Casimir Force for the ${\mathbb C}P^{N-1}$ Model

TL;DR

This paper derives exact solutions for the $ ext{CP}^{N-1}$ model on finite intervals, computes the vacuum energy and Casimir force, and finds the force is always attractive, enhancing understanding of boundary effects in quantum field theories.

Contribution

It provides the first exact self-consistent solutions to the gap equations of the $ ext{CP}^{N-1}$ model with boundary conditions in the large-$N$ limit.

Findings

Casimir force is always attractive.

Solutions reproduce confining phase in the infinite limit.

Vacuum energy computed explicitly.

Abstract

In this work, we derive exact self-consistent solutions to the gap equations of the $\mathbb{C}P^{N-1}$ model on a finite interval with Dirichlet boundary conditions in the large-$N$ approximation. The solution reproduce the confining phase in the infinite system by taking the appropriate limit. We compute the vacuum energy and the Casimir force and observe that the sign of the force is always attractive.