  We study $SU(2)$ Chern-Simons theories at level $k$ coupled to a scalar on
$T^2\times \mathbb{R}$ at large baryon number. We find a homogeneous but
anisotropic ground state configuration for any values of $k$ on the IR
fixed-point of those models. This classical analysis is valid as long as we
take the baryon number large. As a corollary, by comparing the symmetry
breaking pattern at large chemical potential, we find that the theory does not
reduce to the singlet sector of the $O(4)$ Wilson-Fisher fixed-point at
large-$k$, as expected from general grounds. This paper will be one primitive
step towards quantitative analysis of Chern-Simons-matter dualities using the
large charge expansion.
