Hawking fragmentation and Hawking attenuation in Weyl semimetals

We study black and white hole analogs in Weyl semimetals with inhomogeneous nodal tilts. We study how the presence of a microscopic lattice, giving rise to low-energy fermion doubler states at large momenta that are not present for elementary particles, affects the analogy between Weyl Hamiltonians and general relativity. Using a microscopic tight-binding lattice model, we find the doubler states to give rise to Hawking fragmentation and Hawking attenuation of wave packets by the analog event horizon. These phenomena depend on an analog Hawking temperature and can be measured in metamaterials and solids, as we confirm by numerical simulations.

Figure 1

(a) Sketch of a Weyl semimetal with tilt varying along $x_1$ realizing a white hole analog. The asymptotic Hamiltonians far from the central scattering region define the band structures shown below. States with a group velocity towards the horizon (incoming states) are labeled $a$, outgoing states $b$. (b) An incoming wave packet is Hawking fragmented into outgoing wave packets with nonparallel spins. The transmission probability $X$ depends on an effective Hawking temperature, see Eq. (9). Wave packets $|{\mathrm{\Phi }}_i^{\left(j\right)}\rangle$ are labeled by the state they are centered around.

Figure 2

Numerical results for our tight-binding white hole analog at a cut along one node ($p_2=0$ and $p_3=\pi /a$), where we set ${\alpha }_{\mathrm{t}}=0.1$. As indicated by the dotted black line, the $S$-matrix elements $|S_{23}{|}^2$ and $|S_{32}{|}^2$ essentially equal the Hawking-like function $X$ of Eq. (9). Inset: the Hawking fragmentation rate proxy $\tilde{X}$ of Eq. (10) (dots) closely matches $X(1-X)$ (solid line).

Figure 3

Transmission elements $|S_{ij}{|}^2$ of the $S$ matrix for three-dimensional Weyl semimetal black hole analogs in nanowires, where we set ${\alpha }_{\mathrm{t}}=0.1$. Panels (a) and (b) are without, (c) and (d) with, magnetic pseudofield for cross sections of $5\times 5$ [(a), (c)] and $15\times 15$ [(b), (d)] sites. Solid blue lines show $X$ from Eq. (9). In (d) the red line is an example transmission element following $X$.

Figure 4

(a) Black hole analog realized in a Weyl semimetal with inhomogeneous nodal tilt. (b) Sketch of Hawking attenuation. All lables are similar to Fig. 1 in the main text.

Figure 5

Ingoing and outgoing wave packets in real space after scattering through the white hole (a) or black hole (b) horizon. For (a) the incoming wave packet is centered around $p_1=-0.1$ with a standard deviation $s=0.00125$. For (b) the incoming wave packet is centered around $p_1=2.026$ with a standard deviation $s=0.00125$. For both the white and black hole horizons the length of the scattering region is $L=400$, and we use ${\alpha }_{\mathrm{t}}=0.1$. The tilt amplitude changes between $V_{t,1}=0$ at the left end of the scattering region and $V_{t,1}=\pm 2$ at the right end of the scattering region.

Figure 6

Numerical results for the $S$-matrix element $|S_{32}{|}^2$ calculated for the different tilt profile tight-binding white hole analogs at a cut along one node ($p_2=0$ and $p_3=\pi /a$). We set ${\alpha }_{\mathrm{t}}=0.1$ for all tilt profiles. The results are compared to the theoretical function $X$ of Eq. (9) of the main text.