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arXiv:cond-mat/0301433 · Journal of Physics A Mathematical and General

Replicon modes and stability of critical behaviour of disordered systems with respect to the continuous replica symmetry breaking

Andrei A. Fedorenko

Published Jan 22, 2003• 13 citations
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Abstract

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.

DOI
10.1088/0305-4470/36/5/304

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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.
We study the critical properties of the weakly disordered two-dimensional Ising and Baxter models in terms of the renormalization group (RG) theory generalized to take into account replica symmetry breaking (RSB) effects. Recently it has been shown that the traditional replica-symmetric RG flows in the dimension are unstable with respect to the RSB potentials and a new spin-glass type critical phenomena has been discovered (Dotsenko Vik S, Harris B, Sherrington D and Stinchbombe R 1995 J. Phys. A: Math. Gen. 28 3093; Dotsenko Vik S and Feldman D E 1995 J. Phys. A: Math. Gen. 28 5183). In contrast, here it is demonstrated that in the considered two-dimensional systems the renormalization-group flows are stable with respect to the RSB modes. It is shown that the solution of the renormalization group equations with arbitrary starting RSB coupling matrix exhibits asymptotic approach to the traditional replica-symmetric ones. Thus, to leading order the non-perturbative RSB degrees of freedom do not affect the critical phenomena in the two-dimensional weakly disordered Ising and Baxter models studied earlier.
A field-theoretic description of the critical behavior of the weakly disordered systems is given. Directly, for three- and two-dimensional systems a renormalization analysis of the effective Hamiltonian of a model with replica symmetry breaking (RSB) potentials is carried out in the two-loop approximation. For the case with 1-step RSB the fixed points (FP's) corresponding to stability of the various types of critical behavior are identified with the use of the Pade-Borel summation technique. Analysis of FP's has shown a stability of the critical behavior of the weakly disordered systems with respect to RSB effects and realization of a former scenario of disorder influence on critical behavior.
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.
A eld-theoretic description of the critical behaviour of the weakly disordered systems is given.Directly, for three-and two-dimensional systems a renormalization analysis of the eective Hamiltonian of the model with replica symmetry breaking (RSB) potentials is carried out in the two-loop approximation.For the case with rst-step RSB the xed points (FPs) corresponding to the stability of the various types of critical behaviour are identi ed with the use of the Pad e{Borel summation technique.The analysis of FPs has shown a stability of the critical behaviour of the weakly disordered three-and two-dimensional systems with respect to RSB eects and realization of the former scenario of disorder in uence on critical behaviour.For the case of systems with arbitrary dimension from 3 to 4 the ranges of critical behaviour determined by the RSB eects are found for each value of the p-component order parameter without the use of "-expansion.A comparison with the results of the "-expansion calculations is carried out.
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.