Homotopy classification of Floquet band singularities

The spectrum of periodically driven (i.e. “Floquet”) crystalline systems is characterized by energy bands in momentum space. Due to the discrete nature of their time-translation symmetry, Floquet systems conserve energy only modulo the angular frequency of the drive, resulting in the well-known compactification of the real energy axis into a looping “quasi-energy”. The resulting quasi-energy bands can exhibit topologically non-trivial gaps, robust band singularities, and anomalous boundary modes -- including ones that are impossible in static systems.

In this work [1], we derive a classification of singularities of quasi-energy bands. Besides the “band node” singularity known from the static case, we find that Floquet systems in the adiabatic limit further exhibit a novel type of singularity, which we dub “band screw”. We topologically characterize both species of singularities using homotopy theory. This task requires a generalization of the methods used by Ref. [2] in the static case, in particular by considering relative homotopy groups and multi-gap classifying spaces [3]. In this talk, our classification results would be presented.

[1] (in preparation, 2021)
[2] T. Bzdušek and M. Sigrist, PRB 96, 155105 (2017)
[3] A. Bouhon, T. Bzdušek, and R.-J. Slager, PRB 102, 115135 (2020)