# Fracton magnetohydrodynamics

### Submission summary

 As Contributors: Oliver Hart · Rahul Nandkishore Arxiv Link: https://arxiv.org/abs/2205.05695v1 (pdf) Date submitted: 2022-05-16 20:56 Submitted by: Hart, Oliver Submitted to: SciPost Physics Academic field: Physics Specialties: Condensed Matter Physics - Theory Approach: Theoretical

### Abstract

We extend recent work on hydrodynamics with global multipolar symmetries -- known as "fracton hydrodynamics" -- to systems in which the multipolar symmetries are gauged. We refer to the latter as "fracton magnetohydrodynamics", in analogy to magnetohydrodynamics (MHD), which governs systems with gauged charge conservation. Fracton MHD arises naturally from higher-rank Maxwell's equations and in systems with one-form symmetries obeying certain constraints; while we focus on "minimal" higher-rank generalizations of MHD that realize diffusion, our methods may also be used to identify other, more exotic hydrodynamic theories (e.g., with magnetic subdiffusion). In contrast to phenomenological MHD, our approach -- based on higher-form symmetries -- provides a systematic treatment of hydrodynamics beyond "weak coupling", and provides an effective hydrodynamic description, e.g., of quantum spin liquids, whose emergent gauge fields are strongly coupled.

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

### Submission & Refereeing History

Submission 2205.05695v1 on 16 May 2022

## Reports on this Submission

### Strengths

The paper is detailed and easy to follow.
Research topics are exciting and timely.

### Weaknesses

No proposed realistic systems realize the interesting physics discussed in the manuscript.

Section 2 and the results of the matter-free limit in other sections are known (some of them are presented in Ref[32]).

### Report

The authors of the manuscript offered the hydrodynamics description of the systems with higher rank gauge symmetry. The results are interesting and timely, given that fracton systems' dynamics have recently attracted attention from high-energy and condensed matter communities. The paper is scientifically sound and well written.

In conclusion, I recommend the paper for publication after the authors clarify my following questions and concerns

1) Could the author explicitly define the hydrodynamics regime for each case? What is the limit where the hydrodynamics formalism is valid? As my understanding, in graphene, the hydrodynamics regime is where the conserved momentum scattering dominates the momentum relaxation scattering.

2) From the divergent free condition of the magnetic field $\partial_i B_i=0$, by the Gauss theorem, the total magnetic flux through the boundary of any volume vanishes identically rather than conserved as in equation (2.13). Is this correct?

3) In the symmetric rank 2, the “electric conductivity” generally is a 4-indices tensor. What are the argument of the relation $J^{(e)}_{ij}=-\frac{1}{\tau}E_{ij}$ used in equation (3.6) and (5.5). Why the Ohmic conductivities of scalar charge and vector charge are the same? What are the assumed scattering mechanism that lead to this conclusion? Is the proposed “electric conductivity” unique with the assumption of rotational symmetry?

4) In the discussion of one-form symmetries for the conserved magnetic field in equations (3.10) and (5.8), the assumption that there is no magnetic charge was used implicitly. What is the argument for this assumption?

5) It looks like the energy of the photon in the matter-free limit for the traceless symmetric rank $n$ is given by $\omega=\frac{c k_z m}{n}$, is this correct?

6) The explicit forms of functions $f_i$, $f_{ij}$ in sections 2,3,5 remind me the gauge transformations of gauge fields $A_i$ and $A_{ij}$ with time independent gauge transformation. $A_i \to A_i + \partial_i \varphi$ for the usual $U(1)$ gauge theory, $A_{ij} \to A_{ij}+\partial_{i}\partial_j \Phi -\frac{1}{3} \delta_{ij} \partial^2 \Phi$ for scalar charge traceless symmetric tensor gauge theory, and $A_{ij} \to A_{ij} + \partial_i \Psi_j + \partial_j \Psi_i$ for vector charge tensor gauge theory. Is this just a coincident or they are related?

Minors:

1)The magnetic conductivity $\sigma_m$ wasn’t defined before used in equation (2.16)
2) Are $\tilde{J}_k$ in (2.21) and $J_k$ in (2.22) the same?
3) The diffusion constant at rank $n$ given by $D_m=\tau c^2 m^2/n^2$ only works for the traceless symmetric tensor case. It should be clarified explicitly in the introduction section.

### Requested changes

See the report

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