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Composite subsystem symmetries and decoration of subdimensional excitations
by Avi Vadali, Zongyuan Wang, Arpit Dua, Wilbur Shirley, Xie Chen
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Authors (as registered SciPost users):  Xie Chen · Avi Vadali 
Submission information  

Preprint Link:  https://arxiv.org/abs/2312.04467v1 (pdf) 
Date submitted:  20240126 03:21 
Submitted by:  Vadali, Avi 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
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Approach:  Theoretical 
Abstract
Flux binding is a mechanism that is wellunderstood for global symmetries. Given two systems, each with a global symmetry, gauging the composite symmetry instead of individual symmetries corresponds to the condensation of the composite of gauge charges belonging to individually gauged theories and the binding of the gauge fluxes. The condensed composite charge is created by a "short" string given by the new minimal coupling corresponding to the composite symmetry. This paper studies what happens when combined subsystem symmetries are gauged, especially when the component charges and fluxes have different subdimensional mobilities. We investigate $3+1$D systems with planar symmetries where, for example, the planar symmetry of a planon charge is combined with one of the planar symmetries of a fracton charge. We propose the principle of $\textit{Remote Detectability}$ to determine how the fluxes bind and potentially change their mobility. This understanding can then be used to design fracton models with subdimensional excitations that are decorated with excitations having nontrivial statistics or nonabelian fusion rules.
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Reports on this Submission
Report #2 by Anonymous (Referee 2) on 2024430 (Invited Report)
 Cite as: Anonymous, Report on arXiv:2312.04467v1, delivered 20240430, doi: 10.21468/SciPost.Report.8956
Report
The paper investigates the gauging of composite subsystem symmetries, whose parent theories host charge excitations with distinct subdimensional mobilities, e.g., fractons and planons. Notably they find the new, composite fluxes depend on how the original symmetries are combined. The exposition is clear and pedagogical, walking the reader through their findings by contrasting two case examples (their Models FP1 and FP2). The topic is of great interest to researchers working in the topological order community, with special focus on fracton theories.
I am thus very pleased to recommend its publication in the present form. One point I think it could be minimally expanded is the extension to fractal symmetries commented in the conclusions. Maybe the authors could add one sentence or two pointing out possible challenges (if there are any) and how sensitive to details the new fluxes are. Finally, two minor textual details the authors may wish to consider are:
1 the acronym "RD" is already used in page 9, but only defined later (page 15 of main text);
2 in the opening sentence of Sec. IIIB there is a reference to a model "B", which may cause some confusion. Consider clarifying whether this refers to a distinct model or if it is a typo.
Recommendation
Publish (easily meets expectations and criteria for this Journal; among top 50%)
Report #1 by Anonymous (Referee 1) on 2024421 (Invited Report)
 Cite as: Anonymous, Report on arXiv:2312.04467v1, delivered 20240421, doi: 10.21468/SciPost.Report.8915
Report
This paper explores the gauge fluxes of gauge theories that arise from gauging different subsymmetries of a global symmetry (subsystem 1 symmetry) x (subsystem 2 symmetry) x.... In particular, they show how the gauge fluxes that arise from gauging the entire symmetry group are related to those that arise by gauging diagonal subgroups. The paper is well organized and interesting to, at least, both the fracton/topological order community as well as the generalized symmetries community. I am therefore happy to recommend it for publication. However, I hope the authors will consider some of my comments and questions below, which could clarify parts of their paper and provide interesting questions for additions to this paper (or followup work).
Three questions I have regarding gauging:
1) How do the authors expect different types of gauging to affect their results (i.e., various types of twisted gauging implemented by conjugating with an SPT entangler before and after gauging)?
2) I believe the authors always assume the symmetry before gauging is not spontaneously broken (e.g., the starting model is a paramagnet). My understanding of why is that by gauging a nonSSB'd subsystem symmetry, you get a model with a dual subsystem symmetry that is SSB'd, which explains why you have subdimensional gauge charges. Do the authors know how things would change if their starting point differed, especially if the starting model was in a nontrivial SPT phase protected by the symmetry?
3 Do the authors know how the 't Hooft anomalies of the 0form and subsystem symmetries affect their results? In particular, if the symmetry was anomalous and they gauged an anomalyfree subgroup, would they expect to get some sort of subsystem highergroupoid or subsytem noninvertible symmetry?
For 2+1d abelian topological orders, the principle of remote detectability can be nicely stated in terms of 't Hooft anomalies of higherform symmetries. Can the authors phrase their remote detectability principle using 't Hooft anomalies?
To improve clarify, could the authors distinguish between global and gauge symmetries, and global symmetry charges and gauge symmetry charges, more precisely in the introduction? I found the distinction clear in the main text and conclusion section, but I had difficulty following the terminology in the introduction. Relatedly, if the authors could clarify in the introduction what charges in which theory are being condensed (e.g., in the second paragraph), that would also make the introduction clearer.
Recommendation
Publish (easily meets expectations and criteria for this Journal; among top 50%)
Author: Avi Vadali on 20240729 [id 4664]
(in reply to Report 1 on 20240421)
We want to thank the referee for the helpful report. In response to the comments in the report, we have:

added the definition of RD (remote detection) on page 9 when it is first used

replaced "Model B" with "Model FP2", consistent with other parts of the paper

Regarding the extension to fractal symmetries, we have updated the last paragraph in the summary section as follows:
Throughout this work, we considered planar subsystem symmetries. An interesting future direction involves extending this paper's analysis to fractal symmetries and a combination of fractal and planar symmetries. A potential issue arises in obtaining geometrically local fracton models when gauging a combination of fractal and planar symmetries. In particular, it is not obvious how to choose composite subsystem symmetries such that the set of symmetric operators is generated by purely local terms. For all (subsystem) symmetries we consider in this paper, their symmetric operators can be locally generated. For example, in a 2D system with a global Z2 symmetry, the set of symmetric operators can be generated by an onsite transverse field term and a nearestneighbor Ising coupling term. The gauge field DOF's are then placed at the location of the Ising terms. If, on the other hand, the symmetric operators are not locally generated, the gauging process would yield a nonlocal model. For example, consider a 2D system with line symmetries in both the x and y directions and try gauging the composite symmetry of the tensor product of symmetries on row i and column i. In an NxN system, there are N symmetry generators. All twobody Ising coupling terms between sites (j,k) and (k,j) are now symmetric. However, since this set of terms cannot be locally generated, the gauged model would be nonlocal. On the other hand, gauging such composite symmetries can yield interesting klocal Hamiltonians corresponding to quantum lowdensity parity check (LDPC) codes. In general, obtaining LDPC codes via gauging can be insightful in understanding the associated code properties from the perspective of the gauging duality. In order to obtain an LDPC code Hamiltonian, we would want, in the ungauged model, each qubit and each relation to support and involve constant number of klocal terms respectively. Whether this is possible or not would depend on the map from initial to final ungauged symmetric terms obtained after combining the symmetries.
Author: Avi Vadali on 20240729 [id 4663]
(in reply to Report 2 on 20240430)We want to thank the referee for the interesting comments. Here are our responses:
The idea of `twisted gauging' is an interesting one which leads to interesting observations regarding 1+1D theories. However, in this paper, we mainly consider systems with planar symmetries, which upon gauging, turn into systems with longrange entanglement. Because of this, we cannot 'conjugate' the gauging process with an SPT entangler because the systems before and after the gauging have different (generalized) symmetries.
Our conclusion in this paper applies to trivial and symmetryprotected paramagnets alike. In particular, the mobility of the gauge charges does not depend on whether the starting point is a trivial or nontrivial SPT.
If we gauge a nonanomalous subgroup of an anomalous symmetry, a simple case is that the gauged theory breaks the remaining part of the anomalous symmetry. It is not obvious to us how groupoid or noninvertible symmetries can arise. That is definitely an interesting question.
The interpretation of remote detectability in terms of 't Hooft anomalies is an interesting point. The principle should be similarly applicable to subsystem symmetries, although we will need to make use of a more sophisticated version of field theory as discussed in for example arXiv:2004.00015, arXiv:2008.03852, arXiv:2112.05726.
Thanks for pointing out the ambiguity in our terminology. We have modified the introduction section to make clear the distinction between symmetry charges and gauge charges. In particular, in the second paragraph of the Introduction, we discuss the coupling of 2 planar Z2 gauge theories. Here we clarify that forming a composite symmetry between the two Z2 symmetry generators makes the composite of the two symmetry charges no longer a symmetry charge. Additionally, we clarify that the gauge charge pair is condensed after gauging the composite Z2 symmetry.