Quantum Brownian motion in a periodic potential

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 α 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 ${\mu }_l$ is always larger than the corresponding classical mobility which follows from the Langevin equation. In the localized regime α>1, ${\mu }_l$ is an increasing function of temperature, consistent with transport via a thermally assisted hopping mechanism. For α<1, ${\mu }_l$(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 α<1 describes the physics of the observed nonmonotonic temperature dependence of muon diffusion in metals.