Remarks on the large-$N$ ${\mathbb C}P^{N-1}$ model

TL;DR

This paper analyzes the large-N ${ m C}P^{N-1}$ model on a finite interval at finite temperature and chemical potential, deriving a generalized thermodynamic potential and discussing ground state properties and phase transitions.

Contribution

It introduces a mixed-gradient expansion of the effective action for the ${ m C}P^{N-1}$ model, extending previous work to finite chemical potential and large order, with implications for ground state analysis.

Findings

No transition to a massless phase occurs under the studied conditions.

The expansion technique efficiently captures ground state properties.

Clarifies the role of regularization and divergences in relation to the MWHC theorem.

Abstract

In this paper, we consider the ${\mathbb C}P^{N-1}$ model confined to an interval of finite size at finite temperature and chemical potential. We obtain, in the large-N approximation, a mixed-gradient expansion of the one-loop effective action of the order parameter associated with the effective mass of the quantum fluctuations. This expansion gives an expression for the thermodynamic potential density as a functional of the order parameter, generalizing previous calculations to arbitrarily large order and to the case of finite chemical potential and allows one to discuss some generic features of the ground state of the model. The technique used here relies on analytic regularization and provides an efficient scheme to extract the coefficients of the expansion. Once a solution for the ground state is known, these coefficients can be used to deduce some generic properties of the ground state as a function of external conditions. We also show that there can be no transition to a massless phase for any value of the external conditions considered and clarify a seemingly important point regarding the regularization of the effective action connected to the appearance of logarithmic divergences and the Mermin-Wagner-Hoenberg-Coleman (MWHC) theorem.