an:07175768
Zbl 1434.35138
Gomi, Kiyonori; Tauber, Cl??ment
Eigenvalue crossings in Floquet topological systems
EN
Lett. Math. Phys. 110, No. 3, 465-500 (2020).
00447229
2020
j
35Q41 55M25 81Q70 82B20 35P10
Floquet topological insulators; 3d winding number; eigenvalue crossings; local Chern numbers
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 \textit{F. Nathan} and \textit{M. S. Rudner} [``Topological singularities and the general classification of Floquet-Bloch systems'', New J. Phys. 17, No. 12, Article ID 125014, 23 p. (2015; \url{doi:10.1088/1367-2630/17/12/125014})]. 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.