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Charge deficiency, charge transport and comparison of dimensions

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Abstract

We study the relative index of two orthogonal infinite dimensional projections which, in the finite dimensional case, is the difference in their dimensions. We relate the relative index to the Fredholm index of appropriate operators, discuss its basic properties, and obtain various formulas for it. We apply the relative index to counting the change in the number of electrons below the Fermi energy of certain quantum systems and interpret it as the charge deficiency. We study the relation of the charge deficiency with the notion of adiabatic charge transport that arises from the consideration of the adiabatic curvature. It is shown that, under a certain covariance, (homogeneity), condition the two are related. The relative index is related to Bellissard's theory of the Integer Hall effect. For Landau Hamiltonians the relative index is computed explicitly for all Landau levels.

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References

  1. Avron, J.E., Pnueli, A.: Landau Hamiltonians on symmetric spaces. In: Ideas and methods in mathematical analysis, stochastics, and applications. Vol. II. Albeverio, S., Fenstad, J.E., Holden, H., Lindstrøm, T. (eds.), Cambridge: Cambridge University Press 1992

    Google Scholar 

  2. Avron, J.E., Raveh, A., Zur, B.: Adiabatic quantum transport in multiply connected systems. Rev. Mod. Phys.60, 873–916 (1988)

    Article  Google Scholar 

  3. Avron, J.E., Seiler, R., Yaffe, L.G.: Adiabatic theorems and applications to the quantum Hall effect. Commun. Math. Phys.110, 33–49 (1987)

    Article  Google Scholar 

  4. Avron, J.E., Seiler, R.: Quantization of the Hall conductance for general multiparticle Schrödinger Hamiltonians. Phys. Rev. Let.54, 259–262 (1985)

    Article  Google Scholar 

  5. Avron, J.E., Seiler, R., Simon, B.: The index of a pair of projections. Preprint, to appear in Journal of Functional Analysis

  6. Bellissard, J.: Ordinary quantum Hall effect and non-commutative cohomology. In: Localization in disordered systems. Weller, W., Zieche, P. (eds.), Leipzig: Teubner 1988

    Google Scholar 

  7. Birman, M.Sh.: A proof of the Fredholm trace formula as an application of a simple embedding for kernels of integral operators of trace class inL 2(ℝm). Preprint, Department of Mathematics, Linking University, S-581 83 Linkping, Sweden

  8. Block, B., Wen, X.G.: Effective theories of the fractional quantum Hall effect at generic filling fractions. Phys. Rev. B42, 8133–8144 (1990); Effective theories of the Fractional quantum Hall effect: Hierarchy construction. Phys. Rev. B42, 8145–8156 (1990); Structure of the microscopic theory of the hierarchical fractional quantum Hall effect. Phys. Rev. B43, 8337–8349 (1991)

    Article  Google Scholar 

  9. Bregola, M., Marmo, G., Morandi, G. (eds.): Anomalies, phases, defects. Monographs nd Textbooks in Physical Sciences, Lecture Notes, 17; Bibliopolis (Napoli) 1990

  10. Carrey, A.L.: Some homogeneous spaces and representations of the Hilbert Lied group. Rev. Rom. Math. Pure Appl.30, 505–520 (1985)

    Google Scholar 

  11. Combes, J.M., Thomas, L.: Asymptotic behavior of eigenfunctions for multiparticle Schrödinger operators. Commun. Math. Phys.34, 251–270 (1973)

    Article  Google Scholar 

  12. Connes, A.: Noncommutative differential geometry. Pub. Math. IHES62, 257–360 (1986); Geometrie Non Commutative. Paris: Inter Edition 1990

    Google Scholar 

  13. Cuntz, J.: Representations of quantized differential forms in non-commutative geometry. In: Mathematical Physics X. Schmüdgen, K. (ed.), Berlin, Heidelberg, New York: Springer 1992

    Google Scholar 

  14. Cycon, H.L., Froese, R.G., Kirsch, W., Simon, B.: Schrödinger operators, Berlin, Heidelberg, New York: Springer 1987

    Google Scholar 

  15. Dubrovin, B.A., Novikov, S.P.: Ground state of a two-dimensional electron in a periodic magnetic field. Sov. Phys. JETP52, 511–516 (1980)

    Google Scholar 

  16. Efros, E.G.: Why the circle is connected. Math. Intelligencer11, 27–35 (1989)

    Google Scholar 

  17. Fedosov, B.V.: Direct proof of the formula for the index of an elliptic system in Euclidean space. Funct. Anal. Appl.4, 339–341 (1970)

    Article  Google Scholar 

  18. Fradkin, E.: Field theories of condensed matter systems. Reading, MA: Addison-Wesley 1991

    Google Scholar 

  19. Fröhlich, J., Kerler, T.: Universality in quantum Hall systems. Nucl. Phys. B354, 369 (1991); Fröhlich, J., Studer, U.: Gauge invariance in non-relativistic many body theory. In: Mathematical Physics X. Schmüdgen, K. (ed.), Berlin, Heidelberg, New York: Springer 1992

    Article  Google Scholar 

  20. Hörmander, L.: The Weyl calculus of Pseudo-Differential operators. Commun. Pure and Appl. Math.18, 501–517 (1965)

    Google Scholar 

  21. Kanamura, H., Aoki, H.: The physics of interacting electrons in disordered systems. London: Clarendon Press 1989

    Google Scholar 

  22. Kato, T.: Perturbation theory for linear operators, Berlin, Heidelberg, New York: Springer 1966

    Google Scholar 

  23. Kirillov, A.A., Gvishiani, A.D.: Theorems and problems in functional analysis, Berlin, Heidelberg, New York: Springer 1982

    Google Scholar 

  24. Klein, M., Seiler, R.: Power law corrections to the Kubo formula vanish in quantum Hall systems. Commun. Math. Phys.128, 141–160 (1990)

    Article  Google Scholar 

  25. Kohmoto, M.: Topological invariants and the quantization of the Hall conductance. Ann. Phys.160, 343–354 (1985)

    Article  Google Scholar 

  26. Kunz, H.: The quantum Hall effect for electrons in a random potential. Commun. Math. Phys.112, 121 (1987)

    Article  Google Scholar 

  27. Laughlin, R.G.: Elementary theory: The incompressible quantum fluid. In: The quantum Hall effect. Prange, R.E., Girvin, S.M. (eds.), Berlin, Heidelberg, New York: Springer 1987

    Google Scholar 

  28. Matsui, T.: The index of scattering operators of Dirac equations. Commun. Math. Phys.110, 553–571 (1987)

    Article  Google Scholar 

  29. Nakamura, S., Bellissard, J.: Low bands do not contribute to quantum Hall effect. Commun. Math. Phys.131, 283–305 (1990)

    Google Scholar 

  30. Niu, Q.: Towards a quantum pump of electron charge. Phys. Rev. Lett.64, 1812 (1990); Towards an electron load lock. In: Nanostructures of mesoscopic systems. Kirk, W.P., Reed, M.A. (eds.) New York: Academic Press 1991

    Article  Google Scholar 

  31. Niu, Q., Thouless, D.J.: Quantum Hall effect with realistic boundary conditions. Phys. Rev-B35, 2188 (1986)

    Article  Google Scholar 

  32. Niu, Q., Thouless, D.J., Wu, Y.S.: Quantum Hall conductance as a topological invariant. Phys. Rev. B31, 3372–3379 (1985)

    Article  Google Scholar 

  33. Prange, R.E., Girvin, S.M.: The quantum Hall effect. Berlin, Heidelberg, New York: Springer 1987

    Google Scholar 

  34. Russo, S.: The norm of theL p Fourier transform on unimodular groups. Trans. AMS,192, 293–305 (1974); On the Hausdorff-Young theorem for integral operators. Pac. J. Math.28, 1121–1131 (1976)

    Google Scholar 

  35. Seiler, R.: On the quantum Hall effect. In: Recent developments in quantum mechanics. Boutet de Monvel, A., et al. (eds.). Netherlands: Kluwer 1991

    Google Scholar 

  36. Shapere, A., Wilczek, F.: Geometric phases in physics. Singapore: World Scientific 1989

    Google Scholar 

  37. Simon, B.: Trace ideals and their applications. Cambridge: Cambridge Univ. Press 1979

    Google Scholar 

  38. Simon, B.: Schrödinger semigroups. Bull. AMS7, 447–526 (1982)

    Google Scholar 

  39. Stone, M. (ed.): Quantum Hall effect. Singapore: World Scientific 1992

    Google Scholar 

  40. Štreda, P.: Theory of quantized Hall conductivity in two dimensions. J. Phys. C15, L717 (1982)

    Google Scholar 

  41. Thouless, D.J., Kohmoto, M., Nightingale, P., den Nijs, M.: Quantum Hall conductance in a two dimensional periodic potential. Phys. Rev. Lett.49, 40 (1982)

    Google Scholar 

  42. Thouless, D.J.: Quantisation of particle transport. Phys. Rev. B27, 6083 (1983)

    Article  Google Scholar 

  43. Wen, X.: Vacuum degeneracy of chiral spin states in compactified space. Phys. Rev. B40, 7387–7390 (1989); Gapless boundary excitations in the quantum Hall states and in the chiral spin states. Phys. Rev. B43, 11025–11036 (1991)

    Article  Google Scholar 

  44. Wigner, E.P.: Göttinger Nachr.31, 546 (1932); Group Theory. New York: Academic Press 1959

    Google Scholar 

  45. Wilczek, F.: Fractional statistics and Anyon superconductivity. Singapore: World Scientific 1990

    Google Scholar 

  46. Xia, J.: Geometric invariants of the quantum Hall effect. Commun. Math. Phys.119, 29–50 (1988)

    Article  Google Scholar 

  47. Zak, J.: Magnetic translation group. Phys. Rev. A134, 1602–1607 (1964); Magnetic translation group II: Irreducible representations. Phys. Rev. A134, 1607–1611 (1964); In: Solid State Physics 27. Seitz, F., Turnbull, D., Ehrenreich, H. (eds.),59, New York: Academic Press 1972

    Article  Google Scholar 

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Authors and Affiliations

  1. Department of Physics, Technion, Israel Institute of Technology, 32000, Technion City, Haifa, Israel

    Joseph E. Avron

  2. Fachbereich Mathematik, Technische Universität Berlin, Strasse des 17. Juni 136, D-10623, Berlin, Germany

    Ruedi Seiler

  3. Division of Physics Mathematics and Astronomy, Caltech, 91125, Pasadena, CA, USA

    Barry Simon

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  1. Joseph E. Avron
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  2. Ruedi Seiler
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  3. Barry Simon
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Additional information

Communicated by T. Spencer

Dedicated to Elliott Lieb on the occasion of his 60th birthday

Part of this work was written while the authors enjoyed the hospitality of the Landau Center at the Hebrew University. The work is supported by BSF, DFG, GIF, NSF, the fund for the Promotion of Research at teh Technion and the Technion VPR-Steigman research fund. One of the authors (RS) should like to acknowledge the hospitality of the Mittag-Leffler Institute

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Avron, J.E., Seiler, R. & Simon, B. Charge deficiency, charge transport and comparison of dimensions. Commun.Math. Phys. 159, 399–422 (1994). https://doi.org/10.1007/BF02102644

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