Ergodicity and localization in Zn lattice Schwinger model

In this work we analyze the lattice version of the Schwinger model, i.e. the quantum electrodynamics in 1 + 1 dimensions that describes the interactions between fermions via a U(1) gauge field. For discretizing the U(1) gauge field, we employ a recently developed approach that consists in replacing the U(1) group with the finite Zn group. In particular, we focus on the ergodic and localization properties of the Z2 and Z3 cases. We show that the fermionic degrees of freedom can be integrated out from the Hamiltonian by using Gauss law obtaining an effective model that depends only on the gauge fields. This allows us to study analytically and numerically such model. We find that the Z2 Schwinger model can be mapped to a quadratic fermionic Hamiltonian and thus it can be diagonalized by a Bogoliubov rotation, thus exhibiting property of integrability. On the contrary, no such mapping for the model with Z3 symmetry exists, so it has been analyzed numerically. Through the exact diagonalization of the Hamiltonian we compute the energy spectrum, the spacing of the energy levels, the electric field and the inverse participation ratio for each energy level and we find that the phase diagram is characterized by a localized and a delocalized region. Finally, to further investigate the localization properties of the Z3 model, we introduce a random coupling in the Hamiltonian hopping terms. For this case, we analyze how the disorder component modifies the energy level statistics and we show that the localized and delocalized regions behave accordingly to the Poisson and the Wigner-Dyson distributions, respectively. These features can be relevant in future experiments of quantum simulations of gauge theories using ultra-cold atoms.