Quantum Collective Creep: a Quasiclassical Langevin Equation Approach

The dynamics of an elastic medium driven through a random medium by a small applied force is investigated in the low-temperature limit where quantum fluctuations dominate. The motion proceeds via tunneling of segments of the manifold through barriers whose size grows with decreasing driving force $f$. In the limit of small drive, at zero-temperature the average velocity has the form $v\propto\exp[-{\rm const.}/\hbar^{\alpha} f^{\mu}]$. For strongly dissipative dynamics, there is a wide range of forces where the dissipation dominates and the velocity--force characteristics takes the form $v\propto\exp[-S(f)/\hbar]$, with $S(f)\propto 1/ f^{(d+2\zeta)/(2-\zeta)}$ the action for a typical tunneling event, the force dependence being determined by the roughness exponent $\zeta$ of the $d$-dimensional manifold. This result agrees with the one obtained via simple scaling considerations. Surprisingly, for asymptotically low forces or for the case when the massive dynamics is dominant, the resulting quantum creep law is {\it not} of the usual form with a rate proportional to $\exp[-S(f)/\hbar]$; rather we find $v\propto \exp\{-[S(f)/\hbar]^2\}$ corresponding to $\alpha=2$ and $\mu= 2(d+2\zeta-1)/(2-\zeta)$, with $\mu/2$ the naive scaling exponent for massive dynamics. Our analysis is based on the quasi-classical Langevin approximation with a noise obeying the quantum fluctuation--dissipation theorem. The many space and time scales involved in the dynamics are treated via a functional renormalization group analysis related to that used previously to treat the classical dynamics of such systems. Various potential difficulties with these approaches to the multi-scale dynamics -- both classical and quantum -- are raised and questions about the validity of the results are discussed.