On the Bogoliubov-Valatin transformation for fermionic Hamiltonians without a linear part
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915900630761472 |
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| author | Bonaretti, Davide |
| author_facet | Bonaretti, Davide |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_12100 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the Bogoliubov-Valatin transformation for fermionic Hamiltonians without a linear part Bonaretti, Davide Other Condensed Matter Quantum Physics 81Q99 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. |
| title | On the Bogoliubov-Valatin transformation for fermionic Hamiltonians without a linear part |
| topic | Other Condensed Matter Quantum Physics 81Q99 |
| url | https://arxiv.org/abs/2512.12100 |