Solitonic modulation and Lifshitz point in an external magnetic field within Nambu--Jona-Lasinio model

We study the inhomogeneous solitonic modulation of a chiral condensate within the effective Nambu--Jona-Lasinio model when a constant external magnetic field is present. The self-consistent Pauli-Villars regularization scheme is adopted to manipulate the ultraviolet divergence encountered in the thermodynamic quantities. In order to determine the chiral restoration lines efficiently, a new kind of Ginzburg-Landau expansion approach is proposed here. At zero temperature, we find that both the upper and lower boundaries of the solitonic modulation oscillate with the magnetic field in the $\mu$--$B$ phase diagram which is actually the de Hass-van Alphan (dHvA) oscillation. It is very interesting to find out how the tricritical Lifshitz point $(T_L,\mu_L)$ evolves with the magnetic field: There are also dHvA oscillations in the $T_L$--$B$ and $\mu_L$--$B$ curves, though the tricritical temperature $T_L$ increases monotonically with the magnetic field.