Domain wall between the Dirac sea and the "anti-Dirac sea"

It was shown in work \cite{vergeles2021note} that in the theory of gravity coupled with the Dirac field, each state $\mathrm{\mid }\lambda ⟩$ has its own twin $\mathrm{\mid }\lambda ;PT⟩$, which is obtained by a discrete PT transformation. If in the state $\mathrm{\mid }\lambda ⟩$ the Dirac sea is filled, then in the state $\mathrm{\mid }\lambda ;PT⟩$ there is an “anti-Dirac” filling (in terms of the state $\mathrm{\mid }\lambda ⟩$). It is important that the energies of these states are the same. Therefore, there may be domains with different filling of the Dirac sea. Here we study a domain wall connecting two such adjacent domains.