Optimized energy calculation in lattice systems with long-range interactions

TL;DR

This paper presents an optimized method to compute the internal energy in long-range interacting lattice systems, reducing computational complexity from O(N^2) to O(N log N), enabling new physical insights.

Contribution

It introduces an efficient energy calculation algorithm that significantly speeds up simulations of long-range spin systems, facilitating studies of previously inaccessible phenomena.

Findings

Reduced energy calculation complexity to O(N log N)

Enabled detailed analysis of specific heat in long-range models

Demonstrated effectiveness with Ising and Potts models

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.