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arXiv:cond-mat/0011306 · Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

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

Katarina Uzelac, Zvonko Glumac, A. Aničić

Published Feb 20, 2001• 11 citations
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Abstract

Monte Carlo simulations of the one-dimensional Ising model with ferromagnetic interactions decaying with distance r as 1/r(1+sigma) are performed by applying the Swendsen-Wang cluster algorithm with cumulative probabilities. The critical behavior in the nonclassical critical regime corresponding to 0.5<sigma<1 is derived from finite-size scaling analysis of the largest cluster.

DOI
10.1103/physreve.63.037101

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Monte Carlo simulations of the one-dimensional Ising model with ferromagnetic interactions decaying with distance r as 1/r(1+sigma) are performed by applying the Swendsen-Wang cluster algorithm with cumulative probabilities. The critical behavior in the nonclassical critical regime corresponding to 0.5<sigma<1 is derived from finite-size scaling analysis of the largest cluster.
Critical exponents for a one-dimensional general continuous Ising model with long-range ferromagnetic interactions decaying as 1/r1+σ are calculated using a histogram Monte Carlo technique. A continuous Ising model means that a spin can take any value between −1 and 1. The critical point behavior is investigated. It is found that the system exhibits a second-order phase transition with nonstandard critical exponents; the singularities in the specific heat and susceptibility depend on σ. For σ=1, there are extremely weak finite size effects: the first-order and the second-order cumulants of the order parameter yield ν=2.42(1). The susceptibility exponent γ=2.259(9). Results for σ=0.7 will be shown and discussed.
The critical behaviour of the one-dimensional Ising model with long-range ferromagnetic interactions decaying with distance r as 1/r 1+ sigma has been studied by scaling the range of interactions. Exact calculations have been done for a system with finite ranges up to the tenth neighbour for 0< sigma <or=1. Applying the range scaling, the critical temperature, critical exponent nu and the anomalous dimension of the order parameter have been calculated. Additional analysis of the convergence of the method has been performed by applying the Vanden Broeck and Schwartz extrapolation procedure in addition to the simple least-squares approximation and by evaluating the convergence exponents.
The critical behavior of the Ising model on a one-dimensional network, which has long-range connections at distances l>1 with the probability theta(l) approximately l(-m), is studied by using Monte Carlo simulations. Through analyzing the Ising model on networks with different m values, this paper discusses the impact of the global correlation, which decays with the increase of m, on the phase transition of the Ising model. Adding the analysis of the finite-size scaling of the order parameter [M], it is observed that in the whole range of 0<m<2, a finite-temperature transition exists, and the critical exponents show consistence with mean-field values, which indicates a mean-field nature of the phase transition.
We investigate the universality class of the finite-temperature phase transition of the two-dimensional Ising model with the algebraically decaying ferromagnetic long-range interaction, J_{ij}=|r[over ⃗]_{i}-r[over ⃗]_{j}|^{-(d+σ)}, where d (=2) is the dimension of the system and σ is the decay exponent, by means of the order-N cluster-algorithm Monte Carlo method. In particular, we focus on the upper and lower critical decay exponents, the boundaries between the mean-field-universality, intermediate, and short-range-universality regimes. At the critical decay exponents, it is found that the standard Binder ratio of magnetization at the critical temperature exhibits extremely slow convergence as a function of the system size. We propose more effective physical quantities, namely the combined Binder ratio and the self-combined Binder ratio, both of which cancel the leading finite-size corrections of the conventional Binder ratio. Utilizing these techniques, we clearly demonstrate that in two dimensions, the lower and upper critical decay exponents are σ=1 and 7/4, respectively, contrary to the recent Monte Carlo and renormalization-group studies [M. Picco, arXiv:1207.1018; T. Blanchard et al., Europhys. Lett. 101, 56003 (2013)EULEEJ0295-507510.1209/0295-5075/101/56003].