Random Matrix Theory of the Energy-Level Statistics of Disordered Systems at the Anderson Transition

Abstract

We consider a family of random matrix ensembles (RME) invariant under similarity transformations and described by the probability density $P(\mathbf{H})=\exp [-\mathrm{Tr}V(\mathbf{H})]$. Dyson's mean field theory (MFT) of the corresponding plasma model of eigenvalues is generalized to the case of weak confining potential, $V(\epsilon )\sim \frac{A}{2}{\ln }^2(\epsilon )$. The eigenvalue statistics derived from MFT are shown to deviate substantially from the classical Wigner-Dyson statistics when $A<1$. By performing systematic Monte Carlo simulations on the plasma model, we compute all the relevant statistical properties of the RME with weak confinement. For $A_c\approx 0.4$ the distribution function of the energy-level spacings (LSDF) of this RME coincides in a large energy window with the LSDF of the three dimensional Anderson model at the metal-insulator transition. For the same $A_c$, the variance of the number of levels, $⟨n^2⟩-⟨n{⟩}^2$, in an interval containing $⟨n⟩$ levels on average, grows linearly with $⟨n⟩$, and its slope is equal to $0.32\pm 0.02$, which is consistent with the value found for the Anderson model at the critical point.

32 pages, REVTEX 3.0, 10 postscript (uuencoded) figures included

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• Universality of a family of Random Matrix Ensembles with logarithmic soft-confinement potentials
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