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We study the $\mu$-$\mu_{45}$-$T$ phase diagram of the $2+1$-dimensionalGross-Neveu model, where $\mu$ denotes the ordinary chemical potential,$\mu_{45}$ the chiral chemical potential and $T$ the temperature. We use themean-field approximation and two different lattice regularizations with naivechiral fermions. An inhomogeneous phase at finite lattice spacing is found forone of the two regularizations. Our results suggest that there is noinhomogeneous phase in the continuum limit. We show that a chiral chemicalpotential is equivalent to an isospin chemical potential. Thus, all resultspresented in this work can also be interpreted in the context of isospinimbalance.

Inhomogeneous phases in the chirally imbalanced $2+1$-dimensional Gross-Neveu model and their absence in the continuum limit