Horizon physics of quasi-one-dimensional tilted Weyl cones on a lattice

To simulate the dynamics of massless Dirac fermions in curved space-times with one, two, and three spatial dimensions, we construct tight-binding Hamiltonians with spatially varying hoppings. These models represent tilted Weyl semimetals where the tilting varies with position, in a manner similar to the light cones near the horizon of a black hole. We illustrate the gravitational analogies in these models by numerically evaluating the propagation of wave packets on the lattice and then comparing them to the geodesics of the corresponding curved space-time. We also show that the motion of electrons in these spatially varying systems can be understood through the conservation of energy and the quasiconservation of quasimomentum. This picture is confirmed by calculations of the scattering matrix, which indicate an exponential suppression of any noncontinuous change in the quasimomentum. Finally, we show that horizons in the lattice models can be constructed also at finite energies using specially designed tilting profiles.