Efficient predecision scheme for Metropolis Monte Carlo simulation of long-range interacting lattice systems

TL;DR

This paper introduces a fast, general predecision scheme for Metropolis Monte Carlo simulations of long-range lattice models, significantly reducing computational complexity and enabling more efficient studies of complex spin systems.

Contribution

The authors develop a novel predecision algorithm that decreases the computational cost of simulating long-range interactions in lattice models, applicable across various spin types and models.

Findings

Reduces computational complexity from O(N^2) to O(N^{2-σ/d}) or O(N) depending on σ.

Produces identical Markov chains as explicit summation methods.

Applicable to multiple spin models, including Ising, XY, and spin-glass models.

Abstract

We propose a fast and general predecision scheme for Metropolis Monte Carlo simulation of $d$-dimensional long-range interacting lattice models. For potentials of the form $V(r)=r^{-d-\sigma}$, this reduces the computational complexity from $O\left(N^2\right)$ to $O\left(N^{2-\sigma/d}\right)$ for $\sigma < d$ and to $O\left(N \right)$ for $\sigma > d$, respectively. The algorithm is implemented and tested for several $\mathrm{O}(n)$ spin models ranging from the Ising over the XY to the Edwards-Anderson spin-glass model. With the same random number sequence it produces exactly the same Markov chain as a simulation with explicit summation of all terms in the Hamiltonian. Due to its generality, its simplicity, and its reduced computational complexity it has the potential to find broad application and thus lead to a deeper understanding of the role of long-range interactions in the physics of lattice models, especially in nonequilibrium settings.