  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. The
exact 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 variational
formula, here we present this formula's Crisanti-Sommers representation. In the
process, 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 other
species as well? We provide sufficient conditions for the answer to be yes. For
instance, we show that if two species share any quadratic interaction, then RSB
for one implies RSB for the other. Moreover, the level of symmetry breaking
must be identical, even in cases of full RSB. In the presence of an external
field, any type of interaction suffices.
