A numerical perspective on Hartree-Fock-Bogoliubov theory
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 48 (2014) no. 1, p. 53-86
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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.

DOI : https://doi.org/10.1051/m2an/2013094
Classification:  65K10,  35Q40,  81V35
Keywords: Hartree−Fock−Bogoliubov, fixed point algorithm, relaxed constraint algorithm, nuclear physics
@article{M2AN_2014__48_1_53_0,
     author = {Lewin, Mathieu and Paul, S\'everine},
     title = {A numerical perspective on Hartree-Fock-Bogoliubov theory},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     publisher = {EDP-Sciences},
     volume = {48},
     number = {1},
     year = {2014},
     pages = {53-86},
     doi = {10.1051/m2an/2013094},
     zbl = {1301.82069},
     mrnumber = {3177837},
     language = {en},
     url = {http://www.numdam.org/item/M2AN_2014__48_1_53_0}
}
Lewin, Mathieu; Paul, Séverine. A numerical perspective on Hartree-Fock-Bogoliubov theory. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 48 (2014) no. 1, pp. 53-86. doi : 10.1051/m2an/2013094. http://www.numdam.org/item/M2AN_2014__48_1_53_0/

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