Hollow Lattice Tensor Gauge Theories with Bosonic Matter

Abstract: Higher rank gauge theories are generalizations of electromagnetism where, in addition to overall charge conservation, there is also conservation of higher rank multipoles such as the total dipole moment. In this work we study a four dimensional lattice tensor gauge theory coupled to bosonic matter which has second rank tensor electric and magnetic fields and charge conservation on individual planes. Starting from the Hamiltonian, we derive the lattice action for the gauge fields coupled to $q=1,2$ charged scalars. We use the action formulation to carry out Monte Carlo simulations to map the phase diagram as a function of the gauge ($\beta$) and matter ($\kappa$) couplings. We compute the nature of correlators at strong and weak coupling in the pure gauge theory and compare the results to numerical simulations. Simulations show that the naive weak coupling regime (small $\kappa$, large $\beta$) does not survive in the thermodynamic limit. Instead, the strong coupling confined phase, spans the whole phase diagram. It is a proliferation of instantons that destroys the weak coupling phase and we show, via a duality transformation, that the expected strong confinement is present in the analog of Wilson line correlators. For finite matter coupling at $q=1$ we find a single thermodynamic phase albeit with a first order phase transition terminating in a critical endpoint.For $q=2$ it is known that the the X-cube model with $\mathbb{Z}_2$ fractonic topological order is recovered deep in the Higgs regime. The simulations indeed reveal a distinct Higgs phase in this case.