  In this review, we discuss the physical characteristics of the magnetic dual
chiral density wave (MDCDW) phase of dense quark matter and argued why it is a
promising candidate for the interior matter phase of neutron stars. The MDCDW
condensate occurs in the presence of a magnetic field. It is a single-modulated
chiral density wave characterized by two dynamically generated parameters: the
fermion quasiparticle mass $m$ and the condensate spatial modulation $q$. The
lowest Landau level quasiparticle modes in the MDCDW system are asymmetric
about the zero energy, a fact that leads to the topological properties and
anomalous electric transport exhibited by this phase. The topology makes the
MDCDW phase robust against thermal phonon fluctuations, and as such, it does
not display the Landau-Peierls instability, a stapled feature of
single-modulated inhomogeneous chiral condensates in three dimensions. The
topology is also reflected in the presence of the electromagnetic chiral
anomaly in the effective action and in the formation of hybridized propagating
modes known as an axion-polaritons. Taking into account that one of the
axion-polaritons of this quark phase is gapped, we argued how incident
$\gamma$-ray photons can be converted into gapped axion-polaritons in the
interior of a magnetar star in the MDCDW phase leading the star to collapse, a
phenomenon that can serve to explain the so-called missing pulsar problem in
the galactic center.
