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Modified TAP equations for the SK spin glass

T. Plefka, Theoretische Festkörperphysik Tu Darmstadt

Published Jan 1, 2002• 11 citations
OpenAlex

Abstract

PACS. 75.10.Nr – Spin-glass and other random models. PACS. 84.35.+i – Neural networks. 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. Introduction. – Together with the replica approach, the Thouless-Anderson-Palmer (TAP) approach [1] is the most important method to analyze infinite range spin glass models like the Sherrington-Kirkpatrick (SK) model [2] of Ising spins (for reviews see [3, 4]). The TAP equations are well established [3–5] and are exact in the thermodynamic limit (N → ∞) provided that the local magnetizations {mi} satisfy the condition

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Modified TAP equations for the SK spin glass

T Plefka

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.

arXiv:cond-mat/0108191•Europhysics Letters (EPL)•Jun 1, 2002•20 citations

Fermionic Thouless-Anderson-Palmer equations

Martin Rehker, Reinhold Oppermann

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.

Journal of Physics Condensed Matter•Jan 1, 1999•2 citations

Incompleteness of the Thouless, Anderson, and Palmer mean-field description of the spin-glass phase

V. Janiš

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.

arXiv:cond-mat/0512399•Physical Review B•Aug 16, 2006•4 citations

On the mean field theory of spin glasses

Jairo Rolim Lopes de Almeida

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.

•Jan 1, 1980•0 citations

Mean field theory I: Ising model, equilibrium theory

K. H. Fischer, John Hertz

In this chapter we will put the concepts we have developed in Chapter 2 to work in constructing the mean field theory for an Ising spin glass. The term ‘mean field theory’ (henceforth frequently abbreviated ‘MFT’) can be interpreted in many ways. Here we will take it to mean the exact solution of a model in which the forces are of infinitely long range, so that each spin interacts equally strongly with every other one. For spin glasses, this is the Sherrington–Kirkpatrick (SK) model (1975), which we solve heuristically (though unfortunately incorrectly) in Section 3.1. The error is a somewhat subtle one, as is evident from the fact that we obtain exactly the same result in the more systematic calculations of Sections 3.2 (a direct summation of the leading terms in N −1 in high-temperature perturbation theory) and 3.3 (which uses the replica formalism). In both these approaches, we can also see how the theory itself reveals that it is wrong, since they lead to negative values of quantities which are necessarily positive. We then study the correct mean field theory, with replica symmetry breaking, in Section 3.4, including the remarkable nature of the broken ergodicity it implies and an examination of its stability. Further physical insight into the problem is obtained in Section 3.5 from the mean field equations first introduced by Thouless, Anderson and Palmer (TAP) (1977).

Cambridge University Press eBooks•May 2, 1991•0 citations

Stability of the quantum Sherrington-Kirkpatrick spin glass model

A. P. Young

I study in detail the quantum Sherrington-Kirkpatrick (SK) model, i.e., the infinite-range Ising spin glass in a transverse field, by solving numerically the effective one-dimensional model that the quantum SK model can be mapped to in the thermodynamic limit. I find that the replica symmetric solution is unstable down to zero temperature, in contrast to some previous claims, and so there is not only a line of transitions in the (longitudinal) field-temperature plane (the de Almeida-Thouless, AT, line) where replica symmetry is broken, but also a quantum de Almeida-Thouless (QuAT) line in the transverse field-longitudinal field plane at T=0. If the QuAT line also occurs in models with short-range interactions its presence might affect the performance of quantum annealers when solving spin glass-type problems with a bias (i.e., magnetic field).

arXiv:1707.07107•Physical review. E•Sep 11, 2017•29 citations
PACS. 75.10.Nr – Spin-glass and other random models. PACS. 84.35.+i – Neural networks. 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. Introduction. – Together with the replica approach, the Thouless-Anderson-Palmer (TAP) approach [1] is the most important method to analyze infinite range spin glass models like the Sherrington-Kirkpatrick (SK) model [2] of Ising spins (for reviews see [3, 4]). The TAP equations are well established [3–5] and are exact in the thermodynamic limit (N → ∞) provided that the local magnetizations {mi} satisfy the condition
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.
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.
In this chapter we will put the concepts we have developed in Chapter 2 to work in constructing the mean field theory for an Ising spin glass. The term ‘mean field theory’ (henceforth frequently abbreviated ‘MFT’) can be interpreted in many ways. Here we will take it to mean the exact solution of a model in which the forces are of infinitely long range, so that each spin interacts equally strongly with every other one. For spin glasses, this is the Sherrington–Kirkpatrick (SK) model (1975), which we solve heuristically (though unfortunately incorrectly) in Section 3.1. The error is a somewhat subtle one, as is evident from the fact that we obtain exactly the same result in the more systematic calculations of Sections 3.2 (a direct summation of the leading terms in N −1 in high-temperature perturbation theory) and 3.3 (which uses the replica formalism). In both these approaches, we can also see how the theory itself reveals that it is wrong, since they lead to negative values of quantities which are necessarily positive. We then study the correct mean field theory, with replica symmetry breaking, in Section 3.4, including the remarkable nature of the broken ergodicity it implies and an examination of its stability. Further physical insight into the problem is obtained in Section 3.5 from the mean field equations first introduced by Thouless, Anderson and Palmer (TAP) (1977).
I study in detail the quantum Sherrington-Kirkpatrick (SK) model, i.e., the infinite-range Ising spin glass in a transverse field, by solving numerically the effective one-dimensional model that the quantum SK model can be mapped to in the thermodynamic limit. I find that the replica symmetric solution is unstable down to zero temperature, in contrast to some previous claims, and so there is not only a line of transitions in the (longitudinal) field-temperature plane (the de Almeida-Thouless, AT, line) where replica symmetry is broken, but also a quantum de Almeida-Thouless (QuAT) line in the transverse field-longitudinal field plane at T=0. If the QuAT line also occurs in models with short-range interactions its presence might affect the performance of quantum annealers when solving spin glass-type problems with a bias (i.e., magnetic field).