Characterization of Bulk States in One-Edge Quantum Hall Systems

Peter D. Hislop; Nicolas Popoff; Eric Soccorsi

doi:10.1007/s00023-014-0388-3

Characterization of Bulk States in One-Edge Quantum Hall Systems

Peter D. Hislop, Nicolas Popoff, Eric Soccorsi

Mathematics

Research output: Contribution to journal › Article › peer-review

6 Scopus citations

Abstract

We study magnetic quantum Hall systems in a half-plane with Dirichlet boundary conditions along the edge. Much work has been done on the analysis of the currents associated with states whose energy is located between Landau levels. These edge states carry a non-zero current that remains well-localized in a neighborhood of the boundary. In this article, we study the behavior of states with energies close to a Landau level. Such states are referred to as bulk states in the physics literature. Since the magnetic Schrödinger operator is invariant with respect to translations along the edge, it is a direct integral of operators indexed by a real wave number. We analyse these fiber operators and prove new asymptotics on the band functions and their first derivative as the wave number goes to infinity. We apply these results to prove that the current carried by a bulk state is small compared to the current carried by an edge state. We also prove that the bulk states are small near the edge.

Original language	English
Pages (from-to)	37-62
Number of pages	26
Journal	Annales Henri Poincare
Volume	17
Issue number	1
DOIs	https://doi.org/10.1007/s00023-014-0388-3
State	Published - Jan 1 2016

Bibliographical note

Publisher Copyright:
© 2014, Springer Basel.

Funding

N. Popoff is financially supported by the ARCHIMEDE Labex (ANR-11- LABX- 0033) and the A*MIDEX project (ANR-11-IDEX-0001-02) funded by the \u201CInvestissements d\u2019Avenir\u201D French government program managed by the ANR. He also thanks the University of Kentucky for its invitation in February 2014 to Lexington where this work has been finalized. P.D. Hislop thanks the Centre de Physique Theorique, CNRS, Luminy, Marseille, France, for its hospitality. P.D. Hislop was partially supported by the Universite de Toulon, La Garde, France, and National Science Foundation grant 11-03104 during the time part of the work was done.

Funders	Funder number
University of Kentucky
CNRS Centre National de la Recherche Scientifique
Universite de Toulon
Centre de Physique Theorique de Marseille
Not added	ANR-11-IDEX-0001
A*MIDEX	ANR-11-IDEX-0001-02
U.S. Department of Energy Chinese Academy of Sciences Guangzhou Municipal Science and Technology Project Oak Ridge National Laboratory Extreme Science and Engineering Discovery Environment National Science Foundation National Energy Research Scientific Computing Center National Natural Science Foundation of China	11-03104
ARCHIMEDE Labex	ANR-11- LABX- 0033

ASJC Scopus subject areas

Statistical and Nonlinear Physics
Nuclear and High Energy Physics
Mathematical Physics

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10.1007/s00023-014-0388-3

Cite this

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abstract = "We study magnetic quantum Hall systems in a half-plane with Dirichlet boundary conditions along the edge. Much work has been done on the analysis of the currents associated with states whose energy is located between Landau levels. These edge states carry a non-zero current that remains well-localized in a neighborhood of the boundary. In this article, we study the behavior of states with energies close to a Landau level. Such states are referred to as bulk states in the physics literature. Since the magnetic Schr{\"o}dinger operator is invariant with respect to translations along the edge, it is a direct integral of operators indexed by a real wave number. We analyse these fiber operators and prove new asymptotics on the band functions and their first derivative as the wave number goes to infinity. We apply these results to prove that the current carried by a bulk state is small compared to the current carried by an edge state. We also prove that the bulk states are small near the edge.",

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N2 - We study magnetic quantum Hall systems in a half-plane with Dirichlet boundary conditions along the edge. Much work has been done on the analysis of the currents associated with states whose energy is located between Landau levels. These edge states carry a non-zero current that remains well-localized in a neighborhood of the boundary. In this article, we study the behavior of states with energies close to a Landau level. Such states are referred to as bulk states in the physics literature. Since the magnetic Schrödinger operator is invariant with respect to translations along the edge, it is a direct integral of operators indexed by a real wave number. We analyse these fiber operators and prove new asymptotics on the band functions and their first derivative as the wave number goes to infinity. We apply these results to prove that the current carried by a bulk state is small compared to the current carried by an edge state. We also prove that the bulk states are small near the edge.

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Characterization of Bulk States in One-Edge Quantum Hall Systems

Abstract

Bibliographical note

Funding

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