Chiral symmetry breaking and phase diagram of dual chiral density wave in a rotating quark matter

We study the inhomogeneous phase of a two-flavor quark matter under rotation at finite temperature and density using the Nambu-Jona-Lasinio model. To do this, we consider the chiral broken phase, in particular, described by the so-called dual chiral density wave which is formed as a standing wave of simultaneous scalar and pseudoscalar condensates. The solution of the corresponding Dirac equation as well as the energy spectrum found in the mean-field approximation. We then use the thermodynamic potential calculated for this model, to study the $\mu$ and $\Omega$ dependence of constituent mass and the wave vector at $T = 0$. We find there exist two islands in the $\mu - \Omega$ plane that the dual-chiral density wave survives. The first region lies at intermediate densities and small $\Omega$. We observe, by increasing the angular velocity of matter, dual-chiral density wave forms in regions with smaller chemical potential. On the other hand, in contrast to the former, the second region is located at the large $\Omega$ and small densities. Finally, we study this phase of quark matter at finite temperature and present $T-\mu$, $T-\Omega$, and $\mu-\Omega$ phase portraits of a hot-rotating quark matter at finite density.