Densities of States and Thermodynamics of Mega-Dimensional Sparse Matrices

Abstract

Mega-dimensional matrices occur in a wide range of modern numerical studies.’ For example in ab initio band structure calculations, image reconstruction problems, nuclear shell models, transfer matrices, and exact diagonalization on small clusters, the size of the resulting matrices controls the computer resources needed and thereby limits the possible size of the systems under study. As a typical example we chose to study the thermodynamics of the spin \(\tfrac{1}{2}\) antiferromagnetic Heisenberg model (HAFM) in two dimensions on the square, triangular, and Kagome lattice.These are described by the Hamiltonian $$\mathcal{H} = \mathop{\sum }\limits_{{ }} {{S}_{i}}.{{S}_{j}},$$ (1) where the sum over runs over all nearest neighbor bonds on the respective lattice. In the Heisenberg models the size of the Hilbert space grows as 2N, where N is the number of lattice sites. Even when symmetries are used to block diagonalize the Hamiltonian, the dimension of the resulting matrices is too large to use conventional diagonalization techniques, which can only manage matrices of dimension of a few times 104. One then usually resorts to special algorithms that only use matrix vector multiplications to obtain a few isolated eigenvalues at the edges of the spectrum. The most commonly used one is the Lanczos algorithm.’ Since only a few eigenvalues can be calculated, the exact diagonalization technique has so far been limited to T = 0, except for smaller lattice sizes up to N = 18. Here we present a method that provides a controlled and easy to use approximation to the many body density of states (DOS). In this way we extend the exact diagonalization studies of the Heisenberg antiferromagnets to finite temperatures.