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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Avron, J.E., Seiler, R.: Quantisation of the Hall conductance for general multiparticle Schrödinger Hamiltonians. Phys. Rev. Lett.54, 259–262 (1985)
Avron, Y., Seiler, R., Shapiro, B.: Generic properties of quantum Hall Hamiltonians for finite systems. Nucl. Phys. B265 [FS 15], 364–374 (1986)
Berry, M.V.: Quantal phase factors accompanying adiabatic changes. Proc. R. Soc. Lond. A392, 45–57 (1984)
Born, M., Fock, V.: Beweis des Adiabatensatzes. Z. Phys.51, 165–169 (1928)
Friedrichs, K.: The mathematical theory of quantum theory of fields. New York: Interscience 1953
Garrido, L.M.: Generalized adiabatic invariance. J. Math. Phys.5, 355–362 (1964)
Kato, T.: Perturbation theory of linear operators. Berlin, Heidelberg, New York. Springer 1966
Kato, T.: On the adiabatic theorem of quantum mechanics. J. Phys. Soc. J. Jpn.5, 435–439 (1950)
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).
Krein, S.G.: Linear differential equations in Banach space. Transl. Math. Monog.27 (1972)
Landau, L., Lifshitz, I.M.: Quantum mechanics. Sec. (revised) ed. London: Pergamon 1965
Laughlin, R.B.: Quantized hall conductivity in two dimensions. Phys. Rev. B23 (1981) 5632–5633 (1981)
Lenard, A.: Adiabatic invariants to all orders. Ann. Phys.6, 261–276 (1959)
Milnor, J., Stasheff, J.D.: Characteristic classes. Princeton, NJ: Princeton University Press 1974
Nenciu, G.: Adiabatic theorem and spectral concentration. Commun. Math. Phys.82, 121–135 (1981)
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)
Reed, M., Simon, B.: Methods of modern mathematical physics, Vol. II. Fourier analysis, self-adjointness. New York: Academic Press 1975
Sancho, S.J.:m-th order adiabatic invariance for quantum systems. Proc. Phys. Soc. Lond.89, 1–5 (1966)
Schering, G.: On the adiabatic theorem (in preparation)
Shapiro, B.: Finite size corrections in quantum Hall effect. Technion preprint
Simon, B.: Holonomy, the quantum adiabatic theorem and Berry's phase Phys. Rev. Lett.51, 2167–2170 (1983)
Simon, B.: Hamiltonians defined as quadratic forms. Princeton, NJ: Princeton University Press 1971
Tao, R., Haldane, F.D.M.: Impurity effect, degeneracy and topological invariant in the quantum Hall effect. Phys. Rev. B33, 3844–3855 (1986)
Thouless, D.J., Niu, Q.: Nonlinear corrections to the quantization of Hall conductance. Phys. Rev. B30, 3561–3562 (1984)
Wilczek, F., Zee, A.: Appearance of Gauge structure in simple dynamical systems. Phys. Rev. Lett.52, 2111–2114 (1984)
Yoshida, K.: Functional analysis. Grundlagen der Math. Wissenschaften, Bd.123. Berlin, Heidelberg, New York: Springer
Author information
Authors and Affiliations
Additional information
Communicated by B. Simon
Rights and permissions
About this article
Cite this article
Avron, J.E., Seiler, R. & Yaffe, L.G. Adiabatic theorems and applications to the quantum hall effect. Commun.Math. Phys. 110, 33–49 (1987). https://doi.org/10.1007/BF01209015
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01209015