Multi-shell Dirac fermions in the Einstein--Dirac system

Abstract: We present multi-fermions in the spherically symmetric Einstein--Dirac system. Dirac fermions are self-localized within a spherically symmetric Einstein gravity, i.e., the Schwarzschild-like space-time metric. Most of previous studies of the Einstein--Dirac system are restricted to two neutral fermions or to many fermions with the same high-angular momentum filling a single shell. Our model considers full-filling of fermions in multiple shells, similarly to the conventional nuclear shell model. We solve the model for fermion numbers $N_\textrm{f}=2,6,12$ and $20$, which can realize a spherically-symmetric system. Even single-shell multi-fermions exhibit a multi-peak structure and fragmentation in the high redshift region. The behavior observed in our multi-shell model can be explained by interactions between the shells and resulting delocalization. We also investigate the pressure of the solutions which defines the existence (or absence) of inter-shell interactions. The radial pressure is attractive, supporting compactness of the solutions. Finally, we show the correlations between the nontrivial changes in the Shannon entropy (a logarithmic measure of information content) and the structural deformations of the solutions.