A numerical perspective on Hartree-Fock-Bogoliubov theory

Lewin, Mathieu; Paul, Séverine

ESAIM: Mathematical Modelling and Numerical Analysis , Tome 48 (2014) no. 1, pp. 53-86.

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.

[1] H. Attouch and J. Bolte, On the convergence of the proximal algorithm for nonsmooth functions involving analytic features. Math. Program. 116 (2009) 5-16. | MR | Zbl

[2] V. Bach, Error bound for the Hartree-Fock energy of atoms and molecules. Commun. Math. Phys. 147 (1992) 527-548. | MR | Zbl

[3] V. Bach, J. Fröhlich and L. Jonsson, Bogolubov-Hartree-Fock mean field theory for neutron stars and other systems with attractive interactions. J. Math. Phys. 50 (2009) 22. | MR | Zbl

[4] V. Bach, E.H. Lieb and J.Ph. Solovej, Generalized Hartree-Fock theory and the Hubbard model. J. Statist. Phys. 76 (1994) 3-89. | MR | Zbl

[5] J. Bardeen, L.N. Cooper and J.R. Schrieffer, Theory of superconductivity. Phys. Rev. 108 (1957) 1175-1204. | MR | Zbl

[6] L. Baudouin and J. Salomon, Constructive solution of a bilinear optimal control problem for a Schrödinger equation. Syst. Cont. Lett. 57 (2008) 453-464. | MR | Zbl

[7] P. Billard and G. Fano, An existence proof for the gap equation in the superconductivity theory. Commun. Math. Phys. 10 (1968) 274-279. | Zbl

[8] N.N. Bogoliubov, About the theory of superfluidity. Izv. Akad. Nauk SSSR 11 (1947) 77. | MR

[9] N.N. Bogoliubov, Energy levels of the imperfect Bose gas. Bull. Moscow State Univ. 7 (1947) 43.

[10] N.N. Bogoliubov, On the theory of superfluidity. J. Phys. (USSR) 11 (1947) 23. | Zbl

[11] N.N. Bogoliubov, On a New Method in the Theory of Superconductivity. J. Exp. Theor. Phys. 34 (1958) 58. | Zbl

[12] J. Bolte, A. Daniilidis, O. Ley and L. Mazet, Characterizations of Łojasiewicz inequalities: subgradient flows, talweg, convexity. Trans. Amer. Math. Soc. 362 (2010) 3319-3363. | MR | Zbl

[13] 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. | MR | Zbl

[14] 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. | MR | Zbl

[15] É. Cancès and C. Le Bris, Can we outperform the DIIS approach for electronic structure calculations? Int. J. Quantum Chem. 79 (2000) 82-90.

[16] É. Cancès and C. Le Bris, On the convergence of SCF algorithms for the Hartree-Fock equations. ESAIM: M2AN 34 (2000) 749-774. | Numdam | MR | Zbl

[17] É. Cancès, C. Le Bris and Y. Maday, Méthodes mathématiques en chimie quantique. Une introduction, vol. 53 of Collection Mathématiques et Applications. Springer (2006). | Zbl

[18] E.B. Davies, Spectral theory and differential operators, vol. 42, Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge (1995). | Zbl

[19] J. Dechargé and D. Gogny, Hartree-Fock-Bogolyubov calculations with the D1 effective interaction on spherical nuclei. Phys. Rev. C 21 (1980) 1568-1593.

[20] C. Fefferman and R. De La Llave, Relativistic stability of matter. I. Rev. Mat. Iberoamericana 2 (1986) 119-213. | Zbl

[21] R.L. Frank, C. Hainzl, R. Seiringer and J.P. Solovej, Microscopic Derivation of Ginzburg-Landau Theory. J. Amer. Math. Soc. 25 (2012) 667-713. | Zbl

[22] R.L. Frank, C. Hainzl, S. Naboko and R. Seiringer, The critical temperature for the BCS equation at weak coupling. J. Geom. Anal. 17 (2007) 559-567. | Zbl

[23] G. Friesecke, The multiconfiguration equations for atoms and molecules: charge quantization and existence of solutions. Arch. Ration. Mech. Anal. 169 35-71 (2003). | MR | Zbl

[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] D. Gogny and P.-L. Lions, Hartree-Fock theory in nuclear physics. RAIRO Modél. Math. Anal. Numér. 20 (1986) 571-637. | Numdam | MR | Zbl

[27] C. Hainzl, E. Hamza, R. Seiringer and J.P. Solovej, The BCS functional for general pair interactions. Commun. Math. Phys. 281 (2008) 349-367. | MR | Zbl

[28] C. Hainzl, E. Lenzmann, M. Lewin and B. Schlein, On blowup for time-dependent generalized Hartree-Fock equations. Annal. Henri Poincaré 11 (2010) 1023-1052. | MR | Zbl

[29] C. Hainzl and R. Seiringer, General decomposition of radial functions on Rn and applications to N-body quantum systems. Lett. Math. Phys. 61 (2002) 75-84. | MR | Zbl

[30] C. Hainzl and R. Seiringer, The BCS critical temperature for potentials with negative scattering length. Lett. Math. Phys. 84 (2008) 99-107. | MR | Zbl

[31] M. Hoffmann-Ostenhof and T. Hoffmann-Ostenhof, Schrödinger inequalities and asymptotic behavior of the electron density of atoms and molecules. Phys. Rev. A 16 (1977) 1782-1785. | MR

[32] T. Kato, Perturbation theory for linear operators. Springer (1995). | MR | Zbl

[33] C. Le Bris, Computational chemistry from the perspective of numerical analysis. Acta Numerica 14 (2005) 363-444. | MR | Zbl

[34] E. Lenzmann and M. Lewin, Minimizers for the Hartree-Fock-Bogoliubov theory of neutron stars and white dwarfs. Duke Math. J. 152 (2010) 257-315. | MR | Zbl

[35] A. Levitt, Convergence of gradient-based algorithms for the Hartree-Fock equations. ESAIM: M2AN 46 (2012) 1321-1336. | Numdam | MR | Zbl

[36] M. Lewin, Geometric methods for nonlinear many-body quantum systems. J. Funct. Anal. 260 (2011) 3535-3595. | MR | Zbl

[37] E.H. Lieb, Variational principle for many-fermion systems. Phys. Rev. Lett. 46 (1981) 457-459. | MR

[38] E.H. Lieb and R. Seiringer, The Stability of Matter in Quantum Mechanics. Cambridge Univ. Press (2010). | MR | Zbl

[39] E.H. Lieb and B. Simon, The Hartree-Fock theory for Coulomb systems. Commun. Math. Phys. 53 (1977) 185-194. | MR

[40] E.H. Lieb and W.E. Thirring, Gravitational collapse in quantum mechanics with relativistic kinetic energy. Annal. Phys. 155 (1984) 494-512. | MR

[41] E.H. Lieb and H.-T. Yau, The Chandrasekhar theory of stellar collapse as the limit of quantum mechanics. Commun. Math. Phys. 112 (1987) 147-174. | MR | Zbl

[42] P.-L. Lions, Solutions of Hartree-Fock equations for Coulomb systems. Commun. Math. Phys. 109 (1987) 33-97. | MR | Zbl

[43] S. Łojasiewicz, Une propriété topologique des sous-ensembles analytiques réels. Colloques du CNRS, Les équations aux dérivés partielles (1963) 117. | Zbl

[44] S. Łojasiewicz, Sur la géométrie semi- et sous-analytique. Ann. Inst. Fourier (Grenoble) 43 (1993) 1575-1595. | Numdam | Zbl

[45] J.B. Mcleod and Y. Yang, The uniqueness and approximation of a positive solution of the Bardeen-Cooper-Schrieffer gap equation. J. Math. Phys. 41 (2000) 6007-6025. | Zbl

[46] S. Paul, Modèle de Hartree-Fock-Bogoliubov : une perspective mathématique et numérique. Ph.D. thesis, Univ. Cergy-Pontoise (2012).

[47] P. Quentin and H. Flocard. Self-Consistent Calculations of Nuclear Properties with Phenomenological Effective Forces. Ann. Rev. Nucl. Part. Sci. 28 (1978) 523-594.

[48] P. Ring and P. Schuck, The nuclear many-body problem, volume Texts and Monographs in Physics. Springer Verlag, New York (1980).

[49] C.C.J. Roothaan, New developments in molecular orbital theory. Rev. Mod. Phys. 23 (1951) 69-89. | Zbl

[50] J. Salomon, Convergence of the time-discretized monotonic schemes. ESAIM: M2AN 41 (2007) 77-93. | Numdam | MR | Zbl

[51] S. Consortium, Scilab: The free software for numerical computation. Scilab Consortium, Digiteo, Paris, France (2011).

[52] B. Simon, Geometric methods in multiparticle quantum systems. Commun. Math. Phys. 55 (1977) 259-274. | MR | Zbl

[53] T.H.R. Skyrme. The effective nuclear potential. Nuclear Phys. 9 (1959) 615-634. | Zbl

[54] J.Ph. Solovej, Proof of the ionization conjecture in a reduced Hartree-Fock model. Invent. Math. 104 (1991) 291-311. | MR | Zbl

[55] J.Ph. Solovej, The ionization conjecture in Hartree-Fock theory. Annal. Math. 158 (2003) 509-576. | MR | Zbl

[56] A. Vansevenant, The gap equation in superconductivity theory. Phys. D 17 (1985) 339-344. | MR

[57] Y.S. Yang, On the Bardeen-Cooper-Schrieffer integral equation in the theory of superconductivity. Lett. Math. Phys. 22 (1991) 27-37. | MR | Zbl