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Universal Features of Higher-Form Symmetries at Phase Transitions

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

Submission summary

As Contributors: Cenke Xu
Preprint link: scipost_202102_00023v3
Date accepted: 2021-08-11
Date submitted: 2021-07-17 19:19
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.

Published as SciPost Phys. 11, 033 (2021)

Author comments upon resubmission

Dear editor,

We want to thank the referee for all the suggestions. Here is our response to the new report:

1, We have added a new appendix to clarify all the notions used in the manuscript. We believe it should be clear now. In short, we listed two qualities that a standard “explicit symmetry” should satisfy. An “explicit symmetry” and the dual “inexplicit symmetry” used in this manuscript both satisfy quality (1), which corresponds to the conservation law of excitations, but the dual “inexplicit symmetry” does not satisfy quality (2). Both symmetries can be made explicit (meaning they both satisfy (1) and (2)) by embedding the system to the boundary of higher dimensional topological order (like Ref.16, now Ref.15). In this perspective the conservation laws associated with these two “symmetries” arise from the fusion rules of the anyons in the bulk. But we would like to emphasize that the quantities we are interested in can be defined and computed without the bulk.

2, Ref.16 (now Ref.15) first introduced a new phrase “patch operator” as a generalization of the disorder operator. Indeed, we also feel a new phrase is necessary for more general cases, because the phrase “disorder” operator implies that when it condenses, the original symmetry would be restored or the system should enter a disordered phase of the original symmetry. But in some cases that involve higher form symmetries both the symmetry and the dual symmetry can be spontaneously broken simultaneously, namely both the explicit symmetry and its dual inexplicit symmetry can enter the ordered phase simultaneously under proper generalization. But the phrase “patch operator” is also not satisfactory: it is not clear about the function and purpose of this quantity. And for situations that involve more exotic subsystem symmetries, the desired quantity is not defined on a simple patch of the system. Hence together with one of the authors of Ref.16 (now 15), we proposed the phrase “order diagnosis operator” in our previous work.

We would also like to mention that, overlap between different concepts/notions/phrases, though not ideal, is very common at an early stage of the research on a subject. For example, in the last 10 years the integer quantum Hall state was categorized as either “short range entangled state” (in the sense that it does not have topological entanglement entropy) or “long range entangled state” (in the sense that it cannot be connected to simple product state through finite depth quantum circuit) by different researchers, and later it was also categorized as “invertible topological order”. In more recent literature it seems researchers are converging to the phrase “invertible topological order”, as this phrase more accurately captures the essence of the concept. Also, the phrase “symmetry protected topological states” was also called “symmetry protected trivial states” in some literature (in fact both phrases were used by the same researcher). Time will tell what phrase is the most appropriate; and before that, different authors should be allowed to use the phrase that they consider most appropriate. We do believe the phrase “order diagnosis operator” properly captures the purpose of the series of desired quantities. And we believe the usage of nomenclature is self-consistent in the current version of our work.

3, we only learned about the reference mentioned by the referee (2102.06223) after our work was posted (the reference was posted one month after our work). Indeed, this reference made a general analysis of correlation functions integrated on an area, and a corner contribution is universal. We have added a footnote and cited the main result of this reference.

4, Regarding the question “Is it possible to understand the relation between the coefficient b and the conductivity in some more general way”, we think the procedure we used in the paper is already quite general: the computation of b is reduced to a density-density correlation, which in a system with Lorentz invariance in the IR is proportional to the current-current correlation, which is proportional to the conductivity.

5, in the previous version of our manuscript we actually have cited two references from the high energy physics literature that computed the cusp contribution to the Wilson loop of gauge field. We have now cited these two references in the introduction section, together with the review article mentioned by the referee.

6, We believe we did mention why we choose <(log O)^2> over log <O> in the previous version of our manuscript: for an interacting theory, log <O> can only be evaluated by expanding the quantity as a polynomial of fields, and the second order can be evaluated conveniently. So far we have not proven whether higher order in this expansions would change the corner dependence or not. The definition of <(log O)^2> was further explained in one of the footnotes.

7, the typos mentioned by the referee were corrected. And Ref.9 is removed.

List of changes

The main change is an appendix meant to clarify all the rudimentary notions used in this manuscript. Other changes are marked blue in the text.

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