Modified TAP equations for the SK spin glass

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. The natural requirement of self-consistency leads to modified equations which are in complete agreement with the original ones above the instability. Essentially altered results below the instability are presented. 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 and several alternative derivations are known [3, 4, 5]. These equations are exact in the thermodynamic limit (N → ∞) provided that the local magnetizations {mi} satisfy the condition x ≡ 1− β(1− 2q2 + q4) > 0 (1) where qν = N −1 ∑ i mi ν = 2, 4. (2) The condition (1) represents the central stability condition for the SK spin glass and therefore is also found in other approaches [6, 3, 4]. Within the TAP approach two different arguments for the validity of (1) are known. Bray and Moore [7] found a divergence of the spinglass susceptibility for x → 0. The expansion of Plefka [5], leading to the TAP equations, is limited to x > 0. In next section a further, basically simple, aspect of the stability condition is presented, leading naturally to a modification of the TAP equations. Stability analysis revised With the N site-magnetizations mi and the temperature β as independent variables, the TAP free energy F in the presence of local external fields hi is given by F = − 1 2 ∑ ij Jijmimj − Nβ 4 (1− q2) 2 − ∑