Mean-field equations for spin models with orthogonal interaction matrices

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Abstract
We study the metastable states in Ising spin models with orthogonal interaction matrices. We focus on three realizations of this model, the random case and two non-random cases, i.e. the fully-frustrated model on an infinite-dimensional hypercube and the so-called sine model. We use the mean-field (or TAP) equations which we derive by resumming the high-temperature expansion of the Gibbs free energy. In some special non-random cases, we can find the absolute minimum of the free energy. For the random case we compute the average number of solutions to the TAP equations. We find that the configurational entropy (or complexity) is extensive in the range TRSBM. We also present an apparently unrelated replica calculation which reproduces the analytical expression for the total number of TAP solutions.