  We calculate the properties of a 1D $Z_2$ lattice gauge theory in different
Gauss law sectors, corresponding to different configurations of static charges
set by the orientations of the gauge spins. Importantly, in quantum simulator
experiments these sectors can be accessed without adding any additional
physical particles or changing the Hamiltonian: The Gauss law sectors are
simply set by the initial conditions. We study the interplay between
conservation laws and interactions when the static charges are chosen to form
periodic patterns. We classify the different Gauss law sectors and use the
density matrix renormalization group to calculate the ground state
compressibility, density profiles, charge density wave order parameters, and
single particle correlation functions as a function of matter density. We find
confined and deconfined phases, charge density waves, correlated insulators,
and supersolids.
