Eigenvalue crossings in Floquet topological systems

The topology of electrons on a lattice subject to a periodic driving is captured by the three-dimensional winding number of the propagator that describes time evolution within a cycle. This index captures the homotopy class of such a unitary map. In this paper, we provide an interpretation of this w...

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Bibliographic Details
Published in:Letters in mathematical physics 2020-03, Vol.110 (3), p.465-500
Main Authors: Gomi, Kiyonori, Tauber, Clément
Format: Article
Language:English
Subjects:
Complex Systems
Condensed Matter
Eigenvalues
Geometry
Group Theory and Generalizations
Manifolds (mathematics)
Mathematical and Computational Physics
Mathematical Physics
Mesoscopic Systems and Quantum Hall Effect
Physics
Physics and Astronomy
Theoretical
Topology
Winding
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Summary:The topology of electrons on a lattice subject to a periodic driving is captured by the three-dimensional winding number of the propagator that describes time evolution within a cycle. This index captures the homotopy class of such a unitary map. In this paper, we provide an interpretation of this winding number in terms of local data associated with the eigenvalue crossings of such a map over a three-dimensional manifold, based on an idea from Nathan and Runder (New J Phys 17(12):125014, 2015). We show that, up to homotopy, the crossings are a finite set of points and non-degenerate. Each crossing carries a local Chern number, and the sum of these local indices coincides with the winding number. We then extend this result to fully degenerate crossings and extended submanifolds to connect with models from the physics literature. We finally classify up to homotopy the Floquet unitary maps, defined on manifolds with boundary, using the previous local indices. The results rely on a filtration of the special unitary group as well as the local data of the basic gerbe over it.
ISSN:0377-9017
1573-0530
DOI:10.1007/s11005-019-01225-4