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Journal of Physics A Mathematical and General

Finite-range scaling for a one-dimensional system with long-range interactions

Katarina Uzelac, Zvonko Glumac

Published Apr 7, 1988• 19 citations
Publisher

Abstract

A procedure, similar to the finite-size scaling, is proposed for systems with long-range interactions. It is applied to the one-dimensional Ising model with interactions decaying as 1/ mod i-j mod 1+ sigma with distance mod i-j mod . The critical temperature and exponent nu are calculated for sigma <or=1.
DOI
10.1088/0305-4470/21/7/011

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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.
Small systems consisting of particles interacting with long-range potentials exhibit enormous size effects. The Tsallis conjecture [Tsallis, Fractals 3, 541 (1995)FRACEG0218-348X10.1142/S0218348X95000473], valid for translationally invariant systems with long-range interactions, states a well-known scaling relating different sizes. Here we propose to generalize this conjecture to systems with this symmetry broken, by adjusting one parameter that determines an effective distance to compute the strength of the interaction. We apply this proposal to the one-dimensional Ising model with ferromagnetic interactions that decay as 1/r^{1+σ} in the region where the model has a finite critical temperature. We demonstrate the convenience of using this generalization to study finite-size effects, and we compare this approach with the finite-size scaling theory.
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 .