Fermi-Arc-Induced Vortex Structure in Weyl Beam Shifts

In periodic media, despite the close relationship between geometrical effects in the bulk and topological surface states, the two are typically probed separately. We show that when beams in a Weyl medium reflect off an interface with a gapped medium, the trajectory is influenced by both bulk geometrical effects and the Fermi arc surface states. The reflected beam experiences a displacement, analogous to the Goos-Hänchen or Imbert-Fedorov shifts, that forms a half-vortex in the two-dimensional surface momentum space. The half-vortex is centered where the Fermi arc of the reflecting surface touches the Weyl cone, with the magnitude of the shift scaling as an inverse square root away from the touching point, and diverging at the touching point. This striking feature provides a way to use bulk transport to probe the topological characteristics of a Weyl medium.

Figure 1

(a) Schematic of the reflection setup. A beam is incident from a Weyl medium in the space $z>0$, and reflects off a gapped medium at $z<0$. The vector $\overrightarrow{\mathrm{\Delta }}$ denotes the total displacement of the reflected beam. (b)–(c) Map of the reflection phase $\varphi$ experienced by an incident plane wave, versus the in-plane wave numbers $k_x$ and $k_y$ (normalized to $E/v$). Results are shown for two different values of ${\theta }_b$, which governs the boundary condition. The Fermi arc is denoted by dashes, and $\varphi$ exhibits a $-2\pi$ phase shift during a half-encirclement of the Fermi arc’s touching point (black arrow).

Figure 2

Maps of the beam shift $\overrightarrow{\mathrm{\Delta }}$ versus the mean in-plane wave numbers $K_x$ and $K_y$ (normalized to $E/v$) for the incident beam. Results are shown for two values of the boundary parameter: (a) ${\theta }_b=\pi /3$ and (b) ${\theta }_b=2\pi /3$. In each case, the left panel shows a streamline plot indicating the direction of $\overrightarrow{\mathrm{\Delta }}$ but not its magnitude; the right panel shows a quiver plot indicating both the direction and magnitude of $\overrightarrow{\mathrm{\Delta }}$, over a region of $K$-space surrounding the Fermi arc touching point. The dashes indicate the Fermi arc.

Figure 3

Maps of the beam shift $\overrightarrow{\mathrm{\Delta }}$, for the quadratic Hamiltonian (9) with $v=1$, $m_0=1$, and $m_1=100$. The horizontal and vertical axes are the mean in-plane wave numbers $K_x$ and $K_y$ (normalized to $E/v$) for the incident beam. Results are shown for (a) $E=0.75$, where the Hamiltonian describes a pair of Weyl cones, and (b) $E=1.5$, where the two Weyl cones have merged into a single band. The left panels show a streamline plot of $\overrightarrow{\mathrm{\Delta }}$, while the right panels show a quiver plot of $\overrightarrow{\mathrm{\Delta }}$ near the Fermi arc touching points. The dashes indicate the Fermi arc.