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.
Strong disorder renormalization is an energy-based renormalization that leads to a complicated renormalized topology for the surviving clusters as soon as d > 1. In this paper, we propose to include strong disorder renormalization ideas within the more traditional fixed-cell-size real space RG framework. We first consider the one-dimensional chain as a test for this fixed-cell-size procedure: we find that all exactly known critical exponents are reproduced correctly, except for the magnetic exponent β (because it is related to more subtle persistence properties of the full RG flow). We then apply numerically this fixed-cell-size procedure to two types of renormalizable fractal lattice: (i) the Sierpinski gasket of fractal dimension D = ln3/ln2, where there is no underlying classical ferromagnetic transition, so that the RG flow in the ordered phase is similar to what happens in d = 1; (ii) a hierarchical diamond lattice of fractal dimension D = 4/3, where there is an underlying classical ferromagnetic transition, so that the RG flow in the ordered phase is similar to what happens on hypercubic lattices of dimension d > 1. In both cases, we find that the transition is governed by an infinite disorder fixed point: besides the measure of the activated exponent ψ, we analyze the RG flow of various observables in the disordered and ordered phases, in order to extract the 'typical' correlation length exponents of these two phases which are different from the finite-size correlation length exponent.
In order to study the influence of quenched disorder on second-order phase transitions, high-temperature series expansions of the susceptibility and the free energy are obtained for the quenched bond-diluted Ising model in
d=3−−5 dimensions. They are analyzed using different extrapolation methods tailored to the expected singularity behaviors. In
d=4 and 5 dimensions we confirm that the critical behavior is governed by the pure fixed point up to dilutions near the geometric bond percolation threshold. The existence and form of logarithmic corrections for the pure Ising model in
d=4 are confirmed, and our results for the critical behavior of the diluted system are in agreement with the type of singularity predicted by renormalization group considerations. In three dimensions we find large crossover effects between the pure Ising, percolation, and random fixed points. We estimate the critical exponent of the susceptibility to be $\ensuremath{\gamma}=1.305(5)$ at the random fixed point.
Ferdinand and Fisher have analyzed the finite-size behavior of the two-dimensional Ising model with periodic boundary conditions for
T near
Tc. In the thermodynamic limit, the leading correction term to the specific heat has an unusual and unexpected behavior as the shape of the $m\ifmmode\times\else\texttimes\fi{}n$ lattice is changed. For shape parameter
s=nm=1, the peak in the specific-heat correction term occurs at a reduced temperature ${\ensuremath{\tau}}_{max}=2{K}_{c}n\frac{(T\ensuremath{-}{T}_{c})}{{T}_{c}}>0$. With increasing
s the peak moves to negative $\ensuremath{\tau}$ values and then increases to 0. We show that this effect may be understood by keeping only the two largest eigenvalues of the transfer matrix. To this approximation, which is good for
s=1 and improves exponentially with increasing
s, all the shape dependence of the free energy is due to a factor that can be expressed entirely in terms of a domain-wall energy. Furthermore, the functional form of this factor is the same as that of the finite-length correction to the one-dimensional Ising model. Thus the dependence of ${\ensuremath{\tau}}_{max}$ on
s is qualitatively similar to the dependence of
Tmax1 the location of the one-dimensional Ising-model specific-heat peak on
N, the number of spins. We also argue that the two-dimensional partition function is expressible at least for
s>1 as a factor due to domains, and another due to a one-dimensional Ising array of domain walls. Approximate expressions are obtained for the effective number of spins and the effective coupling of the array. We also point out that there is reason to believe the one-dimensional Ising array of domain walls at fixed $\ensuremath{\tau}$ connects smoothly with a similar array for
T<Tc.
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.