Quantum Brownian motion in a periodic potential

Matthew P.A. Fisher, Wilhelm Zwerger

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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 ± is raised through one. The scaling equations are derived by applying a perturbative renormalization group to the systems partition function. The dynamics is studied using Feynmans influence-functional theory. We compute directly the nonlinear mobility of the Brownian particle in the weak-corrugation limit, for arbitrary temperature. The linear mobility 1/4l is always larger than the corresponding classical mobility which follows from the Langevin equation. In the localized regime ±>1, 1/4l is an increasing function of temperature, consistent with transport via a thermally assisted hopping mechanism. For ±<1, 1/4l(T) shows a nonmonotonic dependence on T with a minimum at a temperature T*. This is due to a crossover between quantum tunneling below T* and thermally assisted hopping above T*. For low friction the crossover occurs when the particles thermal de Broglie wavelength is roughly equal to the distance between minima in the periodic potential. We suggest that the regime ±<1 describes the physics of the observed nonmonotonic temperature dependence of muon diffusion in metals.

Original languageEnglish
Pages (from-to)6190-6206
Number of pages17
JournalPhysical Review B
Issue number10
StatePublished - 1985
Externally publishedYes


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