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There is a rich history of expressing the limiting free energy of mean-fieldspin glasses as a variational formula over probability measures on $[0,1]$,where the measure represents the similarity (or "overlap") of two independentlysampled spin configurations. At high temperatures, the formula's minimum isachieved at a measure which is a point mass, meaning sample configurations areasymptotically orthogonal up to a magnetic field correction. At lowtemperatures, though, a very different behavior emerges known as replicasymmetry breaking (RSB). The deep wells in the energy landscape create morerigid structure, and the optimal overlap measure is no longer a point mass. Theexact size of its support remains in many cases an open problem.  Here we consider these themes for multi-species spherical spin glasses.Following a companion work in which we establish the Parisi variationalformula, here we present this formula's Crisanti-Sommers representation. In theprocess, we gain new access to a problem unique to the multi-species setting.Namely, if RSB occurs for one species, does it necessarily occur for otherspecies as well? We provide sufficient conditions for the answer to be yes. Forinstance, we show that if two species share any quadratic interaction, then RSBfor one implies RSB for the other. Moreover, the level of symmetry breakingmust be identical, even in cases of full RSB. In the presence of an externalfield, any type of interaction suffices.

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