[Paper Review] Anisotropic optics and gravitational lensing of tilted Weyl fermions

Viktor Könye, Lotte Mertens+5 more|arXiv (Cornell University)|Oct 28, 2022

Topological Materials and Phenomena7 citations

TL;DR

This paper proposes that tilted Weyl semimetals with spatially varying Weyl cone tilting enable analogues of anisotropic optics and gravitational lensing for electrons. By mapping electron trajectories to Fermat’s principle in an anisotropic effective refractive index and simulating a synthetic black hole horizon, the study demonstrates electron beam bending and lensing, with experimentally measurable current density patterns in multi-terminal setups.

ABSTRACT

We show that tilted Weyl semimetals with a spatially varying tilt of the Weyl cones provide a platform for studying analogues 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 electrons 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 novel pathways for experimental solid-state electron optics.

Motivation & Objective

• To establish tilted Weyl semimetals with spatially varying tilting as a platform for analogues of anisotropic optics and gravitational physics.
• To connect electron dynamics in such systems to Fermat’s principle in anisotropic media through an effective refractive index.
• To simulate gravitational lensing effects, including synthetic black hole horizons, using electron wave packets and current density profiles.
• To provide a bridge between electronic condensed matter systems and general relativistic phenomena via emergent spacetime geometries.
• To propose experimentally accessible setups using multi-terminal transport to measure lensing effects in current density.

Proposed method

• Model the system using a real-space Hamiltonian with position-dependent tilt vector $\mathbf{t}(\mathbf{r})$, describing tilted Weyl cones with slowly varying tilting.
• Apply the eikonal approximation to derive semiclassical trajectories from Fermat’s principle with an anisotropic effective refractive index.
• Use numerical simulations to evolve electronic wave packets and track their group velocity and wave vector evolution under inhomogeneous tilting.
• Construct a multi-terminal scattering setup using the Kwant package to compute scattering states and electronic current densities.
• Map the effective geometry of the system to a curved spacetime metric, identifying event horizons from critical tilting profiles.
• Analyze the breakdown of the semiclassical approximation when wave packets delocalize or become momentum-space mixed.

Experimental results

Research questions

• RQ1 Can electron trajectories in tilted Weyl semimetals with spatially varying tilting be described by Fermat’s principle in an anisotropic effective medium?
• RQ2 How does inhomogeneous tilting in Weyl cones lead to gravitational lensing effects for electrons, analogous to light bending near a black hole?
• RQ3 What is the role of the effective refractive index in shaping electron current density patterns in a multi-terminal setup?
• RQ4 How do wave packet broadening and delocalization affect the validity of the semiclassical approximation in this system?
• RQ5 Can experimentally measurable current density profiles reveal signatures of synthetic black hole horizons in electronic transport?

Key findings

• Electron trajectories in systems with spatially varying Weyl cone tilting follow Fermat’s principle with an anisotropic effective refractive index, enabling anisotropic optics for electrons.
• The current density in a multi-terminal setup shows clear lensing effects: trajectories bend around a synthetic black hole horizon, with increased current flow to the bottom lead when all modes are included.
• Wave packet dynamics reveal that trajectories near the horizon experience stronger compression and faster delocalization, with the $w_0/w$ ratio decreasing significantly, indicating breakdown of the point-particle approximation.
• The group velocity and wave vector become non-parallel near the horizon due to elliptical Fermi surfaces, leading to trajectory bending via continuous rotation of the effective ellipses.
• The semiclassical description breaks down when wave packets become too delocalized in real space or too spread in momentum space, invalidating the linear dispersion approximation.
• Experimentally, the lensing effect is measurable via current density patterns in transport measurements, with conductance to the bottom lead increasing as the overtilted region's radius grows.