Adiabatic theorems and applications to the quantum Hall effect. (English) Zbl 0626.58033
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 infinitely 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.
MSC:
| 58Z05 | Applications of global analysis to the sciences |
| 81S99 | General quantum mechanics and problems of quantization |
References:
| [1] | Avron, J.E., Seiler, R.: Quantisation of the Hall conductance for general multiparticle Schrödinger Hamiltonians. Phys. Rev. Lett.54, 259-262 (1985) · doi:10.1103/PhysRevLett.54.259 |
| [2] | Avron, Y., Seiler, R., Shapiro, B.: Generic properties of quantum Hall Hamiltonians for finite systems. Nucl. Phys. B265 [FS 15], 364-374 (1986) · doi:10.1016/0550-3213(86)90315-9 |
| [3] | Berry, M.V.: Quantal phase factors accompanying adiabatic changes. Proc. R. Soc. Lond. A392, 45-57 (1984) · Zbl 1113.81306 |
| [4] | Born, M., Fock, V.: Beweis des Adiabatensatzes. Z. Phys.51, 165-169 (1928) · JFM 54.0994.03 · doi:10.1007/BF01343193 |
| [5] | Friedrichs, K.: The mathematical theory of quantum theory of fields. New York: Interscience 1953 · Zbl 0052.44504 |
| [6] | Garrido, L.M.: Generalized adiabatic invariance. J. Math. Phys.5, 355-362 (1964) · doi:10.1063/1.1704127 |
| [7] | Kato, T.: Perturbation theory of linear operators. Berlin, Heidelberg, New York. Springer 1966 · Zbl 0148.12601 |
| [8] | Kato, T.: On the adiabatic theorem of quantum mechanics. J. Phys. Soc. J. Jpn.5, 435-439 (1950) · doi:10.1143/JPSJ.5.435 |
| [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). · doi:10.1103/PhysRevLett.45.494 |
| [10] | Krein, S.G.: Linear differential equations in Banach space. Transl. Math. Monog.27 (1972) · Zbl 0236.47035 |
| [11] | Landau, L., Lifshitz, I.M.: Quantum mechanics. Sec. (revised) ed. London: Pergamon 1965 · Zbl 0178.57901 |
| [12] | Laughlin, R.B.: Quantized hall conductivity in two dimensions. Phys. Rev. B23 (1981) 5632-5633 (1981) |
| [13] | Lenard, A.: Adiabatic invariants to all orders. Ann. Phys.6, 261-276 (1959) · Zbl 0084.44403 · doi:10.1016/0003-4916(59)90082-X |
| [14] | Milnor, J., Stasheff, J.D.: Characteristic classes. Princeton, NJ: Princeton University Press 1974 · Zbl 0298.57008 |
| [15] | Nenciu, G.: Adiabatic theorem and spectral concentration. Commun. Math. Phys.82, 121-135 (1981) · Zbl 0493.47009 · doi:10.1007/BF01206948 |
| [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) · Zbl 1188.81191 |
| [17] | Reed, M., Simon, B.: Methods of modern mathematical physics, Vol. II. Fourier analysis, self-adjointness. New York: Academic Press 1975 · Zbl 0308.47002 |
| [18] | Sancho, S.J.:m-th order adiabatic invariance for quantum systems. Proc. Phys. Soc. Lond.89, 1-5 (1966) · Zbl 0144.23501 · doi:10.1088/0370-1328/89/1/302 |
| [19] | Schering, G.: On the adiabatic theorem (in preparation) · JFM 17.0153.01 |
| [20] | Shapiro, B.: Finite size corrections in quantum Hall effect. Technion preprint |
| [21] | Simon, B.: Holonomy, the quantum adiabatic theorem and Berry’s phase Phys. Rev. Lett.51, 2167-2170 (1983) · doi:10.1103/PhysRevLett.51.2167 |
| [22] | Simon, B.: Hamiltonians defined as quadratic forms. Princeton, NJ: Princeton University Press 1971 · Zbl 0208.38804 |
| [23] | Tao, R., Haldane, F.D.M.: Impurity effect, degeneracy and topological invariant in the quantum Hall effect. Phys. Rev. B33, 3844-3855 (1986) |
| [24] | Thouless, D.J., Niu, Q.: Nonlinear corrections to the quantization of Hall conductance. Phys. Rev. B30, 3561-3562 (1984) |
| [25] | Wilczek, F., Zee, A.: Appearance of Gauge structure in simple dynamical systems. Phys. Rev. Lett.52, 2111-2114 (1984) · doi:10.1103/PhysRevLett.52.2111 |
| [26] | Yoshida, K.: Functional analysis. Grundlagen der Math. Wissenschaften, Bd.123. Berlin, Heidelberg, New York: Springer |
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