Exceptional points and ground-state entanglement spectrum of a fermionic extension of the Swanson oscillator

Motivated by the structure of the Swanson oscillator, which is a well-known example of a non-hermitian quantum system consisting of a general representation of a quadratic Hamiltonian, we propose a fermionic extension of such a scheme which incorporates two fermionic oscillators, together with bilinear-coupling terms that do not conserve particle number. We determine the eigenvalues and eigenvectors, and expose the appearance of exceptional points where two of the eigenstates coalesce with the corresponding eigenvectors exhibiting the self-orthogonality relation. We compute the entanglement spectrum and entanglement entropy of the ground state in two different ways, with one of them being via the Gelfand-Naimark-Segal construction. In addition to the approach involving the usual bi-normalization of the eigenvectors of the non-hermitian Hamiltonian, we also discuss the case where the eigenvectors are normalized with respect to the Dirac norms. It is found that the model exhibits a quantum phase transition due to the presence of a ground-state crossing.