Dirac semimetal and topological phase transitions in $A_3$Bi ($A=\text{Na}$, K, Rb)

Three-dimensional (3D) Dirac point, where two Weyl points overlap in momentum space, is usually unstable and hard to realize. Here we show, based on the first-principles calculations and effective model analysis, that crystalline $A_3$Bi ($A=\text{Na}$, K, Rb) are Dirac semimetals with bulk 3D Dirac points protected by crystal symmetry. They possess nontrivial Fermi arcs on the surfaces and can be driven into various topologically distinct phases by explicit breaking of symmetries. Giant diamagnetism, linear quantum magnetoresistance, and quantum spin Hall effect will be expected for such compounds.

Figure 1

(a) Crystal structure of Na${}_3$Bi with $P{6}_3/mmc$ symmetry. Na(1) is at $2b$ position $\pm$(0,0,$\frac{1}{4}$), and Bi is at $2c$ position $\pm$($\frac{1}{3}$,$\frac{2}{3}$,$\frac{1}{4}$). They form honeycomb lattice layers. Na(2) is at $4f$ position $\pm$($\frac{1}{3}$, $\frac{2}{3}$,$u$) and $\pm$($\frac{2}{3}$,$\frac{1}{3}$,$\frac{1}{2}+u$) with $u=0.583$, threading Bi along the $c$ axis. (b) Brillouin zone of bulk and the projected surface Brillouin zones of (001) and (010) planes.

Figure 2

The calculated electronic structures of Na${}_3$Bi. (a) The total and partial density of states. (b) and (c) The band structures without and with spin-orbit coupling, respectively. The red circles indicate the projection to the Na-3$s$ states. The orbital characters of wave functions at the $\Gamma$ point are labeled in the inset (see Sec. 3b for details).

Figure 3

The projected surface states and their Fermi surfaces of Na${}_3$Bi. (a) and (b) The projected surface density of states for [001] and [010] surfaces, respectively. (c) The Fermi surfaces (Fermi arcs) and their spin textures (in-plane components) for the [010] surface states. (d) The Fermi arcs of the [010] surface obtained from the fitted effective Hamiltonian with additional exchange field $h_1=6$ meV (see Sec. 3c for details). The discontinuity around the singular Fermi points becomes now obvious (enlarged in the insets).

Figure 4

Phase diagrams of Na${}_3$Bi with mass term $m=0$ meV (left panels) and $m=5.6$ meV (right panels). The high-order term of $B\left(\mathbf{k}\right)$ is neglected for the case of $m=5.6$ meV. (a) and (b) Phase diagrams. (c) – (j) Band dispersions, corresponding Fermi surfaces, and their topological charges for some characteristic phases (with $h_1$ fixed to be 10.0 meV). Only the neighborhood around one of the Dirac points is shown with the $\tilde{\mathbf{k}}$ defined as $\mathbf{k}-{\mathbf{k}}^c$. The $|\pm \frac{1}{2}\rangle$, $|\pm \frac{3}{2}\rangle$ are abbreviations (i.e., $J_z$ values) for the four bases, which are used to indicate the main component of the wave functions for the states away from band crossings (see Sec. 3b for details).