Protected Weyl semimetals within 2D chiral classes

Abstract: Weyl semimetals in three dimensions can exist independently of any symmetry apart from translations. In contrast, in two dimensions, Weyl semimetals require additional symmetries, including crystalline symmetries, to exist. Previous research, based on K-theory classification, suggested that chiral symmetry can protect Weyl nodes in two dimensions. According to K-theory, stable Weyl nodes can exist in four chiral classes-AIII, BDI, CII, and DIII-and are classified by $\mathbb{Z}$ (AIII, BDI, DIII) and $\mathbb{Z}_2$ (CII) invariants. However, it was later found that the $\mathbb{Z}_2$ and trivial indices predicted by K-theory do not reliably indicate the presence or absence of Weyl nodes in two dimensions. In this study, we demonstrate that stable Weyl nodes exist in each of the five chiral classes and can be characterized by a $\mathbb{Z}$ winding number in two dimensions. Our conclusion is supported by the explicit solution of the most general Hamiltonian consistent with the symmetry class. We also discuss protected Fermi arc edge states, which always connect the projections of Weyl nodes with opposite topological charges. Unlike the surface states in three-dimensional Weyl semimetals, the edge states in two-dimensional Weyl semimetals within chiral classes are completely dispersionless and remain at zero energy due to the protecting chiral symmetry.