Tunable corner states in topological insulators with long-range hoppings and diverse shapes

TL;DR

This paper presents a theoretical framework for precisely controlling and tuning corner states in higher-order topological insulators with long-range hoppings and various geometries, enabling advanced design of topological materials.

Contribution

It introduces a method to tune corner states via long-range hoppings and analyzes the impact of shape on corner mode presence and location in HOTIs.

Findings

Corner states can be finely tuned by varying long-range hoppings.

Long-range hoppings in different directions affect corner positions differently.

Corner modes depend on the sign of Dirac masses at adjacent edges.

Abstract

In this work we develop a theoretical framework for the control of corner modes in higher-order topological insulators (HOTIs) featuring long-range hoppings and diverse geometries, enabling precise tunability of their spatial positions. First, we demonstrate that the locations of corner states can be finely tuned by varying long-range hoppings in a circular HOTI, as revealed by a detailed edge theory analysis and the condition of vanishing Dirac mass. Moreover, we show that long-range hoppings in different directions (e.g., $x$ and $y$) have distinct effects on the positioning of corner states. Second, we investigate HOTIs with various polygonal geometries and find that the presence and location of corner modes depend sensitively on the shape. In particular, a corner hosts a localized mode if the Dirac masses of its two adjacent edges have opposite signs, while no corner mode emerges if the masses share the same sign. Our findings offer a versatile approach for the controlled manipulation of corner modes in HOTIs, opening avenues for the design and implementation of higher-order topological materials.