On the Bogoliubov–Valatin transformation for fermionic Hamiltonians without a linear part

A self-contained treatment of the Bogoliubov–Valatin transformation for homogeneous fermionic Hamiltonians is presented. The aim is to provide a quick reference that may also serve as supplementary material for a graduate-level course and that can be understood with quantum mechanics knowledge up to the level of the second quantization's rules. The objective of the transformation is to cast a quadratic Hamiltonian into a diagonal form that resembles the Hamiltonian of a system of non-interacting particles. To obtain this, the first step consists of putting its coefficient matrix into its canonical form; the transformation can always be performed on fermionic Hamiltonians, but some care must be taken when this form is singular. Having explained how to cast a general matrix into its standard form, a complete description of the transformation is provided; a novel procedure is proposed here for the singular matrix case.