Absence of long-range order in three-dimensional spherical models

The spherical model is a unique model for which an exact solution at finite temperature exists in three dimensions (3D). In this paper we prove that this model may show an absence of long-range order (LRO) in 3D if a suitable competition between exchange couplings is assumed. In particular we find an absence of LRO in wedge-shaped regions around the ferromagnet- or antiferromagnet-helix transition line or in the vicinity of a degeneration line, where infinite nonequivalent isoenergetic helix configurations are possible. We evaluate explicitly the phase diagram of a tetragonal antiferromagnet with exchange couplings up to third neighbors but our conclusions apply as well to any Bravais lattice. We also discuss the connection of the spherical model or classical Heisenberg Hamiltonian for parameters lying on the degeneration line with more general spin Hamiltonians where the interaction may be written in terms of the adjacency matrix. This seems particularly promising for describing a perturbative approach to the Hubbard Hamiltonian, which is of particular interest in high-Tc superconductivity.