Le papillon de Hofstadter

Jean Bellissard

Séminaire Bourbaki (1991-1992)

  • Volume: 34, page 7-39
  • ISSN: 0303-1179

How to cite

top

Bellissard, Jean. "Le papillon de Hofstadter." Séminaire Bourbaki 34 (1991-1992): 7-39. <http://eudml.org/doc/110164>.

@article{Bellissard1991-1992,
author = {Bellissard, Jean},
journal = {Séminaire Bourbaki},
keywords = {Hofstadter's butterfly; crystalline electrons in uniform magnetic fields; fractional character of the band energy spectrum as a function of magnetic flux; Harper's model; renormalisation group theory},
language = {fre},
pages = {7-39},
publisher = {Société Mathématique de France},
title = {Le papillon de Hofstadter},
url = {http://eudml.org/doc/110164},
volume = {34},
year = {1991-1992},
}

TY - JOUR
AU - Bellissard, Jean
TI - Le papillon de Hofstadter
JO - Séminaire Bourbaki
PY - 1991-1992
PB - Société Mathématique de France
VL - 34
SP - 7
EP - 39
LA - fre
KW - Hofstadter's butterfly; crystalline electrons in uniform magnetic fields; fractional character of the band energy spectrum as a function of magnetic flux; Harper's model; renormalisation group theory
UR - http://eudml.org/doc/110164
ER -

References

top
  1. [AL] S. Alexander -Superconductivity of networks. A percolation approach to the effect of disorder, Phys. Rev., B27, (1983), 1541-1557. MR690539
  2. [AA] S. Aubry, G. Andre -Analyticity breaking and the Anderson localization in incommensurate lattices, Ann. Israeli Phys. Soc., 3, (1980), 133. Zbl0943.82510MR626837
  3. [AU] S. Aubry -The new concept of transition by breaking of analyticity in cristallographic models, Solid State Sci., 8, (1978), 264. MR522493
  4. [AMS] J. Avron, P.H.M. van Mouche, B. Simon - On the Measure of the Spectrum for the Almost Mathieu Operator, Preprint CALTECH, 1990, Commun. Math. Phys.. MR1069202
  5. [AZ] Ya. Azbel -Energy spectrum of a conduction electron in a magnetic field, Sov. Phys. JETP, 19(3), (1964), 634. 
  6. [BEI] J. Bellissard -C*-Algebras in Solid State Physics: 2D electrons in a uniform magnetic field, in Operator Algebras and Application, Vol.2, D.E.Evans & M.Takesaki Eds., Cambridge University Press, (1988). Zbl0677.46055MR996451
  7. [BE2] J. Bellissard -Almost Periodicity in Solid State Physics and C*- Algebras, in The Harald Bohr Centenary, C.Berg & B.Fuglede Eds., The Danish Royal Acad. Sci., 42:3, (1989), 35-75. Zbl0678.42007MR1031737
  8. [BE3] J. Bellissard - The Rotation Algebra, Lecture donnée à l'école d'été de Nantes, Juin 1991, en préparation. 
  9. [BKS] J. Bellissard, C. Kreft, R. Seiler -Analysis of the spectrum of a particle on a triangular lattice with two magnetic fluxes by algebraic and numerical methods, J. Phys., A24, (1991), 2329-2353. Zbl0738.35047MR1118535
  10. [BES] J. Bellissard, B. Simon - Cantor Spectrum for the Almost Mathieu Equation, J. Funct. Anal., 48, (1982), 408-419. Zbl0516.47018MR678179
  11. [BV] J. Bellissard, M. Vittot -Heisenberg's picture and non commutative geometry in the semi-classical limit in qauntum Mechanics, Ann. Inst. Henri Poincaré, 52, (1990), 175-235. Zbl0705.46037MR1057445
  12. [CH] W.G. Chambers - Phys Rev., A140, (1965), 135-143. 
  13. [CEY] M.D. Choi, G. Elliott, K. Yui -Gauss polynomials and the rotation Algebra, Inventiones Math., (1991). Zbl0665.46051
  14. [GE] P.G. de Gennes -Diamagnétisme de grains supraconducteurs près d'un seuil de percolation, C.R. Acad. Sci., B292, (1981), 9-12; Champ critique d'une boucle supraconductrice ramifiée, C.R. Acad.Sci., B292, (1981), 279-282. 
  15. [GH] J.P. Guillement, B. Helffer, P. Treton- Walk inside Hofstadter's butterfly, J. Phys. France, 50, (1989), 2019-2058. 
  16. [HA] P.G. Harper -Single Band Motion of Conduction Electrons in a Uniform Magnetic Field, Proc.Phys. Soc. London, A68, (1955), 874. Zbl0065.23708
  17. [HR] B. Helffer, D. Robert -Calcul fonctionnel de la transformée de Mellin et applications, J. Funct. Anal., 53, (1983). Zbl0524.35103MR724029
  18. [HS0] B. Helffer, J. Sjöstrand -Microlocal Analysis for the Periodic Magnetic Schrödinger Equation and related questions, C.I.M.E Lectures (1989), in Lecture Notes in Math.. Zbl0761.35090
  19. [HS1] B. Helffer, J. Sjöstrand -Analyse semi-classique pour l'équation de Harper (application à l'étude de l'équation de Schrödinger avec champ magnétique), Mémoires de la S.M.F., (1989). Zbl0714.34130
  20. [HS2] B. Helffer, J. Sjöstrand -Analyse semi-classique pour l'équation de Harper II. Comportement semi-classique près d'un rationnel, Mémoires de la S.M.F n°40, 118, (1990), 1-139. Zbl0714.34131MR1052373
  21. [HS3] B. Helffer, J. Sjöstrand -Semi-classical analysis for Harper's equation III, Mémoires de la S.M.F., (1991). Zbl0725.34099
  22. [HO] D.R. Hofstadter -Energy levels and wave functions of Bloch electrons in a rational or irrational magnetic field, Phys. Rev., B14, (1976), 2239. 
  23. [KL] K. von Klitzing, G. Dorda, M. Pepper -Realization of a resistance standard based on fundamental constants, Phys. Rev. Letters, 45, (1980), 494-497. 
  24. [LA] L. Landau -Diamagnetismus der Metalle, Z. für Phys., 64, (1930), 629-637. JFM56.1318.10
  25. [LU] J.M. Luttinger -The Effect of a Magnetic Field on Electrons in a Periodic Potential, Phys. Rev., 84, (1951), 814. Zbl0044.45301MR45052
  26. [MO] P. van Mouche -The coexistence problem for the discrete Mathieu operator, Comm. Math. Phys., 122, (1989), 23-34. Zbl0669.34016MR994494
  27. [PCR] B. Pannetier, J. Chaussy, R. Rammal - Experimental Determination of the (H, T) Phase Diagram of a Superconducting Network, J. de Physique Lettres44, (1983), L853-L858. 
  28. [PCRV] B. Pannetier, J. Chaussy, R. Rammal, J.C. Villegier - Experimental Fine Tuning of the Frustration: 2D Superconducting Networks in a Magnetic Field, Phys. Rev. Letters, 53, (1984), 1845-1848. 
  29. [PED] G. Pedersen -C*-Algebras and their automorphism groups, Academic Press, London, New-York, (1979). Zbl0416.46043MR548006
  30. [PE] R. Peierls -Zur Theorie des Diamagnetismus von Leitungelektronen, Z. für Phys., 80, (1933), 763-791. Zbl0006.19204
  31. [RB1] R. Rammal, J. Bellissard -Ground State of the Fermi Gas in 2D Lattices with a Magnetic Field: New Exact Results, Europhys. Letters, 13, (1990), 205-210. 
  32. [RB2] R. Rammal, J. Bellissard -An algebraic semi-classical approach to Bloch electrons in a magnetic field, J. Phys. France, 51, (1990), 1803-1830. MR1156190
  33. [SI] Ya.G. Sinai -Anderson localization for the one dimensional difference Schrödinger operator with quasiperiodic potentials, in Proc. Int. Congress Math. Phys. Marseille 1986, World Scientific Pub. Co., Singapore, (1987), pp. 870-903. MR915574
  34. [SO] J.B. Sokoloff -Unusual band structure, wave functions and electrical conductance in crystals with incommensurable periodic potentials, Phys. Reports, 126, (1985), 189-244. 
  35. [TH] D. Thouless -Bandwidths for a quasiperiodic tight-binding model, Phys. Rev., B28, (1983), 4272-4276. 
  36. [TKN2] D. Thouless, M. Kohmoto, M. Nightingale, M. den Nijs -Quantized Hall Conductance in two dimensional periodic potential, Phys. Rev. Letters, 49, (1982), 405. 
  37. [WI] M. Wilkinson -Critical Properties of Electron Eigenstates in Incommensurate Systems, Proc. Roy. Soc. London, A391, (1984), 305-350. MR739684
  38. [ZA] J. Zak -Magnetic Translation Group, Phys. Rev., A134, (1964), 1602- 1607; Magnetic Translation Group II: Irreducible Representations, Phys. Rev., A134, (1964), 1607-1611. Zbl0131.45404MR177769

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.