Higher-order topological states mediated by long-range coupling in $D_4$-symmetric lattices

TL;DR

This paper demonstrates how long-range couplings in a $D_4$-symmetric electric circuit induce higher-order topological states, experimentally confirming their topological nature and highlighting differences from tight-binding models.

Contribution

The study designs and experimentally realizes a $D_4$-symmetric resonant circuit with long-range couplings, revealing the role of distant neighbor interactions in creating topological corner states.

Findings

Long-range couplings induce in-gap corner states.

Experimental retrieval of topological invariants confirms topological phases.

Distinction between tight-binding and photonic systems with long-range interactions.

Abstract

Topological physics opens a door towards flexible routing and resilient localization of waves of various nature. Recently proposed higher-order topological insulators provide advanced control over wave localization in the structures of different dimensionality. In many cases, the formation of such higher-order topological phases is governed by the lattice symmetries, with kagome and breathing honeycomb lattices being prominent examples. Here, we design and experimentally realize the resonant electric circuit with $D_4$ symmetry and additional next-nearest-neighbor couplings. As we prove, a coupling of the distant neighbors gives rise to an in-gap corner state. Retrieving the associated invariant directly from the experiment, we demonstrate the topological nature of the designed system, revealing the role of long-range interactions in the formation of topological phases. Our results thus highlight the distinctions between tight-binding systems and their photonic counterparts with long-range couplings.