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The phase structure of magnetized cold quark matter is analyzed in theframework of the two-flavor Nambu-Jona-Lasinio models paying special attentionto its dependence on the model parameters as different values within thephenomenological allowed range are considered. We first discuss the simplerchiral limit case, and then the more realistic situation of finite currentmasses. We show that in spite of the difference in the nature of sometransitions, both cases are alike and exhibit a rather rich phase structure fora significant range of acceptable parameters. A simplification of the phasestructure is obtained as parameters leading to larger values of the dressedquark mass in the vacuum are considered. Finally, we consider the so-called"inverse catalysis effect" showing that in some phases it implies an actualdecrease of the order parameter as the magnetic field increases.

Quark matter under strong magnetic fields in SU(2) NJL-type models: parameter dependence of the cold dense matter phase diagram