Interference in disordered systems: A particle in a complex random landscape

Abstract

We consider a particle in one dimension submitted to amplitude and phase disorder. It can be mapped onto the complex Burgers equation, and provides a toy model for problems with interplay of interferences and disorder, such as the NSS model of hopping conductivity in disordered insulators and the Chalker-Coddington model for the (spin) quantum Hall effect. The model has three distinct phases: (I) a {\em high-temperature} or weak disorder phase, (II) a {\em pinned} phase for strong amplitude disorder, and (III) a {\em diffusive} phase for strong phase disorder, but weak amplitude disorder. We compute analytically the renormalized disorder correlator, equivalent to the Burgers velocity-velocity correlator at long times. In phase III, it assumes a universal form. For strong phase disorder, interference leads to a logarithmic singularity, related to zeroes of the partition sum, or poles of the complex Burgers velocity field. These results are valuable in the search for the adequate field theory for higher-dimensional systems.

16 pages, 7 figures

References in corpus (5)

• Burgers Turbulence
• Numerical Calculation of the Functional renormalization group fixed-point functions at the depinning transition
• On an imaginary exponential functional of Brownian motion
• Exact results and open questions in first principle functional RG
• Interfering directed paths and the sign phase transition

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• Theory and Experiments for Disordered Elastic Manifolds, Depinning, Avalanches, and Sandpiles
• Disordered statistical physics in low dimensions: extremes, glass transition, and localization