Quantized Berry winding from an emergent $\mathcal{PT}$ symmetry

TL;DR

This paper demonstrates that in 2D materials with band crossings, an emergent $ ext{PT}$ symmetry enables the quantization of Berry windings as multiples of $ ext{ extpi}$, linked to 3D Chern numbers via a dimensional extension.

Contribution

It introduces a novel link between 2D Berry windings and 3D Chern numbers using an emergent $ ext{PT}$ symmetry and a dimensional embedding approach.

Findings

Berry windings are quantized as multiples of $ ext{ extpi}$

The quantization relates to 3D Chern numbers via dimensional extension

Illustrations on three-band crossings confirm the mechanism

Abstract

Linear crossing of energy bands occur in a wide variety of materials. In this paper we address the question of the quantization of the Berry winding characterizing the topology of these crossings in dimension $D=2$. Based on the historical example of $2$-bands crossing occuring in graphene, we propose to relate these Berry windings to the topological Chern number of a $D=3$ dimensional extension of these crossings. This dimensional embedding is obtained through a choice of a gap-opening potential. We show that the presence of an (emergent) $\mathcal{PT}$ symmetry, local in momentum and antiunitary, allows us to relate $D=3$ Chern numbers to $D=2$ Berry windings quantized as multiple of $\pi$. We illustrate this quantization mechanism on a variety of three-band crossings.