Dynamics of manifolds in random media: the selfconsistent Hartree approximation

Abstract

We discuss a dynamical description of fluctuating manifolds in random media, using an approximation which becomes exact in the «spherical limit» N→∞, N being the dimension of the transversal fluctuations. The system behaviour is studied in detail for the case of quenched disorder with short range correlations. In the static limit, we recover solutions known from a replica calculation with one step replica symmetry breaking. In the dynamical formulation it turns out that this replica symmetry breaking is related to broken ergodicity for temperatures below a critical point. While the static behaviour in the low temperature phase superficially seems to be not very different from that of the high temperature phase, the dynamic treatment shows that the manifolds get dynamically localized below the critical temperature. The value we find for this «freezing temperature» is higher than the corresponding result from replica theory. Introducing an additional external quadratic potential, we find that the character of the phase transition changes with increasing coupling strength μ. There is a tricitical point μ * , for μ μ * it becomes a second order one. We give the phase diagram for this generalized model and discuss the dynamical behaviour in the different regimes