Chiral Floquet systems and quantum walks at half period

TL;DR

This paper classifies one-dimensional chiral symmetric Floquet systems using half-period operators, establishing a complete framework for understanding their topological properties and edge states.

Contribution

It introduces a complete classification of half-step operators in chiral Floquet systems via five integer indices, linking them to quantum walks and topological invariants.

Findings

Complete classification of half-step operators using five indices

Bulk-edge correspondence established for these systems

Second timeframe distinguishes symmetry-protected edge states

Abstract

We classify chiral symmetric periodically driven quantum systems on a one-dimensional lattice. The driving process is local, can be continuous or discrete in time, and we assume a gap condition for the corresponding Floquet operator. The analysis is in terms of the unitary operator at a half-period, the half-step operator. We give a complete classification of the connected classes of half-step operators in terms of five integer indices. On the basis of these indices it can be decided whether the half-step operator can be obtained from a continuous Hamiltonian driving, or not. The half-step operator determines two Floquet operators, obtained by starting the driving at zero or at half period, respectively. These are called timeframes and are chiral symmetric quantum walks. Conversely, we show under which conditions two chiral symmetric walks determine a common half-step operator. Moreover, we clarify the connection between the classification of half-step operators and the corresponding quantum walks. Within this theory we prove bulk-edge correspondence and show that a second timeframe allows to distinguish between symmetry protected edge states at $+1$ and $-1$ which is not possible for a single timeframe.