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Journal of Physics A Mathematical and General

The square-lattice Ising model with first and second neighbour interactions

J. Oitmaa

Published May 1, 1981• 109 citations
Publisher

Abstract

The critical behaviour of the square-lattice Ising model, with nearest and next-nearest neighbour interactions of either sign, has been investigated by means of high-temperature series. lhe location of the critical lines in the coupling constant plane has been accurately determined. Along the critical line which corresponds to transitions to the layered or superantiferromagnetic state a breakdown of universality is observed and explicit numerical estimates obtained for the exponent of the ordering susceptibility.

DOI
10.1088/0305-4470/14/5/035

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Journal of Physics A Mathematical and General•Apr 21, 1987•13 citations
The critical behaviour of the square-lattice Ising model, with nearest and next-nearest neighbour interactions of either sign, has been investigated by means of high-temperature series. lhe location of the critical lines in the coupling constant plane has been accurately determined. Along the critical line which corresponds to transitions to the layered or superantiferromagnetic state a breakdown of universality is observed and explicit numerical estimates obtained for the exponent of the ordering susceptibility.
Reports studies of the critical behaviour of the Ising model with first and second neighbour interactions J 1 , J 2 on the triangular lattice. High-temperature series have been used to determine the locations of the ferromagnetic critical line and the critical line in the region (J 1 <0, J 2 >0). On the latter line the ordering susceptibility appears to diverge algebraically with the exponent gamma approximately 2.4. The existence of a sinusoidal phase and an associated Lifshitz point are conjectured.
The critical behaviour of the Ising model on the square lattice with ferromagnetic nearest-neighbour interactions (J) and anti-ferromagnetic next-nearest neighbour interactions (J') is discussed for small lambda =J/ mod J' mod . The singular part of the free energy is calculated to second order in lambda by perturbation theory. Near the critical temperature of the unperturbed system (J=0), this expansion is found to have a form which may be exponentiated yielding, for non-zero lambda , a non-universal critical line along which the exponents vary continuously with lambda . For small lambda , the specific heat exponent, alpha = alpha 1 lambda 2 +O( lambda 2 ) with alpha 1 approximately=1.5.
We have calculated the high-temperature series expansion of the zero-field susceptibility of the square-lattice Ising model with first and second neighbour interactions to the 20th order by computer. Our results extend the previous calculation by Hsiao and Lin to two more orders. We use the Padé approximants to estimate the critical exponent γ and the critical temperature. Our result 1.747 < γ < 1.753 supports the universality conjecture that all two-dimensional Ising models have the same critical exponent γ = 1.75.
The critical behaviour of an Ising model with competing first- and third-nearest-neighbour interactions ('biaxial next-nearest-neighbour Ising' or BNNNI model) on the square lattice is investigated by finite lattice methods. In the ferromagnetic region, the phase boundary is located with an accuracy at least equal to that of alternative methods. In the antiphase region, distinctive structure in finite lattice estimators is found over an extended temperature range. However, the nature of the transition remains unclear.