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Physical review. B, Condensed matter

Critical singularities of the random two-dimensional Ising model

Giancarlo Jug

Published Apr 1, 1983• 19 citations
Publisher

Abstract

The critical properties of random ferromagnets with pure specific-heat exponent ${\ensuremath{\alpha}}_{P}=0$ are analyzed using renormalization-group methods. The origin and form of the impurity-induced logarithmic corrections for the random two-dimensional Ising model are discussed. Corrections develop only for the specific heat, which presents a new singularity of the form ${|\mathrm{ln}|T\ensuremath{-}{T}_{c}||}^{1\ensuremath{-}\ensuremath{\mu}}$ as well as a new critical exponent $\ensuremath{\mu}$.

DOI
10.1103/physrevb.27.4518

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The critical properties of random ferromagnets with pure specific-heat exponent \ensuremathαP=0{\ensuremath{\alpha}}_{P}=0\ensuremathαP​=0 are analyzed using renormalization-group methods. The origin and form of the impurity-induced logarithmic corrections for the random two-dimensional Ising model are discussed. Corrections develop only for the specific heat, which presents a new singularity of the form ∣ln∣T\ensuremath−Tc∣∣1\ensuremath−\ensuremathμ{|\mathrm{ln}|T\ensuremath{-}{T}_{c}||}^{1\ensuremath{-}\ensuremath{\mu}}∣ln∣T\ensuremath−Tc​∣∣1\ensuremath−\ensuremathμ as well as a new critical exponent \ensuremathμ\ensuremath{\mu}\ensuremathμ.
Renormalization-group equations are exactly solved for the random Ising model with (i) short-range interaction at d=4d=4d=4, and (ii) dipolar interactions at d=3d=3d=3. In both cases, the leading singularities of the susceptibility $\ensuremath{\chi}$ and of the specific heat CCC are found to be $\ensuremath{\chi}\ensuremath{\propto}{t}^{\ensuremath{-}1}\mathrm{exp}[{(D|\mathrm{ln}t|)}^{\frac{1}{2}}]$ and $C\ensuremath{\propto}\ensuremath{-}{|\mathrm{ln}t|}^{\frac{1}{2}}\mathrm{exp}[{\ensuremath{-}2(D|\mathrm{ln}t|)}^{\frac{1}{2}}]$ as $t=\frac{(T\ensuremath{-}{T}_{c})}{{T}_{c}}\ensuremath{\rightarrow}0$. DDD is a universal constant, equal to 6/53 in case (i) and to 9/[81ln(4/3) + 53] in case (ii). Relations between amplitudes of CCC and of the correlation length, corrections to the leading singularities, crossover effects from the nonrandom region or from the meanfield region to the asymptotic critical region and possible experiments are also discussed.
The two-dimensional random bond Ising model with two distinct ferromagnetic nearest-neighbour couplings, J and J ', distributed randomly over the lattice with concentration 1/2 is studied for various ratios J '/ J at criticality. The critical temperature is exactly known from self-duality. The size-dependent specific heat shows a crossover from the perfect Ising to the randomness-dominated critical behaviour of doubly logarithmic form. In the latter region, our estimates for the critical exponents of the size-dependent magnetization and susceptibility are in very good agreement with analytic results suggesting β/ν = 1/8 and γ/ν = 7/4.
We investigate the effects of quenched bond randomness on the critical properties of the two-dimensional ferromagnetic Ising model embedded in a triangular lattice. The system is studied in both the pure and disordered versions by the same efficient two-stage Wang-Landau method. In the first part of our study, we present the finite-size scaling behavior of the pure model, for which we calculate the critical amplitude of the specific heat's logarithmic expansion. For the disordered system, the numerical data and the relevant detailed finite-size scaling analysis along the lines of the two well-known scenarios-logarithmic corrections versus weak universality--strongly support the field-theoretically predicted scenario of logarithmic corrections. A particular interest is paid to the sample-to-sample fluctuations of the random model and their scaling behavior that are used as a successful alternative approach to criticality.
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.
The effects of random impurities on the various thermodynamic functions are studied with the aid of the two-dimensional Ising model. Of special interest are the limiting forms of these functions near the critical point. This problem is approached by a Green's function formulation as developed by Kadanoff. Since experiments only measure the average effects of impurities, we consider the average of the Green's function over all distributions of impurities and calculate it by perturbation theory. From this function we find the location of the new critical temperature and the temperature dependence of the coherence length. Thermodynamic and correlation functions are calculated by means of perturbation expansions and their functional dependence on the coherence length investigated. It is found that the magnetic properties such as the magnetization, susceptibility, and spin-spin correlations retain the same functional dependence on the coherence length. The specific heat is found to have a finite value at the critical point, and the T = Tc form of the energy-energy correlation c function is changed. The thermodynamic functions exhibit new critical indices very close to Tc. These renormalized exponents have been predicted c in literature and are found to satisfy the scaling laws predictions.
We consider the magnetic properties and spin correlation functions of a two-dimensional Ising model with random impurities near the critical temperature and find that the usual parametrization provided by the exponents $\ensuremath{\gamma}$, $\ensuremath{\delta}$, and $\ensuremath{\eta}$ and the correlation length $\ensuremath{\xi}$ for $T\ensuremath{\gtrsim}{T}_{c}$ does not describe our results. Experimental implications of this breakdown of the critical exponent forms are discussed.