Large-$N$ ${\mathbb{CP}}^{N-1}$ sigma model on a finite interval: 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.

Comments: Latex, 19 figures