Free Energy in Spin Glass Models with Conventional Order

Recently, Baldwin and Swingle (J Stat Phys 190(7):125, 2023) considered spin glass models with additional conventional order parameters characterizing single-replica properties. These parameters are distinct from the standard order parameter, the overlap, used to measure correlations between replicas. A “min-max” formula for the free energy was prescribed in Baldwin and Swingle (2023). We rigorously verify this prescription in the setting of vector spin glass models featuring additional deterministic spin interactions. Notably, our results can be viewed as a generalization of the Parisi formula for vector spin glass models in Panchenko (Ann Probab 46(2):865–896, 2018), where the order parameter for self-overlap is already present.