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Critical behaviour of magnets with quenched disorder studied by operator product expansion

Eugene Kogan, M. Kaveh

Published Feb 22, 2016• 0 citations
OpenAlex

Abstract

We study critical behaviour of disordered magnets near four dimensions. We consider the system with explicit cubic anisotropy and scalar disorder and that with random direction of anisotropy axis. The quenched disorder is taken into account by replica method. Using the method of operator product expansion, we derive in the first order to $\epsilon$ approximation the renormalization group equations taking into account possible replica symmetry breaking.

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Spin-glass effects on critical behavior in systems with extended quenched disorder

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We study critical behaviour of disordered magnets near four dimensions. We consider the system with explicit cubic anisotropy and scalar disorder and that with random direction of anisotropy axis. The quenched disorder is taken into account by replica method. Using the method of operator product expansion, we derive in the first order to ϵ\epsilonϵ approximation the renormalization group equations taking into account possible replica symmetry breaking.
We study critical behaviour of disordered magnets near four dimensions. We consider the system with explicit cubic anisotropy and scalar disorder and that with random direction of anisotropy axis. The quenched disorder is taken into account by replica method. Using the method of operator product expansion, we derive in the first order to εεε approximation the renormalization group equations taking into account possible replica symmetry breaking.
The critical behaviour of an m -vector model with a local anisotropy axis of random orientation is studied within the field-theoretical renormalization group approach for cubic distribution of anisotropy axis.Expressions for the renormalization group functions are calculated up to the two-loop order and investigated both by an ε = 4 -d expansion and directly at space dimension d = 3 by means of the Pad é-Borel resummation.One accessible stable fixed point indicating a 2nd order ferromagnetic phase transition with dilute Ising-like critical exponents is obtained.
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.
We study the purely relaxational critical dynamics with non-conserved order parameter (model A critical dynamics) for three-dimensional magnets with disorder in a form of the random anisotropy axis.For the random axis anisotropic distribution, the static asymptotic critical behaviour coincides with that of random site Ising systems.Therefore the asymptotic critical dynamics is governed by the dynamical exponent of the random Ising model.However, the disorder effects considerably the dynamical behaviour in the non-asymptotic regime.We perform a field-theoretical renormalization group analysis within the minimal subtraction scheme in two-loop approximation to investigate asymptotic and effective critical dynamics of random anisotropy systems.The results demonstrate the non-monotonic behaviour of the dynamical effective critical exponent z eff .
Equilibrium properties of the phase transition of a ferromagnetic spin system under a quenched random magnetic field are briefly summarized. Based upon a path integral formulation for stochastic processes, the n→0 method of replica is extended to treat critical relaxation of quenched random spin systems. As an application, the dynamic scaling hypothesis is examined for the random field problem by means of 1/m-expansion method (m; the number of order parameter components) and the renormalization group method near 6 space dimensions. It is shown that d→d−2 rule does not hold for the dynamic critical exponent of the characteristic frequency.
A renormalization-group approach is used for investigating the effects of the Parisi replica symmetry breaking (``spin-glass effects'') on critical properties of d-dimensional systems with extended quenched disorder along ${\ensuremath{\varepsilon}}_{d}$ directions. Within a double expansion scheme to first order in $(\ensuremath{\varepsilon}=4\ensuremath{-}d, {\ensuremath{\varepsilon}}_{d})$ it is found that for $d<~4+{\ensuremath{\varepsilon}}_{d}$ and order-parameter dimensionalities n< 4,n<~4,n< 4, a full fixed-point instability with a consequent runaway towards a strong-coupling regime occurs in the parameter space while, for d<4d<4d<4 and n>4,n>4,n>4, compatibly with the modified Harris criterion, an additional marginally stable fixed point exists which might control a new anisotropic random critical behavior. In any case, the traditional replica-symmetric Dorogovtsev-like solution for classical and quantum systems is unstable against the replica symmetry-breaking modes.