Non-Hermitian Bulk-Boundary Correspondence in Periodically Driven System

TL;DR

This paper establishes a non-Hermitian bulk-boundary correspondence in periodically driven systems, introducing non-Bloch winding numbers to characterize edge states even when traditional topological invariants are trivial.

Contribution

It constructs a non-Hermitian Floquet model with skin effect and defines non-Bloch invariants to restore bulk-boundary correspondence in these systems.

Findings

Non-Hermitian skin effect causes all eigenstates to localize at boundaries.

Non-Bloch winding numbers characterize edge states with quasienergies 0 and π.

Robust edge states can exist even when the Floquet bands are topologically trivial.

Abstract

Bulk-boundary correspondence, connecting the bulk topology and the edge states, is an essential principle of the topological phases. However, the bulk-boundary correspondence is broken down in general non-Hermitian systems. In this paper, we construct one-dimensional non-Hermitian Su-Schrieffer-Heeger model with periodic driving that exhibits non-Hermitian skin effect: all the eigenstates are localized at the boundary of the systems, whether the bulk states or the zero and the $\pi$ modes. To capture the topological properties, the non-Bloch winding numbers are defined by the non-Bloch periodized evolution operators based on the generalized Brillouin zone. Furthermore, the non-Hermitian bulk-boundary correspondence is established: the non-Bloch winding numbers ($W_{0,\pi}$) characterize the edge states with quasienergies $\epsilon=0, \pi$. In our non-Hermitian system, a novel phenomenon can emerge that the robust edge states can appear even when the Floquet bands are topological trivial with zero non-Bloch band invariant, which is defined in terms of the non-Bloch effective Hamiltonian. We also show that the relation between the non-Bloch winding numbers ($W_{0,\pi}$) and the non-Bloch band invariant ($\mathcal{W}$): $\mathcal{W}= W_{0}- W_{\pi}$.