Artificial event horizons in Weyl semimetal heterostructures and their non-
equilibrium signatures

C. De Beule1*, S. Groenendijk1, T. Meng2, and T. L. Schmidt1

1 Department of Physics and Materials Science,

University of Luxembourg, L-1511 Luxembourg, Luxembourg

2 Institute for Theoretical Physics and Würzburg-Dresden Cluster of Excellence
ct.qmat, Technische Universität Dresden, 01069 Dresden, Germany

*<EMAIL_ADDRESS>
August 27, 2024

## Abstract

We investigate transport in type-I/type-II Weyl semimetal heterostructures
that realize effective black- or white-hole event horizons. We provide an
exact solution to the scattering problem at normal incidence and low energies,
both for a sharp and a slowly-varying Weyl cone tilt profile. In the latter
case, we find two channels with transmission amplitudes analog to those of
Hawking radiation. Whereas the Hawking-like signatures of these two channels
cancel in equilibrium, we demonstrate that one can favor the contribution of
either channel using a non-equilibrium state, either by irradiating the type-
II region or by coupling it to a magnetic lead. This in turn gives rise to a
peak in the two-terminal differential conductance which can serve as an
experimental indicator of the artificial event horizon.

###### Contents

  1. 1 Introduction
  2. 2 Model
  3. 3 Scattering at an effective horizon
    1. 3.1 Slowly-varying tilt profile
      1. 3.1.1 WKB solution
      2. 3.1.2 Solution near a linear horizon
      3. 3.1.3 Connection formulas and S matrix
    2. 3.2 Sharp tilt profile
  4. 4 Hawking effect out of equilibrium
    1. 4.1 Irradiation by circularly-polarized light
    2. 4.2 Coupling to magnetic leads
    3. 4.3 Differential conductance
  5. 5 Conclusion
  6. A Covariant form of the Weyl equation
  7. B Lattice model
  8. C Scattering matrix for sharp horizon

##  1 Introduction

Hawking radiation is the phenomenon whereby black holes slowly evaporate by
emitting thermal radiation due to quantum fluctuations near the event horizon
[1, 2]. It is one of the most exotic predictions of quantum field theory in a
curved spacetime but its experimental verification remains elusive. Indeed,
the corresponding Hawking temperature is inversely proportional to the mass of
the black hole such that the effect is masked by the cosmic microwave
background for generic black holes. However, analogs of Hawking radiation can
arise in other physical systems that are more amenable to experimental
verification, featuring artificial event horizons sharing many similarities to
their gravitational counterpart [3]. This field was pioneered by Unruh who
proposed such an analog at the interface between subsonic and supersonic flow
in a hydrodynamic system [4]. In condensed-matter physics, similar black-hole
analogs have been proposed in Bose-Einstein condensates [5, 6], optical
systems [7], borophene [8], one-dimensional fermionic chains [9], among
others, and recently in Weyl semimetals [10, 11, 12, 13, 14, 15, 16]. In this
work, we study electronic analogs of stimulated Hawking emission in
heterostructures containing an interface of a type-I and type-II Weyl
semimetal, as illustrated in Fig. 1(a). To the best of our knowledge, our work
provides the first explicit calculation of physical observables in Weyl
semimetal black hole analogs, and does so using a minimal model that captures
all salient features of Weyl semimetals.

Weyl semimetals host quasiparticles near generic crossings of the energy bands
whose low-energy physics are captured by a Weyl Hamiltonian ($\hbar=v_{F}=1$)
[17],

$\mathcal{H}_{\chi}=E_{\chi}+V\left(k_{z}-k_{\chi
z}\right)+\chi\left(\bm{k}-\bm{k}_{\chi}\right)\cdot\bm{\sigma}$ (1)

where $\bm{k}=(k_{x},k_{y},k_{z})$ and $\left(E_{\chi},\bm{k}_{\chi}\right)$
is the energy and position of the Weyl node in the Brillouin zone, and
$\bm{\sigma}=\left(\sigma_{x},\sigma_{y},\sigma_{z}\right)$ is the vector of
Pauli matrices. Weyl nodes carry a net Berry flux $\chi=\pm 1$ and necessarily
come in pairs as the total Berry flux in the Brillouin zone vanishes [18].
Depending on the tilt, given by the second term in Eq. (1), one can
distinguish two types of Weyl semimetals based on the Fermi surface topology,
as illustrated in Fig. 1(b). When $V^{2}<1$, the Fermi surface at the Weyl
node is a point (type I). At the critical tilt $V^{2}=1$, there is a Lifshitz
transition and the system is a nodal line semimetal. For $V^{2}>1$, the nodal
line evolves into an electron and hole pocket that touch at the Weyl node
(type II), shown in Fig. 1(b) and (c) [19].

Figure 1: (a) Weyl semimetal (WS) heterostructure where the type-II region is
irradiated, which gives rise to a Hawking current between the type-II and
type-I region. (b) Energy dispersion for $k_{y}=0$. (c) Zero-energy Fermi
surface in the type-II phase for $V/v_{F}=1.2$ where the Weyl nodes are shown
as black dots.

The connection between type-I/type-II heterostructures and Weyl fermions in an
effective curved spacetime is made explicit by writing down the covariant form
of the Weyl equation in a general spacetime with the tetrad formalism [10, 20,
21],

$\sigma^{a}e_{\,\,\,a}^{\mu}\left(\partial_{\mu}+\Omega_{\mu}\right)\psi=0,$
(2)

where we use the Einstein summation convention with $\mu=0,1,2,3$
corresponding to general spacetime coordinates and $a=0,1,2,3$ to local
inertial coordinates and where $e^{\mu}_{a}$ are tetrad components. A short
introduction to the tetrad formalism and more details on the covariant Weyl
equation are given in App. A. Here, we also introduced $\sigma^{a}$ given by
the identity and the Pauli matrices $\chi\bm{\sigma}$, and the spin connection
$\Omega_{\mu}$. The spin connection ensures covariance since
$\partial_{\mu}\psi$ does not transform as a spinor under local Lorentz
transformations. The corresponding Weyl Hamiltonian can be written as
$\mathcal{H}=\bm{\alpha}\cdot\bm{k}-i\Upsilon$ with

$\alpha^{i}(p)=(e_{\,\,\,b}^{0}\sigma^{b})^{-1}e_{\,\,\,a}^{i}\sigma^{a},\qquad\Upsilon(p)=(e_{\,\,\,b}^{0}\sigma^{b})^{-1}e_{\,\,\,a}^{\mu}\sigma^{a}\Omega_{\mu},$
(3)

where $p$ is a point in spacetime. If we compare this to Eq. (1), we identify

$e_{\,\,\,a}^{\mu}(z)=\delta^{\mu}_{a}+V(z)\delta_{3}^{\mu}\delta_{a}^{0},$
(4)

which correspond to the so-called acoustic metric [4]:

$g_{\mu\nu}=\begin{pmatrix}V^{2}-1&0&0&-V\\\ 0&1&0&0\\\ 0&0&1&0\\\
-V&0&0&1\end{pmatrix},$ (5)

and which follows from
$g^{\mu\nu}=e_{\,\,\,a}^{\mu}e_{\,\,\,b}^{\nu}\eta_{ab}$ with
$\eta_{ab}=\textrm{diag}\left(-1,1,1,1\right)$ the Minkowski metric of flat
spacetime (see App. A). The spin connection becomes

$\Omega_{\mu}=\frac{\chi\sigma^{3}}{2}\left(\delta_{\mu}^{3}-V\delta_{\mu}^{0}\right)V^{\prime},$
(6)

with $V^{\prime}=\partial V/\partial z$, and the corresponding Weyl equation
can then be written as

$i\partial_{t}\phi=\left[-i\left(\chi\bm{\sigma}+V\bm{e}_{z}\right)\cdot\nabla-
iV^{\prime}/2\right]\phi\equiv\mathcal{H}_{\chi}\phi.$ (7)

where the extra term proportional to $V^{\prime}$ comes from the spin
connection and ensures that the Hamiltonian is Hermitian for a position-
dependent tilt [14].

We can thus interpret a tilted Weyl cone in terms of a free Weyl fermion in an
effective curved spacetime with line element

$ds^{2}=g_{\mu\nu}dx^{\mu}dx^{\nu}=-dt^{2}+\left[dz-V(z)dt\right]^{2}+dx^{2}+dy^{2},$
(8)

whose null trajectories at normal incidence are given by

$\frac{dx}{dt}=\frac{dy}{dt}=0,\qquad\frac{dz}{dt}=V(z)\pm 1,$ (9)

where $\pm$ corresponds to so-called copropagating and counterpropagating
solutions. Note that one obtains the same equations from the semiclassical
equations of motion [22]

$\dot{\bm{r}}=\frac{\partial
E}{\partial\bm{k}}-\chi\dot{\bm{k}}\times\bm{\Omega},\qquad\dot{\bm{k}}=-\frac{\partial
V}{\partial z}\,k_{z}\,\bm{e}_{z},$ (10)

in the absence of external electromagnetic fields and where
$\bm{\Omega}(\bm{k})=\bm{k}/2k^{3}$ is the Berry curvature and
$E(\bm{k})=Vk_{z}\pm|\bm{k}|$. One can check that at normal incidence
($k_{x}=k_{y}=0$) the first equation of motion yields Eqs. (9).

Now consider the case where $V(z)$ is monotonic and $|V(0)|=1$, corresponding
to an interface at $z=0$ between a type-I and type-II Weyl semimetal [Fig.
1(a)]. While the type-I region ($|V|<1$) supports trajectories that propagate
in both directions, the type-II region ($|V|>1$) only supports unidirectional
trajectories. Semiclassically, the interface gives rise to a turning point
where the group velocity of counterpropagating modes vanishes. Hence, we can
regard the interface as an artificial event horizon, where the type-I and
type-II regions respectively correspond to a normal region of spacetime and a
“black hole” ($V>1$) or “white hole” ($V<-1$) spacetime. However, we note that
the acoustic metric is not a solution of the Einstein equations in four-
dimensional spacetime. Actually, since the effective spacetime is essentially
two dimensional, it is conformally flat [3]. However, due to the horizon, the
effective spacetime is not equivalent to a flat spacetime globally. For these
reasons, the acoustic metric that we consider is not equivalent to a black
hole from general relativity. Instead, the correspondence resides in the fact
that both feature an event horizon. Hence, the effective spacetime has a
similar causal structure as that of a real black hole. We note that an exact
mapping between a type-I/type-II interface to a Schwarzschild black hole in
Gullstrand-Painlevé coordinates is obtained for a radial tilt profile
$V(r)=-\sqrt{r_{S}/r}$ with $r_{S}$ the Schwarzschild radius[10].

In this work, we investigate transport through a type-I/type-II interface
using a minimal model for a Weyl semimetal with a tilt profile $V(z)$. We
consider both fast and slow varying tilt profiles relative to the Fermi
wavelength. In both cases, we obtain low-energy expressions for the scattering
matrix and the tunneling rates at normal incidence. In the case of a slowly-
varying tilt profile with a linear horizon, for energies $\omega$ close to the
Weyl node, counterpropagating particles tunnel through the effective horizon
from inside the black hole region via two channels with probability

$T_{1,2}=\frac{1}{1+e^{\pm 2\pi\omega/\hbar V^{\prime}(0)}},$ (11)

where $cV^{\prime}(0)$ is the effective gravitational field strength at the
horizon with $c$ the speed of light [1, 2, 3]. In equilibrium, both channels
contribute equally and since $T_{1}+T_{2}=1$ there is no net analog of Hawking
radiation [10, 16]. We therefore propose a means of creating a stationary non-
equilibrium distribution by irradiating the type-II region with light or by
injecting a spin-polarized current from a magnetic lead. Both cases favor the
occupation of one of the two channels, yielding a net non-equilibrium Hawking
effect. Summing over all transverse channels, we then find that the
differential conductance is asymmetric about the energy of the Weyl node and
features a peak whose position and height is characterized by the slope of the
tilt profile at the horizon.

This paper is organized as follows: In Sec. 2, we introduce the continuum
model for the Weyl semimetal heterostructure and in Sec. 3 we solve the
scattering problem at normal incidence for the case of a fast or slow varying
tilt profile. In the former case, we employ standard scattering theory where
the horizon only enters through the boundary conditions, while in the latter
we use the WKB approximation in combination with an approximate solution that
is valid close to a linear horizon. In Sec. 4, we discuss how to obtain a net
Hawking effect out of equilibrium. In particular, we show how to favor the
occupation of one of the two counterpropagating modes that tunnel across the
horizon, and we calculate the differential conductance. Finally, we present
our conclusions in Sec. 5.

##  2 Model

We start from a minimal model for a tilted Weyl semimetal with two isotropic
Weyl cones that are cotilted normal to the axis along which the nodes lie
[23]. For a bulk system, the Hamiltonian is given by
$\hat{H}=\sum_{\bm{k}}\hat{c}_{\bm{k}}^{\dagger}\mathcal{H}(\bm{k})\hat{c}_{\bm{k}}$
with $\hat{c}_{\bm{k}}=\left(\hat{c}_{1\bm{k}},\hat{c}_{2\bm{k}}\right)^{t}$
and

$\mathcal{H}(\bm{k})=Vk_{z}\,\sigma_{0}+\frac{1}{2k_{0}}\left(|\bm{k}|^{2}-k_{0}^{2}\right)\sigma_{x}+k_{y}\,\sigma_{y}+k_{z}\,\sigma_{z},$
(12)

with $\bm{k}=(k_{x},k_{y},k_{z})$ and $\sigma_{0}$ the identity matrix. The
Weyl nodes with chirality $\chi=\pm$ are located at momenta $\pm
k_{0}\bm{e}_{x}$ where we set $v_{F}=1$. The tilt $V$ is applied along the $z$
axis and given by the first term of Eq. (12). In general, our model only has a
mirror symmetry about the $yz$ plane:
$\mathcal{H}(-k_{x},k_{y},k_{z})=\mathcal{H}(k_{x},k_{y},k_{z})$ and a chiral
symmetry given by
$\sigma_{z}\mathcal{H}(k_{x},k_{y},-k_{z})\sigma_{z}=-\mathcal{H}(k_{x},k_{y},k_{z})$.

For $|V|<1$, this model gives a type-I Weyl semimetal with a point Fermi
surface at the Weyl nodes, while for $|V|>1$ we obtain a type-II Weyl
semimetal. This is illustrated in Fig. 1(b), where we show the energy
dispersion relation for $k_{y}=0$ in both phases. In the type-II phase, the
zero-energy Fermi surface consists of electron and hole pockets touching at
the Weyl nodes. The connectivity of these pockets depends on the details of
the tilting term. For our model, the pockets form a crescent between the Weyl
nodes, as shown in Fig. 1(c). For example, isolated pairs of electron and hole
pockets are obtained from the tilting term $\propto k_{x}k_{z}$ which tilts
the Weyl cones in opposite directions and preserves inversion symmetry. Note
also that the second-order terms in Eq. (12) break Lorentz covariance and
introduce a length scale $k_{0}^{-1}$ which regularizes divergences in the
limit $|V|\rightarrow 1$, known as the trans-Planckian problem [2, 3].
Moreover, these terms keep the Fermi surface finite in the overtilted regime
and therefore they cannot be neglected in a realistic system, see Fig. 1(c).

We now consider an interface between a type-I and a type-II Weyl semimetal
modeled by a tilt profile $V(z)$ with $V(+\infty)=V_{R}$ and
$V(-\infty)=V_{L}$ such that $|V_{L}|<1$ (type I) and $|V_{R}|>1$ (type II) as
shown in Fig. 1(a). Since translation symmetry is only broken along the tilt
axis, the single-particle wave function can be written as
$\Psi(\bm{r},t)=e^{i\left(\bm{k}_{\perp}\cdot\bm{r}_{\perp}-\omega
t\right)}\phi(z)$ with $\bm{r}_{\perp}=\left(x,\,y\right)$, where
$\bm{k}_{\perp}=\left(k_{x},k_{y}\right)$ and $\omega$ are the conserved
transverse momentum and energy, respectively. The continuum Hamiltonian
becomes

$\hat{H}=\sum_{\bm{k}_{\perp}}\int_{-L/2}^{L/2}dz\,\hat{\psi}_{\bm{k}_{\perp}}^{\dagger}(z)\mathcal{H}(\bm{k}_{\perp},-i\partial_{z})\hat{\psi}_{\bm{k}_{\perp}}(z),$
(13)

with $L$ the system length,
$\hat{\psi}_{\bm{k}_{\perp}}(z)=L^{-1/2}\sum_{k_{z}}\hat{c}_{\bm{k}}e^{ik_{z}z}$,
and

$\mathcal{H}=\frac{1}{2i}\frac{\partial V}{\partial
z}\,\sigma_{0}-i\left[V(z)\sigma_{0}+\sigma_{z}\right]\partial_{z}-\partial_{z}^{2}\,\sigma_{x}+\left(k_{\perp}^{2}-1/4\right)\sigma_{x}+k_{y}\,\sigma_{y},$
(14)

in dimensionless form such that now $\mathcal{H}$ is given in units of $2\hbar
v_{F}k_{0}$, $z$ in units of $1/2k_{0}$, momenta in units of $2k_{0}$, and the
tilt $V$ in units of $v_{F}$ such that the transition between the type-I and
type-II phase corresponds to $V^{2}=1$. Similarly as before, the first term in
the Hamiltonian (14) ensures Hermiticity. We now first consider the special
case $\bm{k}_{\perp}=\left(\pm 1/2,0\right)$ which we call the _Hawking
channel_. Later we return to the full problem which we solve numerically using
the Kwant Python package [24].

Figure 2: Dispersion relation for $\bm{k}_{\perp}=\left(\pm k_{0},0\right)$ in
the undertilted ($0\leq V<1$) and overtilted case ($V>1$) where the
copropagating and counterpropagating branch correspond to the red and blue
solid lines, respectively.

One can show that any solution $\psi(z,t)$ of the wave equation
$\mathcal{H}_{0}\psi=i\partial_{t}\psi$, where
$\mathcal{H}_{0}=\mathcal{H}(\pm 1/2,0,-i\partial_{z})$, obeys a continuity
equation

$\partial_{t}(\psi^{\dagger}\psi)+\partial_{z}j=0,$ (15)

with current density

$j=\psi^{\dagger}\left[V(z)\sigma_{0}+\sigma_{z}\right]\psi+2\,\textrm{Im}\big{(}\psi^{\dagger}\sigma_{x}\partial_{z}\psi\big{)},$
(16)

which is constant in space and time for stationary states
$\psi(z,t)=e^{-i\omega t}\phi(z)$ as our problem is effectively one
dimensional. For constant tilt, the eigenstates of $\mathcal{H}_{0}$ are given
by plane waves, $\phi(z)=e^{ikz}\varphi_{k\lambda}$, with $\lambda=\pm$ and

$\displaystyle\varphi_{k+}$
$\displaystyle=\frac{1}{\sqrt{2c\left(1+c\right)}}\begin{pmatrix}1+c\\\
k\end{pmatrix},$ (17) $\displaystyle\varphi_{k-}$
$\displaystyle=\frac{1}{\sqrt{2c\left(1+c\right)}}\begin{pmatrix}-k\\\
1+c\end{pmatrix},$ (18)

where $c(k)=\sqrt{1+k^{2}}$. Note that we dropped the subscript $z$ for
$k_{z}$ and that $\\{\mathcal{H}_{0},\sigma_{y}\\}=0$ such that
$\varphi_{k-}=-i\sigma_{y}\varphi_{k+}$. For concreteness, we now take $V\geq
0$ in the remainder of this work. In this case, the $\lambda=+$ ($\lambda=-$)
branch is referred to as the copropagating (counterpropagating) branch [3].
This nomenclature follows from the dispersion relations

$\omega=k\left[V+\lambda c(k)\right].$ (19)

Note that the same dispersion is also obtained for a Bose-Einstein condensate
with macroscopic velocity $V$ and contact interactions in mean-field
approximation [5]. Here, we have one copropagating mode and three
counterpropagating modes for a given energy, see Fig. 2. Closed-form
expressions for the momenta of these modes exist, but they are unwieldy and
not very insightful. Up to first order in $\omega$, we find

$\displaystyle k_{v}$
$\displaystyle=\frac{\omega}{V+1}+\mathcal{O}(\omega^{3}),$ (20)
$\displaystyle k_{u}$
$\displaystyle=\frac{\omega}{V-1}+\mathcal{O}(\omega^{3}),$ (21)
$\displaystyle k_{w_{1,2}}$
$\displaystyle=\pm\sqrt{V^{2}-1}+\frac{V\omega}{1-V^{2}}+\mathcal{O}(\omega^{2}),$
(22)

where co- and counterpropagating modes are labeled by $v$ and
$\\{u,w_{1},w_{2}\\}$, respectively, and the corresponding spinors are given
in lowest order by

$\displaystyle\varphi_{v}$
$\displaystyle\simeq\left(1,\,\,\frac{\omega}{2\left(1+V\right)}\right)^{t},$
(23) $\displaystyle\varphi_{u}$
$\displaystyle\simeq\left(\frac{\omega}{2\left(1-V\right)},\,\,1\right)^{t},$
(24) $\displaystyle\varphi_{w_{1,2}}$
$\displaystyle\simeq\frac{1}{\sqrt{2V}}\left(\mp\sqrt{V-1},\,\,\sqrt{V+1}\right)^{t},$
(25)

which are normalized up to first order in $\omega$. It follows that the $w$
modes are evanescent for $0\leq V<1$, while for $V>1$ all counterpropagating
modes are scattering states for

$|\omega|<\omega_{c}=k_{c}\left[V-c(k_{c})\right],$ (26)

with $k_{c}=(V^{2}-4+V\sqrt{V^{2}+8}\,)^{1/2}/(2\sqrt{2})$ [5]. Here, the
group velocity of the counterpropagating modes vanishes, corresponding to a
classical turning point (Fig. 2). The low-energy expansions of the wavevectors
and spinors are valid away from these extrema.

##  3 Scattering at an effective horizon

In this section, we solve the scattering problem for the Hawking channel
analytically at low energies in two limits, namely, when the tilt profile
$V(z)$ varies fast or slow compared to the Fermi wave length. In the former
case, we use plane-wave scattering modes together with a boundary condition
that conserves the current. In the latter case, we calculate the WKB wave
function away from the classical turning point in combination with an
approximate solution near a linear horizon valid at low energies close to the
Weyl node, in order to match the WKB wave functions at either side of the
horizon.

###  3.1 Slowly-varying tilt profile

####  3.1.1 WKB solution

We first consider the slowly-varying limit for which the tilt profile is
slowly varying on the scale of the Fermi wavelength. To this end, we use a WKB
_ansatz_

$\phi(z)=e^{i\int^{z}dz^{\prime}k(z^{\prime})}\varphi(z),$ (27)

with [25]

$\varphi(z)=a(z)\varphi_{k+}+b(z)\varphi_{k-},$ (28)

where $k(z)$, $a(z)$, and $b(z)$ are to be determined from the wave equation.
Here, the spinors $\varphi_{k\pm}$ are eigenstates for constant tilt [Eqs.
(17) and (18)], in which case $k$ corresponds to the momentum in the $z$
direction. If we plug the _ansatz_ in the wave equation, we obtain

$\displaystyle-i\left[k\left(V\sigma_{0}+\sigma_{z}\right)+k^{2}\sigma_{x}-\omega\right]\varphi$
$\displaystyle=\frac{1}{2}\left(V^{\prime}\sigma_{0}+2k^{\prime}\sigma_{x}\right)\varphi$
(29)
$\displaystyle+\left(V\sigma_{0}+\sigma_{z}+2k\sigma_{x}\right)\varphi^{\prime}-i\sigma_{x}\varphi^{\prime\prime},$

where primes indicate derivatives with respect to $z$. So far, everything is
exact. We now make a WKB approximation for a slowly-varying tilt profile by
only keeping terms up to first order in $V^{\prime}$ and dropping all terms
proportional to $\left(V^{\prime}\right)^{2}$ and $V^{\prime\prime}$. Next, we
multiply with the spinor $\left(\varphi_{k\pm}\right)^{t}$ from the left which
yields two coupled equations for the spinor coefficients $a(z)$ and $b(z)$,

$\displaystyle-ia\left[k\left(V+c\right)-\omega\right]$
$\displaystyle=\frac{av_{+}^{\prime}}{2}+a^{\prime}v_{+}+\left(\frac{1}{c}-V\right)\frac{bk^{\prime}}{2c^{2}}+\frac{b^{\prime}k}{c},$
(30) $\displaystyle-ib\left[k\left(V-c\right)-\omega\right]$
$\displaystyle=\frac{bv_{-}^{\prime}}{2}+b^{\prime}v_{-}+\left(\frac{1}{c}+V\right)\frac{ak^{\prime}}{2c^{2}}+\frac{a^{\prime}k}{c},$
(31)

where

$v_{\pm}(k)=V\pm\frac{d}{dk}\,kc(k)=V\pm\left(2c-\frac{1}{c}\right),$ (32)

is the group velocity where we used $dc/dk=k/c$.

In lowest order, we have $a=0$ ($b=0$) for counterpropagating (copropagating)
modes since this corresponds to the case of constant tilt. Furthermore, any
terms containing derivatives of the tilt will be small. Hence, we find
$\omega=k(z)\left[V(z)+\lambda c(k(z))\right]$ where $k(z)$ is the
semiclassical momentum. We now assume that first-order corrections to $a$
($b$) for counterpropagating (copropagating) modes are proportional to
$V^{\prime}$. For example, for counterpropagating modes ($\lambda=-1)$, Eqs.
(30) and (31) become

$\displaystyle 2ikca$
$\displaystyle=\left(V-\frac{1}{c}\right)\frac{bk^{\prime}}{2c^{2}}-\frac{b^{\prime}k}{c},$
(33) $\displaystyle\frac{b^{\prime}}{b}$
$\displaystyle=-\frac{v_{-}^{\prime}}{2v_{-}},$ (34)

where we dropped higher-order terms taking into account $a\propto V^{\prime}$
for counterpropagating modes. The second equation is solved by $b(z)\propto
1/\sqrt{v_{-}(z)}$ and plugging this back into the first equation yields

$\frac{ia}{b}=\frac{1}{4c^{2}}\left[\left(V-\frac{1}{c}\right)\frac{k^{\prime}}{kc}+\frac{v_{-}^{\prime}}{v_{-}}\right].$
(35)

Up to first order, the WKB solution of the counterpropagating branch thus
becomes

$\phi_{\mu}(z)=c_{\mu}\,\frac{e^{i\int^{z}dz^{\prime}k_{\mu}(z^{\prime})}}{\sqrt{v_{\mu}(z)}}\left(\sigma_{0}+\frac{ia}{b}\,\sigma_{y}\right)\varphi_{\mu}(z),$
(36)

where $c_{\mu}$ is a constant and $\mu\in\\{u,w_{1},w_{2}\\}$. Here
$k_{\mu}(z)$ is a solution of the local dispersion relation,
$v_{\mu}(z)=v_{-}(k_{\mu}(z))$ is the corresponding group velocity defined in
Eq. (32), and $\varphi_{\mu}(z)=\varphi_{k_{\mu}(z)-}$ is the wavefunction,
given in Eq. (18). The WKB solution breaks down at the classical turning point
where the group velocity vanishes, which is located across the horizon ($V>1$)
at finite energies [Eq. (26)]. The group velocity of copropagating ($v$) modes
never vanishes such that the WKB solution is valid everywhere and the $v$
modes are therefore perfectly transmitted in the slowly-varying limit. Indeed,
in lowest order an incident $v$ mode has nowhere else to go since it is
decoupled from the $u$ and $w$ modes. Observe also that the first-order
correction in Eq. (36) couples the $\\{u,w_{1},w_{2}\\}$ and $v$ modes since
$\varphi_{k+}=i\sigma_{y}\varphi_{k-}$.

In the following, we are mostly interested in the lowest-order result, where
the copropagating and counterpropagating branches are decoupled. From Eq.
(35), we find this generally holds for

$\left|k^{\prime}/k\right|,\,\,\left|v^{\prime}/v\right|\ll 1,$ (37)

which by definition is satisfied for a slowly-varying tilt profile away from
the turning point.

We now consider a slowly-varying tilt profile with $V(-\infty)=V_{L}$ with
$0\leq V_{L}<1$ and $V(+\infty)=V_{R}>1$ that increases monotonically, such
that the WKB solutions are valid sufficiently far away from the horizon, where
they eventually reduce to the solutions for constant tilt. To solve the
scattering problem for the counterpropagating modes, we need to connect wave
functions on opposite sides of the horizon. Hence, we need to find a solution
that is valid close to the horizon and match it to the WKB solution in a
region where both solutions hold simultaneously.

####  3.1.2 Solution near a linear horizon

Let us place the horizon at $z=0$ such that $V(0)=1$. We further restrict
ourselves to a linear horizon, i.e., close to the origin we assume the tilt
profile can be approximated as $V(z)=1+\alpha z$ with $\alpha>0$. In this
case, the classical turning point is located at
$z_{c}=3|\omega|^{2/3}/2\alpha$. In the linear regime, the wave equation
becomes

$\frac{1}{2}\left(\alpha-2i\omega\right)\phi+\begin{pmatrix}2+\alpha z&0\\\
0&\alpha z\end{pmatrix}\phi^{\prime}-i\sigma_{x}\phi^{\prime\prime}=0,$ (38)

or explicitly

$\displaystyle\frac{1}{2}\left(\alpha-2i\omega\right)\phi_{1}+\left(2+\alpha
z\right)\phi_{1}^{\prime}-i\phi_{2}^{\prime\prime}=0,$ (39)
$\displaystyle\frac{1}{2}\left(\alpha-2i\omega\right)\phi_{2}+\alpha
z\phi_{2}^{\prime}-i\phi_{1}^{\prime\prime}=0.$ (40)

This yields a fourth-order linear differential equation for $\phi_{1}$ or
$\phi_{2}$. If we further assume that $|\alpha z|\ll 1$ and
$\alpha,|\omega|\ll 1$, we find

$\phi_{2}^{(4)}+2\alpha
z\phi_{2}^{(2)}+\left(3\alpha-2i\omega\right)\phi_{2}^{(1)}\simeq 0,$ (41)

and $\phi_{1}\simeq\left(i/2\right)d\phi_{2}/dz$. Equation (41) can be solved
exactly, giving

$\phi_{2}(z)=c_{0}+\sum_{n=1}^{3}c_{n}z^{n-1}{}_{1}F_{2}\left(a_{n};b_{n};\frac{-2\alpha
z^{2}}{9}\right),$ (42)

where ${}_{1}F_{2}$ is a generalized hypergeometric function with
$a_{n}=\left(2n-1\right)/6-i\omega/3\alpha$,
$b_{1}=\left\\{\frac{1}{3},\frac{2}{3}\right\\}$,
$b_{2}=\left\\{\frac{2}{3},\frac{4}{3}\right\\}$, and
$b_{3}=\left\\{\frac{4}{3},\frac{5}{3}\right\\}$, and where $c_{0}$ and
$c_{n}$ are constants.

####  3.1.3 Connection formulas and S matrix

To determine the $S$ matrix, we have to match the asymptotic forms of Eq. (42)
to the WKB solutions. To simplify this calculation, we choose a particularly
convenient solution that is purely evanescent outside of the horizon ($z<0$)
[26, 27]. Physically, this solution corresponds to a specific linear
combination of modes that interfere in such a way that there is no
transmission to the normal region. This requirement fixes the coefficients
$c_{n}$ in (42). We will see that this solution is already sufficient to
determine the whole $S$ matrix. We find up to an overall constant factor,

$\phi_{2}(z\rightarrow-\infty)\simeq\frac{e^{-\frac{2}{3}\sqrt{-2\alpha
z^{3}}}(-z)^{-\frac{i\omega}{2\alpha}}}{\sqrt{-2\alpha z}},$ (43)

for $c_{0}=0$ and

$\displaystyle c_{1}$
$\displaystyle=\frac{\left(2\alpha/9\right)^{\frac{i\omega}{6\alpha}}}{\sqrt{3\pi}}\left(\frac{4}{3\alpha}\right)^{\frac{1}{3}}\Gamma\left(a_{1}\right)\cos\left[\frac{\pi}{3}\left(1+\frac{i\omega}{\alpha}\right)\right],$
(44) $\displaystyle c_{2}$
$\displaystyle=\frac{\left(2\alpha/9\right)^{\frac{i\omega}{6\alpha}}}{\sqrt{3\pi}}\,2\,\Gamma\left(a_{2}\right)\cosh\frac{\pi\omega}{3\alpha},$
(45) $\displaystyle c_{3}$
$\displaystyle=\frac{\left(2\alpha/9\right)^{\frac{i\omega}{6\alpha}}}{\sqrt{3\pi}}\left(6\alpha\right)^{\frac{1}{3}}\Gamma\left(a_{3}\right)\cos\left[\frac{\pi}{3}\left(1-\frac{i\omega}{\alpha}\right)\right].$
(46)

With the integration constants fixed, we find at the other side of the
horizon,

$\phi_{2}(z\rightarrow+\infty)\simeq
c_{u}\,\frac{z^{\frac{i\omega}{\alpha}}}{\sqrt{\alpha
z}}+c_{w_{1}}\,\frac{e^{+i\frac{2}{3}\sqrt{2\alpha
z^{3}}}z^{-\frac{i\omega}{2\alpha}}}{\sqrt{-2\alpha
z}}+c_{w_{2}}\,\frac{e^{-i\frac{2}{3}\sqrt{2\alpha
z^{3}}}z^{-\frac{i\omega}{2\alpha}}}{\sqrt{-2\alpha z}},$ (47)

with

$c_{u}=\frac{\sqrt{2\pi}\left(2\alpha\right)^{\frac{i\omega}{2\alpha}}}{\Gamma\left(\frac{1}{2}+\frac{i\omega}{\alpha}\right)},\qquad
c_{w_{1,2}}=\pm e^{\pm\frac{\pi\omega}{2\alpha}},$ (48)

which correspond to the matching coefficients of the WKB modes. Indeed, in the
linear regime, away from the turning point, the WKB modes are given by

$\displaystyle\frac{e^{i\int^{z}dz^{\prime}k_{u}(z^{\prime})}}{\sqrt{v_{u}}}\,\varphi_{u}$
$\displaystyle\simeq\frac{z^{\frac{i\omega}{\alpha}}}{\sqrt{\alpha
z}}\begin{pmatrix}-\frac{\omega}{2\alpha z}\\\ 1\end{pmatrix},$ (49)
$\displaystyle\frac{e^{i\int^{z}dz^{\prime}k_{w_{1,2}}(z^{\prime})}}{\sqrt{v_{w_{1,2}}}}\,\varphi_{w_{1,2}}$
$\displaystyle\simeq\frac{e^{\pm i\frac{2}{3}\sqrt{2\alpha
z^{3}}}z^{-\frac{i\omega}{2\alpha}}}{\sqrt{-2\alpha
z}}\begin{pmatrix}\mp\sqrt{\frac{\alpha z}{2}}\\\ 1\end{pmatrix},$ (50)

where Eq. (50) holds for $\ln z\gg\alpha z/2$. Note that we used the low-
energy forms of the wavevectors, which hold away from the turning point $z\gg
z_{c}$.

Figure 3: Real and imaginary part of the second spinor component $\psi_{2}$
for the case of zero transmission outside of the horizon and the tilt profile
$V(z)=1+\tanh\left(\alpha z\right)$ with $\alpha=0.1$ and $\omega=0.05$. Solid
curves give the exact solution, while the circles and crosses give the WKB
solutions and the approximate solution near the linear horizon, respectively.
The vertical dotted line marks the classical turning point.

Hence, we find that the WKB solution and the approximate solution in the
vicinity of the horizon both hold in a region where $|z-z_{c}|\gg 1$ and
$|z|\ll 1/\alpha$ are satisfied simultaneously. We demonstrate this explicitly
in Fig. 3 where we show the exact solution, obtained from numerically solving
the stationary wave equation, together with the approximate solution near the
horizon and the WKB solution. Here, the total WKB solution is given by

$\phi_{s}^{(\textrm{\tiny WKB})}(z)=\sum_{\mu}c_{\mu}\phi_{\mu
s}(z_{0})\,\frac{\sqrt{v_{\mu}(z_{0})}}{\varphi_{\mu
s}(z_{0})}\frac{e^{i\int_{z_{0}}^{z}dz^{\prime}k_{\mu}(z^{\prime})}}{\sqrt{v_{\mu}(z)}}\,\varphi_{\mu
s}(z),$ (51)

where $s=1,2$ corresponds to the spinor components and the sum runs over
$\mu=\\{w_{2}\\}$ with $c_{w_{2}}=1$ outside the horizon and
$\mu=\\{u,w_{1},w_{2}\\}$ with the $c_{\mu}$ given in Eq. (48) inside the
horizon. Here, $\phi_{\mu s}(z_{0})$ are the WKB modes in the linear regime
and the fitting parameter $z_{0}<0$ ($z_{0}>0$) outside (inside) the horizon.
For simplicity, we take the same $z_{0}$ for all modes in a given region. We
then optimize its value by hand in a region where both solutions should
approximately hold. In the figure, we see that the approximations match
reasonably well to the exact solution. While the WKB solution breaks down near
the classical turning point and the horizon at the origin, the approximate
solution near the horizon fits perfectly in the type-I region, but starts to
fail in the type-II region away from the horizon. In general, matching becomes
worse with increasing $|\omega|$ and $\alpha$ as expected from our
assumptions. Inside the horizon, the wave function features an envelope from
the $u$ mode which has a relatively long wavelength, while the oscillations
inside the envelope are due to the short-wavelength $w$ modes (Fig. 2).

The solution that we obtained in the previous section is purely decaying in
the type-I region such that $b_{Lu}=0$ and the scattering coefficients are
related by

$\begin{pmatrix}0\\\ b_{Ru}\\\
b_{Rv}\end{pmatrix}=\begin{pmatrix}t_{uw_{1}}&t_{uw_{2}}&0\\\
r_{uw_{1}}&r_{uw_{2}}&0\\\ 0&0&1\end{pmatrix}\begin{pmatrix}a_{Rw_{1}}\\\
a_{Rw_{2}}\\\ a_{Lv}\end{pmatrix},$ (52)

where we already took into account that the $v$ modes are decoupled from the
$u$ and $w$ modes for a slowly-varying tilt profile. From the first equation
of (52), we obtain

$\frac{t_{uw_{2}}}{t_{uw_{1}}}=-\frac{a_{Rw_{1}}}{a_{Rw_{2}}}=e^{\frac{\pi\omega}{\alpha}},$
(53)

where we used Eq. (48). Together with the unitarity of the $S$ matrix, we find

$S_{\textrm{counter}}=\frac{\left(2\alpha\right)^{\frac{i\omega}{2\alpha}}e^{-\frac{\pi\omega}{2\alpha}}\Gamma\left(\frac{1}{2}-\frac{i\omega}{\alpha}\right)}{\sqrt{2\pi}}\begin{pmatrix}1&e^{\frac{\pi\omega}{\alpha}}\\\
e^{\frac{\pi\omega}{\alpha}}&-1\end{pmatrix},$ (54)

with scattering probabilities

$T_{u\leftarrow w_{1,2}}=R_{u\leftarrow
w_{2,1}}=\frac{1}{1+e^{\pm\frac{2\pi\omega}{\alpha}}},$ (55)

which have the form of a Fermi-Dirac distribution with an effective Hawking
temperature $T_{H}=\hbar v\alpha/2\pi k_{B}$. Corrections to this low-energy
result yield an energy-dependent Hawking temperature
$T_{H}(\omega)=T_{H}(-\omega)$. These analytical results agree well with
numerical lattice calculations (see App. B).

Note that the transmission saturates for $|\omega|\approx\alpha$ such that
tunneling through the horizon occurs only for $|\omega|<\alpha$. We can
understand this by noting that the $S$ matrix has simple poles at

$\omega=-i\alpha\left(n+\frac{1}{2}\right),\qquad\left(n=0,1,2,\ldots\right),$
(56)

which correspond to quasi-bound states [28]. Classically, an incoming $w_{1}$
or $w_{2}$ particle is completely reflected at the turning point and the
classical contribution to the transmission is a step function
$\Theta(\mp\omega)$. However, quantum-mechanically the $w$ particles can
tunnel through the horizon via a transient state with lifetime $1/\alpha$.

Before we proceed with the implications of these results in a two-terminal
transport setup, we first consider the opposite limit where the tilt profile
is sharp relative to the Fermi wavelength. We will demonstrate that in this
case the co- and counterpropagating modes are coupled by the horizon.
Nevertheless, one can still define a Hawking temperature at low energies, even
though the transmission is not a thermal distribution in this case.

###  3.2 Sharp tilt profile

When the tilt profile is sharp on the scale of the Fermi wavelength (i.e., the
limit $\alpha\gg 1$), an incoming wave packet cannot resolve the precise
details of the interface and we can model the tilt profile with a step
function

$V(z)=V_{L}\Theta(-z)+V_{R}\Theta(z),$ (57)

where $\Theta$ is the Heaviside step function. Assuming the wave function is
continuous at $z=0$, we integrate the wave equation
$\mathcal{H}_{0}\phi=\omega\phi$ over an infinitesimal region of length
$2\epsilon$ centered at the origin. This gives

$\left[-i\partial_{z}\phi\right]_{-\epsilon}^{+\epsilon}+\frac{V_{R}-V_{L}}{2}\,\sigma_{x}\,\phi(0)=0,$
(58)

where we used $\partial V/\partial z=\left(V_{R}-V_{L}\right)\delta(z)$. These
boundary conditions are physically sound since they keep the current density
continuous

$\displaystyle j_{R}$
$\displaystyle=\left[\phi^{\dagger}\left(V_{R}\sigma_{0}+\sigma_{z}\right)\phi+2\,\textrm{Im}\big{(}\phi^{\dagger}\sigma_{x}\partial_{z}\phi\big{)}\right]_{z=+\epsilon}$
(59)
$\displaystyle=\left[\phi^{\dagger}\left(V_{R}\sigma_{0}+\sigma_{z}\right)\phi+2\,\textrm{Im}\big{(}\phi^{\dagger}\sigma_{x}\partial_{z}\phi\big{)}\right]_{z=-\epsilon}-\left(V_{R}-V_{L}\right)|\phi(0)|^{2}$
(60)
$\displaystyle=\left[\phi^{\dagger}\left(V_{L}\sigma_{0}+\sigma_{z}\right)\phi+2\,\textrm{Im}\big{(}\phi^{\dagger}\sigma_{x}\partial_{z}\phi\big{)}\right]_{z=-\epsilon}=j_{L}.$
(61)

We now consider a black hole horizon, i.e., we take $0\leq V_{L}<1$ and
$V_{R}>1$. Since the tilt is constant in each region, the wave function is
given by a superposition of plane waves. In the type-I region ($z<0$), we
obtain

$\Phi_{L}(z)=\frac{a_{Lv}}{\sqrt{v_{Lv}}}\,\varphi_{Lv}e^{ik_{Lv}z}+\frac{b_{Lu}}{\sqrt{-v_{Lu}}}\,\varphi_{Lu}e^{ik_{Lu}z}+c_{L}\,\varphi_{Lw}e^{ik_{Lw}z},$
(62)

with $\textrm{Im}\,k_{Lw}<0$ and where $a$, $b$, and $c$ are coefficients of
incoming, outgoing, and evanescent modes. Here, we also normalized the
scattering states such that each mode contributes unit current. Up to first
order in $\omega$, the group velocities are given by

$\displaystyle v_{v}$ $\displaystyle\simeq V+1,$ (63) $\displaystyle v_{u}$
$\displaystyle\simeq V-1,$ (64) $\displaystyle v_{w_{1,2}}$
$\displaystyle\simeq\frac{1-V^{2}}{V}\pm\frac{2+V^{-2}}{\sqrt{V^{2}-1}}\,\omega.$
(65)

In the type-II region ($z>0$), we find for $|\omega|<\omega_{c}$,

$\displaystyle\Phi_{R}(z)$
$\displaystyle=\frac{b_{Rv}}{\sqrt{v_{Rv}}}\,\varphi_{Rv}e^{ik_{Rv}z}+\frac{b_{Ru}}{\sqrt{v_{Ru}}}\,\varphi_{Ru}e^{ik_{Ru}z}$
(66)
$\displaystyle+\frac{a_{Rw_{1}}}{\sqrt{-v_{Rw_{1}}}}\,\varphi_{Rw_{1}}e^{ik_{Rw_{1}}z}+\frac{a_{Rw_{2}}}{\sqrt{-v_{Rw_{2}}}}\,\varphi_{Rw_{2}}e^{ik_{Rw_{2}}z},$

such that in this case the $S$ matrix can be written as

$\begin{pmatrix}b_{Lu}\\\ b_{Ru}\\\
b_{Rv}\end{pmatrix}=\begin{pmatrix}t_{uw_{1}}&t_{uw_{2}}&r_{uv}\\\
r_{uw_{1}}&r_{uw_{2}}&t_{uv}\\\
r_{vw_{1}}&r_{vw_{2}}&t_{vv}\end{pmatrix}\begin{pmatrix}a_{Rw_{1}}\\\
a_{Rw_{2}}\\\ a_{Lv}\end{pmatrix}.$ (67)

On the other hand, for $|\omega|>\omega_{c}$, the solution in the overtilted
region also consists of two scattering states and one evanescent mode. We do
not discuss this regime here, as we are mostly interested in the low-energy
physics.

The $S$ matrix is then determined as usual. Namely, by setting all but one of
the incoming coefficients zero, and calculating the outgoing coefficients with
the boundary conditions at the origin. Note that one must include the
evanescent modes to obtain a unique solution. In principle, one can obtain
closed form expressions for the scattering coefficients but this is cumbersome
as the wavevectors are given by the roots of a fourth-order polynomial. When
the incoming mode comes from the effective black hole region behind the
horizon, we find up to first order in $\omega$,

$\displaystyle T_{u\leftarrow w_{1,2}}$
$\displaystyle\simeq\frac{8\left(V_{L}+V_{R}\right)}{\left(V_{L}+V_{R}+2\right)^{2}}\left(\frac{1}{2}\mp\frac{\omega}{4k_{B}T_{u}}\right),$
(68) $\displaystyle R_{v\leftarrow w_{1,2}}$
$\displaystyle\simeq\frac{\left(V_{L}-V_{R}\right)^{2}}{\left(V_{L}+V_{R}+2\right)^{2}}\left(\frac{1}{2}\pm\frac{\omega}{4k_{B}T_{v}}\right),$
(69)

and $R_{u\leftarrow w_{1,2}}=1-T_{u\leftarrow w_{1,2}}-R_{v\leftarrow
w_{1,2}}$ with

$\displaystyle T_{u}$
$\displaystyle=\frac{1}{2k_{B}}\frac{\left(1-V_{L}^{2}\right)\left(V_{R}-V_{L}\right)\left(V_{R}^{2}-1\right)^{3/2}}{2+V_{R}^{4}+3V_{L}^{3}V_{R}+V_{L}^{2}\left(1-4V_{R}^{2}\right)+V_{L}V_{R}\left(2V_{R}^{2}-5\right)},$
(70) $\displaystyle T_{v}$
$\displaystyle=\frac{1}{2k_{B}}\frac{\left(V_{R}-V_{L}\right)\left(V_{R}^{2}-1\right)^{3/2}}{3V_{R}\left(V_{R}^{2}-2\right)+V_{L}\left(V_{R}^{2}+2\right)},$
(71)

which are the effective Hawking temperatures of intrabranch and interbranch
processes, respectively, for the sharp tilt profile ($\alpha\gg 1$) [3]. This
interpretation rests on our results for the slowly-varying limit ($\alpha\ll
1)$ and demonstrates that some aspect of analog Hawking radiation survives for
the sharp tilt profile at low energies.

Figure 4: Transmission and reflection probabilities for an incoming mode
incident on the sharp horizon from the black hole (type II) region for
$(V_{L},V_{R})=(0.2,2)$. Dashed lines correspond to the first-order analytical
results given in Eqs. (68) and (69).

These low-energy expressions are compared to the exact results in Fig. 4. Note
that the sharp horizon couples co- and counterpropagating modes, e.g., through
scattering processes such as $w_{1}(w_{2})\rightarrow v$ although these
processes are generally suppressed. As we discussed in the previous section,
such processes become negligible in the slowly-varying tilt profile. Moreover,
unlike in the slowly-varying limit, the transmission functions now depend
explicitly on the asymptotic values of the tilt profile. Similar expressions
for the transmission functions can be obtained when the incoming mode is
incident on the horizon from the normal (type-I) region, although the first-
order term vanishes in this case. The complete $S$ matrix for the sharp
horizon at low energies is given in App. C.

##  4 Hawking effect out of equilibrium

In equilibrium, there is no net Hawking current as the particle number is
conserved in our system, unlike for an actual black hole which provides an
energy source for particle creation [29, 10]. Furthermore, as long as the
type-II region is in local equilibrium, the total current out of the black
hole region, summing contributions from $w_{1}$ and $w_{2}$ modes, is always
ballistic (in the absence of disorder) and the two-terminal conductance is
simply a measure of the density of states. In order to obtain a net Hawking
current, we require a non-equilibrium occupation in the type-II region. For
example, if the type-II phase is induced in a quenched way [10],
$V(t)=V_{0}\Theta(t)$ with $V_{0}>1$, one obtains a transient state with
excited $w_{1}$ modes above the Fermi level and empty $w_{2}$ states below the
Fermi level, giving rise to a net transmission of $w$ modes.

Here, we propose two alternative ways of achieving a non-equilibrium situation
by taking advantage of the spin structure of the $w$ modes. In particular, we
demonstrate that one can favor populating $w_{1}$ over $w_{2}$ by exciting a
photocurrent with circularly-polarized light or by injecting a spin-polarized
current from a magnetic lead. In both cases, the occupation of one of the $w$
modes is favored, which will then tunnel through the horizon from the type-II
region, giving rise to a net Hawking current in the type-I region, assuming
relaxation due to, e.g., disorder, within the type-II region occurs over a
sufficiently long time scale such that the favored $w$ mode can be transmitted
across the horizon. In fact, transport lifetimes up to $\tau\approx 45$ ps
have been measured in the type-I Weyl semimetal TaAs [30], giving rise to a
rather long mean free path of 5.2 $\mu$m.

###  4.1 Irradiation by circularly-polarized light

We can favor the occupation of one of the $w$ modes by irradiating the type-II
region. This gives rise to optical transitions inside the overtilted region
from $v$ modes to $w_{2}$ modes for $V/v_{F}>1$ (Fig. 2). In this case, there
are no states available for optical transitions to the $w_{1}$ modes (we
approximate optical transitions to be effectively momentum-conserving). This
continues to hold in the presence of more bands as long as the energy
separation to the lower bands is of a different energy range than the photon
energy of interest.

Figure 5: Out-of-equilibrium occupation $n_{\bm{k}+}$ of the conduction band,
integrated over the photon energies, in units of
$4\pi\tau(ev_{F}A_{0})^{2}/\hbar$ for $k_{y}=0$ (left) and at $k_{x}=-k_{0}$
(right). The black region for negative (positive) $k_{z}$ denotes an electron
(hole) pocket where optical transitions are forbidden for $E_{F}=0$. The
occupied states with negative $k_{z}$ away from the electron pockets have
negative group velocities and are transmitted across the horizon.

For concreteness, we consider circularly-polarized light of frequency $\Omega$
traveling in the $z$ direction, as illustrated in Fig. 1(a). Here, we assume
that the light hits the type-II Weyl semimetal sufficiently far away from the
horizon where the tilt $V/v_{F}>1$ is essentially constant. The light-matter
interaction is treated classically with the vector potential

$\bm{A}(t)=\bm{A}_{+}e^{-i\Omega t}+\bm{A}_{-}e^{i\Omega t},$ (72)

where $\bm{A}_{\pm}=A_{0}\left(\bm{e}_{x}\pm i\bm{e}_{y}\right)/\sqrt{2}$.
Note that we discard Zeeman coupling which is suppressed by a factor $10^{-3}$
compared to orbital coupling in generic Weyl semimetals [31]. Letting
$\bm{k}\to\bm{k}+e\bm{A}/\hbar$ in (12) gives the light-matter interaction
$V_{\pm}=e\bm{J}\cdot\bm{A}_{\pm}$ to first order in the electron charge $-e$.
These terms describe transitions between occupied and unoccupied modes via
absorption ($V_{+}$) and emission ($V_{-}$). Here, we defined the current
operator $\bm{J}$ with components

$J_{x}=\frac{v_{F}}{k_{0}}k_{x}\sigma_{x},\quad
J_{y}=\frac{v_{F}}{k_{0}}k_{y}\sigma_{x}+v_{F}\sigma_{y},\quad
J_{z}=\frac{v_{F}}{k_{0}}k_{z}\sigma_{x}+v_{F}\sigma_{z}+V\sigma_{0}.$ (73)

The rate of change of the distribution function $n_{\bm{k}\lambda}$ of the
$\lambda=\pm$ band is obtained from the Boltzmann equation in the relaxation-
time approximation [32],

$\frac{\partial n_{\bm{k}\lambda}}{\partial t}+\dot{\bm{k}}\cdot\frac{\partial
n_{\bm{k}\lambda}}{\partial\bm{k}}+\dot{\bm{r}}\cdot\frac{\partial
n_{\bm{k}\lambda}}{\partial\bm{r}}=\Gamma_{\bm{k}}(\Omega)\left(n_{\bm{k}-\lambda}-n_{\bm{k}\lambda}\right)-\frac{n_{\bm{k}\lambda}-n_{\bm{k}\lambda}^{0}}{\tau},$
(74)

where the relaxation time $\tau$ takes into account intraband impurity and
phonon scattering [33] and $\Gamma_{\bm{k}}(\Omega)$ gives the rates for
vertical transitions, calculated with Fermi’s golden rule,

$\Gamma_{\bm{k}}(\Omega)=\frac{2\pi}{\hbar}\left|\left<\varphi_{\bm{k}+}\right|V_{+}\left|\varphi_{\bm{k}-}\right>\right|^{2}\delta\left(E_{+}(\bm{k})-E_{-}(\bm{k})-\hbar\Omega\right),$
(75)

where the dispersion relation is given by

$E_{\pm}(\bm{k})=2\hbar v_{F}k_{0}\left[\frac{V}{v_{F}}\frac{k_{z}}{2k_{0}}\pm
d(\bm{k})\right].$ (76)

with
$d=\sqrt{\left[(k/2k_{0})^{2}-1/4\right]^{2}+(k_{y}/2k_{0})^{2}+(k_{z}/2k_{0})^{2}}$
and $k=|\bm{k}|$. If we Fourier transform Eq. (74) and consider the DC limit,
as well as the long wavelength limit ($\partial_{\bm{r}}n_{\bm{k}}\approx 0$),
the non-equilibrium occupation is given in lowest order of $A_{0}$ by

$n_{\bm{k}\lambda}\simeq
n^{0}_{\bm{k}\lambda}+\tau\Gamma_{\bm{k}}(\Omega)\left(n^{0}_{\bm{k}-\lambda}-n^{0}_{\bm{k}\lambda}\right).$
(77)

The transition matrix elements writes:

$\left|\left<\varphi_{\bm{k}+}\right|V_{+}\left|\varphi_{\bm{k}-}\right>\right|^{2}=2(ev_{F}A_{0})^{2}\,F(\bm{k}),\qquad
F(\bm{k})=\frac{\left[d+k_{x}k_{z}/(2k_{0}^{2})\right]^{2}}{4d^{2}},$ (78)

where $0\leq F\leq 1$ and which at the Weyl nodes $\bm{k}_{\perp}^{(\pm)}=(\pm
k_{0},0)$ reduces to

$F(k_{z})=\frac{\left[c(k_{z})\pm\operatorname{sgn}(k_{z})\right]^{2}}{4c(k_{z})^{2}}.$
(79)

where $c(k_{z})=\sqrt{1+(k_{z}/2k_{0})^{2}}$. Since there are no states below
the $w_{1}$ modes, optical transitions are absent and the occupation above the
Fermi energy vanishes at zero temperature, i.e., $\Delta n_{w_{1}}=0$. On the
other hand, for $w_{2}$ modes, at zero temperature, the difference in
occupation between the two bands vanishes inside the electron pocket which
corresponds to $k_{w_{2}}(E_{F})\leq k_{z}\leq 0$ where $E_{F}$ is the Fermi
energy and $k_{w_{2}}$ is given by Eq. (22). The occupation of $w_{2}$ modes
for $\bm{k}=(\bm{k}^{(\pm)}_{\perp},k_{z})$ is thus given by

$\Delta
n_{w_{2}}=\frac{4\pi\tau(ev_{F}A_{0})^{2}}{\hbar}\,F(k_{z})\,\Theta\left[k_{w_{2}}(E_{F})-k_{z}\right]\delta\left(\hbar
v_{F}|k_{z}|c(k_{z})-\hbar\Omega\right),$ (80)

such that a net occupation imbalance is generated. Away from
$\bm{k}_{\perp}=\bm{k}_{\perp}^{(\pm)}$, we plot in Fig. 5 the non-equilibrium
occupation, integrated over the energy, in units of
$4\pi\tau(ev_{F}A_{0})^{2}/\hbar$. As we see in the right figure, for the Weyl
node at $k_{x}=-k_{0}$, the occupation probability of the $w_{2}$ modes with
negative group velocity along the $z$ axis increases. Hence, provided the
energy of the photons matches the transition energy, only these $w_{2}$ modes
are transmitted across the horizon. On the other hand, since
$n_{(-k_{x},k_{y},k_{z})+}=n_{(k_{x},k_{y},-k_{z})+}$, the occupation
probability for the second Weyl node at $k_{x}=+k_{0}$ will decrease for
negative $k_{z}$, as is already apparent from the left hand side of Fig. 5. As
such, only one of the overtilted Weyl cones contributes to the fermionic
Hawking effect out of equilibrium. The non-equilibrium occupation of the $u$
modes above the Fermi energy in the undertilted region is then given by

$n_{L,u}=T_{u\leftarrow w_{1}}n_{R,w_{1}}+T_{u\leftarrow
w_{2}}n_{R,w_{2}}=T_{u\leftarrow w_{2}}\Delta n_{w_{2}},$ (81)

where the first equality holds only for a slowly-varying tilt profile.

###  4.2 Coupling to magnetic leads

An out-of-equilibrium distribution can also be induced by coupling the
overtilted region to magnetic leads. Here, we assume that the Pauli matrices
in Hamiltonian (12) effectively correspond to the physical spin degrees of
freedom of the electrons. In fact, for Weyl semimetals with broken time-
reversal symmetry, but with inversion and cubic symmetries, this
correspondence is exact [34]. In general, they also contain other degrees of
freedom like orbital or lattice degrees of freedom. Moreover, models for which
$\bm{\sigma}$ corresponds to the real spin have been shown to simulate spin
textures of Weyl semimetals observed in experiments [35, 36].

For simplicity, we consider the 1D effective model for $\bm{k}_{\perp}=(\pm
k_{0},0)$. In this case, the electrons will be effectively polarized in the
$xz$ plane with different orientations for the $w_{1}$ and $w_{2}$ modes. This
is already apparent in Eq. (25) where the modes have different spin
projections along $k_{z}$. The magnetic leads can be modeled, for instance, by
a 1D chiral fermion with spin-dependent group velocities, giving rise to
constant but different density of states for the two spin bands.

We further assume that the spin-up lead electrons have the same polarization
as the $w_{1}$ electrons at $\omega=0$. In this case, the spinors
$\psi_{\sigma=\pm}$ of the magnetic lead can be written as
$\psi_{+}=\varphi_{w_{1}}$ and $\psi_{-}=i\sigma_{y}\varphi_{w_{1}}$, with
$\varphi_{w_{1}}$ given in Eq. (25). A bias voltage $U_{DC}$ is applied
between the magnetic lead and the overtilted region by setting the chemical
potential to $\mu_{R}=eU_{DC}$ inside the lead for both spin species, and to
$\mu_{L}=0$ inside the type-II Weyl semimetal. The overlap between the lead
spinors and the Weyl semimetal spinor will then give rise to an effective
tunneling Hamiltonian [37],

$H_{T}=\sum_{k,k^{\prime}}\sum_{\sigma\lambda}t_{k\lambda}^{\sigma}\hat{c}^{\dagger}_{k\lambda}\hat{f}_{k^{\prime}\sigma}+h.c.,$
(82)

where $\hat{f}_{k^{\prime}\sigma}$ destroys a spin-$\sigma$ electron with
momentum $k^{\prime}$ inside the lead, $\hat{c}_{k\lambda}^{\dagger}$ creates
a Weyl mode inside the overtilted region, and
$t_{k\lambda}^{\sigma}=t_{0}\varphi_{k\lambda}^{\dagger}\psi_{\sigma}$ are the
tunneling matrix elements with tunneling strength $t_{0}$. By construction,
the spin-up electrons couple much stronger to the $w_{1}$ modes than to the
$w_{2}$ modes. Hence, tunneling between the magnetic lead and the overtilted
Weyl semimetal will give rise to a non-equilibrium occupation. Following the
previous section, the occupation can be modeled with the Boltzmann equation in
the relaxation-time approximation, which essentially leads to Eq. (77).
Assuming the density of states of the spin-up lead electrons $g_{\uparrow}\gg
g_{\downarrow}$, one finds at zero temperature,

$n_{k\lambda}=n^{0}_{k\lambda}+\tau\frac{2\pi}{\hbar}g_{\uparrow}|t^{\uparrow}_{k\lambda}|^{2}\Big{[}\Theta(eU_{DC}-E_{\lambda}(k))-\Theta\left(-E_{\lambda}(k)\right)\Big{]},$
(83)

where the dispersion is given by Eq. (76). In particular, for our choice of
polarization of the magnetic lead, we have
$|t^{\uparrow}_{w_{2}}|^{2}/|t^{\uparrow}_{w_{1}}|^{2}\approx v_{F}^{2}/V^{2}$
for states close to $\omega=0$. Hence, the occupation of the $w_{2}$ modes is
suppressed by the tilt $V$. This again gives rise to a population imbalance
between the $w_{1,2}$ modes, such that the Hawking signature can in principle
be observed.

Figure 6: (a) $T_{u\leftarrow w_{1}}$ (solid) and $T_{u\leftarrow w_{2}}$
(dashed) for $\bm{k}_{\perp}=(\pm k_{0},0)$ calculated with the lattice model
for the tilt profile $V(z)=v_{F}\left[1+\tanh(\alpha z)\right]$ where
$\omega_{c}\approx 0.6\times 2\hbar v_{F}k_{0}$ in the type-II region. (b)
Zero-bias differential conductance at zero temperature $G_{1}$ (solid) and
$G_{2}$ (dashed) for the same parameters as in (a).

###  4.3 Differential conductance

In Section 3, we considered the case of normal incidence with
$\bm{k}_{\perp}=(\pm k_{0},0)$ whose contribution dominates at low energies.
However, in a transport experiment all transverse channels contribute to the
current. Hence, it is not clear what remains of the Hawking effect even if one
can excite an $I_{w_{i}}$ ($i=1,2$) current by means of a non-equilibrium
occupation of $w$ modes, as described above. We therefore calculate the
contribution of the $w_{i}$ mode to the two-terminal zero-bias differential
conductance at zero temperature,

$\displaystyle G_{i}(E_{F})\equiv\left.\frac{dI_{w_{i}}}{dV}\right|_{V=0,T=0}$
$\displaystyle=\frac{e^{2}}{h}\sum_{\bm{k}_{\perp}}T_{u\leftarrow
w_{i}}(\omega=E_{F},\bm{k}_{\perp})$ (84)
$\displaystyle=\frac{e^{2}}{h}\frac{W^{2}}{2\pi^{2}}\int_{0}^{\infty}dk_{x}\int_{-\infty}^{\infty}dk_{y}\,T_{u\leftarrow
w_{i}}(\omega=E_{F},\bm{k}_{\perp}),$ (85)

where we take a sample with transverse dimensions $W\times W$ and we used
$\mathcal{H}(k_{x})=\mathcal{H}(-k_{x})$ in the second line. Here, we reverted
to dimensionful units for clarity. The transmission functions for general
$\bm{k}_{\perp}$ are calculated with the Kwant Python package (see App. B).

The transmission of the Hawking channel ($k_{x}=\pm k_{0}$ and $k_{y}=0$)
obtained with the lattice model is shown in Fig. 6(a). In the slowly-varying
limit ($\alpha\ll 1$), the transmission matches perfectly our analytical
results given in Eq. (55). On the other hand, for $\alpha\gg 1$, we also find
agreement with our results for the sharp horizon. The conductance for the
$w_{1}$ and $w_{2}$ current is shown in Fig. 6(b) as a function of the Fermi
energy. Here, the quadratic behavior near the Weyl node is attributed to the
low-energy density of states in the type-I region, given by
$g(\omega)=v_{F}\omega^{2}/[2\pi^{2}\hbar^{3}\left(v_{F}^{2}-V_{L}^{2}\right)^{2}]$.
Interestingly, we observe that the conductance $G_{1}$ ($G_{2}$) features a
local maximum at positive (negative) energies, due to tunneling of $w_{1}$
($w_{2}$) modes through the effective horizon. The peak position in the
$(\alpha,E_{F})$ plane is shown in Fig. 7 for three different tilt profiles.
In all cases, the peak position increases linearly for small $\alpha$ at
first, with the same slope $E_{F}/\omega_{c}\approx 0.7\alpha/2k_{0}$.
Furthermore, in the limit of a sharp horizon ($\alpha\gg 1$), the same
constant is attained, given by $E_{F}/\omega_{c}\approx 0.6$ and whose precise
value depends only on the asymptotic values of the tilt profile. Surprisingly,
we find that the position of the maxima as a function of $\alpha$ are fitted
reasonably well to functions that are similar to the tilt profile, as
illustrated in Fig. 7, except in the case with the purely linear tilt profile.

Figure 7: Position of the local maximum in the zero-bias differential
conductance for the $w_{1}$ mode, as a function of the Fermi energy $E_{F}$
and the slope of the tilt profile $\alpha$, for different tilt profiles (in
units of $v_{F}$) as detailed in App. B. Here, the solid blue and green curves
are fits of the data to $a\tanh b\alpha$ and $\tfrac{2a}{\pi}\arctan\tfrac{\pi
b\alpha}{2}$, respectively, with $a\approx 0.6$ and $b\approx 1.2$.

We thus conclude that it is in principle possible to induce a stationary non-
equilibrium occupation of $w$ modes, either by irradiation of light in the
type-II region, or by coupling the type-II region to a magnetic lead. In both
cases, we assume this takes place in the asymptotic tilt region far away from
the horizon. For sufficiently long mean-free paths this then gives rise to a
net $w_{1}$ or $w_{2}$ current. The corresponding two-terminal differential
conductance features a peak due to tunneling of $w_{1}$ or $w_{2}$ modes
across the horizon whose height and position depends in general on all the
details of the tilt profile. However, for a slowly-varying (sharp) tilt
profile relative to the Fermi wavelength, the properties of the peak depend
only on the slope $\alpha$ (the asymptotic values of the tilt).

##  5 Conclusion

In this work, we investigated mesoscopic transport across effective event
horizons at the interface of a type-I and type-II Weyl semimetal. To this end,
we used a minimal model that captures all salient features of a Weyl
semimetal, and studied type-I/type-II interfaces with different Weyl node tilt
profiles. We solved the scattering problem analytically at normal incidence in
the low-energy limit for two cases, a sharp horizon and a slowly varying
tilts. More precisely, these two cases are distinguished by the length scale
of the tilt profile relative to the Fermi wavelength.

For a slowly-varying tilt profile, we employed the WKB formalism together with
an approximate solution near a linear horizon. We find that co- and
counterpropagating modes are decoupled in this limit and we calculated the $S$
matrix, which depends only on the energy and the slope of the tilt profile at
the horizon. The irrelevance of further microscopic details is reminiscent of
a “no-hair theorem” for real black holes. Moreover, the transmission functions
of counterpropagating modes are given by a thermal distribution with effective
Hawking temperature inversely proportional to the slope. Adding the different
contributions of the counter-propagating modes to transport in a Landauer-
Büttiker picture, however, masks all analogs of Hawking effects.

For the sharp horizon, we solved the scattering problem by first deriving
appropriate boundary conditions for the wavefunction at the horizon. In this
case, the $S$ matrix explicitly depends on the asymptotic values of the tilt
profile. Hence, the black hole analogy breaks down in this case, even though
one can still define a temperature scale.

To circumvent the ballistic nature of transport in the slowly-varying limit,
we considered means to drive a non-equilibrium occupation of $w$ modes, i.e.,
those modes that tunnel through the horizon from inside the type-II region. We
showed that one can favor populating one of the $w$ modes over the other by
irradiating the type-II region with circularly polarized light, or by coupling
it to a magnetic lead. Given this non-equilibrium occupation, we then
calculated the differential conductance for a single $w$ mode which displays a
peak as a function of the Fermi energy. The peak position becomes universal in
the limit of very slowly varying tilts: it only depends on the slope of the
tilt profile at the horizon. In the opposite limit of rapidly changing tilts,
the peak position saturates to a value that only depends on the asymptotic
values of the tilt. However, in the intermediate regime, the details of the
whole tilt profile become important.

The transport experiment we propose can serve as a proof of principle for
analog event horizons in fermionic systems. Additionally, to better
distinguish different tilting regimes, one may apply a magnetic field along
the tilting direction, to freeze out all transverse degrees of freedom save
the Hawking channel. In conclusion, we have proposed how to detect signatures
of analog Hawking radiation in a type-I and type-II Weyl semimetal
heterostructure by driving the system out of equilibrium. The proposed setup
has potential further applications in terms of electron lensing, which can be
readily understood through the effective spacetime and associated
gravitational lensing analogies.

## Acknowledgements

The authors are indebted to Christian Schmidt and Andreas Haller for
insightful discussions.

##### Funding information

CDB, SG, and TLS acknowledge support by the National Research Fund Luxembourg
under the grants ATTRACT 7556175 and PRIDE/15/10935404. TM acknowledges
financial support by the Deutsche Forschungsgemeinschaft via the Emmy Noether
Programme ME4844/1-1 (project id 327807255), the Collaborative Research Center
SFB 1143 (project id 247310070), and the Cluster of Excellence on Complexity
and Topology in Quantum Matter ct.qmat (EXC 2147, project id 390858490).

_Note:_ A related publication in Ref. [38] provides a numerical analysis of
wavepacket dynamics in microscopic lattice models of Weyl black and white hole
analogs with a focus on lattice effects, and a discussion of realizations in
metamaterials.

Appendix

##  Appendix A Covariant form of the Weyl equation

In this appendix, we explicitly demonstrate how the effective Weyl equation
for a tilt profile $V=V(z)$ describing low-energy excitations of a tilted Weyl
semimetal can be cast into a manifestly covariant form using the tetrad
formalism [10, 20].

Unlike vectors and tensors, the Lorentz transformation rule for spinors in
flat spacetime does not generalize to curved spacetime, because the group of
real invertible matrices $GL(4,\mathds{R})$ has no spinor representations [21,
39]. However, we can introduce local Lorentz frames at each point in spacetime
and define spinors with respect to these frames. To this end, we choose an
orthonormal basis of the tangent space $\hat{e}_{a}(p)$ ($a=0,1,2,3$) at each
point $p$ of spacetime, i.e.,

$g(\hat{e}_{a},\hat{e}_{b})=\eta_{ab},$ (86)

where the vector fields $\hat{e}_{a}(p)$ are called frame fields or tetrads
and $\eta_{ab}$ is the Minkowski metric of flat spacetime. Hence, we can think
of tetrads as arising from a set of coordinate transformations
$x^{\mu}\rightarrow\xi^{a}_{p}$, one for each point $p$, such that
$g_{ab}(p)=\eta_{ab}$ or

$\eta_{ab}=\left.\frac{\partial x^{\mu}}{\partial\xi_{p}^{a}}\frac{\partial
x^{\nu}}{\partial\xi_{p}^{b}}\,g_{\mu\nu}\right|_{p}=e^{\mu}_{\,\,\,a}(p)e^{\nu}_{\,\,\,b}(p)\,g_{\mu\nu}(p),$
(87)

where $g_{\mu\nu}$ is the metric in the coordinate basis
$\hat{e}_{\mu}=\partial_{\mu}$ and we use the Einstein summation convention.
The $\xi^{a}_{p}$ coordinates are called local inertial coordinates and the
tetrads $\hat{e}_{a}(p)=e^{\mu}_{\,\,\,a}(p)\,\hat{e}_{\mu}$ constitute a
local Lorentz frame [40]. Note that tetrads are not unique, since a different
choice of local inertial coordinates at each point $p$ of spacetime,
$\xi^{a}_{p}\rightarrow\xi^{a^{\prime}}_{p}=\Lambda^{a^{\prime}}_{\,\,\,b}(p)\xi^{b}_{p}$
induces a local Lorentz transformation $\Lambda(p)$ such that
$e_{\,\,\,a}^{\mu}\rightarrow
e_{\,\,\,a^{\prime}}^{\mu}=\Lambda^{b}_{\,\,\,a^{\prime}}e_{\,\,\,b}^{\mu}$ ,
where $\Lambda^{b}_{\,\,\,a^{\prime}}$ is the inverse transformation, and
which leaves Eq. (86) invariant.

To construct the covariant Dirac equation, we require a covariant derivative
$\mathcal{D}_{a}\psi$ which is a local Lorentz vector that transforms as a
spinor. It follows that [21]

$\gamma^{a}\mathcal{D}_{a}\psi=0,$ (88)

where
$\mathcal{D}_{a}=e_{\,\,\,a}^{\mu}\left(\partial_{\mu}+\Omega_{\mu}\right)$,
$\gamma^{a}$ are the Dirac matrices with
$\\{\gamma^{a},\gamma^{b}\\}=2\eta^{ab}$, and $\Omega_{\mu}$ is called the
spin connection. The spin connection ensures covariance since
$\partial_{\mu}\psi$ does not transform as a spinor under local Lorentz
transformations. In general, we can write Eq. (88) as

$i\partial_{t}\psi=\left(-i\alpha^{i}\partial_{i}-i\Upsilon\right)\psi\equiv\mathcal{H}\psi,$
(89)

with

$\alpha^{i}(p)=(e_{\,\,\,b}^{0}\gamma^{b})^{-1}e_{\,\,\,a}^{i}\gamma^{a},\qquad\Upsilon(x)=(e_{\,\,\,b}^{0}\gamma^{b})^{-1}e_{\,\,\,a}^{\mu}\gamma^{a}\Omega_{\mu},$
(90)

where the spin connection ensures that the Hamiltonian $\mathcal{H}$ is
Hermitian [14]. If we compare this to the Hamiltonian of the tilted Weyl
semimetal (1), we identify

$e_{\,\,\,a}^{\mu}(z)=\delta^{\mu}_{a}+V(z)\delta_{3}^{\mu}\delta_{a}^{0},$
(91)

which correspond to the so-called acoustic metric [4]:

$g_{\mu\nu}=\eta_{ab}e^{a}_{\,\,\,\mu}e^{b}_{\,\,\,\nu}=\begin{pmatrix}V^{2}-1&0&0&-V\\\
0&1&0&0\\\ 0&0&1&0\\\ -V&0&0&1\end{pmatrix},$ (92)

where $e^{a}_{\,\,\,\mu}$ are the inverse tetrads. Equivalently, it can be
obtained from the inverse metric
$g^{\mu\nu}=\eta^{ab}e_{\,\,\,a}^{\mu}e_{\,\,\,b}^{\nu}$. Explicitly, the spin
connection is given by

$\Omega_{\mu}=\frac{1}{8}[\gamma^{a},\gamma^{b}]g_{\kappa\lambda}e_{\,\,\,a}^{\kappa}(\partial_{\mu}e_{\,\,\,b}^{\lambda}+\Gamma_{\mu\nu}^{\lambda}e_{\,\,\,b}^{\nu}),$
(93)

which can be derived from infinitesimal local Lorentz transformations [21,
39], and where $\Gamma_{\kappa\mu\nu}$ and
$\Gamma_{\mu\nu}^{\lambda}=g^{\kappa\lambda}\Gamma_{\kappa\mu\nu}$ are
Christoffel symbols of the first and second kind, respectively. In our case,
the only nonzero Christoffel symbols are

$\Gamma_{003}=\Gamma_{030}=-\Gamma_{300}=VV^{\prime},\qquad\Gamma_{033}=-V^{\prime},$
(94)

with $V^{\prime}=\partial V/\partial z$ and

$\Gamma_{0\nu}^{\mu}=VV^{\prime}\begin{pmatrix}V&-1\\\
V^{2}-1&-V\end{pmatrix},\qquad\Gamma_{3\nu}^{\mu}=V^{\prime}\begin{pmatrix}-V&1\\\
-V^{2}&V\end{pmatrix},$ (95)

where $\mu,\nu=0,3$. After some tedious algebra, the spin connection becomes

$\Omega_{\mu}=\frac{1}{4}[\gamma^{3},\gamma^{0}]\left(\delta_{\mu}^{3}-V\delta_{\mu}^{0}\right)V^{\prime},$
(96)

such that

$\displaystyle e_{\,\,\,a}^{\mu}\gamma^{a}\Omega_{\mu}$
$\displaystyle=\gamma^{a}\left(e_{\,\,\,a}^{0}\Omega_{0}+e_{\,\,\,a}^{3}\Omega_{3}\right)$
(97)
$\displaystyle=\gamma^{0}\left(\Omega_{0}+V\Omega_{3}\right)+\gamma^{3}\Omega_{3}=\gamma^{3}\Omega_{3}=\frac{1}{2}\gamma^{0}V^{\prime}.$
(98)

In our specific case, we have
$\alpha^{i}=\gamma^{i}\gamma^{0}+V\delta_{3}^{i}$ and $\Upsilon=V^{\prime}/2$.
In the Weyl representation, we have

$\gamma^{i}\gamma^{0}=\begin{pmatrix}\sigma^{i}&0\\\
0&-\sigma^{i}\end{pmatrix},$ (99)

with $\sigma^{i}$ the Pauli matrices and we obtain the Weyl Hamiltonian

$\mathcal{H}_{\chi}=-i\left(\chi\bm{\sigma}+V\bm{e}_{z}\right)\cdot\nabla-
iV^{\prime}/2.$ (100)

##  Appendix B Lattice model

To solve the scattering problem numerically, we perform a lattice simulation
with the Kwant Python package [24]. To this end, we discretize the continuum
Hamiltonian (14) giving a one-dimensional chain along the $z$ direction with
lattice constant $a$ (in units of $1/2k_{0}$) and two orbitals per cell. The
dimensionless lattice Hamiltonian becomes

$\displaystyle\hat{H}$
$\displaystyle=\sum_{\bm{k}_{\perp}}\sum_{n}\hat{c}_{\bm{k}_{\perp}n}^{\dagger}\left[\left(k_{\perp}^{2}-\frac{1}{4}+\frac{2}{a^{2}}\,\right)\sigma_{x}+k_{y}\,\sigma_{y}\right]\hat{c}_{\bm{k}_{\perp}n}$
(101)
$\displaystyle+\left\\{\hat{c}_{\bm{k}_{\perp}n+1}^{\dagger}\left[\frac{1}{2ia}\left(V_{n+1/2}\,\sigma_{0}+\sigma_{z}\right)-\frac{1}{a^{2}}\,\sigma_{x}\right]\hat{c}_{\bm{k}_{\perp}n}+\textrm{h.c.}\right\\},$
(102)

where $n=0,1,\ldots,N-1$ labels the cells of the chain and

$\hat{c}_{\bm{k}_{\perp}n}=\frac{1}{\sqrt{N}}\sum_{k_{z}}e^{ik_{z}na}\hat{c}_{\bm{k}},$
(103)

where
$\hat{c}_{\bm{k}_{\perp}n}=\left(\hat{c}_{\bm{k}_{\perp}n1},\,\hat{c}_{\bm{k}_{\perp}n2}\right)^{t}$
and $V_{n}=V(na)$ is the value of the tilt at site $n$. Here, we take the
average value of the tilt for hopping between sites $n$ and $n+1$. For all
calculations, we take a scattering region of length $L=a\left(N-1\right)$ with
$N=201$ that is sufficiently long so that the tilt profile is practically
constant at the boundaries which are connected to semi-infinite leads. In our
simulations, we considered three different tilt profiles, all with linear
horizons. Firstly, we used

$V(z)=A+B\tanh\left(Cz+D\right),$ (104)

with

$\displaystyle A$ $\displaystyle=\frac{V_{R}+V_{L}}{2},$ $\displaystyle
C=\frac{\alpha}{2}\frac{V_{R}-V_{L}}{\left(V_{0}-V_{L}\right)\left(V_{R}-V_{0}\right)},$
(105) $\displaystyle B$ $\displaystyle=\frac{V_{R}-V_{L}}{2},\qquad$
$\displaystyle D=\frac{1}{2}\ln\left(\frac{V_{0}-V_{L}}{V_{R}-V_{0}}\right),$
(106)

where $V(0)=V_{0}$ and $V^{\prime}(0)=\alpha$ and which is valid for real $D$,
i.e., either $V_{L}<V_{0}$ and $V_{R}>V_{0}$ or $V_{L}>V_{0}$ and
$V_{R}<V_{0}$. Here, $V_{0}=\pm 1$ for a black hole and white hole horizon at
the origin, respectively and $V_{R,L}=\lim_{z\rightarrow\pm\infty}V(z)$. Note
that this tilt profile is symmetric about the origin for $V_{R}+V_{L}=2V_{0}$.
Secondly, we considered the tilt profile
$V(z)=1+\left(2/\pi\right)\arctan\left(\alpha z\pi/2\right)$ for the case
$V_{L}=0$ and $V_{R}=2$. For this profile, the asymptotic values are
approached linearly instead of exponentially. Finally, we also looked at a
purely linear tilt profile, given by

$V(z)=\begin{cases}V_{L}&\quad z\leq 0,\\\ V_{L}+\alpha z&\quad 0<z<L_{0},\\\
V_{R}&\quad z\geq L_{0},\end{cases}$ (107)

with $L_{0}=\left(V_{R}-V_{L}\right)/\alpha$.

##  Appendix C Scattering matrix for sharp horizon

Here, we give the complete $S$ matrix at low energies for the sharp black hole
horizon where we take $V_{L}=0$ and $V_{R}=V$. Expressions exist for the
general case but they are too unwieldy. Here, we obtained the low-energy
expressions by expanding the spinors, wavevectors, and group velocity up to
third order in $\omega$.

When the incoming mode comes from the type-I region ($v$ mode), the scattering
coefficients up to second order in $\omega$ are found to be given by

$\displaystyle r_{uv}$
$\displaystyle=i\,\frac{2-V}{2+V}-\frac{V^{4}-5V^{2}+8}{\left(2+V\right)^{2}\left(V^{2}-1\right)}\,2\omega$
(108)
$\displaystyle+\frac{V^{7}+3V^{6}-4V^{5}-18V^{4}+5V^{3}+43V^{2}+6V-32}{\left(2+V\right)^{3}\left(V^{2}-1\right)^{2}}\,2i\omega^{2},$
(109) $\displaystyle t_{uv}$
$\displaystyle=i\,\frac{2\sqrt{V-1}}{2+V}+\frac{V\left(V-1\right)\left(3V+8\right)-16}{\left(2+V\right)^{2}\left(1+V\right)\sqrt{V-1}}\,\omega$
(110)
$\displaystyle-\frac{7V^{7}+27V^{6}-17V^{5}-159V^{4}+6V^{3}+432V^{2}+176V-256}{4\sqrt{V-1}\left(2+V\right)^{3}\left(V^{2}-1\right)^{2}}\,i\omega^{2},$
(111) $\displaystyle t_{vv}$
$\displaystyle=\frac{2\sqrt{1+V}}{2+V}-\frac{iV^{2}\sqrt{1+V}}{\left(2+V\right)^{2}\left(V-1\right)}\,\omega+\frac{V^{5}+3V^{4}-7V^{3}-7V^{2}+58V+80}{4\left(2+V\right)^{3}\left(1+V\right)^{5/2}\left(V-1\right)^{2}}\,V^{2}\omega^{2},$
(112)

such that whenever the zeroth-order term is real (imaginary), the first-order
term is imaginary (real). Hence, the first-order term enters only as a phase
such that the transmission probability is constant up to first order.
Numerically, we find that this holds at all odd orders such that the
transmission functions $R_{u\leftarrow v}$, $T_{u\leftarrow v}$, and
$T_{v\leftarrow v}$ are even functions of the energy.

On the other hand, when the incoming mode comes from the type-II region, there
are two possibilities: $w_{1}$ and $w_{2}$. Up to first order in $\omega$, we
find

$\displaystyle t_{uw_{1,2}}$ $\displaystyle=\sqrt{2}\,\frac{V^{2}+V-2\pm
i\sqrt{V^{2}-1}\left(2-V\right)}{V\left(2+V\right)\sqrt{V-1}}$ (113)
$\displaystyle\pm\frac{P_{1}(V)\mp
i\sqrt{V^{2}-1}\left(16+12V-16V^{2}-7V^{3}+3V^{4}+V^{5}\right)}{\sqrt{2}\,V\left(1+V\right)^{3/2}\left(V^{2}+V-2\right)^{2}}\,\omega,$
(114) $\displaystyle r_{uw_{1,2}}$ $\displaystyle=\frac{V^{2}-4\pm
4i\sqrt{V^{2}-1}}{\sqrt{2}\,V\left(2+V\right)}$ (115)
$\displaystyle\pm\frac{P_{2}(V)\mp
i\sqrt{V^{2}-1}\left(32+40V-12V^{2}-20V^{3}-4V^{4}\right)}{2\sqrt{2}\,V\left(2+V\right)^{2}\left(V^{2}-1\right)^{3/2}}\,\omega,$
(116) $\displaystyle r_{vw_{1,2}}$
$\displaystyle=\mp\frac{V}{\sqrt{2}\left(2+V\right)}$ (117)
$\displaystyle\pm\frac{4+8V-8V^{3}-4V^{4}\mp
i\sqrt{V^{2}-1}\left(12+6V-6V^{2}-3V^{2}\right)}{2\sqrt{2}\left(V-1\right)^{2}\left(1+V\right)^{2}\left(2+V\right)^{2}}\,iV\omega,$
(118)

where $P_{1}=16+12V-28V^{2}-9V^{3}+12V^{4}-3V^{6}$, and
$P_{2}=32+40V-44V^{2}-54V^{3}-6V^{4}+5V^{5}$. Taking the expressions of the
scattering amplitudes up to first order in $\omega$, one can check that
$SS^{\dagger}=S^{\dagger}S=1+\mathcal{O}(\omega^{2})$ where $S$ is defined in
Eq. (67).

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