Self-Averaging, Distribution of Pseudo-Critical Temperatures and Finite Size Scaling in Critical Disordered Systems

TL;DR

This paper investigates the distribution of thermodynamic quantities and pseudo-critical temperatures in disordered systems at criticality, confirming theoretical predictions and revealing different self-averaging behaviors in models with varying disorder relevance.

Contribution

It provides the first detailed Monte Carlo analysis of pseudo-critical temperature distributions and their scaling, confirming and extending renormalization group predictions for disordered critical systems.

Findings

Relative squared width $R_X$ scales with $l^{rac{eta}{ u}}$ in weakly disordered models.

In strongly disordered models, $R_X$ tends to a universal constant, differing between disorder types.

Variance of pseudo-critical temperatures scales as $l^{-2/ u}$, not $l^{-d}$.

Abstract

The distributions $P(X)$ of singular thermodynamic quantities in an ensemble of quenched random samples of linear size $l$ at the critical point $T_c$ are studied by Monte Carlo in two models. Our results confirm predictions of Aharony and Harris based on Renormalization group considerations. For an Ashkin-Teller model with strong but irrelevant bond randomness we find that the relative squared width, $R_X$, of $P(X)$ is weakly self averaging. $R_X\sim l^{\alpha/\nu}$, where $\alpha$ is the specific heat exponent and $\nu$ is the correlation length exponent of the pure model fixed point governing the transition. For the site dilute Ising model on a cubic lattice, known to be governed by a random fixed point, we find that $R_X$ tends to a universal constant independent of the amount of dilution (no self averaging). However this constant is different for canonical and grand canonical disorder. We study the distribution of the pseudo-critical temperatures $T_c(i,l)$ of the ensemble defined as the temperatures of the maximum susceptibility of each sample. We find that its variance scales as $(\delta T_c(l))^2 \sim l^{-2/\nu}$ and NOT as $\sim l^{-d}. We find that $R_\chi$ is reduced by a factor of $\sim 70$ with respect to $R_\chi (T_c)$ by measuring $\chi$ of each sample at $T_c(i,l)$. We analyze correlations between the magnetization at criticality $m_i(T_c,l)$ and the pseudo-critical temperature $T_c(i,l)$ in terms of a sample independent finite size scaling function of a sample dependent reduced temperature $(T-T_c(i,l))/T_c$. This function is found to be universal and to behave similarly to pure systems.