Crisanti-Sommers formula and simultaneous symmetry breaking in multi-species spherical spin glasses

Understand

There is a rich history of expressing the limiting free energy of mean-field spin glasses as a variational formula over probability measures on $[0,1]$, where the measure represents the similarity (or "overlap") of two independently sampled spin configurations.

• At high temperatures, the formula's minimum is achieved at a measure which is a point mass, meaning sample configurations are asymptotically orthogonal up to a magnetic field correction.
• At low temperatures, though, a very different behavior emerges known as replica symmetry breaking (RSB).
• The deep wells in the energy landscape create more rigid structure, and the optimal overlap measure is no longer a point mass.