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Computational Evidence for the Transition Point of a Square Ising Lattice with First and Second Neighbour Interactions

W. Bitterlich, Rainer J. Jelitto

Published Jan 1, 1968• 4 citations

Abstract

Abstract From numerical calculations for a toroidally closed Ising lattice of finite diameter it is seen that it is possible to obtain the transition point of a square Ising lattice with nearest and next‐nearest neighbour interactions to a high degree of accuracy. In the case of a lattice with pure nearest neighbour interactions the curves of the internal energy for all diameters of the torus intersect in one point, which is the transition point of the infinite lattice. When next‐nearest neighbour coupling is present, this is no longer exactly valid, but the dependence of the situation of the different points of intersection on the diameter is so exceedingly weak that a rapid convergence to the Curie point may be assumed with increasing size of the lattice and the transition point can be deduced thereby. The assumptions involved in this method and its consequences are discussed and the result of the investigation is compared with a theory of Vaks et al., which deals with a similar type of interaction.

DOI: 10.1002/pssb.19680280137

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Abstract From numerical calculations for a toroidally closed Ising lattice of finite diameter it is seen that it is possible to obtain the transition point of a square Ising lattice with nearest and next‐nearest neighbour interactions to a high degree of accuracy. In the case of a lattice with pure nearest neighbour interactions the curves of the internal energy for all diameters of the torus intersect in one point, which is the transition point of the infinite lattice. When next‐nearest neighbour coupling is present, this is no longer exactly valid, but the dependence of the situation of the different points of intersection on the diameter is so exceedingly weak that a rapid convergence to the Curie point may be assumed with increasing size of the lattice and the transition point can be deduced thereby. The assumptions involved in this method and its consequences are discussed and the result of the investigation is compared with a theory of Vaks et al., which deals with a similar type of interaction.

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Abstract. We use improved Monte-Carlo algorithms to study the antiferromagnetic 2D-Ising model with competing interactions J1 on nearest neighbour and J2 on next-nearest neighbour bonds. The finite-temperature phase diagram is divided by a critical point at J2 = J1/2 where the groundstate is highly degenerate. To analyse the phase boundaries we look at the specific heat and the energy distribution for various ratios of J2/J1. We find a first order transition for small J2> J1/2 and the transition temperature suppressed to TC = 0 at the critical point. 1.

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.

We use improved Monte-Carlo algorithms to study the antiferromagnetic 2D-Ising model with competing interactions J 1 on nearest neighbour and J 2 on next-nearest neighbour bonds. The finite-temperature phase diagram is divided by a critical point at J 2 = J 1 /2 where the groundstate is highly degenerate. To analyse the phase boundaries we look at the speciflc heat and the energy distribution for various ratios of J 2 / 1 . We find a first order transition for small J 2 > J 1 /2 and the transition temperature suppressed to T C = 0 at the critical point.