# Systematic Constructions of Fracton Theories

### Submission summary

 As Contributors: Djordje Radicevic Preprint link: scipost_201912_00043v6 Date submitted: 2020-06-27 02:00 Submitted by: Radicevic, Djordje Submitted to: SciPost Physics Discipline: Physics Subject area: Condensed Matter Physics - Theory Approach: Theoretical

### Abstract

Fracton theories possess exponentially degenerate ground states, excitations with restricted mobility, and nontopological higher-form symmetries. This paper shows that such theories can be defined on arbitrary spatial lattices in three dimensions. The key element of this construction is a generalization of higher-form gauge theories to so-called $\mathfrak{F}_p$ gauge theories, in which gauge transformations of rank-$k$ fields are specified by rank-$(k - p)$ gauge parameters. The $\mathbb{Z}_2$ rank-two theory of type $\mathfrak{F}_2$, placed on a cubic lattice and coupled to scalar matter, is shown to have a topological phase exactly dual to the well-known X-cube model. Generalizations of this example yield novel fracton theories. In the continuum, the U(1) rank-two theory of type $\mathfrak{F}_2$ is shown to have a perturbatively gapless fracton regime that cannot be consistently interpreted as a tensor gauge theory of any kind. The compact scalar fields that naturally couple to this $\mathfrak{F}_2$ theory also show gapless fracton behavior; on a cubic lattice they have a conserved U(1) charge and dipole moment, but these particular charges are not necessarily conserved on more general lattices. The construction generalizes to $\mathfrak{F}_2$ theories of nonabelian rank-two gauge fields, giving first examples of pure nonabelian higher-rank theories.

###### Current status:
Editor-in-charge assigned

### List of changes

Small changes and clarifications throughout the text.

### Submission & Refereeing History

Resubmission scipost_201912_00043v6 on 27 June 2020
Resubmission scipost_201912_00043v5 on 2 April 2020
Resubmission scipost_201912_00043v4 on 10 March 2020
Submission scipost_201912_00043v1 on 11 December 2019

## Reports on this Submission

### Report

It seems my previous report was mostly ignored by the author (with the exception that the word "straightforward" was removed from the abstract). So my report and conclusion remain mostly unchanged. But I will elaborate some more in this report. I still think this work deserves eventual publication, but there are some important validity and misrepresentation issues that should be fixed before publication.

Point 6 below is about a result in the paper that seems incorrect. 6b and 6d are pointing out claims made by the author that I think are misleading. 6b is especially important because it is about the last sentence of the abstract, which makes a very significant claim that is not substantiated. That sentence (or Sec 4.5) should be heavily edited. 6c and 6e are asking for clarification. Point 5 is an optional suggestion.

### Requested changes

5) I previously misread the text below Eq 55. I now understand the author's notation in Eq 75. However, instead of writing $\varphi(r+e_i)$, I would instead suggest writing $\varphi(r) + a \partial_i \varphi(r)$, with some dimensionful coupling constant '$a$'. I think this would help prevent confusion. (This is an optional suggestion.)

These points are all regarding Sec 4.5:

6) Since v5, the issue of gauge invariance is resolved by dropping half of the vertex operators and 2/3 of the belt operators. In this new model, I think there is a ground state degeneracy that grows exponentially with the volume of the system since there aren't enough operators. This is usually very bad because it almost always means that arbitrarily small perturbations to the Hamiltonian can result in an instability to almost any phase of matter (which occurs e.g. when the Hamiltonian is zero, H=0). So it seems like this model is incomplete: either more terms need to be added to split the extensive degeneracy, or somehow the degeneracy isn't as large as I expect. Alternatively, the author could point out this issue much more explicitly in the abstract and introduction, which would also address point 6b.

6b) The abstract claims that: "The construction generalizes to F2 theories of nonabelian rank-two gauge fields, giving first examples of pure nonabelian higher-rank theories." A similar statement was also made in the introduction. I think this claim is misleading since the generalized F2 theory in Sec 4.5 has serious issues, including point 6 above and the "patch" complication discussed below Eq 91 in order to achieve partial gauge invariance. As such, I would not call this a "nonabelian higher-rank theory". Furthermore, the discussion of the field theory is very brief, which makes it difficult to know if there are other issues with this field theory that I have not noticed yet. (Sec 4.5 has already evolved considerably throughout this review process to address a number of issues.) If these issues are not fixed, my suggestion is to remove the last sentence of the abstract and point out these issues when this section is mentioned in the main text.

6c) I think Eq 91 is missing the 'a' superscripts in $F_{\mu\nu}^a$.

6d) A new sentence was added below Eq 91:
"This situation is reminiscent of the original X-cube model, which features three different types of gauge transformation."
This seems misleading. In the X-cube model, all terms are gauge invariant everywhere. But in Eq 91, only some terms are gauge invariant under certain gauge transformations, which are supposed to reside in different "patches" of space. This is a complication that I've never seen considered before. I suggest removing this sentence.

6e) Could you please specify the belt operator sign convention on the left half of Fig 6?

• validity: ok
• significance: ok
• originality: ok
• clarity: good
• formatting: perfect
• grammar: perfect

### Report

The author has implemented most of the changes I had requested in my initial report, and to a large extent, has addressed the issues regarding the non-Abelian generalization raised by other referees as well. The current version of the manuscript is thus suitable for publication in SciPost.

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

### Report

In addition to the concerns from my original report, which were largely addressed in the previous resubmission, the manuscript has now adequately addressed all issues raised by the other referees, including concerns related to ground state degeneracy and notation. In light of the updates to the manuscript, I feel that this work is ready for publication in SciPost.

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