  We study the $\mu$-$\mu_{45}$-$T$ phase diagram of the $2+1$-dimensional
Gross-Neveu model, where $\mu$ denotes the ordinary chemical potential,
$\mu_{45}$ the chiral chemical potential and $T$ the temperature. We use the
mean-field approximation and two different lattice regularizations with naive
chiral fermions. An inhomogeneous phase at finite lattice spacing is found for
one of the two regularizations. Our results suggest that there is no
inhomogeneous phase in the continuum limit. We show that a chiral chemical
potential is equivalent to an isospin chemical potential. Thus, all results
presented in this work can also be interpreted in the context of isospin
imbalance.
