Strong-coupling analysis of two-dimensional O(N) sigma models with N<=2 on square, triangular, and honeycomb lattices

The critical behavior of two-dimensional (2D) O(N) sigma models with N less than or equal to 2 on square, triangular, and honeycomb lattices is investigated by an analysis of the strong-coupling expansion of the two-point fundamental Green's function G(x), calculated up to 21st order on the square lattice, 15th order on the triangular lattice, and 30th order on the honeycomb lattice. For N<2 the critical behavior is of power-law type, and the exponents gamma and nu extracted from our strong-coupling analysis confirm exact results derived assuming universality with solvable solid-on-solid models. At N=2, i.e., for the 2D XY model, the results from all lattices considered are consistent with the Kosterlitz-Thouless exponential approach to criticality, characterized by an exponent sigma=1/2, and with universality. The value sigma=1/2 is confirmed within an uncertainty of few percent. The prediction eta=1/4 is also roughly verified. For various values of N less than or equal to 2, we determine some ratios of amplitudes concerning the two-point function G(x) in the critical limit of the symmetric phase. This analysis shows that the low-momentum behavior of G(x) in the critical region is essentially Gaussian at all values of N less than or equal to 2. Exact results for the long-distance behavior of G(x) when N=1 (Ising model in the strong-coupling phase) confirm this statement.