Are the surface Fermi arcs in Dirac semimetals topologically protected?

Abstract

Motivated by recent experiments probing anomalous surface states of Dirac semimetals (DSMs) Na3Bi and Cd3As2, we raise the question posed in the title. We find that, in marked contrast to Weyl semimetals, the gapless surface states of DSMs are not topologically protected in general, except on time-reversal-invariant planes of surface Brillouin zone. We first demonstrate this finding in a minimal four-band model with a pair of Dirac nodes at [Formula: see text] where gapless states on the side surfaces are protected only near [Formula: see text] We then validate our conclusions about the absence of a topological invariant protecting double Fermi arcs in DSMs, using a K-theory analysis for space groups of Na3Bi and Cd3As2 Generically, the arcs deform into a Fermi pocket, similar to the surface states of a topological insulator, and this pocket can merge into the projection of bulk Dirac Fermi surfaces as the chemical potential is varied. We make sharp predictions for the doping dependence of the surface states of a DSM that can be tested by angle-resolved photoemission spectroscopy and quantum oscillation experiments.

Keywords: Dirac semimetals; Fermi arcs; Weyl semimetals; topological insulators.

Fig. 1.

(A) Schematic k-space picture of a Dirac semimetal showing Dirac nodes along the $k_z$ axis in bulk BZ and possible double Fermi arcs on the surface BZs, shown as blue squares. Note that surfaces perpendicular to the z axis have no arcs. A 2D slice of the bulk BZ perpendicular to the $k_z$ axis is shown as a green square, which projects to a green dashed line on a side surface. (B) Surface spectral density of model in [1], which clearly shows the existence of double Fermi arcs on the (100) surface. (C and D) Continuous deformation of double Fermi arcs on the (100) surface by adding the perturbation $\delta H_4\left(\mathbf{k}\right)$ to [1]. C shows the effect at $m\prime =-0.4t,$ whereas D corresponds to $m\prime =-0.8t,$ showing that the Fermi arcs are progressively destroyed by increasing strength of perturbation. The red solid circles in B–D correspond to the projection of bulk nodes and we set $E_F=0$ to line up the Fermi level with bulk Dirac nodes. (E and F) Electron-doped systems with $m\prime =-0.8t$ (case in D) by raising the Fermi energy to $E_F=0.1t$ and $E_F=0.15t,$ respectively. The large red blobs in E and F mark the projection of bulk states onto the surface BZ. The Fermi contour of the surface states is disconnected from the blobs in E, whereas it merges into the bulk states in F.

Fig. S1.

Surface spectral density for a finite-size system along the x direction described by a four-band model in [1] with ${\epsilon }_{\mathbf{k}}^0=0.$ In A–C, from Left to Right corresponds to adding a perturbation preserving all symmetries of the system with $m\prime =0,$ $m\prime =0.4t,$ and $m\prime =0.8t,$ respectively. In the absence of perturbation the surface states are a line (A) connecting projected bulk nodes. As soon as the perturbation is turned on, the surface states start to disappear from the surface in B and C.

Fig. S2.

(Top) Spectral density of a four-band model with $\delta H_4=0$ at the $\omega -k_y$ plane. Each panel corresponds to a fixed $k_z$ as shown. (Bottom) The same panels but with $\delta H_4\ne 0.$ The dashed line corresponds to $\omega =0.$ It is clearly seen that the in the absence of bulk perturbation $\delta H_4$ gapless states exist for all values of $k_z$ between nodes, whereas in the presence of perturbation $\delta H_4$ the gapless edge states only at $k_z=0$ and $k_z=\pi /2$ remain. The latter one corresponds to the projection of bulk nodes onto the surface.