Topological Berry phase and semiclassical quantization of cyclotron orbits for two dimensional electrons in coupled band models

TL;DR

This paper investigates how topological Berry phases influence semiclassical quantization of cyclotron orbits in 2D electron systems, revealing a topological index shift in Landau levels in coupled band models like graphene and boron nitride.

Contribution

It identifies a topological component of the Berry phase responsible for Landau level shifts in coupled band systems, clarifying the role of winding numbers over the full Berry phase.

Findings

Landau levels are shifted due to a topological Berry phase component.

The index shift relates to a winding number of the pseudo-spin orientation.

Careful computation of Berry curvature is essential for accurate results.

Abstract

The semiclassical quantization of cyclotron orbits for two-dimensional Bloch electrons in a coupled two band model with a particle-hole symmetric spectrum is considered. As concrete examples, we study graphene (both mono and bilayer) and boron nitride. The main focus is on wave effects -- such as Berry phase and Maslov index -- occurring at order $\hbar$ in the semiclassical quantization and producing non-trivial shifts in the resulting Landau levels. Specifically, we show that the index shift appearing in the Landau levels is related to a topological part of the Berry phase -- which is basically a winding number of the direction of the pseudo-spin 1/2 associated to the coupled bands -- acquired by an electron during a cyclotron orbit and not to the complete Berry phase, as commonly stated. As a consequence, the Landau levels of a coupled band insulator are shifted as compared to a usual band insulator. We also study in detail the Berry curvature in the whole Brillouin zone on a specific example (boron nitride) and show that its computation requires care in defining the "k-dependent Hamiltonian" H(k), where k is the Bloch wavevector.