Spectral theory of extended Harper’s model and a question by Erdős and Szekeres

Title:
Spectral theory of extended Harper’s model and a question by Erdős and Szekeres
Authors:
Avila, A.; Jitomirskaya, S.; Marx, Chris
Abstract:
The extended Harper’s model, proposed by D.J. Thouless in 1983, generalizes the famous almost Mathieu operator, allowing for a wider range of lattice geometries (parametrized by three coupling parameters) by permitting 2D electrons to hop to both nearest and next nearest neighboring (NNN) lattice sites, while still exhibiting its characteristic symmetry (Aubry–André duality). Previous understanding of the spectral theory of this model was restricted to two dual regions of the parameter space, one of which is characterized by the positivity of the Lyapunov exponent. In this paper, we complete the picture with a description of the spectral measures over the entire remaining (self-dual) region, for all irrational values of the frequency parameter (the magnetic flux in the model). Most notably, we prove that in the entire interior of this regime, the model exhibits a collapse from purely ac spectrum to purely sc spectrum when the NNN interaction becomes symmetric. In physics literature, extensive numerical analysis had indicated such “spectral collapse,” however so far not even a heuristic argument for this phenomenon could be provided. On the other hand, in the remaining part of the self-dual region, the spectral measures are singular continuous irrespective of such symmetry. The analysis requires some rather delicate number theoretic estimates, which ultimately depend on the solution of a problem posed by Erdős and Szekeres (On the product ∏nk=1(1−zak)∏k=1n(1−zak) , Publ. de l’Institut mathématique, Paris, 1950).
Citation:
Avila, A., S. Jitomirskaya, and C.A. Marx. 2017. "Spectral theory of extended Harper’s model and a question by Erdős and Szekeres." Inventiones Mathematicae 210(1): 283-339.
Publisher:
Springer Heidelberg
DATE ISSUED:
2017-10
Department:
Mathematics
Type:
Article
PUBLISHED VERSION:
10.1007/s00222-017-0729-1
Additional Links:
http://link.springer.com/10.1007/s00222-017-0729-1
PERMANENT LINK:
http://hdl.handle.net/11282/620536

Full metadata record

DC FieldValue Language
dc.contributor.authorAvila, A.en
dc.contributor.authorJitomirskaya, S.en
dc.contributor.authorMarx, Chrisen
dc.date.accessioned2017-10-10T14:44:56Z-
dc.date.available2017-10-10T14:44:56Z-
dc.date.issued2017-10-
dc.identifier.citationAvila, A., S. Jitomirskaya, and C.A. Marx. 2017. "Spectral theory of extended Harper’s model and a question by Erdős and Szekeres." Inventiones Mathematicae 210(1): 283-339.en
dc.identifier.issn0020-9910-
dc.identifier.urihttp://hdl.handle.net/11282/620536-
dc.description.abstractThe extended Harper’s model, proposed by D.J. Thouless in 1983, generalizes the famous almost Mathieu operator, allowing for a wider range of lattice geometries (parametrized by three coupling parameters) by permitting 2D electrons to hop to both nearest and next nearest neighboring (NNN) lattice sites, while still exhibiting its characteristic symmetry (Aubry–André duality). Previous understanding of the spectral theory of this model was restricted to two dual regions of the parameter space, one of which is characterized by the positivity of the Lyapunov exponent. In this paper, we complete the picture with a description of the spectral measures over the entire remaining (self-dual) region, for all irrational values of the frequency parameter (the magnetic flux in the model). Most notably, we prove that in the entire interior of this regime, the model exhibits a collapse from purely ac spectrum to purely sc spectrum when the NNN interaction becomes symmetric. In physics literature, extensive numerical analysis had indicated such “spectral collapse,” however so far not even a heuristic argument for this phenomenon could be provided. On the other hand, in the remaining part of the self-dual region, the spectral measures are singular continuous irrespective of such symmetry. The analysis requires some rather delicate number theoretic estimates, which ultimately depend on the solution of a problem posed by Erdős and Szekeres (On the product ∏nk=1(1−zak)∏k=1n(1−zak) , Publ. de l’Institut mathématique, Paris, 1950).en
dc.language.isoen_USen
dc.publisherSpringer Heidelbergen
dc.identifier.doi10.1007/s00222-017-0729-1-
dc.relation.urlhttp://link.springer.com/10.1007/s00222-017-0729-1en
dc.subject.departmentMathematicsen_US
dc.titleSpectral theory of extended Harper’s model and a question by Erdős and Szekeresen_US
dc.typeArticleen
dc.identifier.journalInventiones mathematicaeen
dc.subject.keywordSingular continuous-spectrumen_US
dc.subject.keywordPeriodic Schrodinger-operatorsen_US
dc.subject.keywordAbsolutely continuous-spectrumen_US
dc.subject.keywordPhase-diagramen_US
dc.subject.keywordMathieu operatoren_US
dc.identifier.volume210en_US
dc.identifier.issue1en_US
dc.identifier.startpage283en_US
dc.rightsArchived with thanks to Inventiones mathematicaeen
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