Critical behavior of the long-range Ising chain from the largest-cluster probability distribution

Abstract

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...

Relations between short-range and long-range Ising models

“…[18], and for the LR one-dimensional model, from Ref. [29] and [30]. For the LR d = 1 and d = 2 model, our results are reported…”

Long-range multi-scalar models at three loops

“…Their long-range versions have instead been studied much less, and the renormalization group analysis has been halted at the two-loop computations done in the 1970s [32,33]. Similarly, other methods have also been underdeveloped as compared to the short-range case, with Monte Carlo simulations being mostly limited to the Ising model in one or two dimensions [6,[34][35][36][37][38][39][40][41], and with only occasional excursions from other methods, such as the functional renormalization group [8] or the conformal bootstrap [42].…”

“…First, in section 3.1, we study the long-range Ising model. We give the fixed points and critical exponents in the expansion up to order 3 and compare them with numerical simulations at d = 1 [34][35][36][37] and d = 2 [6].…”

Fast flat-histogram method for generalized spin models

“…These include, e.g., the effect of dimensionality [4], the crossover from shortrange to long-range behavior [5,6,7], mean-field driven phase transitions [8], and possible connections with Tsallis's non-extensive thermodynamics [9,10,11]. Monte Carlo (MC) methods have now gained a prominent role in the investigation of phase transitions in these models [12,13,14,15,16,17,18]. In particular, a major breakthrough was recently initiated by the introduction of a (canonical) cluster algorithm able to overcome the algorithm complexity inherent to long-range (LR) models, namely, the need to take a huge number of interactions into account at each Monte Carlo step (MCS) [12].…”