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arXiv:cond-mat/0108191 · Europhysics Letters (EPL)

Modified TAP equations for the SK spin glass

T Plefka

Published Jun 1, 2002• 20 citations
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Abstract

The stability of the TAP mean-field equations is reanalyzed with the conclusion that the exclusive reason for the breakdown at the spin glass instability is an inconsistency for the value of the local susceptibility. A new alternative approach leads to modified equations which are in complete agreement with the original ones above the instability. Essentially altered results below the instability are presented and the consequences for the dynamical mean-field equations are discussed.

DOI
10.1209/epl/i2002-00457-7

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The stability of the TAP mean-field equations is reanalyzed with the conclusion that the exclusive reason for the breakdown at the spin glass instability is an inconsistency for the value of the local susceptibility. A new alternative approach leads to modified equations which are in complete agreement with the original ones above the instability. Essentially altered results below the instability are presented and the consequences for the dynamical mean-field equations are discussed.
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We derive the TAP equations for the fermionic Ising spin glass. It is found that, just as in the non-fermionic model, the conditions for stability and for validity of the free energy are equivalent. We determine the breakdown of the paramagnetic phase. Numeric solutions of the fermionic TAP-equations at T=0 allowed to localize a first order transition between the spin glass phase and the paramagnetic phase at μ≈0.8μ\approx 0.8μ≈0.8. We computed at zero temperature the filling factor ν(μ)ν(μ)ν(μ) and the distribution of internal fields. The saddle-point equations resulting from the calculation of the number of solutions to the TAP-equations were found to be much more complicated as in the non-fermionic case.
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.
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.
We derive the Thouless-Anderson-Palmer (TAP) equations for the fermionic Ising spin glass. It is found that, just as in the classical Sherrington-Kirkpatrick spin-glass model, the conditions for stability and for validity of the free energy are equivalent. We determine the breakdown of the paramagnetic phase. Numerical solutions of the fermionic TAP equations at T = 0 allowed us to localize a first-order transition between the spin-glass phase and the paramagnetic phase at µ≈0.8. We computed at zero temperature the filling factor ν(µ) and the distribution of the internal fields. The saddle-point equations resulting from the calculation of the number of solutions to the TAP equations were found to be much more complicated, as in the classical case.
Within the replica approach to mean-field spin-glasses the transition from ergodic high-temperature behaviour to the glassy low-temperature phase is marked by the instability of the replica-symmetric saddle-point. For general spin-glass models with non-Gaussian field distributions the corresponding Hessian is a 2n×2n2^n\times 2^n2n×2n matrix with the number nnn of replicas tending to zero eventually. We block-diagonalize this Hessian matrix using representation theory of the permutation group and identify the blocks related to the spin-glass susceptibility. Performing the limit n→0n\to 0n→0 within these blocks we derive expressions for the de~Almeida-Thouless line of general spin-glass models. Specifying these expressions to the cases of the Sherrington-Kirkpatrick, Viana-Bray, and the Lévy spin glass respectively we obtain results in agreement with previous findings using the cavity approach.