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Journal of Physics A Mathematical and General

Finite-range scaling study of the 1D long-range Ising model

Zvonko Glumac, Katarina Uzelac

Published Oct 21, 1989• 41 citations
Publisher

Abstract

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.

DOI
10.1088/0305-4470/22/20/020

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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.
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 behaviour of the one-dimensional q-state Potts model with long-range interactions decaying with distance r as r −(1+σ) has been studied in the wide range of parameters 0 < σ ≤ 1 and 1 16 ≤ q ≤ 64. A transfer matrix has been constructed for a truncated range of interactions for integer and continuous q, and finite range scaling has been applied. Results for the phase diagram and the correlation length critical exponent are presented. Short title: Critical behaviour of the 1D LR q-state Potts model Recently, finite range scaling (FRS) appeared as an efficient method to deal with various discrete one-dimensional models with long range (LR) interactions (Glumac and Uzelac, 1989, 1991). The aim of the present study is to apply it to the onedimensional q-state Potts model with LR interactions, defined by the hamiltonian