Quantum Brownian motion in a periodic potential

Abstract

We study the statics and dynamics of a quantum Brownian particle moving in a periodic potential and coupled to a dissipative environment in a way which reduces to a Langevin equation with linear friction in the classical limit. At zero temperature there is a transition from an extended to a localized ground state as the dimensionless friction \ensuremath{\alpha} is raised through one. The scaling equations are derived by applying a perturbative renormalization group to the system's partition function. The dynamics is studied using Feynman's influence-functional theory. We compute directly the nonlinear mobility of the Brownian particle in the weak-corrugation limit, for arbitrary temperature. The linear mobility ${\ensuremath{\mu}}_{l}$ is always larger than the corresponding classical mobility which follows from the Langevin equation. In the localized regime \ensuremath{\alpha}>1, ${\ensuremath{\mu}}_{l}$ is an increasing function of temperature, consistent with transport via a thermally assisted hopping mechanism. For \ensuremath{\alpha}1, ${\ensuremath{\mu}}_{l}$(T) shows a nonmonotonic dependence on T with a minimum at a temperature ${T}^{\mathrm{*}}$. This is due to a crossover between quantum tunneling below ${T}^{\mathrm{*}}$ and thermally assisted hopping above ${T}^{\mathrm{*}}$. For low friction the crossover occurs when the particle's thermal de Broglie wavelength is roughly equal to the distance between minima in the periodic potential. We suggest that the regime \ensuremath{\alpha}1 describes the physics of the observed nonmonotonic temperature dependence of muon diffusion in metals.