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Physical review. B, Condensed matter

Critical properties of S>12 Ising chains with long-range interactions

M. Barati, Abdolali Ramazani

Published Nov 1, 2000• 10 citations
Publisher

Abstract

The phase diagram and critical exponent of the correlation length for an Ising chain with a long-range interaction in the form of ${1/r}^{1+\ensuremath{\sigma}}$ for $S=1/2,1,3/2,2$ is calculated by the finite-range scaling technique. The spin dependence of the critical temperature and critical exponent $\ensuremath{\nu}$ in the classical and nonclassical regions is also studied. It is observed that, with a good approximation, the critical exponent of the correlation length is independent of the magnitude of spin in both the classical and nonclassical regions.

DOI
10.1103/physrevb.62.12130

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The phase diagram and critical exponent of the correlation length for an Ising chain with a long-range interaction in the form of 1/r1+\ensuremathσ{1/r}^{1+\ensuremath{\sigma}}1/r1+\ensuremathσ for S=1/2,1,3/2,2S=1/2,1,3/2,2S=1/2,1,3/2,2 is calculated by the finite-range scaling technique. The spin dependence of the critical temperature and critical exponent \ensuremathν\ensuremath{\nu}\ensuremathν in the classical and nonclassical regions is also studied. It is observed that, with a good approximation, the critical exponent of the correlation length is independent of the magnitude of spin in both the classical and nonclassical regions.
The classical phase diagram of an Ising chain for S> 12S>~\frac{1}{2}S> 21​ with long-range interaction in the form of ${1/r}^{1+\ensuremath{\sigma}}$ is determined. The calculation is performed by using the exact value of the critical exponent of the correlation length, $\ensuremath{\nu}=1/\ensuremath{\sigma},$ given by the renormalzation-group technique in the classical region, and applying a searching program based on the finite-range-scaling method. The results for the critical temperature in this approach are in excellent agreement with those calculated by the extensive Monte Carlo technique. This is strong support for the range-scaling method in the classical region.
The phase diagram of a two-sited magnetic Ising chain, with a long-range interaction in the form of ${1/r}^{1+\ensuremath{\sigma}},$ is studied. In this investigation, the finite-range scaling technique is employed and a proper transfer matrix is developed. The critical temperature of the chain, as a function of each type of interaction, i.e., the interaction between the similar sites and different sites, in both the classical and nonclassical regions is calculated. The results indicate that the critical temperature is strongly affected by the interaction range parameter \ensuremath{\sigma} for both types of interaction, but its behavior is quite different for each of the interactions. The behavior of the system is explained by calculating the strength of each interaction using the mean-field approximation. The study of the critical exponent of the correlation length indicates that the exponent is, within a good approximation, independent of the interaction constants.
A long-range Ising chain, which has ferromagnetic 1/r 3 type interactions, has been studied by exact numerical calculation of thermodynamic properties of the sequence of systems of N spins (N<or=30). By expressing the sequence in terms of a series and using extrapolation methods such as the Pade approximant, the authors have found a consistent way to obtain critical temperatures which is accurate to 0.4% for the very long-range equivalent neighbour model. The authors have estimated the high-temperature susceptibility and the low- temperature long-range correlations of the infinite chain and thence have made estimates for the critical exponents gamma and beta . Although the nature of the phase transition appears to be classical for 1<s<or approximately=1.3, it is certainly not classical for 1.6<s<2; in particular, as s approaches 2 from below, gamma increases regularly to about 2 or greater and beta decreases regularly to near 0.
The coherent anomaly method and a cluster mean-field approach combined with finite-sizescaling extrapolation were used to determine the critical temperature and the critical exponents of aS =1 Ising chain with a long-range interaction in the form of 1/ r1+σ . The results of criticaltemperature are in good agreement with the recent results of the finite-range scaling method and aremore accurate in the classical region. The critical exponents are in agreement with the results of spin-21Ising chains and indicate that the critical exponents are independent of the magnitude of spin in Isingmodels with long-range interactions.
Recently, it has been found that an effective long-range interaction is realized among local bistable variables (spins) in systems where the elastic interaction causes ordering of the spins. In such systems, we generally expect both long-range and short-range interactions to exist. In the short-range Ising model, the correlation length diverges at the critical point. In contrast, in the long-range interacting model the spin configuration is always uniform and the correlation length is zero. As long as a system has nonzero long-range interactions, it shows criticality in the mean-field universality class, and the spin configuration is uniform beyond a certain scale. Here we study the crossover from the pure short-range interacting model to the long-range interacting model. We investigate the infinite-range model (Husimi-Temperley model) as a prototype of this competition, and we study how the critical temperature changes as a function of the strength of the long-range interaction. This model can also be interpreted as an approximation for the Ising model on a small-world network. We derive a formula for the critical temperature as a function of the strength of the long-range interaction. We also propose a finite-size scaling form for the spin correlation length at the critical point, which is finite as long as the long-range interaction is included, though it diverges in the limit of the pure short-range model. These properties are confirmed by extensive Monte Carlo simulations.
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.