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We study topological indices of Fermionic time-reversal invariant topological insulators in two dimensions, in the regime of strong Anderson localization. We devise a method to interpolate between certain Fredholm operators arising in the context of these systems. We use this technique to prove the bulk-edge correspondence for mobility-gapped 2D topological insulators possessing a (Fermionic) time-reversal symmetry (class AII) and provide an alternative route to a theorem by Elgart-Graf-Schenker (Commun Math Phys 259(1):185–221, 2005) about the bulk-edge correspondence for strongly-disordered integer quantum Hall systems. We furthermore provide a proof of the stability of the Z2 index in the mobility gap regime. These two-dimensional results serve as a model for the study of higher dimensional Z2 indices.
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A “lifting” method for exponential large deviation estimates and an application to certain non-stationary 1D lattice Anderson models
Proofs of localization for random Schrödinger operators with sufficiently regular distribution of the potential can take advantage of the fractional moment method introduced by Aizenman–Molchanov [Commun. Math. Phys. 157(2), 245–278 (1993)] or use the classical Wegner estimate as part of another method, e.g., the multi-scale analysis introduced by Fröhlich–Spencer [Commun. Math. Phys. 88, 151–184 (1983)] and significantly developed by Klein and his collaborators. When the potential distribution is singular, most proofs rely crucially on exponential estimates of events corresponding to finite truncations of the operator in question; these estimates in some sense substitute for the classical Wegner estimate. We introduce a method to “lift” such estimates, which have been obtained for many stationary models, to certain closely related non-stationary models. As an application, we use this method to derive Anderson localization on the 1D lattice for certain non-stationary potentials along the lines of the non-perturbative approach developed by Jitomirskaya–Zhu [Commun. Math. Physics 370, 311–324 (2019)] in 2019.
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- PAR ID:
- 10509751
- Publisher / Repository:
- JMP
- Date Published:
- Journal Name:
- Journal of Mathematical Physics
- Volume:
- 64
- Issue:
- 6
- ISSN:
- 0022-2488
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1063/5.0150430
