  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.
