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 $|\lambda\rangle$ has its own twin $|\lambda;PT\rangle$, which is obtained by a discrete PT transformation. If in the state $|\lambda\rangle$ the Dirac sea is filled, then in the state $|\lambda;PT\rangle$ there is an "anti-Dirac" filling (in terms of the state $|\lambda\rangle$). 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.