Dispersion of Fermi arcs in Weyl semimetals and their evolutions to Dirac cones

We study dispersions of Fermi arcs in the Weyl semimetal phase by constructing an effective model. We calculate how the surface Fermi-arc dispersions for the top and bottom surfaces merge into the bulk Dirac cones in the Weyl semimetal at both ends of the arcs, and show that they have opposite velocities. This result is common to general Weyl semimetals, and is also confirmed by a calculation using a tight-binding model. Furthermore, by changing a parameter in the system while preserving time-reversal symmetry, we show that two Fermi arcs evolve into a surface Dirac cone when the system transits from the Weyl semimetal to the topological insulator phase. We also demonstrate that choices of surface terminations affect the pairing of Weyl nodes, from which the Fermi arcs are formed.

Figure 1

Bulk and surface states of the model for the WSM. The parameters are set as $v=\gamma =m_0=1$. (a) shows the bulk states, and (b) shows the surface Fermi arcs and the bulk states. (c) and (d) The Fermi surface for the Fermi energy at (c) $E=0.5$ and at (d) $E=1.5$.

Figure 2

(a) Phase diagram in the $\delta t_{+}$-${\lambda }_v$ plane, with $\delta t_{-}=0.1t$ and ${\lambda }_{\mathrm{so}}=0.1t$, where $\delta t_{\pm }=\delta t_1\pm \delta t_2$. The axes are in the unit of $t$. When ${\lambda }_v=0$, the system is I-symmetric, and no WSM phase appears. (b)–(d) Surface Fermi surface at $E=0$ around the point $M_2$ near the Fermi level with ${\lambda }_v=0.2t$ and the following values for $\delta t_{+}$: (b) $\delta t_{+}=0$ (WSM), (c) $\delta t_{+}=0.03t$ (WSM), and (d) $\delta t_{+}=0.05t$ (STI). The axes are in the unit of $2\pi /b$. (b) and (c) show Fermi arcs in the WSM phase, and (d) shows a surface Dirac cone in the STI phase. We note that the end points of the surface Fermi arcs in the WSM phase are the gapless points of the bulk bands.

Figure 3

Side views of the Fermi arcs around point $M_2=\frac{2\pi }{b}(0,1/\sqrt{3})$ with ${\lambda }_v=0.2t$, $\delta t_{+}=0$ (WSM phase) for (a) the top surface and (b) the bottom surface. The $k_x$ and $k_y$ axes are in the unit of $2\pi /b$. The red (blue) points are the gapless points, which have positive (negative) monopole charges for the Berry curvature.

Figure 4

Schematic drawing of the surface energy bands around a TRIM. The red (green) cone is the top (bottom) surface states. (a) In the STI phase, the two Dirac cones on the top and bottom surfaces are split in energy when the I symmetry is broken. (b) In the WSM phase, there are a pair of Fermi arcs on each surface. These Fermi arcs will evolve into a Dirac cone shown in (a) in the STI phase.

Figure 5

The surface Fermi surfaces at $E=0$ in the whole BZ when ${\lambda }_v=0.2t$. The values of $\delta t_{+}$ is $\delta t_{+}=0$ (WSM) for (a1) and (a2), and $\delta t_{+}=0.05t$ (STI) for (b1) and (b2). In (a1) and (b1), the surfaces are terminated without dangling bonds, and in (a2) and (b2) with dangling bonds. The insets show the magnified images of the surface Fermi surface around the $M_2$ point.