Abstract
We study the two-dimensional Hall effect with a random potential. The Hall conductivity is identified as a geometric invariant associated with an algebra of observables. Using the pairing betweenK-theory and cyclic cohomology theory, we identify this geometric invariant with a topological index, thereby giving the Hall conductivity a new interpretation.
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Avron, J., Seiler, R.: Quantization of the Hall conductance for general multiparticle Schrödinger Hamiltonians. Phys. Rev. Lett.54, 259–262 (1985)
Avron, J., Seiler, R., Simon, B.: Homotopy and quantization in condensed matter physics. Phys. Rev. Lett.51, 51–53 (1983)
Avron, J., Seiler, R., Yaffe, L.: Adiabatic theorems and applications to the quantum Hall effect. Commun. Math. Phys.110, 33–49 (1987)
Bellissard, J.: Schrödinger operators with an almost periodic potential. Lecture Notes in Physics, Vol.153, Berlin, Heidelberg, New York: Springer 1982
Bellissard, J.:K-Theory ofC*-algebras in solid state physics. Statistical mechanics and field theory, Mathematical aspects. Dorlas, T., Hugenholtz, M., Winnink, M. (eds.). Lecture Notes in Physics, Vol.257, pp. 99–156 Berlin, Heidelberg, New York: Springer 1986
Bellissard, J.: Ordinary quantum Hall effect and noncommutative cohomology. Lecture Notes, 1986
Bellissard, J.:C*-Algebra in solid state physics, 2D electrons in a uniform magnetic field. Lecture Notes, 1987
Bellisard, J.: Private communication, 1988
Bellisard, J., Lima R., Testard, D.: Almost periodic Schrödinger operators, in “Mathematics + Physics”,1, pp. 1–64. Streit, L. (ed), Philadelphia, Singapore: World Scientific 1985
Blackadar, B.:K-Theory for Operator Algebras, MSRI Publications, 1985
Claro, F., Wannier, W.: Magnetic subband structure of electrons in hexagonal lattices. Phys. Rev.B19, 6068–6074 (1979)
Coburn, L., Moyer, R., Singer, I.:C*-Algebras of almost periodic pseudo-differential operators. Acta Math.139, 279–307 (1973)
Connes, A.: An analogue of the Thom isomorphism ofC*-algebras by an action of ℝ. Adv. Math.39, 31–55 (1981)
Connes, A.: A survey of foliations and operator algebras. Proceedings of Symposia in Pure Mathematics, Vol.38, pp. 521–628. Providence, RI: AMS 1982
Connes, A.: Noncommutative differential geometry. Publ. Math. I.H.E.S.,62, 257–360 (1986)
Elliott, G.: Gaps in the spectrum of an almost periodic Schrödinger operator. C.R. Math. Rep. Acad. Sci. Canada4, 255–259 (1982)
Elliott, G., Natsume, T., Nest, R.: Cyclic cohomology for one parameter smooth crossed products. Acta Math. (to appear)
Gudmundsson, V., Gerhardts, R.: Interpretation of experiments implying density of states between Landau levels of a two-dimensional electron gas by a statistical model for inhomogeneity, Phys. Rev.B35, 8005–8014 (1987)
Halperin, B.: Quantized Hall conductance current-carrying edge states, and the existence of extended states in a two-dimensional disordered potential. Phys. Rev.B25, 2185 (1982)
Kaminker, J., Xia, J.: The spectrum of operators elliptic along the orbits of ℝn action. Commun. Math. Phys.110, 427–438 (1987)
von Klitzing, K., Dorda, G., Pepper, M.: Realization of a resistance standard based on fundamental constants. Phys. Rev. Lett.45, 494–497 (1980)
Kunz, H.: The quantum Hall effect for electrons in a random potential. Commun. Math. Phys.112, 121–145 (1987)
Laughlin, R.: Quantized Hall conductivity in two dimensions. Phys. Rev.B23, 5632–5634 (1981)
Mott, N., Jones, H.: The Theory of the Properties of Metals and Alloys. Oxford, University Press 1962
Naimark, M.: Linear differential operators, part II, linear differential operators in Hilbert space. New York: Frederick Ungar Publishing 1968
Pedersen, G.:C*-Algebras and their automorphism groups. London: Academic Press 1979
Shubin, M.: The spectral theory and the index of elliptic operators with almost periodic potential. Russ. Math. Surv.34, 109–157 (1979)
Thouless, D. Kohmoto, M., Nightingale P., Deh-Nijs, M.: Quantized Hall conductance in a two-dimensional periodic potential. Phys. Rev. Lett.49, 405–408 (1982)
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Communicated by B. Simon
Supported in part by the National Science Foundation under Grant No. DMS-8717185
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Xia, J. Geometric invariants of the quantum Hall effect. Commun.Math. Phys. 119, 29–50 (1988). https://doi.org/10.1007/BF01218259
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DOI: https://doi.org/10.1007/BF01218259