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arXiv:cond-mat/9802102 · Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Self-averaging, distribution of pseudocritical temperatures, and finite size scaling in critical disordered systems

Shai Wiseman, Eytan Domany

Published Sep 1, 1998• 129 citations
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Abstract

We evaluate by Monte Carlo simulations various singular thermodynamic quantities $X$ for ensembles of quenched random Ising and Ashkin-Teller models. The measurements are taken at ${T}_{c}$ and we study how the distributions $P(X)$ (and, in particular, their relative squared width, ${R}_{X})$ over the ensemble depend on the system size $l.$ 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 ${R}_{X}$ 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 $i,$ ${T}_{c}(i,l),$ as the temperature at which the susceptibility reaches its maximal value ${\ensuremath{\chi}}^{\mathrm{max}}.$ The distribution of these sample dependent ${T}_{c}(i,l)$ 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 $\ensuremath{\chi}{(T}_{c}^{\ensuremath{\infty}}).$

DOI
10.1103/physreve.58.2938

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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; RX\ensuremath∼l\ensuremathα/\ensuremathν\ensuremath→0,{R}_{X}\ensuremath{\sim}{l}^{\ensuremath{\alpha}/\ensuremath{\nu}}\ensuremath{\rightarrow}0,RX​\ensuremath∼l\ensuremathα/\ensuremathν\ensuremath→0, where \ensuremathα<0\ensuremath{\alpha}<0\ensuremathα<0 and \ensuremathν\ensuremath{\nu}\ensuremathν 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 \ensuremathχmax.{\ensuremath{\chi}}^{\mathrm{max}}.\ensuremathχmax. 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 [\ensuremathδTc(l)]2\ensuremath∼l\ensuremath−2/\ensuremathν.[\ensuremath{\delta}{T}_{c}(l){]}^{2}\ensuremath{\sim}{l}^{\ensuremath{-}2/\ensuremath{\nu}}.[\ensuremathδTc​(l)]2\ensuremath∼l\ensuremath−2/\ensuremathν. 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 \ensuremathχmax{\ensuremath{\chi}}^{\mathrm{max}}\ensuremathχmax than the commonly used \ensuremathχ(Tc\ensuremath∞).\ensuremath{\chi}{(T}_{c}^{\ensuremath{\infty}}).\ensuremathχ(Tc\ensuremath∞​).
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