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,
VP, 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<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
P which scales as ${l}^{\ensuremath{\rho}}$ at criticality, its variance
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. The theory may also be compatible with the data for the variance of the magnetization
VM and the variance of the specific heat
VC, but evidence for this is less convincing.
We investigate and contrast, via the Wang-Landau (WL) algorithm, the effects of quenched bond randomness on the self-averaging properties of two Ising spin models in 2d. The random bond version of the superantiferromagnetic (SAF) square model with nearest- and next-nearest-neighbor competing interactions and the corresponding version of the simple ferromagnetic Ising model are studied. We find that, the random bond SAF model shows a strong violation of self-averaging, much stronger than that observed in the case of the random bond Ising model. Our analysis of the asymptotic scaling behavior of the variance of the distribution of the sample-dependent pseudocritical temperatures is found to be consistent with the renormalization group prediction of Aharony and Harris. Using this alternative approach, we find estimates of the correlation length exponent
ν in agreement with results obtained from the usual finite-size scaling (FSS) methodology.
Abstract To understand the finite-size-scaling properties of phases transitions in classical and quantum models in the presence of quenched disorder, it has proven to be fruitful to introduce the notion of a finite-size-pseudo-critical point in each disordered sample and to analyze its sample-to-sample fluctuations as a function of the size. For the many-body-localization transition, where very strong eigenstate-to-eigenstate fluctuations have been numerically reported even within a given disordered sample at a given energy density (Yu, Luitz and Clark 2016 arXiv: 1606.01260 and Khemani, Lim, Sheng and Huse 2016 arXiv: 1607.05756 ), it seems thus useful to introduce the notion of a finite-size-pseudo-critical point for each individual eigenstate and to study its eigenstate-to-eigenstate fluctuations governed by the correlation length exponent ν . The scaling properties of critical eigenstates are also expected to appear much more clearly if one considers each eigenstate at its finite-size-pseudo-critical point, where it is ‘truly critical’, while standard averages over eigenstates and samples in the critical region actually see a mixture of states that are effectively either localized or delocalized.
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
Tc and we study how the distributions
P(X) (and, in particular, their relative squared width,
RX) 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
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, 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) 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 consider the two-dimensional randomly site diluted Ising model and the random-bond +/-J Ising model (also called the Edwards-Anderson model), and study their critical behavior at the paramagnetic-ferromagnetic transition. The critical behavior of thermodynamic quantities can be derived from a set of renormalization-group equations, in which disorder is a marginally irrelevant perturbation at the two-dimensional Ising fixed point. We discuss their solutions, focusing in particular on the universality of the logarithmic corrections arising from the presence of disorder. Then, we present a finite-size scaling analysis of high-statistics Monte Carlo simulations. The numerical results confirm the renormalization-group predictions, and in particular the universality of the logarithmic corrections to the Ising behavior due to quenched dilution.