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Quadratic Hamiltonians in Fermionic Fock Spaces

Jean-Bernard Bru, Nathan Metraud

TL;DR

This work provides a rigorous treatment of fermionic quadratic Hamiltonians, establishing essential self-adjointness for unbounded one-particle data and linking Berezin’s operator-based definitions with Bach–Lieb–Solovej’s Bogoliubov-transformation framework. It introduces an elliptic, operator-valued Brockett–Wegner flow on the one-particle space that induces a non-autonomous Bogoliubov flow on the Fock space, enabling ($N$-)diagonalization under significantly weaker hypotheses than classical results. The authors prove the existence and asymptotics of the flow, show that the interacting Hamiltonians $H_t$ remain unitarily equivalent to the initial one, and derive conditions under which the limit $H_ ty$ is N-diagonal and explicitly computable in terms of the limiting one-particle data $Υ_ ty$. The analysis further clarifies the relationship between Berezin’s definition and BLS’s approach, situating Bogoliubov transformations within the Shale–Stinespring framework, and discusses the Araki $C^{*}$-algebraic perspective. Collectively, these results advance the mathematical foundations of fermionic quadratic Hamiltonians and provide practical pathways to diagonalization in physically relevant models such as the BCS Hamiltonian.

Abstract

Quadratic Hamiltonians are important in quantum field theory and quantum statistical mechanics. Their general studies, which go back to the sixties, are relatively incomplete for the fermionic case studied here. Following Berezin, they are quadratic in the fermionic field and in this way well-defined self-adjoint operators acting on the fermionic Fock space. We analyze their diagonalization by applying a novel elliptic operator-valued differential equations studied in a companion paper. This allows for their ($\mathrm{N}$-) diagonalization under much weaker assumptions than before. Last but not least, in 1994 Bach, Lieb and Solovej defined them to be generators of strongly continuous unitary groups of Bogoliubov transformations. This is shown to be an equivalent definition, as soon as the vacuum state belongs to the domain of definition of these Hamiltonians. This second outcome is demonstrated to be reminiscent to the celebrated Shale-Stinespring condition on Bogoliubov transformations.

Quadratic Hamiltonians in Fermionic Fock Spaces

TL;DR

This work provides a rigorous treatment of fermionic quadratic Hamiltonians, establishing essential self-adjointness for unbounded one-particle data and linking Berezin’s operator-based definitions with Bach–Lieb–Solovej’s Bogoliubov-transformation framework. It introduces an elliptic, operator-valued Brockett–Wegner flow on the one-particle space that induces a non-autonomous Bogoliubov flow on the Fock space, enabling (-)diagonalization under significantly weaker hypotheses than classical results. The authors prove the existence and asymptotics of the flow, show that the interacting Hamiltonians remain unitarily equivalent to the initial one, and derive conditions under which the limit is N-diagonal and explicitly computable in terms of the limiting one-particle data . The analysis further clarifies the relationship between Berezin’s definition and BLS’s approach, situating Bogoliubov transformations within the Shale–Stinespring framework, and discusses the Araki -algebraic perspective. Collectively, these results advance the mathematical foundations of fermionic quadratic Hamiltonians and provide practical pathways to diagonalization in physically relevant models such as the BCS Hamiltonian.

Abstract

Quadratic Hamiltonians are important in quantum field theory and quantum statistical mechanics. Their general studies, which go back to the sixties, are relatively incomplete for the fermionic case studied here. Following Berezin, they are quadratic in the fermionic field and in this way well-defined self-adjoint operators acting on the fermionic Fock space. We analyze their diagonalization by applying a novel elliptic operator-valued differential equations studied in a companion paper. This allows for their (-) diagonalization under much weaker assumptions than before. Last but not least, in 1994 Bach, Lieb and Solovej defined them to be generators of strongly continuous unitary groups of Bogoliubov transformations. This is shown to be an equivalent definition, as soon as the vacuum state belongs to the domain of definition of these Hamiltonians. This second outcome is demonstrated to be reminiscent to the celebrated Shale-Stinespring condition on Bogoliubov transformations.
Paper Structure (18 sections, 35 theorems, 284 equations)

This paper contains 18 sections, 35 theorems, 284 equations.

Table of Contents

  1. Introduction
  2. Fermionic Quadratic Hamiltonians
  3. Preliminary Definitions
  4. Fermionic Quadratic Hamiltonians
  5. Diagonalization of Quadratic Hamiltonians
  6. Araki's Approach on the Level of $C^{\ast }$-Algebras
  7. Brockett-Wegner Flow and Elliptic Operator Flows
  8. Bach, Lieb and Solovej's Approach to Quadratic Hamiltonians
  9. Technical Results: Implementation of the Brockett-Wegner Flow
  10. Self-adjointness of Quadratic Operators
  11. Continuous Flow of Bogoliubov Transformations
  12. Solution to the Brockett-Wegner Flow on Quadratic Hamiltonians
  13. Asymptotics of the Solution to the Brockett-Wegner Flow
  14. Diagonalization of Quadratic Hamiltonians
  15. Technical Results: BLS's Approach to Quadratic Operators
  16. ...and 3 more sections

Key Result

Theorem 2.3

Let $E_{0}\in \mathbb{R}$. Take $D_{0}=\bar{D}_{0}=-D_{0}^{\top }\in \mathcal{L}^{2}(\mathfrak{h})$ and $\Upsilon _{0}=\Upsilon _{0}^{\top }=\Upsilon _{0}^{\ast }\geq \alpha \mathbf{1}$ with strictly positive $\alpha \in \mathbb{R}^{+}$. Assume additionally that the commutator $[D_{0},\Upsilon _{0}] Then, there is a unitary transformation $\mathrm{U}$ such that $\mathrm{U\mathrm{H}}_{0}\mathrm{U}^

Theorems & Definitions (47)

  • Remark 2.1
  • Remark 2.2: Alternative definition
  • Theorem 2.3: Berezin
  • Theorem 2.4: Diagonalization of quadratic Hamiltonians
  • Remark 2.5
  • Remark 2.6
  • Lemma 2.7: From quadratic Hamiltonians to quasi-free dynamics
  • Theorem 2.8: Implementation of Bogoliubov transformations
  • Remark 2.9: Implementation of non-autonomous flows of Bogoliubov transformations
  • Definition 2.10: Quadratic Hamilonians BLS
  • ...and 37 more