## Topology and Localization: Mathematical Aspects of Electrons in Strongly-Disordered Media

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Author

Date

2018Type

- Doctoral Thesis

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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 non-commutative 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 non-commutative C-star algebra, which is called the non-commutative Brillouin zone.
Simply put, this dissertation is about what happens when the Fermi projection does not belong to the non-commutative C-star algebra which is the Brillouin torus. This is called the mobility gap regime, which happens when there is either no spectral gap at all (strong-disorder) 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 C-star algebra, one uses (algebraic) K-theory. However, the K-theory groups of the W-star algebras which contain the Fermi projections (as they are defined via the Borel bounded calculus, they belong to the enveloping W-star 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 C-star algebra generated by a given Hamiltonian, and K-theory applies.
The mobility gap regime has been studied previously, first in the context of the IQHE by the Bellissard school. They defined a non-commutative Sobolev space (Bellissard 1994) which is smaller than the aforementioned W-star algebra. Elgart, Graf and Schenker in 2005 provided the first bulk-edge 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 one-dimensional systems and Floquet two-dimensional systems. We also study localization for these chiral one-dimensional systems, which is crucial to have a well-defined 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 strongly-disordered topological insulators from our mathematical physics perspective. This is a description of the problem as an algebraic topology of disordered insulating single-particle Hamiltonians. Our approach is deterministic (it doesn't use ergodicity) though its assumptions are modeled after almost-sure statements one can make about ergodic random operators which exhibit localization.
In the next two chapters, based on (Graf-Shapiro 2018) we study the chiral one-dimensional case of the Kitaev table. We prove its complete dynamical localization at all non-zero energies. We connect its topological invariants to localization (via the zero energy Lyapunov exponents), and prove a bulk-edge duality for this system in the strongly-disordered, mobility gap regime.
In the last chapter, which is based on (Shapiro-Tauber 2018), we turn to study Floquet systems with no spectral gap but with a mobility gap, prove the bulk-edge 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. Show more

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https://doi.org/10.3929/ethz-b-000300657Publication status

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Publisher

ETH ZurichSubject

Mathematical physics; Quantum mechanics; Topological insulators; RANDOM MEDIA (PROBABILITY THEORY); functional analysisOrganisational unit

03355 - Graf, Gian Michele / Graf, Gian Michele
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