  Numerous theoretical studies using various equation of state models have
shown that quark matter may exist at the extreme densities in the cores of
high-mass neutron stars. It has also been shown that a phase transition from
hadronic matter to quark matter would result in an extended mixed phase region
that would segregate phases by net charge to minimize the total energy of the
phase, leading to the formation of a crystalline lattice. The existence of
quark matter in the core of a neutron star may have significant consequences
for its thermal evolution, which for thousands of years is facilitated
primarily by neutrino emission. In this work we investigate the effect a
crystalline quark-hadron mixed phase can have on the neutrino emissivity from
the core. To this end we calculate the equation of state using the relativistic
mean-field approximation to model hadronic matter and a nonlocal extension of
the three-flavor Nambu-Jona-Lasinio model for quark matter. Next we determine
the extent of the quark-hadron mixed phase and its crystalline structure using
the Glendenning construction, allowing for the formation of spherical blob,
rod, and slab rare phase geometries. Finally we calculate the neutrino
emissivity due to electron-lattice interactions utilizing the formalism
developed for the analogous process in neutron star crusts. We find that the
contribution to the neutrino emissivity due to the presence of a crystalline
quark-hadron mixed phase is substantial compared to other mechanisms at fairly
low temperatures ($\lesssim 10^9$ K) and quark fractions ($\lesssim 30\%$), and
that contributions due to lattice vibrations are insignificant compared to
static-lattice contributions.
