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

TL;DR

This paper extends the Crisanti-Sommers variational formula to spherical spin glasses with vector spins, addressing the challenge of preserving matrix path monotonicity in the functional order parameters.

Contribution

It derives a new variational formula for vector spin models, adapting the classical Crisanti-Sommers approach to account for matrix path monotonicity constraints.

Findings

Derived the vector spin analogue of the Crisanti-Sommers formula

Connected the formula to the Parisi variational principle for free energy

Addressed the monotonicity constraint in matrix path variations

Abstract

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.