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.
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