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

It was shown in work (Vergeles 2021 Class. Quantum Grav. 38 085022) that in the theory of gravity coupled with the Dirac field, each state |λ⟩ has its own twin |λ; PT⟩, which is obtained by a discrete PT transformation. If in the state |λ⟩ the Dirac sea is filled, then in the state |λ; PT⟩ there is an 'anti-Dirac' filling (in terms of the state |λ⟩). 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.