Fractons from Vector Gauge Theory

Motivated by the prediction of fractonic topological defects in a quantum crystal, we utilize a reformulated elasticity duality to derive a description of a fracton phase in terms of coupled vector U(1) gauge theories. The fracton order and restricted mobility emerge as a result of an unusual Gauss law where electric field lines of one gauge field act as sources of charge for others. At low energies this vector gauge theory reduces to the previously studied fractonic symmetric tensor gauge theory. We construct the corresponding lattice model and a number of generalizations, which realize fracton phases via a condensation of stringlike excitations built out of charged particles, analogous to the $p$-string condensation mechanism of the gapped $X$-cube fracton phase.

Figure 1

Geometry of the lattice coupled vector gauge theory Hamiltonian Eq. (12). Gray solid lines represent the underlying square lattice. On the left, a single plaquette is shown for the $x$ lattice (dashed red lines) and the $y$ lattice (dotted blue lines). On the right, the thick solid line represents a loop of $\mathbf{e}$ electric field on a plaquette of the underlying lattice, with the arrows indicating the direction of $\mathbf{e}$. Because the $e_k$ electric field lines carry gauge charge of the ${\mathbf{E}}_k$ electric field as expressed by the Gauss law Eq. (2), there are also necessarily lines of ${\mathbf{E}}_x$ (dotted blue line with arrow) and ${\mathbf{E}}_y$ (dashed red line with arrow) electric field.