raising this point. In the revised paper, we add comments “Note that this dual dipole algebra is valid when the gauged theory is in the ground state subspace, where there is no excess magnetic flux rather than on the full tensor-factorized Hilbert space.” below eq. 21.

3." In (32), there are two zero-form symmetry listed after gauging, while we have only one zero-form symmetry U(0)Z before gauging other than the U(0)X symmetry that were gauged. It would be helpful if the authors can clarify why the symmetry operators are doubled. One explanation the referee sees is that the U(0)Z does not commute with the Gauss law in (25) but it can be made gauge invariant in two different ways, which then lead to the two symmetry operators listed in (32)." 4. "Related to point 3, in the field theory analysis in section 4.3 (which is related to the lattice model discussed section 3.1), there is only one zero-form symmetry after gauging the LSM anomaly, while on the lattice, there are two zero-form symmetry after gauging as listed in (32). Can the author give a field theoretic explanation on this point?"

--There is in fact a single zero-form symmetry, although it admits two equivalent operator expressions. (Eq.(32) is also useful for the explicit demonstration of dipole algebra Eq.(34) )

To clarify this point, let us first recall the standard example of a U(1) one-form symmetry in 2+1 dimensions. Given a conserved two-form current $j^{(2)}$, the corresponding one-form symmetry charge is defined as

Q_1[Σ_1] = \int_{Σ_1} * j^{(2)} ,

where Σ_1 is a one-dimensional manifold. On a torus, Σ_1 can be chosen along either of the noncontractible cycles in the x-or y-direction. These different choices give rise to distinct loop operators on the lattice, but they generate the same underlying one-form symmetry. Correspondingly, in our lattice model discussed in Sec. 3.2.1, there exist Z_N one-form loop operators winding around the x- or y-direction of the torus.

By analogy, in Eq.~(32) we argue that there is only one zero-form symmetry. As discussed in the field-theoretical analysis in Sec.~4.3 (Eqs.~(77), (78), (84), and (85)), the dipole algebra between zero-form and one-form symmetries depends on the choice of the spatial manifoldsΣ_1, Σ_2, as well as on the foliation field e^I. In particular, depending on (i) the choice of Σ_ 1 on which the one-form current is supported, (ii) the foliation field e^I along which the one-form currents are stacked, and (iii) how Σ_2 is decomposed into Σ_1 and e^I, one obtains two different operator expressions for the same zero-form symmetry. Also, due to the flatness condition of the gauge fields, the two expressions are related as discussed below eq. 32.