Fractons from covariant higher-rank three-dimensional BF theory

In this paper we study the three-dimensional (3D) gauge theory of two tensor gauge fields: ${a}_{\ensuremath{\mu}\ensuremath{\nu}}(x)$, which we take symmetric, and ${B}_{\ensuremath{\mu}\ensuremath{\nu}}(x)$, 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 behavior 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 Kitaev's toric code. In the last part of the paper we analyze the case in which the field ${B}_{\ensuremath{\mu}\ensuremath{\nu}}(x)$ 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.