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Universal Features of HigherForm Symmetries at Phase Transitions
by XiaoChuan Wu, ChaoMing Jian, Cenke Xu
This is not the current version.
Submission summary
As Contributors:  Cenke Xu 
Preprint link:  scipost_202102_00023v1 
Date submitted:  20210216 17:20 
Submitted by:  Xu, Cenke 
Submitted to:  SciPost Physics 
Academic field:  Physics 
Specialties: 

Approach:  Theoretical 
Abstract
We investigate the behavior of higherform symmetries at various quantum phase transitions. We consider discrete 1form symmetries, which can be either part of the generalized concept ``categorical symmetry" (labelled as $\tilde{Z}_N^{(1)}$) introduced recently, or an explicit $Z_N^{(1)}$ 1form 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 $Z_N^{(1)}$ 1form 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 microscopicindependent 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 manybody states with different topological nature.
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Reports on this Submission
Anonymous Report 2 on 202145 (Invited Report)
 Cite as: Anonymous, Report on arXiv:scipost_202102_00023v1, delivered 20210405, doi: 10.21468/SciPost.Report.2754
Strengths
The authors identify the universal scaling behavior of the disorder operator in various quantum phase transitions in 2+1d.
Weaknesses
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 $\mathbb{Z}_N$ oneform or zeroform 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.
Report
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 twopoint function. This is an interesting result that deserves to be published.
Requested changes
1. The authors should clarify the role played by the $\mathbb{Z}_N$ zeroform and oneform 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 $\mathbb{Z}_N$ zeroform or oneform 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.
Anonymous Report 1 on 2021323 (Invited Report)
 Cite as: Anonymous, Report on arXiv:scipost_202102_00023v1, delivered 20210323, doi: 10.21468/SciPost.Report.2729
Strengths
1. interesting and timely subject
2. clear presentation of results
Weaknesses
1. some discussion is a bit terse and could be expanded
Report
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 welldefined charges under these other "symmetries". Is it clear this is a welldefined 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 followup) 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.