Critical temperature for two-dimensional ising ferromagnets with quenched bond disorder

Abstract

The configuration-averaged free energy of a quenched, random bond Ising model on a square lattice which contains an equal mixture of two types of ferromagnetic bonds J₁ and J₂ is shown to obey the same duality relation as the ordered rectangular model with the same two bond strengths. If the random.system has a single, sharp critical point, the critical temperature T_c must be identical to that of the ordered system, i.e., sinh(2J ₁/kT _c) sinh(2J ₂/kT _c) = 1. Since ^(B)_c = 1/2, we can takeJ ₂ → 0 and use Bergstresser-type inequalities to obtain(ρ/ρdp) exp(−2J ₁/kT_c¦_p=p_c + = 1, in agreement with Bergstresser's rigorous result for the diluted ferromagnet near the percolation threshold.