Spinless particle in rapidly fluctuating random magnetic field

TL;DR

This paper investigates the quantum dynamics of a spinless particle in a rapidly fluctuating random magnetic field, revealing different diffusion regimes and scaling behaviors depending on the spatial correlation exponent of the magnetic field.

Contribution

It introduces a detailed evolution equation for the density matrix in a disordered magnetic field with short time fluctuations, analyzing how spatial correlations affect diffusion and growth regimes.

Findings

For h<2, the particle's center of mass diffuses with a coefficient scaling as x^{-h}.

At h=2, the system exhibits a transition to logarithmic growth in the density matrix width.

The density matrix's width in the center of mass coordinate grows proportionally to log(t/x^2) for large times.

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 $<x^{(1)} |\hat{\rho} (t)| x^{(2)}>$; in this description the two coordinates are associated with the retarded and advanced paths respectively. The static part of the vector potential correlator is assumed to grow with distance with a power $h$; the case $h = 0$ corresponds to a $\delta$-correlated magnetic field, and $h = 2$ to free massless field. The value $h = 2$ separates two different regimes, diffusion and logarithmic growth respectively. When $h < 2$ the baricentric coordinate $r = (1/2)(x^{(1)} + x^{(2)})$ diffuses with a coefficient $D_{r}$ proportional to $x^{-h}$, where $x$ is the relative coordinate: $x = x^{(1)} - x^{(2)}$. As $h > 2$ the correlator of the magnetic field is a power of distance with positive exponent; then the coefficient $D_{r}$ scales as $x^{-2}$. The density matrix is a function of $r$ and $x^2/t$,and its width in $r$ grows for large times proportionally to $log(t/x^2)$.