Diagonalisation of the quadratic fermion Hamiltonian with a linear part

Abstract

It is well known that the spectrum of a homogeneous quadratic Hamiltonian in m fermion construction-operator pairs is characterised by m mode energies. In this paper it is proved that the spectrum of a quadratic Hamiltonian with a linear part presents the same feature. Simple methods are given for the calculation of the mode energies and of the ground-state energy of the Hamiltonian. It is shown that, in contrast to the boson case, the diagonalisation cannot be carried out by a linear transformation of the fermion construction operators. Application of such a transformation can only result in a diagonalisation after the introduction of a 'ghost' particle (fermion) and a corresponding pair of Hermitian-conjugate construction operators.