High-root topological edge-state bands

TL;DR

This paper analyzes the edge-state bands of high-root topological insulators using complex band analysis, revealing their relation to impurity bands and providing a method to determine edge state levels without diagonalization.

Contribution

It introduces a simplified topological characterization of high-root topological insulators and links edge states to impurity states in a uniform chain, enabling easier analysis.

Findings

Edge-state bands are slices of impurity bands in a uniform chain.

Edge states in finite systems are subsets of evanescent states.

Edge state levels can be determined without diagonalizing Hamiltonians.

Abstract

This paper presents a complex band analysis of one-dimensional (1D) square and high-root topological insulators (HRTIs). We show that edge-state bands of HRTIs are sliced sections of impurity bands of a uniform tight-binding chain. A simplified topological characterization of HRTIs with generalized boundary conditions is carried out based on the existence of edge-state bands in the infinite HRTI and the restrictions imposed by the boundary conditions. Edge states in finite or semi-infinite 1D HRTIs are shown to be a subset of evanescent states of the infinite system and mapped onto impurity states of the uniform chain with effective energy-dependent edge potentials. The latter result allows the determination of the edge state levels without needing the diagonalization of real space or bulk Hamiltonians.