  In high density quark matter under a strong external magnetic field, possible
phases are investigated by using the two-flavor Nambu-Jona-Lasinio model with
tensor-type four-point interaction between quarks, as well as the
axial-vector-type four-point interaction. In the tensor-type interaction under
the strong external magnetic field, it is shown that a quark spin polarized
phase is realized in all regions of the quark chemical potential under
consideration within the lowest Landau level approximation. In the
axial-vector-type interaction, it is also shown that the quark spin polarized
phase appears in the wide range of the quark chemical potential. In both the
interactions, the quark mass in zero and small chemical potential regions
increases which indicates that the chiral symmetry breaking is enhanced, namely
the magnetic catalysis occurs.
