Level spacings at the metal-insulator transition in the Anderson Hamiltonians and multifractal random matrix ensembles

Details Level spacings at the metal-insulator transition in the Anderson Hamiltonians and multifractal random matrix ensembles Level spacings at the metal-insulator transition in the Anderson Hamiltonians and multifractal random matrix ensembles

1998-09-10

arxiv.org/abs/cond-mat/9809147v3

Phys.Rev. E59 (1999) 2853-2862

Physics

Condensed Matter

Disordered Systems and Neural Networks

12 pages, 8 figures, REVTeX. Additional figure and text on the level number variance, to appear in Phys.Rev.E

Scientific paper

10.1103/PhysRevE.59.2853

We consider orthogonal, unitary, and symplectic ensembles of random matrices with (1/a)(ln x)^2 potentials, which obey spectral statistics different from the Wigner-Dyson and are argued to have multifractal eigenstates. If the coefficient $a$ is small, spectral correlations in the bulk are universally governed by a translationally invariant, one-parameter generalization of the sine kernel. We provide analytic expressions for the level spacing distribution functions of this kernel, which are hybrids of the Wigner-Dyson and Poisson distributions. By tuning the single parameter, our results can be excellently fitted to the numerical data for three symmetry classes of the three-dimensional Anderson Hamiltonians at the metal-insulator transition, previously measured by several groups using exact diagonalization.

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