DENSITIES OF STATES OF MEGA-DIMENSIONAL HAMILTONIAN MATRICES

Abstract

We propose a statistical method to estimate densities of states (DOS) and thermodynamic functions of very large Hamiltonian matrices. Orthogonal polynomials are defined on the interval between lower and upper energy bounds. The DOS is represented by a kernel polynomial constructed out of polynomial moments of the DOS and modified to damp the Gibbs phenomenon. The moments are stochastically evaluated using matrixvector multiplications on Gaussian random vectors and the polynomial recurrence relations. The resulting kernel estimate is a controlled approximation to the true DOS, because it also provides estimates of statistical and systematic errors. For a given fractional energy resolution and statistical accuracy, the required cpu time and memory scale linearly in the number of states for sparse Hamiltonians. The method is demonstrated for the two-dimensional Heisenberg anti-ferromagnet with the number of states as large as 226. Results are compared to exact diagonalization where available.