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 efficiently determine the chiral restoration lines, 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.

Figure 1

The $\mu –B$ phase diagram at zero temperature. The upper and lower boundaries correspond to the transition from the solitonic modulation phase to the chiral symmetry restoration phase and from the chiral symmetry breaking phase to the solitonic modulation phase, respectively.

Figure 2

The phase transition lines for the cases with $B=0$ (black dashed lines) and $\sqrt{eB}=0.6\mathrm{GeV}$ (red dotted lines). The solitonic modulation phase is in the small closed regions at large chemical potential and the bullets are the Lifshitz points. The inverse magnetic catalysis effect to the case with $\sqrt{eB}=0.6\mathrm{GeV}$ is also illuminated (blue dot-dashed line).

Figure 3

The evolutions of the temperature $T_L$ and the chemical potential ${\mu }_L$ at the Lifshitz point with the magnetic field $B$.