  An interesting application of the gauge/gravity duality to condensed matter
physics is the description of a lattice via breaking translational invariance
on the gravity side. By making use of global symmetries, it is possible to do
so without scarifying homogeneity of the pertinent bulk solutions, which we
thus term as "homogeneous holographic lattices." Due to their technical
simplicity, these configurations have received a great deal of attention in the
last few years and have been shown to correctly describe momentum relaxation
and hence (finite) DC conductivities. However, it is not clear whether they are
able to capture other lattice effects which are of interest in condensed
matter. In this paper we investigate this question focusing our attention on
the phenomenon of commensurability, which arises when the lattice scale is
tuned to be equal to (an integer multiple of) another momentum scale in the
system. We do so by studying the formation of spatially modulated phases in
various models of homogeneous holographic lattices. Our results indicate that
the onset of the instability is controlled by the near horizon geometry, which
for insulating solutions does carry information about the lattice. However, we
observe no sharp connection between the characteristic momentum of the broken
phase and the lattice pitch, which calls into question the applicability of
these models to the physics of commensurability.
