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Physical review. B, Solid state

Comments on the critical behavior of random systems

Amnon Aharony A, Y. Imry, Shang-keng Ma

Published Jan 1, 1976• 64 citations
Publisher

Abstract

The renormalization-group treatment of the critical behavior of random systems is augmented by including all the cumulants of the disorder distribution. The results substaniate Lubensky's argument that the $n\ensuremath{\rightarrow}0$ isotropic fixed point is "unphysical," and that the phase transition of random $m$-component systems which have a positive specific-heat exponent will be sharp, with exponents determined by the "random" fixed point. The renormalization-group transformations of a random Gaussian model lead to a "runaway," which is shown to be unrelated to a first-order transition. This is related to the problem of a particle in a random potential with and without an absorptive part.

DOI
10.1103/physrevb.13.466

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The critical behavior of spin systems with quenched disorder is studied by renormalization-group methods. For the randomly dilute $m$-vector model, the $n=0$ limit is used to construct a translationally invariant effective Hamiltonian which describes the original disordered system. This Hamiltonian is analyzed in the $\ensuremath{\epsilon}$ expansion to order ${\ensuremath{\epsilon}}^{2}$. Sharp second-order phase transitions with exponents which do not depend continuously on impurity concentration are predicted. For $m>{m}_{c}\ensuremath{\equiv}4\ensuremath{-}4\ensuremath{\epsilon}+O({\ensuremath{\epsilon}}^{2})$ the isotropic $m$-component fixed point, which characterizes the critical behavior of the pure system, is stable. For $m<{m}_{c}$, a new random fixed point becomes stable. The exponents corresponding to this fixed point are $\ensuremath{\eta}=[\frac{(5{m}^{2}\ensuremath{-}8m)}{256{(m\ensuremath{-}1)}^{2}}]{\ensuremath{\epsilon}}^{2}+O({\ensuremath{\epsilon}}^{3})$, $\ensuremath{\nu}=\frac{1}{2}+[\frac{3m}{32(m\ensuremath{-}1)}]\ensuremath{\epsilon}+[\frac{m(127{m}^{2}\ensuremath{-}572m\ensuremath{-}32)}{4096{(m\ensuremath{-}1)}^{3}}]{\ensuremath{\epsilon}}^{2}+O({\ensuremath{\epsilon}}^{3})$ for $m\ensuremath{\ne}1$, and $\ensuremath{\eta}=\ensuremath{-}\frac{\ensuremath{\epsilon}}{106}+O({\ensuremath{\epsilon}}^{\frac{3}{2}})$, $\ensuremath{\nu}=\frac{1}{2}+\frac{{(\frac{6\ensuremath{\epsilon}}{53})}^{\frac{1}{2}}}{4}+O(\ensuremath{\epsilon})$ for $m=1$. More general random systems are qualitatively discussed from the effective-Hamiltonian viewpoint.

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Critical behavior of a random m-vector model

G. Grinstein

The critical behavior of an m‐vector model with random impurities is investigated through the use of renormalization group techniques near four dimensions. For almost all values of m, this disordered model has two stable fixed points and a single nontrivialy unstable fixed point. The model exhibits sharp phase transitions with exponents which do not depend continuously on the concentration of impurities. The familiar m‐component fixed point, which is normally assumed to characterize the critical behavior of the pure m‐vector model, is unstable whenever m%4.

AIP conference proceedings•Jan 1, 1975•14 citations
The renormalization-group treatment of the critical behavior of random systems is augmented by including all the cumulants of the disorder distribution. The results substaniate Lubensky's argument that the n\ensuremath→0n\ensuremath{\rightarrow}0n\ensuremath→0 isotropic fixed point is "unphysical," and that the phase transition of random mmm-component systems which have a positive specific-heat exponent will be sharp, with exponents determined by the "random" fixed point. The renormalization-group transformations of a random Gaussian model lead to a "runaway," which is shown to be unrelated to a first-order transition. This is related to the problem of a particle in a random potential with and without an absorptive part.
The critical behavior of spin systems with quenched disorder is studied by renormalization-group methods. For the randomly dilute mmm-vector model, the n=0n=0n=0 limit is used to construct a translationally invariant effective Hamiltonian which describes the original disordered system. This Hamiltonian is analyzed in the $\ensuremath{\epsilon}$ expansion to order ${\ensuremath{\epsilon}}^{2}$. Sharp second-order phase transitions with exponents which do not depend continuously on impurity concentration are predicted. For $m>{m}_{c}\ensuremath{\equiv}4\ensuremath{-}4\ensuremath{\epsilon}+O({\ensuremath{\epsilon}}^{2})$ the isotropic mmm-component fixed point, which characterizes the critical behavior of the pure system, is stable. For m<mcm<{m}_{c}m<mc​, a new random fixed point becomes stable. The exponents corresponding to this fixed point are $\ensuremath{\eta}=[\frac{(5{m}^{2}\ensuremath{-}8m)}{256{(m\ensuremath{-}1)}^{2}}]{\ensuremath{\epsilon}}^{2}+O({\ensuremath{\epsilon}}^{3})$, $\ensuremath{\nu}=\frac{1}{2}+[\frac{3m}{32(m\ensuremath{-}1)}]\ensuremath{\epsilon}+[\frac{m(127{m}^{2}\ensuremath{-}572m\ensuremath{-}32)}{4096{(m\ensuremath{-}1)}^{3}}]{\ensuremath{\epsilon}}^{2}+O({\ensuremath{\epsilon}}^{3})$ for $m\ensuremath{\ne}1$, and $\ensuremath{\eta}=\ensuremath{-}\frac{\ensuremath{\epsilon}}{106}+O({\ensuremath{\epsilon}}^{\frac{3}{2}})$, $\ensuremath{\nu}=\frac{1}{2}+\frac{{(\frac{6\ensuremath{\epsilon}}{53})}^{\frac{1}{2}}}{4}+O(\ensuremath{\epsilon})$ for m=1m=1m=1. More general random systems are qualitatively discussed from the effective-Hamiltonian viewpoint.
The phase transitions in a large class of physical systems are described by $n\ensuremath{\ge}4$ component order parameters. Here, the critical behavior of quenched random $n\ensuremath{\ge}4$ vector models is studied by means of renormalization-group theory in $4\ensuremath{-}\ensuremath{\epsilon}$ dimensions. Recursion relations for average potentials are constructed following the methods derived by Lubensky. For several Hamiltonians describing homogeneous $n\ensuremath{\ge}4$ systems there exist no stable fixed points in $4\ensuremath{-}\ensuremath{\epsilon}$ dimensions which explains the first-order transitions actually observed in these systems. It is shown that the recursion relations for the corresponding quenched random systems are also unstable. However, the runaway in this case is of a fundamentally different nature. The fluctuations of the local mean-field transition temperature diverge, and this behavior is interpreted as a "smeared" transition. This interpretation is consistent with existing experiments. On the other hand, in the cases where the homogeneous Hamiltonian possesses a stable fixed point in $4\ensuremath{-}\ensuremath{\epsilon}$ dimensions, this fixed point remains stable against random perturbations, so no change in the critical behavior is expected. For most of the models studied there is at least one fixed point of order ${\ensuremath{\epsilon}}^{\frac{1}{2}}$. These fixed points are all unstable. It is suggested that experiments should be performed to determine the critical behavior of random $n\ensuremath{\ge}4$ systems. Of particular interest are the systems with no stable fixed points, such as Cr, Eu, MnO, and UO2{\mathrm{UO}}_{2}UO2​, where crossover from first-order to a smeared transition is predicted.
In quenched random-spin systems, the renormalization group can be used to develop recursion relations for the probability distribution for random potentials. Alternatively, recursion relations for the average values of potentials and their higher cumulants can be obtained. In this paper, the above technique is used to study phase transitions in quenched random nnn-component classical spin systems using the $\ensuremath{\epsilon}$ expansion to second order in $\ensuremath{\epsilon}$. It there are long-range correlations in the random potentials (e.g., all potentials along a line are equal), there are no stable physical fixed points within the $\ensuremath{\epsilon}$ expansion. This is interpreted as a smeared transition. If there are no long-range correlations in the random potentials, there is a sharp transition with pure system exponents if the specific-heat exponent ${\ensuremath{\alpha}}^{H}$ of the pure system is negative. If ${\ensuremath{\alpha}}^{H}>0$ and n>1n>1n>1, there is a sharp transition with new exponents $\ensuremath{\eta}=[\frac{(5{n}^{2}\ensuremath{-}8n)}{256{(n\ensuremath{-}1)}^{2}}]{\ensuremath{\epsilon}}^{2}$ and $\ensuremath{\nu}=\frac{1}{2}+[\frac{3n}{32(n\ensuremath{-}1)}]\ensuremath{\epsilon}+[\frac{n(127{n}^{2}\ensuremath{-}572n\ensuremath{-}32)}{4096{(n\ensuremath{-}1)}^{3}}]{\ensuremath{\epsilon}}^{2}$. For n=1n=1n=1, there is no stable fixed point, which is again interpreted as a smeared transition.
The renormalization-group (RG) method is applied to the problem of formation of a localized state of a particle moving in a given potential. It is shown that RG transformation on the particle's Green's function can be performed exactly. The fixed-point equations yield information on the critical binding strength, while the transformation equations near the fixed points give scaling laws and critical exponents, e.g., for the dependence of the localization radius on the energy and the potential strength. The general theory is illustrated by considering the simplest case of the Slater-Koster problem in detail. We find that it is possible to take all the irrelevant variables into account and thus obtain the exact result for the critical binding point. Critical exponents and scaling laws are, however, universal in character. The method is then extended to the case of motion in random potentials, and some applications are pointed out. In particular it is shown that the RG properties of the averaged absolute value squared (〈∣G∣2〉〈{|G|}^{2}〉〈∣G∣2〉) of the Green's function do not correspond to a simple second-order phase transition. The analysis illustrates certain features of RG methods which have not heretofore been encountered in other problems and which are likely to be found in treatments of disordered materials. In particular, the runaway behavior observed in the RG transformation of 〈G〉〈G〉〈G〉 and 〈∣G∣2〉〈{|G|}^{2}〉〈∣G∣2〉 as well as in many random-spin systems, might be associated with the formation of bound states in the band tail for any amount of randomness.
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.
The critical behavior of an m‐vector model with random impurities is investigated through the use of renormalization group techniques near four dimensions. For almost all values of m, this disordered model has two stable fixed points and a single nontrivialy unstable fixed point. The model exhibits sharp phase transitions with exponents which do not depend continuously on the concentration of impurities. The familiar m‐component fixed point, which is normally assumed to characterize the critical behavior of the pure m‐vector model, is unstable whenever m%4.