Advertisement

Communications in Mathematical Physics

, Volume 110, Issue 1, pp 33–49 | Cite as

Adiabatic theorems and applications to the quantum hall effect

  • J. E. Avron
  • R. Seiler
  • L. G. Yaffe
Article

Abstract

We study an adiabatic evolution that approximates the physical dynamics and describes a natural parallel transport in spectral subspaces. Using this we prove two folk theorems about the adiabatic limit of quantum mechanics: 1. For slow time variation of the Hamiltonian, the time evolution reduces to spectral subspaces bordered by gaps. 2. The eventual tunneling out of such spectral subspaces is smaller than any inverse power of the time scale if the Hamiltonian varies infinitly smoothly over a finite interval. Except for the existence of gaps, no assumptions are made on the nature of the spectrum. We apply these results to charge transport in quantum Hall Hamiltonians and prove that the flux averaged charge transport is an integer in the adiabatic limit.

Keywords

Neural Network Quantum Mechanic Quantum Computing Folk Hall Effect 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Avron, J.E., Seiler, R.: Quantisation of the Hall conductance for general multiparticle Schrödinger Hamiltonians. Phys. Rev. Lett.54, 259–262 (1985)CrossRefGoogle Scholar
  2. 2.
    Avron, Y., Seiler, R., Shapiro, B.: Generic properties of quantum Hall Hamiltonians for finite systems. Nucl. Phys. B265 [FS 15], 364–374 (1986)CrossRefGoogle Scholar
  3. 3.
    Berry, M.V.: Quantal phase factors accompanying adiabatic changes. Proc. R. Soc. Lond. A392, 45–57 (1984)Google Scholar
  4. 4.
    Born, M., Fock, V.: Beweis des Adiabatensatzes. Z. Phys.51, 165–169 (1928)Google Scholar
  5. 5.
    Friedrichs, K.: The mathematical theory of quantum theory of fields. New York: Interscience 1953Google Scholar
  6. 6.
    Garrido, L.M.: Generalized adiabatic invariance. J. Math. Phys.5, 355–362 (1964)CrossRefGoogle Scholar
  7. 7.
    Kato, T.: Perturbation theory of linear operators. Berlin, Heidelberg, New York. Springer 1966Google Scholar
  8. 8.
    Kato, T.: On the adiabatic theorem of quantum mechanics. J. Phys. Soc. J. Jpn.5, 435–439 (1950)Google Scholar
  9. 9.
    von-Klitzing, K., Dorda, G., Pepper, M.: New method for high accuracy determination of the fine structure constant based on the quantized Hall effect. Phys. Rev. Lett.45, 494–497 (1980).CrossRefGoogle Scholar
  10. 10.
    Krein, S.G.: Linear differential equations in Banach space. Transl. Math. Monog.27 (1972)Google Scholar
  11. 11.
    Landau, L., Lifshitz, I.M.: Quantum mechanics. Sec. (revised) ed. London: Pergamon 1965Google Scholar
  12. 12.
    Laughlin, R.B.: Quantized hall conductivity in two dimensions. Phys. Rev. B23 (1981) 5632–5633 (1981)Google Scholar
  13. 13.
    Lenard, A.: Adiabatic invariants to all orders. Ann. Phys.6, 261–276 (1959)CrossRefGoogle Scholar
  14. 14.
    Milnor, J., Stasheff, J.D.: Characteristic classes. Princeton, NJ: Princeton University Press 1974Google Scholar
  15. 15.
    Nenciu, G.: Adiabatic theorem and spectral concentration. Commun. Math. Phys.82, 121–135 (1981)CrossRefGoogle Scholar
  16. 16.
    Niu, Q., Thouless, D.J.: Quantised adiabatic charge transport in the presence of substrate disorder and many body interactions. J. Phys. A17, 30–49 (1984)Google Scholar
  17. 17.
    Reed, M., Simon, B.: Methods of modern mathematical physics, Vol. II. Fourier analysis, self-adjointness. New York: Academic Press 1975Google Scholar
  18. 18.
    Sancho, S.J.:m-th order adiabatic invariance for quantum systems. Proc. Phys. Soc. Lond.89, 1–5 (1966)Google Scholar
  19. 19.
    Schering, G.: On the adiabatic theorem (in preparation)Google Scholar
  20. 20.
    Shapiro, B.: Finite size corrections in quantum Hall effect. Technion preprintGoogle Scholar
  21. 21.
    Simon, B.: Holonomy, the quantum adiabatic theorem and Berry's phase Phys. Rev. Lett.51, 2167–2170 (1983)Google Scholar
  22. 22.
    Simon, B.: Hamiltonians defined as quadratic forms. Princeton, NJ: Princeton University Press 1971Google Scholar
  23. 23.
    Tao, R., Haldane, F.D.M.: Impurity effect, degeneracy and topological invariant in the quantum Hall effect. Phys. Rev. B33, 3844–3855 (1986)Google Scholar
  24. 24.
    Thouless, D.J., Niu, Q.: Nonlinear corrections to the quantization of Hall conductance. Phys. Rev. B30, 3561–3562 (1984)Google Scholar
  25. 25.
    Wilczek, F., Zee, A.: Appearance of Gauge structure in simple dynamical systems. Phys. Rev. Lett.52, 2111–2114 (1984)Google Scholar
  26. 26.
    Yoshida, K.: Functional analysis. Grundlagen der Math. Wissenschaften, Bd.123. Berlin, Heidelberg, New York: SpringerGoogle Scholar

Copyright information

© Springer-Verlag 1987

Authors and Affiliations

  • J. E. Avron
    • 1
  • R. Seiler
    • 2
  • L. G. Yaffe
    • 3
  1. 1.Department of Physics, TechnionHaifaIsrael
  2. 2.Fachbereich MathematikTechnische UniversitätBerlin 12
  3. 3.Princeton UniversityPrincetonUSA

Personalised recommendations