Effect of random defects on the critical behaviour of Ising models

TLDR

The study employs a cumulant expansion to compute the transition temperature of Ising models with random‑bond defects. The results show that random‑bond defects increase the transition‑temperature sensitivity (−Tc⁻¹dTc/dx|₀=1.329) and modify the critical specific‑heat behavior, broadening its divergence over a defect‑concentration–dependent interval and producing a maximum of order x⁻¹, effects that appear only when the specific‑heat exponent α is positive.

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 -Tc-1 dTc/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 -Tc-1 dTc/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/Tc 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.