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A numerical perspective on Hartree−Fock−Bogoliubov theory

Published online by Cambridge University Press:  15 November 2013

Mathieu Lewin  and
Séverine Paul
Mathieu Lewin
Affiliation:
CNRS and Laboratoire de Mathématiques (CNRS UMR 8088), Université de Cergy-Pontoise, 95000 Cergy-Pontoise, France.. Mathieu.Lewin@math.cnrs.fr
Séverine Paul
Affiliation:
Laboratoire de Mathématiques (CNRS UMR 8088), Université de Cergy-Pontoise, 95000 Cergy-Pontoise, France.
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Abstract

The method of choice for describing attractive quantum systems is Hartree−Fock−Bogoliubov (HFB) theory. This is a nonlinear model which allows for the description of pairing effects, the main explanation for the superconductivity of certain materials at very low temperature. This paper is the first study of Hartree−Fock−Bogoliubov theory from the point of view of numerical analysis. We start by discussing its proper discretization and then analyze the convergence of the simple fixed point (Roothaan) algorithm. Following works by Cancès, Le Bris and Levitt for electrons in atoms and molecules, we show that this algorithm either converges to a solution of the equation, or oscillates between two states, none of them being solution to the HFB equations. We also adapt the Optimal Damping Algorithm of Cancès and Le Bris to the HFB setting and we analyze it. The last part of the paper is devoted to numerical experiments. We consider a purely gravitational system and numerically discover that pairing always occurs. We then examine a simplified model for nucleons, with an effective interaction similar to what is often used in nuclear physics. In both cases we discuss the importance of using a damping algorithm.

Keywords

Hartree−Fock−Bogoliubovfixed point algorithmrelaxed constraint algorithmnuclear physics
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 48 , Issue 1 , January 2014 , pp. 53 - 86
Copyright
© EDP Sciences, SMAI 2013

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