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

We consider a family of random matrix ensembles (RME) invariant under similarity transformations and described by the probability density P (H) = exp[−TrV (H)]. Dyson’s mean field theory (MFT) of the corresponding plasma model of eigenvalues is generalized to the case of weak confining potential, V (ǫ) ∼ A2 ln(ǫ). 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 Ac ≈ 0.4 the distribution function of the energylevel 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 Ac, the variance of the number of levels, 〈n2〉− 〈n〉2, in an interval containing 〈n〉 levels on average, grows linearly with 〈n〉, and its slope is equal to 0.32 ± 0.02, which is consistent with the value found for the Anderson model at the critical point. PACS numbers: 71.30.+h, 72.15.Rn, 05.60.+w Typeset using REVTEX