Exact results for the spherical model with competing interactions on the Bethe lattice

We extend to next-nearest-neighbor (NNN) interactions a technique which allows the exact solution of the spherical model of Berlin and Kac on a general discrete geometrical structure (a graph). We give the solution when the graph is a Bethe lattice. The model shows collinear (ferromagnetic or antiferromagnetic) long-range order at low temperature when NNN interactions favor the same order as nearest-neighbor ones, while it is disordered at any finite temperature when competition exceeds a critical value. For vanishing nearest-neighbor interaction the lattice decouples in two independent Cayley cacti; if the exchange on each sublattice is ferromagnetic, the model becomes ordered at a nonzero temperature, while antiferromagnetic exchange gives again disorder at any temperature.