Random Matrix Theory and Anderson Localisation
The random statistical ensembles of short-ranged tight-binding lattice Hamiltonian matrices are introduced and studied in connection with the Anderson metal-insulator transition. The short and long range spectral fluctuation-correlation properties are examined for systems classified into three symmetry universality classes. In the metallic regime for two and three dimensions the results are consistent with the properties of level repulsion and spectral rigidity known from the Gaussian random matrix ensembles of Wigner-Dyson. In the insulating regime the spectrum is instead uncorrected. The eigenvalue statistics is directly related to the universal conductance fluctuations observed in small metallic (mesoscopic) systems. The corresponding statistical behaviour of the wave functions is discussed and found to exhibit multifractal scaling at the transition. Similarities of the Anderson transition with what is known as the transition to “quantum chaos” are also discussed.
KeywordsUniversality Class Random Matrix Theory Mobility Edge Conductance Fluctuation Random Matrix Ensemble
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