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Physical review. B, Condensed matter

Effect of long-range interactions on the critical behavior of the continuous Ising model

E. Bayong, H. T. Diep

Published May 1, 1999• 28 citations
Publisher

Abstract

Critical behavior of the one- and two-dimensional general continuous Ising models with long-range ferromagnetic interactions decaying as ${1/r}^{d+\ensuremath{\sigma}}$ is studied using a histogram Monte Carlo technique. A continuous Ising model means that a spin can take any value between $\ensuremath{-}1$ and 1. It is found that the system exhibits a second-order phase transition with nonstandard critical exponents which depend on $\ensuremath{\sigma}.$ Results for various values of $\ensuremath{\sigma}$ will be shown and compared to predictions from renormalization-group theory. Though there is an agreement with the overall tendency predicted, there are several fundamental differences. Discussion is given.

DOI
10.1103/physrevb.59.11919

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Critical behavior of the one- and two-dimensional general continuous Ising models with long-range ferromagnetic interactions decaying as 1/rd+\ensuremathσ{1/r}^{d+\ensuremath{\sigma}}1/rd+\ensuremathσ is studied using a histogram Monte Carlo technique. A continuous Ising model means that a spin can take any value between \ensuremath−1\ensuremath{-}1\ensuremath−1 and 1. It is found that the system exhibits a second-order phase transition with nonstandard critical exponents which depend on \ensuremathσ.\ensuremath{\sigma}.\ensuremathσ. Results for various values of \ensuremathσ\ensuremath{\sigma}\ensuremathσ will be shown and compared to predictions from renormalization-group theory. Though there is an agreement with the overall tendency predicted, there are several fundamental differences. Discussion is given.
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.
We study the nature of the phase transition of the qqq-state Potts model with long-range ferromagnetic interactions decaying as $1/{r}^{d+\ensuremath{\sigma}}$, in dimension d=1d\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1d=1, using a histogram Monte Carlo (MC) technique. The model can exhibit a first-order transition or a second-order phase transition with nonstandard critical exponents. The critical value of qqq above which a first-order transition occurs decreases with decreasing $\ensuremath{\sigma}$, from qc=8{q}_{c}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}8qc​=8 for $\ensuremath{\sigma}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1$ to qc=2{q}_{c}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}2qc​=2 for $\ensuremath{\sigma}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0.3$. Detailed results for various $\ensuremath{\sigma}$ will be shown and discussed. Mean-field calculation confirms the tendency of our MC results.
We present the results of extensive Monte Carlo simulations of Ising models with algebraically decaying ferromagnetic interactions in the regime where classical critical behavior is expected for these systems. We corroborate the values for the exponents predicted by renormalization theory for systems in one, two, and three dimensions and accurately observe the predicted logarithmic corrections at the upper critical dimension. We give both theoretical and numerical evidence that above the upper critical dimension the decay of the critical spin-spin correlation function in finite systems consists of two different regimes. For one-dimensional systems our estimates for the critical couplings are more than two orders of magnitude more accurate than existing estimates. In two and three dimensions we are unaware of any other results for the critical couplings.
Extensive Monte Carlo simulations are employed in order to study the dynamic critical behavior of the one-dimensional Ising magnet, with algebraically decaying long-range interactions of the form 1rd+σ\frac{1}{r^{d+σ}}rd+σ1​, with σ=0.75σ=0.75σ=0.75. The critical temperature, as well as the critical exponents, is evaluated from the power-law behavior of suitable physical observables when the system is quenched from uncorrelated states, corresponding to infinite temperature, to the critical point. These results are compared with those obtained from the dynamic evolution of the system when it is suddenly annealed at the critical point from the ordered state. Also, the critical temperature in the infinite interaction limit is obtained by means of a finite-range scaling analysis of data measured with different cutoffs of the interaction range. All the estimated static critical exponents (γ/νγ/νγ/ν, β/νβ/νβ/ν, and 1/ν1/ν1/ν) are in good agreement with Renormalization Group (RG) predictions and previously reported numerical data obtained under equilibrium conditions. It is found that the dynamic exponent zzz is different for quenching and annealing experiments, most likely due to the influence of the Kosterlitz-Thouless transition occurring at relatively similar algebraic decay of the interactions with σ=1σ=1σ=1. However, for annealing experiments the measured exponent zzz is close to the RG predictions. On the other hand, the relevant exponents of the dynamic behavior (zzz and θθθ) are slightly different than the RG predictions, most likely due to the fact that they may depend on the especific dynamics used (Metropolis in the present paper).
We consider Ising models with long-range ferromagnetic pair interactions decaying as for . We first find approximate values for the critical temperature. We use a cluster mean-field approach combined with finite-size scaling and Vanden Broeck and Schwartz transformations. For we find which can be compared with recent results of Luijten and Blöte who found , and which is two orders of magnitude more accurate than any previous results. Since we use a mean-field cluster approximation as part of our approach, the accuracy for larger values of decreases significantly. In addition to we obtain approximate values for the critical exponents , and using the coherent anomaly method. For we obtain , , and - all extremely close to the predictions of renormalization group calculations which say that these exponents should take on their classical values for this value of .
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.