Order-N Cluster Monte Carlo Method for Spin Systems with Long-range Interactions

TL;DR

This paper introduces an efficient O(N) cluster Monte Carlo algorithm for long-range Ising models that maintains detailed balance and outperforms existing methods, enabling accurate studies of phase transitions.

Contribution

The paper presents a novel O(N) cluster Monte Carlo method for long-range interactions that does not require cutoff and is more efficient than previous algorithms.

Findings

The algorithm achieves O(N) complexity for energy and specific heat calculations.

It outperforms traditional O(N^2) and O(N log N) algorithms in efficiency.

Confirmed the occurrence of a Kosterlitz-Thouless transition in long-range Ising chains.

Abstract

An efficient O(N) cluster Monte Carlo method for Ising models with long-range interactions is presented. Our novel algorithm does not introduce any cutoff for interaction range and thus it strictly fulfills the detailed balance. The realized stochastic dynamics is equivalent to that of the conventional Swendsen-Wang algorithm, which requires O(N^2) operations per Monte Carlo sweep if applied to long-range interacting models. In addition, it is shown that the total energy and the specific heat can also be measured in O(N) time. We demonstrate the efficiency of our algorithm over the conventional method and the O(N log N) algorithm by Luijten and Bloete. We also apply our algorithm to the classical and quantum Ising chains with inverse-square ferromagnetic interactions, and confirm in a high accuracy that a Kosterlitz-Thouless phase transition, associated with a universal jump in the magnetization, occurs in both cases.