Hasse Diagrams for Gapless SPT and SSB Phases with Non-Invertible Symmetries

1. This work generalizes the concept of gSPT and igSPT to non-invertible symmetries and provides explicit examples for them via SymTFT. This extends our understanding of phases of matter and the SymTFT construction can potentially guide lattice model and real-world realization of such novel phases.

2. This work reviews the structure of gSPT in a very clear way that makes the appearance of SymTFT and condensable algebra very natural.

Some of the statements in the work are assumed to hold in arbitrary dimension. I do not find this very obvious and in some cases I think they don't hold in higher-D. See report for details.

In this work the authors generalizes the notion of gSPT to non-invertible symmetries and study them via condensable algebras in the SymTFT . By analyzing the structure of symmetry charges in a system, they define gSPT, igSPT and gSSB for a general fusion categorical symmetry, which I believe to be solid in 1+1D. This analysis also makes it clear that the structure of a SPT/gSPT is naturally captured by a condensable algebra in the SymTFT. After establishing the general theory, the authors study numerous examples.

This work provides a systematic study of gapless phases in 1+1D with generalized symmetries via SymTFT, and the general principles proposed in this work can potentially be generalized to higher-D, shedding light on the long-standing problem of understanding gapless quantum phases. The SymTFT construction of gSPTs can potentially guide lattice model and real-world realization of such novel phases. Therefore I recommend publication in Scipost, provided the requested changes below are addressed.

1. In the "Generalized Superconductivity Interpretation." on page3, the authors seem to call all objects of the SymTFT "local". Maybe locality here is defined differently, but I believe traditionally, e.g. the operator that creates a single domain wall in the Z2 Ising chain, is not considered local.

2. Page 4, "For example, for a non-anomalous group symme- try in (1+1)d, two SPT phases are distinguished by the charges of ground states on a circle in twisted sectors for the group symmetry." There is an example that a nontrivial SPT has trivial charge of ground states on a circle in any twisted sector. This SPT has a symmetry group of order 128.

3. On the same page the authors introduce pivotal fusion D-category as symmetry category in D+1-dimensions. Such pivotal structure does not seem to be discussed often in literature. I wonder what is the motivation for the pivotal structure, and is it important for a fusion higher category to qualify as a physical symmetry?

4. On page 9 the authors discuss "symmetry protected criticality". I believe such protection is rare in higher dimensions. Even if a symmetry lacks a fiber functor, it does not protect gaplessness in D>3, the system can develop topological order to match the anomaly.

5. On page 8 the authors say "Different gSPT phases are distinguished by the corresponding sets of confined charges." There are two problems with this statement.

5.1 In 1+1D, is it always true that just the "set" of confined charges is enough to distinguish different gSPTs? Even for SPT this does not seem obvious to me. Although for the examples studied in this work this is definitely the case. Therefore I only raise this question for intellectual curiosity and the authors do not have to make changes about this.

5.2 In D>2 , I think there are examples that the set of confined symmetry charges is not enough to distinguish phases. E.g. a toric code can be enriched nontrivially by Z2 in two ways, called e_Cm and em_C respectively. They have different symmetry fractionalization pattern therefore are different SETs. However it is clear that their symmetry charges have the same behaviour as far as being confined/condensed/deconfined is concerned. Similarly I believe there can certainly be gSETs in D>2, and they can not be distinguished by the set of confined symmetry charges in general.

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1. Presents an explicitly dictionary between types of (1+1)d symmetric gapless phases and condensable algebras.
2. Illustrates this dictionary with numerous non-trivial examples.

1. ) My main concern pertains to the premise that condensable algebras should have anything to do with gaplessness. I have not quite found a clear explanation or motivation, whether mathematical or physical, for this crucial question. Firstly, is gaplessness the fundamental notion here, or rather that of phase transition between gapped phases? If so, are condensable algebras inherently and uniquely related to phase transitions between gapped phases? It is my understanding that the authors rather think of condensable algebras as means to construct `maps’ between inequivalent symmetries, in such a way that phase transitions between gapped phases associated with a given symmetry can be lifted to phase transitions between gapped phases associated with a larger symmetry. This would mean that gaplessness is added by hand and is not fundamentally related to working with condensable algebras. Does this mean that this construction cannot shed light for instance on the gapless point of the transverse field Ising model? That said, I do not believe this puts in question the results of table I, but additional motivation is required.

2.) As alluded to above, the central mathematical concept around which this work is organised is that of condensable algebra. Throughout this manuscript, condensable algebras are merely specified as objects in the Drinfel’d center of the symmetry fusion category. What about the multiplication rule of these algebras? It is well documented that even for invertible symmetries, it is possible for two objects to be equipped with distinct condensable/Lagrangian structures. Now, I do not believe that such a situation occurs for the specific examples considered by the authors. But, how would they deal with this scenario as it occurs? Would any of the arguments need to be revised? If it happens that this aspect can be largely ignored, it seems important to clarify it early in the text.

3.) Finally, this paper focus solely on (1+1)d theories. In a few places, the authors make statements that are meant to be valid in any dimension, but these are lacunar and in my opinion unnecessary. There are too many unknowns about the physics about gapless phases in arbitrary dimensions as well as the mathematical framework of fusion n-categories for these statements to be made very concrete. In my opinion, it does not add much to the main discussion and potentially harm the overall rigour of the manuscript. Maybe these conjectural statements could be relegated to the discussion?

Given a (possibly non-invertible) symmetry) in (1+1)d encoded into a fusion category, there is an accepted classification of gapped symmetric phases in terms of indecomposable module categories over the symmetry fusion category. Alternatively, these can be classified by Lagrangian algebras in the Drinfel’d centre of the fusion category. In this manuscript, the authors discuss gapless symmetric phases from the viewpoint of (non-Lagrangian) condensable algebras. In spite of the comments highlighted above, it is my opinion that this manuscript is a valuable addition to the literature, providing a useful guiding principle for the exploration of gapless phases, which is illustrated by numerous examples that are thoroughly investigated. Provided that the authors address the comments above as well as the requested changes below, I would recommend publication in Scipost.

I also list below more specific questions/remarks:

In spite of gaplessness being inherently a statement about low-energy excitations, it seems to me that clear statements about the nature of the excitations are lacking. Given the representative of a gapped symmetric phase, a classification of excitations is understood algebraically, and one can physically demonstrate that these do give rise to gapped spectra. What is the additional ingredient in the gapless case that physically explains the nature of the spectrum?

A crucial role is played by a notion of functor between symmetry categories. It is not always clear whether additional properties are required. Sometimes, a pivotal tensor functor is mentioned, which could be more carefully motivated, while in many places no additional structure seems to be required.

Related to the previous comment, in the abstract one can read “gSSB phases are classified by functors from fusion to multi-fusion categories”. Should I understand that one can associate a gSSB phase to any functor between any fusion category and any other multi-fusion category?

Related to a previous comment, there exist general formulas for condensable algebras in $\mathcal Z(\text{Vect}_G)$ for any finite group $G$, which have the merit of specifying the multiplication rule of the algebra. Wouldn’t it be beneficial to rely on such formulas?

1. ) I would encourage the authors to revise some of their citations. For instance, in page 1, [1,2] should include the earlier references https://arxiv.org/abs/1912.02817 and https://arxiv.org/abs/2008.07567. In addition to [6-8], numerous references by Kong et al. about gapless phases seem especially relevant. Early in the text, some mathematical references regarding the notions of condensable algebras are lacking. On page 3, in addition to [21-23], earlier references shed light on the role of the Drinfel’d centre as a classifying tool for sectors/charges. Page 7, numerous earlier references than [40] discusses the correspondence between the topological orders $\mathcal Z(\mathsf{Rep}(D_8))$ and $\mathcal Z(\mathsf{Vec}_{\mathbb Z_2^3}^\omega)$. On page 14, when discussing SSB phases, the references pasted above should be highlighted as they established a direct connection between module categories and gapped phases.

2. ) Page 1: According to the authors, “the symTFT description allows for a systematic exploration of all gapped phases”. Are any assumptions required about the nature of these gapped phases?

3. ) Page 3: There, the authors start referring to “charges”. I think it would be very helpful to make this notion more precise. In particular, when considering the Drinfel’d center of a fusion category such as $\mathsf{Vect}_G$, one typically makes distinction between (magnetic) fluxes and (electric charges), while the authors seem to refer to any object in the Drinfel’d center as a charge.

4. ) Page 4: The notion of “confined charges” could be more carefully defined. The authors refer here to a “pivotal” structure when they only seem to require rigidity. As commented above, most of the notions referred to in this paragraph are difficult to define in the context of d-categories.

5.) Page 6: Since there is a simple general formula for the anyons associated with SPT order parameters, it may be useful to provide it here. Rather than being equal or being “the same” the Drinfeld center of $\mathsf{Rep}(D_8)$ and $\mathsf{Vect}_{D_8}$ are equivalent, which is an important distinction, especially in this context. Below, these are simple objects of the center that are labelled by pairs $([g],\rho)$.

6.) Page 8: It seems crucial to me to make more precise at the beginning of section III, the explicit (or implicit) relation between gaplessness and the notion of confined charges. More generally, what is the physical reasoning underlying all these statements?

7.) Page 9: The authors mention that a gapless system is obtained “by inserting a suitable physical boundary”. What are the requirements on this physical boundary, is it where the gaplessness is inserted? If so, under which assumptions “the problem of classification of gapless SPTs in (1+1)d is essentially the problem of classification of functors between fusion categories”?

8.) Page 10: Graphical notation in (III.12) and related equations may be slightly confusing as the horizontal line may be confused with a domain wall.

9.) Page 11: References should accompany the mention that there is a “well-known example” of a igSPT for $\mathbb Z_4$.

10.) Page 14: Are the statements about Euler terms boil down to having ground states labelled by simple objects whose quantum dimensions are larger than 1, or is there more to this terminology?

11.) Page 15: The paragraph starting with “Mathematically, …” may be confusing for the following reason: It is well-documented that SPTs can be classified in terms of fiber functors from the symmetry category to $\mathsf{Vec}$, where $\mathsf{Vec}$ is here an indecomposable module category. In this context, SSB phases are associated with indecomposable module categories that are not equivalent to $\mathsf{Vec}$. Keeping this classification of SPTs in mind, the statements of this paragraph may be misinterpreted as module categories over the symmetry category do not in general have the structure of fusion or multi-fusion categories. I would encourage the authors to highlight the distinctions between these various classifying schemes. Later in the page, could the authors clarify whether charges and confined charges correspond to order parameters and excitations, respectively?

12.) Page 16: The authors seem to rule out the possibility to obtain a gapless domain wall excitation. Could the authors clarify why this is the case? Further, the authors comment that “An igSSB phase exhibits symmetry protected criticality”. More generally, could the authors clarify whether all the gapless phases considered in this manuscript are critical? If so, it seems like an important point to highlight in connection to some of my previous comments about the origin of gaplessness.

13.) Page 17: Shouldn’t $\mathcal Z(\mathcal A) \boxtimes \mathcal Z(\mathcal S’)$ read $\mathcal Z(\mathcal A) \boxtimes \overline{\mathcal Z(\mathcal S’)}$?

14.) Page 42: Why are the manipulations considered in this appendix referred to as Kennedy-Tasaki transformations? These transformations were introduced in the context of the Haldane phase to relate a model with (non-trivial) SPT order and a model with SSB order with respect to the same symmetry. The manipulations presented by the authors do not seem to be an obvious generalisation of this concept.

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