# Ground state modulations in the ${\mathbb C}P^{N-1}$ model

## TL;DR

This paper investigates the ground state properties of the ${
m C}P^{N-1}$ model on a finite interval at finite temperature, revealing boundary condition-dependent modulations and crystal-like structures in the ground state.

## Contribution

It develops a systematic method to analyze boundary effects and ground state modulations in the ${
m C}P^{N-1}$ model, including inhomogeneous phases and boundary condition responses.

## Key findings

- Inhomogeneous phases are disfavored for large intervals with periodic boundaries.
- Boundary conditions induce a crystal-like modulation in the ground state.
- The formalism can be applied to experimental and numerical studies of nonlinear sigma models.

## Abstract

In this work we examine a system consisting of a confined one-dimensional arrangement of atoms that we describe by using the 2-dimensional ${\mathbb C}P^{N-1}$ model, restricted to an interval and at finite temperature. We develop a method to obtain the bulk and boundary parts of the one-loop effective action as a function of the effective mass of the fluctuations. The formalism has the advantage of allowing for a systematic analysis of a large class of boundary conditions and to model the (adiabatic) response of the ground state to changes in the boundary conditions. In the case of periodic boundary conditions, we find that inhomogeneous phases are disfavored for intervals of large size. Away from periodic boundary conditions, our numerical results show that the ground state has a generic crystal-like structure that can be modulated by variations of the boundary conditions. The results presented here could be relevant for experimental implementations of nonlinear sigma models and could be tested by lattice numerical simulations.