Theory of stretched exponential relaxation and critical behavior at depinning for charge density waves

We study analytically and numerically the critical behavior at the depinning transition and the stretched exponential relaxation in the pinned region of a charge density wave (CDW) under the influence of an applied field. The original model of an elastic string pinned by random sinusoidal impurity potentials is transformed, by a coarse graining operation, into a simpler model for which analytical expressions for the exponents corresponding to the critical behavior and to the stretched exponential relaxation can be derived in any dimension. In particular we can identify different correlation lenghts (static and dynamic) that diverge at the depinning transition with different exponents. Computer simulations on the coarse grained model and on the original CDW model turn out to be in very good agreement with our analytical expressions. These results clarify the nature of the critical behavior of the depinning transition and of the glassy-type relaxation in the pinned region. In addition our theoretical analysis may provide a new point of view for the description of other disordered and glassy systems.