Anisotropic optics and gravitational lensing of tilted Weyl fermions

We show that tilted Weyl semimetals with a spatially varying tilt of the Weyl cones provide a platform for studying analogs to problems in anisotropic optics as well as curved spacetime. Considering particular tilting profiles, we numerically evaluate the time evolution of electronic wave packets and their current densities. We demonstrate that electron trajectories in such systems can be obtained from Fermat's principle in the presence of an inhomogeneous, anisotropic effective refractive index. On the other hand, we show how the electron dynamics reveal gravitational features and use them to simulate gravitational lensing around a synthetic black hole. These results bridge optical and gravitational analogies in Weyl semimetals, suggesting different pathways for experimental solid-state electron optics.

Figure 1

Current density emanating from a lead in the upper left corner, flowing through a scattering region without [(a) and (c)] and with [(b) and (d)] a horizon (black circle). In (a) and (b) only a single mode is shown, while in (c) and (d) all modes are present. Color indicates the absolute value of the current density, red arrows its direction (see SM for details).

Figure 2

Definition of the vectors of importance for the variational problem based on Fermat's principle. $\mathbf{v}$ is the group velocity tangential to the trajectory (blue dashed line) with angle $\vartheta$. $\mathbf{k}$ is the wave vector with angle $\alpha$. $r$ and $\phi$ are the polar coordinates.

Figure 3

Geodesics of light around a black hole (gray sphere) in the Painlevé-Gullstrand coordinates. The red lines represent light with a different initial position and zero radial velocity. The light gets deflected inward by the gravitational potential. The dashed black line is the photon sphere, the stable orbit of light. The blue dots and lines give a schematic representation of the lattice in a corresponding Weyl semimetal implementation, with thickness indicating the strength of hopping.

Figure 4

(a) Calculated trajectories of wave packets using the Chebyshev expansion method (blue), and trajectories obtained using Fermat's principle (red), in an inhomogeneously tilted Weyl system. The black solid line delineates the transition between type-I and type-II Weyl cones, which corresponds to the black hole horizon in the analogous gravitational system. The dotted black line corresponds to the photon sphere. The shade of blue denotes the inverse of the width $w$ of the wave packet relative to the initial width $w_0$. (b) Constant energy lines in the local Bloch Hamiltonians along five different trajectories. The black and cyan lines indicate the wave vector and group velocity, respectively. (c) Constant energy lines for local Bloch Hamiltonians throughout the whole system. In (b) and (c) the $\mathrm{\Gamma }$ point of the local Brillouin zone is represented by the black point and the color of the energy lines denotes the sign of $v_y$ red (blue) is positive (negative). (d) Probability density of a propagating wave packet shown at five different time steps.