Critical properties of dilute quasi-low-dimensional Ising magnets

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Abstract
A scaling transformation is developed for the pure square-lattice-spin-1/2 Ising model with different exchange couplings J1, J2 in the two lattice directions. The transformation preserves Onsager's exact duality relationship and so can treat cases of strong lattice anisotropy. It is generalised to the dilute case and used to obtain critical exponents and critical curves (transition temperature Tc as a function of concentration p of magnetic bonds for various exchange ratios gamma =J2/J1). For gamma small, corresponding to weakly coupled chains, the transition temperature falls rapidly with dilution, then levels off, finally vanishing at the two-dimensional percolation concentration. Extensions are made to three-dimensional dilute magnets, including weakly coupled layer and chain magnets as special cases. Critical curves and exponents are calculated, and a quantitative description is obtained of various crosscovers between low- and high-dimensional behaviour which occur in the quasi-low-dimensional magnets as the transition temperature falls with dilution and the weak exchanges become significant.