Higher-order Topological Knots and the classification of non-Hermitian lattices under $C_n$ symmetry

TL;DR

This paper explores the topological properties of non-Hermitian lattices, revealing new higher-order topological phases with circulating edge states and their relation to complex Chern insulators, expanding understanding of non-Hermitian topological matter.

Contribution

It uncovers novel non-Hermitian topological phases with higher-order topological knots protected by crystalline symmetries and links them to complex Chern insulator phases.

Findings

Discovery of higher-order topological knots in non-Hermitian lattices

Circulating nonreciprocal edge states around the entire boundary

Phase transitions between HOTK and complex Chern insulator phases

Abstract

In two dimensions, Hermitian lattices with non-zero Chern numbers and non-Hermitian lattices with a higher-order skin effect (HOSE) bypass the constraints of the Nielsen-Ninomiya no-go theorem at their one-dimensional boundaries. This allows the realization of topologically-protected one-dimensional edges with nonreciprocal dynamics. However, unlike the edge states of Chern insulators, the nonreciprocal edges of HOSE phases exist only at certain edges of the two-dimensional lattice, not all, leading to corner-localized states. In this paper, we investigate the topological connections between these two systems and uncover novel non-Hermitian topological phases possessing higher-order topological knots (HOTKs). These phases arise from multiband topology protected by crystalline symmetries and host nonreciprocal edge states that circulate the entire boundary of the two-dimensional lattice. We show that phase transitions typically separate HOTK phases from complex Chern insulator phases - non-Hermitian lattices with nonzero Chern numbers protected by imaginary line gaps in the presence of time-reversal symmetry.