Statistical mechanics of random matrices: Application to disordered metals

An ensemble of N x N hermitians is constructed which satisfies representation covariance and translation invariance on the basis of maximum entropy and level dynamics, concluding that the resulting reduced distribution of N eigenvalues of a sample hermitian, P(x(1),x(2),..,x(N)), must be of the form Pi (j <k) c(x(j) - x(k)) in terms of a single function c(x - y) of the difference of a pair of eigenvalues, x and y. It provides us with a concrete view of the level-statistics of quantum systems as the statistical mechanics of a 1-dimensional gas whose distribution function on {x(j)} may be represented as P({x(j)}) = C(N)exp (-beta Sigma (j <k) phi (x(j) - x(k))), where beta is the symmetry parameter and [GRAPHICS] An additional restriction on the average level density provides its relation to the single-level potential V(x) and the two-level correlation function R(x - y), and furthermore, the possibility of attractiveness of the pair potential. The scheme allows one to incorporate the scaling theory of the metal-insulator transition by means of a dimensionless conductance g(L) depending on dimensionality, which is outlined. As a result, the universal spectral statistics for 3D metals in the limit L --> infinity are indicated.