Quantum Phases and Transitions in Spin Chains with Non-Invertible Symmetries

We thank both the referees for their careful review of our paper and for their recommendations for publication. We are also indebted to Referee 1 for their helpful comments for improving the clarity of our presentation. Below, we reproduce the specific comments of Referee 1 as an enumerated list, to which each item below is our corresponding response. We have also updated our manuscript following these comments. We hope that our manuscript can now be published.

1. \emph{from the abstract: “Such symmetries appear generically in gapless states of quantum matter, constraining their low-energy dynamics.” I expect that this statement about non-invertible symmetries might not be true. We know that they appear in rational CFT and in a few other examples of gapless field theories (including nonlinear sigma models of goldstone modes [64]). But I don’t see any reason to believe that they will appear even in generic 1+1d CFTs (such as Calabi-Yau sigma models at generic points in the moduli space).}

• Indeed, that statement appears a bit off-handed. What we wanted to emphasize was that there are many families of gapless states of quantum matter which exhibit non-invertible symmetries. We mention these constructions in the section on “Relation to prior work”, as the referee noted. We have replaced the statement quoted by the referee to “Such symmetries appear in large families of gapless states of quantum matter and constrain their low-energy dynamics”.

1. \emph{The section called “relation to previous work” is very helpful to the reader and much appreciated.}

• We thank the referee for their appreciation.

1. \emph{When one says that a category is “anomaly-free” (e.g. top of page 4) I guess this means just that all the quantum dimensions are integers? Or is it a stronger condition? Some explanation would be helpful. [I eventually see that this notion is defined in footnote 33.]}

• We thank the referee for pointing out our oversight. We have added footnote 3 defining the notion of an “anomaly-free” symmetry that generalizes to non-invertible symmetries.

1. \emph{ would like to discourage the use of the adjective “holographic” to describe the ideas related to “SymTO” or “SymTFT”. The word “holographic” has been extensively used in the context of gauge/gravity duality, which is (conjectured to be) an exact equivalence between a d-dimensional theory without gravity and a d+1 (or more) dimenisonal theory with gravity (the Hilbert spaces are supposed to be the same; the dynamics are supposed to be the same). As far as I understand, the relation between SymTO and the lower-dimensional theory whose symmetries it encodes is not such an equivalence at all.}

• The referee is absolutely correct that the bulk and the boundary are not equivalent in SymTO/SymTFT. However, the terminology of “topological holography” is now used in published papers, including Ref. [33] published in SciPost. We precisely state in what sense we use the word “holographic” in great detail in our Introduction and include references to past usage of the terminology, in order to avoid any misunderstanding. With that in mind, we feel that the usage of this terminology is not unjustified and hope that the referee would understand.

1. \emph{It would be good if the authors explain their use of the term “incommensurate” early on in the paper (around page 7). I see that a definition appears eventually on page 44. Is there actually spontaneous breaking of internal and translation symmetry in these states?}

• We thank the referee for the suggestion. We have added footnote 8 to explain the meaning of “gapless incommensurate phase”. In such a phase, translation symmetry is not spontaneously broken, i.e., there is a nondegenerate ground state in which the expectation value of translation operator is the non-vanishing phase $e^{i k_0}$ , where $k_0$ is the ground state momentum. The word “gapless” refers to the fact that the lowest energy excitations above the ground state have a vanishing energy gap in the thermodynamic limit. The word “incommensurate” refers to the feature that there are low-energy modes with non-zero quasi-momentum in the many-body spectrum. These modes couple to generic local operators whose correlation functions, as a result, decay algebraically with an oscillatory factor of period incommensurate with the lattice spacing. Such an oscillatory behavior is shown in Fig. 4 of our manuscript. Previous usage of this terminology is also indicated in the newly added footnote 8, as a guide to the reader.

1. \emph{The generalization of the work of Ref [74], making completely explicit duality operators for the Z3 case, is nice.}

• We thank the referee for their appreciation of this generalization.

1. \emph{I didn’t understand what is being shown in Figure 9.}

• Figure 9 shows the so-called “sandwich” decomposition in the SymTFT/SymTO framework; see Ref. [9] for the first use of this terminology and Ref. [45] (Fig. 2 therein) for two of the present authors’ discussion of its implications for gapless states. The idea of such an “isomorphic holographic decomposition” was introduced earlier in Ref. [27] (Eq. 4.3 therein), although it was only referred to by this name first in Ref. [45]. In the present case, Figure 9 depicts the enlarged SymTO obtained by including the non-invertible symmetries (2.15a) and (3.18) for the “self-dual” $S_3$-symmetric (sub-figure (a)) and $\mathsf{Rep}(S_3)$-symmetric (sub-figure (b)) Hamiltonians respectively. The enlarged SymTO is the same for both cases, consistent with the fact that the enlarged symmetry categories are Morita equivalent. The top boundary denoted by $\tilde{R}$ – often referred to as “symmetry boundary” or “reference boundary” – specifies which of the two symmetries is realized on the “physical boundary” at the bottom.

1. \emph{I didn’t understand yet the explanation on page 47 for why, in this “incommensurate phase”, the numerically-inferred central charge to varies across the phase diagram. I see that a variation with parameters of the 1/L term in the ground state energy would lead to a variation of the inferred cv, but wouldn’t that simply contradict equation 5.9? What does the groundstate quasimomentum have to do with it?}

• This is a subtle point; we thank the referee for bringing it to our attention. There is no contradiction. The ground state energy density $\epsilon$ in Eq. (5.9) itself depends on L in the incommensurate gapless phases we considered. Because of this, the scheme outlined on page 47 for computing the central charge shows an L dependence. To demonstrate the dependence of the ground state energy on the system size, we carried out the numerical procedure described on page 47 for the Hamiltonian (5.3), which is exactly solvable. In the supplement linked at the end of our response, we plotted $cv$ for parameters corresponding to the c = 1 incommensurate phase (left) and the c = 1/2 Ising CFT (right). We have added more discussion on this point below Eq. (5.9) in our manuscript.

Figure caption: cv extracted from the 1/L term in the ground state energy, as described in the numerical method below Eq. (5.9) of the manuscript for the exactly solvable model in Sec 5.3. The model parameters are h = J = 0.4 (left) and h = J = 1 (right).

1. \emph{“A related discussion on the classification of gravitational anomalies and anomalies of group- like symmetries can be found in Refs. [26,36])” I think there is no open parenthesis.}

• Indeed, there is a stray closing parenthesis which we have removed.

1. \emph{page 11: “Therefore, the duality holds when both conditions (2.12a) and (2.12b).” should be “Therefore, the duality holds when both conditions (2.12a) and (2.12b) hold.”}

• We have updated the incomplete sentence.

1. \emph{page 48: “this gapless phase turns into and incommensurate phase” should be “this gapless phase turns into an incommensurate phase”} • We have corrected the typo.

Link to supplement: https://drive.google.com/file/d/10dUpF_V0QQ_-WLPBXBqZgvbFV1HdpZ3C/view?usp=sharing

Summary of changes:

• We modified one sentence in the abstract following referee 1's first comment.
• We added two footnotes, 3 and 8 following referee 1's third and fifth comments, respectively.
• We added more discussion around Eq. (5.9) following referee 1's eighth comment.
• We fixed the typos pointed out by the referees and some others.
• We made small changes to the Appendix G on fermionic non-invertible symmetry.