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arXiv:cond-mat/0011028 · Physical review. B, Condensed matter

Quantum Thouless-Anderson-Palmer equations for glassy systems

Giulio Biroli, Leticia F. Cugliandolo

Published Jun 15, 2001• 59 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevb.64.014206

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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 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.
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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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 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.