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Physical Review Letters

Absence of Self-Averaging and Universal Fluctuations in Random Systems near Critical Points

Amnon Aharony A, A. Brooks Harris

Published Oct 28, 1996• 234 citations
Publisher

Abstract

The distributions $P(X)$ of singular thermodynamic quantities, on an ensemble of $d$-dimensional quenched random samples of linear size $L$ near a critical point, are analyzed using the renormalization group. For $L$ much larger than the correlation length $\ensuremath{\xi}$, we recover strong self-averaging (SA): $P(X)$ approaches a Gaussian with relative squared width ${R}_{X}\ensuremath{\sim}(L/\ensuremath{\xi}{)}^{\ensuremath{-}d}$. For $L\ensuremath{\ll}\ensuremath{\xi}$ we show weak SA ( ${R}_{X}$ decays with a small power of $L$) or no SA [ $P(X)$ approaches a non-Gaussian, with universal $L$-independent relative cumulants], when the randomness is irrelevant or relevant, respectively.

DOI
10.1103/physrevlett.77.3700

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Finite-size scaling in disordered systems

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The distributions P(X)P(X)P(X) of singular thermodynamic quantities, on an ensemble of ddd-dimensional quenched random samples of linear size LLL near a critical point, are analyzed using the renormalization group. For LLL much larger than the correlation length \ensuremathξ\ensuremath{\xi}\ensuremathξ, we recover strong self-averaging (SA): P(X)P(X)P(X) approaches a Gaussian with relative squared width RX\ensuremath∼(L/\ensuremathξ)\ensuremath−d{R}_{X}\ensuremath{\sim}(L/\ensuremath{\xi}{)}^{\ensuremath{-}d}RX​\ensuremath∼(L/\ensuremathξ)\ensuremath−d. For L\ensuremath≪\ensuremathξL\ensuremath{\ll}\ensuremath{\xi}L\ensuremath≪\ensuremathξ we show weak SA ( RX{R}_{X}RX​ decays with a small power of LLL) or no SA [ P(X)P(X)P(X) approaches a non-Gaussian, with universal LLL-independent relative cumulants], when the randomness is irrelevant or relevant, respectively.
We simulated site dilute Ising models in d=3d\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}3d=3 dimensions for several lattice sizes LLL. For each LLL singular thermodynamic quantities XXX were measured at criticality and their distributions P(X)P(X)P(X) were determined for ensembles of several thousand random samples. For $L\ensuremath{\rightarrow}\ensuremath{\infty}$ the relative width of P(X)P(X)P(X) tends to a universal constant: there is no self-averaging. The width of the distribution of the sample (i)(i)(i) dependent pseudocritical temperatures Tc(i,L){T}_{c}(i,L)Tc​(i,L) scales as $\ensuremath{\delta}{T}_{c}(L)\ensuremath{\sim}{L}^{\ensuremath{-}1/\ensuremath{\nu}}$ and not as $\mathrm{\ensuremath{\sim}}{L}^{\ensuremath{-}d/2}$. The sample dependence of Xi(T,L){X}_{i}(T,L)Xi​(T,L) enters dominantly, but not exclusively, via Tc(i,L){T}_{c}(i,L)Tc​(i,L).
We evaluate by Monte Carlo simulations various singular thermodynamic quantities XXX for ensembles of quenched random Ising and Ashkin-Teller models. The measurements are taken at Tc{T}_{c}Tc​ and we study how the distributions P(X)P(X)P(X) (and, in particular, their relative squared width, RX){R}_{X})RX​) over the ensemble depend on the system size l.l.l. The Ashkin-Teller model was studied in the regime where bond randomness is irrelevant and we found weak self-averaging; ${R}_{X}\ensuremath{\sim}{l}^{\ensuremath{\alpha}/\ensuremath{\nu}}\ensuremath{\rightarrow}0,$ where $\ensuremath{\alpha}<0$ and $\ensuremath{\nu}$ are the exponents (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 RX{R}_{X}RX​ tends to a constant, as predicted by Aharony and Harris. We tested whether this constant is universal. However, this constant is different for canonical and grand canonical disorder. We identify the pseudocritical temperature of each sample i,i,i, Tc(i,l),{T}_{c}(i,l),Tc​(i,l), as the temperature at which the susceptibility reaches its maximal value ${\ensuremath{\chi}}^{\mathrm{max}}.$ The distribution of these sample dependent Tc(i,l){T}_{c}(i,l)Tc​(i,l) was investigated; we found that its variance scales as $[\ensuremath{\delta}{T}_{c}(l){]}^{2}\ensuremath{\sim}{l}^{\ensuremath{-}2/\ensuremath{\nu}}.$ Our previously proposed finite size scaling ansatz for disordered systems was tested and found to hold. We did observe deviations from a single function, which imply that sample dependent scaling functions are needed. These deviations are, however, relatively small and hence to obtain a fixed statistical error it may be more computationally efficient to measure ${\ensuremath{\chi}}^{\mathrm{max}}$ than the commonly used $\ensuremath{\chi}{(T}_{c}^{\ensuremath{\infty}}).$
We simulated site dilute Ising models in d=3d=3d=3 dimensions for several lattice sizes LLL. For each LLL singular thermodynamic quantities XXX were measured at criticality and their distributions P(X)P(X)P(X) were determined, for ensembles of several thousand random samples. For L→∞L \to \inftyL→∞ the width of P(X)P(X)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 Tc(i,L)T_c(i,L)Tc​(i,L) scales as δTc(L)∼L−1/νδT_c(L) \sim L^{-1/ν}δTc​(L)∼L−1/ν and NOT as ∼L−d/2\sim L^{-d/2}∼L−d/2. Finite size scaling holds; the sample dependence of Xi(Tc)X_i(T_c)Xi​(Tc​) enters predominantly through Tc(i,L)T_c(i,L)Tc​(i,L).
Abstract In this paper, we review recent results on sample-to-sample fluctuations in a critical Ising model with quenched random ferromagnetic couplings. In particular, in terms of the renormalized replica Ginzburg–Landau Hamiltonian in dimensions D < 4 an explicit expression for the probability distribution function (PDF) of the critical free energy fluctuations is derived. Next, using known fixed-point values for the renormalized coupling parameters the universal curve for such PDF in the dimension D = 3 is obtained. For the specific case of the two-dimensional Ising model, using replica calculations in the renormalization group framework, we derive explicit expressions for the PDF of the critical internal energy and for the specific heat fluctuations. It is shown that the disorder distribution of internal energy is Gaussian, and its typical sample-to-sample fluctuations as well as its average value scale with the system size L like ~ Llnln(L). In contrast, the specific heat is shown to be self-averaging with a distribution function that tends to a δ-peak in the thermodynamic limit L → ∞.
We consider the sample to sample fluctuations that occur in the value of a thermodynamic quantity PPP in an ensemble of finite systems with quenched disorder, at equilibrium. The variance of PPP, VP{V}_{P}VP​, which characterizes these fluctuations is calculated as a function of the systems' linear size lll, 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=2d=2d=2 is the dimension). At criticality though, the results indicate that the magnetization MMM 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 EEE at criticality is clearly weakly self-averaging, that is ${V}_{E}\ensuremath{\sim}{l}^{\ensuremath{-}{y}_{v}}$ with 0<yv<d0<{y}_{v}<d0<yv​<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 PPP which scales as ${l}^{\ensuremath{\rho}}$ at criticality, its variance VP{V}_{P}VP​ 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 VE{V}_{E}VE​. The theory may also be compatible with the data for the variance of the magnetization VM{V}_{M}VM​ and the variance of the specific heat VC{V}_{C}VC​, but evidence for this is less convincing.