Low-energy effective Hamiltonian and end states of an inverted HgTe nanowire

TL;DR

This paper derives a low-energy effective Hamiltonian for an inverted HgTe nanowire, revealing the conditions for topological end states and the importance of multiple subbands near the band gap.

Contribution

It introduces a three-subband effective Hamiltonian for HgTe nanowires, capturing the band inversion and end state formation near the fundamental gap.

Findings

Effective Hamiltonian is block diagonal with 3x3 blocks.

End states appear in the inverted regime with open boundary conditions.

Multiple subbands are essential for accurate low-energy modeling.

Abstract

The band inversion transition in a cylindrical HgTe nanowire is inducible via varying the nanowire radius. Here we derive the low-energy effective Hamiltonian describing the band structure of the HgTe nanowire close to the fundamental band gap. Because both the $E_{1}$ and $H_{1}$ subbands have quadratic dependence on $k_{z}$ when the gap closes, we need to consider at least three subbands, i.e., the $E_{1}$, $H_{1}$, and $H_{2}$ subbands, in building the effective Hamiltonian. The resulting effective Hamiltonian is block diagonal and each block is a $3\times3$ matrix. End states are found in the inverted regime when we solve the effective Hamiltonian with open boundary condition.