Scattering approach to Anderson localisation

Abstract

We develop a novel approach to the Anderson localisation problem in a $d$-dimensional disordered sample of dimension $L\times M^{d-1}$. Attaching a perfect lead with the cross-section $M^{d-1}$ to one side of the sample, we derive evolution equations for the scattering matrix and the Wigner-Smith time delay matrix as a function of $L$. Using them one obtains the Fokker-Planck equation for the distribution of the proper delay times and the evolution equation for their density at weak disorder. The latter can be mapped onto a non-linear partial differential equation of the Burgers type, for which a complete analytical solution for arbitrary $L$ is constructed. Analysing the solution for a cubic sample with $M=L$ in the limit $L\to \infty$, we find that for $d<2$ the solution tends to the localised fixed point, while for $d>2$ to the metallic fixed point and provide explicit results for the density of the delay times in these two limits.

arXiv comment4+3 pages

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Cited by in corpus (11)

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• Chaotic Scattering with Localized Losses: S-Matrix Zeros and Reflection Time Difference for Systems with Broken Time Reversal Invariance
• Reflection Time Difference as a probe of S-matrix Zeroes in Chaotic Resonance Scattering
• Symmetry constraints for vector scattering and transfer matrices containing evanescent components: energy conservation, reciprocity and time reversal
• Softening of Majorana edge states by long-range couplings
• Wigner-Smith matrix, exponential functional of the matrix Brownian motion and matrix Dufresne identity
• Resonance distribution in the quantum random Lorentz gas
• Delay time of waves performing Lévy walks in 1D random media
• Anderson localisation in two dimensions: insights from Localisation Landscape Theory, exact diagonalisation, and time-dependent simulations
• Directional Localization in Disordered 2D Tight-Binding Systems: Insights from Single Particle Entanglement Measures