# Linearity of the edge states energy spectrum in the 2D topological insulator

## TL;DR

This paper investigates the linearity of edge state energy spectra in 2D topological insulators across different models, finding that some models exhibit ideal linearity while others show weak non-linearity, depending on boundary conditions.

## Contribution

It provides a comparative analysis of edge state spectrum linearity in various models, clarifying the conditions under which linearity is preserved or weakly broken.

## Key findings

- VP and BHZ1 models show ideally linear edge states
- Linearity in BHZ1 persists up to spectrum endpoints
- BHZ2 and tight-binding models exhibit weak non-linearity

## Abstract

Linearity of the topological insulator edge state spectrum plays the crucial role for various transport phenomena. The previous studies found that this linearity exists near the spectrum crossing point, but did not determine how perfect the linearity is. The purpose of the present study is to answer this question in various edge states models. We examine Volkov and Pankratov (VP) model [1] for the Dirac Hamiltonian and the model of [2,3] (BHZ1) for the Bernevig, Hughes and Zhang (BHZ) Hamiltonian [4] with zero boundary conditions. It is found that both models yield ideally linear edge states. In the BHZ1 model the linearity is conserved up to the spectrum ending points corresponding to the tangency of the edge spectrum with the boundary of 2D states. In contrast, the model of[5] (BHZ2) with mixed boundary conditions for BHZ Hamiltonian and the 2D tight-binding (TB) model from [4] yield weak non-linearity.