Emergent generalized symmetries in ordered phases and applications to quantum disordering

We thank both referees for their careful reading of our manuscript, encouraging comments, and useful suggestions. We detail below the changes we made and address the referees’ comments:

Referee 1:

We thank the referee for their report. We have implemented many of their suggestions and are grateful for the chance to improve our paper.

Report:

The manuscript discuss emergent symmetries in ordered phase with spontaneously symmetry breaking. The emergent symmetries are organized in terms of homotopy groups of the sigma model.

Response:

In our paper, we organize the emergent symmetries not by the homotopy groups of the sigma model’s target space but rather by its homotopy type. This subtle difference in terminology reflects a crucial technical difference since the homotopy type of a topological space includes much more data than just its homotopy groups. Indeed, as described by Eq. 13, the homotopy type includes the homotopy groups, an action of $\pi_1$ on each homotopy group, and twisted cocycles called Postnikov invariants. To avoid confusion, we have added a paragraph after Eq. 13 to state this explicitly and include an example of two topological spaces with the same homotopy groups but different homotopy types.

Report:

The author does not discuss possible topological action such as theta term or Wess-Zumino term: even when the symmetry breaking pattern is the same, there are different sigma models distinguished by topological actions, and they can have different symmetries.

Response:

In this paper, we have focused on conventional ordered phases whose low-energy effective field theory does not include topological terms. The referee's suggestion of considering more exotic ordered phases is very interesting. We do believe that both $\theta$ terms and WZW terms would affect the symmetries studied in our paper. The effect of WZW was briefly explored in the follow-up paper "Pace, Zhu, Beaudry, and Wen, arXiv:2310.08554," where, using the SymTFT framework, we found the Morita-equivalence class of the symmetry. We feel that these aspects lie outside the scope of the current paper, but we have added a remark in the conclusion section emphasizing this as an important follow-up direction to this paper.

Report:

The author discuss symmetry in terms of homotopy groups instead of cohomology. However, homotopy group does not always give the correct symmetry, see e.g. https://arxiv.org/pdf/1707.05448.pdf
https://arxiv.org/abs/2210.13780

Response:

While we appreciate the referee’s comment, it is untrue that we discuss the symmetry in terms of “homotopy groups instead of cohomology.” The referee is correct that homotopy groups are incomplete in describing the magnetic symmetries of nonlinear sigma models. In fact, this is true for the cohomology groups too. One of the main messages of our paper, as we mentioned earlier in our response and stress throughout the paper, is that we organize the symmetries using the homotopy type of the target space, not the homotopy groups. This data includes the homotopy groups, but also the data $\alpha_n$ and $\beta^{n+1}$. Indeed, the general monoidal higher category describing the symmetry we study is given by Equation 19, which manifestly goes beyond the Pontryagin dual of homotopy/cohomology groups.  The examples considered throughout the main text (i.e., sections II-IIII) were aimed to emphasize this pedagogically and explore aspects of the higher category Eq. 19. For example, in section III, we started in IIIA and IIIB by considering the case where the homotopy groups do work, which is when $\alpha_n$ and $\beta^{n+1}$ are all trivial. In section IIIC, we then investigated in detail the most straightforward scenario where the Postnikov data is nontrivial (i.e., $\alpha_1$ being nontrivial), showing how it gives rise to a non-invertible symmetry and goes beyond the homotopy/cohomology group formalism explored previously in the literature. 

Report:

The author discuss whether homotopy defects are invertible. But fusing two homotopy defects can produce nontrivial non-topological defects with topological charge zero (e.g. most elementary excitations have zero topological charges). Can the author clarify how the fusion is defined?

Response:

We use the standard definition of fusing homotopy defects presented in the original works on the subject (e.g., in N. D. Mermin, “The topological theory of defects in ordered media,” Rev. Mod. Phys. 51, 591 (1979), we follow the discussion on page 599 and Figures 6 and 7.). We presented this definition in Figure 1 of our paper, which was referred to after introducing the notation of fusing homotopy defects. We have modified the relevant text such that when referring to Figure 1, we emphasize that it includes this definition.

Report:

There is a discussion using Postnikov system. What is the physical meaning in terms of defects, e.g. does it imply some relations between correlation function? (as the defects are generally not topological, it is hard to imagine there is such universal relation just from homotopy groups)

Response:

This is a very interesting point, and we thank the referee for bringing it up! The discussion using the Postnikov truncation of the topological space and its homotopy type is crucial for stating the general higher category structure formed by the symmetry defects studied in this paper (i.e., Eq. 19). The physical meaning of the entire homotopy-type data in terms of the homotopy defects is only partially answered in the literature, and we summarize our understanding below. The effect of the group homomorphism $\alpha_n$ has been known since the early papers from the late 70s on the topic (see, e.g., Kobayashi et al., “Abe homotopy classification of topological excitations under the topological influence of vortices”, Nucl. Phys. B 856, 577 (2012) and references within). For example, braiding a $\pi_2$ defect around a $\pi_1$ defect in $D=3$ causes the $pi_2$ defect charge to change according to the group homomorphism $\alpha_2$. The effect of the Postnikov invariants $[\beta^{n+1}]$ has received much less attention (if any at all). We stated in section IIA our general expectation of its physical effect: non-trivial Postnikov invariants physically reflect how lower codimension homotopy defects can carry the topological charge of higher codimension homotopy defects. For example, when $\alpha_2$ is trivial, as a 3-cocycle, the Postnikov 3-invariant algebraically describes a map $\pi_1^3 \to \pi_2$, physically reflecting how a configuration with three $\pi_1$ defects can be deformed into a configuration with a $\pi_2$ defect. Since $\beta^3$ is the associator of the 2-group describing the topological space’s homotopy 2-type, the homotopy between a configuration with three $\pi_1$ defects and the one with a $\pi_2$ defect is implemented by having the $\pi_1$ defects perform an $F$-move. Therefore, the $\pi_1$ defects “know” about the $\pi_2$ defects. In particular, the topological defect surface detecting the $\pi_2$ defects must also detect the $\pi_1$. It would be interesting to explore further the physical interpretation of the Postnikov data on the homotopy defects. We have updated Section IIA to emphasize the physical features we currently understand, including the example for $\beta^3$. We thank the referee for suggesting this improvement! However, we feel that any further investigation of the Postnikov data lies outside the scope of this work.

Referee 2:

We thank the referee for their report. We have used their suggestions to improve our paper and are grateful for the opportunity.

Report:

This article presents an interesting perspective on where generalized symmetries can appear in quantum systems. The author describes how ordered systems of conventional or more generally invertible/higher group symmetries could provide a pretty general avenue to find more exotic higher categorical symmetries. While this is an interesting and less explored work in the literature thus far, I think the work could benefit from having more examples formulated as conventional quantum field theories and Hamiltonian lattice models to convey the theoretical ideas.

Response:

We agree that our work has demonstrated how conventional ordered phases are a general setting for exotic higher categorical symmetries. This scenario is rarely explored in the literature, and our work represents one of the first explorations into the general structure and physical applications (i.e., through quantum disordering) of these exotic symmetries.  We appreciate the referee's comment on demonstrating the theoretical ideas discussed in concrete models. Throughout the paper, we considered various examples of Euclidean spacetime models, including both continuum nonlinear sigma models and various Euclidean lattice regularizations of such nonlinear sigma models. While these are not perhaps "conventional quantum field theories," they are the simplest examples in which the generalized symmetry's topological defects could be written down explicitly. Indeed, in conventional continuum field theories, the topological invariants such defects are constructed from do not typically have local expressions in terms of the continuum nonlinear sigma model's fields (e.g., the Hopf invariant characterizing $\pi_3(S^2)$ of the $S^2$ nonlinear sigma model). Considering Villain regularizations of such continuum models allowed us to explicitly discuss the homotopy defects and construct the related topological defects. From this perspective, our approach represents a considerable generalization of the Euclidean Villain models constructed in "Gorantla, Lam, Seiberg, Shao, J. Math. Phys. 62, 102301 (2021)." We agree with the referee that it would be interesting to understand the construction of these Villain Euclidean lattice models as Hamiltonian lattice models. There have been recent papers doing so for previously constructed Euclidean Villain models (e.g., "Cheng, Seiberg, SciPost Phys. 15, 051 (2023)," and "Fazza, Sulejmanpasic, JHEP 2023, 17 (2023)”), and we feel such a direction lies outside of the scope of this work.

Report:

Before recommending this work, I also have some slightly more specific questions:

Response:

We thank the referee for their interesting questions! Many of these questions regard higher categorical symmetry in general, whether such generalized symmetries are realized in an ordered phase or not. We remark that the primary purpose of our paper was to identify conventional settings for higher categorical symmetries to emerge and related applications, rather than investigate the structures of higher categorical symmetries in greater detail. We hope our results can be used to shed light on general aspects of higher-categorical symmetries in future works. However, since, as we show, the symmetries from homotopy defects are equivalent to the magnetic symmetries of higher-gauge theories, it would perhaps be better suited to explore such questions directly in the context of higher lattice topological gauge theory instead of regularized nonlinear sigma models which are much more complicated.

Report:

Can the author describe the structure of 3Rep(G) symmetries concretely in some G symmetry broken phase. This category has infinitely many simple objects that presumably act identically on the charged operators. How are these simple objects represented within a concrete model?

Response:

In our paper, we find that the fusion 3-category $3\text{-}\mathsf{Rep}(\mathbb{G}^{(3)})$ describes the symmetries arising from homotopy defects in $D=4$ dimensions. While it would be interesting to describe the simple objects of this 3-representation category, it is not a purpose of this paper. We know that 2-representation categories of particular 2-groups have been studied in the physics literature, but to our knowledge, there is not yet a study on 3-representations of nontrivial 3-groups from a physics perspective.  The models we construct act as simple demonstrations of the theoretical ideas presented. Consequentially, when identifying and constructing the symmetries of these models, we considered only the “elementary” topological defects and not the condensation defects, which can be constructed from the “elementary” topological defects by higher-gauging. It would be interesting to design physical models with the goal of learning more about higher categories, but it is not something we have attempted to do in this paper.

Report:

1. In examples where the symmetry is 2-group with a non-trivial Postnikov class, how does the Postnikov class appear in the properties of the homotopy defects?

Response:

We thank the referee for this interesting question. While we discussed the physical consequences of the Postnikov class of the 2-group $\mathbb{G}^{(2)}$, defined in Eq. 19, on the homotopy defects, we did not consider, nor claim to consider, any examples of ordered phases where the symmetry is a nontrivial 2-group. From the formalism developed in our work, for a model in $D=3$, this would require the 2-group $\mathbb{G}^{(2)}$ to obey $2\text{-}\mathsf{Rep}(\mathbb{G}^{(2)}) = 2\text{-}\mathsf{Vec}(\tilde{\mathbb{G}}^{(2)})$ for some dual 2-group $\tilde{\mathbb{G}}^{(2)}$. If the $pi_1$ action on $pi_2$ is trivial, we believe this dual 2-group exists only when the Postnikov 3-invariant $\beta^3$ is a stable cohomology operation. When this is the case, the Postnikov class of $\tilde{\mathbb{G}}^{(2)}$ would affect the homotopy defects in the same way as any 2-group symmetry would affect its charged operators. There is nothing special about the charged operators being homotopy defects. 

Report:

1. What are the forms of the condensation defects in 2Rep(S3) or 2Rep(D8) concretely realised within an ordered model? How do these defects act the homotopy defects?

Response:

While we do not consider any models with $2\text{-}\mathsf{Rep}(S_3)$ or $2\text{-}\mathsf{Rep}(D_8)$ symmetry, we thank the referee for this interesting question, which seems related to their first question. However, we are unsure what ordered model the referee refers to as we do not claim to construct such models. Such symmetries would arise in $D=3$ models with symmetry breaking pattern $G \to H$ whose fundamental group $\pi_1(G/H)$ is $S_3$ or $D_8$ while $\pi_2(G/H) = 0$. However, we do not claim to identify such symmetry breaking patterns in this paper and, therefore, do not know of such ordered models for which these are symmetries. From the perspective of $D=3$ nonlinear sigma models, these symmetries can occur when the target space is the classifying spaces $B S_3$ or $B D_8$. But in these cases, the nonlinear sigma model is just $S_3$ and $D_8$ gauge theory, respectively, which is not an ordered phase. Regarding how these topological defects would act on the homotopy defects, similar to what we wrote above, they would act in the same way that $2\text{-}\mathsf{Rep}(S_3)$ or $2\text{-}\mathsf{Rep}(D_8)$ symmetry would act on its charged operators. In other words, there is nothing special about the charged operators being homotopy defects.

Report:

1. Similarly is it possible to write the form of Q8 non-invertible 1-form symmetry generators, say wrapping a non-contractible cycle of space, in a concrete model displaying SO(3)—> Z2 x Z2 SSB? What determines the vacuum expectation value of such an operator?

Response: In section IVA, we constructed a Euclidean lattice model, which upon setting $\tilde{H} = Q_8$, realizes this ordered phase and the nontrivial disordered phase (which is $Q_8$ gauge theory). While this example uses the Euclidean lattice model formalism, perhaps the referee is instead asking about the Hamiltonian formalism and conventional lattice model of spins. For these typical spin models, the Rep(Q_8) higher-form symmetry would not be an exact lattice symmetry. This is similar to how winding symmetries in U(1) SSB phases are not exact lattice symmetry of conventional spin models, and how the magnetic symmetry in gauge theory is not an exact symmetry of conventional Hamiltonian lattice gauge theory models (see "Gorantla, Lam, Seiberg, Shao, J. Math. Phys. 62, 102301 (2021)," "Cheng, Seiberg, SciPost Phys. 15, 051 (2023)," and "Fazza, Sulejmanpasic, JHEP 2023, 17 (2023)"). In the context of such quantum lattice models, the symmetries we study emerge upon taking the continuum limit. Writing down the symmetry operator in such a Hamiltonian lattice model requires constructing a generalized Villain Hamiltonian model. While it would be interesting to do so and extend our results to the Hamiltonian formalism, it is outside the scope of this paper.