×

zbMATH — the first resource for mathematics

Fractal spectrum of periodic quantum systems in a magnetic field. (English) Zbl 1116.81313
Summary: Models of two-dimensional periodic quantum-mechanical systems in a uniform magnetic field are considered. Results of the numerical analysis of the energy spectrum for these models are presented. The flux-energy diagrams for the magnetic Bloch bands are obtained. Evidence for a fractal structure of these diagrams is given.

MSC:
81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Azbel, M.Ya., Energy spectrum of a conductivity electron in a magnetic field, Sov. phys. ZETP, 46, 634-657, (1964)
[2] Hofstadter, D.R., Energy levels and wave functions of Bloch electrons in rational and irrational magnetic fields, Phys.rev. B, 14, 2239-2249, (1976)
[3] Wannier, G.H., A remark not dependent on rationality for Bloch electrons in a magnetic field, Phys. stat. sol. B, 88, 757-765, (1978)
[4] Weiss, D.; von Klitzing, K.; Ploog, K.; Weimann, G., Magnetoresistance oscillations in a two – dimensional electron gas induced by a submicrometer periodic potential, Europhys. lett., 8, 179-184, (1989)
[5] Gerhardts, R.R.; Weiss, D.; Wulf, U., Magnetoresistance oscillations in a grid potential: indication of a Hofstadter-type energy spectrum, Phys. rev. B, 43, 5192-5195, (1991)
[6] Gudmundsson, V.; Gerhardts, R.R., Manifestation of the Hofstadter butterfly in far-infrared absorption, Phys. rev. B, 54, 5223-5227, (1996)
[7] Helffer, B.; Sjöstrand, J., Semi-classical analysis for Harper’s equation III, Memories soc. math. France, 39, 1-124, (1989) · Zbl 0725.34099
[8] Avron, J.; van Mouche, P.H.M.; Simon, B., On the measure of the spectrum for almost-Mathieu operator, Commun. math. phys., 132, 103-118, (1990) · Zbl 0724.47002
[9] Choi, M.D.; Elliot, G.A.; Yui, N., Gauss potentials and the rotation algebra, Invent. math., 99, 225-246, (1990) · Zbl 0665.46051
[10] Bellisard, J.; vanElst, A.; Schulz-Baldes, H., The noncommutative geometry and quantum Hall effect, J. math. phys., 35, 5373-5451, (1994) · Zbl 0824.46086
[11] Bellissard, J., La papillon de Hofstadter, Astérisque, 206, 7-39, (1992) · Zbl 0791.46057
[12] Guillement, J.P.; Helffer, B.; Treton, P., Walk inside Hofstadter’s butterfly, J. phys. France, 50, 2019-2058, (1989)
[13] Helffer, B.; Kerdelhué, P.; Sjöstrand, J., La papillon de Hofstadter revesité, Bull. soc. math. France, 118, 1-87, (1990) · Zbl 0732.44004
[14] Bellisard, J.; Kreft, C.; Seiler, R., Analysis of the spectrum of a particle on a triangular lattice with two magnetic fluxes by algebraic and numerical methods, J. phys. A, 24, 2329-2353, (1991) · Zbl 0738.35047
[15] Kreft, C.; Seiler, R., Model of the Hofstadter-type, J. math. phys., 37, 5207-5243, (1996) · Zbl 0867.58051
[16] Wilkinson, A.M., Critical properties of electron eigenstates in incommensurate systems, Proc. R. soc. London, ser., A 391, 305-350, (1984)
[17] Hatsugai, Y.; Kohmoto, M., Energy spectrum and the quantum Hall effect on the square lattice with next-nearest-neighbor hopping, Phys. rev. B, 42, 8282-8294, (1990)
[18] Andrade Neto, M.A.; Schulz, P.A., Hofstadter spectra in two-dimensional superlattice with arbitrary modulation strength, Phys. rev. B, 52, 14093-14097, (1995)
[19] Kühn, O.; Fessatidis, V.; Cui, H.L.; Sellbmann, P.E.; Horing, N.J.M., Energy spectrum for two-dimensional periodic potentials in a magnetic field, Phys. rev. B, 47, 13019-13022, (1993)
[20] Petschel, G.; Geisel, T., Bloch electron in magnetic fields: classical chaos and Hofstadter’s butterfly, Phys. rev. lett., 71, 239-242, (1993)
[21] Gredeskul, S.; Avishai, Ya.; Azbel, M.Y., Two-dimensional electron gas in a magnetical field and point potentials, Low-temp. physics, 23, 21-33, (1997)
[22] Geyler, V.A., The two-dimensional Schrödinger operator with a uniform magnetic field, and its perturbation by periodic zero-range potentials, Saint |St. Petersburg math. J., 3, 489-532, (1992) · Zbl 0791.35025
[23] Geyler, V.A.; Popov, I.Yu., The spectrum of a magneto-Bloch electron in a periodic array of quantum dots: explicitly solvable model, Z. phys. B, 93, 437-439, (1994)
[24] Geyler, V.A.; Popov, I.Yu., Periodic array of quantum dots in a magnetic field: irrational flux; honeycomb lattice, Z. phys. B, 98, 473-477, (1995)
[25] Geyler, V.A.; Pavlov, B.S.; Popov, I.Yu., Spectral properties of a charged particle in antidot array: a limiting case of quantum billiard, J. math. phys., 37, 5171-5194, (1996) · Zbl 0892.47066
[26] V.A. Geyler, A.V. Popov, Hofstadter butterfly for a periodic array of quantum dots, in: Proc. Fourth Int. Conf. on Integral Methods in Science and Engineering, 17-20 June 1996, Oulu, Finland, 1997
[27] Weiss, D.; Roukes, M.L.; Menschig, A.; Grambow, P.; von Klitzing, K.; Weimann, G., Electron pinball and commensurate orbits in a periodic array of scatters, Phys. rev. lett., 66, 2790-2793, (1991)
[28] Pavlov, B.S., The theory of extensions and explicitly-solvable models, Russ. math. surv., 42, 127-168, (1987) · Zbl 0665.47004
[29] S. Albeverio, F. Gesztesy, R. Hoegh-Krohn, H. Holden, Solvable Models in Quantum Mechanics, Springer, Berlin, 1988 · Zbl 0679.46057
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.