Semi-classical Limit of Confined Fermionic Systems in Homogeneous Magnetic Fields

  • Søren FournaisEmail author
  • Peter S. Madsen


We consider a system of N interacting fermions in \( {\mathbb {R}}^3 \) confined by an external potential and in the presence of a homogeneous magnetic field. The intensity of the interaction has the mean-field scaling 1/N. With a semi-classical parameter \( \hbar \sim N^{-1/3} \), we prove convergence in the large N limit to the appropriate magnetic Thomas–Fermi-type model with various strength scalings of the magnetic field.

Mathematics Subject Classification

Primary 81Q20 Secondary 35P20 



The authors were partially supported by the Sapere Aude Grant DFF–4181-00221 from the Independent Research Fund Denmark. Part of this work was carried out while both authors visited the Mittag-Leffler Institute in Stockholm, Sweden.


  1. 1.
    Braun, W., Hepp, K.: The Vlasov dynamics and its fluctuations in the \(1/N\) limit of interacting classical particles. Commun. Math. Phys. 56, 101–113 (1977)ADSMathSciNetCrossRefGoogle Scholar
  2. 2.
    Caglioti, E., Lions, P.-L., Marchioro, C., Pulvirenti, M.: A special class of stationary flows for two-dimensional Euler equations: a statistical mechanics description. Commun. Math. Phys. 143, 501–525 (1992)ADSMathSciNetCrossRefGoogle Scholar
  3. 3.
    de Finetti, B.: Funzione caratteristica di un fenomeno aleatorio. Atti della R. Accademia Nazionale dei Lincei, 1931. Ser. 6, Memorie, Classe di Scienze Fisiche, Matematiche e NaturaliGoogle Scholar
  4. 4.
    Diaconis, P., Freedman, D.: Finite exchangeable sequences. Ann. Probab. 8, 745–764 (1980)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Fournais, S., Lewin, M., Solovej, J.P.: The semi-classical limit of large fermionic systems. Calc. Var. Partial Differ. Equ. 57, 105 (2018)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Hainzl, C., Seiringer, R.: Bounds on one-dimensional exchange energies with application to lowest Landau band quantum mechanics. Lett. Math. Phys. 55, 133–142 (2001)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Hainzl, C., Seiringer, R.: A discrete density matrix theory for atoms in strong magnetic fields. Commun. Math. Phys. 217, 229–248 (2001)ADSMathSciNetCrossRefGoogle Scholar
  8. 8.
    Hauksson, B., Yngvason, J.: Asymptotic exactness of magnetic Thomas–Fermi theory at nonzero temperature. J. Stat. Phys. 116, 523–546 (2004)ADSMathSciNetCrossRefGoogle Scholar
  9. 9.
    Hewitt, E., Savage, L.J.: Symmetric measures on Cartesian products. Trans. Am. Math. Soc. 80, 470–501 (1955)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Hudson, R.L., Moody, G.R.: Locally normal symmetric states and an analogue of de Finetti’s theorem. Z. Wahrscheinlichkeitstheor. und Verw. Gebiete 33, 343–351 (1975/76)Google Scholar
  11. 11.
    Ivrii, V.: Asymptotics of the ground state energy of heavy molecules in a strong magnetic field. I. Russ. J. Math. Phys. 4, 29–74 (1996)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Ivrii, V.: Heavy molecules in the strong magnetic field. Russ. J. Math. Phys. 4, 449–455 (1996)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Ivrii, V.: Asymptotics of the ground state energy of heavy molecules in a strong magnetic field. II. Russ. J. Math. Phys. 5, 321–354 (1997)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Ivrii, V.: Heavy molecules in a strong magnetic field. III. Estimates for ionization energy and excessive charge. Russ. J. Math. Phys. 6, 56–85 (1999)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Kiessling, M.K.-H.: Statistical mechanics of classical particles with logarithmic interactions. Commun. Pure. Appl. Math. 46, 27–56 (1993)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Lewin, M., Madsen, P.S., Triay, A.: Semi-classical limit of large fermionic systems at positive temperature. ArXiv e-prints (2019)Google Scholar
  17. 17.
    Lewin, M., Nam, P.T., Rougerie, N.: Derivation of Hartree’s theory for generic mean-field Bose systems. Adv. Math. 254, 570–621 (2014)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Lewin, M., Nam, P.T., Rougerie, N.: Remarks on the quantum de Finetti theorem for bosonic systems. Appl. Math. Res. Express (AMRX) 2015, 48–63 (2015)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Lewin, M., Nam, P.T., Rougerie, N.: The mean-field approximation and the non-linear Schrödinger functional for trapped Bose gases. Trans. Am. Math. Soc 368, 6131–6157 (2016)CrossRefGoogle Scholar
  20. 20.
    Lieb, E.H., Loss, M.: Analysis. Graduate Studies in Mathematics, vol. 14, 2nd edn. American Mathematical Society, Providence, RI (2001)zbMATHGoogle Scholar
  21. 21.
    Lieb, E.H., Simon, B.: The Hartree–Fock theory for Coulomb systems. Commun. Math. Phys. 53, 185–194 (1977)ADSMathSciNetCrossRefGoogle Scholar
  22. 22.
    Lieb, E.H., Simon, B.: The Thomas–Fermi theory of atoms, molecules and solids. Adv. Math. 23, 22–116 (1977)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Lieb, E.H., Solovej, J.P., Yngvason, J.: Asymptotics of heavy atoms in high magnetic fields: I. Lowest Landau band regions. Commun. Pure Appl. Math. 47, 513–591 (1994)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Lieb, E.H., Solovej, J.P., Yngvason, J.: Asymptotics of heavy atoms in high magnetic fields: II. Semiclassical regions. Commun. Math. Phys. 161, 77–124 (1994)ADSMathSciNetCrossRefGoogle Scholar
  25. 25.
    Lieb, E.H., Solovej, J.P., Yngvason, J.: Ground states of large quantum dots in magnetic fields. Phys. Rev. B 51, 10646–10665 (1995)ADSCrossRefGoogle Scholar
  26. 26.
    Lieb, E.H., Thirring, W.E.: Gravitational collapse in quantum mechanics with relativistic kinetic energy. Ann. Phys. 155, 494–512 (1984)ADSMathSciNetCrossRefGoogle Scholar
  27. 27.
    Lieb, E.H., Yau, H.-T.: The Chandrasekhar theory of stellar collapse as the limit of quantum mechanics. Commun. Math. Phys. 112, 147–174 (1987)ADSMathSciNetCrossRefGoogle Scholar
  28. 28.
    Madsen, P.: In preparation, Ph.D. thesis, Aarhus University (2019)Google Scholar
  29. 29.
    Messer, J., Spohn, H.: Statistical mechanics of the isothermal Lane–Emden equation. J. Stat. Phys. 29, 561–578 (1982)ADSMathSciNetCrossRefGoogle Scholar
  30. 30.
    Rougerie, N.: De Finetti theorems, mean-field limits and Bose–Einstein condensation. ArXiv e-prints (2015)Google Scholar
  31. 31.
    Seiringer, R.: On the maximal ionization of atoms in strong magnetic fields. J. Phys. A Math. General 34, 1943–1948 (2001)ADSMathSciNetCrossRefGoogle Scholar
  32. 32.
    Sobolev, A.V.: The quasi-classical asymptotics of local Riesz means for the Schrödinger operator in a strong homogeneous magnetic field. Duke Math. J. 74, 319–429 (1994)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Spohn, H.: On the Vlasov hierarchy. Math. Methods Appl. Sci. 3, 445–455 (1981)ADSMathSciNetCrossRefGoogle Scholar
  34. 34.
    Størmer, E.: Symmetric states of infinite tensor products of \(C^{\ast } \)-algebras. J. Funct. Anal. 3, 48–68 (1969)MathSciNetCrossRefGoogle Scholar
  35. 35.
    Thirring, W.: A lower bound with the best possible constant for Coulomb Hamiltonians. Commun. Math. Phys. 79, 1–7 (1981)ADSMathSciNetCrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.Department of MathematicsAarhus UniversityAarhus CDenmark
  2. 2.CNRS & CEREMADEParis-Dauphine University, PSL UniversityParisFrance

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