DENSITIES OF STATES OF MEGA-DIMENSIONAL HAMILTONIAN MATRICES

We propose a statistical method to estimate densities of states (DOS) and thermodynamic functions of very large Hamiltonian matrices. Orthog onal polynomials are defined on the interval between lower and upper e nergy bounds. The DOS is represented by a kernel polynomial constructe d out of polynomial moments of the DOS and modified to damp the Gibbs phenomenon. The moments are stochastically evaluated using matrix-vect or multiplications on Gaussian random vectors and the polynomial recur rence relations, The resulting kernel estimate is a controlled approxi mation to the true DOS, because it also provides estimates of statisti cal and systematic errors. For a given fractional energy resolution an d statistical accuracy, the required cpu time and memory scale linearl y in the number of states for sparse Hamiltonians. The method is demon strated for the two-dimensional Heisenberg anti-ferromagnet with the n umber of states as large as 2(26). Results are compared to exact diago nalization where available.