  In this paper, the deconfinement and chiral restoration transitions in strong
magnetic field is realized at zero temperature in the Polyakov
Nambu$-$Jona-Lasinio model. We provide the thermodynamic treatment to mimic the
deconfinement phase transition at zero temperature together with the entangled
scalar and vector interactions coupled with the Polyakov loop. The magnetic
catalysis is found by a rising behavior of the critical chemical potential for
the first-order deconfinement phase transition. While the magnetic catalysis on
the chiral restoration could convert to inverse magnetic catalysis under the
running coupling interaction ansatz. Furthermore, the stronger magnetic field
makes the possible quarkyonic phase window to be enlarged under the running
coupling interaction.
