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Universal Features of Higher-Form Symmetries at Phase Transitions

by Xiao-Chuan Wu, Chao-Ming Jian, Cenke Xu

This is not the current version.

Submission summary

As Contributors: Cenke Xu
Preprint link: scipost_202102_00023v2
Date submitted: 2021-05-09 22:40
Submitted by: Xu, Cenke
Submitted to: SciPost Physics
Academic field: Physics
  • Condensed Matter Physics - Theory
Approach: Theoretical


We investigate the behavior of higher-form symmetries at various quantum phase transitions. We consider discrete 1-form symmetries, which can be either part of the generalized concept ``categorical symmetry" (labelled as $\tilde{Z}_N^{(1)}$) introduced recently, or an explicit $Z_N^{(1)}$ 1-form symmetry. We demonstrate that for many quantum phase transitions involving a $Z_N^{(1)}$ or $\tilde{Z}_N^{(1)}$ symmetry, the following expectation value $ \langle \left( \log O_\mathcal{C} \right)^2 \rangle$ takes the form $\langle \left( \log O_\mathcal{C} \right)^2 \rangle \sim - \frac{A}{\epsilon} P+ b \log P $, where $O_\mathcal{C} $ is an operator defined associated with loop $\mathcal{C} $ (or its interior $\mathcal{A} $), which reduces to the Wilson loop operator for cases with an explicit $Z_N^{(1)}$ 1-form symmetry. $P$ is the perimeter of $\mathcal{C} $, and the $b \log P$ term arises from the sharp corners of the loop $\mathcal{C} $, which is consistent with recent numerics on a particular example. $b$ is a universal microscopic-independent number, which in $(2+1)d$ is related to the universal conductivity at the quantum phase transition. $b$ can be computed exactly for certain transitions using the dualities between $(2+1)d$ conformal field theories developed in recent years. We also compute the ``strange correlator" of $O_\mathcal{C} $: $S_{\mathcal{C} } = \langle 0 | O_\mathcal{C} | 1 \rangle / \langle 0 | 1 \rangle$ where $|0\rangle$ and $|1\rangle$ are many-body states with different topological nature.

Current status:
Has been resubmitted

Author comments upon resubmission

Response to report 1:

We want to thank referee 1 for very careful reading of our manuscript, and also very helpful comments.

Response to the questions:

  1. For Gaussian theory, one can simply take the exponential of <(log(O_C))^2> to reproduce <O_C>. However, for strongly interacting field theories, due to the lack of the Wick theorem, one can only say that <(log(O_C))^2> coincides with the second order expansion of the field operator of <O_C>. But one can alternatively view <(log(O_C))^2> as another quantity that diagnoses the phase transitions.

  2. About the numerical results in Figure 2, we simply mean that we performed a numerical integral of Eq.6 (for example) along the loop object. The numerical integral has no expansion of 1/L or epsilon, while the analytical evaluation in this paper extracts the dominant orders in the expansion of 1/L and epsilon.

  3. The referee raised very good question about orbifold. We will do our best to give a brief review of (our understanding of) categorical symmetry and ODO here. In recent years the concept of symmetries has been significantly generalized, and many concepts that were thought to be beyond the Landau’s paradigm (such as emergent photon, topological order), can now be interpreted in the language of spontaneous symmetry breaking (SSB) of generalized symmetries. The categorical symmetry is one step further of generalizing the notion of symmetry. For example, there are two 1d boundary states of the 2d toric code separated by an unavoidable phase transition. Neither boundary phase has ground state degeneracy, but they can still be described as SSB of two “Z2 symmetries”, but the physical ground state can be viewed as the orbifold of the corresponding Z2 symmetry. In fact in the first paper that introduced the categorical symmetry (Ref.16), the authors were specifically discussing the “symmetric sector” of the Ising model, which is the orbifold of the Z2 symmetry. The situation is similar to the “dual inexplicit symmetries” in higher dimensions, where the ground state is an orbifold of the dual symmetry. Many usual concepts of symmetry and symmetry breaking no longer apply here. This is why we introduced the notion of ODO to describe systems with a symmetry, or orbifolds of symmetry (using the referee’s language). As long as we use the correct notion, we believe the concept of categorical symmetry is indeed well-defined: there is still an unavoidable phase transition between the disordered and ordered phases, even if we only consider the orbifold of a symmetry, and the ODO is the quantity to diagnose the two phases.

Response to the minor comments:

  1. Yes, the “ODO" in the manuscript indeed corresponds to the disorder operator in the case of Ising model. The concept of ODO was developed for more general cases, including cases with higher symmetries, or subsystem symmetries as well. We have added a footnote to explain that for the simple Ising model the ODO is the same as the disorder operator.

  2. Yes we agree the linear divergences in all ODOs studied here can be eliminated, and the universal log contribution is the most interesting object to study.

  3. Indeed, the connection to the Renyi entropy is very interesting. We are in the process of further study, and we would prefer to leave it for future work.

Response to report 2:

We really appreciate the helpful comments and suggestions from referee 2.

Response to the questions:

1, About Z_N v.s. U(1)

This is a very good question. Indeed most of the calculations can go through with just U(1) symmetry, and we are aware that Ref. 78 (now Ref.79) studied the case that kept the U(1) symmetry. However, we have concerns about the lattice definition of categorical symmetry with respect to an original U(1) symmetry. When the original symmetry is Z_N, the ODO of the dual 1-form symmetry is a line/loop object defined on the lattice, whose behavior diagnoses the order and disordered phase of Z_N; but for the cases with U(1) symmetry, we know that the “disorder” is driven by vortices, rather than a line object. This may be just an unnecessary philosophical concern, but we feel the physical picture is most clear for the Z_N categorical symmetries. Generalization of categorical symmetries to continuous symmetry is possible, but we prefer to leave this to more careful future study. We have added a footnote to explain this point.

  1. We are very grateful to the referee for bringing the reference to our attention, we have added it to our reference list.

List of changes

1, we added a footnote to explain the relation to the disorder operator defined previously, as suggested by referee 1;

2, we added a footnote to explain the relation between Z_N and U(1) symmetries, raised by the referee 2. In general, generalization of categorical symmetries to continuous symmetry is possible, but we prefer to leave this to more careful future study.

2, references suggested by the referees were added (Ref.48 and Ref.80).

Reports on this Submission

Anonymous Report 2 on 2021-6-30 (Contributed Report)

  • Cite as: Anonymous, Report on arXiv:scipost_202102_00023v2, delivered 2021-06-30, doi: 10.21468/SciPost.Report.3153


Interesting and useful work


Confusingly written


A conservative point of view is that this paper is doing a simple and interesting thing, namely studying the universal bits of the behavior of disorder operators (with corners) at certain continuous phase transitions.

But they are describing what they are doing in a way that I find very confusing.

Specifically, I have to admit that I do not understand at all what the authors mean by "inexplicit dual symmetry". In fact, my understanding is that (for example in the Ising model example studied as example 1 in section IIA) it is only actually a symmetry at all when the system arises on the boundary of a higher dimensional system with topological order.
This apparently crucial point was made explicit in the earlier work of Wen and Ji (ref 16) but seems to not be mentioned at all here!
The authors know very well that duality does not preserve global data like the number of groundstates
(the 2d $Z_N$ Ising model in the broken phase has $N$ groundstates on the plane; this phase is dual to the deconfined phase of the $Z_N$ gauge theory, which has a unique groundstate on the plane).

No one will disagree that the correlations of the order parameter and disorder operator can be used to characterize these different phases, or that it is interesting to study their behavior at the critical point.

But since this obscure concept of "inexplicit dual symmetry" plays such a central role in the narrative structure of the present manuscript (particularly the abstract), I feel that it needs serious revision.

Furthermore, the authors' response to the very reasonable question 3 of Referee 1 I find completely mystifying and unhelpful. Part of the problem is that there are currently various competing definitions in the literature of the meaning of the term "categorical symmetry".

I also agree with another referee that what the authors call "ODO" is just a disorder operator. I don't understand the need to proliferate nomenclature here.

The work is clearly publishable, but I think the authors need to make more of an effort to explain their ideas clearly.

smaller comments:

-- Is it possible to understand the relation between the coefficient $b$ and the conductivity in some more general way? Why should there be such a connection?

-- Most of the calculation on page 3 of the manuscript is done in section VI of the famous review by Kogut:

-- I think the authors should at least make some comments about the difference between $ \langle \log^2 O\rangle $ and
the better-defined object $ \log \langle O \rangle $ in their interacting theories.

-- Corner contributions to Wilson (and 't Hooft) loop expectation values have been studied in gauge theory for several decades, for example under the name "cusp anomalous dimension".

-- The recent paper
gives a nice super-universal geometric explanation for the universal behavior of the angle-dependence of the corner contribution to various measures of fluctuations. I believe this result explains why the same function is seen in the entanglement entropy as in the behavior of Wilson loops.

-- Reference 9 (cited in the batch-reference in the first paragraph) is not about higher-form symmetry.

-- "Tt is known" should be "It is known"

-- "The angle dependence of the ODO is still give by Eq. 14"
should be "The angle dependence of the ODO is still given by Eq. 14"

Requested changes

Improve the presentation along the lines discussed in the report.

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Anonymous Report 1 on 2021-5-21 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:scipost_202102_00023v2, delivered 2021-05-20, doi: 10.21468/SciPost.Report.2947


In the new footnote 83, the authors say that they are not certain about what the U(1) categorical symmetry is. I have a comment regarding this point.

It has long been known that if we gauge a $\mathbb{Z}_N$ 0-form symmetry of a 1+1d QFT $\cal T$, then the gauged theory ${\cal T}'$ also has a $\mathbb{Z}_N$ 0-form symmetry. The latter is known as the "quantum symmetry". If we gauge the quantum symmetry of ${\cal T}'$, we get back to $\cal T$. See "Quantum Symmetries of String Vacua" of C. Vafa in 1989.

This was further generalized in 1412.5148 to the following statement: if we gauge a $\mathbb{Z}_N$ $q$-form symmetry of a $d+1$-dimensional QFT $\cal T$, then the gauged theory ${\cal T}'$ has a $\mathbb{Z}_N$ $(d-q-1)$-form symmetry. If we gauge the $\mathbb{Z}_N$ $(d-q-1)$-form symmetry of ${\cal T}'$, we get back to $\cal T$.

In the past couple of years, this notion has been discussed under the name of "categorical symmetry" in the condensed matter literature.

What about continuous symmetries? It was famously known that if we gauge a $U(1)$ 0-form symmetry of a 2+1d QFT $\cal T$, then the gauged theory ${\cal T}'$ has a magnetic $U(1)$ 0-form symmetry. This is the $S$ of Witten's $SL(2,\mathbb{Z})$ action in hep-th/0307041. If we gauge the magnetic $U(1)$ 0-form symmetry of ${\cal T}'$, we get back to $\cal T$.

It appears to me that this is the proper notion of a "$U(1)$ categorical symmetry" in the condensed matter sense.

In fact, the explicit calculations in this paper are mostly about the Noether currents for the magnetic $U(1)$ symmetry. It therefore appears to me that this paper is really about the "$U(1)$ categorical symmetry" (in the sense I described above), rather than the "$\mathbb{Z}_N$ categorical symmetry".

I'll leave it to the authors' decision if they would like to further comment on this point.

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