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Generalization of Lieb's variational principle to Bogoliubov–Hartree–Fock theory

### Abstract

In its original formulation, Lieb's variational principle holds for fermion systems with purely repulsive pair interactions. As a generalization we prove for both fermion and boson systems with semi-bounded Hamiltonian that the infimum of the energy over quasifree states coincides with the infimum over pure quasifree states. In particular, the Hamiltonian is not assumed to preserve the number of particles. To shed light on the relation between our result and the usual formulation of Lieb's variational principle in terms of one-particle density matrices, we also include a characterization of pure quasifree states by means of their generalized one-particle density matrices.

© 2014 AIP Publishing LLC

Received 02 May 2013
Accepted 08 December 2013
Published online 02 January 2014

Article outline:

I. INTRODUCTION
II. SECOND QUANTIZATION AND BOGOLIUBOV–HARTREE–FOCK THEORY
A. Second quantization
1. Bogoliubov transformation
2. States and quasifree states
3. One-particle and generalized one-particle density matrices
4. Two-particle density matrix and representability
5. Further generalized one-particle density matrix and generalized two-particle density matrix for bosons
B. Bogoliubov–Hartree–Fock theory
1. Boson Bogoliubov–Hartree–Fock theory
2. Fermion Bogoliubov–Hartree–Fock theory
III. VARIATION OVER PURE QUASIFREE STATES AND BOGOLIUBOV–HARTREE–FOCK ENERGY
A. Bosons
B. Fermions
IV. PURE QUASIFREE STATES AND THEIR GENERALIZED ONE-PARTICLE DENSITY MATRIX
A. Fermions
B. Bosons