Bogolubov–Hartree–Fock Theory for Strongly Interacting Fermions in the Low Density Limit

Abstract

We consider the Bogolubov-Hartree-Fock functional for a fermionic many-body system with two-body interactions. For suitable interaction potentials that have a strong enough attractive tail in order to allow for two-body bound states, but are otherwise sufficiently repulsive to guarantee stability of the system, we show that in the low-density limit the ground state of this model consists of a Bose-Einstein condensate of fermion pairs. The latter can be described by means of the Gross-Pitaevskii energy functional. * c 2015 by the authors. This work may be reproduced, in its entirety, for non-commercial purposes. arXiv:1511.08047v1 [math-ph] 25 Nov 2015 1 2h . This assumes, of course, that the two-body interaction potential V allows for a negative energy bound state, which is part of the following assumption.and −2∆ + V has a normalized ground state α 0 with corresponding ground state energy −E b < 0.

Condensation of fermion pairs in a domain

“…The BCS model that we consider has received considerable interest in recent years in mathematical physics. Most closely related to our paper are the derivations of effective GP theories for periodic boundary conditions in [22] and for a system in R 3 at fixed particle number [4]. The time-dependent analogue of this derivation was performed in [21].…”

“…We emphasize that all of these papers assume that the system has no boundary (either by working on the torus or on the whole space) and the same holds true for the resulting effective GP or GL theories. (We also mention that the derivation in [4] depends on W L ∞ (R d ) < ∞ and so one cannot obtain the Dirichlet boundary conditions as the limiting case of a sufficiently deep potential well from [4]. )…”

“…While the overall strategy is as in [4,14], there are some additional technical difficulties, mainly due to the boundary conditions:…”

“…Moreover, the macroscopic variations of the condensate density are given in terms of the nonlinear Gross-Pitaevskii (GP) theory [11,32,33]. An effective GP theory was recently derived mathematically starting from the microscopic BCS theory, see [4,22] for the stationary case and [21] for the time-dependent case. This is in the spirit of Gorkov's paper [19] on how Ginzburg-Landau theory arises from BCS theory for superconductors at positive temperature.…”

Persistence of Translational Symmetry in the BCS Model with Radial Pair Interaction

“…Based on a characterization of the critical temperature in terms of this linear operator, its behavior has been investigated in the limit of small couplings and in the low-density limit, see [6,16] and [14], respectively. Recently, there has also been considerable interest in the BCS functional with external fields, and in particular, in its connection to the Ginzburg-Landau theory of superconductivity, see [17,4,18,7,9,5,8,20].…”