Large-$N$ $\mathbb{CP}^{N-1}$ sigma model on a finite interval: general Dirichlet boundary conditions

TL;DR

This paper investigates the large-$N$ $ ext{CP}^{N-1}$ sigma model on a finite interval with general Dirichlet boundary conditions, revealing how boundary orientations affect the model's energy and properties.

Contribution

It extends previous studies by analyzing general boundary orientations and providing numerical solutions, highlighting distinctive features of the $ ext{CP}^{N-1}$ model.

Findings

Energy minimized when boundary orientations are aligned

Distinctive features of $ ext{CP}^{N-1}$ model compared to $O(N)$ model

Numerical solutions for general Dirichlet boundary conditions

Abstract

This is the third of the series of articles on the large-$N$ two-dimensional $\mathbb{CP}^{N-1}$ sigma model, defined on a finite space interval $L$ with Dirichlet boundary conditions. Here the cases of the general Dirichlet boundary conditions are studied, where the relative $\mathbb{CP}^{N-1}$ orientations at the two boundaries are generic, and numerical solutions are presented. Distinctive features of the $\mathbb{CP}^{N-1}$ sigma model, as compared e.g., to an $O(N)$ model, which were not entirely evident in the basic properties studied in the first two articles in the large $N$ limit, manifest themselves here. It is found that the total energy is minimized when the fields are aligned in the same direction at the two boundaries.