Fracton magnetohydrodynamics

We thank the referees for their careful reading and detailed feedback on our manuscript, and the editor for arranging this review of our manuscript. We have provided a point-by-point response to each of the referees on the submission page. A detailed list of the corresponding changes made to the manuscript in response to the referees' feedback is given below. We hope that these revisions have satisfied the referees' concerns and that the current manuscript will be judged suitable for publication in SciPost Physics.

In response to Report 1:

1) the first paragraph of the introduction has been modified to clarify our use of the term "hydrodynamics"

> "the universal properties of these theories can be characterized within the framework of hydrodynamics, which is the coarse-grained effective theory of the long-time and long-wavelength dynamics of systems as they relax to equilibrium"

2) Additionally, we have added a footnote after the sentence ending "...these long-lived modes are associated with conserved densities (or Goldstone bosons), and their dynamics is dubbed "hydrodynamics""

> "Note that our use of the term "hydrodynamics"—the coarse-grained description of systems as they relax to equilibrium—does not require that momentum be conserved, and need not correspond to the Navier-Stokes equations, for example."

3) Above Eq. (3.6), we have explained that the current tensor and the electric field tensor being proportional to one another is, in fact, the most general form of Ohm's law permitted by symmetry, if the electric field tensor transforms in the 5 of SO(3).

4) Similarly, below Eq. (4.7), we have made analogous arguments for the rank-n case, where the electric field tensor is a symmetric, traceless, rank-n quantity.

5) For the traceful scalar charge theory in Sec. 5, Eqs. (5.5) through (5.7) have been modified to include the two time scales that are in general permitted by SO(3) rotational invariance, since the trace part and the traceless symmetric part can decay on different time scales.

6) We have added the following sentence below Eq. (3.10) to remind the reader that the vanishing of magnetic charge is only ever an approximation in emergent theories:

> "We remind the reader that the vanishing of magnetic charge density can at best only be expected to hold approximately in emergent theories; see Sec. 3.2.3 for a discussion of the length and time scales over which (3.10) provides an accurate description of the dynamics."

7) Below Eq. (3.13) we have added the following sentences to clarify that the structure of time-independent gauge transformations and the conserved moments of the magnetic field tensor coincide only in the case of self-dual theories:

> "It is worth noting that (3.13) coincides with the structure of time-independent gauge transformations acting on the vector potential $A_{ij}$, canonically conjugate to $E_{ij}$. This apparent equivalence derives from the self-dual nature of the traceless scalar charge theory—i.e., the derivative and tensor structure of the electric and magnetic Gauss’s laws is identical."

8) The first sentence of Sec. 2.2.5 has been modified in order to define the "magnetic conductivity".

9) The tilde has been removed in (2.21) and in the subsequent paragraph.

10) The sentence in the introduction that states the result for the diffusion constant for rank-n theories has been modified to clarify that the result applies only to rank-n traceless symmetric theories.

In response to Report 2:

11) The abstract has been modified to better represent the utility of a symmetry-based approach:

> "In contrast to semi-microscopic derivations of MHD, our approach elucidates the origin of the hydrodynamic modes by identifying the corresponding higher-form symmetries. Being rooted in symmetries, the hydrodynamic modes may persist even when the semi-microscopic equations no longer provide an accurate description of the system."

12) Similarly, in the introduction, we have emphasized that the main advantage of the "symmetry-based" approach is primarily conceptual in nature:

> "Somewhat surprisingly, a first-principles derivation of magnetohydrodynamics using one-form symmetries was not done until the past decade..."

and

> "This approach, based on higher-form symmetries, has significant conceptual advantages over more familiar semi-microscopic derivations of MHD. Specifically, the symmetry-based approach highlights the underlying symmetries responsible for the observed long-wavelength modes, while also being less limited in its regime of validity than the semi-microscopic approach. For example, in conventional (rank-one) MHD, the semi-microscopic derivation invokes approximate separability of the electromagnetic and matter stress tensors. In the symmetry-based approach [48], one invokes hydrodynamic principles to recover a coarse-grained theory of the long-time and long-wavelength dynamics of the fields in the most interesting and physically relevant regimes, where there may not be a clean separation between the two tensors. This approach also gives predictions for particular limits of conventional U(1) spin-liquids in which the relevant symmetries are weakly broken. In the case of emergent electromagnetism in fractonic spin liquids, the emergent gauge fields are higher rank, leading to additional subtleties and new universality classes. The hydrodynamic description of these higher-rank theories is the subject of this work."

13) In the conclusion,

> "The theories we present describe the generic long-time description of the quantum dynamics of fractonic phases (at nonzero charge density) exhibiting gauged—as opposed to global—multipolar symmetries (as relevant to spin liquids and fracton phases). We develop a symmetry-based approach describing arbitrary higher-rank theories of this type, and showcase the subdiffusion of magnetic fields as an example of the exotic universal dynamics that may arise in this context."

14) At the top of page 9, we have reminded the reader that SO(3) rotational invariance is assumed throughout the manuscript.

15) Below (2.29), we have explained that spatial inversion can alternatively be used to forbid the term proportional to the vector density, i.e., to require $\alpha = 0$.

In response to Report 3:

[See points 1) and 2) in response to Report 1, which address the manner in which we use the term "hydrodynamics"]

16) To highlight the physical systems that our results apply to, at the start of each section we have made more prominent the citations that discuss concrete examples of lattice models giving rise to emergent electromagnetism of the corresponding rank.

17) Since there are few examples in the literature, we have added a new subsection [Section 4.2] in which we illustrate an emergent rank-three variant of QED in a lattice O(2) quantum rotor system.

18) We have also included an additional figure (Fig. 2) to accompany the introduction of the rank-three lattice model.

[See points 3), 4), 5) in response to Report 1, which employ a symmetry-based analysis to constrain the form of Ohm's law]