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

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Abstract

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.

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References

  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)

    Article  ADS  MathSciNet  Google Scholar 

  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)

    Article  ADS  MathSciNet  Google Scholar 

  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 Naturali

  4. Diaconis, P., Freedman, D.: Finite exchangeable sequences. Ann. Probab. 8, 745–764 (1980)

    Article  MathSciNet  Google Scholar 

  5. Fournais, S., Lewin, M., Solovej, J.P.: The semi-classical limit of large fermionic systems. Calc. Var. Partial Differ. Equ. 57, 105 (2018)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  7. Hainzl, C., Seiringer, R.: A discrete density matrix theory for atoms in strong magnetic fields. Commun. Math. Phys. 217, 229–248 (2001)

    Article  ADS  MathSciNet  Google Scholar 

  8. Hauksson, B., Yngvason, J.: Asymptotic exactness of magnetic Thomas–Fermi theory at nonzero temperature. J. Stat. Phys. 116, 523–546 (2004)

    Article  ADS  MathSciNet  Google Scholar 

  9. Hewitt, E., Savage, L.J.: Symmetric measures on Cartesian products. Trans. Am. Math. Soc. 80, 470–501 (1955)

    Article  MathSciNet  Google Scholar 

  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)

  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)

    MathSciNet  MATH  Google Scholar 

  12. Ivrii, V.: Heavy molecules in the strong magnetic field. Russ. J. Math. Phys. 4, 449–455 (1996)

    MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  15. Kiessling, M.K.-H.: Statistical mechanics of classical particles with logarithmic interactions. Commun. Pure. Appl. Math. 46, 27–56 (1993)

    Article  MathSciNet  Google Scholar 

  16. Lewin, M., Madsen, P.S., Triay, A.: Semi-classical limit of large fermionic systems at positive temperature. ArXiv e-prints (2019)

  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)

    Article  MathSciNet  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  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)

    Article  Google Scholar 

  20. Lieb, E.H., Loss, M.: Analysis. Graduate Studies in Mathematics, vol. 14, 2nd edn. American Mathematical Society, Providence, RI (2001)

    MATH  Google Scholar 

  21. Lieb, E.H., Simon, B.: The Hartree–Fock theory for Coulomb systems. Commun. Math. Phys. 53, 185–194 (1977)

    Article  ADS  MathSciNet  Google Scholar 

  22. Lieb, E.H., Simon, B.: The Thomas–Fermi theory of atoms, molecules and solids. Adv. Math. 23, 22–116 (1977)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

    Article  ADS  MathSciNet  Google Scholar 

  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)

    Article  ADS  Google Scholar 

  26. Lieb, E.H., Thirring, W.E.: Gravitational collapse in quantum mechanics with relativistic kinetic energy. Ann. Phys. 155, 494–512 (1984)

    Article  ADS  MathSciNet  Google Scholar 

  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)

    Article  ADS  MathSciNet  Google Scholar 

  28. Madsen, P.: In preparation, Ph.D. thesis, Aarhus University (2019)

  29. Messer, J., Spohn, H.: Statistical mechanics of the isothermal Lane–Emden equation. J. Stat. Phys. 29, 561–578 (1982)

    Article  ADS  MathSciNet  Google Scholar 

  30. Rougerie, N.: De Finetti theorems, mean-field limits and Bose–Einstein condensation. ArXiv e-prints (2015)

  31. Seiringer, R.: On the maximal ionization of atoms in strong magnetic fields. J. Phys. A Math. General 34, 1943–1948 (2001)

    Article  ADS  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  33. Spohn, H.: On the Vlasov hierarchy. Math. Methods Appl. Sci. 3, 445–455 (1981)

    Article  ADS  MathSciNet  Google Scholar 

  34. Størmer, E.: Symmetric states of infinite tensor products of \(C^{\ast } \)-algebras. J. Funct. Anal. 3, 48–68 (1969)

    Article  MathSciNet  Google Scholar 

  35. Thirring, W.: A lower bound with the best possible constant for Coulomb Hamiltonians. Commun. Math. Phys. 79, 1–7 (1981)

    Article  ADS  MathSciNet  Google Scholar 

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Acknowledgements

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.

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Authors and Affiliations

  1. Department of Mathematics, Aarhus University, Ny Munkegade 118, DK-8000, Aarhus C, Denmark

    Søren Fournais

  2. CNRS & CEREMADE, Paris-Dauphine University, PSL University, 75016, Paris, France

    Peter S. Madsen

Authors
  1. Søren Fournais
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  2. Peter S. Madsen
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Correspondence to Søren Fournais.

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Communicated by Jan Derezinski.

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Fournais, S., Madsen, P.S. Semi-classical Limit of Confined Fermionic Systems in Homogeneous Magnetic Fields. Ann. Henri Poincaré 21, 1401–1449 (2020). https://doi.org/10.1007/s00023-019-00880-6

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