Monte Carlo simulations of the 1D Ising model with ferromagnetic interactions decaying with distance r as 1/r 1+σ are performed by applying the Swendsen-Wang cluster algorithm with cumulative probabilities. The critical behavior in the non-classical critical regime corresponding to 0.5 < σ < 1 is derived from the finite-size scaling analysis of the largest cluster.PACS numbers: 05.50.+q, 64.60.CnThe calculation of the critical behavior of the Ising model with long-range (LR) interactions decaying with distance with a power law 1/r d+σ is not an easy task, even in one dimension, where phase transition at finite temperature occurs for 0 < σ ≤ 1 [1,2]. Exact analytical expressions for the critical exponents may be derived for σ ≤ 0.5 [3], corresponding to the classical regime, while for σ > 0.5 only the approximate results exist. Various analytical and numerical approaches have been applied, from direct numerical calculations on finite chains [4], to several approaches based on the renormalization group (RG) and scaling [5][6][7][8][9][10]. The nonlocal character of interactions reduces the efficiency of most of the standard approaches so that the values of critical exponents obtained by all these methods differ considerably.In numerous cases of phase transitions in systems with short-range interactions, a useful complementary tool for obtaining both qualitative and quantitative results is provided by Monte Carlo (MC) simulations in combination with finite-size scaling.Such systematic studies were lacking for the LR models until recently. Namely, when applied to models with LR interactions, the standard MC approaches based either on Metropolis or on various cluster algorithms are particularly time consuming, since the number of operations per spin-flip is proportional there to the size of the system. Recently, this problem was successfully resolved by Luijten and Blöte [11] who used the cumulative probabilities within the Wolff cluster algorithm [12], which they applied to the Ising and similar models [13][14][15] reducing the computing time by several orders of magnitude.Their very exhaustive studies concentrate on questions related to the mean-field (MF) regime, while very little or no interest has been dedicated yet to the regime of the non-classical critical behavior corresponding to σ > 0.5.The purpose of this work is to extend the MC studies of the critical behavior of the LR Ising model to the non-classical regime. At the same time this is a suitable example to examine the efficiency of using only the cluster statistics in deriving the critical properties of the LR model. The 1D Ising model with LR interactions written in form of a special case (q = 2) of the Potts model is described by the Hamiltonianwhere J > 0, s i is a two-state Potts variable at the site i, δ is the Kronecker symbol and the summation is over all pairs of the system. By the substitution δ si,sj = (S i · S j + 1)/2, and J I = J/2, where S i = ±1 and J I denote the Ising spins and the interaction constant, respectively, one recovers the st...
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.