The Crisanti-Sommers Formula for Spherical Spin Glasses with Vector Spins

We obtain the analogue of the Crisanti-Sommers variational formula for spherical spin glasses with vector spins. This formula is derived from the discrete Parisi variational formula for the limit of the free energy of constrained copies of spherical spin glasses. In vector spin models, the variations of the functional order parameters must preserve the monotonicity of matrix paths which introduces a new challenge in contrast to the derivation of the classical Crisanti-Sommers formula.