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Communications in Theoretical Physics

Replica Symmetry Breaking in Critical Phenomena in Long-Range Disordered Ferromagnets

Wu Xintian, Zhiwen Pan, Gong ChangDe

Published Jun 15, 1999• 0 citations
Publisher

Abstract

The replica symmetry breaking (RSB) is investigated in -component ferromagnetic spin systems with a long-range disorder, with a correlation function obeying a power law . The Landau–Ginzburg Hamiltonian with an RSB quartic interaction term is studied, with the renormalization-group expansion in and , where is the spatial dimension. It is shown that the long-range disorder fixed point found previously is unstable under the perturbation of RSB for and stable for . The RSB fixed points are calculated and there is no physical stable fixed point for .

DOI
10.1088/0253-6102/31/4/561

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The replica symmetry breaking (RSB) is investigated in -component ferromagnetic spin systems with a long-range disorder, with a correlation function obeying a power law . The Landau–Ginzburg Hamiltonian with an RSB quartic interaction term is studied, with the renormalization-group expansion in and , where is the spatial dimension. It is shown that the long-range disorder fixed point found previously is unstable under the perturbation of RSB for and stable for . The RSB fixed points are calculated and there is no physical stable fixed point for .
We study the critical properties of the weakly disordered p-component ferromagnet in terms of the renormalization group (RG) theory generalized to take into account the replica symmetry breaking (RSB) effects coming from the multiple local minima solutions of the mean-field equations. Recently it has been shown that for p<4 the traditional RG flows at dimensions D=4- epsilon , which are usually considered as describing the disorder-induced universal critical behaviour, are unstable with respect to the RSB potentials as found in spin glasses, and a new type of stable one-step RSB fixed point has been discovered. Here it is demonstrated that for a general type of the Parisi RSB structures there exist no stable fixed points, and the RG flows lead to the strong-coupling regime at the finite scale R * approximately exp(1/u), where u is the small parameter describing the disorder. The physical consequences of the obtained RG solutions are discussed. In particular, we argue that discovered RSB strong-coupling phenomena indicate the onset of a new spin-glass-type critical behaviour in the temperature interval tau < tau * approximately exp(-1/u) near Tr. The possible relevance of the considered RSB effects for the Griffith phase is also discussed.
We study the critical properties of the weakly disordered p-component random Heisenberg ferromagnet. It is shown that if the specific-heat critical exponent of the pure system is positive, the traditional renormalization group (RG) flows at dimensions D=4-ε, which are usually considered as describing the disorder-induced universal critical behaviour, are unstable with respect to replica-symmetry breaking (RSB) potentials such as those found in spin glasses. It is demonstrated that the RG flows involving RSB potentials lead to fixed points which have a structure known as the one-step RSB, and there exists a whole spectrum of such fixed points. It is argued that spontaneous RSB can occur due to the interactions of the fluctuating fields with the local non-perturbative degrees of freedom coming from the multiple local minima solutions of the mean-field equations. However, it is not clear whether or not RSB occurs for infinitesimally weak disorder. Physical consequences of these conclusions are discussed.
We study the critical properties of the weakly disordered p-component random Heisenberg ferromagnet. It is shown that if the specific-heat critical exponent of the pure system is positive, the traditional renormalization group (RG) flows at dimensions D=4- epsilon , which are usually considered as describing the disorder-induced universal critical behaviour, are unstable with respect to replica-symmetry breaking (RSB) potentials such as those found in spin glasses. It is demonstrated that the RG flows involving RSB potentials lead to fixed points which have a structure known as the one-step RSB, and there exists a whole spectrum of such fixed points. It is argued that spontaneous RSB can occur due to the interactions of the fluctuating fields with the local non-perturbative degrees of freedom coming from the multiple local minima solutions of the mean-field equations. However, it is not clear whether or not RSB occurs for infinitesimally weak disorder. Physical consequences of these conclusions are discussed.
The effects of quenched disorder on the critical properties of itinerant quantum antiferromagnets and ferromagnets are considered. Particular attention is paid to locally ordered spatial regions that are formed in the presence of quenched disorder even when the bulk system is still in the paramagnetic phase. These rare regions or local moments are reflected in the existence of spatially inhomogeneous saddle points of the Landau-Ginzburg-Wilson functional. We derive an effective theory that takes into account small fluctuations around all of these saddle points. The resulting free energy functional contains a new term in addition to those obtained within the conventional perturbative approach, and it comprises what would be considered nonperturbative effects within the latter. A renormalization group analysis shows that in the case of antiferromagnets, the previously found critical fixed point is unstable with respect to this new term, and that no stable critical fixed point exists at one-loop order. This is contrasted with the case of itinerant ferromagnets, where we find that the previously found critical behavior is unaffected by the rare regions due to an effective long-ranged interaction between the order parameter fluctuations.
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.
A field-theory approach is used to investigate the 'spin-glass effects' on the critical behaviour of systems with weak temperature-like quenched disorder. The renormalization group (RG) analysis of the effective Hamiltonian of a model with replica symmetry breaking (RSB) potentials of a general type is carried out in the two-loop approximation. The fixed point (FP) stability, recently found within the one-step RSB RG treatment, is further explored in terms of replicon eigenvalues. We find that the traditional FPs, which are usually considered to describe the disorder-induced universal critical behaviour, remain stable when the continuous RSB modes are taken into account.