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.
@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},
volume = {48},
year = {2014},
pages = {53-86},
doi = {10.1051/m2an/2013094},
mrnumber = {3177837},
zbl = {1301.82069},
language = {en},
url = {http://dml.mathdoc.fr/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, Tome 48 (2014) pp. 53-86. doi : 10.1051/m2an/2013094. http://gdmltest.u-ga.fr/item/M2AN_2014__48_1_53_0/
[1] and , On the convergence of the proximal algorithm for nonsmooth functions involving analytic features. Math. Program. 116 (2009) 5-16. | MR 2421270 | Zbl 1165.90018
[2] , Error bound for the Hartree-Fock energy of atoms and molecules. Commun. Math. Phys. 147 (1992) 527-548. | MR 1175492 | Zbl 0771.46038
[3] , and , Bogolubov-Hartree-Fock mean field theory for neutron stars and other systems with attractive interactions. J. Math. Phys. 50 (2009) 22. | MR 2573100 | Zbl 1248.81277
[4] , and , Generalized Hartree-Fock theory and the Hubbard model. J. Statist. Phys. 76 (1994) 3-89. | MR 1297873 | Zbl 0839.60095
[5] , and , Theory of superconductivity. Phys. Rev. 108 (1957) 1175-1204. | MR 95694 | Zbl 0090.45401
[6] and , Constructive solution of a bilinear optimal control problem for a Schrödinger equation. Syst. Cont. Lett. 57 (2008) 453-464. | MR 2413741 | Zbl 1153.49023
[7] and , An existence proof for the gap equation in the superconductivity theory. Commun. Math. Phys. 10 (1968) 274-279. | Zbl 0164.57002
[8] , About the theory of superfluidity. Izv. Akad. Nauk SSSR 11 (1947) 77. | MR 22177
[9] , Energy levels of the imperfect Bose gas. Bull. Moscow State Univ. 7 (1947) 43.
[10] , On the theory of superfluidity. J. Phys. (USSR) 11 (1947) 23. | Zbl 1165.82028
[11] , On a New Method in the Theory of Superconductivity. J. Exp. Theor. Phys. 34 (1958) 58. | Zbl 0090.45501
[12] , , and , Characterizations of Łojasiewicz inequalities: subgradient flows, talweg, convexity. Trans. Amer. Math. Soc. 362 (2010) 3319-3363. | MR 2592958 | Zbl 1202.26026
[13] , 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. | MR 1855573 | Zbl 0992.81103
[14] , , , and , Computational quantum chemistry: a primer, in Handbook of numerical analysis, vol. X, Handb. Numer. Anal. North-Holland, Amsterdam (2003) 3-270. | MR 2008386 | Zbl 1070.81534
[15] and , Can we outperform the DIIS approach for electronic structure calculations? Int. J. Quantum Chem. 79 (2000) 82-90.
[16] and , On the convergence of SCF algorithms for the Hartree-Fock equations. ESAIM: M2AN 34 (2000) 749-774. | Numdam | MR 1784484 | Zbl 1090.65548
[17] , and , Méthodes mathématiques en chimie quantique. Une introduction, vol. 53 of Collection Mathématiques et Applications. Springer (2006). | Zbl 1167.81001
[18] , Spectral theory and differential operators, vol. 42, Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge (1995). | Zbl 0893.47004
[19] and , Hartree-Fock-Bogolyubov calculations with the D1 effective interaction on spherical nuclei. Phys. Rev. C 21 (1980) 1568-1593.
[20] and , Relativistic stability of matter. I. Rev. Mat. Iberoamericana 2 (1986) 119-213. | Zbl 0602.58015
[21] , , and , Microscopic Derivation of Ginzburg-Landau Theory. J. Amer. Math. Soc. 25 (2012) 667-713. | Zbl 1251.35156
[22] , , and , The critical temperature for the BCS equation at weak coupling. J. Geom. Anal. 17 (2007) 559-567. | Zbl 1137.82025
[23] , The multiconfiguration equations for atoms and molecules: charge quantization and existence of solutions. Arch. Ration. Mech. Anal. 169 35-71 (2003). | MR 1996268 | Zbl 1035.81069
[24] D. Gogny, in Proceedings of the International Conference on Nuclear Physics, edited by J. de Boer and H.J. Mang. (1973) 48.
[25] D. Gogny, in Proceedings of the International Conference on Nuclear Self-Consistent Fields, edited by M. Porneuf and G. Ripka. Trieste (1975) 333.
[26] and , Hartree-Fock theory in nuclear physics. RAIRO Modél. Math. Anal. Numér. 20 (1986) 571-637. | Numdam | MR 877058 | Zbl 0607.35078
[27] , , and , The BCS functional for general pair interactions. Commun. Math. Phys. 281 (2008) 349-367. | MR 2410898 | Zbl 1161.82027
[28] , , and , On blowup for time-dependent generalized Hartree-Fock equations. Annal. Henri Poincaré 11 (2010) 1023-1052. | MR 2737490 | Zbl 1209.85009
[29] and , General decomposition of radial functions on Rn and applications to N-body quantum systems. Lett. Math. Phys. 61 (2002) 75-84. | MR 1930084 | Zbl 1016.81059
[30] and , The BCS critical temperature for potentials with negative scattering length. Lett. Math. Phys. 84 (2008) 99-107. | MR 2415542 | Zbl 1164.81006
[31] and , Schrödinger inequalities and asymptotic behavior of the electron density of atoms and molecules. Phys. Rev. A 16 (1977) 1782-1785. | MR 471726
[32] , Perturbation theory for linear operators. Springer (1995). | MR 1335452 | Zbl 0836.47009
[33] , Computational chemistry from the perspective of numerical analysis. Acta Numerica 14 (2005) 363-444. | MR 2168346 | Zbl 1119.65390
[34] and , Minimizers for the Hartree-Fock-Bogoliubov theory of neutron stars and white dwarfs. Duke Math. J. 152 (2010) 257-315. | MR 2656090 | Zbl 1202.49013
[35] , Convergence of gradient-based algorithms for the Hartree-Fock equations. ESAIM: M2AN 46 (2012) 1321-1336. | Numdam | MR 2996329 | Zbl 1269.82008
[36] , Geometric methods for nonlinear many-body quantum systems. J. Funct. Anal. 260 (2011) 3535-3595. | MR 2781970 | Zbl 1216.81180
[37] , Variational principle for many-fermion systems. Phys. Rev. Lett. 46 (1981) 457-459. | MR 601336
[38] and , The Stability of Matter in Quantum Mechanics. Cambridge Univ. Press (2010). | MR 2583992 | Zbl 1179.81004
[39] and , The Hartree-Fock theory for Coulomb systems. Commun. Math. Phys. 53 (1977) 185-194. | MR 452286
[40] and , Gravitational collapse in quantum mechanics with relativistic kinetic energy. Annal. Phys. 155 (1984) 494-512. | MR 753345
[41] and , The Chandrasekhar theory of stellar collapse as the limit of quantum mechanics. Commun. Math. Phys. 112 (1987) 147-174. | MR 904142 | Zbl 0641.35065
[42] , Solutions of Hartree-Fock equations for Coulomb systems. Commun. Math. Phys. 109 (1987) 33-97. | MR 879032 | Zbl 0618.35111
[43] , Une propriété topologique des sous-ensembles analytiques réels. Colloques du CNRS, Les équations aux dérivés partielles (1963) 117. | Zbl 0234.57007
[44] , Sur la géométrie semi- et sous-analytique. Ann. Inst. Fourier (Grenoble) 43 (1993) 1575-1595. | Numdam | Zbl 0803.32002
[45] and , The uniqueness and approximation of a positive solution of the Bardeen-Cooper-Schrieffer gap equation. J. Math. Phys. 41 (2000) 6007-6025. | Zbl 1054.82036
[46] , Modèle de Hartree-Fock-Bogoliubov : une perspective mathématique et numérique. Ph.D. thesis, Univ. Cergy-Pontoise (2012).
[47] and . Self-Consistent Calculations of Nuclear Properties with Phenomenological Effective Forces. Ann. Rev. Nucl. Part. Sci. 28 (1978) 523-594.
[48] and , The nuclear many-body problem, volume Texts and Monographs in Physics. Springer Verlag, New York (1980).
[49] , New developments in molecular orbital theory. Rev. Mod. Phys. 23 (1951) 69-89. | Zbl 0045.28502
[50] , Convergence of the time-discretized monotonic schemes. ESAIM: M2AN 41 (2007) 77-93. | Numdam | MR 2323691 | Zbl 1124.65059
[51] , Scilab: The free software for numerical computation. Scilab Consortium, Digiteo, Paris, France (2011).
[52] , Geometric methods in multiparticle quantum systems. Commun. Math. Phys. 55 (1977) 259-274. | MR 496073 | Zbl 0413.47008
[53] . The effective nuclear potential. Nuclear Phys. 9 (1959) 615-634. | Zbl 0083.44004
[54] , Proof of the ionization conjecture in a reduced Hartree-Fock model. Invent. Math. 104 (1991) 291-311. | MR 1098611 | Zbl 0732.35066
[55] , The ionization conjecture in Hartree-Fock theory. Annal. Math. 158 (2003) 509-576. | MR 2018928 | Zbl 1106.81081
[56] , The gap equation in superconductivity theory. Phys. D 17 (1985) 339-344. | MR 826974
[57] , On the Bardeen-Cooper-Schrieffer integral equation in the theory of superconductivity. Lett. Math. Phys. 22 (1991) 27-37. | MR 1121846 | Zbl 0729.45009