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arXiv:cond-mat/0504132 · physica status solidi (b)

Thermodynamic origin of order parameters in mean‐field models of spin glasses

V. Janiš, Lenka Zdeborová

Published Jan 5, 2006• 5 citations
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Abstract

Abstract We analyze the thermodynamic behavior of general n ‐component mean‐field spin glass models in order to identify the origin of the hierarchical structure of the order parameters from the replica‐symmetry breaking solution. We derive a configurationally dependent free energy with local magnetizations and averaged local susceptibilities as order parameters. Using an example of the replicated Ising spin glass we demonstrate that the hierarchy of order parameters in mean‐field models results from the structure of inter‐replica susceptibilities. These susceptibilities serve for lifting the degeneracy due to the existence of many metastable states and for recovering the thermodynamic homogeneity of the free energy. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

DOI
10.1002/pssb.200541070

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Abstract We analyze the thermodynamic behavior of general n ‐component mean‐field spin glass models in order to identify the origin of the hierarchical structure of the order parameters from the replica‐symmetry breaking solution. We derive a configurationally dependent free energy with local magnetizations and averaged local susceptibilities as order parameters. Using an example of the replicated Ising spin glass we demonstrate that the hierarchy of order parameters in mean‐field models results from the structure of inter‐replica susceptibilities. These susceptibilities serve for lifting the degeneracy due to the existence of many metastable states and for recovering the thermodynamic homogeneity of the free energy. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)
We analyze the low-temperature behavior of mean-field equations of Thouless, Anderson, and Palmer (TAP). We demonstrate that degeneracy in free energy makes the low-temperature TAP states unstable. Different solutions of the TAP equations, independent in the TAP approach, become coupled if an infinitesimal interaction between them is introduced. By means of real spin replicas we derive a self-averaging free energy free of unstable states with local magnetizations and homogeneous overlap susceptibilities between different spin replicas as order parameters. We thereby extend the TAP approach to a consistent description of the spin-glass phase for all configurations of spin exchange with (marginally) stable and thermodynamically homogeneous free energy.
This thesis is concerned with the mean field theory of spin glasses within the replica method approach. We have studied the Sherrington-Kirkpatrick model [33,4δ] both with Ising and m-component spins. It is shown that the replica symmetric solution gives an instability in the condensed phase which requires a breaking of replica symmetry in the spin glass phase. The various broken replica symmetry solutions proposed thus far (Bray and Moore [52], Blandin et al [53], Parisi [54]) are reviewed and particular attention is given to the one proposed by Parisi. The order parameter susceptibility and the stability of the solution are studied within this breaking of replica symmetry scheme which give results in agreement with the predictions of TAP [47] and Bray and Moore [52]. The breakdown of linear response theory in the spin glass phase is also discussed.
Parisi's replica-symmetry-breaking scheme is applied to the quantum Sherrington-Kirkpatrick Ising spin glass in a transverse magnetic field. The order-parameter function q(x) is calculated near Tf{\mathit{T}}_{\mathit{f}}Tf​, while for all temperatures the first stage of replica-symmetry breaking is calculated. Internal energy, entropy, and magnetic susceptibility are calculated in order to investigate the quality of both the replica-symmetric and the replica-symmetry-broken theories.
The n to 0 replica method is used to study the Ising spin glass below its transition temperature. The order parameter correlation functions are shown to contain massless modes ('replicon' modes) within mean-field theory. These become unstable at first order in perturbation theory about the mean-field solution. An alternative formulation of perturbation theory, however, yields a family of new solutions which break, in general, the symmetry between the replicas. Stability requirements single out a unique member of this family which exhibits (i) massless moded, and (ii) a restoration of replica symmetry. Within the framework of the new solution, perturbation theory for the order parameter (Q alpha beta ) breaks down for dimensionalities d<or=4 due to an infrared divergence. Four dimensions is interpreted as the 'lower critical dimensionality' of the theory, below which there can be no order of the Edwards-Anderson kind (1975).
We construct a cluster-mean-field theory for short-range spin glasses in the replica space, in which only the effective fields which are conjugate to the order parameters are taken into account. In the one-step replica-symmetry breaking (RSB) scheme just below the critical temperature, the solutions of the self-consistent equations for general clusters are given in terms of the thermal and quenched random averages of spin products. From the comparison of the free energies we have introduced, we find the (one-step) RSB solution is more stable than the replica-symmetric (RS) solution for arbitrary clusters so long as the former exists. Analytic solutions of the Bethe cluster approximation suggest that the RSB effects are rather enhanced by short-range fluctuations.
A mean-field spherical model with random couplings between pairs, quartets, and possibly higher multiplets of spins is considered. It has the same critical behavior as the Sherrington-Kirkpatrick model. It thus exhibits replica symmetry breaking. The order parameter function is solved exactly in the whole low temperature phase. The zero-field-cooled susceptibility remains finite at low TTT. Next a quantum version of the system is considered. Whereas the magnetic properties are not altered qualitatively, the thermodynamics is now regular at small temperatures.