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. 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 $\chi$ are non self averaging, i.e. $\frac{V_{\chi }}{{\chi }^2},\frac{V_M}{M^2}\nrightarrow 0$. 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 $\alpha <0$ ($\alpha$ of the disordered model) then, for a quantity $P$ which scales as $l^{\rho }$ at criticality, its variance $V_P$ will scale asymptotically as $l^{2\rho +\frac{\alpha }{\nu }}$. we found very good agreement between the theory and the data for $V_{\chi }$ and $V_E$.

33 pages, RevTex, 16 figures in tar compressed form included, Submitted to Phys. Rev. E The figures which were missing are now included, in a uuencoded tar compressed form

References in corpus (1)

• Critical behaviour of the Random--Bond Ashkin--Teller Model, a Monte-Carlo study

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