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A numerical perspective on Hartree−Fock−Bogoliubovtheory

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.. [email protected]
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 ofpairing effects, the main explanation for the superconductivity ofcertain materials at very low temperature. This paper is the first study ofHartree−Fock−Bogoliubov theory from the point of view of numerical analysis. We start bydiscussing its proper discretization and then analyze the convergence of the simple fixedpoint (Roothaan) algorithm. Following works by Cancès, Le Bris and Levitt for electrons inatoms and molecules, we show that this algorithm either converges to a solution of theequation, or oscillates between two states, none of them being solution to the HFBequations. We also adapt the Optimal Damping Algorithm of Cancès and Le Bris to the HFBsetting 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 alwaysoccurs. We then examine a simplified model for nucleons, with an effective interactionsimilar to what is often used in nuclear physics. In both cases we discuss the importanceof 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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References

Attouch, H. and Bolte, J., On the convergence of the proximal algorithm for nonsmooth functions involving analytic features. Math. Program. 116 (2009) 5–16. Google Scholar
Bach, V., Error bound for the Hartree-Fock energy of atoms and molecules. Commun. Math. Phys. 147 (1992) 527–548. Google Scholar
Bach, V., Fröhlich, J. and Jonsson, L., Bogolubov-Hartree-Fock mean field theory for neutron stars and other systems with attractive interactions. J. Math. Phys. 50 (2009) 22. Google Scholar
Bach, V., Lieb, E.H. and Solovej, J.Ph., Generalized Hartree-Fock theory and the Hubbard model. J. Statist. Phys. 76 (1994) 3–89. Google Scholar
Bardeen, J., Cooper, L.N. and Schrieffer, J.R., Theory of superconductivity. Phys. Rev. 108 (1957) 1175–1204. Google Scholar
Baudouin, L. and Salomon, J., Constructive solution of a bilinear optimal control problem for a Schrödinger equation. Syst. Cont. Lett. 57 (2008) 453–464. Google Scholar
Billard, P. and Fano, G., An existence proof for the gap equation in the superconductivity theory. Commun. Math. Phys. 10 (1968) 274–279. Google Scholar
Bogoliubov, N.N., About the theory of superfluidity. Izv. Akad. Nauk SSSR 11 (1947) 77. Google Scholar
Bogoliubov, N.N., Energy levels of the imperfect Bose gas. Bull. Moscow State Univ. 7 (1947) 43. Google Scholar
Bogoliubov, N.N., On the theory of superfluidity. J. Phys. (USSR) 11 (1947) 23. Google Scholar
Bogoliubov, N.N., On a New Method in the Theory of Superconductivity. J. Exp. Theor. Phys. 34 (1958) 58. Google Scholar
Bolte, J., Daniilidis, A., Ley, O. and Mazet, L., Characterizations of Łojasiewicz inequalities: subgradient flows, talweg, convexity. Trans. Amer. Math. Soc. 362 (2010) 3319–3363. Google Scholar
É. Cancès, SCF algorithms for HF electronic calculations, in Mathematical models and methods for ab initio quantum chemistry, vol. 74, in Lect. Notes Chem., Chapt. 2. Springer, Berlin (2000) 17–43.
É. Cancès, M. Defranceschi, W. Kutzelnigg, C. Le Bris and Y. Maday, Computational quantum chemistry: a primer, in Handbook of numerical analysis, vol. X, Handb. Numer. Anal. North-Holland, Amsterdam (2003) 3–270.
É., Cancès and C., Le Bris, Can we outperform the DIIS approach for electronic structure calculations? Int. J. Quantum Chem. 79 (2000) 82–90. Google Scholar
É., Cancès and C., Le Bris, On the convergence of SCF algorithms for the Hartree-Fock equations. ESAIM: M2AN 34 (2000) 749–774. Google Scholar
É. Cancès, C. Le Bris and Y. Maday, Méthodes mathématiques en chimie quantique. Une introduction, vol. 53 of Collection Mathématiques et Applications. Springer (2006).
E.B. Davies, Spectral theory and differential operators, vol. 42, Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge (1995).
Dechargé, J. and Gogny, D., Hartree-Fock-Bogolyubov calculations with the D1 effective interaction on spherical nuclei. Phys. Rev. C 21 (1980) 1568–1593. Google Scholar
Fefferman, C. and de la Llave, R., Relativistic stability of matter. I. Rev. Mat. Iberoamericana 2 (1986) 119–213. Google Scholar
Frank, R.L., Hainzl, C., Seiringer, R. and Solovej, J.P., Microscopic Derivation of Ginzburg-Landau Theory. J. Amer. Math. Soc. 25 (2012) 667–713. Google Scholar
Frank, R.L., Hainzl, C., Naboko, S. and Seiringer, R., The critical temperature for the BCS equation at weak coupling. J. Geom. Anal. 17 (2007) 559–567. Google Scholar
G., Friesecke, The multiconfiguration equations for atoms and molecules: charge quantization and existence of solutions. Arch. Ration. Mech. Anal. 169 35–71 (2003). Google Scholar
D. Gogny, in Proceedings of the International Conference on Nuclear Physics, edited by J. de Boer and H.J. Mang. (1973) 48.
D. Gogny, in Proceedings of the International Conference on Nuclear Self-Consistent Fields, edited by M. Porneuf and G. Ripka. Trieste (1975) 333.
Gogny, D. and Lions, P.-L., Hartree-Fock theory in nuclear physics. RAIRO Modél. Math. Anal. Numér. 20 (1986) 571–637. Google Scholar
Hainzl, C., Hamza, E., Seiringer, R. and Solovej, J.P., The BCS functional for general pair interactions. Commun. Math. Phys. 281 (2008) 349–367. Google Scholar
Hainzl, C., Lenzmann, E., Lewin, M. and Schlein, B., On blowup for time-dependent generalized Hartree-Fock equations. Annal. Henri Poincaré 11 (2010) 1023–1052. Google Scholar
Hainzl, C. and Seiringer, R., General decomposition of radial functions on Rn and applications to N-body quantum systems. Lett. Math. Phys. 61 (2002) 75–84. Google Scholar
Hainzl, C. and Seiringer, R., The BCS critical temperature for potentials with negative scattering length. Lett. Math. Phys. 84 (2008) 99–107. Google Scholar
Hoffmann-Ostenhof, M. and Hoffmann-Ostenhof, T., Schrödinger inequalities and asymptotic behavior of the electron density of atoms and molecules. Phys. Rev. A 16 (1977) 1782–1785. Google Scholar
T. Kato, Perturbation theory for linear operators. Springer (1995).
C., Le Bris, Computational chemistry from the perspective of numerical analysis. Acta Numerica 14 (2005) 363–444. Google Scholar
Lenzmann, E. and Lewin, M., Minimizers for the Hartree-Fock-Bogoliubov theory of neutron stars and white dwarfs. Duke Math. J. 152 (2010) 257–315. Google Scholar
Levitt, A., Convergence of gradient-based algorithms for the Hartree-Fock equations. ESAIM: M2AN 46 (2012) 1321–1336. Google Scholar
Lewin, M., Geometric methods for nonlinear many-body quantum systems. J. Funct. Anal. 260 (2011) 3535–3595. Google Scholar
Lieb, E.H., Variational principle for many-fermion systems. Phys. Rev. Lett. 46 (1981) 457–459. Google Scholar
E.H. Lieb and R. Seiringer, The Stability of Matter in Quantum Mechanics. Cambridge Univ. Press (2010).
Lieb, E.H. and Simon, B., The Hartree-Fock theory for Coulomb systems. Commun. Math. Phys. 53 (1977) 185–194. Google Scholar
Lieb, E.H. and Thirring, W.E., Gravitational collapse in quantum mechanics with relativistic kinetic energy. Annal. Phys. 155 (1984) 494–512. Google Scholar
Lieb, E.H. and Yau, H.-T., The Chandrasekhar theory of stellar collapse as the limit of quantum mechanics. Commun. Math. Phys. 112 (1987) 147–174. Google Scholar
Lions, P.-L., Solutions of Hartree-Fock equations for Coulomb systems. Commun. Math. Phys. 109 (1987) 33–97. Google Scholar
S. Łojasiewicz, Une propriété topologique des sous-ensembles analytiques réels. Colloques du CNRS, Les équations aux dérivés partielles (1963) 117.
Łojasiewicz, S., Sur la géométrie semi- et sous-analytique. Ann. Inst. Fourier (Grenoble) 43 (1993) 1575–1595. Google Scholar
McLeod, J.B. and Yang, Y., The uniqueness and approximation of a positive solution of the Bardeen-Cooper-Schrieffer gap equation. J. Math. Phys. 41 (2000) 6007–6025. Google Scholar
S. Paul, Modèle de Hartree-Fock-Bogoliubov : une perspective mathématique et numérique. Ph.D. thesis, Univ. Cergy-Pontoise (2012).
P., Quentin and H., Flocard. Self-Consistent Calculations of Nuclear Properties with Phenomenological Effective Forces. Ann. Rev. Nucl. Part. Sci. 28 (1978) 523–594. Google Scholar
P. Ring and P. Schuck, The nuclear many-body problem, volume Texts and Monographs in Physics. Springer Verlag, New York (1980).
Roothaan, C.C.J., New developments in molecular orbital theory. Rev. Mod. Phys. 23 (1951) 69–89. Google Scholar
Salomon, J., Convergence of the time-discretized monotonic schemes. ESAIM: M2AN 41 (2007) 77–93. Google Scholar
S. Consortium, Scilab: The free software for numerical computation. Scilab Consortium, Digiteo, Paris, France (2011).
Simon, B., Geometric methods in multiparticle quantum systems. Commun. Math. Phys. 55 (1977) 259–274. Google Scholar
T.H.R., Skyrme. The effective nuclear potential. Nuclear Phys. 9 (1959) 615–634. Google Scholar
Solovej, J.Ph., Proof of the ionization conjecture in a reduced Hartree-Fock model. Invent. Math. 104 (1991) 291–311. Google Scholar
Solovej, J.Ph., The ionization conjecture in Hartree-Fock theory. Annal. Math. 158 (2003) 509–576. Google Scholar
Vansevenant, A., The gap equation in superconductivity theory. Phys. D 17 (1985) 339–344. Google Scholar
Yang, Y.S., On the Bardeen-Cooper-Schrieffer integral equation in the theory of superconductivity. Lett. Math. Phys. 22 (1991) 27–37. Google Scholar