Fractons from covariant higher-rank 3D BF theory

TL;DR

This paper introduces a covariant higher-rank 3D BF theory with tensor gauge fields, revealing fracton behavior, mapping to the Rank-2 Toric Code, and generalizing 3D BF theory to higher ranks with connections to quantum spin Hall insulators.

Contribution

It develops a covariant higher-rank 3D BF-like model, establishes its relation to the Rank-2 Toric Code, and extends the theory to include symmetric tensors and higher-rank Chern-Simons actions.

Findings

Fracton behavior emerges with matter coupling.

The theory maps to the low-energy R2TC.

Generalization to symmetric tensors yields higher-rank Chern-Simons actions.

Abstract

In this paper we study the 3D gauge theory of two tensor gauge fields: $a_{\mu\nu}(x)$, which we take symmetric, and $B_{\mu\nu}(x)$, with no symmetry on its indices. The corresponding invariant action is a higher-rank BF-like model, which is first considered from a purely field theoretical point of view, and the propagators with their poles and the degrees of freedom are studied. Once matter is introduced, a fracton behaviour naturally emerges. We show that our theory can be mapped to the low-energy effective field theory describing the Rank-2 Toric Code (R2TC). This relation between our covariant BF-like theory and the R2TC is a higher-rank generalization of the equivalence between the ordinary 3D BF theory and the Kitaev's Toric Code. In the last part of the paper we analyze the case in which the field $B_{\mu\nu}(x)$ is a symmetric tensor. It turns out that the obtained BF-like action can be cast into the sum of two rank-2 Chern-Simons actions, thus generalizing the ordinary abelian case. Therefore, this represents a higher-rank generalization of the ordinary 3D BF theory, which well describes the low-energy physics of quantum spin Hall insulators in two spatial dimensions.