Ising model with two-, three- and four-spin interactions

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The two-state Ising model with nearest-neighbour bilinear exchange constant J(2), three-spin coupling constant J(3) and four-spin coupling constant J(4) is studied using the effective-field approximation. The temperature variation in spontaneous magnetization is studied for different coordination numbers z and for various values of the parameters alpha =J(3)/J(2) and alpha '=J(4)/J(2). It is seen that, in the absence of three-spin coupling, there exists a critical value of alpha ' (say, alpha 'c) above which the magnetization becomes double valued at and beyond the Curie temperature Tc, forming a bulge near Tc which indicates the occurrence of a first-order transition. The expressions for alpha 'c for z=3, 4, 6 are derived and it is found that alpha 'c increases as z increases. In the absence of four-spin coupling it is found that first-order transitions appear for alpha >or=0. The magnetization curves are also studied for the case when all the interactions are present. The results are discussed with reference to those obtained from the molecular-field approximation.