Anomalous Floquet Higher-order Topological Insulators

Introduction

Floquet systems refer to a particular subset of the non-equilibrium (quantum) systems where the Hamiltonian of the system is drived periodically within the temporal domain. As the Floquet theorem states, Hamiltonians of the form

$H(t)=H(t+nT), n\in\mathbb Z$,

owns eigenstates that can be labelled by a family of good quantum numbers $\varepsilon$ called quasienergies. This theorem is exactly the temperal analogus of the Bloch theorem in spatial domain. It turns out that such quasienergies as the charge of the discrete time translational symmetry of Floquet dynamics will bring exotic non-equilibrium but solvable physics in many systems, such as Floquet engineering in cold-atom systems with tunable laser excitations (e.g. optical lattice), time crystal, and Floquet topological materials.

In Floquet topological materials, Floquet drive will give rise to multiple quasienergy bands, each of which hosts topological attributes such as non-zero Chern numbers. For one species of this class of materials, namely anomalous Floquet topological insulators (AFTIs), people has predicted and confirmed that when the Floquet drive frequency is comparable to the characterisitic frequency of the static topological materials, an anomalous new phase will emerge on the space-time manifold where the total Chern number of the quasienergy band is zero, but two channels of edge states will still be hosted robustly on the boundary of the materials. These two channels, dubbed as the zero-mode and the anomalous Floquet $\pi$-mode, is the distinct feature of AFTIs.

Researchers have formulated new topological classification theory to address these new kind of materials that combining periodic dynamics with the traditional topological insulators, and a new Floquet version of ten-fold way table is constructed, see Phys. Rev. B 96, 195303 (2017). Yet there are still a variety of Floquet topological insulators that do not obey the as-developed formalism, such as anomalous Floquet higher-order topological insulators (AFHOTIs) that their static version (HOTIs) are SPTs that are not protected by any CPT symmetries rather more complicated crystalline symmetries. Novel theories are constructed to describe the general topological features and classify all bulk-boundary information of such HOTIs when driven by a Floquet engine, such as the recent dynamical polarization theory (Phys. Rev. Lett. 124, 216601 (2020)) and dynamical singularity theory (Phys. Rev. Lett. 124, 057001 (2020)). However, these theories are limited to only a few small simple AFHOTI species and needs to be generalized to a broader framework of anomalous Floquet quantum many-body topological matter.

Our Work

We proposed and investigated a new species of AFHOTIs, dubbed as Floquet boundary obstructed topological insulators (FBOTIs). We compared different current dynamical strategies to resolve the Floquet anomaly in such systems, and generalized the dynaimcal polarization theory as a universal and robust framework of classifying any FBOTIs. Please read our manuscript below, some parts are still revising.

Codes and Data Availability

Please see all codes and data on my GitHub repository.