Second-order topological modes in two-dimensional continuous media

Jan Košata,Oded Zilberberg

doi:10.1103/physrevresearch.3.l032029

Abstract

We present a symmetry-based scheme to create 0D second-order topological modes in continuous 2D systems. We show that a metamaterial with a \textit{p6m}-symmetric pattern exhibits two Dirac cones, which can be gapped in two distinct ways by deforming the pattern. Combining the deformations in a single system then emulates the 2D Jackiw-Rossi model of a topological vortex, where 0D in-gap bound modes are guaranteed to exist. We exemplify our approach with simple hexagonal, Kagome and honeycomb lattices. We furthermore formulate a quantitative method to extract the topological properties from finite-element simulations, which facilitates further optimization of the bound mode characteristics. Our scheme enables the realization of second-order topology in a wide range of experimental systems.

Highlights

The development of topological insulators (TIs), while originating in electronic systems, has made a profound impact in the field of classical metamaterials
Compared with standard 0D modes formed at nontopological lattice defects, topological vortex modes offer several advantages [39]: (i) Their frequency is near midgap, resulting in better spatial confinement and quality factors. (ii) The number of modes formed is fixed by the topology. (iii) The modal area is scalable, as the modes form independently of the vortex size
The presented symmetry principles and formulas for numerical calculations enable a systematic exploration of a plethora of structures, which feature high-order topological defects

Summary

Introduction

The development of topological insulators (TIs), while originating in electronic systems, has made a profound impact in the field of classical metamaterials. We consider two distinct perturbations of the primitive cell, namely the breaking of inversion and translation symmetries, and find their matrix representations in the 4D eigenspace [42,43]. The perturbation matrices are shown to anticommute and correspond to different gap-opening terms; these constitute the Jackiw-Rossi model.

Results

Conclusion