  A recent experiment [Nature Physics 10, 1 (2019)] has realized a dynamical
gauge system with $\mathbb{Z}_2$ gauge symmetry in a double-well potential. In
this work we propose a method to generalize this model from a single double
well to a one-dimensional chain. We show that although there is no disordered
potential in the original model, the phenomenon of many-body localization can
occur. The key ingredient is that different symmetry sectors with different
local gauge charges play the role of different disorder configurations, which
becomes clear after exactly mapping our model to a transverse Ising model in a
random longitudinal field. We show that both the ergodic regime and the
many-body localized regime exist in this model from four different metrics,
which include level statistics, volume law versus area law of entanglement
entropy of eigenstates, quench dynamics of entanglement entropy and physical
observables.
