Abstract
This is abrief and informal introduction to a differential geometric interpetation of adiabatic charge transport in quantum mechanics. It involves the study of afamily of Schrödinger operators. For compact multiply connected surfaces the charge transported around the “holes” is related to the first Chern character of spectral bundles. For noncompact surfaces the charge transported to infinity is related to the index of a certain Fredholm operator which involves the comparison of appropriate spectral projections. There are also relations to Connes noncommutative differential geometry. Simple examples are given.
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References
[AS] J. Avron and L. Sadun,Ann. Phys. 206, 440 (1991).
[ASS] J. Avron, R. Seiler and B. Simon,Phys. Rev. Lett. 65, 2185 (1990).
[Be] J. Bellissard,Ordinary quantum Hall effect and noncommutative cohomology, Bad-Schandau Conference on Localization (W. Weller and P. Ziesche, eds.), Teubner, Leipzig, 1988.
[Co] A. Connes,Geometrie Non Commutative, InterEdition, Paris 1990.
[Du] B. A. Dubrovin and S. P. Novikov,Sov. Phys. JETP 52, 511 (1980).
[Ef] E. G. Effros,Math. Intelligencer 11, 27 (1989).
[Fr] J. Fröhlich and T. Kerler,Nucl. Phys. B354, 369 (1991).
[FrSt] J. Fröhlich and U. M. Studer, ETH preprint (1991).
[KlSe] M. Klein and R. Seiler,Comm. Math. Phys. 128, 141 (1990).
[Ku] H. Kunz,Comm. Math. Phys. 112, 121 (1987).
[La] R. Laughlin, inThe Quantum Hall Effect (R. E. Prange and S. M. Girvin, eds.), Springer, Berlin, 1987.
[NiTh] Q. Niu and D. J. Thouless,Phys. Rev. B 35, 2188 (1987).
[ShWi] A. Shapere and F. Wilczek,Geometric Phases in Physics, World Scientific, Singapore, 1989.
[TKNN] D. J. Thouless, M. Kohmoto, P. Nightingale and M. den-Nijs,Phys. Rev. Lett. 49, 405 (1982).
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Dedicated to Professor Shmuel Agmon
This research is supported by BSF, the Israeli Academy of Sciences, and the Fund for the Promotion of Research at the Technion.
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Avron, J.E. Geometry and quantum transport. J. Anal. Math. 58, 1–7 (1992). https://doi.org/10.1007/BF02790354
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DOI: https://doi.org/10.1007/BF02790354