Exact solution of the 2D « brickwork » Ising model with second neighbour interactions in the horizontal direction

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Abstract
Boehm and Bak [11] have analysed, within the mean field approximation, a model which leads to a phase diagram including multiple phase transitions between commensurate phases. It happens that an exactly soluble two-dimensional Ising model exhibiting similar mean field features, is available. This model is built on a brickwork lattice, and its exact partition function is obtained within the following assumptions : i) In the vertical direction, nearest neighbours are coupled by ferromagnetic interactions 2 J' (J' > 0). ii) In the horizontal direction, nearest neighbours are coupled by ferromagnetic interactions J (J > 0) while nextnearest neighbours are coupled by arbitrary interactions J2. Results are discussed with respect to the value of the ratio J2/J. It is found in particular that : 1) For J2/J > — 1/2, the system orders ferromagnetically at a critical temperature Tc = 1/kB βc defined by tanh 2 βc J'. sinh 2 βc J. exp 4 βc J 2 = 1. 2) For J2/J ≤ - 1/2, no ordering occurs even at zero temperature. Ground states and their degeneracy are examined. These results are compared with those obtained for the same system in the mean field approximation, which predicts the existence of a devil's staircase behaviour for the periodicity versus temperature curve, and with those given for the two-dimensional ANNNI model by a recent Monte-Carlo study [13]