  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.
