CP(N) sigma model on a finite interval revisited

TL;DR

This paper analyzes the large N solution of the CP(N) sigma model on a finite interval, identifying boundary conditions that allow analytical solutions and revealing phase structures depending on the interval length.

Contribution

It introduces a family of boundary conditions enabling analytical large N solutions and explores phase transitions in the model based on interval length.

Findings

Existence of boundary conditions with analytical saddle points

Identification of a single phase for all interval lengths under certain conditions

Discovery of phase transition between unbroken and Higgs phases depending on L

Abstract

In this short note we will revisit the large $N$ solution of $\mathbb{C}P^N$ sigma model on a finite interval of length $L$. We will find a family of boundary conditions for which the large $N$ saddle point can be found analytically. For a certain choice of the boundary conditions the theory has only one phase for all values of $L$. Also, we will provide an example when there are two phases: for large $L$ there is a standard phase with an unbroken $U(1)$ gauge symmetry and for small $L$ there is Higgs phase with a broken gauge symmetry.