## SciPost Submission Page

# Foliated Field Theory and String-Membrane-Net Condensation Picture of Fracton Order

### by Kevin Slagle, David Aasen, Dominic Williamson

#### - Published as SciPost Phys. 6, 043 (2019)

### Submission summary

As Contributors: | David Aasen · Kevin Slagle |

Arxiv Link: | https://arxiv.org/abs/1812.01613v3 (pdf) |

Date accepted: | 2019-04-08 |

Date submitted: | 2019-03-21 |

Submitted by: | Slagle, Kevin |

Submitted to: | SciPost Physics |

Discipline: | Physics |

Subject area: | Condensed Matter Physics - Theory |

Approach: | Theoretical |

### Abstract

Foliated fracton order is a qualitatively new kind of phase of matter. It is similar to topological order, but with the fundamental difference that a layered structure, referred to as a foliation, plays an essential role and determines the mobility restrictions of the topological excitations. In this work, we introduce a new kind of field theory to describe these phases: a foliated field theory. We also introduce a new lattice model and string-membrane-net condensation picture of these phases, which is analogous to the string-net condensation picture of topological order.

### Ontology / Topics

See full Ontology or Topics database.Published as SciPost Phys. 6, 043 (2019)

### Author comments upon resubmission

Thank you for considering and carefully reviewing our work and for the helpful comments and suggestions. Below are detailed responses to the referee reports.

Best regards,

Kevin Slagle

David Aasen

Dominic Williamson

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Response to both Referees:

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-- Weakness 1 from both Referees:

Both referees noted that our work does not use our formalism to study physics beyond what has already been currently studied. (Although we do not emphasize it, Appendix A actually does include new models.) Indeed, this was a deliberate choice. We think our current work already introduces a lot of new ideas with a focused theme and that including further generalizations would obfuscate the message of this work, along with lengthening the manuscript and delaying publication. We are currently working on exploring generalizations in a forthcoming work.

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Response to Referee 1:

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-- Weakness 2, Report, and Requested change 1:

Indeed, the gauge transformation is rather exotic. We now make note of this and point out that it is due to the third term in Eq. 3, which couples the b and A fields together. We also added footnote 6 to make more connections.

We prefer to think of A^k and a (and B^k and b) as separate gauge fields, but either viewpoint may be fine. Even in nonabelian BF theory where we treat A as a nonabelian gauge field, when A transforms we must also transform B; but we do not interpret A and B as the same gauge field.

That is an interesting observation that trivial (i.e. closed lambda) gauge transformations on b can be absorbed into the chi guage transformation on B. The consequence of this fact is that these trivial gauge transformations do not impose any additional mobility constraints on the currents beyond the constraint implied by the chi transformation. This can be seen by taking the divergence (\partial_mu) of both sides of the i^{nu,mu} current constraint (middle equality in Eq. 8) and noting that the left-hand side is zero due to the antisymmetry of i^{nu,mu} and the right-hand side is zero due to the constraint imposed by the chi transformation (left equality in Eq. 8).

-- Requested change 2:

We added a paragraph at the end of Section 2.1.1 and a sentence to Section 3.4.1 to discuss the creation of four fractons.

-- Requested change 3:

We note in Section 4.1.1 that the line integral of e^k is a dimensionless number (when e^k is dimensionless, which we think is the only sensible choice). If one considers the line integral along a non-contractable loop, then it is tempting to interpret that as an integer number of layers along a periodic direction. As such, it is feasible that imposing a cutoff may not be necessary. We modified Section 4.1.1 to try to make this more clear.

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Response to Referee 2:

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-- Report:

We only mentioned p-strings to note a connection to previous work. We moved this comment to a footnote. Thank you for mentioning the Z_N qubit and "equal+weight" typos; we have fixed them.

-- Requested change 1:

We removed the comment of large gauge transformations since it is indeed rather speculative.

-- Requested change 2:

We removed the mention of soft modes since we're only trying to make a connection to the large ground state degeneracy in linearized gravity that was argued to exist in Ref 90.

Although a finite-energy photon has a finite probability of being detected by a finite-sized detector during a finite amount of time, this probability approaches zero as the energy of the photon approaches zero. Thus, low-energy photon states can be arbitrarily locally indistinguishable in the low-energy limit. (Ref 90 also argues that U(1) gauge theory has a large ground state degeneracy.)

-- Communication via editor:

The referee asked the following question via a communication with the editor:

"In their lagrangian (3), why do they not consider an additional coupling of the form ∑kek∧b∧Bk ? I suppose that, if its coefficient is properly quantized, it can be removed by a rotation between Ak and Bk, but this acts on the charge lattice of the theory on the leaves. I think this merits some comment."

This kind of term is briefly mentioned in Eq. 38 as a possible future direction. Simply adding it to our Lagrangian (Eq. 3) could drastically change the theory since it is not invariant under the lambda gauge transformation in Eq. 6. We did not consider it for this reason and because it was not motivated by the scope of this work.

### List of changes

1) We added a summary of the string-membrane-net model in the introduction.

2) We elaborate on the gauge transformation after Eq. 6.

3) We reordered the equalities in Eq. 6-8 and 12 to make the ordering more intuitively simple.

4) We added discussion of the 4-fold fracton creation at the end of Section 2.1.1 and 3.4.1.

5) We moved the p-string discussion at the beginning of Section 3.2 to a footnote.

6) We clarified the cutoff/quantization issues in 4.1.1.

7) We defined k in Eq. 76 more explicitly.

8) We removed discussion of soft modes at the end of Appendix C.

A detailed markup of the changes can be found at:

https://drive.google.com/file/d/13HtnUac8-dna3EG8CTwv4Kr3nu961QpC