Free energy landscapes in spherical spin glasses
We introduce and analyze free energy landscapes defined by associating to any point inside the sphere a free energy calculated on a thin spherical band around it, using many orthogonal replicas. This allows us to reinterpret, rigorously prove and extend for general spherical models the main ideas of the Thouless-Anderson-Palmer (TAP) approach originally introduced in the 70s for the Sherrington-Kirkpatrick model. In particular, we establish a TAP representation for the free energy, valid for any overlap value which can be sampled as many times as we wish in an appropriate sense. We call such overlaps multi-samplable. The correction to the Hamiltonian in the TAP representation arises in our analysis as the free energy of a certain model on an overlap dependent band. For the largest multi-samplable overlap it coincides with the Onsager reaction term from physics. For smaller multi-samplable overlaps the formula we obtain is new. We also derive the corresponding TAP equation for critical points. We prove all the above without appealing to the celebrated Parisi formula or the ultrametricity property. We prove that any overlap value in the support of the Parisi measure is multi-samplable. For generic models, we further show that the set of multi-samplable overlaps coincides with a certain set that arises in the characterization for the Parisi measure by Talagrand. The ultrametric tree of pure states can be embedded in the interior of the sphere in a natural way. For this embedding, we show that the points on the tree uniformly maximize the free energies we define. From this we conclude that the Hamiltonian at each point on the tree is approximately maximal over the sphere of same radius, and that points on the tree approximately solve the TAP equations for critical points.