The quenched-disordered Ising model in two and four dimensions

TL;DR

This paper reviews the critical behavior of the quenched-disordered Ising model in two and four dimensions, focusing on universality hypotheses, logarithmic corrections, and scaling relations through finite-size scaling analysis.

Contribution

It provides a detailed finite-size scaling analysis supporting strong universality in 2D and clarifies the nature of corrections to scaling in 4D.

Findings

Supports strong universality hypothesis in 2D with logarithmic corrections.

Identifies and discusses corrections to scaling in 4D.

Clarifies the debate on the critical behavior of the disordered Ising model.

Abstract

We briefly review the Ising model with uncorrelated, quenched random-site or random-bond disorder, which has been controversial in both two and four dimensions. In these dimensions, the leading exponent alpha, which characterizes the specific-heat critical behaviour, vanishes and no Harris prediction for the consequences of quenched disorder can be made. In the two-dimensional case, the controversy is between the strong universality hypothesis which maintains that the leading critical exponents are the same as in the pure case and the weak universality hypothesis, which favours dilution-dependent leading critical exponents. Here the random-site version of the model is subject to a finite-size scaling analysis, paying special attention to the implications for multiplicative logarithmic corrections. The analysis is fully supportive of the scaling relations for logarithmic corrections and of the strong scaling hypothesis in the 2D case. In the four-dimensional case unusual corrections to scaling characterize the model, and the precise nature of these corrections has been debated. Progress made in determining the correct 4D scenario is outlined.