Metastable states in the solvable spin glass model

Free-energy barriers in the Sherrington-Kirkpatrick model

The free-energy landscape of the Sherrington-Kirkpatrick (SK) Ising spin glass is simple in the framework of the Thouless-Anderson-Palmer (TAP) equations as each solution (which are minima of the free energy) has associated with it a nearby index-one saddle point. The free-energy barrier to escape the minimum is just the difference between the saddle point free energy and that at its associated minimum. This difference is calculated for the states with free energies f>f_{c}. It is very small for these states, decreasing as 1/N^{2}, where N is the number of spins in the system. These states are not marginally stable. We argue that such small barriers are why numerical studies never find these states when N is large. Instead, the states that are found are those that have marginal stability. For them the barriers are at least of O(1). f_{c} is the free energy per spin below which the states develop broken replica-symmetry-like overlaps with each other. In the regime f<f_{c} we can only offer some possibilities based around scaling arguments. One of these suggest that the barriers might become as large as N^{1/3}. That might be consistent with recent numerical studies on the Viana-Bray model, which were at variance with the expectations of Cugliandolo and Kurchan for the SK model.

Realizable solutions of the Thouless-Anderson-Palmer equations

We show that the only solutions of the Thouless-Anderson-Palmer (TAP) equations for the Sherrington-Kirkpatrick model of Ising spin glasses which can be found by iteration are those whose free energy lies on the border between replica-symmetric and broken-replica-symmetric states, when the number of spins N is large. Convergence to this same borderline also happens in quenches from a high-temperature initial state to a locally stable state where each spin is parallel to its local field; both are examples of self-organized criticality. At this borderline the band of eigenvalues of the Hessian associated with a solution extends to zero, so the states reached have marginal stability. We have also investigated the factors which determine the free-energy difference between a stationary solution corresponding to a saddle point and its associated minimum, which is the barrier which has to be surmounted to escape from the vicinity of a TAP minimum or pure state.

Spin-glass complexity

We study the quenched complexity in spin-glass mean-field models satisfying the Becchi-Rouet-Stora-Tyutin supersymmetry. The outcome of such study, consistent with recent numerical results, allows, in principle, to conjecture the absence of any supersymmetric contribution to the complexity in the Sherrington-Kirkpatrick model. The same analysis can be applied to any model with a full replica symmetry breaking phase, e.g., the Ising p-spin model below the Gardner temperature. The existence of different solutions, breaking the supersymmetry, is also discussed.

Supersymmetric complexity in the Sherrington-Kirkpatrick model

By using a supersymmetric approach we compute the complexity of the metastable states in the Sherrington-Kirkpatrick spin-glass model. We prove that the supersymmetric complexity is exactly equal to the Legendre transform of the thermodynamic free energy, thus providing a recipe to find the complexity once the free energy is known. Our results suggest that the supersymmetry may be a useful tool for the calculation of the entropy of metastable states in generic glassy systems.