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

We consider orthogonal, unitary, and symplectic ensembles of random matrices with (1/a)(lnx)(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. [S1063-651X(99)06503-4].