  The quark anomalous magnetic moment (AMM) is dynamically generated through
the spontaneous chiral symmetry breaking. It has been revealed that even though
its exact form is still unknown, the quark AMM is essential to explore quark
matter properties and QCD phase structure under external magnetic fields. In
this study, we take three different forms of the quark AMM and investigate its
influence on the chiral phase transition under magnetic field. In general, a
negative quark AMM plays the role as magnetic catalyzer and a positive quark
AMM plays the role of magnetic inhibition. It is found that a constant quark
AMM drives an unexpected 1st order chiral phase transition; a quark AMM
proportional to the chiral condensate gives a flip of the sign on the chiral
condensate; and a quark AMM proportional to the square of chiral condensate can
produce results of chiral condensate as functions of the temperature and the
magnetic field in good agreement with the lattice result.
