Lack of Self Averaging and Finite Size Scaling in Critical Disordered Systems

TL;DR

This study investigates the behavior of disordered 3D Ising models at criticality, revealing the absence of self-averaging and clarifying the finite size scaling of pseudocritical temperatures.

Contribution

It demonstrates the lack of self-averaging in critical disordered systems and clarifies the scaling of pseudocritical temperature distributions.

Findings

Width of distribution of critical quantities tends to a constant as system size increases.

Sample-dependent pseudocritical temperatures scale as L^{-1/ν}, not L^{-d/2}.

Finite size scaling remains valid despite lack of self-averaging.

Abstract

We simulated site dilute Ising models in $d=3$ dimensions for several lattice sizes $L$. For each $L$ singular thermodynamic quantities $X$ were measured at criticality and their distributions $P(X)$ were determined, for ensembles of several thousand random samples. For $L \to \infty$ the width of $P(X)$ tends to a universal constant, i.e. there is no self averaging. The width of the distribution of the sample dependent pseudocritical temperatures $T_c(i,L)$ scales as $\delta T_c(L) \sim L^{-1/\nu}$ and NOT as $\sim L^{-d/2}$. Finite size scaling holds; the sample dependence of $X_i(T_c)$ enters predominantly through $T_c(i,L)$.