Surface theory of a second-order topological insulator beyond the Dirac approximation

TL;DR

This paper investigates the surface and hinge states of a 3D second-order topological insulator under magnetic fields, revealing deviations from Dirac theory and extending the surface Hamiltonian to include quadratic terms for accurate modeling.

Contribution

It introduces an extended surface theory with quadratic Hamiltonian terms to explain Landau level asymmetries in second-order topological insulators beyond Dirac approximation.

Findings

Landau levels on pierced surfaces form as expected, but parallel surfaces are unaffected.

Higher-order Hamiltonian terms explain asymmetries in Landau levels.

The extended theory aligns better with lattice model results.

Abstract

We study the surface states and chiral hinge states of a 3D second-order topological insulator in the presence of an external magnetic gauge field. Surfaces pierced by flux host Landau levels, while surfaces parallel to the applied field are not significantly affected. The chiral hinge modes mediate spectral flow between neighbouring surfaces. As the magnetic field strength is increased, the surface Landau quantization deviates from that of a massive Dirac cone. Quantitatively, the $n = 0$ Landau level falls inside the surface Dirac gap, and not at the gap edge. The $n \ne 0$ levels exhibit a further, qualitative discrepancy: while the massive Dirac cone is expected to produce pairs of levels ($\pm n$) which are symmetric around zero energy, the $n$ and $-n$ levels become asymmetric in our lattice model -- one of the pair may even be absent from the spectrum, or hybridized with the continuum. In order to resolve the issue, we extend the standard 2D massive Dirac surface theory, by including additional Hamiltonian terms at $\mathcal{O} (k^2)$. While these terms do not break particle-hole symmetry in the absence of magnetic field, they lead to the aforementioned Landau level asymmetry once the magnetic field is applied. We argue that similar $\mathcal{O}(k^2)$ correction terms are generically expected in lattice models containing gapped Dirac fermions, using the BHZ model of a 2D topological insulator as an example.