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N-electron Slater determinants from non-unitary canonical transformations of fermion operators

Mean-field methods such as Hartree-Fock (HF) or Hartree-Fock-Bogoliubov (HFB) constitute the building blocks upon which more elaborate many-body theories are based on. The HF and HFB wavefunctions are built out of independent quasi-particles resulting from a unitary linear canonical transformation of the elementary fermion operators. Here, we discuss the possibility of allowing the HF transformation to become non-unitary. The properties of such HF vacua are discussed, as well as the evaluation of matrix elements among such states. We use a simple ansatz to demonstrate that a non-unitary transformation brings additional flexibility that can be exploited in variational approximations to many-fermion wavefunctions. The action of projection operators on non-unitary based HF states is also discussed and applied to the one-dimensional Hubbard model with periodic boundary conditions.