Monte Carlo Simulation of Spin Models with Long-Range Interactions

Abstract

An efficient Monte Carlo algorithm for the simulation of spin models with long-range interactions is discussed. Its central feature is that the number of operations required to flip a spin is independent of the number of interactions between this spin and the other spins in the system. In addition, critical slowing down is strongly suppressed. In order to illustrate the range of applicability of the algorithm, two specific examples are presented. First, some aspects of the Kosterlitz-Thouless transition in the one-dimensional Ising chain with inverse-square interactions are calculated. Secondly, the crossover from Ising-like to classical critical behavior in two-dimensional systems is studied for several different interaction profiles.

The fate of Ernst Ising and the fate of his model

“…Already from the Curie-Weiss model it was known that taking into account the interaction of all the spins by an effective field a phase transition came about even in the 1D case. However an interaction between two positions in the chain i, j with a decay according to a power law like 1/|i − j| 1+α leads to a phase transition for a sufficient weak decay, α < 1 [40] (see also [41]).…”

Fast flat-histogram method for generalized spin models

“…We now give a sketchy outline of the method in the context of longrange chains. Extensive details may otherwise be found in [12,58]. First of all, the probability to add a bond is split up into two parts, namely, (i) a provisional probability π l (E) (hereafter simply denoted π l ) depending on the distance l = |i − j| between spins and on the lattice energy E, and (ii) a factor f (σ i , σ j ) controlled by the spin values, e.g., a Kronecker delta symbol in the case of a Potts model.…”