The generalized TAP free energy

Wei-Kuo Chen, Dmitry Panchenko, Eliran Subag

Published 2018-12-12Version 1

In this paper, we define and compute the generalized TAP correction for the energy of the mixed $p$-spin models with Ising spins, and give the corresponding generalized TAP representation for the free energy. We study the generalized TAP states, which are the maximizers of the generalized TAP free energy, and show that their order parameter matches the order parameter of the ancestor states in the Parisi ansatz. We compute the critical point equations of the TAP free energy that generalize the classical TAP equations for pure states. Furthermore, we give an exact description of the region where the generalized TAP correction is replica symmetric, in which case it coincides with the classical TAP correction, and show that Plefka's condition is necessary for this to happen. In particular, the generalized TAP correction is not always replica symmetric on the points corresponding to the Edwards-Anderson parameter.