The Bogoliubov Free Energy Functional I: Existence of Minimizers and Phase Diagram

Abstract

The Bogoliubov free energy functional is analysed. The functional serves as a model of a translation-invariant Bose gas at positive temperature. We prove the existence of minimizers in the case of repulsive interactions given by a sufficiently regular two-body potential. Furthermore, we prove existence of a phase transition in this model and provide its phase diagram.

Spin Symmetry Breaking in the Translation-Invariant Hartree--Fock Electron Gas

“…In the model we study, it is the competition between the (concave) exchange term and the (convex) entropy term which is responsible for this non uniqueness. Similar effects have been found recently for instance for the Bogoliubov model describing an infinite translation-invariant Bose gas but the phase transition is there due to the interplay between pairing and Bose-Einstein condensation [NRS18a,NRS18b,NRS18c].…”

Bogoliubov Excitation Spectrum for Bose–einstein Condensates

“…of (4.8). Already from [10] and, more recently, from [19], it follows that Bogoliubov states, i.e. in our setting states of the form U * N T (µ)Ω for some µ ∈ ℓ 2 (Λ * + ), can only approximate the ground state energy up to an error of order one, even after optimizing the choice of the function µ.…”

Ground state energy of a dilute two-dimensional Bose gas from the Bogoliubov free energy functional

“…Knowing that minimizers of the functional exist, one can ask the question about their structure. It turns out that if the model exhibits a phase transition, then it does not distinguish between BEC and superfluidity as follows from Theorem 4 (Theorem 2.5. in [27]). Let (γ, α, ρ 0 ) be a minimizing triple for either (2.1) or (2.10).…”

“…This means that the fundamental question that one needs to ask first is the one about the existence of minimizers of the functional. The positive answer to that question has been given in [27]). Then the grand-canonical (canonical) minimization problem has a minimizer for any (T, µ) (or (T, ρ)).…”