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Bounds on the Minimal Energy of Translation Invariant N-Polaron Systems

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Abstract

For systems of N charged fermions (e.g. electrons) interacting with longitudinal optical quantized lattice vibrations of a polar crystal we derive upper and lower bounds on the minimal energy within the model of H. Fröhlich. The only parameters of this model, after removing the ultraviolet cutoff, are the constants U > 0 and α > 0 measuring the electron-electron and the electron-phonon coupling strengths. They are constrained by the condition \({\sqrt{2}\alpha < U}\), which follows from the dependence of U and α on electrical properties of the crystal. We show that the large N asymptotic behavior of the minimal energy E N changes at \({\sqrt{2}\alpha=U}\) and that \({\sqrt{2}\alpha\leq U}\) is necessary for thermodynamic stability: for \({\sqrt{2}\alpha > U}\) the phonon-mediated electron-electron attraction overcomes the Coulomb repulsion and E N behaves like −N 7/3.

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References

  1. Alexandrov A.S., Mott N.: Polarons and Bipolarons. World Scientific, Singapore (1996)

    Google Scholar 

  2. Ammari Z.: Asymptotic completeness for a renormalized nonrelativistic Hamiltonian in quantum field theory: The Nelson model. Math. Phys. Anal. Geom. 3, 217–285 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  3. Bach V.: Error bound for the Hartree-Fock energy of atoms and molecules. Commun. Math. Phys. 147, 527–548 (1992)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  4. Brosens F., Klimin S.N., Devreese J.T.: Variational path-integral treatment of a translation invariant many-polaron system. Phys. Rev. B 71, 214301 (2005)

    Article  ADS  Google Scholar 

  5. Brosens F., Klimin S.N., Devreese J.T.: Path-integral approach to the ground-state energy of a homogeneous polaron gas. Phys. Rev. B 77, 085308 (2008)

    Article  ADS  Google Scholar 

  6. Cannon J.T.: Quantum field theoretic properties of a model of Nelson: Domain and eigenvector stability for perturbed linear operators. J. Funct. Anal. 8, 101–152 (1971)

    Article  MATH  MathSciNet  Google Scholar 

  7. Devreese, J.T.: Polarons. In: Encyclopedia of Applied Physics, G. L. Trigg, E. H. Immergut, eds., Vol. 14, Weinhein: Wiley-VCH, 1996, pp. 383–409

  8. Devreese J.T., Tempere J.: Large-polaron effects in the infrared spectrum of high-T c cuprate superconductors. Solid State Commun. 106, 309–313 (1998)

    Article  ADS  Google Scholar 

  9. Donsker M.D., Varadhan S.R.: Asymptotics for the polaron. Comm. Pure Appl. Math. 36, 505–528 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  10. Feynman R.P.: Slow electrons in a polar crystal. Phys. Rev. 97, 660–665 (1955)

    Article  MATH  ADS  Google Scholar 

  11. Feynman, R.P.: Statistical Mechanics. A Set of Lectures, Frontiers in Physics, Reading, MA: W. A. Benjamin, Inc., 1972

  12. Fröhlich H.: Electrons in lattice fields. Adv. in Phys. 3, 325–362 (1954)

    Article  ADS  Google Scholar 

  13. Fröhlich J.: Existence of dressed one-electron states in a class of persistent models. Fortschr. Phys. 22, 159–198 (1974)

    Article  Google Scholar 

  14. Gerlach B., Löwen H.: Analytical properties of polaron systems or: Do polaronic phase transitions exist or not?. Rev. Mod. Phys. 63, 63–90 (1991)

    Article  ADS  Google Scholar 

  15. Griesemer M.: Exponential decay and ionization thresholds in non-relativistic quantum electrodynamics. J. Funct. Anal. 210, 321–340 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  16. Haga E.: Note on the slow electrons in a polar crystal. Prog. Theoretical Phys. 11, 449–460 (1954)

    Article  MATH  ADS  Google Scholar 

  17. Hartinger Ch., Mayr F., Deisenhofer J., Loidl A., Kopp T.: Large and small polaron excitations in L a2/3(S r/C a)1/3M n O3 films. Phys. Rev. B 69, 100403 (2004)

    Article  ADS  Google Scholar 

  18. Klimin S.N., Fomin V.M., Brosens F., Devreese J.T.: Characterization of shell-filling of interacting polarons in a quantum dot through their optical absorption. Physica E 22, 494–497 (2004)

    Article  ADS  Google Scholar 

  19. Lee T.D., Low F.E., Pines D.: The motion of slow electrons in a polar crystal. Phys. Rev. 90, 297–302 (1953)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  20. Lévy-Leblond J.-M.: Nonsaturation of gravitational forces. J. Math. Phys. 10, 806–812 (1968)

    Article  Google Scholar 

  21. Lieb E.H.: Existence and uniqueness of the minimizing solution of Choquard’s nonlinear equation. Stud. Appl. Math. 70A, 444–446 (1977)

    MathSciNet  Google Scholar 

  22. Lieb, E.H.: Thomas-Fermi and related theories of atoms and molecules, Rev. Mod. Phys. 53, 603–604 (1981), Erratum 54, 311 (1981)

  23. Lieb, E.H.: Variational principle for many-fermion systems. Phys. Rev. Lett. 46, 457–459 (1981), Erratum 47, 69 (1981)

    Google Scholar 

  24. Lieb, E.H., Loss, M.: Analysis, 2nd ed., Graduate Studies in Mathematics, Vol. 14, Providence, RI: Amer. Math. Soc., 2001

  25. Lieb, E.H., Thomas, L.E.: Exact ground state energy of the strong-coupling polaron. Commun. Math. Phys. 183, 511–519 (1997), Erratum 188, 499–500 (1997)

    Google Scholar 

  26. Miyao T., Spohn H.: The bipolaron in the strong coupling limit. Ann. Henri Poincaré 8, 1333–1370 (2007)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  27. Nelson E.: Interaction of non-relativistic particles with a quantized scalar field. J. Math. Phys. 5, 1190–1197 (1964)

    Article  ADS  Google Scholar 

  28. Verbist G., Peeters F.M., Devreese J.T.: Large bipolarons in two and three dimensions. Phys. Rev. B 43, 2712–2720 (1991)

    Article  ADS  Google Scholar 

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Authors and Affiliations

  1. Fachbereich Mathematik, Universität Stuttgart, 70550, Stuttgart, Germany

    Marcel Griesemer

  2. Department of Mathematical Sciences, Aarhus University, 8000, Århus C, Denmark

    Jacob Schach Møller

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  1. Marcel Griesemer
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  2. Jacob Schach Møller
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Correspondence to Marcel Griesemer.

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Communicated by H. Spohn

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Griesemer, M., Møller, J.S. Bounds on the Minimal Energy of Translation Invariant N-Polaron Systems. Commun. Math. Phys. 297, 283–297 (2010). https://doi.org/10.1007/s00220-010-1013-z

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