Introducing interactions in quantum dimer, vertex and loop models at the RK point

We present a generalized class of Rokhsar-Kivelson (RK) Hamiltonians that are in one-to-one correspondence with generic stochastic classical systems described by a Master equation in matrix form. We show that the ground state of the quantum system can be computed exactly and the zero-temperature phase diagram is captured by the phase diagram of the corresponding classical system at equilibrium. Moreover, the excitation spectrum over the ground state is given by the relaxation rates of the Master equation for the classical system. We then show how this framework allows to introduce interactions in known systems at the RK point, such as quantum dimer and vertex models, and to study phase transitions along lines of RK points. We also illustrate how one can construct exotic quantum Hamiltonians that are incapable of equilibrating to their ground state when coupled to a local thermal bath, and that exhibit relaxation time scales characteristic of a quantum glass.