Dyson-Schwinger equations for the non-linear σ-model

Abstract

Dyson-Schwinger equations for the O( N)-symmetric non-linear σ-model are derived. They are polynomials in N, hence 1/ N-expanded ab initio. A finite, closed set of equations is obtained by keeping only the leading term and the first correction term in this 1/ N-series. These equations are solved numerically in two dimensions on square lattices measuring 50 × 50, 100 × 100, 200 × 200, and 400 × 400. They are also solved analytically at strong coupling and at weak coupling in a finite volume. In these two limits the solution is asymptotically identical to the exact strong- and weak-coupling series through the first three terms. Between these two limits, results for the magnetic susceptibility and the mass gap are identical to the Monte Carlo results available for N = 3 and N = 4 within a uniform systematic error of O(1/N 3) , i.e. the results seem good to O(1/N 2) , though obtained from equations that are exact only to O(1/N) . This is understood by seeing the results as summed infinite subseries of the 1/ N-series for the exact susceptibility and mass gap. We conclude that the kind of 1/ N-expansion presented here converges as well as one might ever hope for, even for N as small as 3.