  Despite recent advancements in the study and understanding of the phase
diagram of strongly interacting matter, the region of high baryonic densities
and low temperatures has remained difficult to reach in the lab. Things are
expected to change with the planned HIC experiments at FAIR in Germany and NICA
in Russia, which will open a window to the high-density-low-temperature segment
of the QCD phase map, providing a unique opportunity to test the validity of
model calculations that have predicted the formation of spatially inhomogeneous
phases with broken chiral symmetry at intermediate-to-high densities. Such a
density region is also especially relevant for the physics of neutron stars, as
they have cores that can have several times the nuclear saturation density. On
the other hand, strong magnetic fields, whose presence is fairly common in HIC
and in neutron stars, can affect the properties of these exotic phases and lead
to signatures potentially observable in these two settings. In this paper, I
examine the anomalous transport properties produced by the spectral asymmetry
of the lowest Landau level (LLL) in a QCD-inspired NJL model with a background
magnetic field that exhibits chiral symmetry breaking at high density via the
formation of a Dual Chiral Density Wave (DCDW) condensate. It turns out that in
this model the electromagnetic interactions are described by the axion
electrodynamics equations and there is a dissipationless Hall current.
