Le papillon de Hofstadter
Séminaire Bourbaki (1991-1992)
- Volume: 34, page 7-39
- ISSN: 0303-1179
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topBellissard, 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- [AL] S. Alexander -Superconductivity of networks. A percolation approach to the effect of disorder, Phys. Rev., B27, (1983), 1541-1557. MR690539
- [AA] S. Aubry, G. Andre -Analyticity breaking and the Anderson localization in incommensurate lattices, Ann. Israeli Phys. Soc., 3, (1980), 133. Zbl0943.82510MR626837
- [AU] S. Aubry -The new concept of transition by breaking of analyticity in cristallographic models, Solid State Sci., 8, (1978), 264. MR522493
- [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
- [AZ] Ya. Azbel -Energy spectrum of a conduction electron in a magnetic field, Sov. Phys. JETP, 19(3), (1964), 634.
- [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
- [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
- [BE3] J. Bellissard - The Rotation Algebra, Lecture donnée à l'école d'été de Nantes, Juin 1991, en préparation.
- [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
- [BES] J. Bellissard, B. Simon - Cantor Spectrum for the Almost Mathieu Equation, J. Funct. Anal., 48, (1982), 408-419. Zbl0516.47018MR678179
- [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
- [CH] W.G. Chambers - Phys Rev., A140, (1965), 135-143.
- [CEY] M.D. Choi, G. Elliott, K. Yui -Gauss polynomials and the rotation Algebra, Inventiones Math., (1991). Zbl0665.46051
- [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.
- [GH] J.P. Guillement, B. Helffer, P. Treton- Walk inside Hofstadter's butterfly, J. Phys. France, 50, (1989), 2019-2058.
- [HA] P.G. Harper -Single Band Motion of Conduction Electrons in a Uniform Magnetic Field, Proc.Phys. Soc. London, A68, (1955), 874. Zbl0065.23708
- [HR] B. Helffer, D. Robert -Calcul fonctionnel de la transformée de Mellin et applications, J. Funct. Anal., 53, (1983). Zbl0524.35103MR724029
- [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
- [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
- [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
- [HS3] B. Helffer, J. Sjöstrand -Semi-classical analysis for Harper's equation III, Mémoires de la S.M.F., (1991). Zbl0725.34099
- [HO] D.R. Hofstadter -Energy levels and wave functions of Bloch electrons in a rational or irrational magnetic field, Phys. Rev., B14, (1976), 2239.
- [KL] K. von Klitzing, G. Dorda, M. Pepper -Realization of a resistance standard based on fundamental constants, Phys. Rev. Letters, 45, (1980), 494-497.
- [LA] L. Landau -Diamagnetismus der Metalle, Z. für Phys., 64, (1930), 629-637. JFM56.1318.10
- [LU] J.M. Luttinger -The Effect of a Magnetic Field on Electrons in a Periodic Potential, Phys. Rev., 84, (1951), 814. Zbl0044.45301MR45052
- [MO] P. van Mouche -The coexistence problem for the discrete Mathieu operator, Comm. Math. Phys., 122, (1989), 23-34. Zbl0669.34016MR994494
- [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.
- [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.
- [PED] G. Pedersen -C*-Algebras and their automorphism groups, Academic Press, London, New-York, (1979). Zbl0416.46043MR548006
- [PE] R. Peierls -Zur Theorie des Diamagnetismus von Leitungelektronen, Z. für Phys., 80, (1933), 763-791. Zbl0006.19204
- [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.
- [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
- [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
- [SO] J.B. Sokoloff -Unusual band structure, wave functions and electrical conductance in crystals with incommensurable periodic potentials, Phys. Reports, 126, (1985), 189-244.
- [TH] D. Thouless -Bandwidths for a quasiperiodic tight-binding model, Phys. Rev., B28, (1983), 4272-4276.
- [TKN2] D. Thouless, M. Kohmoto, M. Nightingale, M. den Nijs -Quantized Hall Conductance in two dimensional periodic potential, Phys. Rev. Letters, 49, (1982), 405.
- [WI] M. Wilkinson -Critical Properties of Electron Eigenstates in Incommensurate Systems, Proc. Roy. Soc. London, A391, (1984), 305-350. MR739684
- [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
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