Abstract
Topological properties of physical systems can lead to robust behaviors that are insensitive to microscopic details. Such topologically robust phenomena are not limited to static systems but can also appear in driven quantum systems. In this paper, we show that the Floquet operators of periodically driven systems can be divided into topologically distinct (homotopy) classes and give a simple physical interpretation of this classification in terms of the spectra of Floquet operators. Using this picture, we provide an intuitive understanding of the well-known phenomenon of quantized adiabatic pumping. Systems whose Floquet operators belong to the trivial class simulate the dynamics generated by time-independent Hamiltonians, which can be topologically classified according to the schemes developed for static systems. We demonstrate these principles through an example of a periodically driven two-dimensional hexagonal lattice tight-binding model which exhibits several topological phases. Remarkably, one of these phases supports chiral edge modes even though the bulk is topologically trivial.
Highlights
Following the discovery of the quantized Hall effect in 19801, there has been great excitement about the possibility of observing extremely robust, “topologically protected” quantum phenomena in solid state systems[2,3]
We demonstrate the general relation between the topology captured by homotopy groups of Floquet operators and Chern numbers in Appendix C
The phenomena associated with non-trivial topology of Floquet operators under the homotopy groups are unique to periodically-driven systems
Summary
Following the discovery of the quantized Hall effect in 19801, there has been great excitement about the possibility of observing extremely robust, “topologically protected” quantum phenomena in solid state systems[2,3]. Throughout the main text, we will focus primarily on systems with discrete (lattice) translational symmetry in d-dimensions, which are subjected to spatially homogenous, periodic, time-dependent driving In this case, we obtain simple expressions for topological invariants associated with the first and third homotopy groups, written in terms of Floquet operators parameterized by the conserved crystal momentum k. When the homotopy group classification returns a trivial result, i.e. for systems without quasi-energy winding, the Floquet operator can be expressed in terms of a local effective Hamiltonian Heff through U (T ) = e−iHeff T. The characterization of periodically driven systems in terms of the topological structure of Floquet operators constitutes the major result of this paper This approach provides a natural description of topologically-quantized pumping, and reveals a simple and intuitive picture in which to understand this phenomenon.
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