Non-Abelian properties of electron wave packets in the Dirac semimetals $A_3\mathrm{Bi}$ ($A=\mathrm{Na},\mathrm{K},\mathrm{Rb})$

The motion of electron wave packets in the Dirac semimetals $A_3\text{Bi}$ ($A=$ Na,K,Rb) is studied in a semiclassical approximation. Because of the twofold degeneracy of the Dirac points and a momentum-dependent gap term in the low-energy Hamiltonian, the associated Berry curvature is non-Abelian. In the presence of background electromagnetic fields, such a Berry curvature leads to a splitting of trajectories for the wave packets that originate from different Dirac points and chiral sectors. The nature of the splitting strongly depends on the background fields as well as the initial chiral composition of the wave packets. In parallel electric and magnetic fields, while a well-pronounced valley splitting is achieved for any chirality composition, the chirality separation takes place predominantly for the initially polarized states. On the other hand, in perpendicular electric and magnetic fields, there are clear deviations from the conventional Abelian trajectories, albeit without a well-pronounced valley splitting.

Figure 1

The energy spectrum of the low-energy model (1) at $\alpha =0$ (left panel) and $\alpha =10{\alpha }^{*}$ (right panel), where ${\alpha }^{*}$ is the characteristic value defined in Eq. (5) and ${\epsilon }_{+d}\left(0\right)$ is given in Eq. (10).

Figure 2

The positions ${\mathbf{r}}^{(\pm )}$ of the wave packets as a function of time $t$ (left panel) and the splitting $\mathrm{\Delta }\mathbf{r}={\mathbf{r}}^{(+)}-{\mathbf{r}}^{(-)}$ between the coordinates of the wave packets from different Dirac points (right panel). The red, blue, and green lines correspond to the $x$, $y$, and $z$ components, respectively. The solid and dashed lines represent the results for the wave packets described by Hamiltonians (7) and (8), respectively. We used $\alpha =0.5{\alpha }^{*}$, $\mathbf{E}=E\widehat{\mathbf{y}}$, and $\mathbf{B}=B\widehat{\mathbf{y}}$, where $E=200\text{V}/\text{m}$ and $B=10\text{G}$.

Figure 3

Left panel: the trajectories of the wave packets from different Dirac points for $t\le t_{\max }=5\text{ns}$. The red and blue lines represent the wave packets described by Hamiltonians (7) and (8) at $\alpha =0.5{\alpha }^{*}$, respectively. The black line corresponds to the case $\alpha =0$, where the wave packets do not split. Right panel: the time dependence of the probabilities $|{\eta }_{\pm }{|}^2$ to find the wave packets in certain ${\mathrm{\Gamma }}_5$-chirality states. The red and blue lines correspond to the wave packets described by Hamiltonians (7) and (8). The solid and dashed lines describe $|{\eta }_{+}{|}^2$ and $|{\eta }_{-}{|}^2$, respectively. We used $\mathbf{E}=E\widehat{\mathbf{y}}$ and $\mathbf{B}=B\widehat{\mathbf{y}}$, where $E=200\text{V}/\text{m}$ and $B=10\text{G}$.

Figure 4

The projections of the wave-packet trajectories onto the following planes: $x\text{−}y$ (a), $x\text{−}z$ (b), and $y\text{−}z$ (c). The red and blue lines correspond to the wave packets described by Hamiltonians (7) and (8), respectively. We used $t\le t_{\max }=5\text{ns}$, $\alpha =0.5{\alpha }^{*}$, $\mathbf{E}=E\widehat{\mathbf{y}}$, and $\mathbf{B}=B\widehat{\mathbf{y}}$, where $E=200\text{V}/\text{m}$ and $B=10\text{G}$.

Figure 5

The positions ${\mathbf{r}}^{(\pm )}$ of the wave packets as a function of time $t$ (left panel) and the splitting $\mathrm{\Delta }\mathbf{r}={\mathbf{r}}^{(+)}-{\mathbf{r}}^{(-)}$ of the coordinates of the wave packets from different Dirac points (right panel). The red, blue, and green lines correspond to the $x$, $y$, and $z$ components, respectively. The solid and dashed lines represent the results for the wave packets described by Hamiltonians (7) and (8), respectively. We used $\alpha =0.5{\alpha }^{*}$, $\mathbf{E}=E\widehat{\mathbf{y}}$, and $\mathbf{B}=B\widehat{\mathbf{z}}$, where $E=200\text{V}/\text{m}$ and $B=10\text{G}$.

Figure 6

Left panel: the trajectories of the wave packets corresponding to different Dirac points or valleys for $t\le t_{\max }=0.35\text{ns}$. The red and blue lines represent the wave packets described by Hamiltonians (7) and (8) at $\alpha =0.5{\alpha }^{*}$, respectively. The black line corresponds to the case $\alpha =0$, where the wave packets are not split. Right panel: the probabilities $|{\eta }_{\pm }{|}^2$ to find the wave packets in certain ${\mathrm{\Gamma }}_5$ states for $\alpha =0.5{\alpha }^{*}$. The red and blue lines correspond to the wave packets described by Hamiltonians (7) and (8). The solid and dashed lines describe $|{\eta }_{+}{|}^2$ and $|{\eta }_{-}{|}^2$, respectively. We used $\mathbf{E}=E\widehat{\mathbf{y}}$ and $\mathbf{B}=B\widehat{\mathbf{z}}$, where $E=200\text{V}/\text{m}$ and $B=10\text{G}$.

Figure 7

The projections of the wave-packet trajectories onto the following planes: $x\text{−}y$ (a), $x\text{−}z$ (b), and $y\text{−}z$ (c). The red solid and blue dashed lines correspond to the wave packets described by Hamiltonians (7) and (8), respectively. We used $t\le t_{\max }=0.35\text{ns}$, $\alpha =0.5{\alpha }^{*}$, $\mathbf{E}=E\widehat{\mathbf{y}}$, and $\mathbf{B}=B\widehat{\mathbf{z}}$, where $E=200\text{V}/\text{m}$ and $B=10\text{G}$.

Figure 8

The trajectories of the wave packets in parallel electric and magnetic fields. The red and blue lines correspond to the wave packets for the initial weights (33) described by Hamiltonians (7) and (8), respectively. The green and brown lines correspond to the wave packets for the initial weights (34) described by Hamiltonians (7) and (8), respectively. We used $t\le t_{\max }=1\mathrm{ns}$, $\alpha =0.5{\alpha }^{*}$, $\mathbf{E}=E\widehat{\mathbf{y}}$, and $\mathbf{B}=B\widehat{\mathbf{y}}$, where $E=200\text{V}/\text{m}$ and $B=10\text{G}$.

Figure 9

The projections of the wave-packet trajectories onto the following planes: $x\text{−}y$ (a), $x\text{−}z$ (b), and $y\text{−}z$ (c). The red solid and blue dashed lines correspond to the wave packets for the initial weights (33) described by Hamiltonians (7) and (8), respectively. The green and brown lines correspond to their counterparts with the initial weights (34). We used $t\le t_{\max }=1\mathrm{ns}$, $\alpha =0.5{\alpha }^{*}$, $\mathbf{E}=E\widehat{\mathbf{y}}$, and $\mathbf{B}=B\widehat{\mathbf{y}}$, where $E=200\text{V}/\text{m}$ and $B=10\text{G}$.