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Homogeneous spaces in Hartree-Fock-Bogoliubov theory

Claudia D. Alvarado, Eduardo Chiumiento

Abstract

We study the action of Bogoliubov transformations on admissible generalized one-particle density matrices arising in Hartree-Fock-Bogoliubov theory. We show that the orbits of this action are reductive homogeneous spaces, and we give several equivalences that characterize when they are embedded submanifolds of natural ambient spaces. We use Lie theoretic arguments to prove that these orbits admit an invariant symplectic form. If, in addition, the operators in the orbits have finite spectrum, or infinite spectrum and trivial kernel, then we obtain that the orbits are actually Kähler homogeneous spaces.

Homogeneous spaces in Hartree-Fock-Bogoliubov theory

Abstract

We study the action of Bogoliubov transformations on admissible generalized one-particle density matrices arising in Hartree-Fock-Bogoliubov theory. We show that the orbits of this action are reductive homogeneous spaces, and we give several equivalences that characterize when they are embedded submanifolds of natural ambient spaces. We use Lie theoretic arguments to prove that these orbits admit an invariant symplectic form. If, in addition, the operators in the orbits have finite spectrum, or infinite spectrum and trivial kernel, then we obtain that the orbits are actually Kähler homogeneous spaces.
Paper Structure (13 sections, 25 theorems, 163 equations)

This paper contains 13 sections, 25 theorems, 163 equations.

Table of Contents

  1. Introduction
  2. Reductive homogeneous spaces
  3. Lie group structure of Bogoliubov transformations
  4. A reductive structure on orbits of g1-pdms and quasi-free states
  5. Embedded submanifolds
  6. Derivations induced by g1-pdms
  7. Embedded submanifod structure of orbits of g1-pdms
  8. Kähler homogeneous spaces
  9. An invariant symplectic form on orbits of g1-pdms
  10. Orbits of g1-pdms as Kähler homogeneous spaces
  11. Appendix
  12. Hartree-Fock-Bogoliubov theory
  13. Banach manifolds basics

Key Result

Proposition 2.2

$\mathrm{U}_{\mathrm{Bog}}$ is a real Lie subgroup of $\mathrm{G}$, whose Lie algebra is given by In particular, the topology of $\mathrm{U}_{\mathrm{Bog}}$ is defined by the norm Furthermore, the following assertions hold:

Theorems & Definitions (63)

  • Remark 2.1
  • Proposition 2.2
  • proof
  • Remark 2.3
  • Remark 2.4
  • Lemma 2.5
  • proof
  • Remark 2.6
  • Definition 2.7
  • Remark 2.8
  • ...and 53 more