Absence of glassy behaviour in the deterministic spherical and XY models

We consider the infinite-range spin models with Hamiltonian H = ∑^N_{i, j = 1} J_{i, j}σ_iσ_j, where J is the quantization of a map of the torus. Although deterministic, these models are known to exhibit glassy behaviour. We show, through explicit computation of the Gibbs free energy, that unlike the random case this behaviour disappears in the corresponding spherical and continuous XY models. The only minimum of the Gibbs free energy is indeed the trivial one, even though the ground state is highly degenerate.