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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_00023v1
Date submitted: 2021-02-16 17:20
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

Reports on this Submission

Anonymous Report 2 on 2021-4-5 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:scipost_202102_00023v1, delivered 2021-04-05, doi: 10.21468/SciPost.Report.2754


The authors identify the universal scaling behavior of the disorder operator in various quantum phase transitions in 2+1d.


In most of the examples (including the ones in Section II and in Section III.A), it is not clear to me the role of the $\mathbb{Z}_N$ one-form or zero-form global symmetry in the microscopic systems. In these examples, the expectation value of the disorder operator, which is the main objective of this paper, is completely determined by the emergent $U(1)$ global symmetry data (which include $\sigma$) in the low energy CFT.


The authors compute the expectation value of the disorder operator. The authors identify an interesting subleading log term when there is a corner, and show that the coefficient is determined by the current two-point function. This is an interesting result that deserves to be published.

Requested changes

1. The authors should clarify the role played by the $\mathbb{Z}_N$ zero-form and one-form global symmetry. See my comments in Weakness.

Concretely, I don't understand the significance of the $\cos(N\hat\theta_i)$ term in (1) (and the related $U(1)$-breaking, $\mathbb{Z}_N$-preserving terms in (2)). Since all the subsequent discussion and calculation assume that the $U(1)$ global symmetry reemerges in the low energy, why do we need to break it in the UV? It appears to me that the main result (which I find very interesting) has more to do with the $U(1)$ global symmetry at the critical point, and less to do with the $\mathbb{Z}_N$ zero-form or one-form global symmetry realized in the microscopic system. Indeed, in a closely related paper ref. 78, the authors there present similar results while keeping the $U(1)$ symmetry in the microscopic model.

2. In the discussion below (5), perhaps the authors can include the bootstrap reference 1912.03324, which reports the state of the art value of $C_J$ in the 2+1d O(2) CFT.

  • validity: high
  • significance: high
  • originality: high
  • clarity: good
  • formatting: excellent
  • grammar: good

Anonymous Report 1 on 2021-3-23 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:scipost_202102_00023v1, delivered 2021-03-23, doi: 10.21468/SciPost.Report.2729


1. interesting and timely subject
2. clear presentation of results


1. some discussion is a bit terse and could be expanded


This is overall a good paper, and I think it is fit to be published in this journal.

I have a few questions:

1. In the introduction (and also in these related works Refs 19 and 78) the authors discuss the behavior of <O_C>. Is there a simple relation between this operator and the operator <(log(O_C))^2>, which is studied in the paper?

2. What numerical methods were used to obtain figure 2? In Ref 19 the authors studied a square loop on a square lattice (using QMC), but I don't see how to get the shape depicted in figure 1.

3. I am a little puzzled by the notion of "categorical symmetry". Suppose I have a system with a global Zn symmetry. I guess you would like to say that the symmetries of the Zn orbifold should be considered as "symmetries" of the original theory? But operators with Zn charge in the original theory are projected out when we go to the orbifold, so they don't have well-defined charges under these other "symmetries". Is it clear this is a well-defined concept?

Some minor comments:

1. It seems to me that what is called an "ODO" operator here is known as a disorder operator since Kadanoff.

2. The divergent 1/epsilon dependent piece of eqn (8) can be eliminated by a local redefinition of the loop operator, since it is linear in R, so the constant in R and epsilon piece is universal. This redefinition also eliminates the linear divergences in (12) and (13).

3. The connection to the Renyi entropy seems particularly nice. I would appreciate if the authors (perhaps in a follow-up) would explain the details of this connection.

Requested changes

1. If the authors wish, they can choose to address my questions and comments in the paper.

  • validity: top
  • significance: good
  • originality: good
  • clarity: good
  • formatting: perfect
  • grammar: excellent

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