Lattice Perturbation Theory for $O(N)$-Symmetric $\sigma$-Models with General Nearest-Neighbour Action I. Conventional Perturbation Theory

TL;DR

This paper calculates high-order perturbative corrections for O(N)-symmetric sigma models with general nearest-neighbour actions, revealing large corrections in certain models and providing detailed asymptotic expansions for key physical quantities.

Contribution

It provides the first three-loop beta-function and anomalous dimensions for the most general nearest-neighbour O(N) models, including corrections to long-distance observables.

Findings

Large corrections in $RP^{N-1}$ models at small N

Asymptotic scaling observed only at very large beta

First three terms of asymptotic expansion for vector and tensor energies

Abstract

We compute the beta-function and the anomalous dimension of all the non-derivative operators of the theory up to three-loops for the most general nearest-neighbour O(N)-invariant action together with some contributions to the four-loop beta-function. These results are used to compute the first analytic corrections to various long-distance quantities as the correlation length and the general spin-$n$ susceptibility. It is found that these corrections are extremely large for $RP^{N-1}$ models (especially for small values of N), so that asymptotic scaling can be observed in these models only at very large values of beta. We also give the first three terms in the asymptotic expansion of the vector and tensor energies.