Maxwell theory of fractons

Abstract

We show that the main properties of the fracton quasiparticles can be derived from a generalized covariant Maxwell-like action. Starting from a rank-2 symmetric tensor field ${A}_{\ensuremath{\mu}\ensuremath{\nu}}(x)$, we build a partially symmetric rank-3 tensor field strength ${F}_{\ensuremath{\mu}\ensuremath{\nu}\ensuremath{\rho}}(x)$ which obeys a kind of Bianchi identity. The most general action invariant under the covariant ``fracton'' transformation ${\ensuremath{\delta}}_{\text{fract}}{A}_{\ensuremath{\mu}\ensuremath{\nu}}(x)={\ensuremath{\partial}}_{\ensuremath{\mu}}{\ensuremath{\partial}}_{\ensuremath{\nu}}\mathrm{\ensuremath{\Lambda}}(x)$ consists of two independent terms: one describing linearized gravity (LG) and the other referable to fractons. The whole action can be written in terms of ${F}_{\ensuremath{\mu}\ensuremath{\nu}\ensuremath{\rho}}(x)$, and the fracton part of the invariant Lagrangian writes as ${F}^{2}(x)$, in analogy with Maxwell theory. The canonical momentum derived from the fracton Lagrangian coincides with the tensor electric field appearing in the fracton literature, and the field equations of motion, which have the same form as the covariant Maxwell equations (${\ensuremath{\partial}}^{\ensuremath{\mu}}{F}_{\ensuremath{\alpha}\ensuremath{\beta}\ensuremath{\mu}}(x)=0$), can be written in terms of the generalized electric and magnetic fields and yield two of the four Maxwell equations (generalized electric Gauss and Amp\`ere laws), while the other two (generalized magnetic Gauss and Faraday laws) are consequences of the ``Bianchi identity'' for the tensor ${F}_{\ensuremath{\mu}\ensuremath{\nu}\ensuremath{\rho}}(x)$, as in Maxwell theory. In the covariant generalization of the fracton theory, the equations describing the fracton limited mobility, i.e., the charge and dipole conservation, are not external constraints, but rather consequences of the field equations of motion, hence of the invariant action and, ultimately, of the fracton covariant symmetry. Finally, we increase the known analogies between LG and fracton theory by noting that both satisfy the generalized Gauss constraint which underlies the limited mobility property, which one would not expect in LG.