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

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) $\mathcal{P}\mathcal{T}$ 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.

25 pages including appendices and references, 4 figures

References in corpus (8)

• Electric Field Effect in Atomically Thin Carbon Films
• Chiral tunneling and the Klein paradox in graphene
• Unconventional quantum Hall effect and Berry's phase of 2pi in bilayer graphene
• Topological Semimetals
• Merging of Dirac points in a two-dimensional crystal
• Magneto-optics of massless Kane fermions: Role of the flat band and unusual Berry phase
• Topological and geometrical aspects of band theory
• Topological Lasing from Thouless Pumping in Bilayer Photonic Crystal