Kramers nodal line metals

Abstract

Recently, it was pointed out that all chiral crystals with spin-orbit coupling (SOC) can be Kramers Weyl semimetals (KWSs) which possess Weyl points pinned at time-reversal invariant momenta. In this work, we show that all achiral non-centrosymmetric materials with SOC can be a new class of topological materials, which we term Kramers nodal line metals (KNLMs). In KNLMs, there are doubly degenerate lines, which we call Kramers nodal lines (KNLs), connecting time-reversal invariant momenta. The KNLs create two types of Fermi surfaces, namely, the spindle torus type and the octdong type. Interestingly, all the electrons on octdong Fermi surfaces are described by two-dimensional massless Dirac Hamiltonians. These materials support quantized optical conductance in thin films. We further show that KNLMs can be regarded as parent states of KWSs. Therefore, we conclude that all non-centrosymmetric metals with SOC are topological, as they can be either KWSs or KNLMs.

Classification of Weyl points and nodal lines based on magnetic point groups for spin-$\frac{1}{2}$ quasiparticles

“…Noting that the Γ point of any material has the full symmetry of the magnetic point group our results can be summarized as follows. For gray point groups, our work emphasizes the conclusion drawn by Xie et al [43]: All non-centrosymmetric non-magnetic (semi-) metals are topological in this sense. Furthermore, as revealed by our analysis, any material that belongs to a monochromatic point group is topological if the point group is either chiral and has more than one rotation axis or if it is achiral and has at least one rotation axis that is located in a mirror plane.…”

“…Our findings for the achiral gray point groups agree with the results of Xie et al [43]. The authors report so-called Kramers nodal lines, which are protected by a combination of time-reversal symmetry and achiral symmetries, in achiral non-centrosymmetric materials with spin-orbit coupling [43]. Kramers nodal lines are doubly degenerate band-touching lines that connect time-reversal invariant momenta.…”

“…C 3h ⊗ {e, Θ} has an open nodal line in its mirror plane similar to C 1h ⊗ {e, Θ}, and possesses an open nodal line on its rotation axis, which is also an improper rotation axis, similar to S 4 ⊗ {e, Θ}. Our findings for the achiral gray point groups agree with the results of Xie et al [43]. The authors report so-called Kramers nodal lines, which are protected by a combination of time-reversal symmetry and achiral symmetries, in achiral non-centrosymmetric materials with spin-orbit coupling [43].…”

“…In addition to catalogs of emergent particles for the 230 space groups [34] and of topological materials [35,36], various other studies provide symmetry-based classifications of different band touchings including Dirac and Weyl points [37][38][39], Kramers-Weyl points in chiral materials [40], nodal lines [37,[41][42][43], and three-, six-, and eightfold degenerate band-touching points giving rise to unconventional quasiparticles [44]. In all of these investi-gations, the presence of time-reversal symmetry plays a crucial role.…”

Topological charge distributions of an interacting two-spin system

“…As electric charge is the source of electric field, we define topological charge as the source of Berry curvature. Fine-tuning or the presence of symmetries can lead to anomalous, nongeneric situations when degeneracy points in a three-dimensional parameter space are (i) isolated but the energy splitting is not linear, but of higher order [15][16][17][18] or (ii) not isolated but they form a continuous line or surface [16,[19][20][21][22][23][24][25][26]. These anomalous features have been demonstrated in electronic band structure models of three-dimensional solids, where the parameters are the Cartesian components of crystal momentum, and also in interacting spin systems with a threedimensional magnetic parameter space [11].…”