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

Abstract

Linear crossings 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=2D=2. Based on the historical example of 22-bands crossing occuring in graphene, we propose to relate these Berry windings to the topological Chern number within a D=3D=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}𝒫𝒯 symmetry, local in momentum and antiunitary, allows the quantization of the Berry windings as multiples of \piπ. We illustrate this quantization mechanism on a variety of three-band crossings.