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

Abstract

The critical behavior of two-dimensional ${\rm O}(N)$ $\sigma$ models with $N\leq 2$ 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 $\gamma$ 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 2-$d$ 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 per cent. The prediction $\eta=1/4$ is also roughly verified. For various values of $N\leq 2$, 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 $N\leq 2$. New exact results for the long-distance behavior of $G(x)$ when $N=1$ (Ising model in the strong-coupling phase) confirm this statement.