Universität zu Köln

Klevtsov, Semyon, Ma, Xiaonan ORCID: 0000-0001-8960-7623, Marinescu, George ORCID: 0000-0001-6539-7860 and Wiegmann, Paul (2017). Quantum Hall Effect and Quillen Metric. Commun. Math. Phys., 349 (3). S. 819 - 856. NEW YORK: SPRINGER. ISSN 1432-0916

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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-Soul, 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.

Item Type:	Journal Article
Creators:	Creators Email ORCID ORCID Put Code Klevtsov, Semyon UNSPECIFIED UNSPECIFIED UNSPECIFIED Ma, Xiaonan UNSPECIFIED orcid.org/0000-0001-8960-7623 UNSPECIFIED Marinescu, George UNSPECIFIED orcid.org/0000-0001-6539-7860 UNSPECIFIED Wiegmann, Paul UNSPECIFIED UNSPECIFIED UNSPECIFIED
URN:	urn:nbn:de:hbz:38-241092
DOI:	10.1007/s00220-016-2789-2
Journal or Publication Title:	Commun. Math. Phys.
Volume:	349
Number:	3
Page Range:	S. 819 - 856
Date:	2017
Publisher:	SPRINGER
Place of Publication:	NEW YORK
ISSN:	1432-0916
Language:	English
Faculty:	Faculty of Mathematics and Natural Sciences
Divisions:	Faculty of Mathematics and Natural Sciences > Department of Mathematics and Computer Science > Mathematical Institute
Subjects:	no entry
Uncontrolled Keywords:	Keywords Language HOLOMORPHIC DETERMINANT BUNDLES; ANALYTIC-TORSION; LANDAU HAMILTONIANS; INDEX THEOREM; QUANTIZATION; CONDUCTANCE; INVARIANCE; FAMILIES; GEOMETRY; LAPLACIANS Multiple languages Physics, Mathematical Multiple languages
Refereed:	Yes
URI:	http://kups.ub.uni-koeln.de/id/eprint/24109