Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Journal of Physics Condensed Matter

Fermionic Thouless-Anderson-Palmer equations

Martin Rehker, Reinhold Oppermann

Published Jan 1, 1999• 2 citations
Publisher

Abstract

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.

DOI
10.1088/0953-8984/11/6/018

Related papers

Fermionic TAP-equations

Martin Rehker, Oppermann, Reinhold

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 $μ\approx 0.8$. We computed at zero temperature the filling factor $ν(μ)$ 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.

arXiv:cond-mat/9806092•Jun 7, 1998•0 citations

Modified TAP equations for the SK spin glass

T. Plefka, Theoretische Festkörperphysik Tu Darmstadt

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

•Jan 1, 2002•11 citations

Quantum Thouless-Anderson-Palmer equations for glassy systems

Giulio Biroli, Leticia F. Cugliandolo

We derive Thouless-Anderson-Palmer (TAP) equations for quantum disordered systems. We apply them to the study of the paramagnetic and glassy phases of a quantum extension of the spherical p spin-glass model. We generalize several useful quantities (complexity, threshold level, etc.) and various ideas (configurational entropy crisis, etc.) that have been developed within the classical TAP approach to quantum systems. The analysis of the quantum TAP equations allows us to show that the phase diagram (temperature versus quantum parameter) of the p spin-glass model should be generic. In particular, we argue that a crossover from a second-order thermodynamic transition close to the classical critical point to a first-order thermodynamic transition close to the quantum critical point is to be expected in a large class of systems.

arXiv:cond-mat/0011028•Physical review. B, Condensed matter•Jun 15, 2001•59 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

Equivalence of statistical-mechanical and dynamic descriptions of the infinite-range Ising spin-glass

Chandan Dasgupta, Haim Sompolinsky

We prove the equivalence between the dynamic mean-field theory of the Ising spin-glass and the statistical-mechanical theory of Thouless, Anderson, and Palmer (TAP). Individual low-free-energy TAP solutions describe short-time properties, whereas thermodynamic equilibrium corresponds to averaging over all such solutions. The square of the staggered magnetization associated with the largest eigenvalue of the interaction matrix scales as ${N}^{\frac{5}{6}}$ ($N$ is the number of spins). Results are confirmed by Monte Carlo simulation and numerical solution of the TAP equations.

Physical review. B, Condensed matter•Apr 1, 1983•24 citations

Replica symmetry breaking solution for the fermionic Ising spin-glass and the Ghatak-Sherrington model

H Feldmann, R Oppermann

We solve the fermionic version of the Ising spin glass for arbitrary filling µ and temperature T taking into account replica symmetry breaking. Using a simple exact mapping from µ to the anisotropy parameter D , we also obtain the solution of the S = 1 Sherrington-Kirkpatrick model. An analytic expression for T = 0 gives an improved critical value for the first-order phase transition. We revisit the question of stability against replica-diagonal fluctuations and find that the appearance of complex eigenvalues of the Almeida-Thouless matrix is not an artefact of the replica-symmetric approximation.

arXiv:cond-mat/9910034•Journal of Physics A Mathematical and General•Feb 10, 2000•15 citations
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 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.
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 Thouless-Anderson-Palmer (TAP) equations for quantum disordered systems. We apply them to the study of the paramagnetic and glassy phases of a quantum extension of the spherical p spin-glass model. We generalize several useful quantities (complexity, threshold level, etc.) and various ideas (configurational entropy crisis, etc.) that have been developed within the classical TAP approach to quantum systems. The analysis of the quantum TAP equations allows us to show that the phase diagram (temperature versus quantum parameter) of the p spin-glass model should be generic. In particular, we argue that a crossover from a second-order thermodynamic transition close to the classical critical point to a first-order thermodynamic transition close to the quantum critical point is to be expected in a large class of systems.
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 prove the equivalence between the dynamic mean-field theory of the Ising spin-glass and the statistical-mechanical theory of Thouless, Anderson, and Palmer (TAP). Individual low-free-energy TAP solutions describe short-time properties, whereas thermodynamic equilibrium corresponds to averaging over all such solutions. The square of the staggered magnetization associated with the largest eigenvalue of the interaction matrix scales as N56{N}^{\frac{5}{6}}N65​ (NNN is the number of spins). Results are confirmed by Monte Carlo simulation and numerical solution of the TAP equations.
We solve the fermionic version of the Ising spin glass for arbitrary filling µ and temperature T taking into account replica symmetry breaking. Using a simple exact mapping from µ to the anisotropy parameter D , we also obtain the solution of the S = 1 Sherrington-Kirkpatrick model. An analytic expression for T = 0 gives an improved critical value for the first-order phase transition. We revisit the question of stability against replica-diagonal fluctuations and find that the appearance of complex eigenvalues of the Almeida-Thouless matrix is not an artefact of the replica-symmetric approximation.