  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.
