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' 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 [Phys. Rev. 64, 178 (1943)]. Using extensive 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 all thermodynamic quantities which were examined (energy, magnetization, specific heat, susceptibility) are strongly self-averaging, that is ${V}_{P}\ensuremath{\sim}{l}^{\ensuremath{-}d}$ (where $d=2$ is the dimension). At criticality though, the results indicate that the magnetization $M$ and the susceptibility $\ensuremath{\chi}$ are non-self-averaging, i.e., $\frac{{V}_{\ensuremath{\chi}}}{{\ensuremath{\chi}}^{2}}, \frac{{V}_{M}}{{M}^{2}}\ensuremath{\nrightarrow}0$. The energy $E$ at criticality is clearly weakly self-averaging, that is ${V}_{E}\ensuremath{\sim}{l}^{\ensuremath{-}{y}_{v}}$ 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, based on physical considerations similar to those leading to the Harris criterion. Its main prediction is that when the specific heat exponent $\ensuremath{\alpha}<0$ ($\ensuremath{\alpha}$ of the disordered model) then, for a quantity $P$ which scales as ${l}^{\ensuremath{\rho}}$ at criticality, its variance ${V}_{P}$ will scale asymptotically as ${l}^{2\ensuremath{\rho}+\frac{\ensuremath{\alpha}}{\ensuremath{\nu}}}$. The theory is not applicable in the asymptotic limit ($l\ensuremath{\rightarrow}\ensuremath{\infty}$) to the bond-disordered Ashkin-Teller model where $\frac{\ensuremath{\alpha}}{\ensuremath{\nu}}={0}_{+}$. Nonetheless in the accessible range of lattice sizes we found very good agreement between the theory and the data for ${V}_{\ensuremath{\chi}}$ and ${V}_{E}$. The theory may also be compatible with the data for the variance of the magnetization ${V}_{M}$ and the variance of the specific heat ${V}_{C}$, but evidence for this is less convincing.