Effective vacua for Floquet topological phases: A numerical perspective on switch-function formalism

We propose a general edge index definition for two-dimensional Floquet topological phases based on a switch-function formalism. When the Floquet operator has a spectral gap the index covers both clean and disordered phases, anomalous or not, and does not require the bulk to be fully localized. It is interpreted as a non-adiabatic charge pumping that is quantized when the sample is placed next to an effective vacuum. This vacuum is gap-dependent and obtained from a Floquet Hamiltonian. The choice of a vacuum provides a simple and alternative gap-selection mechanism. Inspired by the model from Rudner et al. we then illustrate these concepts on Floquet disordered phases. Switch-function formalism is usually restricted to infinite samples in the thermodynamic limit. Here we circumvent this issue and propose a numerical implementation of the edge index that could be adapted to any bulk or edge index expressed in terms of switch functions, already existing for many topological phases.