Numerical studies of the Ising chain with long-range ferromagnetic interactions

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Abstract
A long-range Ising chain, which has ferromagnetic 1/r3 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.