Generalized Fermion Doubling Theorems: Classification of 2D Nodal Systems in Terms of Wallpaper Groups

TL;DR

This paper generalizes the Nielsen-Ninomiya Theorem to classify minimal nodal points in 2D systems with wallpaper group symmetries, aiding the discovery of new topological materials and extending to non-Hermitian systems.

Contribution

It provides a comprehensive classification of minimal nodal points for all wallpaper groups considering chiral and space-time-inversion symmetries, and extends the theorem to non-Hermitian systems.

Findings

Minimal nodal points depend on wallpaper group symmetries.

Examples of new topological materials are provided.

Extension of the theorem to non-Hermitian systems.

Abstract

The Nielsen-Ninomiya Theorem has set up a ground rule for the minimal number of the topological points in a Brillouin zone. Notably, in the 2D Brillouin zone, chiral symmetry and space-time inversion symmetry can properly define topological invariants as charges characterizing the stability of the nodal points so that the non-zero charges protect these points. Due to the charge neutralization, the Nielsen-Ninomiya Theorem requires at least two stable topological points in the entire Brillouin zone. However, additional crystalline symmetries might duplicate the points. In this regard, for the wallpaper groups with crystalline symmetries, the minimal number of the nodal points in the Brillouin zone might be more than two. In this work, we determine the minimal numbers of the nodal points for the wallpaper groups in chiral-symmetric and space-time-inversion-symmetric systems separately and provide examples for new topological materials, such as topological nodal time-reversal-symmetric superconductors and Dirac semimetals. This generalized Nielsen-Ninomiya Theorem serves as a guide to search for 2D topological nodal materials and new platforms for twistronics. Furthermore, we show the Nielsen-Ninomiya Theorem can be extended to 2D non-Hermitian systems hosting topologically protected exceptional points and Fermi points for the 17 wallpaper groups and use the violation of the theorem on the surface to classify 3D Hermitian and non-Hermitian topological bulks.