Exact Ground State Energy of the Strong-Coupling Polaron

  • Elliott H. Lieb
  • Lawrence E. Thomas


The polaron has been of interest in condensed matter theory and field theory for about half a century, especially the limit of large coupling constant, a. It was not until 1983, however, that a proof of the asymptotic formula for the ground state energy was finally given by using difficult arguments involving the large deviation theory of path integrals. Here we derive the same asymptotic result, E 0~ —Cα2,and with explicit error bounds, by simple, rigorous methods applied directly to the Hamiltonian. Our method is easily generalizable to other settings, e.g., the excitonic and magnetic polarons.


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  1. 1.
    Fröhlich, H.: Proc. Phys. Soc. A160, 230 (1937); Electrons in lattice fields. Adv. in Phys. 3, 325–361 (1954)Google Scholar
  2. 2.
    Pekar, S.I.: Untersuchung iiber die Elektronentheorie der Kristalle. Berlin: Akademie Verlag, 1954Google Scholar
  3. 3.
    Bogolyubov, N.N.: On a new form of the adiabatic theory of disturbances in the problem of interaction of particles with a quantum field (Russian). Ukr. Mat. Zh 2, 3–24 (1950)Google Scholar
  4. 4.
    Tyablikov, S. V.: An adiabatic form of perturbation theory in the problem of the interaction of a particle with a quantum field (Russian). Zh. Eksper. Teor. Fiz. 21, 377–388 (1951)MathSciNetGoogle Scholar
  5. 5.
    Gross, E.P.: Small oscillation theory of the interaction of a prticle in a scalar field. Phys. Rev. 100, 1571–1578 (1955); Strong coupling polaron theory and translation invariance. Ann. Phys. (NY) 99, 1–29 (1976)Google Scholar
  6. 6.
    Fröhlich, J.: On the infrared problem in a model of scalar electrons and massless scalar bosons. Ann. Inst. Poincaré A 19, 1–103; Existence of dressed one electron states in a class of persistent models. Fortschr. Phys. 22, 159–198 (1974)Google Scholar
  7. 7.
    Spohn, H.: Roughening and pinning transition for the polaron, J. Phys. A 19 533–545 (1986); Effective mass of the polaron: A functional integral approach. Ann. Phys. (NY), 175 278–318(1987); The polaron at large total momentum. J. Phys. A 21 1199–1211(1988)Google Scholar
  8. 8.
    Lee, T.D., Low, E and Pines, D.: The motion of slow electrons in a polar crystal. Phys. Rev. 90, 297–302 (1953)MathSciNetADSMATHCrossRefGoogle Scholar
  9. 9.
    Schultz, T.D.: “Electron-Lattice Interactions in Polar crystals.” Technical Report no. 9 Solid State and Molecular Theory Group, M.I.T., August 20, (1956). See also article in Polarons and Excitons, Scottish Summer School (1962), C.G. Kuper and G.D. Whitfield, Edinburgh, London: Oliver and Boyd, 1963Google Scholar
  10. 10.
    Landau, L.D.: Electron motion in crystal lattices. Phys. Z. Sowjet 3, 664–665 (1933) Il. Gerlach, B. and Löwen, H.: Analytical properties of polaron systems or: Do polaronic phase transitions exist or not? Rev. Mod. Phys. 63, 63–90 (1991)Google Scholar
  11. 12.
    Peeters, E. M. and Devreese, J. T.: On the existence of a phase transition for the Fröhlich polaron. Phys. Status Solidi B 112, 219–229 (1982)ADSCrossRefGoogle Scholar
  12. 13.
    Lieb, E.H.: Existence and uniqueness of the minimizing solution of Choquard’s nonlinear equation. Stud. Appl. Math. 57, 93–105 (1977)MathSciNetADSGoogle Scholar
  13. 14.
    Lieb, E.H. and Yamazaki, K.: Ground-state energy and effective mass of the polaron. Phys. Rev. 111, 728–733 (1958)ADSMATHCrossRefGoogle Scholar
  14. 15.
    Adamowski, J.: Gerlach, B. and Leschke, H.: Strong-coupling limit of polaron energy, revisited. Phys. Lett. 79A, 249–251 (1980)Google Scholar
  15. 16.
    Donsker, M. and Varadhan, S.R.S.: Asymptotics for the polaron. Commun. Pure Appt Math. 36 Google Scholar
  16. -528 (1983); Asymptotic evaluation of certain Markov process expectations for large times, IV. Comm. Pure Appl. Math. 36, 183–212 (1983)Google Scholar
  17. 17.
    Varadhan, S.R.S.: Private communicationGoogle Scholar
  18. 18.
    Feynman, R.P.: Slow electrons in a polar crystal. Phys. Rev. 97, 660–665 (1955)ADSMATHCrossRefGoogle Scholar
  19. 19.
    Ginibre, J.: Applications of functional integration. In: Statistical Mechanics and Quantum Field Theory,C. DeWitt and R. Stora , London-New York: Gordon and Breach, 1971, pp. 327–427. See pages 420–426Google Scholar
  20. 20.
    Roepstorff, G.: Path integral approach to quantum physics. Berlin-Heidelberg-New York: Springer, 1994. See Sect. 5. 3Google Scholar
  21. 21.
    Fischer, W., Leschke, H. and Müller, P.: In Path Integrals in Physics, Bangkok,1993,. V. Sa-yakanit, J.O. Berananda and W. Sritrakool, Singapore: World Scientific, 1994, pp. 347–354Google Scholar
  22. 22.
    Schrödinger, E.: Zum Heisenbergschen Unscharfe Prinzip, Sitzungsber. der Preuss. Akad. der Wiss., Berlin: Phys-Math Klasse XIX, 1930, pp. 348–356Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1997

Authors and Affiliations

  • Elliott H. Lieb
    • 1
  • Lawrence E. Thomas
    • 2
  1. 1.Departments of Physics and Mathematics, Jadwin HallPrinceton UniversityPrincetonUSA
  2. 2.Department of MathematicsUniversity of VirginiaCharlottesvilleUSA

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