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 C₀/(1+x gamma ²C₀), where gamma is a constant and C₀ 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.