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

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.