Abstract
We study the charge transport of the noninteracting electron gas in a two-dimensional quantum Hall system with Anderson-type impurities at zero temperature. We prove that there exist localized states of the bulk order in the disordered-broadened Landau bands whose energies are smaller than a certain value determined by the strength of the uniform magnetic field. We also prove that, when the Fermi level lies in the localization regime, the Hall conductance is quantized to the desired integer and shows the plateau of the bulk order for varying the filling factor of the electrons rather than the Fermi level.
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Prange, R.E., Girvin, S.M. (Eds.): The Quantum Hall Effect, 2nd edn. Springer, Berlin (1990)
Thouless, D.J., Kohmoto, M., Nightingale, M.P., den Nijs, M.: Quantized Hall conductance in a two-dimensional periodic potential. Phys. Rev. Lett. 49, 405–408 (1982)
Kohmoto, M.: Topological invariant and the quantization of the Hall conductance. Ann. Phys. 160, 343–354 (1985)
Aoki, H., Ando, T.: Effect of localization on the Hall conductivity in the two-dimensional system in strong magnetic fields. Solid State Commun. 38, 1079–1082 (1981)
Avron, J.E., Seiler, R., Yaffe, L.G.: Adiabatic theorems and applications to the quantum Hall effect. Commun. Math. Phys. 110, 33–49 (1987)
Koma, T.: Insensitivity of quantized Hall conductance to disorder and interactions. J. Stat. Phys. 99, 383–459 (2000)
Elgart, A., Schlein, B.: Adiabatic charge transport and the Kubo formula for Landau type Hamiltonian. Commun. Pure Appl. Math. 57, 590–615 (2004)
Koma, T.: Revisiting the charge transport in quantum Hall systems. Rev. Math. Phys. 16, 1115–1189 (2004)
Bouclet, J.-M., Germinet, F., Klein, A., Schenker, J.H.: Linear response theory for magnetic Schrödinger operators in disordered media. J. Funct. Anal. 226, 301–372 (2005)
Avron, J.E., Seiler, R., Simon, B.: Quantum Hall effect and the relative index for projections. Phys. Rev. Lett. 65, 2185–2188 (1990)
Avron, J.E., Seiler, R., Simon, B.: Charge deficiency, charge transport and comparison of dimensions. Commun. Math. Phys. 159, 399–422 (1994)
Dorlas, T.C., Macris, N., Pulé, J.V.: Localisation in a single-band approximation to random Schroedinger operators in a magnetic field. Helv. Phys. Acta 68, 329–364 (1995)
Dorlas, T.C., Macris, N., Pulé, J.V.: Localization in single Landau bands. J. Math. Phys. 37, 1574–1595 (1996)
Dorlas, T.C., Macris, N., Pulé, J.V.: The nature of the spectrum for a Landau Hamiltonian with delta impurities. J. Stat. Phys. 87, 847–875 (1997)
Combes, J.M., Hislop, P.D.: Landau Hamiltonians with random potentials: localization and the density of states. Commun. Math. Phys. 117, 603–629 (1996)
Wang, W.-M.: Microlocalization, percolation, and Anderson localization for the magnetic Schrödinger operator with a random potential. J. Funct. Anal. 146, 1–26 (1997)
Germinet, F., Klein, A.: Explicit finite volume criteria for localization in continuous random media and applications. Geom. Funct. Anal. 13, 1201–1238 (2003)
Barbaroux, J.M., Combes, J.M., Hislop, P.D.: Localization near band edges for random Schrödinger operators. Helv. Phys. Acta 70, 16–43 (1997)
Kunz, H.: The quantum Hall effect for electrons in a random potential. Commun. Math. Phys. 112, 121–145 (1987)
Bellissard, J., Van Elst, A., Schulz-Baldes, H.: The noncommutative geometry of the quantum Hall effect. J. Math. Phys. 35, 5373–5451 (1994)
Aizenman, M., Graf, G.M.: Localization bounds for an electron gas. J. Phys. A 31, 6783–6806 (1998)
Richter, T., Schulz-Baldes, H.: Homotopy arguments for quantized Hall conductivity. J. Math. Phys. 42, 3439–3444 (2001)
Elgart, A., Graf, G.M., Schenker, J.H.: Equality of the bulk and edge Hall conductances in a mobility gap. Commun. Math. Phys. 259, 185–221 (2005)
Nakamura, S., Bellissard, J.: Low energy bands do not contribute to quantum Hall effect. Commun. Math. Phys. 131, 283–305 (1990)
Germinet, F., Klein, A., Schenker, J.H.: Dynamical delocalization in random Landau Hamiltonians. Preprint arXiv:math-ph/0412070
Wang, W.-M.: Asymptotic expansion for the density of states of the magnetic Schrödinger operator with a random potential. Commun. Math. Phys. 172, 401–425 (1995)
Reed, M., Simon, B.: Methods of Modern Mathematical Physics, vol. IV: Analysis of Operators. Academic Press, New York (1978)
Hislop, P.D., Müller, P.: A lower bound for the density of states of the lattice Anderson model. Preprint arXiv:math-ph/0705.1707
Fröhlich, J., Spencer, T.: Absence of diffusion in the Anderson tight binding model for large disorder or low energy. Commun. Math. Phys. 88, 151–184 (1983)
von Dreifus, H., Klein, A.: A new proof of localization in the Anderson tight binding model. Commun. Math. Phys. 124, 285–299 (1989)
Carmona, R., Lacroix, J.: Spectral theory of random Schrödinger operators. Birkhäuser, Boston (1990)
Aizenman, M., Elgart, A., Naboko, S., Schenker, J.H., Stolz, G.: Moment analysis for localization in random Schrödinger operators. Invent. Math. 163, 343–413 (2006)
Zak, J.: Magnetic translation group. Phys. Rev. A 134, 1602–1606 (1964)
Zak, J.: Magnetic translation group, II: irreducible representations. Phys. Rev. A 134, 1607–1611 (1964)
Kato, T.: Perturbation Theory for Linear Operators, 2nd edn. Springer, Berlin (1980)
Koma, T.: Spectral gaps of quantum Hall systems with interactions. J. Stat. Phys. 99, 313–381 (2000)
Kesten, H.: Percolation Theory for Mathematicians. Birkhäuser, Boston (1982)
Grimmett, G.: Percolation. Springer, New York (1989)
Wegner, F.: Bonds on the density of states for disordered systems. Z. Phys. 44, 9–15 (1981)
Combes, J.M., Thomas, L.: Asymptotic behaviour of eigenfunctions for multiparticle Schrödinger operators. Commun. Math. Phys. 34, 251–270 (1973)
Combes, J.M., Hislop, P.D.: Localization for some continuous, random Hamiltonians in d-dimensions. J. Funct. Anal. 124, 149–180 (1994)
Kirsch, W., Stollmann, P., Stolz, G.: Localization for random perturbations of periodic Schrödinger operators. Random Oper. Stoch. Equ. 6(3), 241–268 (1998)
de Branges, L.: Perturbations of self-adjoint transformations. Am. J. Math. 84, 543–560 (1962)
Kirillov, A.A., Gvishiani, A.D.: Theorems and Problems in Functional Analysis. Springer, Berlin (1982)
Connes, A.: Noncommutative Geometry. Academic Press, San Diego (1994)
Reed, M., Simon, B.: Methods of Modern Mathematical Physics, vol. I: Functional Analysis. Academic Press, New York (1972)
Kotani, S., Simon, B.: Localization in general one-dimensional random systems, II: continuum Schrödinger operators. Commun. Math. Phys. 112, 103–119 (1987)
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Koma, T. Widths of the Hall Conductance Plateaus. J Stat Phys 130, 843–934 (2008). https://doi.org/10.1007/s10955-007-9432-8
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DOI: https://doi.org/10.1007/s10955-007-9432-8