1/N-expansion of the non-linear σ-model: The first three orders

The two-point function of the O(N)-symmetric non-linear σ-model is expanded in 1/N, keeping terms of three leading orders. The mass gap and the magnetic susceptibility are obtained from the two-point function. They are evaluated on square lattices for N=3 and N=4. The systematic errors of the 1/N-series truncated after the first, second, or third term are found by using recent high precision Monte Carlo results as bench marks. For all three truncations, we find systematic errors which are smaller than the expected magnitude of neglected terms, both for the mass gap and for the susceptibility. This result is uniform in the inverse coupling β, and valid for N as small as 3. We conclude that the 1/N-series approach the exact results as rapidly as one could ever hope for. (orig.)