Advertisement

Journal of Statistical Physics

, Volume 67, Issue 3–4, pp 675–780 | Cite as

Chaotic polaronic and bipolaronic states in the adiabatic Holstein model

  • Serge Aubry
  • Gilles Abramovici
  • Jean -Luc Raimbault
Articles

Abstract

A rigorous proof for the existence of bipolaronic states is given for the adiabatic Holstein model for any lattice at any dimension, periodic or not, and for an arbitrary band filling, provided that the electron-phonon coupling (in dimensionless units) is large enough. The existence of mixed polaronic-bipolaronic states is also proven, but for larger electron-phonon coupling. These states consist of arbitrary distributions of bipolarons (or of bipolarons and polarons) localized in real space which can be simply labeled by pseudospin configurations as for a lattice gas model. The theory not only applies to periodic crystals, but also to quasicrystals, amorphous structures, polymer network, etc.

When these bipolaronic and mixed polaronic-bipolaronic states exist, it is proven that: (1) These bipolaronic (and mixed polaronic-bipolaronic) states exhibit a nonzero phonon gap with a nonvanishing lower bound and an electronic gap at the Fermi energy. (2) These structures are insulating. The perturbation generated by any local change in the bipolaronic or polaronic distribution or by any charged impurity or defect decays exponentially at long distance. (3) These bipolaronic (and mixed polaronic-bipolaronic) states persist for any uniform magnetic field. (4) For large enough electron-phonon coupling, the ground state of the extended adiabatic Holstein model is a bipolaronic state when there is no uniform magnetic field or when it is small enough. It becomes a mixed polaronic-bipolaronic state for large enough magnetic field (note that the mixed polaronic-bipolaronic states are magnetic).

In one-dimensional models, the ground state is an incommensurate (or commensurate) charge density wave (CDW) as predicted by Peierls (this result is not rigorous, but has been confirmed numerically). It is proven that the ground state becomes a “bipolaronic charge density wave” (BCDW) at large enough electron-phonon coupling. The existence of a transition by breaking of analyticity (TBA), which was numerically observed as a function of the electron-phonon coupling, is then confirmed. In that case, the shape of the effective bipolaron can be numerically calculated. It is observed that its size diverges at the TBA. The physical properties of BCDWs are rather different from those predicted by standard charge density wave theory. Bipolaronic charge density waves can also exist in models which are not only low-dimensional, but purely two- or three-dimensional.

The technique for proving these theorems is an application of the concept of anti-integrability initially developed for Hamiltonian dynamical systems. It consists in proving that the eigenstates of the (trivial) Hamiltonian (called antiintegrable) obtained by canceling all electronic and lattice kinetic terms survive as a uniformly continuous function of the electronic kinetic energy terms in the Hamiltonian up to a certain threshold.

Key words

Chaos anti-integrability bipolaron polaron charge density wave breaking of analyticity 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    S. Aubry and G. Abramovici, Chaotic trajectories in the standard map: The concept of anti-integrability,Physica D43:199–219 (1990); S. Aubry, The concept of antiintegrability: Definition, theorems and applications, inProceedings of the IM A Workshop on “Twist Mappings and their Applications” (March 12–16, 1990, Minneapolis, Minnesota, in press).Google Scholar
  2. 2.
    S. Aubry, J. P. Gosso, G. A. Bramovici, J. L. Raimbault, and P. Quemerais, Effective discommensurations in the incommensurate ground-states of the extended Frenkel-Kontorowa models,Physica D47:461–497 (1991).Google Scholar
  3. 3.
    P. Y. Le Daeron and S. Aubry,J. Phys. C 16:4827–4838 (1983).Google Scholar
  4. 4.
    P. Y. Le Daeron and S. Aubry,J. Phys. (Paris)44:C3-1573–1577 (1983).Google Scholar
  5. 5.
    P. Y. Le Daeron, Transition Metal-Isolant dans les Chaînes de Peierls, Ph.D. thesis, Université Paris-Sud, Orsay, France (1983).Google Scholar
  6. 6.
    S. Aubry and P. Quemerais, inLow Dimensional Electronic Properties of Molybdenum Bronzes and Oxides, C. Schlenker, eds. (Kluwer, 1989), pp. 295–405.Google Scholar
  7. 7.
    S. Aubry, The new concept of transition by breaking of analyticity..., inSoliton and Condensed Matter (Symposium, Oxford, 1978), A. R. Bishop and T. Schneider, eds. (Solid State Sciences, Vol. 8, 1978), p. 264–278.Google Scholar
  8. 8.
    E. Zeidler,Nonlinear Functional Analysis and Its Applications I.Fixed-Point Theorems (Springer, 1986), Theorem 1.A, p. 17, and Proposition 1.2, p. 19.Google Scholar
  9. 9.
    S. Aubry and P. Quemerais, inSingular Behavior and Nonlinear Dynamics, St. and Sp. Pnevmatikos and T. Bountis, eds. (Riedel, 1989), pp. 342–363.Google Scholar
  10. 10.
    S. Aubry, G. Abramovici, D. Feinberg, P. Quemerais, and J. L. Raimbauit, Nonlinear coherent structures in physics, mechanics and biological systems,Lectures Notes in Physics, Vol. 353 (Springer, 1989), pp. 103–116.Google Scholar
  11. 11.
    S. Aubry, P. Quemerais, and J. L. Raimbauit, in Proceeding of Third European Conference on Low Dimensional Conductors and Superconductors, S. Barisić, ed.,Fisica 21 (Supp. 3):98–101, 106–108 (1990).Google Scholar
  12. 12.
    S. Aubry, Bipolaronic Charge Density Waves, in “MicroscopicAspects of Non-Linearity in Condensed Physics” A. R. Bishop, V. L. Pokrovsky, and V. Tognetti, NATO ASI Series B, Vol. 264 (Plenum, 1991), pp. 105–111.Google Scholar
  13. 13.
    G. Abramovici, Structures chaotiques dans les phases incommensurables et ondes de densité de charge, Ph.D. thesis, Université Paris VI, Paris (1990).Google Scholar
  14. 14.
    L. Landau,Phys. Z. Sowjet. 3:664 (1933).Google Scholar
  15. 15.
    C. G. Kuper and G. D. Whitfield,Polarons and Excitons (Oliver and Boyd, Edinburgh, 1963).Google Scholar
  16. 16.
    J. Appel,Solid State Phys. 21:193 (1968).Google Scholar
  17. 17.
    T. Holstein,Ann. Phys. 8:325, 343 (1959).Google Scholar
  18. 18.
    W. P. Su, J. R. Schrieffer, and A. J. Heeger,Phys. Rev. Lett. 25:1968 (1979);Phys. Rev. B 22:2099 (1980).Google Scholar
  19. 19.
    C. Kittel,Quantum Theory of Solids (Wiley, New York) (1963).Google Scholar
  20. 20.
    Y. Hatsugai and M. Kohmoto,Phys. Rev. B 42:82 (1990).Google Scholar
  21. 21.
    V. J. Goldman, M. Santos, M. Shayegan, and J. E. Cunningham,Phys. Rev. Lett. 65:2189 (1990).Google Scholar
  22. 22.
    D. R. Hofstadter,Phys. Rev. B 14:2239 (1976).Google Scholar
  23. 23.
    A. S. Alexandrov and J. Ranninger,Phys. Rev. B 24:1164 (1981).Google Scholar
  24. 24.
    A. S. Alexandrov, J. Ranninger, and S. Robaszkiewicz,Phys. Rev. B 33:4526–4542 (1986).Google Scholar
  25. 25.
    P. Quemerais, Une nouvelle approche pour l'étude des composés à onde de densité de charge: Conséquences de la brisure d'analyticité, Ph.D. thesis, Université de Nantes (1987).Google Scholar
  26. 26.
    J. L. Raimbauit, Configurations bipolaroniques et fluctuations quantiques du réseau dans les chaînes de Peierls, Ph.D. thesis, Université de Nantes (1990).Google Scholar
  27. 27.
    F. Vallet, R. Schilling, and S. Aubry, Hierarchical low-temperature behavior of onedimensional incommensurate structures,Europhys. Lett. 2:815–822 (1986); Low temperature excitations, specific heat and hierarchical melting of a one-dimensional incommensurate structure,J. Phys. C 21:67–105 (1988).Google Scholar
  28. 28.
    F. Vallet, Thermodynamique unidimensionnelle et structures bidimensionnelle de quelques modèles pour des systèmes incommensurables, Ph.D. thesis, Université Paris I (1986).Google Scholar
  29. 29.
    R. Schilling and S. Aubry, Static structure factor of one-dimensional non-analytic incommensurate structures,J. Phys. C 20:4881–4889 (1987).Google Scholar
  30. 30.
    S. Aubry, Devil's staircase and order without periodicity,J. Phys. (Paris)44:147–162 (1983); Weakly periodic structures and example,J. Phys. (Paris)50:C3-97–106 (1989); Weakly periodic structures with a singular continuous spectrum, inGeometry and Thermodynamics, J. C. Toledano, ed., NATO ASI Series B, Vol. 229 (Plenum Press, New York, 1991), pp. 281–300.Google Scholar
  31. 31.
    P. Monceau, ed.,Electronic Properties of Inorganic Quasi-One-Dimensional Compounds (Reidel, Boston, 1985).Google Scholar
  32. 32.
    G. Grüner and A. Zettl,Phys. Rep. 119:117, (1985).Google Scholar
  33. 33.
    G. Grüner,Rev. Mod. Phys. 60:1129 (1988).Google Scholar
  34. 34.
    C. Schlenker, ed.,Low Dimensional Electronic Properties of Molybdenum Bronzes and Oxides (Kluwer, 1989).Google Scholar
  35. 35.
    P. Monceau, Recent developments in charge density wave systems, inApplication of Statistical and Field Theory Methods to Condensed Matter, D. Baeriswylet al., eds. (Plenum Press, New York, 1990), p. 357.Google Scholar
  36. 36.
    S. Aubry, R. MacKay, and C. Baesens, Equivalence of uniform hyperbolicity for symplectic twist maps and phonon gap for Frenkel-Kontorova models,Physica D (1991).Google Scholar
  37. 37.
    P. Nozieres and S. Schmitt-Rink,J. Low Temp. Phys. 59:195 (1985).Google Scholar
  38. 38.
    S. A. Brazovskii, I. E. Dzyaloshinskii, and I. M. Krichever,Sov. Phys. JETP 56:212 (1982).Google Scholar
  39. 39.
    A. H. Moudden, J. L. Raimbault, S. Aubryet al., in preparation.Google Scholar
  40. 40.
    C. Noguera and J. P. Pouget, Temperature dependence of the Peierls wave-vector in quasi one dimensional conductors,J. Physique I (France)1:1035–1054 (1991).Google Scholar
  41. 41.
    M. Floria, P. Quemerais, and S. Aubry,J. Phys. C (1992), in press.Google Scholar
  42. 42.
    M. Ido, Y. Okayama, T. Ijiri, and Y. Okajina,J. Phys. Soc. Jpn. 59:1341–1347 (1990).Google Scholar
  43. 43.
    M. D. Nunez Regueiro, J. M. Mignot, and D. Castello, Superconductivity under pressure in NbSe3, Preprint (1991).Google Scholar
  44. 44.
    R. V. Coleman, G. Eiserman, M. P. Everson, and A. Johnson,Phys. Rev. Lett. 55:863 (1985).Google Scholar
  45. 45.
    G. Mihaly,Physica Scripta T29:67 (1989).Google Scholar
  46. 46.
    P. Littlewood and R. Rammal,Phys. Rev. B 38:2675 (1988).Google Scholar
  47. 47.
    P. Monceau and J. Richard,Phys. Rev. B 37:7982 (1988).Google Scholar
  48. 48.
    K. Biljakovic, J. C. Lasjaunias, P. Monceau, and F. Levy,Phys. Rev. Lett. 62:1512 (1989);Europhys. Lett. 8:771 (1989); J. C. Lasjaunias, K. Biljakovic, and P. Monceau,Physica B 165&166 (1990).Google Scholar
  49. 49.
    A. Fournel, J. P. Sorbier, M. Konczykowski, P. Monceau, and F. Levy,Physica 143B:177 (1986).Google Scholar
  50. 50.
    R. S. MacKay and C. Baesens, “A new paradigm in quantum chaos ...” to appear in Proceeding of “The international school of quantum chaos” Enrico Fermi Institute, Varenna, Italy, August 1991.Google Scholar

Copyright information

© Plenum Publishing Corporation 1992

Authors and Affiliations

  • Serge Aubry
    • 1
  • Gilles Abramovici
    • 1
  • Jean -Luc Raimbault
    • 1
  1. 1.Laboratoire Léon Brillouin (Laboratoire commun CEA-CNRS)CEN SaclayGif-sur-Yvette CédexFrance

Personalised recommendations