  Spin glass models involving multiple replicas with constrained overlaps have
been studied in [FPV92; PT07; Pan18a]. For the spherical versions of these
models [Ko19; Ko20] showed that the limiting free energy is given by a Parisi
type minimization. In this work we show that for Sherrington-Kirkpatrick (i.e.
2-spin) interactions, it can also be expressed in terms of a
Thouless-Andersson-Palmer (TAP) variational principle. This is only the second
spin glass model where a mathematically rigorous TAP computation of the free
energy at all temperatures and external fields has been achieved. The
variational formula we derive here also confirms that the model is replica
symmetric, a fact which is natural but not obviously deducible from its Parisi
formula.
