Subthreshold behavior and avalanches in an exactly solvable Charge Density Wave system

We present a toy charge density wave (CDW) model in 1d exhibiting a depinning transition with threshold force and configurations that are explicit. Due to the periodic boundary conditions imposed, the threshold configuration has a set of topological defects whose location and number depend on the realization of the random phases. Approaching threshold, these defects are relocated by avalanches whose size dependence on the external driving force $F$ is described by a record-breaking process. We find that the depinning transition in this model is a critical phenomenon, with the cumulative avalanche size diverging near threshold as $(F_{\rm th} - F)^{-2}$. The exact avalanche size distributions and their dependence on the control parameter $(F_{\rm th} - F)$ are obtained. Remarkably, the scaling exponents associated with the critical behavior depend on (1) the initial conditions and (2) the relationship between the system size and the pinning strength.