On the Bogoliubov-Valatin transformation for fermionic Hamiltonians without a linear part
Davide Bonaretti
TL;DR
The paper develops a self-contained Bogoliubov-Valatin transformation for homogeneous fermionic quadratic Hamiltonians without a linear term, aiming to diagonalize the Hamiltonian $\hat{H}$ into a form of non-interacting fermions. It constructs a unitary $U$ with Van Hemmen structure to redefine the fermionic operators $\hat{A}$ into $\hat{B}$, yielding $\hat{H}=c+\sum_{\mu} d_{\mu} \hat{B}_{\mu} \hat{B}_{\mu}^\dagger$ with $\{\hat{B}_\mu,\hat{B}_\nu^\dagger\}=\delta_{\mu\nu}$, and enforces the ground-state condition through $d_{j+N} \ge d_j$. The method relies on casting the coefficient matrix into a standard form $M_\ominus$ and uses a spectral-based construction of $U$, with a novel procedure to handle singular cases via a $J$-invariant kernel basis and a map $\tilde{R}$. A detailed numerical example for $N=2$ demonstrates the steps from computing $H_\ominus$ to obtaining the diagonalized Hamiltonian and the explicit $U$, making the approach practical for teaching and quick reference. The work solidifies the BV transformation as a robust algebraic tool in superconductivity, mean-field theories, and spin-model mappings, providing a clear, self-contained pathway from a general quadratic Hamiltonian to a diagonal, non-interacting-fermion description.
Abstract
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 in putting its coefficient matrix into its canonical form; the transformation can always be performed on fermionic Hamiltonians, only 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.