Anomalous diffusion in disordered media and random quantum spin chains

TL;DR

This paper establishes a fundamental link between anomalous diffusion in disordered media and the critical behavior of random quantum spin chains, providing new exact results and analytical expressions for key properties.

Contribution

It introduces a novel correspondence that allows deriving exact results for anomalous diffusion and quantum spin chain dynamics, including the dynamical exponent in Griffiths-McCoy regions.

Findings

Derived exact space and time correlations of surviving random walks

Obtained the distribution of gaps of the Focker-Planck operator

Analytically calculated the dynamical exponent in Griffiths-McCoy regions

Abstract

Using exact expressions for the persistence probability and for the leading eigenvalue of the Focker-Planck operator of a random walk in a random environment we establish a fundamental relation between the statistical properties of anomalous diffusion and the critical and off-critical behavior of random quantum spin chains. Many new exact results are obtained from this correspondence including the space and time correlations of surviving random walks and the distribution of the gaps of the corresponding Focker-Planck operator. In turn we derive analytically the dynamical exponent of the random transverse-field Ising spin chain in the Griffiths-McCoy region.