  In this note we consider a non-linear, large-$N$ ${\mathbb C}P^{N-1}$ sigma
model on a finite size interval with periodic boundary conditions, at finite
temperature and chemical potential in the regime of $\beta \mu$ large. Our goal
is to extend previous calculations and obtain the coefficients of the
derivative expansion of the one-loop effective action in the region of $\beta
\mu$ large by carrying out the appropriate analytical continuation. This
calculation complements previous results and allows us to conclude that the
ground state remains homogeneous in this regime as long as it is assumed to be
a slowly varying function of the spatial coordinates. While this is reasonable
at the two extremes of small or large chemical potential, for intermediate
values of the chemical potential and small enough temperature, one might expect
(by analogy with other models) that lower energy crystalline solutions may
exist. In this case a simple derivative expansion, like the one discussed here,
would need to be modified in order to capture these features.
