Topology and Localization: Mathematical Aspects of Electrons in StronglyDisordered Media
dc.contributor.author
Shapiro, Jacob
dc.contributor.supervisor
Graf, Gian Michele
dc.contributor.supervisor
Müller, P.
dc.contributor.supervisor
SchulzBaldes, Hermann
dc.date.accessioned
20181102T09:50:02Z
dc.date.available
20181101T16:00:34Z
dc.date.available
20181101T16:13:26Z
dc.date.available
20181101T16:23:57Z
dc.date.available
20181102T08:12:13Z
dc.date.available
20181102T09:50:02Z
dc.date.issued
2018
dc.identifier.uri
http://hdl.handle.net/20.500.11850/300657
dc.identifier.doi
10.3929/ethzb000300657
dc.description.abstract
Topological insulators are usually studied in physics under the assumption of translation invariance, which allows for the usage of Bloch decomposition. Mathematically vector bundle theory over the Brillouin zone is employed. Yet more sophisticated (and already deep within the realms of mathematical physics) are the methods of noncommutative geometry, championed by Bellissard and coworkers (see Bellissard 1994), which dispose of translation invariance. Instead of classifying vector bundles over the Brillouin zone, they classify (Fermi) projections in a noncommutative Cstar algebra, which is called the noncommutative Brillouin zone.
Simply put, this dissertation is about what happens when the Fermi projection does not belong to the noncommutative Cstar algebra which is the Brillouin torus. This is called the mobility gap regime, which happens when there is either no spectral gap at all (strongdisorder) or there is one but the Fermi energy is placed within the localized part of the spectrum. Physically this is the most interesting case (it is the only way to explain the plateaus in the integer quantum Hall effect (IQHE) for example), and mathematically it is the most difficult. Indeed, usually to classify projections in a Cstar algebra, one uses (algebraic) Ktheory. However, the Ktheory groups of the Wstar algebras which contain the Fermi projections (as they are defined via the Borel bounded calculus, they belong to the enveloping Wstar algebra of a given Hamiltonian) are trivial. This problem is avoided in the spectral gap regime since then the Fermi projection may be deformed into a continuous function of the Hamiltonian, whence it belongs to the Cstar algebra generated by a given Hamiltonian, and Ktheory applies.
The mobility gap regime has been studied previously, first in the context of the IQHE by the Bellissard school. They defined a noncommutative Sobolev space (Bellissard 1994) which is smaller than the aforementioned Wstar algebra. Elgart, Graf and Schenker in 2005 provided the first bulkedge correspondence proof in the mobility gap regime of the IQHE, and also provided a definition for the edge index in this regime. Prodan et al in 2016 studied the mobility gap regime of chiral systems in all dimensions, although always within the probabilistic ergodic framework.
This work is another step forward in studying the mobility gap regime, this time of chiral onedimensional systems and Floquet twodimensional systems. We also study localization for these chiral onedimensional systems, which is crucial to have a welldefined topology. Beyond the technical achievements, the main message of the present work is the connection between topology and localization, which is established in a quantitative way, and not just in terms of one being a prerequisite for the other.
This dissertation has four chapters as well as a technical appendix. In the introductory chapter we give a brief overview of the field of stronglydisordered topological insulators from our mathematical physics perspective. This is a description of the problem as an algebraic topology of disordered insulating singleparticle Hamiltonians. Our approach is deterministic (it doesn't use ergodicity) though its assumptions are modeled after almostsure statements one can make about ergodic random operators which exhibit localization.
In the next two chapters, based on (GrafShapiro 2018) we study the chiral onedimensional case of the Kitaev table. We prove its complete dynamical localization at all nonzero energies. We connect its topological invariants to localization (via the zero energy Lyapunov exponents), and prove a bulkedge duality for this system in the stronglydisordered, mobility gap regime.
In the last chapter, which is based on (ShapiroTauber 2018), we turn to study Floquet systems with no spectral gap but with a mobility gap, prove the bulkedge duality for systems in this regime, and formulate a new definition for the topological invariant, which is shown to coincide with the old one, but is perhaps easier to understand or calculate than previous definitions in the literature.
en_US
dc.format
application/pdf
en_US
dc.language.iso
en
en_US
dc.publisher
ETH Zurich
en_US
dc.rights.uri
http://rightsstatements.org/page/InCNC/1.0/
dc.subject
Mathematical physics
en_US
dc.subject
Quantum mechanics
en_US
dc.subject
Topological insulators
en_US
dc.subject
RANDOM MEDIA (PROBABILITY THEORY)
en_US
dc.subject
functional analysis
en_US
dc.title
Topology and Localization: Mathematical Aspects of Electrons in StronglyDisordered Media
en_US
dc.type
Doctoral Thesis
dc.rights.license
In Copyright  NonCommercial Use Permitted
ethz.size
164 p.
en_US
ethz.code.ddc
DDC  DDC::5  Science::530  Physics
ethz.identifier.diss
25455
en_US
ethz.publication.place
Zurich
en_US
ethz.publication.status
published
en_US
ethz.leitzahl
ETH Zürich::00002  ETH Zürich::00012  Lehre und Forschung::00007  Departemente::02010  Dep. Physik / Dep. of Physics::02511  Institut für Theoretische Physik / Institute for Theoretical Physics::03355  Graf, Gian Michele / Graf, Gian Michele
en_US
ethz.leitzahl.certified
ETH Zürich::00002  ETH Zürich::00012  Lehre und Forschung::00007  Departemente::02010  Dep. Physik / Dep. of Physics::02511  Institut für Theoretische Physik / Institute for Theoretical Physics::03355  Graf, Gian Michele / Graf, Gian Michele
en_US
ethz.relation.cites
20.500.11850/295712
ethz.relation.cites
handle/20.500.11850/317116
ethz.date.deposited
20181101T16:00:50Z
ethz.source
FORM
ethz.eth
yes
en_US
ethz.availability
Open access
en_US
ethz.rosetta.installDate
20181102T09:51:05Z
ethz.rosetta.lastUpdated
20210215T02:23:17Z
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true
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