Symmetry Protected Topological Criticality: Decorated Defect Construction, Signatures and Stability

In this work, the authors study gapless (or critical) analogs of symmetry protected topological (SPT) systems. They present a decorated domain wall construction, where they differentiate two cases: the 'weak' case where the decoration consistency condition is akin to that for gapped SPT phases, and the 'strong' case where it is not (the latter case has been called 'intrinsically gapless SPT phases' in other works). Some concrete examples and numerical results are discussed as well.

Although the topic of gapless SPT phases is timely and this work discusses examples of interest, there are a variety of serious shortcomings, ranging from certain incorrect statements to unjustified claims of novelty. For these reasons I cannot recommend publication of this work in its present form. The issues are as follows, in no particular order:

1. [Insufficient analysis of numerical examples and incorrect conclusions.] In the discussion of stability, the authors present some small-scale numerical studies, involving numerically calculating quantum numbers of certain (un)twisted boundary conditions. While I agree that these quantum numbers can give some insight into the potential SPT-like properties of the state, it is inadequate for gaining insight into what the actual state is of the phase (e.g., is it gapless or gapped etc). Since the authors aim to discuss the stability of the topological properties of critical/gapless states, a natural requirement is to check the stability of the criticality/gaplessness. For instance, the authors consider a 'weak' version perturbed with V in Eq.(2.24). From the finite-size quantum numbers in Fig. 4, it is apparently concluded that the "results suggest that there is a topological phase transition at h = hc, between the gapless Z2_A × Z2_G weak SPTC region and Z2_A SSB phase" (where hc was determined to be roughly 1.2), while I find that there is no reason to infer this. In fact, from counting relevant perturbations of the original Ising CFT, it is clear that one instead expects that a generic (Z2-preserving) perturbation should gap out the system for any h \neq 0, without preserving some duality. To confirm this, I have independently performed density matrix renormalization group calculations which clearly show that at e.g. h = 0.5 the system is in a gapped SPT phase (see attached DMRG plot), in apparent conflict with the above claim.

There is a similar numerical discussion for the 'strong' case later in the manuscript, where an appropriate analysis of the model is similarly lacking.

2. [Claims of novelty.] There are various implied claims of novelty, ranging from a framework that ties together weak and strong gapless SPTs, to claiming a novel probe (given by quantum numbers of twisted sectors) and a systematic way to generate Hamiltonians. I find that some of these claims are unjustified:

2a. [Claiming novel framework.] This manuscript claims to provide (for the first time) a framework that treats the weak and strong cases in a single description, namely by generalizing the notion of the decorated domain wall (DDW) construction to gapless systems. First, the DDW construction was already explicitly used and discussed in one of the first works on gapless SPT phases: Ref.1 in the manuscript (Scaffidi et al); this captures all the weak cases that the authors discuss. Second, the first paper on intrinsically gapless SPT phases (called 'strong SPTC' in the present work) already presented a framework that treats the weak and strong cases in a single language: Ref.3 in the manuscript (Thorngren et al.). For instance, see the discussion around Eq.6 of that work, and Appendix C.1 of that work for a detailed discussion of how the strong case corresponds to certain fluxes having an anomaly constraint (e.g. Eq.C6) and how in the non-anomalous case the discussion reduces to the weak case. In fact, it is not apparent to me whether the discussion in the present manuscript goes beyond the framework discussed in that appendix, or whether it is simply a reformulation of it; this is quite important to clarify when claiming novelty. On a related note, at some point the authors claim that their framework goes beyond that of Ref.3 since "This provides an example where the SSB symmetry G is strictly smaller than the anomalous symmetry \hat \Gamma, which generalizes the construction in [3]". Are the authors referring only to the Z4 example? Ref. 3 also discussed other examples (see Table I and Appendix C.2), where one does not need to spontaneously break the full symmetry group to satisfy the anomaly. Moreover, a follow-up work by Ma, Zou and Wang ("Edge physics at the deconfined transition between a quantum spin Hall insulator and a superconductor", SciPost Phys. 12, 196 (2022), not cited in this manuscript) discusses one of these 2+1D examples in more detail, where the emergent anomaly can be satisfied by breaking only a strict subgroup of the symmetry.
In conclusion, the way the authors present their framework as a first unifying description seems unjustified, and in fact no comparison has been made to the existing framework in Ref. 3.

2b. [Claiming to generate novel examples using framework.] One interesting claim that the authors make is that their framework allows for the systematic generation of novel models. As we have already noted for the weak case this is not novel and was explicitly used in Ref. 1 by Scaffidi, Parker and Vasseur (indeed the particular weak example that is extensively discussed in this manuscript was already discussed in great detail in Ref.1, which is perhaps not given due credit). However, for the strong case this is a more interesting claim: Ref.3 did also provide a constructive method for the strong case, but it required building strips of higher-dimensional SPTs and then making one direction finite, making it somewhat cumbersome. It seems that the authors of the present work imply that their formulation allows for a more direct construction? This would be interesting if correct. However, the current manuscript does not seem to justify this claim (at least it is not clear to me). For instance, the model presented in Eq.3.7 had been studied before (as the authors note) and it is not shown how it is supposedly derived from the DDW construction; instead, the authors start with this model and show how by rewriting it, that at low energies "The symmetry operator is realized in a non-on-site way, which is demanded by the anomaly (3.4) at the low energy. This justifies that the proposed Hamiltonian (3.7) comes from the prescribed construction in section 3.1.1". I agree with the first sentence (this shows it is a strong gapless SPT), but I do not see why this implies the second sentence: it is not clear why this counts as an application of the DDW construction. Exactly the same argument had already been given in Ref.49 (Borla et al), e.g., see Eq.51 in that work, or the expression for the U(1) generator Q = \sum XX above Eq.52. It is not clear whether one would actually constructively derive this model by simply applying the DDW construction, rather than retro-actively interpreting the low-energy symmetry action as being anomalous (the latter perspective is already a well-known general way of generating examples, see Ref 3). Similarly, it is not clear how the spin-1 example (which certainly looks interesting) is constructively derived by starting from the general DDW picture (rather than retro-actively interpreting it).

2c. [Claiming a novel probe.] In Sec.1.4, the authors two possible characteristics for gapless SPT phases: edge modes and non-trivial quantum numbers in twisted sectors. They note that the former has been extensively explored, but that the latter is a novel bulk probe proposed in this manuscript. There is a brief footnote referring to v2 of Ref.2 (its preprint was updated several months before this manuscript appeared) where the authors acknowledge that the same probe was introduced in Ref.2. However, a striking omission is the discussion of another well-studied bulk probe: namely that of string order parameters (which is only mentioned in a single passing sentence on p21). Already version 1 of Ref.2 (a preprint from 2019) showed that this can be used to identify gapless SPT phases in the bulk. In fact, this seems to be equivalent to the bulk probe proposed by the authors in the present work, although this does not seem to be discussed in this work. Indeed, the v2 update of Ref.2 explicitly states that the quantum numbers with twisted boundary conditions is *equivalent* to the string order parameters due to state-operator correspondence. Moreover, Ref.3 (studying the 'strong SPTC' case, i.e., intrinsically gapless SPT phases) also points out how string orders can be used to detect them, where in this case it was found that the charges of the endpoints are incompatible with any gapped SPT phase; again, by state-operator correspondence, this is equivalent to studying the quantum numbers of twisted sectors. It is rather surprising that the authors do no discuss these previous insights which (to the best of my knowledge) are equivalent to those discussed by the present authors.

2d. In addition, let me note that the authors say that "We will find that in general, the strong SPTC is more stable than the weak SPTC as well as conventional Landau transition, in the sense that there may not be a relevant perturbation in strong SPTC that triggers the flow towards a Gamma gapped SPT phase." It is rather misleading to represent this as a new finding: since their conception in Ref.3, intrinsically gapless SPT phases rely on emergent anomalies, which directly implies that there can be no nearby (i.e., no relevant operator perturbing to a) symmetric gapped phase, and in particular no nearby gapped SPT phase.

3. [Meaning and consistency of definition(s).] In this work, the authors introduce the terminology of Symmetry Protected Topological Criticality (SPTC). However, there seem to be some issues with the definition(s) given in the manuscript:

3a. [No stability criterion] A list of desired criteria is given on p4, one of them is the existence of degeneracy with open boundaries. However, there seems to be no criterion for how (un)stable this can be. Presumably, one would not want a fine-tuned property that is not related to any symmetry protection, say. But if we allow for perturbations, then one has to determine what is the allowed set of perturbations (i.e., how does one define one's equivalence class)? E.g., for gapped SPT phases, we allow for all symmetric perturbations which do not close the bulk gap. Do the authors require that the bulk remains gapless? Or remains in a particular type of universal criticality? E.g., Ref 2 defines a given CFT to be in two distinct symmetry-enriched versions if one cannot continuously transform one into the other, which in certain cases is indeed due to bulk topological invariants. This is not a mere academic discussion: whatever criterion one decides on is also what one has to check in numerical studies (see point 1 above).

3b. [Two distinct notions/definitions] It is rather confusing that the authors first list a set of criteria on p4, and say that they would like that these characterize SPTCs, but that at the same time they say they use a 'working definition' for SPTC which is what is produced by the DDW construction discussed later in the work. This is quite confusing, since the latter is much more restrictive (as the authors themselves note the latter always comes with exponentially decaying degeneracy whereas the former does not, etc). Indeed, it seems this manuscript is essentially about the DDW construction, in which case it is perhaps better to devise a name for that, rather than the somewhat vague general notion, especially since the general case has been discussed before (with more precise definitions); see point 4.

4. [Confusing terminology.] Lastly, the above terminology of SPTC introduced by the authors is rather confusing and unfortunate for several reasons:

4a. [Prone to misinterpretation.] The name "symmetry-protected topological criticality" can easily be misinterpreted to mean that the criticality is the protected property (as opposed to the intended meaning of there being criticality with protected edge modes). As a case in point, one of the authors of the present work has a work from several years ago [Yao, Hsieh, Oshikawa, Phys. Rev. Lett. 123, 180201 (2019)] titled "Anomaly matching and symmetry-protected critical phases in SU(N) spin systems in 1+1 dimensions", where "symmetry protected criticality" was indeed used to denote situations where the criticality is symmetry-protected. Hence, the name "symmetry-protected (topological) criticality" seems like a misnomer. From private communications with other physicists, I have heard similar concerns about this terminology.

4b. [Conflicts with existing terminology.] Gapless analogs of SPT phases have been studied for several years now, and so naturally some terminology already exists. The most common name is perhaps simply 'gapless SPT phases', with Ref.1 being one of the first to explicitly define/introduce this term. Ref.2 considered 'symmetry-enriched quantum criticality' which differs in two slight respects [the word 'criticality' is less committing to whether one has a stable phase or not, and 'symmetry-enriched' allows for a slightly broader set of circumstances where a given universality class subdivides into distinct symmetry-enriched versions in the presence of symmetries (which cannot be connected with one another without destroying said universality class), where discrete topological invariants are only one way of achieving this result]. Most works perhaps use the minimal terminology of gapless SPT phases, and it is then understood by context whether additional symmetries (or dualities) are necessary to give rise to an extended phase, or whether there is only a fine-tuned critical point. It seems ill-advised to introduce a whole new term (SPTC) to essentially discuss the same physics that has already been discussed. Indeed, many of the spin chain examples discussed by the authors have already been discussed before as examples of gapless SPT phases. Giving these an entirely new name might incorrectly signal to the community that a new phenomenon is being studied, and is likely to seed confusion rather than bring clarity (here the fact that the terminology SPTC is prone to misinterpretation (see point 4a) does not help).
Two further problems are that on p4 the authors claim to compare to the existing terminology of "gapless SPT phases", "symmetry-enriched criticality", and "intrinsically gaples SPT phases" (I think the latter is simply a stricter subset of the first two, which the present work rechristens as 'strong SPTC'), but such a comparison is never explicitly actually made. Again, I think this would likely cause confusion in the literature. The second strange aspect is that on p4 the authors claim they want to reserve the (already established) name of symmetry-enriched criticality for the case when it is no longer true that "there is no ground state degeneracy under periodic boundary conditions". The authors seem to give the gauged Ising transition (from toric code to paramagnet) as an example, but this does in fact have a unique ground state on the torus [e.g. see Schuler et al., Phys. Rev. Lett. 117, 210401 (2016)]. Either way, reserving a particular name for a (yet-to-be-discovered) circumstance when that name is already in use for the same systems one is studying seems rather puzzling to this referee.

The authors discuss symmetry protected topological critical states, which they define as gapless systems with gapless edge excitations that, especially by finite size considerations, can be well distinguished from gapless bulk excitations. Several one-dimensional examples are discussed whose constructions follow the decorated defect approach. For these models, detailed considerations regarding ground state degeneracies for open and periodic boundary conditions, symmetry charges, and the effects of twists and other perturbations are presented.
A weak and a strong SPTC are being distinguished, where the former can be “trivialized” by “adding” an SPT.

The paper is conceptually innovative, clear, and well reasoned. I only have a few questions and comments:

In section 2.4 “Stability of Weak SPTC”, I found no arguments that the perturbation Eq. 2.24 preserves the criticality. If it does, can arguments be given? If it doesn’t, then how does this perturbation really help making an argument about stability of a critical system?

Section 3.2.1 finds a somewhat intricate pattern of ground state degeneracies as function of system size for OBCs. This is somewhat in contradiction with the authors’ own definition of SPTCs. My reading of this section is that this contradiction is not fully resolved, and the authors are relatively open about that. As a suggestion, I think it could help to point out that such patterns of ground state degeneracies with system size, as well as their interplay with fluxes, are common even in quite ordinary gapless 1D systems, such as Luttinger and Luther-Emery liquids, as discussed, e.g., throughout A. Seidel, D.-H. Lee, Phys. Rev. B 71, 045113 (2005).

Also, in section 3.2, the authors begin by observing that “An immediate fact to realize is that there is no Z^Gamma _4 gapped SPT in (1 + 1)d.” To the uninitiated reader, as which I’m counting myself here, a little elaboration or reference would be of help. I’m reading between the lines that the authors are appealing to the fact that the short exact sequence 1.1 does not split in this case. However, if the authors can say more about that without forcing the reader to dig through the literature on SPTs, that would be appreciated.

Finally, Refs. [25] and [38] appear to be the same.

In short, I enjoyed reading this paper would definitely recommend publication in SciPost with a few optional improvements.

The manuscript presents a unified construction to understand a variety of symmetry-protected topological order which can occur in critical systems. Whereas gapped topological phases are fairly well understood by now, the study of topological effects in gapless systems is still work in progress, with an ongoing program to understand how topological effects enrich the classification of quantum phase transitions. As such, this manuscript is an important advance on a timely topic.

The authors were able to unify both weak and strong Symmetry-protected topological criticality within the same framework. Whereas the weak case is by now fairly well understood, the strong case remained a bit more mysterious, so it is nice to see it treated within a similar framework. I would say this is the most important contribution of the paper. It is also nice to see an implementation of both weak and strong cases within a single model of spin 1 chain.

The paper is extremely well written and pedagogical, and I strongly recommend publication after the authors have addressed my comments below:

* The authors have this constraint in the definition of SPTC: “The criticality is confined. In particular, if the criticality has a 1-form symmetry, it should not be spontaneously broken”. I did not understand why they imposed that constraint, and what it actually meant (what does deconfined mean if the CFT is not described in terms of a gauge theory?). This constraint did not appear anywhere else in the paper as far as I can tell, same goes for 1-form symmetries. Some clarification would be good.

* In Fig 5, the authors study numerically what happens to a strong SPTC when adding a trivial paramagnet term proportional to h, and try to locate a quantum phase transition. Based on their numerics, the authors are not able to locate the transition in the thermo. limit. I was a bit confused by this section. Since the point h=0 is a known CFT (it maps to free-fermions, so even easier), shouldn’t it be possible to identify the h term as one of the CFT operators, and then check whether it is relevant, marginal or irrelevant.
In particular, the point h=0 has a U(1) symmetry (fermion number), and I imagine the h term breaks it, so I would deem it likely that the h term is relevant and gaps out the SPTC for any h>0.