Multi-state Dirac stars

In this paper, we construct the multi-state Dirac stars (MSDSs) consisting of two pairs of Dirac fields. The two pairs of Dirac fields are in the ground state and the first excited state, respectively. Each pair consists of two fields with opposite spins, ensuring spherical symmetry of the system. We discuss the solutions of the MSDSs under synchronized and nonsynchronized frequencies. By varying the mass $\tilde{\mu}_1$ of the excited state Dirac field and the frequency $\tilde{\omega}_0$ of the ground state Dirac field, we obtain different types of solutions, including single-branch and double-branch solutions. These two types of solutions do not smoothly transition into each other as the parameters $\tilde{\mu}_1$ and $\tilde{\omega}_0$ continuously change, but undergo a sudden transition when $\tilde{\mu}_1$ ($\tilde{\omega}_0$) is greater than or less than the threshold value of $0.7694$ ($0.733$). Furthermore, we analyze the characteristics of the various MSDSs solutions and analyze the relationship between the ADM mass $M$ of the MSDSs and the synchronized and nonsynchronized frequencies. Subsequently, we calculate the binding energy $E_B$ of the MSDSs and discuss the stability of the solutions. Finally, we discuss the feasibility of simulating the dark matter halos using MSDSs.