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Canadian Journal of Physics

Static critical behaviors in random-spin systems with short-range interaction

Yoshitake Yamazaki

Published Jan 1, 1978• 13 citations
Publisher

Abstract

Critical behaviors in quenched random-spin systems with N-spin component are studied in the limit M → 0 of the non-random MN-component models by means of the renormalization group theory. As the static critical phenomena the stability of the fixed points is investigated and the critical exponents η[~ O(ε3); ε ≡ 4 – d], γ, α, and crossover index [Formula: see text] and the equation of state [~ O(ε)] are obtained. Within the approximation up to the order ε2, even the random-spin systems with N = 2 or 3 are unstable in the three dimensions and the pure systems are stable there.

DOI
10.1139/p78-020

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Physical review. B, Solid state•Feb 1, 1976•529 citations
Critical behaviors in quenched random-spin systems with N-spin component are studied in the limit M → 0 of the non-random MN-component models by means of the renormalization group theory. As the static critical phenomena the stability of the fixed points is investigated and the critical exponents η[~ O(ε3); ε ≡ 4 – d], γ, α, and crossover index [Formula: see text] and the equation of state [~ O(ε)] are obtained. Within the approximation up to the order ε2, even the random-spin systems with N = 2 or 3 are unstable in the three dimensions and the pure systems are stable there.
By means of the renormalisation-group approach, static and dynamic critical behaviours are studied for the XY spin system with cubic anisotropy and with quenched random nonmagnetic impurities. The stability of the quenched-random-impurity fixed point in d ( identical to 4- epsilon ) dimensions, the flow diagram of the interactions and the critical exponents are investigated, within the two-loop approximation, in terms of the epsilon 1/2 expansion. It is shown that the transition that appears is a first-order transition.
Tricriticdal behavior, in particular, critical exponents (η, γ, φ, ν, α), of classical N-component spin systems with a short-range or long-range interaction are studied in the dimension d (≪ 3 σ/2; σ: potential range parameter, 1 ≪ σ≤2) by means of the renormalization theory. The obtained expressions for all these critical exponents in the long-range interaction system and for the crossover exponent φ in the short-range one are new. The corrections of these tricritical exponents to the classical ones are shown to be smaller than those in the corresponding systems undergoing the second-order phase transition.
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.
The fixed points of the exact renormalisation group equations with smooth cut-off are determined in the epsilon -expansion for systems characterised by mn-component vector order-parameters. The critical behaviour of quenched random n-component spin systems is obtained by taking the limit m=O. The critical exponent is obtained to order epsilon 2 , and the other exponents to order epsilon . A derivation of Nelson's trajectory-integral expression for the free energy, appropriate for the case of smooth cut-off renormalisation group equations, is presented.
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 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.