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

Abstract

The Ising and $\mathrm{O}\mathrm{}\left(n\right)\mathrm{}$, $2\le n\le 4$, 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 $\beta$ (Callan-Symanzik) functions are computed and are used to search for phase transitions. The critical point and critical index $\nu$ of the Ising model are found exactly. The critical point of the O(2) model is found ($g^{*}=1.08\mathrm{}$), 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 Padé approximants) for these theories' $\beta$ (Callan-Symanzik) functions are computed for all $g$. The transition regions between weak and strong coupling are seen to be quite narrow.