We propose a new picture of the renormalization group (RG) approach in the presence of disorder, which considers the RG trajectories of each random sample (realization) separately instead of the usual renormalization of the averaged free energy. The main consequence of the theory is that the average over randomness has to be taken after finding the critical point of each realization. To demonstrate these concepts, we study the finite-size scaling properties of the two-dimensional random-bond Ising model. We find that most of the previously observed finite-size corrections are due to the sample-to-sample fluctuation of the critical temperature and scaling predictions are fulfilled only by the new average.
Finite-size scaling above the upper critical dimension is a long-standing puzzle in the field of statistical physics. Even for pure systems various scaling theories have been suggested, partially corroborated by numerical simulations. In the present manuscript we address this problem in the even more complicated case of disordered systems. In particular, we investigate the scaling behavior of the random-field Ising model at dimension D=7, i.e., above its upper critical dimension D_{u}=6, by employing extensive ground-state numerical simulations. Our results confirm the hypothesis that at dimensions D>D_{u}, linear length scale L should be replaced in finite-size scaling expressions by the effective scale L_{eff}=L^{D/D_{u}}. Via a fitted version of the quotients method that takes this modification, but also subleading scaling corrections into account, we compute the critical point of the transition for Gaussian random fields and provide estimates for the full set of critical exponents. Thus, our analysis indicates that this modified version of finite-size scaling is successful also in the context of the random-field problem.
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 simulated site dilute Ising models in
d=3 dimensions for several lattice sizes
L. For each
L singular thermodynamic quantities
X were measured at criticality and their distributions
P(X) were determined, for ensembles of several thousand random samples. For
L→∞ the width of
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) scales as
δTc(L)∼L−1/ν and NOT as
∼L−d/2. Finite size scaling holds; the sample dependence of
Xi(Tc) enters predominantly through
Tc(i,L).
The disorder-driven phase transition of the random-field Ising model is observed using exact ground-state computer simulations for hypercubic lattices in
d=5,6,7 dimensions. Finite-size scaling analyses are used to calculate the critical point and the critical exponents of the specific heat, magnetization, susceptibility, and of the correlation length. For dimensions
d=6,7 which are larger or equal to the assumed upper critical dimension,
du=6, mean-field behavior is found, i.e., $\ensuremath{\alpha}=0, \ensuremath{\beta}=1/2, \ensuremath{\gamma}=1, \ensuremath{\nu}=1/2$. For the analysis of the numerical data, it appears to be necessary to include recently proposed corrections to scaling at and beyond the upper critical dimension.
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 (
RX 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.