Higher-Order Nodal Points in Two Dimensions

Abstract

A two-dimensional (2D) topological semimetal is characterized by the nodal points in its lowenergy band structure. While the linear nodal points have been extensively studied, especially in the context of graphene, the realm beyond linear nodal points remains largely unexplored. Here, we explore the possibility of higher-order nodal points, i.e., points with higher-order energy dispersions, in 2D systems. We perform an exhaustive search over all 80 layer groups both with and without spin-orbit coupling (SOC), and reveal all possible higher-order nodal points. We show that they can be classified into two categories: the quadratic nodal point (QNP) and the cubic nodal point (CNP). All the 2D higher-order nodal points have twofold degeneracy, and the order of dispersion cannot be higher than three. QNPs only exist in the absence of SOC, whereas CNPs only exist in the presence of SOC. Particularly, the CNPs represent a new topological state not known before. We show that they feature nontrivial topological charges, leading to extensive topological edge bands. Our work completely settles the problem of higher-order nodal points, discovers novel topological states in 2D, and provides detailed guidance to realize these states. Possible material candidates and experimental signatures are discussed.

Encyclopedia of emergent particles in type-IV magnetic space groups

“…2(d) We find that the cubic Weyl point can exist in spinful type-II MLGs, consistent with the prediction in Ref. [52]. In addition, our result shows that it also exists in spinful type-I MLGs, but not in type-III and type-IV MLGs.…”

Ideal topological Weyl complex phonons in two dimensions

“…Such linear points locate at non-time-reversal invariant momentum (TRIM), and are enforced by C 3z and the spacetime inversion PT ð Þ symmetries. Most recently, there have been theoretical efforts 23,24 to conceive topological nodal points beyond the scope of linear nodal points. It has been discussed that a rotation symmetry (C 3z , na2) together with time-reversal symmetry T ð Þ can enforce a quadratic nodal point at TRIM in 2D spinless systems.…”

“…Akin to the 3D case, Weyl points in spinless systems also have a higher dispersion. Most recently, Wu et al 23 and Yu et al 24 have demonstrated that the highest order of 2D Weyl points in a spinless system is quadratic. Inspired by the coexistence of different types of Weyl points in 3D, one may consider: Is it possible to stabilize a Weyl complex in a 2D phononic system?…”

A two-dimensional tunable double Weyl fermion in BL-α borophene

“…In addition, it is necessary to be noted that the highest order for the 2D nodal point is not limited to the linear dispersion; it could be up to the second-order in the absence of spin-orbital coupling (SOC). 14 The prominent example may be the bilayer graphene, [15][16][17][18][19][20][21] and it could host a nodal point with quadratic dispersion close to the Fermi level, giving rise to a new 2D quasiparticle excitation, dubbed as a double Weyl fermion, 22 underlying many novel physical consequences of bilayer graphene. Hence, it is of great interest to explore 2D materials that host novel topological excitations.…”