  There is substantial evidence that the ground state of the 2d chiral
Gross-Neveu model, in the presence of a $U(1)$ fermion number chemical
potential $\mu$ and in the large $N$ limit, is given by a {\it chiral spiral}
phase, namely an inhomogeneous phase with a chiral condensate having a
spatially periodic phase. We show that the chiral spiral configuration persists
at finite $N$ and $T=0$ for any $\mu>0$. Our analysis is based on non-abelian
bosonization, that relates the model to a $U(N)_1$ WZW model deformed by
current-current interactions. In this description the appearance of the
inhomogeneous phase is surprisingly simple. We also rederive the phase diagram
of the large $N$ chiral Gross-Neveu model via a direct diagrammatic
computation, finding agreement with previous results in the literature.

High Energy Physics - Theory;High Energy Physics - Lattice