The $1/N$ expansion of two-dimensional spin models

TL;DR

This paper investigates a two-dimensional U(N)-invariant spin model interpolating between PN and O(2N) models, using the 1/N expansion to analyze scaling behavior, derive physical predictions, and establish universality in lattice and continuum schemes.

Contribution

It provides a detailed analysis of the 1/N expansion on the lattice, including second-nearest-neighbor interactions, and develops techniques for extracting scaling behavior and proving universality.

Findings

Derived quantitative O(1/N) predictions for physical quantities.

Established universality among different lattice and continuum schemes.

Set up the 1/N expansion with effective propagators and vertices, including asymptotic expansion techniques.

Abstract

A general two-dimensional spin model with U$(N)$ invariance, interpolating between $\CPN$ and ${\rm O}(2N)$ models, is studied in detail in order to illustrate both the general features of the $1/N$ expansion on the lattice and the specific techniques devised to extract scaling (field-theoretical) behavior. The continuum version of the model is carefully analyzed deriving quantitative $O(1/N)$ physical predictions in order to establish a benchmark for lattice computations. The $1/N$ expansion on the lattice, including second-nearest-neighbor interactions, is set up by constructing explicitly effective propagators and vertices. The technique of asymptotic expansion of the lattice propagators, basic to the derivation of analytical results in the scaling domain, is presented in full detail and applied to the model. Physical quantities, like the free energy and different definitions of correlation length, are evaluated. The lattice renormalization-group trajectories are identified and universality among different lattice (and continuum) schemes in the scaling region is explicitly proven. A review of other developments based on the lattice $1/N$ expansion is presented.