Hopf characterization of two-dimensional Floquet topological insulators

F Nur Ünal,André Eckardt,Robert-Jan Slager

doi:10.1103/physrevresearch.1.022003

F Nur Ünal, André Eckardt + Show 1 more

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https://doi.org/10.1103/physrevresearch.1.022003

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Abstract

We present a topological characterization of time-periodically driven two-band models in 2+1 dimensions as Hopf insulators. The intrinsic periodicity of the Floquet system with respect to both time and the underlying two-dimensional momentum space constitutes a map from a three dimensional torus to the Bloch sphere. As a result, we find that the driven system can be understood by appealing to a Hopf map that is directly constructed from the micromotion of the drive. Previously found winding numbers are shown to correspond to Hopf invariants, which are associated with linking numbers describing the topology of knots in three dimensions. Moreover, after being cast as a Hopf insulator, not only the Chern numbers, but also the winding numbers of the Floquet topological insulator become accessible in experiments as linking numbers. We exploit this description to propose a feasible scheme for measuring the complete set of their Floquet topological invariants in optical lattices.

Highlights

With the advent of topological insulators [1, 2], the past decades have witnessed a rekindling of interest in band theory
We have shown that two-dimensional Floquet topological insulators with two bands are characterized by two Hopf constructions rooted in the inherent periodicity of micromotion
The periodic table for Floquet states [14], predicts a Z×n classification, where n refers to the number of gaps, showing e.g. that there is a 0- and πgap topological invariant for the case of two bands. These numbers directly coincide with our framework and the Hopf characterization of the micromotion, highlighting the difference between (2 + 1)dimensional Floquet states and 3 dimensional static insulators, as reflected in the periodic tables themselves

Summary

Introduction

With the advent of topological insulators [1, 2], the past decades have witnessed a rekindling of interest in band theory. We present a topological characterization of time-periodically driven two-band models in 2 + 1 dimensions as Hopf insulators. We find that upon varying the period of the drive, in combination with a flattening procedure, the Hopf characterization can be experimentally invoked to deduce the full set of Floquet topological invariants of the two-dimensional quantum system.

Results

Conclusion