Topological Fermi-arc-like surface states in Kramers nodal line metals

Abstract

The discovery of Kramers nodal line metals (KNLMs) and Kramers Weyl semimetals (KWSs) has significantly expanded the range of metallic topological materials to all noncentrosymmetric crystals. However, a key characteristic of this topology - the presence of topologically protected surface states in KNLMs - is not well understood. In this work, we use a model of a $C_{1v}$ KNLM with curved Kramers nodal lines (KNLs) to demonstrate that Fermi-arc-like surface states (FALSSs), which have a $\mathbb{Z}_2$ topological origin, appear on surfaces parallel to the mirror plane. These states connect two surface momenta, corresponding to the projections of two touching points on the Fermi surfaces. Notably, as achiral symmetries (mirrors and roto-inversions) are gradually broken, the KNLM transitions into a KWS, allowing the FALSSs to evolve continuously into the Fermi arc states of the KWS. We also explore the conditions under which FALSSs emerge in KNLMs with straight KNLs. Through bulk-boundary correspondence, we clarify the topological nature of KNLMs.

Figures (4)

• Figure 1: The $C_{1v}$ KLNM. (a) A domatic lattice upon which the $C_{1v}$ tight-binding model is established. (b) The band structure of the Hamiltonian from Eq. \ref{['eq:H_k']}. Specific tight-binding parameters are given in Appendix A. (c) The Brillouin zone (BZ) of the $C_{1v}$ model. The KNLs upon the $k_{z}=0$$(\pi)$ plane are plotted in the red (blue) solid curves. (d) The surface projection of KNLs is the boundary between the topologically non-trivial (cyan) and trivial regions (white). The octdong Fermi surface is also shown with the semi-transparent pink (blue) color representing the electron (hole) pockets. Their touching points, $e.g.$, $P_1$ and $P_2$, are on the KNLs. (e) and (f) show the surface spectral function $A^{t/b}(\bm{k}_\parallel,E)$ on the (001) surface of the $C_{1v}$ tight-binding model. The energy level in (e) is set as $E=-1.1$, as indicated by the horizontal black dashed line in (f).
• Figure 2: The $\mathbb{Z}_2$ topological invariant protected by the mirror symmetry. (a) and (b) show how the spin rotates upon the unit sphere when $k_z$ of the 1D Hamitonian $H_{\bm{k}_\parallel}(k_z)$ changes from $-\pi$ to $\pi$, respectively for topologically trivial ($\mathcal{Z}=0$) and non-trivial case ($\mathcal{Z}=\pi$).
• Figure 3: The FALSSs of the KNLM evolve into the Fermi arc states of the KWS. (a) shows the (001)-surface spectral weight $A(\bm{k}_\parallel,E=0)$ of the $C_{1v}$ KNLM $k\cdot p$ Hamiltonian $h(\bm{k})$ of Eq. \ref{['eq:kp_model']} with $m=0.2, t_z=0.02, \mu=0.2, \alpha=1.5, \beta=1, a_1=4, a_2=2, b_1=2, b_2=4$. The red (blue) dashed line is the projection of the KNL in the $k_z=0\,(\pi)$ plane, which separates the topologically trivial and nontrivial regions with 0 and $\pi$ Zak phase, respectively. (b) shows the (001)-surface spectral weight when the mirror-breaking term $h^\prime(\bm{k})$ from Eq. \ref{['eq:mirror_breaking']} with $c=1$ is added. Now the chirality of Kramers Weyl nodes at $\Gamma$ and Z are both negative, and the Chern number defined on the torus surface whose cross-section with constant $k_z$ planes is indicated by the green circle is -2. (c) and (d) show the (001)-surface spectral function $A(\bm{k}_{\parallel}, E=-1.1)$ of the lattice models, respectively for the $C_{1v}$ KNLM and the $C_1$ KWS. The parameters of the mirror-breaking term in Eq. \ref{['eq:mirror-breaking']} are set as $\beta_1=0.2, \beta_2=-0.1$. In (d), the surface projections of the Weyl nodes with chirality +1(-1) are labeled by red(blue) dots. For the Kramers Weyl nodes at the TRIMs, the chirality of the Weyl node on the $k_z=0(\pi)$ plane is denoted by the outer (inner) circles.
• Figure 4: The $C_{2v}$ KNLM. (a,b) show the KNLs (blue) and the extra nodal lines (green) in the first Brillouin zone. (c,d,e) show the surface spectrum under three different parameter regimes. When there is no extra nodal line other than the KNLs, the whole (001)-surface Brillouin zone is either completely trivial (as shown in (c)) or completely topological (as shown in (d)). When the extra nodal lines are present, their surface projection will cut the (001)-surface Brillouin zone into trivial and topological (yellow areas in (b)) regions.