A strong - coupling analysis of two - dimensional O ( N ) σ models with N ≤ 2 on square , triangular and honeycomb lattices

The critical behavior of two-dimensional O(N) σ models with N ≤ 2 on the 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 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 2-d XY model, the results from all lattices considered are consistent with the Kosterlitz-Thouless exponential approach to criticality, characterized by an exponent σ = 12 , and with universality. The value σ = 12 is confirmed within an uncertainty of few per cent. The prediction η = 14 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. New exact results for the long-distance behavior of G(x) when N = 1 (Ising model in the strong-coupling phase) confirm this statement. PACS numbers: 75.10 Hk, 05.50.+q, 11.10 Kk, 64.60 Fr. Typeset using REVTEX