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arXiv:cond-mat/9304025 · Journal de Physique I

Lack of self-averaging in weakly disordered one-dimensional systems

A. Crisanti, G. Paladin, Maurizio Serva, Angelo Vulpiani

Published Oct 1, 1993• 2 citations
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Abstract

We introduce a one dimensional disordered Ising model which at zero temperature is characterized by a non-trivial, non-self-averaging, overlap probability distribution when the impurity concentration vanishes in the thermodynamic limit. The form of the distribution can be calculated analytically for any realization of disorder. For non-zero impurity concentration the distribution becomes a self-averaging delta function centered on a value which can be estimated by the product of appropriate transfer matrices.

DOI
10.1051/jp1:1993227

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We introduce a one dimensional disordered Ising model which at zero temperature is characterized by a non-trivial, non-self-averaging, overlap probability distribution when the impurity concentration vanishes in the thermodynamic limit. The form of the distribution can be calculated analytically for any realization of disorder. For non-zero impurity concentration the distribution becomes a self-averaging delta function centered on a value which can be estimated by the product of appropriate transfer matrices.
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.
In a disordered system, a quantity is self-averaging when the ratio between its variance for disorder realizations and the square of its mean decreases as the system size increases. Here, we consider a chaotic disordered many-body quantum system and search for a relationship between self-averaging behavior and the properties of the distributions over disorder realizations of various quantities and at different timescales. An exponential distribution, as found for the survival probability at long times, explains its lack of self-averaging, since the mean and the dispersion are equal. Gaussian distributions, however, are obtained for both self-averaging and non-self-averaging quantities. Our studies show also that one can make conclusions about the self-averaging behavior of one quantity based on the distribution of another related quantity. This strategy allows for semianalytical results, and thus circumvents the limitations of numerical scaling analysis, which are restricted to few system sizes.
Current research on the disordered topological quantum phases primarily focuses on the uncorrelated and short-range correlated disorder regime. These topological Anderson insulators are typically self-averaging. However, topological quantum systems with long-range correlated disorder have received limited attention due to the absence of tractable analytical methods. In fact, the long-range correlated disorder introduces more complex effects on topological quantum states. Here, we demonstrate that the long-range correlated disorder could induce anomalously statistical feature where the topological Anderson states become non-self-averaging, and the phase diagram is strongly dependent on individual disorder configurations. We term this statistical phase the non-self-averaging topological Anderson insulator. The non-self-averaging property is identified by the non-vanishing finite values of the relative variance of the Lyapunov exponent in the thermodynamic limit, alongside the non-Gaussian distributions of the Lyapunov exponent. Consequently, the topological properties of a single disordered sample deviate from the ensemble average, causing a breakdown of the central limit theorem. The non-self-averaging topological Anderson insulator provides insights into the interplay between correlated disorder and topology.
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 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).