Influence of vector interaction and Polyakov loop dynamics on inhomogeneous chiral symmetry breaking phases

We investigate the role of the isoscalar-vector interaction and the dynamics of the Polyakov loop on inhomogeneous phases in the phase diagram of the two-flavor Nambu-Jona--Lasinio model. Thereby we concentrate on inhomogeneous phases with a one-dimensional modulation, explicitly domain-wall solitons and, for comparison, the chiral spiral. While the inclusion of the Polyakov loop merely leads to quantitative changes compared to the original Nambu-Jona--Lasinio model, the inclusion of a repulsive vector-channel interaction has significant qualitative effects: Whereas for homogeneous phases the first-order phase transition gets weakened and eventually turns into a second-order transition or a crossover, the domain of inhomogeneous phases is less affected. In particular the location of the Lifshitz point in terms of temperature and density is not modified. Consequently, the critical point disappears from the phase diagram and only a Lifshitz point (showing a different critical behavior) remains. In particular, susceptibilities remain finite.