Lack of Self-Averaging in Critical Disordered Systems

Abstract

We consider the sample to sample fluctuations that occur in the value of a thermodynamic quantity $P$ in an ensemble of finite systems with quenched disorder, at equilibrium. The variance of $P$, $V_{P}$, which characterizes these fluctuations is calculated as a function of the systems&#39; linear size $l$, focusing on the behavior at the critical point. The specific model considered is the bond-disordered Ashkin-Teller model on a square lattice. Using Monte Carlo simulations, several bond-disordered Ashkin-Teller models were examined, including the bond-disordered Ising model and the bond-disordered four-state Potts model. It was found that far from criticality the energy, magnetization, specific heat and susceptibility are strongly self averaging, that is $V_{P}\sim l^{-d}$ (where $d=2$ is the dimension). At criticality though, the results indicate that the magnetization $M$ and the susceptibility $χ$ are non self averaging, i.e. $\frac{V_χ}{χ^{2}}, \frac{V_{M}}{M^{2}}\not \rightarrow 0$. The energy $E$ at criticality is weakly self averaging, that is $V_{E}\sim l^{-y_{v}}$ with $0<y_{v}<d$. Less conclusively, and possibly only as a transient behavior, the specific heat too is found to be weakly self averaging. A phenomenological theory of finite size scaling for disordered systems is developed. Its main prediction is that when the specific heat exponent $α<0$ ($α$ of the disordered model) then, for a quantity $P$ which scales as $l^ρ$ at criticality, its variance $V_{P}$ will scale asymptotically as $l^{2ρ+\fracαν}$. we found very good agreement between the theory and the data for $V_χ$ and $V_{E}$.