Effect of random defects on the critical behaviour of Ising models

Abstract

A cumulant expansion is used to calculate the transition temperature of Ising models with random-bond defects. For a concentration, x, of missing interactions in the simple-square Ising model the author finds -T c -1 dT c /dx mod x=0 =1.329 compared with the mean-field value of one. If the interactions are independent random variable with a width delta J/J identical to epsilon , the result is -T c -1 dT c /d epsilon 2 mod epsilon =0 =0.312 compared with the mean-field results of zero. An approximation yields the specific heat in the critical regime as C approximately C 0 /(1+x gamma 2 C 0 ), where gamma is a constant and C 0 is the unperturbed specific heat at a renormalized temperature. Thus, the specific heat divergence is broadened over a temperature interval Delta T, with Delta T/T c approximately x (1 alpha )/, where alpha is the critical exponent for the specific heat, and a maximum value of order x -1 is attained. Heuristic arguments show that this smoothing effect occurs if alpha >0.