  We investigate the phase structure of the Nambu--Jona-Lasinio model at zero
temperature, allowing for a two-dimensional spatial dependence of the chiral
condensate. Applying the mean-field approximation, we consider various periodic
structures with rectangular and hexagonal geometries, and minimize the
corresponding free energy. We find that these two-dimensional chiral crystals
are favored over homogeneous phases in a certain window in the region where the
phase transition would take place when the analysis was restricted to
homogeneous condensates. It turns out, however, that in this regime they are
disfavored against a phase with a one-dimensional modulation of the chiral
condensate. On the other hand, we find that square and hexagonal lattices
eventually get favored at higher chemical potentials. Although stretching the
limits of the model to some extent, this would support predictions from
quarkyonic-matter studies.
