Classification of stable Dirac and Weyl semimetals with reflection and rotational symmetry

Abstract

Three dimensional (3D) Dirac semimetal is a novel state of quantum matter, characterized by the gapless bulk four-fold degeneracy near Fermi energy. Soon after its discovery, the classification of stable 3D Dirac semimetals with inversion and rotational symmetry have been studied. However, only ten out of thirty-two point groups have both inversion and rotational symmetry, and we need a more complete classification of stable 3D Dirac semimetals. Here we classify stable 3D Dirac semimetals with reflection symmetry and rotational symmetry in the presence of time reversal symmetry, which belong to seventeen different point groups. These systems include the systems preserving inversion symmetry except C3i. They have two classes of reflection symmetry, with the mirror plane parallel to rotation axis and the mirror plane perpendicular to rotation axis. In both cases two types of Dirac semimetals are determined by four different reflection symmetries. The first type of Dirac semimetals will appear through accidental band crossing (ABC). The second type of Dirac semimetals have a Dirac point at a time reversal invariant momentum (TBC). We show that in both mirror parallel and perpendicular cases, C2,3 symmetry can only protect stable Dirac points via TBC, while C4,6 symmetry can have stable Dirac points as ABC or TBC. We further discuss that Weyl line nodes and Dirac semimetal can exist in Brillouin zone at the same time using C4v symmetry as an example. Finally we classify Dirac line nodes and Weyl line nodes to show in which types of mirror plane they can exist.

Filling-Enforced Gaplessness in Band Structures of the 230 Space Groups

“…Compared to the interacting results presented in [18], the current Letter serves as an independent verification of the tight filling constraints for 218 SGs using band-theory analysis, and provides the tight bounds for the remaining 12 SGs in the noninteracting limit. In addition, the band-theory arguments presented here form the basis for further k · p effective Hamiltonian analysis, which constrains the generic dispersion about the degeneracy point [13,[21][22][23]. In contrast, our previous interacting argument does not constrain the spectrum of low-energy excitation.…”

Topological Semimetals from First Principles

“…While all these materials are centrosymmetric, the topological Dirac semimetals can also exist in noncentrosymmetric crystals without inversion symmetry. A general analysis of such semimetals was presented by Gao et al (69), who showed that rotation axes with little groups isomorphic to the C 4v and C 6v point groups allow for stable fourfold-degenerate crossings. Material realizations of noncentrosymmetric Dirac semimetals were proposed in a family of hexagonal ABC materials with LiGaGe-type structure and polar space group 186 (P6 3 mc).…”

Topological Dirac semimetal phase in Pd and Pt oxides

“…In this case, we can express the Hamiltonian for k x = k y = 0 as H(k z )| kx=ky=0 = d 0 +d 1 σ 3 +d 2 τ 3 σ 3 +d 3 τ 3 with σ i and τ i being the Pauli matrices in spin and orbital spaces, d i are real coefficients [4,28,40,41].…”