  In this paper, we study the phenomenology of a Dirac dark matter in the
$L_\mu-L_\tau$ model and investigate the neutrino oscillation in the dark halo.
Since dark matter couples to the muon neutrino and the tau neutrino with
opposite sign couplings, it contributes effective potentials, $\pm A_\chi$, to
the evolution equation of the neutrino flavor transition amplitude, which can
be significant for high energy neutrino oscillations in a dense dark matter
environment. We discuss neutrino masses, lepton mixing angles, Dirac CP phase,
and neutrino oscillation probabilities in the dark halo using full numerical
calculations. Results show that neutrinos can endure very different matter
effects. When the potential $A_\chi$ becomes ultra-large, three neutrino
flavors decouple from each other.
