Frozen Bardeen-Dirac stars and light ball

In this paper, we study solutions of a static spherically symmetric system, which is composed of the coupling with the Bardeen action and two Dirac fields. For the case where only the Bardeen action is present, the magnetic charge $q$ can be infinite, then when the magnetic charge is greater than a certain value $q_b$, there exists a black hole solution, which is called the Bardeen black hole (BBH). However, if the Dirac field is introduced, we find that the magnetic charge can only be smaller than the critical value $q_b$, in which there is no black hole solution. Moreover, in the region $q<q_b$, we find that if the magnetic charge exceeds another critical value $q_c$ (i.e., $q_c<q<q_b$), the frequency of the Dirac field can approach zero, and the solution where a critical horizon appears is similar to an extremal black hole outside the critical horizon but has a nonsingular interior. The Dirac fields are also almost concentrated within it. In fact, this is a frozen star solution, we call such solutions frozen Bardeen-Dirac stars (FBDSs). We analyze the light rings of FBDSs and find that there exists a ``true" light ring outside the critical horizon, but inside it, the velocity of photons is very close to zero, which leads to the formation of a ``light ball" inside the critical horizon.