Strong-coupling expansions and phase diagrams for the O(2), O(3), and O(4) Heisenberg spin systems in two dimensions

The Ising and O (n), 24, models in two dimensions are studied using a quantum-mechanical Hamiltonian formalism in which a "time" axis is continuous and a spatial axis is discrete. Strong-coupling series for the theory's mass gaps and (Callan-Symanzik) functions are computed and are used to search for phase transitions. The critical point and critical index of the Ising model are found exactly. The critical point of the O (2) model is found (g^*=1. 08), and the series suggest that the theory's correlation length possesses an essential singularity with the behavior predicted by Kosterlitz. The critical points of the O (3) and O (4) models are predicted to be at zero coupling, i. e. , no evidence for a phase transition at nonzero g is found for the non-Abelian models. Interpolating forms (two-point Pad\'e approximants) for these theories' (Callan-Symanzik) functions are computed for all g. The transition regions between weak and strong coupling are seen to be quite narrow.