Illuminating the bulk-boundary correspondence of a non-Hermitian stub lattice with Majorana stars

TL;DR

This paper investigates the topological properties of a non-Hermitian stub lattice with nonreciprocal hopping, introducing a $Z_2$ invariant based on Majorana stars to predict edge states despite the lack of chiral symmetry.

Contribution

It proposes a novel $Z_2$ topological invariant derived from Majorana's stellar representation for non-Hermitian systems lacking chiral symmetry.

Findings

Majorana's stellar representation predicts edge states via azimuthal winding parity.

The system's topological features are characterized without relying on a quantized Zak phase.

The model is not a square-root topological insulator despite its relation to a sawtooth lattice.

Abstract

Topological characterization of non-Hermitian band structures demands more than a straightforward generalization of the Hermitian cases. Even for one-dimensional tight-binding models with nonreciprocal hopping, the appearance of point gaps and the skin effect leads to the breakdown of the usual bulk-boundary correspondence. Luckily, the correspondence can be resurrected by introducing a winding number for the generalized Brillouin zone for systems with even number of bands and chiral symmetry. Here, we analyze the topological phases of a nonreciprocal hopping model on the stub lattice, where one of the three bands remains flat. Due to the lack of chiral symmetry, the biorthogonal Zak phase is no longer quantized, invalidating the winding number as a topological index. Instead, we show that a $Z_2$ invariant can be defined from Majorana's stellar representation of the eigenstates on the Bloch sphere. The parity of the total azimuthal winding of the entire Majorana constellation correctly predicts the appearance of edge states between the bulk gaps. We further show that the system is not a square-root topological insulator, despite the fact that its parent Hamiltonian can be block diagonalized and related to a sawtooth lattice model. The analysis presented here may be generalized to understand other non-Hermitian systems with multiple bands.