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.

Figure 1

Energy dispersion of a tilted Weyl Hamiltonian in 1D as in Eq. (2) for three different tilting parameters $t$.

Figure 2

Energy dispersion of (a) $H_1^{\text{1D}}$ as in Eq. (5a) and (b) $H_2^{\text{1D}}$ as in Eq. (5b) for three different tilting parameters $t$. The three tiltings correspond to the untilted type-I node, the node at the horizon, and the overtilted type-II node.

Figure 3

Energy dispersion of (a) $H_1^{\text{2D}}$ in Eq. (6a) and (b) $H_2^{\text{2D}}$ in Eq. (6b) for three different tilting parameters in the $x$ direction $t_x$, and $t_y=0$. The three tiltings correspond to the untilted type-I node, the node at the horizon, and the overtilted type-II node.

Figure 4

Propagation of wave packets in a 1D lattice with a horizon. The figure shows the center of mass of the wave packet as a function of time for different starting momenta in systems $H_1^{\text{1D}}$ and $H_2^{\text{1D}}$ in Eq. (4). The rescaled position and time are defined as $\xi =1-2x/L$ and $\tau =2T/L$. The dashed line shows the geodesic calculated from Eq. (15). The length of the system is $L=1000$.

Figure 5

Schematic representation for the propagation of wave packets with $E=0.5$ in systems (a) $H_1^{\text{1D}}$ and (b) $H_2^{\text{1D}}$. The local dispersion relation is shown at three points along the trajectory of the wave packets. The red dot represents the momentum of the wave packet, and the red arrow indicates the direction of change in this momentum. The black arrow shows the direction of the group velocity of the wave packet.

Figure 6

Propagation of wave packets in the 2D systems without tilt. The figure shows snapshots of the wave packet for different starting momenta in the system $H_1^{\text{2D}}$. The length and width of the system are $L=500$ and $W=500$, and the initial wave packet has a width of 40 sites. The anisotropy in the top row is due to the specific pseudospin configuration of the initial wave packet.

Figure 7

Propagation of wave packets in a 2D lattice with a horizon. The figure shows the center of mass of the wave packet along the $x$ axis as a function of time for different starting momenta in the systems $H_1^{\text{2D}}$ and $H_2^{\text{2D}}$ in Eq. (6). The rescaled position and time are defined as $\xi =1-2x/L$ and $\tau =2T/L$. The dashed line shows the geodesic calculated from Eq. (15). The length and width of the system are $L=500$ and $W=500$.

Figure 8

Propagation of wave packets in a 3D lattice with a horizon. The figure shows the center of mass of the wave packet along the $x$ axis as a function of time for different starting momenta in systems $H_1^{\text{2D}}$ and $H_2^{\text{2D}}$ in Eq. (6). The rescaled position and time are defined as $\xi =1-2x/L$ and $\tau =2T/L$. The dashed line shows the geodesic calculated from Eq. (15). The length, width, and height of the system are $L=400$ (we only simulated the left half of the system), $W=300$, and $H=300$.

Figure 9

Propagation of wave packets in a 1D lattice with different types of horizons using the $H_1^{\text{1D}}$ system in Eq. (4). Panels (a, b) show the results for the tilting profile with different power laws as in Eq. (16) for three different $\gamma$ exponents. Panels (c, d) show the results for the artificial horizons at finite momenta as in Eq. (20) for three different $k_0$ values. Panels (a) and (c) show the position dependence of the tilting parameter in the different horizon models. Panels (b) and (d) show the center of mass of the wave packet as a function of time for the models corresponding to panels (a) and (c). In panel (b) solid lines are with $k=0$ and dashed lines with $k=0.1$. In panel (d) solid lines are at tilting profiles and momenta corresponding to panel (c), while dashed lines are at momenta corresponding to panel (c) but with the linear tilting profile. The rescaled position and time are defined as $\xi =1-2x/L$ and $\tau =2T/L$. The length of the system in all panels is $L=1000$.

Figure 10

Propagating modes and scattering matrix of the $H_1^{\text{1D}}$ (a, c) and $H_2^{\text{1D}}$ (b, d) lattice models with a horizon. In panels (a) and (b) the dispersion relations in the left and right leads are shown. At the energy indicated by the dashed line, the possible propagating modes are indicated by the filled circles. In panels (c) and (d), the scattering probabilities between the modes at zero energy are shown. The length of the scattering region is $L=1000$.

Figure 11

Energy dependence of the scattering probability from the incoming mode $i_l$ to the outgoing modes $o_{r1}$ and $o_{r2}$ from Fig. 10. The colored lines show the numerical results for different system lengths. The solid black lines show the analytic formulas in Eq. (22).