Rabi lattice models with discrete gauge symmetry: Phase diagram and implementation in trapped-ion quantum simulators

We study a spin-boson chain that exhibits a local Z(2) symmetry. We investigate the quantum phase diagram of the model by means of perturbation theory, mean-field theory, and the density matrix renormalization group method. Our calculations show the existence of a first-order phase transition in the region where the boson quantum dynamics is slow compared to the spin-spin interactions. Our model can be implemented with trapped-ion quantum simulators, leading to a realization of minimal models showing local gauge invariance and first-order phase transitions.