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Subsystem Symmetry Fractionalization and Foliated Field Theory
by PoShen Hsin, David T. Stephen, Arpit Dua, Dominic J. Williamson
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Authors (as registered SciPost users):  PoShen Hsin 
Submission information  

Preprint Link:  https://arxiv.org/abs/2403.09098v1 (pdf) 
Date submitted:  20240410 19:46 
Submitted by:  Hsin, PoShen 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
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Approach:  Theoretical 
Abstract
Topological quantum matter exhibits a range of exotic phenomena when enriched by subdimensional symmetries. This includes new features beyond those that appear in the conventional setting of global symmetry enrichment. A recently discovered example is a type of subsystem symmetry fractionalization that occurs through a different mechanism to global symmetry fractionalization. In this work we extend the study of subsystem symmetry fractionalization through new examples derived from the general principle of embedding subsystem symmetry into higherform symmetry. This leads to new types of symmetry fractionalization that are described by foliation dependent higherform symmetries. This leads to field theories and lattice models that support previously unseen anomalous subsystem symmetry fractionalization. Our work expands the range of exotic topological physics that is enabled by subsystem symmetry in field theory and on the lattice.
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In this paper, which appears to be a follow up of references [22] and [23], the authors study subsystem symmetries which naturally arise in systems with ordinary higherform symmetries. Their analysis occurs both on the lattice and in the continuum, and in the latter context they use a foliated field theory approach. The continuum perspective is one of the ways this paper distinguishes itself from its antecedents.
One of the main examples is a 2+1d topological order with 1form symmetry: the 1dimensional symmetry operators of such a system, when inserted on lines which foliate space, can be thought of as furnishing a linear subsystem symmetry. They analyze the possible patterns of fractionalization of the “global relation” enjoyed by models with subsystem symmetries. One of the reasons this is interesting is because the global relation appears to be one of the main features which distinguishes a nontrivial subsystemsymmetric phase from one which is obtained by trivially layering lowerdimensional systems. The continuum perspective taken in many parts of this paper will hopefully make fractons and fractonadjacent physics more accessible to those in the high energy community who think about generalized global symmetries. I recommend this paper for publication.
I offer a few comments and ask a few questions below.
The use of the terminology “subsystem oneform symmetry” (e.g. around equation 1.2) is somewhat confusing on first read. The authors seem to mean “0form subsystem symmetry which is a subgroup of an ordinary oneform symmetry” but the language used might leave readers confusing it with what is sometimes called a “oneform subsystem symmetry” which is, for example, the kind of symmetry that the Xcube model has. I think the authors can be more careful and consistent with their language throughout.
This paragraph is regarding the toric code example analyzed in Section 1.1.1. The standard gauging of the $\mathbb{Z}_2$ oneform symmetry of the 2+1d toric code looks very different from what is carried out in the text. For example, in the standard approach, the end result of the gauging is essentially the 2+1d transverse field Ising model on the dual lattice. The difference between these two gauging procedures appears to stem from the Gauss law terms used in Figure 1, which differ from the more conventional choice $\tilde{X}_p X_e \tilde{X}_{p’}$, where $e$ is an edge and $p$ and $p’$ are the two plaquettes which share $e$ on their boundaries. Can the authors comment on the relationship between their approach and the more standard approach? Perhaps the difference is due to the fact that the authors say they are gauging the “contractible part of the oneform symmetry”, but it is not entirely clear to me what gauging a contractible oneform symmetry means in general.
Some small typos/comments:
 Incomplete sentence on page 4: “Starting from a model where the higherform symmetry is only topological on the states without higherform charge.”
 Around equation 1.2, three sentences start with “For instance, …”
 “Systems with fully mobile excitations generically possess nform higher symmetries that act on the extended operators that create these excitations [10].” I don’t know that I’d use the word “generically” to describe systems with a symmetry. Perhaps “often”?
 There should not be a period in equation 2.10.
 On page 13: “subsystem system symmetries”
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This work explores enrichment of topological order with subsystem symmetries in various dimensions. This is done by embedding subsystem symmetries into higherform symmetries as rigid subgroups. The general constructions are demonstrated neatly in several examples with explicit Hamiltonians.
While the results are very interesting and are worth publishing in this journal, I have some concerns/questions listed below:
1. My main concern is in the section on "gauging contractible oneform symmetry". In the example discussed by the authors, the starting point is the 2+1d toric code with a vertex and a plaquette term, whereas the resulting model has only the plaquette term (on the dual lattice). While the 1form symmetry is topological in the new model, its spectrum contains a lot more ground states (because of the absence of vertex terms in the Hamiltonian) than the toric code. In other words, the new model is not described by the $\mathbb Z_2$ gauge theory at low energies. In what sense is this topological 1form symmetry of the new model related to the topological 1form symmetry in field theory?
2. I am puzzled by the operators in the lower part of fig 2. The untruncated versions of these operators are trivial because they can be written as product of Gauss law operators in fig 1, which are imposed exactly. The nontrivial 1form symmetry operators are still $\prod X$ over curves on dual lattice, which are now fully topological. So, it's not clear to me what the authors mean by "[the operators in the lower part of fig 2] satisfy a global relation" when they are in fact trivial.
3. Related to the point above, the authors say that the edgequbits can be removed to get the new model with only facequbits. In ordinary gauging, this works fine because the Gauss law involves only one $X$ so the edgequbits can all be gaugefixed. On the other hand, the Gauss laws in fig 1 involve two $X$'s each, so gaugefixing needs more care. In fact, on a finite lattice with periodic b.c., I do not believe they can all be gaugedfixed because of the nontrivial 1form symmetry operators mentioned in the previous point. This should be clarified.
While the above concerns should be addressed, I believe the main results of the paper are not significantly affected by them. There are also some minor issues listed below:
4. On page 7, I think $(n1)$ and $n$ should be exchanged in the sentence: "Furthermore, since the fully topological $(n1)$form symmetry on any contractible submanifolds $\delta$ is trivial, the new $n$form symmetry on such a submanifold $\Sigma$ does not depend on the foliation."
5. Below eq (2.4), the authors write: "The quotient is the product of contractible oneform symmetries." It is not clear to me why this is the case. For example, I don't think the line operator $Q^x$ at a fixed $x$ is contractible. Can the authors please explain?
6. In eq (2.6), why does $\lambda^k$ not depend on $x^k$? I would assume that the gauge fields $A^k$ at different layers labelled by $x^k$ have their own gauge parameters, so I would expect that $\lambda^k$ depends on $x^k$ too.
7. There are several typos: For example, in eqs (6.19)(6.21), the superscript $a$ on $B_p$ is missing. Also, in the first paragraph of sec 5.4, there is a broken sentence.
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