Symmetry breaking and localization in a random Schwinger model with commensuration

Abstract

We numerically investigate a lattice regularized version of quantum electrodynamics in one spatial dimension (Schwinger model). We work at a density where lattice commensuration effects are important, and preclude analytic solution of the problem by bosonization. We therefore numerically investigate the interplay of confinement, lattice commensuration, and disorder, in the form of a random chemical potential. We begin by pointing out that the ground state at commensurate filling spontaneously breaks the translational symmetry of the lattice. This feature is absent in the conventional lattice regularization, which breaks the relevant symmetry explicitly, but is present in an alternative (symmetric) regularization that we introduce. Remarkably, the spontaneous symmetry breaking survives the addition of a random chemical potential (which explicitly breaks the relevant symmetry) in apparent contradiction of the Imry-Ma theorem, which forbids symmetry breaking in one dimension with this kind of disorder. We identify the long range interaction as the key ingredient enabling the system to evade Imry-Ma constraints. We examine spatially resolved energy level statistics for the disordered system, and demonstrate that the low energy Hilbert space exhibits ergodicity breaking, with level statistics that fail to follow random matrix theory. A careful examination of the structure of low lying excited states reveals that disorder induced localization is responsible for the deviations from random matrix theory, and further reveals that the elementary excitations are charge neutral, and therefore not long range interacting.