Strong-coupling analysis of two-dimensional O(N) σ models withN⩽2 on square, triangular, and honeycomb lattices

The critical behavior of two-dimensional (2D) O(N) {σ} models with N{≤}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 N2 the critical behavior is of power-law type, and the exponents {γ} and {ν} 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 {σ}=1/2, and with universality. The value {σ}=1/2 is confirmed within an uncertainty of few percent. The prediction {η}=1/4 is also roughly verified. For various values of N{≤}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{≤}2. Exact results for the long-distance behavior of G(x) when N=1 (Ising model in the strong-coupling phase) confirm this statement. {} 1996 The American Physical Society.