The 2D± J Ising spin glass: exact partition functions in polynomial time

Abstract

We describe an exact integer algorithm to compute the partition function of a two-dimensional ± J Ising spin glass. The algorithm takes as input a set of quenched random bonds on the square lattice and returns the density of states as a function of energy. Unlike the transfer-matrix method, the algorithm is limited to two dimensions; the computation time, however, is polynomial in the lattice size. The algorithm is used to study the ± J spin glass on L × L lattices with periodic boundary conditions. The lattices vary in size from L=4 to L=36. We investigate scaling laws for properties of the ground state and low-level excitations. We also examine the roots of the partition function in the complex plane. Quenched averages are performed by statistically sampling a large number of realizations of randomness. The potential to handle two-dimensional Ising models with different types of quenched randomness is also discussed.