Optimized energy calculation in lattice systems with long-range interactions

Abstract

We discuss an efficient approach to the calculation of the internal energy in numerical simulations of spin systems with long-range interactions. Although, since the introduction of the Luijten-Bl\"ote algorithm, Monte Carlo simulations of these systems no longer pose a fundamental problem, the energy calculation is still an O(N^2) problem for systems of size N. We show how this can be reduced to an O(N logN) problem, with a break-even point that is already reached for very small systems. This allows the study of a variety of, until now hardly accessible, physical aspects of these systems. In particular, we combine the optimized energy calculation with histogram interpolation methods to investigate the specific heat of the Ising model and the first-order regime of the three-state Potts model with long-range interactions.