Topological Micromotion of Floquet Quantum Systems

TL;DR

This paper introduces a novel approach to describe Floquet quantum systems by incorporating the micro-motion parameter as an extra dimension, enabling accurate topological classification and understanding of edge states.

Contribution

It proposes representing a $d$-dimensional Floquet system as a $d+1$-dimensional static Hamiltonian using the micro-motion parameter as a synthetic dimension.

Findings

Floquet systems can be described by a $d+1$-dimensional Hamiltonian.

Topological invariants in the extended space predict edge states.

The scheme uses a three-dimensional Hopf invariant for classification.

Abstract

The Floquet Hamiltonian has often been used to describe a time-periodic system. Nevertheless, because the Floquet Hamiltonian depends on a micro-motion parameter, the Floquet Hamiltonian with a fixed micro-motion parameter cannot faithfully represent a driven system, which manifests as the anomalous edge states. Here we show that an accurate description of a Floquet system requires a set of Hamiltonian exhausting all values of the micro-motion parameter, and this micro-motion parameter can be viewed as an extra synthetic dimension of the system. Therefore, we show that a $d$-dimensional Floquet system can be described by a $d+1$-dimensional static Hamiltonian, and the advantage of this representation is that the periodic boundary condition is automatically imposed along the extra-dimension, which enables a straightforward definition of topological invariants. The topological invariant in the $d+1$-dimensional system can ensure a $d-1$-dimensional edge state of the $d$-dimensional Floquet system. Here we show two examples where the topological invariant is a three-dimensional Hopf invariant. We highlight that our scheme of classifying Floquet topology on the micro-motion space is different from the previous classification of Floquet topology on the time space.