Spinless particle in a rapidly fluctuating random magnetic field

Abstract

We study a two-dimensional spinless particle in a disordered Gaussian magnetic field with short-time fluctuations, by means of the evolution equation for the density matrix $〈{\mathbf{x}}^{(1)}|\stackrel{^}{\ensuremath{\rho}}(t)|{\mathbf{x}}^{(2)}〉$; in this description the two coordinates are associated with the retarded and advanced paths, respectively. In the classical limit the baricentric coordinate $\mathbf{r}=(1/2)({\mathbf{x}}^{(1)}+{\mathbf{x}}^{(2)})$ is the particle position and the dual of the relative coordinate $\mathbf{x}={\mathbf{x}}^{(1)}\ensuremath{-}{\mathbf{x}}^{(2)}$ its momentum. The vector potential correlator is assumed to grow with distance with a power $h$: when $h=0$ it corresponds to a $\ensuremath{\delta}$-correlated magnetic field, when $h=2$ to a magnetic field with infinite range fluctuations. We find that the value $h=2$ separates two different propagation regimes, of diffusion and logarithmic growth, respectively. When $h<2$, $\mathbf{r}$ undergoes diffusion with a coefficient ${D}_{r}$ proportional to ${x}^{\ensuremath{-}h}$. As $h>2$, the magnetic-field fluctuations grow with distance and ${D}_{r}$ scales as ${x}^{\ensuremath{-}2}$. The width in $r$ of the density matrix then grows for large times proportionally to $\mathrm{ln}{(t/x}^{2})$.