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Quantum Hall Effect and Quillen Metric

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Abstract

We study the generating functional, the adiabatic curvature and the adiabatic phase for the integer quantum Hall effect (QHE) on a compact Riemann surface. For the generating functional we derive its asymptotic expansion for the large flux of the magnetic field, i.e., for the large degree k of the positive Hermitian line bundle L k. The expansion consists of the anomalous and exact terms. The anomalous terms are the leading terms of the expansion. This part is responsible for the quantization of the adiabatic transport coefficients in QHE. We then identify the non-local (anomalous) part of the expansion with the Quillen metric on the determinant line bundle, and the subleading exact part with the asymptotics of the regularized spectral determinant of the Laplacian for the line bundle L k, at large k. Finally, we show how the generating functional of the integer QHE is related to the gauge and gravitational (2+1)d Chern–Simons functionals. We observe the relation between the Bismut-Gillet-Soulé curvature formula for the Quillen metric and the adiabatic curvature for the electromagnetic and geometric adiabatic transport of the integer Quantum Hall state. We then obtain the geometric part of the adiabatic phase in QHE, given by the Chern–Simons functional.

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Authors and Affiliations

  1. Mathematisches Institut, Universität zu Köln, Weyertal 86-90, 50931, Cologne, Germany

    Semyon Klevtsov & George Marinescu

  2. Université Paris Diderot-Paris 7, UFR de Mathématiques, Case 7012, 75205, Paris Cedex 13, France

    Xiaonan Ma

  3. Institute of Mathematics ‘Simion Stoilow’, Romanian Academy, Bucharest, Romania

    George Marinescu

  4. Department of Physics, University of Chicago, 929 57th St, Chicago, IL, 60637, USA

    Paul Wiegmann

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  1. Semyon Klevtsov
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  2. Xiaonan Ma
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Correspondence to George Marinescu.

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Communicated by S. Zelditch

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Klevtsov, S., Ma, X., Marinescu, G. et al. Quantum Hall Effect and Quillen Metric. Commun. Math. Phys. 349, 819–855 (2017). https://doi.org/10.1007/s00220-016-2789-2

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