Renormalisation group attemts to obtain the transition line of the square-lattice bond-dilute Ising model

Abstract

Two different renormalisation-group (RG) approaches are used to determine approximate solutions for the paramagnetic-ferromagnetic transition line of the square-lattice bond-dilute first-neighbour-interaction Ising model. The first approach (in two different versions, named RG1 and RG2) consists of substituting a single bond for an H-shaped cluster. In the second approach (RG3, RG4, RG5 and RG6) the authors take advantage of the self-duality of the square-lattice and define a duality renormalisation operation. All six renormalisation groups are defined as operations on the p-t space, where p is the independent occupancy probability, and t=tanh(J/k_bT). Both approaches yield very good results, including the exact values t_c= square root 2-1 and p_c=¹/₂ (for all six RGs) and dt₀/dp mod _p=1=8-6 square root 2 (for RG5 and RG6), as well as the correct asymptotical behaviour in the neighbourhood of t=1 (that is, T=0). The transition line obtained by RG5 might be extremely close (probably better than 0.5% in the most unfavourable case) to the unknown exact solution.