The renormalization group approach is used to study the effects of a “canonical” constraint (e.g., a fixed number of occupied bonds) on critical quenched disordered systems. The constraint is found to be always irrelevant, even near the “random” fixed point. This proves that α<0, or that ν>2/d. “Canonical” and “grand canonical” averages thus belong to the same universality class. Related predictions concerning the universality of non-self-averaging distributions are tested by Monte Carlo simulations of the site-diluted Ising model on the cubic lattice. In this case, the approach to the asymptotic distribution for “canonical” averaging is slow, resulting in effectively smaller fluctuations.
The renormalization group approach is used to study the effects of a ``canonical'' constraint (e.g., a fixed number of occupied bonds) on critical quenched disordered systems. The constraint is found to be always irrelevant, even near the ``random'' fixed point. This proves that $\ensuremath{\alpha}<0$, or that $\ensuremath{\nu}>2/d$. ``Canonical'' and ``grand canonical'' averages thus belong to the same universality class. Related predictions concerning the universality of non-self-averaging distributions are tested by Monte Carlo simulations of the site-diluted Ising model on the cubic lattice. In this case, the approach to the asymptotic distribution for ``canonical'' averaging is slow, resulting in effectively smaller fluctuations.
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 $\ensuremath{\chi}{(T}_{c}^{\ensuremath{\infty}}).$
Self-averaging of singular thermodynamic quantities at criticality for randomly and thermally diluted three-dimensional Ising systems has been studied by the Monte Carlo approach. Substantially improved self-averaging is obtained for critically clustered (critically thermally diluted) vacancy distributions in comparison with the observed self-averaging for purely random diluted distributions. Critically thermal dilution, leading to maximum relative self-averaging, corresponds to the case when the characteristic vacancy ordering temperature (theta) is made equal to the magnetic critical temperature for the pure three-dimensional (3D) Ising systems (T(3D)(c)). For the case of a high ordering temperature (theta>>T(3D)(c)), the self-averaging obtained is comparable to that in a randomly diluted system.
We perform a high-statistics simulation of the three-dimensional randomly dilute Ising model on cubic lattices L3 with L< or =256. We choose a particular value of the density, x=0.8, for which the leading scaling corrections are suppressed. We determine the critical exponents, obtaining nu=0.683(3), eta=0.035(2), beta=0.3535(17), and alpha=-0.049(9), in agreement with previous numerical simulations. We also estimate numerically the fixed-point values of the four-point zero-momentum couplings that are used in field-theoretical fixed-dimension studies. Although these results somewhat differ from those obtained using perturbative field theory, the field-theoretical estimates of the critical exponents do not change significantly if the Monte Carlo result for the fixed point is used. Finally, we determine the six-point zero-momentum couplings, relevant for the small-magnetization expansion of the equation of state, and the invariant amplitude ratio R(+)(xi) that expresses the universality of the free-energy density per correlation volume. We find R(+)(xi)=0.2885(15).
The random transverse-field Ising chain is the simplest disordered model presenting a quantum phase transition at
T=0. We compare analytically its finite-size scaling properties in two different ensembles for the disorder: (i) the canonical ensemble, where the disorder variables are independent and (ii) the microcanonical ensemble, where there exists a global constraint on the disorder variables. The observables under study are the surface magnetization, the correlation of the two surface magnetizations, the gap and the end-to-end spin-spin correlation
C(L) for a chain of length L. At criticality, each observable decays typically as ${e}^{\ensuremath{-}w\sqrt{L}}$ in both ensembles, but the probability distributions of the rescaled variable w are different in the two ensembles, in particular, in their asymptotic behaviors. As a consequence, the dependence on L of averaged observables differs in the two ensembles. For instance, the correlation
C(L) decays algebraically as
1/L in the canonical ensemble, but subexponentially as ${e}^{\ensuremath{-}{\mathrm{cL}}^{1/3}}$ in the microcanonical ensemble. Off criticality, probability distributions of rescaled variables are governed by the critical exponent $\ensuremath{\nu}=2$ in both ensembles, but the following observables are governed by the exponent $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\nu}}=1$ in the microcanonical ensemble, instead of the exponent $\ensuremath{\nu}=2$ in the canonical ensemble: (a) In the disordered phase, the averaged surface magnetization, the averaged correlation of the two surface magnetizations and the averaged end-to-end spin-spin correlation; (b) in the ordered phase, the averaged gap. In conclusion, the measure of the rare events that dominate various averaged observables can be very sensitive to the microcanonical constraint.