$1/N$ Expansion of Two-Dimensional Models in the Scaling Region

TL;DR

This paper explores the lattice $1/N$ expansion in two-dimensional spin models, detailing the asymptotic analysis of propagators, and explicitly constructing the lattice renormalization group $eta$ function to order $1/N$, bridging $CP^{N-1}$ and $O(2N)$ models.

Contribution

It provides a detailed method for performing the $1/N$ expansion in the scaling region of two-dimensional models, including the explicit construction of the $eta$ function to order $1/N$.

Findings

Asymptotic expansion of effective propagators for small mass gap

Explicit construction of the lattice $eta$ function to $O(1/N)$

Analysis of models interpolating between $CP^{N-1}$ and $O(2N)$

Abstract

The main technical and conceptual features of the lattice $1/N$ expansion in the scaling region are discussed in the context of a two-parameter two-dimensional spin model interpolating between $CP^{N-1}$ and $O(2N)$ $\sigma$ models, with standard and improved lattice actions. We show how to perform the asymptotic expansion of effective propagators for small values of the mass gap and how to employ this result in the evaluation of physical quantities in the scaling regime. The lattice renormalization group $\beta$ function is constructed explicitly and exactly to $O({1/N})$.