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Physical review. B, Solid state

New singularities in the critical behavior of random Ising models at marginal dimensionalities

Amnon Aharony A

Published Mar 1, 1976• 111 citations
Publisher

Abstract

Renormalization-group equations are exactly solved for the random Ising model with (i) short-range interaction at $d=4$, and (ii) dipolar interactions at $d=3$. In both cases, the leading singularities of the susceptibility $\ensuremath{\chi}$ and of the specific heat $C$ 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 $t=\frac{(T\ensuremath{-}{T}_{c})}{{T}_{c}}\ensuremath{\rightarrow}0$. $D$ 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 $C$ 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.

DOI
10.1103/physrevb.13.2092

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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 $\ensuremath{\mu}$.
Renormalization-group techniques are used to study a model of nnn-coupled mmm-component spin systems, in the limit of large dipole-dipole interactions. As recently shown by Emery, this model describes the critical behavior of a constrained dipolar system in the limit $n\ensuremath{\rightarrow}\ensuremath{\infty}$ and that of a dipolar system with a quenched random perturbation in the limit $n\ensuremath{\rightarrow}0$. In both cases, the unperturbed dipolar fixed point is unstable, and there is a crossover to a new behavior. For the constrained system, this leads to another dipolar fixed point (if $\ensuremath{\alpha}<0$, $\ensuremath{\alpha}$ being the dipolar specific-heat exponent) with the same thermodynamic critical exponents, or to one with renormalized dipolar exponents (if $\ensuremath{\alpha}>0$). These results are different from those of previous "spherical" dipolar models. For the random case, the crossover is either to a new fixed point, with very different exponents, e.g., $2\ensuremath{\nu}\ensuremath{\simeq}1+1.183\ensuremath{\epsilon}$ for $m=d=4\ensuremath{-}\ensuremath{\epsilon}$, or away from all the fixed points found to order $\ensuremath{\epsilon}$. One of the new fixed points in this case has complex eigenvalues of the linearized recursion relations. This is related to the fact that the recursion-relation flow is not of a gradient type for random systems.
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Scaling and renormalization-group theories of crossover in criticality induced by random interactions are searchingly tested for Ising (n=1) systems. Extrapolation of 15th-order series in two variables for the susceptibility of the \ifmmode\pm\else\textpm\fi{}J simple-cubic Ising model yields a crossover exponent \ensuremath{\varphi}=0.125\ifmmode\pm\else\textpm\fi{}0.025. This agrees well with the specific-heat exponent, \ensuremath{\alpha}\ensuremath{\simeq}0.11. Consistency between free-energy and susceptibility scaling with \ensuremath{\varphi}=\ensuremath{\alpha} is cross-checked and leading universal scaling-function coefficients and nonuniversal scaling field amplitudes are determined.