Phonon or electron friction of a quantum heavy particle

Abstract

In this paper we analyse, with the path integral method, the diffusion of a quantum heavy particle moving in a strongly corrugated periodic potential both in the case when the particle is interacting with a thermal bath of phonons or of electrons. In the first case, the integration over the phonon degrees of freedom is performed exactly and in the large mass limit of the heavy particle it gives rise to an ohmic effective action which includes a nonlocal self-interacting term whose strength is the classical friction coefficient. In the second case, the integration over the electronic degrees of freedom is more difficult; we are able to derive an approximate effective action for the heavy particle in two different limiting cases: i) arbitrary large coupling between heavy particle and electrons and linear dissipation; ii) weak coupling and nonlinear dissipation. In i) we obtain an effective action for the particle equal to that found for the phonons but with a friction coefficient given by that of a classical heavy particle in a fermionic bath. In ii) we obtain a nonlinear, but still ohmic, dissipative term. Using an instanton approach we evaluate the mobility (and the diffusion coefficient) of the particle, whose temperature dependence shows a crossover from diffusive to localized behaviour at a critical value of the friction. Finally we discuss whether the electronic and phononic frictions can reach such a critical value.