#  $\mathcal{PT}$ symmetry-protected exceptional cones and analogue Hawking
radiation

Marcus Stålhammar1 Jorge Larana-Aragon1 Lukas Rødland1 Flore K. Kunst2,3
1Department of Physics, Stockholm University, AlbaNova University Center, 106
91 Stockholm, Sweden  2Max Planck Institute of Quantum Optics, Hans-
Kopfermann-Straße 1, 85748 Garching, Germany  3Max Planck Institute for the
Science of Light, Staudtstraße 2, 91058 Erlangen, Germany

###### Abstract

Non-Hermitian Hamiltonians, which effectively describe dissipative systems,
and analogue gravity models, which simulate properties of gravitational
objects, comprise seemingly different areas of current research. Here, we
investigate the interplay between the two by relating parity-time-symmetric
dissipative Weyl-type Hamiltonians to analogue Schwarzschild black holes
emitting Hawking radiation. We show that the exceptional points of these
Hamiltonians form tilted cones mimicking the behavior of the light cone of a
radially infalling observer approaching a black hole horizon. We further
investigate the presence of tunneling processes, reminiscent of those
happening in black holes, in a concrete example model. We interpret the non-
trivial result as the purely thermal contribution to analogue Hawking
radiation in a Schwarzschild black hole. Assuming that our particular
Hamiltonian models a photonic crystal of experimental relevance, we argue that
the loss from the latter in the form of thermal radiation can be interpreted
as the blackbody contribution to analogue black hole radiation when measuring
at the exceptional cone. As such, these systems are promising candidates for
black hole analogue models.

##  I Introduction

Since the experimental discovery of the quantum Hall effect in the 1980s QHE ,
topology has played a prominent role in condensed matter physics. Gapped
topological insulators hasankane ; qizhang  and gapless semimetals, such as
graphene goerbig  and Weyl semimetals (WSMs) weylreview , have Bloch bands
that are characterized by distinct topological invariants, which play a
crucial role in establishing a bulk-boundary correspondence jantopreview .
Notably, the elusive Weyl fermions, despite evading discovery for nearly a
century after being theoretically predicted in the context of particle physics
W1929 , are realized as quasi-particles near the point-like band crossings in
WSMs XBAN2015 ; LWFM2015 ; LWYRFJS2015 . This has led to further observations
of theoretical concepts known from high-energy physics, such as the chiral
anomaly, which has been observed in experiments Chiral1 ; Chiral2 ; Chiral3 ;
Chiral4 .

The general idea of establishing analogies between different areas of physics
is especially relevant in the context of black holes. Even though, the study
of black holes has provided key theoretical insights into the fundamental
nature of spacetime, direct experimental input is difficult to obtain.
However, through investigations in analogue models, black hole physics can be
studied in laboratory setups Unruh ; Barcelo . In this context, WSMs have
received great attention due to the behavior of their cone-like dispersion,
called the Weyl cone. In particular, the tilting of Weyl cones resembles the
tilting of spacetime light cones of observers in the presence of a black hole
beenakker . This resemblance has been formalized via the construction of a
relation between the transition from type-I to type-II WSMs, where the Weyl
cones are overtilted [cf. Fig. 1(d)], and the tilting of an observer’s light
cone when crossing the black hole horizon in the infalling coordinate frame
Volovik2016 . With this as a starting point, recent studies volovik ;
zobkovblackholes ; Zubkovold ; Liu2018 ; Yaron2020 ; Nissinen  have focused on
investigating potential analogues of Hawking radiation in WSMs: In 1975,
Hawking postulated that despite the classical intuition, black holes emit
thermal radiation at a characteristic temperature, the Hawking temperature,
when taking quantum effects into account, and thus evaporate Hawking1975 .
While some of these works predict the existence of analogue Hawking radiation
in WSMs volovik ; Liu2018 ; Zubkovold , others find no radiative processes
Yaron2020 . Indeed, in Ref. Yaron2020,  the absence of analogue Hawking
radiation is attributed to the stability of the Hermitian Weyl model thus
motivating studies of black hole analogue models beyond these systems.

Inspired by the fact that the emission of thermal radiation can be associated
with the loss of energy from a system to its environment, we here instead
consider a dissipative Weyl-type model with parity-time ($\mathcal{PT}$)
symmetry described by a non-Hermitian (NH) Hamiltonian. NH Hamiltonians serve
as effective descriptions of systems subject to gain and loss NHreview , and
find applications both in classical Lin2011 ; Regensburger2012 ; Peng2014 ;
Feng2014 ; Hodaei2014 ; Peng2016 ; Hodaei2017 ; Chen2017 ; Ozdemir2019 ;
Fleury2015 ; Helbig2020 ; Ghatak2020  and quantum setups Kreibich2014 ;
koziifu ; Yoshida2018 ; Yoshida2020 ; Bergholtz2019 . Naturally, these
Hamiltonians are fundamentally different from their Hermitian counterparts,
the most prominent distinction being that the eigenvalues are generally
complex with the imaginary part associated with finite lifetimes and the sets
of left and right eigenvectors may be unequal. Furthermore, the nodal
structures of NH systems consist of exceptional points (EPs) at which not only
the eigenvalues, but also the eigenvectors coalesce brody14 . The sets of EPs
exhibit fascinating topological properties BerryDeg ; carlstroembergholtz ;
molina ; disorderlinesribbons ; koziifu ; ourknots ; Ronnyknot1 ; ourknots2 ;
symprotnod , which can be investigated in experiments at remarkable precision
Lin2011 ; Regensburger2012 ; Peng2014 ; Feng2014 ; Hodaei2014 ; Peng2016 ;
Hodaei2017 ; Chen2017 ; expknots . In particular, NH systems find high
experimental relevance in optics, where they effectively describe photonic
systems subject to $\mathcal{PT}$ symmetry Ozdemir2019 ; topphot ; speclat .
$\mathcal{PT}$-symmetric systems host distinct regions with real and complex
eigenvalues, which are separated by EPs, where $\mathcal{PT}$ symmetry is
spontaneously broken Bender1998 . Consequently, these regions are called the
$\mathcal{PT}$-unbroken and the $\mathcal{PT}$-broken phase, respectively.

Our dissipative Weyl-type model features cones in the spectrum consisting of
EPs, which we call _exceptional cones_ (ECs), whose intersection point, i.e.,
where the vertices of two cones touch, corresponds to an ordinary Weyl node.
For this setup, we investigate whether a possible relation exists between ECs
and spacetime light cones near a black hole. In particular, we show that ECs
of order two, where the order refers to the number of eigenvectors that
coalesce, appear generically in a family of two-band, three-dimensional (3D)
$\mathcal{PT}$-symmetric models. These ECs look similar to Weyl cones but
possess additional features. For example, the EC separates a region, where the
eigenvalues are real, from a region, where the eigenvalues are imaginary thus
mimicking the space-like and time-like regions, respectively, that are
separated by an observer’s spacetime light cone.

By mapping the EC onto a 2D Weyl node, we make a formal connection between the
EC and an observer’s light cone in the vicinity of a stationary (3+1)D
Schwarzschild black hole. We then investigate the existence of analogue
Hawking radiation by calculating the semiclassical quantum tunneling rate.
Indeed, following previous approaches where Hawking radiation is regarded as a
quantum tunneling process Volovik1999 ; PW2000 , we calculate the leading
order contributions to analogue Hawking radiation and find them to be nonzero.
To address the experimental relevance of our findings, we show that our
Hamiltonian model can be thought of as an effective description derived from
the Lindblad master equation describing a photonic crystal. We argue that the
analogue Hawking radiation is manifest in photonic crystals as thermal loss
from the EC, which indicates that these systems constitute promising
candidates in which the suggested analogy can be tested.

This paper is setup as follows: Due to the interdisciplinary character of this
article, we start with a brief discussion on Hawking radiation, WSMs and NH
systems in Sec. II, which are concepts necessary for understanding the present
work. This is followed by the introduction of our model and the establishment
of a relation to the spacetime light cone in Sec. III. In Sec. IV, we apply
the quantum tunneling method in order to compute the Hawking temperature, and
we discuss possible experimental realizations of our model in Sec. V. We
conclude in Sec. VI.

Throughout this article, we use Planck units, i.e.,
$\hbar=c=k_{\text{B}}=G_{\text{N}}=1$, such that all physical quantities
become dimensionless.

##  II Preliminaries

Here we introduce several concepts important for our work. In Sec. II.1, we
briefly discuss black holes and Hawking radiation serving as complementary
motivation for constructing analogue gravity models. In Sec. II.2, we present
the relation between _Hermitian_ two-band models and curved spacetimes,
especially highlighting WSMs as gravitational analogue models. Finally, we
discuss NH two-band models and the role played by $\mathcal{PT}$ symmetry in
Sec II.3. The notation and conventions introduced here are used throughout the
paper.

###  II.1 Hawking radiation and black holes

Black holes are regions in spacetime composed of a singularity, which is a
point at which the curvature of spacetime becomes infinite, and a boundary
surface, known as the event horizon. This is clearly illustrated in the case
of the most general static, spherically symmetric solution to Einstein’s
equations in vacuum, i.e., the asymptotically flat (3+1)D _Schwarzschild black
hole_ Schwarzschild . The most common way of writing the metric for this type
of black hole is in terms of Schwarzschild coordinates, but these cover only
the region outside the black hole horizon and lack a description of the
interior. For later purposes, which will become clear in Sec. IV, we here
instead choose to represent the metric with a set of coordinates valid both in
the inner and outer region of the black hole as well as at its horizon, such
as the Painlevé-Gullstrand coordinates Painleve ; Lemaitre . In this
coordinate frame, the line element reads

$ds^{2}=-\left(1-\frac{2M}{r}\right)dT^{2}+2\sqrt{\frac{2M}{r}}drdT+dr^{2}+r^{2}d\Omega^{2},$
(1)

where $M$ is the mass of the black hole, $r$ is the Painlevé radial
coordinate, $T$ is the Painlevé time and $d\Omega$ is the differential element
of solid angle of the 2-sphere. Evidently, Eq. (1) has a mathematical
singularity at $r=0$ corresponding to the physical singularity of the black
hole. The event horizon, on the other hand, is located at $r=2M$ where the
time component of Eq. (1) vanishes. In Schwarzschild coordinates, this point
corresponds to a coordinate singularity, while in the chosen coordinate frame
the metric is regular at that point. The intuition for the Painlevé-Gullstrand
coordinates naturally arises from considering radial null geodesics of
infalling observers into the black hole Vanzo2011 . These can be obtained from
the metric above and read

$\dot{r}_{\pm}=\pm 1-\sqrt{\frac{2M}{r}},$ (2)

where $\dot{r}:=dr/dT$. By studying the asymptotic behavior of
$\dot{r}_{\pm}$, we see that $\dot{r}_{+}$ and $\dot{r}_{-}$ correspond to
outgoing and ingoing solutions as $\lim_{r\to+\infty}\dot{r}_{+}>0$ and
$\lim_{r\to+\infty}\dot{r}_{-}<0$, respectively. Indeed, these coordinates
describe infalling observers whose light cones tilt as they approach the black
hole horizon. The light cone reaches a critical tilting value at the horizon
and it overtilts in the interior, where the roles of time and space are
interchanged. This occurs generically for infalling observers approaching a
black hole horizon, and can also be seen in, e.g., Eddington-Finkelstein
coordinates carroll .

Classical physics dictates that nothing, not even light, can escape from a
black hole once it has crossed the event horizon. In sharp contrast, when
taking quantum fluctuations into account, black holes are predicted to emit
thermal radiation at a characteristic temperature, called Hawking radiation
and the Hawking temperature, respectively, resulting in their evaporation. Let
us illustrate this by considering the case of a Schwarzschild black hole. The
phenomenon of Hawking radiation can be seen explicitly from the expression for
the spectrum of Hawking quanta Hawking1975 , which resembles a Planckian
spectrum

$N(\omega)=\frac{\gamma_{\omega}}{e^{\beta\omega}-1},$ (3)

where $N$ represents the total number of emitted quanta of energy $\omega$,
$\beta=1/T_{\text{BH}}$ is the inverse Hawking temperature for a Schwarzschild
black hole

$T_{\text{BH}}=\frac{1}{8\pi M},$ (4)

and $\gamma_{\omega}$ is a classical absorption coefficient, known as the
greybody factor, which makes the spectrum deviate slightly from the one
characteristic of a blackbody. For sufficiently massive black holes the
thermal approximation is justified since the greybody factor can be neglected
JennieTraschen , i.e., $\gamma_{\omega}\rightarrow 1$ when
$\left|\omega\right|M\sim\left|\omega\right|/T_{\text{BH}}\gg 1$, and Eq. (3)
can be approximated by the Boltzmann factor $e^{-\beta\omega}$. Since the
Hawking temperature $T_{\text{BH}}$ is inversely proportional to the black
hole mass $M$, the evaporation process is an accelerated process in the sense
that the black hole becomes hotter as it emits (almost purely) thermal
radiation. Following this reasoning, it is possible to consider black holes as
thermal systems with associated thermodynamic variables. The quantity of
interest when it comes to Hawking radiation is the semiclassical emission rate
KrausKeski

$\Gamma(\omega)\propto N(\omega)\sim e^{-\beta\omega},$ (5)

which quantifies the rate of emission of thermal radiation away from the black
hole.

The phenomenon of Hawking radiation is very elusive in the sense that its
direct detection in astrophysical black holes is highly masked by the cosmic
microwave background, and quantities such as the emission rate can thus not be
measured. As such, Hawking radiation has never been directly observed. This
motivates the field of analogue gravity in the search for analogue black hole
models with the aim to gain insight into the nature of Hawking radiation.

###  II.2 Weyl semimetals and analogue spacetime light cones

Figure 1: Illustration of the eigenvalues corresponding to Eq. (9) for
$\boldsymbol{\kappa}=\left(\kappa,0,0\right)$ in the plane $k_{z}=0$ for
different values of $\kappa$: (a) $\kappa=0$, (b) $0.5$, (c) $1$, and (d)
$1.5$. When $\kappa>1$, the cone is overtilted in analogy to the light cone of
an observer that has crossed the black hole horizon. Thus, the two phases,
type I ($\kappa<1$) (a,b) and type II ($\kappa>1$) (d), intuitively resemble
the exterior and the interior of a black hole, respectively.

We continue by describing the connection between Weyl cones and spacetime
light cones in Hermitian systems. A non-interacting Hermitian two-band model
in its most general form is given by the following Bloch Hamiltonian

$\mathcal{H}(\mathbf{k})=d_{0}(\mathbf{k})\sigma^{0}+\mathbf{d}(\mathbf{k})\cdot\boldsymbol{\sigma},$
(6)

where $\mathbf{k}$ is the lattice momentum with the appropriate dimensions,
$\sigma^{0}$ is the $2\times 2$ identity matrix,
$\boldsymbol{\sigma}=\left(\sigma^{x},\sigma^{y},\sigma^{z}\right)$ is the
vector of Pauli matrices,
$\mathbf{d}(\mathbf{k})=\left[d_{x}(\mathbf{k}),d_{y}(\mathbf{k}),d_{z}(\mathbf{k})\right]$,
and $d_{\mu}(\mathbf{k})$ are continuously differentiable real-valued
functions of $\mathbf{k}$ for $\mu=0,x,y,z$. From here on, we use the notation
$d_{\mu}(\mathbf{k})=:d_{\mu}$ for simplicity, while restoring the
$\mathbf{k}$-dependence when appropriate. The corresponding energy eigenvalues
are

$E_{\pm}=d_{0}\pm\sqrt{d_{x}^{2}+d_{y}^{2}+d_{z}^{2}}.$ (7)

Nodal points or higher-dimensional degenerate structures are given by the
intersections of the eigenvalues, which arise from solving

$\sqrt{d_{x}^{2}+d_{y}^{2}+d_{z}^{2}}=0.$ (8)

Solutions to this equation are of codimension 3 meaning that nodal points
occur generically in three dimensions or higher. Well-studied examples, where
these nodal points appear, include WSMs, whose dispersions form cones close to
the nodal points, referred to as Weyl cones. This intriguing feature, together
with the relativistic aspect of the Weyl nodes, led to WSMs being proposed as
potential analogue gravity systems by relating Weyl cones to spacetime light
cones Volovik2016 ; volovik ; zobkovblackholes ; Zubkovold ; Liu2018 ;
Yaron2020 ; Nissinen . The starting point of this analogy is the Weyl
Hamiltonian

$\mathcal{H}=-\boldsymbol{\kappa}\cdot\mathbf{k}\sigma^{0}+\mathbf{k}\cdot\boldsymbol{\sigma},$
(9)

where $\boldsymbol{\kappa}\in\operatorname{\mathbb{R}}^{3}$ is a tilting
parameter and $\mathbf{k}:=\left(k_{x},k_{y},k_{z}\right)$. The corresponding
energy eigenvalues read

$E_{\pm}=-\boldsymbol{\kappa}\cdot\mathbf{k}\pm\sqrt{k_{x}^{2}+k_{y}^{2}+k_{z}^{2}},$
(10)

and constitute a cone in energy-momentum space tilted by
$\boldsymbol{\kappa}$, which is displayed in Fig. 1. $|\boldsymbol{\kappa}|<1$
and $|\boldsymbol{\kappa}|>1$ correspond to type I and type II WSMs,
respectively. The transition between these two different types, i.e., when the
cone overtilts, is intuitively similar to the tilting of the light cone when
the corresponding radially infalling observer crosses the horizon of a
Schwarzschild black hole.

This intuition is made formal by relating the Hamiltonian (9) to the metric of
a curved spacetime. We note that such a relation exists for any Dirac-like
operator, and we refer to Refs. Volovik2014, ; Horova2005,  for a complete and
detailed treatment. The key ingredient is the so-called _vielbeins_
$e\indices{{}^{\mu}_{\alpha}}$ and their inverses
$e\indices{{}_{\mu}^{\alpha}}$, which in general relativity are used to define
local patches of orthonormal frames in spacetime Ortin ; carroll . They relate
to the (inverse) spacetime metric as
$g^{\mu\nu}=e\indices{{}^{\mu}_{\alpha}}e\indices{{}^{\nu}_{\beta}}\eta^{\alpha\beta}$,
with $\eta^{\alpha\beta}=\text{diag}\left(-1,1,1,1\right)$. Following Ref.
Volovik2014, , the Hamiltonian in Eq. (9) can be put on the form

$\mathcal{H}=e\indices{{}^{i}_{a}}k_{i}\sigma^{a}+e\indices{{}^{i}_{0}}k_{i}\sigma^{0},$
(11)

with $i,a=x,y,z$, $e\indices{{}^{i}_{a}}=\delta\indices{{}^{i}_{a}}$, and
$e\indices{{}^{i}_{0}}=-\kappa^{i}$. Setting $e\indices{{}^{0}_{0}}=1$
Volovik2014 ; Volovik2016 , we find that the line element of the corresponding
analogue spacetime metric, $ds^{2}=g_{\mu\nu}dx^{\mu}dx^{\nu}$, reads

$ds^{2}=-\left(1-\left|\boldsymbol{\kappa}\right|^{2}\right)dt^{2}+\left|d\mathbf{x}\right|^{2}+2\boldsymbol{\kappa}\cdot
d\mathbf{x}\,dt,$ (12)

where $d\mathbf{x}=\left(dx,dy,dz\right)$ and
$|d\mathbf{x}|^{2}=dx^{2}+dy^{2}+dz^{2}$. Eq. (12) describes an event horizon
at $|\kappa|=1$, which coincides exactly with when the Weyl cone overtilts,
cf. Fig. 1. Next, we present a short description of NH systems and the role
played by $\mathcal{PT}$ symmetry.

###  II.3 NH systems and $\mathcal{PT}$ symmetry

The mathematical description of systems effectively modelled by NH
Hamiltonians is a straightforward generalization of the reasoning presented in
Sec. II.2. We again consider the Hamiltonian in Eq. (6), where $d_{\mu}$ now
denote _complex-valued_ continuously differentiable functions of the lattice
momentum $\mathbf{k}$. Decomposing
$\mathbf{d}=\mathbf{d}_{\text{R}}+i\mathbf{d}_{\text{I}}$ with
$\mathbf{d}_{\text{R}}$ and $\mathbf{d}_{\text{I}}$ denoting the real and
imaginary parts of $\mathbf{d}$, respectively, the corresponding complex
eigenvalues are

$E_{\pm}=d_{0}\pm\sqrt{d_{\text{R}}^{2}-d_{\text{I}}^{2}+2i\mathbf{d}_{\text{R}}\cdot\mathbf{d}_{\text{I}}}.$
(13)

Hence, the nodal structure is given by solutions to the following system of
equations,

$d_{\text{R}}^{2}-d_{\text{I}}^{2}=0,\qquad\mathbf{d}_{\text{R}}\cdot\mathbf{d}_{\text{I}}=0.$
(14)

In contrast to the eigenvalue degeneracies in Hermitian systems, the nodal
points in NH systems are _exceptional points_ (except for when
$|\mathbf{d}|=0$). The exceptional nodal structure is determined by the
simultaneous solution of two equations meaning that it has codimension 2.
Stable EPs thus occur already in 2D systems, while in 3D solutions to Eq. (14)
attain the form of closed lines NHreview . The EPs furthermore constitute the
boundary of topologically protected bulk states, called Fermi states (FS) and
i-Fermi states (iFS). These correspond to the regions in momentum space
defined by

$\displaystyle\text{Re}\left(E_{+}\right)=\text{Re}\left(E_{-}\right)\Rightarrow\mathbf{d}_{\text{R}}\cdot\mathbf{d}_{\text{I}}$
$\displaystyle=0,\quad d_{\text{R}}^{2}-d_{\text{I}}^{2}\leq 0,$ (15)
$\displaystyle\text{Im}\left(E_{+}\right)=\text{Im}\left(E_{-}\right)\Rightarrow\mathbf{d}_{\text{R}}\cdot\mathbf{d}_{\text{I}}$
$\displaystyle=0,\quad d_{\text{R}}^{2}-d_{\text{I}}^{2}\geq 0,$ (16)

respectively. The presence of these (i)FSs in the bulk of the spectrum is a
characteristic signature of dissipation, and has been experimentally observed
in photonic lattices bfermiarcs .

Imposing $\mathcal{PT}$ symmetry constrains the Hamiltonian as follows.
Following Refs. BLC1, ; BLC2, , we represent $\mathcal{P}$ symmetry by a
$\mathcal{P}$-matrix and $\mathcal{T}$ symmetry by a $\mathcal{K}$-matrix,
where $\mathcal{KK}^{*}=\pm\mathbb{I}$, $\mathcal{P}^{2}=\mathbb{I}$ and
$\mathcal{KP}^{*}=\pm\mathcal{PK}$. We focus on the symmetry class where
$\mathcal{KK}^{*}=\mathbb{I}$ and $\mathcal{KP}^{*}=\mathcal{PK}$, and choose
the following representation

$\mathcal{P}=\sigma^{y},\qquad\mathcal{K}=\sigma^{z}.$ (17)

Imposing this choice of $\mathcal{P}\mathcal{T}$ symmetry on the generic two-
band Hamiltonian in Eq. (6) results in the following constraint

$\mathcal{H}_{\mathcal{P}\mathcal{T}}=\mathcal{P}\mathcal{K}H_{\mathcal{P}\mathcal{T}}^{*}\left(\mathcal{P}\mathcal{K}\right)^{-1},$
(18)

such that $d_{0},d_{x},d_{y}\in\mathbb{R}$ and $d_{z}\in i\mathbb{R}$. This
representation of $\mathcal{PT}$ symmetry is used throughout the present work.
A $\mathcal{PT}$-symmetric two-band Hamiltonian is thus necessarily on the
form

$\mathcal{H}_{\mathcal{PT}}=d_{0}\sigma^{0}+d_{x}\sigma^{x}+d_{y}\sigma^{y}+i\tilde{d}_{z}\sigma^{z},$
(19)

with $d_{0},d_{x},d_{y},\tilde{d}_{z}\in\operatorname{\mathbb{R}}$ and
$i\tilde{d}_{z}:=d_{z}$. The corresponding eigenvalues then read

$E_{\pm,\mathcal{PT}}=d_{0}\pm\sqrt{d_{x}^{2}+d_{y}^{2}-\tilde{d}_{z}^{2}}.$
(20)

As such, the exceptional nodal structures are defined by solutions to only one
equation,

$d_{x}^{2}+d_{y}^{2}=\tilde{d}_{z}^{2},$ (21)

and stable EPs protected by $\mathcal{PT}$ symmetry occur already in 1D
systems, exceptional lines in 2D and exceptional surfaces in 3D symprotnod .
This increased dimension of nodal structures can also be found in Hermitian
systems subject to certain symmetries herring ; branchCutSemimetals ;
Nodallinksemimetals ; explink ; floquetlinks ; nodalknotsemimetals . The EPs
in the $\mathcal{PT}$-symmetric systems separate the eigenvalue spectrum into
two different phases: the $\mathcal{PT}$-unbroken phase for
$\tilde{d}_{z}^{2}<d_{x}^{2}+d_{y}^{2}$, where the eigenvalues are purely
real, and the $\mathcal{PT}$-broken phase with
$\tilde{d}_{z}^{2}>d_{x}^{2}+d_{y}^{2}$, where the eigenvalues are complex and
appear in complex conjugate pairs. Exactly on the EPs, $\mathcal{PT}$ symmetry
is spontaneously broken.

##  III $\mathcal{PT}$-Symmetry Protected Exceptional Cones and Analogue Light
Cones

Figure 2: (a)-(e) FS ($E^{2}<0$) and iFS ($E^{2}>0$), or equivalently, the
$\mathcal{PT}$-broken and $\mathcal{PT}$-unbroken phases, respectively, of the
model given by Eq. (23) at $k_{z}=0$ for different values of $\kappa$. For
$\kappa<1$, the iFS constitute the whole Brillouin zone slice with a clear
transition occurring at $\kappa=1$. This indicates the overtilting of the EC.
In (f)-(j), the same is displayed at $k_{z}=\frac{\pi}{2}$.

We now extend the analogy between spacetime light cones and WSMs (cf. Sec.
II.2) beyond the Hermitian case to the NH realm. In doing so, it is important
to note that the relation between Dirac operators and spacetime metrics cannot
be naively extended to NH operators. Additionally, the eigenvalues of NH
operators are complex, such that there is no clear intuitive picture to follow
as in the case of the (Hermitian) Weyl Hamiltonian. Interestingly, we show
here that these challenges can be circumvented by studying the degeneracies in
the eigenvalues of the NH $\mathcal{PT}$-symmetric Hamiltonian in Eq. (19).
This model hosts exceptional nodal structures determined by solving Eq. (21),
as well as ordinary nodal structures, which follow from solving
$d_{x}=d_{y}=\tilde{d}_{z}=0$. If we now consider the $\mathcal{PT}$-symmetric
NH Hamiltonian to be in three dimensions and at most _linear_ in momentum
making the Hamiltonian Weyl-like, we find that (possibly tilted) cones appear,
which are composed of EPs (we refer to Appendix A for a general proof). In
particular, solving Eq. (21) for $\tilde{d}_{z}$ gives

$\tilde{d}_{z}=\pm\sqrt{d_{x}^{2}+d_{y}^{2}},$ (22)

which is on the same form as the eigenvalues of a _Hermitian_ Dirac operator.
It further indicates that the EC could intuitively be interpreted as an
analogue light cone. In fact, this relation can be readily formalized at the
level of equations, which we illustrate below for a concrete model with a
generalization presented in Appendix A.

Let us now turn to a specific example. Consider the NH
$\mathcal{PT}$-symmetric two-band model

$\mathcal{H}=k_{x}\sigma^{x}+k_{y}\sigma^{y}+i\left(k_{z}+\kappa
k_{x}\right)\sigma^{z},$ (23)

with its corresponding eigenvalues

$E_{\pm}=\pm\sqrt{k_{x}^{2}+k_{y}^{2}-\left(k_{z}+\kappa k_{x}\right)^{2}}.$
(24)

Here, the term proportional to the identity matrix $d_{0}$ has been neglected
since it does not affect the set of EPs. The EPs are explicitly given by

$k_{z}=-\kappa k_{x}\pm\sqrt{k_{x}^{2}+k_{y}^{2}}.$ (25)

The latter constitutes a cone of EPs in momentum space tilted in the
$k_{x}$-direction by the parameter $\kappa$. The corresponding squared
eigenvalues are displayed in Fig. 2, highlighting the $\mathcal{PT}$-broken
and $\mathcal{PT}$-unbroken phases. When $|\kappa|>1$, the corresponding
(i)FSs become infinite [cf. Figs. 2 (d),(e),(i),(j)], indicating that the EC
is overtilted. Furthermore, similar to how an observer’s spacetime light cone
divides space-like and time-like regions, the EC separates regions of real and
imaginary eigenvalues.

We note that $k_{z}$ in Eq. (25) takes the same form as the eigenvalues of a
Dirac-like operator, for instance on the form,

$\hat{k}_{z}=-\kappa k_{x}\sigma^{0}+k_{x}\sigma^{x}+k_{y}\sigma^{y}.$ (26)

Importantly, this equation and other Dirac-like operators obtained through
this method are necessarily $\mathcal{PT}$-symmetric. The connection between
the EC and the eigenvalues of a Dirac-like operator, such as $\hat{k}_{z}$,
allows us to relate the EC to a spacetime light cone. Recalling Sec. II.2,
$\hat{k}_{z}$ can be associated to a spacetime metric via the vielbein
formalism,

$\hat{k}_{z}=e\indices{{}^{i}_{a}}k_{i}\sigma^{a}+e\indices{{}^{i}_{0}}k_{i}\sigma^{0},$
(27)

with $i,a=x,y$, $e\indices{{}^{i}_{a}}=\delta\indices{{}^{i}_{a}}$,
$e\indices{{}^{x}_{0}}=-\kappa$. From this and using
$e\indices{{}^{0}_{0}}=1$, we find the following line element

$ds^{2}=-\left(1-\kappa^{2}\right)dt^{2}+dx^{2}+dy^{2}+2\kappa dxdt,$ (28)

which is the analogue spacetime metric associated to the EC described by Eq.
(25). Importantly, Eq. (28) describes an event horizon at $|\kappa|=1$, which
coincides with the overtilting of the EC. Since there are no relevant dynamics
in the $y$ direction, this spurious degree of freedom can be eliminated from
the analogue metric, and hence we set $dy=0$. This is a direct consequence of
aligning the tilt of the EC in the $k_{x}$-direction. This metric has the same
form as the one describing the spacetime of a radially infalling observer in
the vicinity of a Schwarzschild black hole in Painlevé-Gullstrand coordinates.
Indeed, we retrieve precisely the metric written in Eq. (1) when keeping the
angular degrees of freedom constant, i.e., $d\Omega=0$, and identifying
$x:=r$, $t:=T$ and $\kappa:=+\sqrt{\frac{2M}{r}}$. We note that this relation
between the EC and the spacetime light cone can be readily generalized to any
3D $\mathcal{PT}$-symmetric NH model with terms linear in momentum as
presented in Appendix A.

At this point, we need to comment on a caveat. To make the connection to the
line element in Eq. (1), we have promoted the tilting parameter to a function
with explicit spatial dependence $\kappa=\kappa(r)$, and consequently, neither
$\kappa(r)$ nor $k_{r}$ commute with the Hamiltonian. In principle this gives
commutator contributions of the form
$\left[k_{r},\kappa(r)\right]\propto\frac{\partial\kappa(r)}{\partial r}$ when
solving the eigenvalue equation of the Hamiltonian in Eq. (23). In the
following section, we provide a rigorous argument for why these contributions
can be consistently neglected in the current study.

We observe that the expression for the Dirac-like operator $\hat{k}_{z}$ in
Eq. (26) is identical to the Hamiltonian of a Hermitian 2D Weyl cone with a
tilting term in the $k_{x}$-direction mediated by $\kappa$, and thus
constitutes a 2D version of the Hamiltonian in Eq. (9). Indeed, the line
element in Eq. (28) is identical to the line element in Eq. (12) for ${\bf
x}=(x,y)$ and $\boldsymbol{\kappa}=\kappa\hat{x}$. This is not a surprising
result, as we are relating one cone to another cone similar to what is done in
Hermitian models. However, the conceptual difference between the EC and an
ordinary Weyl cone leads to a very different interpretation of the physics
they display.

##  IV Spontaneous Particle Emission from Quantum Tunneling

Having formalized the analogy between ECs and spacetime light cones in the
previous section, we now turn to investigate physical properties of
$\mathcal{H}$ in Eq. (23). Recalling the motivation to extend the already
existing analogies between Hermitian WSMs and light cones to include NH
systems, we are primarily interested in lossy or dissipative features
resembling analogue Hawking radiation.

To leading order in $M$ the contribution to Hawking radiation comes from
light-like modes PageI ; PageII . For astrophysical black holes, for which $M$
is very large, these are dominating, while the contribution from massive modes
are subleading massHR . The former can be calculated using the quantum
tunneling method near the horizon $r\sim 2M$. This treatment was first shown
by Parikh and Wilczek, where the semiclassical emission rate is computed as a
quantum tunneling rate PW2000 . Drawing intuition from that massless modes
propagate along the light cone, and that the current analogy relates the light
cone to the EC, we here apply the tunneling technique to the EC of the model
in Eq. (23). Thus, we calculate the semiclassical tunneling rate at the EC and
interpret it as analogue Hawking radiation. The non-trivial result indicates
that tunneling processes mimicking those sourcing the thermal contribution to
Hawking radiation in the analogue black hole are present in the model.

###  IV.1 Commutator contributions near the horizon

Before we apply the quantum tunneling method to our model, we first want to
comment on the fact that the explicit spatial dependence of the tilting
parameter $\kappa$ makes the commutators $\left[\mathcal{H},\kappa(r)\right]$
and $\left[\mathcal{H},k_{r}\right]$ non-vanishing. Consequently, when solving
the corresponding eigenvalue equation additional contributions of the form
$\left[k_{r},\kappa(r)\right]$ will appear, which were neglected when deriving
Eqs. (24) and (25). The neglected terms evaluate to

$k_{r}\left[k_{r},\kappa(r)\right]\propto
k_{r}\frac{\partial\kappa(r)}{\partial r}\propto
k_{r}\frac{M}{r^{2}\sqrt{\frac{2M}{r}}},$ (29)

and they are suppressed as $r^{-3/2}$ when $r$ increases. Since we intend to
apply the quantum tunneling technique to investigate the possible existence of
Hawking radiation, we are interested in how Eq. (29) scales when approaching
the event horizon in the semiclassical limit, i.e., when $r\to 2M$ for large
$M$. Taking the limit explicitly yields

$k_{r}\left.\left[k_{r},\kappa(r)\right]\right|_{r\to 2M}\propto k_{r}\
M^{-1}.$ (30)

Neglecting such terms amounts to assuming $M\gg|k_{r}|$, which is satisfied in
the semiclassical approximation as long as $k_{r}$ is small [in accordance to
the linear form of $\mathcal{H}$ in Eq. (23)]. Thus, the commutator
contributions to the eigenvalues are subleading in $M$, and since we are
focusing on the leading-order contribution to analogue Hawking radiation, they
are neglected in this treatment.

###  IV.2 Particle and antiparticle contributions near the horizon

In the quantum tunneling method, there are two processes that give massless
contributions to the spontaneous black hole emission: a particle tunneling
outwards from the interior of the black hole, and an antiparticle tunneling
into the black hole. The quantity of interest describing classically forbidden
trajectories is the imaginary part of the action of a radially infalling
light-like shell of energy PW2000 . As anticipated, we consider the region in
momentum space corresponding to the EC and since the spatial dependence of the
tilting term is taken to be in the $x$-coordinate, we solve Eq. (25), which
are the eigenvalues of the model in Eq. (26), for $k_{x}$. Recalling the
identifications $x:=r$ and $\kappa:=+\sqrt{\frac{2M}{r}}$, we find two
solutions

$k_{r}^{\pm}=\frac{-\sqrt{\frac{2M}{r}}k_{z}}{\frac{2M}{r}-1}\pm\sqrt{\frac{k_{z}^{2}}{\left(\frac{2M}{r}-1\right)^{2}}+\frac{k_{y}^{2}}{\frac{2M}{r}-1}}.$
(31)

Here, the expression inside the square-root is bound to be positive, since
$k_{r}^{\pm}$ has to be real, such that
$k_{z}^{2}\geq-(\frac{2M}{r}-1)k_{y}^{2}$.

To determine which of these two solutions is to be interpreted as the momentum
of the analogue particles and antiparticles, we investigate the behavior of
Eq. (31) when approaching asymptotic infinity, i.e., when $r\to+\infty$, in
close analogy with the discussion on geodesics presented in Sec. II.1. In this
limit, we find

$\displaystyle\lim_{r\to+\infty}k_{r}^{\pm}$
$\displaystyle=\pm\sqrt{k_{z}^{2}-k_{y}^{2}},$ (32)

with $k_{z}^{2}\geq k_{y}^{2}$. Consequently, $k_{r}^{+}\geq 0$ and
$k_{r}^{-}\leq 0$ when $r\to+\infty$. Recalling how the radial null geodesics
$\dot{r}^{\pm}$, and thus $p_{r}^{\pm}$, which is the momentum conjugate to
$r$, behaves at asymptotic infinity PW2000 , we see that $k_{r}^{\pm}$ shares
the behavior of $p_{r}^{\pm}$. Hence, we interpret $k_{r}^{\pm}$ as analogue
to the conjugate momentum $p_{r}$ meaning that $k_{r}^{+}$ and $k_{r}^{-}$
describe the momentum of an analogue particle and antiparticle, respectively.
To make sure that the conclusion above is consistent, we also investigate the
behavior of $k_{r}^{\pm}$ when approaching the horizon, i.e., when $r\to 2M$.
Here, the momentum of a particle should classically reach $+\infty$ and the
momentum of an antiparticle $-\infty$. Indeed, the limits of interest read

$\displaystyle k_{r}^{+}$
$\displaystyle\to\begin{cases}+\infty\quad\text{as}&r\to 2M^{-}\text{ and
}k_{z}<0\\\ +\infty\quad\text{as}&r\to 2M^{+}\text{ and }k_{z}>0,\end{cases}$
(33) $\displaystyle k_{r}^{-}$
$\displaystyle\to\begin{cases}-\infty\quad\text{as}&r\to 2M^{-}\text{ and
}k_{z}>0\\\ -\infty\quad\text{as}&r\to 2M^{+}\text{ and }k_{z}<0.\end{cases}$
(34)

Note that the point $k_{z}=0$ is absent in Eqs. (33) and (34). This point
constitutes a trivial limit, since being on the EC at $k_{z}=0$ gives
$k_{y}=0$, and consequently $k_{r}^{\pm}=0$. The signs of the infinities are
consistent with identifying $k_{r}^{+}$ and $k_{r}^{-}$ as the radial momentum
of the analogue particle and antiparticle in the vicinity of an analogue black
hole, respectively. The different infinities should be interpreted as follows:
The quantity $\lim_{r\to 2M^{-}}k_{r}^{+}$ reflects that, classically, the
analogue particle needs to acquire infinite momentum, and hence infinite
energy, to cross the horizon from the interior of the black hole and escape to
infinity. Similarly, $\lim_{r\to 2M^{+}}k_{r}^{-}$ reflects that the analogue
antiparticle needs negative infinite momentum, and hence negative infinite
energy, to pass through the horizon from the exterior of the black hole. These
two processes are thus classically forbidden, and quantum tunneling is
required for them to happen. On the contrary, $\lim_{r\to 2M^{+}}k_{r}^{+}$
merely reflects the infinite redshift of a particle approaching the analogue
black hole horizon from the outside, as seen by an asymptotic observer. A
similar process is also reflected for the antiparticle in $\lim_{r\to
2M^{-}}k_{r}^{-}$. These processes are not classically forbidden, despite
resulting in divergences in the momentum.

The conceptual reasoning above is also reflected in the corresponding actions.
Classically forbidden processes are described by the imaginary part of the
action, which for an analogue radially infalling shell of energy reads
Volovik1999 ; PW2000 ,

$\text{Im}\left(I\right)=\text{Im}\left(\int dr\,k_{r}\right),$ (35)

which is to be evaluated on the EC. Formally, the actions for each of the
channels on the EC are given by

$\displaystyle\text{Im}\left(I^{\pm}\right)$
$\displaystyle=\text{Im}\left\\{\int dr\times\right.$
$\displaystyle\quad\left.\left[\frac{-\sqrt{\frac{2M}{r}}k_{z}}{\frac{2M}{r}-1}\pm\sqrt{\frac{k_{z}^{2}}{\left(\frac{2M}{r}-1\right)^{2}}+\frac{k_{y}^{2}}{\frac{2M}{r}-1}}\right]\right\\},$
(36)

and lead to

$\displaystyle\text{Im}\left[I^{+}(k_{z})\right]$
$\displaystyle=\begin{cases}2\pi M|k_{z}|\quad&k_{z}<0\\\ 0\quad&k_{z}\geq
0,\end{cases}$ (37) $\displaystyle\text{Im}\left[I^{-}(k_{z})\right]$
$\displaystyle=\begin{cases}2\pi M|k_{z}|\quad&k_{z}>0\\\ 0\quad&k_{z}\leq
0,\end{cases}$ (38)

where detailed calculations are presented in Appendix B. We thus find nonzero
contributions to the imaginary part of the action allowing us to conclude that
processes analogue to classically forbidden processes near a black hole
horizon are indeed present in our model. These results also confirm the
conceptual statements that only $\lim_{r\to 2M^{-}}k_{r}^{+}$ and $\lim_{r\to
2M^{+}}k_{r}^{-}$ reflect classically forbidden processes.

###  IV.3 Semiclassical emission rate

The probability of a classically forbidden process is estimated by the
semiclassical quantum tunneling rate. Since Hawking radiation corresponds to
both particles tunneling out of and antiparticles tunneling into the black
hole, both channels contribute to the total probability of the tunneling
process. The associated tunneling rate can then be interpreted as the analogue
black hole emission rate as long as the respective tunneling amplitudes are
added up and then squared PW2000 . Using the imaginary parts of the actions
derived in Sec. IV.2, the respective amplitudes read

$\displaystyle\mathcal{A}^{+}$ $\displaystyle=e^{-2\pi
M|k_{z}|}\theta(-k_{z}),$ (39) $\displaystyle\mathcal{A}^{-}$
$\displaystyle=e^{-2\pi M|k_{z}|}\theta(k_{z}),$ (40)

where $\theta(k_{z})$ is the Heaviside theta-function defined as

$\theta(x)=\begin{cases}1\quad&x>0\\\ 0\quad&x\leq 0.\end{cases}$ (41)

From the amplitudes, the semiclassical emission rate reads

$\Gamma\sim\left(\mathcal{A}^{+}+\mathcal{A}^{-}\right)^{2}=\left(\mathcal{A}^{+}\right)^{2}+\left(\mathcal{A}^{-}\right)^{2}+\mathcal{A}^{+}\mathcal{A}^{-}.$
(42)

Since $\text{Im}\left(I^{+}\right)$ and $\text{Im}\left(I^{-}\right)$ do not
have support on each others domains [cf. Eqs. (39) and (40)], the cross term
$\mathcal{A}^{+}\mathcal{A}^{-}$ vanishes. We thus find

$\Gamma\sim e^{-4\pi M|k_{z}|}\qquad k_{z}\neq 0,$ (43)

which estimates the total probability for a tunneling process to occur. Given
the interpretation of $\mathcal{H}$ in Eq. (23) as an analogue model for a
black hole, the quantity in Eq. (43) can be thought of as an emission rate,
the nature of which is system dependent. Note that while the probability is
finite and non-zero for $|k_{z}|\neq 0$, it is exponentially suppressed with
increasing values of $|k_{z}|$. This means that a tunneling process is
unlikely unless $|k_{z}|$ is small.

##  V Towards Experimental Realizations

Here, we connect our model to possible experimental realizations in order to
specify the concrete nature of the analogue Hawking radiation.

One promising experimental platform for realizing our model is in photonic
crystals, where gain and loss can be readily implemented and which are well
described by non-Hermitian Hamiltonians Ozdemir2019 ; bfermiarcs ; NHreview ;
NHPC .

The Hermitian part of our NH Hamiltonian $\mathcal{H}$ in Eq. (23) looks like
a Weyl Hamiltonian, which has been previously realized in photonic crystals in
both the Hermitian and non-Hermitian form bfermiarcs ; Chen2016 ; LWYRFJS2015
; Yang2017 . The anti-Hermitian part of $\mathcal{H}$ provides a bigger
challenge for realization. While such momentum-dependent imaginary terms can
be realized using single-photon interferometry expknots , we are not aware of
a realization in a photonic crystal. Let us suggest a possible path towards
realizing such terms. We start by coupling the photonic crystal to its
environment in such a way that the dynamics of the photonic lattice are well-
described by the Lindblad master equation Lindblad1976 ,

$\partial_{t}\rho=i(\rho
H_{\textrm{eff}}^{\dagger}-H_{\textrm{eff}}\rho)+\sum_{n}(\kappa L_{x,n}\rho
L_{x,n}^{\dagger}+L_{z,n}\rho L_{z,n}^{\dagger}),$ (44)

where $\rho$ is the density matrix, $L_{i,n}$ are the jump operators,
$H_{\textrm{eff}}$ is the effective non-Hermitian Hamiltonian which yields

$H_{\textrm{eff}}=H_{\textrm{H}}-i\sum_{n}(\kappa
L^{\dagger}_{x,n}L_{x,n}+L^{\dagger}_{z,n}L_{z,n})/2,$ (45)

with $H_{\textrm{H}}=\sum_{\bf k}{\bf b}^{\dagger}_{\bf
k}\mathcal{H}_{\textrm{H}}{\bf b}_{\bf k}$ the (Hermitian) system Hamiltonian,
where ${\bf b}_{\bf k}=(b_{1,\bf k},b_{2,\bf k})^{T}$ with $b_{i,\bf k}$
annihilates a particle at site $i$ with momentum ${\bf k}$, and
$\sum_{n}(\kappa L_{x,n}\rho L_{x,n}^{\dagger}+L_{z,n}\rho L_{z,n}^{\dagger})$
is known as the quantum jump or recycling term. This assumption, namely that
the bath acts as a Markovian bath, is well justified for the majority of
optical experiments Daley2014 . As the photonic lattice is effectively a
classical system, the quantum jump terms do not contribute and the dynamics
from the master equation coincide with the dynamics generated by the effective
non-Hermitian Hamiltonian $H_{\textrm{eff}}$ Ozdemir2019 ; bfermiarcs ;
NHreview ; NHPC . We are thus left with having to engineer the correct jump
operators such that $\mathcal{H}_{\textrm{eff}}$ in
$H_{\textrm{eff}}=\sum_{\bf k}{\bf b}^{\dagger}_{\bf
k}\mathcal{H}_{\textrm{eff}}{\bf b}_{\bf k}$ resembles $\mathcal{H}$ in Eq.
(23).

If the photonic crystal whose system Hamiltonian is modelled by
$\mathcal{H}_{\textrm{H}}=k_{x}\sigma^{x}+k_{y}\sigma^{y}$, i.e., the
Hermitian part of $\mathcal{H}$ (23), is coupled to the environment such that
its jump operators read

$L_{x,n}=b_{2,n}-ib_{2,n+\hat{x}}\ \ \text{and}\ \
L_{z,n}=b_{2,n}-ib_{2,n+\hat{z}},$ (46)

where $b_{2,n+\hat{e}}$ is a real-space annihilation operator at site $2$ in
unit cell $n+\hat{e}$, we find that the effective Fourier-transformed
Hamiltonian in the linearized form reads

$\mathcal{H}_{\textrm{eff}}=k_{x}\sigma^{x}+k_{y}\sigma^{y}-i(2+\kappa
k_{x}+k_{z})(\sigma_{0}-\sigma_{z}),$ (47)

(see Appendix C for a derivation of this effective Hamiltonian). This
effective description has eigenvalues

$E_{\textrm{eff},\pm}=-i(2+k_{z}+\kappa
k_{x})\pm\sqrt{k_{x}^{2}+k_{y}^{2}-(2+k_{z}+\kappa k_{x})^{2}},$ (48)

from which we see that the effective Hamiltonian thus indeed realizes ECs
albeit at a shifted $k_{z}$ value $(k_{z}\rightarrow k_{z}-2)$. Let us now
address possible ways to realize nonlocal jump operators $L_{i,n}$. To this
end, we draw inspiration from earlier work on nonreciprocal couplings in
optical cavity arrays, where these types of non-local jump operators have been
realized through coupling two optical cavities to a third auxiliary cavity,
which is then adiabatically eliminated Metelmann2015 ; Bernier2017 ; Malz2018
. It is a tantalizing prospect to investigate whether a similar technique can
be used in photonic lattices.

Next, we investigate the significance of the emission rate in Eq. (43) in the
setting of an optical crystal modelled by $\mathcal{H}_{\textrm{eff}}$, where
we note that the emission rate is valid for momenta on the ECs and
$\kappa\rightarrow 1$. In other words, this equation is relevant for the zero-
energy solutions of $\mathcal{H}(\kappa\rightarrow 1)$ or $-i(2+k_{z}+\kappa
k_{x})$ solutions of $\mathcal{H}_{\textrm{eff}}(\kappa\rightarrow 1)$, where
we use our previous result that the Weyl nodes, which also sit at these
energies and are solutions to ${\bf d}={\bf 0}$, do not contribute to the
emission rate. In Ref. NHPC, , the authors access both the real and imaginary
parts of the eigenvalues of a photonic crystal by measuring thermal emission
using angle-resolved thermal emission spectroscopy (ARTES). This in principle
makes it possible to identify the thermal emission coming from the EPs, or
more concretely the ECs of $\mathcal{H}$ or $\mathcal{H}_{\text{eff}}$.
Therefore, we continue along these lines.

To retrieve the information needed to determine the analogue Hawking
temperature $T_{\text{BH}}$, we suggest the following approach. First, perform
an ARTES-measurement on a photonic crystal modelled by
$\mathcal{H}_{\text{eff}}(\kappa\to 1)$. This provides an intensity spectrum
of emitted photons with different frequencies as a function of lattice
momentum. In all, one retrieves three pieces of information: the frequency of
the emitted photons, where in momentum space they were emitted from, and the
corresponding intensities. Second, inspired by the fact that photonic crystals
have been experimentally shown to emit radiation well approximated by that of
a blackbody Lin2003 ; Florescu2005 , we think of the photonic crystal as an
effective blackbody. Then, the probability of emission of photons from the
photonic crystal can be estimated by

$\Gamma_{\text{PL}}\sim e^{-\beta_{\text{eff}}\ \omega^{\prime}},$ (49)

where $\beta_{\text{eff}}=T_{\text{eff}}^{-1}$ is the inverse effective
temperature, and $\omega^{\prime}$ the frequency of the emitted photon. By
plotting the intensity of the emitted photons as a function of the
frequencies, i.e., using parts of the data obtained in the ARTES measurement,
$T_{\text{eff}}$ can be determined using Wien’s displacement law Wien . Third,
identify the location of the EC by using the data from the ARTES measurement,
as shown in Ref. NHPC, . The photons emitted from the EC will resonate at one
frequency, which we denote $\omega_{0}$. For this particular frequency Eq.
(49) estimates the probability for a photon of frequency $\omega_{0}$ to be
emitted from the EC. Recalling that Eq. (43) can be thought of as an emission
rate (cf. Sec. IV.3), both Eqs. (43) and (49) (with
$\omega^{\prime}=\omega_{0}$) describe thermal emission from the EC. Motivated
by this, we identify the two,

$e^{-4\pi M|k_{z}|}\sim e^{-\beta_{\text{eff}}\ \omega_{0}},$ (50)

where $\sim$ means that the two sides have the same exponential behavior, and
thus,

$M=\frac{\beta_{\text{eff}}\ \omega_{0}}{4\pi|k_{z}|}.$ (51)

The analogue Hawking temperature is then given by

$T_{\text{BH}}=\frac{T_{\text{eff}}\ |k_{z}|}{2\omega_{0}},$ (52)

where we have used Eq. (4). This suggests that the analogue black hole mass,
and hence the analogue Hawking temperature, are controlled by where on the EC
the ARTES-measurement is performed. However, we recall that the present
analogy only holds in the semiclassical limit. Let us recall here that the
assumption $M\gg|k_{r}|$ was done to justify neglecting commutator
contributions in Eqs. (24) and (25), cf. Sec. IV.1. On the EC, and for small
values of momentum (justified by the linearized descriptions of $\mathcal{H}$
and $\mathcal{H}_{\text{eff}}$), this can be translated to $M\gg|k_{z}|$, by
using Eq. (25). Thus, for a particular measurement, we demand

$\frac{\omega_{0}}{4\pi
T_{\text{eff}}|k_{z}|}\gg|k_{z}|\Rightarrow|k_{z}|^{2}\ll\frac{\omega_{0}}{4\pi
T_{\text{eff}}}.$ (53)

This means that our analogy holds if the thermal emission from the EC is
measured at $k_{z}$ satisfying Eq. (53).

Thus, we propose that an ARTES-measurement on a photonic crystal modelled by
$\mathcal{H}_{\text{eff}}(\kappa\to 1)$ can be used to interpret the thermal
radiation with frequency $\omega_{0}$ as the purely thermal contribution to
analogue Hawking radiation, for sufficiently small $|k_{z}|$.

##  VI Conclusions and Outlook

In this work, we study analogies between the band structure of a
$\mathcal{PT}$-symmetric NH two-band model and curved spacetime geometries. As
an extension of analogies between Weyl cones and spacetime light cones, we
find a relation between the $\mathcal{PT}$ symmetry-protected cone of EPs,
dubbed the EC, and the light cone of a radially infalling observer in the
presence of a (3+1)D Schwarzschild black hole. By calculating the
semiclassical emission rate, we show that radiative processes reminiscent of
those happening in black holes are present in the NH model. We show that this
particular model can be retrieved as an effective description originating from
a Lindblad master equation describing the dynamics of a photonic crystal. In
this context, we argue that the aforementioned emission rate characterizes
thermal loss from the EC in a photonic crystal. To identify the origin of the
thermal loss, which constitutes effective blackbody radiation, we suggest that
the ARTES technique can be used. In this way, the thermal loss from a photonic
crystal can be interpreted as analogue Hawking radiation.

We note that our toy model in Eq. (23) not only hosts ECs, but that the
eigenvalues themselves also form cones. Indeed, both the real and imaginary
parts of the eigenvalues in Eq. (23) display cone structures, which are
separated from each other in the complex energy plane by EPs. By tuning
$\kappa$, these cones do not tilt unlike the ECs, but instead rotate in the
energy plane. However, by adding an additional ($\mathcal{PT}$ symmetry-
preserving) term to our Hamiltonian, e.g., $\kappa_{1}k_{x}\sigma^{0}$, these
cones can tilt similar to the spacetime light cones upon tuning $\kappa_{1}$.
In fact, this is a generic feature of $\mathcal{PT}$-symmetric NH models in
three dimensions that consists of at most linear terms in momentum. While we
thus set out in this paper to investigate the connection between ECs and
spacetime light cones (due to the attractive similarities between the two
beyond those between Hermitian Weyl cones and light cones), a natural
additional question to ask is whether such a connection can also exist between
these NH eigenvalue cones and light cones. We note that to investigate this
question, the method we have employed in this paper may not be applicable. At
this point, it is not clear how the vielbein construction can be extended to
NH Dirac-like operators, which explicitly include imaginary terms, and we
leave this question for further studies.

Even though the analogy constructed in this work is an extension of the
previously known Hermitian analogies, it is fundamentally different. This is
especially clear in the interpretation of the analogue Hawking radiation.
Concretely, it allows for the interpretation of thermal loss measured on the
EC in a photonic crystal as analogue Hawking radiation. This is favourable,
both theoretically and experimentally, since the analogue Hawking temperature
$T_{\text{BH}}$ is to be measured indirectly through measurements of system
parameters of the photonic crystal, cf. Eq. (52). A direct consequence of this
is that $T_{\text{BH}}$ is not directly restricted by the temperature of the
experimental sample, nor its surroundings, meaning that arbitrarily small
$T_{\text{BH}}$ in principle can be achieved. Therefore, the analogy suggests
a novel way to model realistic astrophysical black holes, which are massive
(large $M$) and hence, cold (small $T_{\text{BH}}$). This is furthermore
consistent with the semiclassical treatment, which motivated neglecting
subleading powers of $M$ in Eq. (29). A natural continuation along these lines
is, however, to investigate the meaning of these subleading corrections. We
note that these amount to commutator contributions to the eigenvalue equation
and thus the equation of the EC, Eq. (25). Keeping those terms would make the
relation to the analogue metric Eq. (28) less apparent, as Eq. (25) could host
imaginary terms. Therefore, studies and interpretations of such corrections
are left for future works.

Another interesting quantity that could be investigated further, apart from
the previously computed emission rate, is the entropy. One could calculate the
entanglement entropy associated to the tilted NH Weyl-like Hamiltonian and see
if it matches the expected result of the Bekenstein-Hawking entropy for the
analogue semiclassical black hole Bekenstein ; Entanglemententropy . As the
present analogy focuses on finding a meaningful analogue of Hawking radiation,
such a computation goes beyond the scope of this particular study and is left
for future works.

The present study investigates analogue Hawking radiation on the EC, where
$\mathcal{PT}$ symmetry is spontaneously broken. One could wonder if further
generalizations of this analogy can lead to more abstract connections between
ECs and analogue event horizons, where the $\mathcal{PT}$ phase transition may
be related to the transition between the time-like (exterior) and space-like
(interior) regions of the black hole experienced by an infalling observer
crossing the black hole horizon. Furthermore, it is interesting to point out
that the interior of the EC satisfying $E^{2}<0$ can be identified as a
tachyon-like phase, where excitations travel faster than the effective speed
of light Feinberg1967 . We note that such a connection between
$\mathcal{PT}$-symmetric models and tachyonic physics was the focus of a very
recent study, where a tachyonic phase was identified in the phase diagram of a
1D Dirac model with real and imaginary mass terms Sticlet2021 . However, to
our knowledge, connections between spontaneous $\mathcal{PT}$-symmetry
breaking and horizon physics have not been investigated enough to establish
any precise relations. We nevertheless note that in the context of gauge-
gravity duality, recent work has made concrete connections between
$\mathcal{PT}$-symmetric quantum theory and black holes in asymptotic AdS
spacetimes NHholography . A possible intriguing relation between these ideas
realized on holographic setups dual to $\mathcal{PT}$ phase transitions and
our work is left for further studies.

Lastly, the importance of multiband models in similar NH models has been
studied in recent works 3EPemil ; HOEP ; 4EPMarcus . In particular, the
appearance of higher-order EPs is investigated. To this point, we note that
considering additional bands, the ECs of order two can be accompanied by
third-order exceptional lines and fourth-order EPs in 3D
$\mathcal{PT}$-symmetric systems 4EPMarcus . The impact of the higher-order
exceptional structure on the physics discussed in this paper may be
interesting but is outside the scope of this work. Here we assume additional
bands to be located far away from the ones considered in this work, and leave
further investigation for future studies.

_Note added:_ After the publication of this paper, we have become aware of
Ref. Sundaram2021, , where a connection is made between
$\mathcal{PT}$-symmetry breaking transitions and the inverse-square potential
relevant for black hole physics near the horizon. The content of that work
thus forms another motivation to search for connections between near-horizon
black hole physics and $\mathcal{PT}$-symmetric systems.

## Acknowledgements

We would like to thank Emil J. Bergholtz, Johan Carlström and Watse Sybesma
for useful discussions. Special thanks to Thors Hans Hansson, Fawad Hassan, Bo
Sundborg and Lárus Thorlacius for useful comments on the manuscript. M.S. is
supported by the Swedish Research Council (VR) and the Wallenberg Academy
Fellows program of the Knut and Alice Wallenberg Foundation. L.R. is supported
by the Knut and Alice Wallenberg foundation under grant no. 2017.0157. F.K.K.
is supported by the Max Planck Institute of Quantum Optics (MPQ) and Max-
Planck-Harvard Research Center for Quantum Optics (MPHQ). The work was
finished while J.L.A. and L.R. were participating in a mathematical physics
conference, organized by Uppsala University and funded by the Göran Gustafsson
foundation, whose stimulating environment is also acknowledged.

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##  Appendix A Cones of exceptional points in
$\mathcal{P}\mathcal{T}$-symmetric systems and their analogue event horizons

In this Appendix, we show that any 3D NH $\mathcal{PT}$-symmetric two-band
model linear in momentum host cones of EPs, which can be mapped to a curved
spacetime metric. Since we are only interested in the EPs, and not the
eigenvalues themselves, we can set the term proportional to the identity
matrix to $0$. Generally, in an appropriate choice of coordinates, the
Hamiltonian attains the form

$\displaystyle H$
$\displaystyle=\left(a_{x}k_{x}+a_{y}k_{y}+a_{0}\right)\sigma^{x}+\left(b_{x}k_{x}+b_{y}k_{y}+b_{0}\right)\sigma^{y}+i\left(c_{x}k_{x}+c_{y}k_{y}+c_{z}k_{z}+c_{0}\right)\sigma^{z},$
(54)

with all parameters being real. We note that the constant terms $a_{0}$,
$b_{0}$ and $c_{0}$ can be set to $0$, since they can be removed by shifting
the momentum components. Explicitly, the corresponding EPs are given by

$\displaystyle\quad\left(a_{x}^{2}+b_{x}^{2}-c_{x}^{2}\right)k_{x}^{2}+\left(a_{y}^{2}+b_{y}^{2}-c_{y}^{2}\right)k_{y}^{2}-c_{z}^{2}k_{z}^{2}+2k_{x}k_{y}\left(a_{x}a_{y}+b_{x}b_{y}-c_{x}c_{y}\right)-2c_{y}c_{z}k_{y}k_{z}-2c_{x}c_{z}k_{x}k_{z}=0,$
(55)

and solving for $k_{z}$ yields

$\displaystyle k_{z}$
$\displaystyle=-\frac{1}{c_{z}}\left(c_{x}k_{x}+c_{y}k_{y}\right)\pm\sqrt{\left(a_{x}k_{x}+a_{y}k_{y}\right)^{2}+\left(b_{x}k_{x}+b_{y}k_{y}\right)^{2}}.$
(56)

with $c_{z}\neq 0$. As in the simple example dealt with in the main text,
$k_{z}$ can be thought of as the eigenvalue of some Dirac-like operator, in
this case, e.g.,

$\hat{k}_{z}=-\frac{1}{c_{z}}\left(c_{x}k_{x}+c_{y}k_{y}\right)\sigma^{0}+\left(a_{x}k_{x}+a_{y}k_{y}\right)\sigma^{x}+\left(b_{x}k_{x}+b_{y}k_{y}\right)\sigma^{y},$
(57)

whose eigenvalues form a cone. It should be stressed that adding a term
proportional to $\sigma^{z}$ Eq. (57) would correspond to adding an equivalent
term to the initial Hamiltonian Eq. (54), which would break the implemented
$\mathcal{PT}$-symmetry. Hence, such terms are forbidden, and we note more
accurate that $k_{z}$ can be thought of the eigenvalues of a Hermitian Dirac-
operator subject to $\mathcal{PT}$ symmetry.

Since a Dirac-operator can be expressed in terms of vielbeins Volovik2014 ;
Horova2005 , we can write $\hat{k}_{z}$ on the form

$\hat{k}_{z}=e\indices{{}^{i}_{a}}k_{i}\sigma^{a}+e\indices{{}^{i}_{0}}k_{i}\sigma^{0},$
(58)

with

$\displaystyle e\indices{{}^{0}_{0}}$ $\displaystyle=1,\quad
e\indices{{}^{x}_{0}}=-\frac{c_{x}}{c_{z}},\quad
e\indices{{}^{y}_{0}}=-\frac{c_{y}}{c_{z}},\quad e\indices{{}^{0}_{x}}=0,$
$\displaystyle e\indices{{}^{0}_{y}}$ $\displaystyle=0,\quad
e\indices{{}^{x}_{x}}=a_{x},\quad e\indices{{}^{x}_{y}}=b_{x},\quad
e\indices{{}^{y}_{x}}=a_{y},\quad e\indices{{}^{y}_{y}}=b_{y},$ (59)

from which a curved spacetime metric can be defined through
$g_{\mu\nu}=e\indices{{}_{\mu}^{\alpha}}e\indices{{}_{\nu}^{\beta}}\eta_{\alpha\beta}$.

##  Appendix B Imaginary parts of the actions: Detailed calculations

Figure 3: The contours used when integrating (a) Eq. (64) and (b) Eq. (65),
respectively. The contours along the real line has to be deformed differently
to reflect the correct physics of particles and antiparticles. Singularities
in the description of particles are pushed to the lower half complex plane,
and hence the contour is deformed to a quarter circle in the first quadrant in
(a). Similarly, the singularities in the description of antiparticles are
pushed to the upper half complex plane, giving a quarter circle in the fourth
quadrant (b).

In this Appendix, we provide the details for the calculation of the imaginary
parts of the actions of the particle and antiparticle channels. The imaginary
part of the action of the analogue radially infalling shell of energy PW2000
reads

$\text{Im}\left(I\right)=\text{Im}\left(\int k_{r}dr\right),$ (60)

which we want to evaluate on the EC. Formally, the actions for each of the
channels on the EC are given by

$\displaystyle\text{Im}\left(I^{\pm}\right)$
$\displaystyle=\text{Im}\left\\{\int
dr\left[\frac{-\sqrt{\frac{2M}{r}}k_{z}}{\frac{2M}{r}-1}\pm\sqrt{\frac{k_{z}^{2}}{\left(\frac{2M}{r}-1\right)^{2}}+\frac{k_{y}^{2}}{\frac{2M}{r}-1}}\right]\right\\}.$
(61)

We know that these solutions correspond to the momentum of a particle $(+)$
and antiparticle $(-)$, respectively. For the momentum of a particle, we
consider a spontaneous radiation process in which a positive energy $\omega>0$
is assumed to radiate out from the interior of the black hole such that the
mass of the black hole changes from $M$ to $M-\omega$. Thus, the action of the
particle channel reads

$\displaystyle\text{Im}\left(I^{+}\right)$
$\displaystyle=\text{Im}\left\\{\int_{2M}^{2\left(M-\omega\right)}dr\left[\frac{-\sqrt{\frac{2M}{r}}k_{z}}{\frac{2M}{r}-1}+\sqrt{\frac{k_{z}^{2}}{\left(\frac{2M}{r}-1\right)^{2}}+\frac{k_{y}^{2}}{\frac{2M}{r}-1}}\right]\right\\}.$
(62)

The antiparticle case, on the other hand, can be interpreted as a particle
with a negative energy $\omega^{\prime}<0$ tunneling inwards such that the
mass changes from $M$ to $M+\omega^{\prime}$. Therefore, the action of the
antiparticle channel is.

$\displaystyle\text{Im}\left(I^{-}\right)$
$\displaystyle=\text{Im}\left\\{\int_{2M}^{2\left(M+\omega^{\prime}\right)}dr\left[\frac{-\sqrt{\frac{2M}{r}}k_{z}}{\frac{2M}{r}-1}-\sqrt{\frac{k_{z}^{2}}{\left(\frac{2M}{r}-1\right)^{2}}+\frac{k_{y}^{2}}{\frac{2M}{r}-1}}\right]\right\\}.$
(63)

Re-writing and making a change of variables, $-r^{\prime}=r-2M$,
$dr=-dr^{\prime}$, gives

$\displaystyle\text{Im}\left(I^{+}\right)$
$\displaystyle=\text{Im}\left\\{\int_{0}^{2\omega}dr^{\prime}\left[\frac{\sqrt{2M\left(2M-r^{\prime}\right)}k_{z}}{r^{\prime}}-\sqrt{\frac{k_{z}^{2}\left(2M-r^{\prime}\right)^{2}}{r^{\prime
2}}+\frac{k_{y}^{2}\left(2M-r^{\prime}\right)}{r^{\prime}}}\right]\right\\},$
(64) $\displaystyle\text{Im}\left(I^{-}\right)$
$\displaystyle=\text{Im}\left\\{\int_{0}^{-2\omega^{\prime}}dr^{\prime}\left[\frac{\sqrt{2M\left(2M-r^{\prime}\right)}k_{z}}{r^{\prime}}+\sqrt{\frac{k_{z}^{2}\left(2M-r^{\prime}\right)^{2}}{r^{\prime
2}}+\frac{k_{y}^{2}\left(2M-r^{\prime}\right)}{r^{\prime}}}\right]\right\\}.$
(65)

When evaluating these integrals, we have to proceed with caution. First, note
that there are branch points at $r^{\prime}=2M:=r^{\prime}_{1}$, which is
positive, and
$r^{\prime}=2M\frac{k_{z}^{2}}{k_{z}^{2}-k_{y}^{2}}:=r^{\prime}_{2}$, which is
negative when $k_{z}^{2}<k_{y}^{2}$, and positive when $k_{z}^{2}>k_{y}^{2}$.
Thus, when $k_{z}^{2}<k_{y}^{2}$, we make branch cuts from
$r^{\prime}=r_{1}^{\prime}$ to positive infinity, and from
$r^{\prime}=r^{\prime}_{2}$ to negative infinity. When $k_{z}^{2}>k_{y}^{2}$,
we first note that $r_{2}^{\prime}\geq r_{1}^{\prime}$, and hence we make a
branch cut from $r^{\prime}=r_{1}^{\prime}$ to $r^{\prime}=r_{2}^{\prime}$.
When $k_{z}^{2}=k_{y}^{2}$, $r_{2}^{\prime}\to\pm\infty$, depending on if
$k_{z}^{2}\to k_{y}^{2}$ from above $(+)$ or below $(-)$. These cases are thus
included in the cases above.

Noting that in both $\omega\leq M$ and $-\omega^{\prime}\leq M$, the energy of
the radiated quanta cannot be larger than the initial mass of the black hole,
the square root functions are analytic on the interval that is integrated
over. Second, both integrals start at a pole of order 1, and evaluation of the
corresponding integrals requires a deformation of the integral contour. Since
$I^{+}$ and $I^{-}$ are actions of particles and antiparticles, respectively,
the corresponding deformations are necessarily different as required by
causality; the poles of the particle action $I^{+}$ are to be located in the
lower half complex $r^{\prime}$-plane, while the poles of the antiparticle
action $I^{-}$ are to be located in the upper half complex $r^{\prime}$-plane.
Therefore, the contour of integration in $I^{+}$ is deformed to a quarter
circle of radius $\epsilon$ in the first quadrant, connected to a line from
$\epsilon$ to $2\omega$, while the contour of integration in $I^{-}$ is
deformed to a quarter circle of radius $\epsilon$ in the fourth quadrant,
connected to a line from $\epsilon$ to $-2\omega^{\prime}$. The appearance of
the different contours are displayed in Fig. 3. By taking the limit
$\epsilon\to 0$, the original integrals are obtained. Hence,

$\displaystyle\text{Im}\left(I^{+}\right)$
$\displaystyle=\text{Im}\left\\{\lim_{\epsilon\to
0}\left(\int_{\gamma^{+}_{\epsilon}}+\int_{\epsilon}^{2\omega}\right)dr^{\prime}\left[\frac{\sqrt{2M\left(2M-r^{\prime}\right)}k_{z}}{r^{\prime}}-\sqrt{\frac{k_{z}^{2}\left(2M-r^{\prime}\right)^{2}}{r^{\prime
2}}+\frac{k_{y}^{2}\left(2M-r^{\prime}\right)}{r^{\prime}}}\right]\right\\},$
(66) $\displaystyle\text{Im}\left(I^{-}\right)$
$\displaystyle=\text{Im}\left\\{\lim_{\epsilon\to
0}\left(\int_{\gamma^{-}_{\epsilon}}+\int_{\epsilon}^{-2\omega^{\prime}}\right)dr^{\prime}\left[\frac{\sqrt{2M\left(2M-r^{\prime}\right)}k_{z}}{r^{\prime}}+\sqrt{\frac{k_{z}^{2}\left(2M-r^{\prime}\right)^{2}}{r^{\prime
2}}+\frac{k_{y}^{2}\left(2M-r^{\prime}\right)}{r^{\prime}}}\right]\right\\},$
(67)

where $\gamma_{\epsilon}^{+}$ and $\gamma^{-}_{\epsilon}$ denote the quarter
circles of radii $\epsilon$ located in the first and fourth quadrant,
respectively [cf. Fig 3]. The second integrals integrate over a domain where
the corresponding integrand is real valued. Hence, they do not contribute to
the imaginary parts and therefore we only evaluate the integrals over
$\gamma_{\epsilon}^{+}$ and $\gamma^{-}_{\epsilon}$, respectively. Choosing a
parametrization as $r^{\prime}=\epsilon e^{i\phi}$, $dr^{\prime}=i\epsilon
e^{i\phi}d\phi$, where $\phi\in(-\pi,\pi]$ is the argument of $r^{\prime}$ in
the principal branch, the imaginary parts of the actions read

$\displaystyle\text{Im}\left(I^{+}\right)$ $\displaystyle=\lim_{\epsilon\to
0}\int_{\pi/2}^{0}d\phi\left[\sqrt{2M\left(2M-\epsilon
e^{i\phi}\right)}k_{z}-\sqrt{k_{z}^{2}\left(2M-\epsilon
e^{i\phi}\right)^{2}+\epsilon e^{i\phi}\left(2M-\epsilon
e^{i\phi}\right)k_{y}^{2}}\right],$ (68)
$\displaystyle\text{Im}\left(I^{-}\right)$ $\displaystyle=\lim_{\epsilon\to
0}\int_{-\pi/2}^{0}d\phi\left[\sqrt{2M\left(2M-\epsilon
e^{i\phi}\right)}k_{z}+\sqrt{k_{z}^{2}\left(2M-\epsilon
e^{i\phi}\right)^{2}+\epsilon e^{i\phi}\left(2M-\epsilon
e^{i\phi}\right)k_{y}^{2}}\right].$ (69)

Since the integrands are bounded from above in the limit $\epsilon\to 0$, the
order of the limit and the integration can be interchanged. Doing so, we
finally arrive at

$\displaystyle\text{Im}\left[I^{+}(k_{z})\right]$
$\displaystyle=\begin{cases}2\pi M|k_{z}|\quad&k_{z}<0\\\ 0\quad&k_{z}\geq
0,\end{cases}$ (70) $\displaystyle\text{Im}\left[I^{-}(k_{z})\right]$
$\displaystyle=\begin{cases}2\pi M|k_{z}|\quad&k_{z}>0\\\ 0\quad&k_{z}\leq
0.\end{cases}$ (71)

##  Appendix C Derivation of the effective Hamiltonian

Here we derive the effective Hamiltonian $H_{\textrm{eff}}$ that appears in
the Lindblad master equation in Sec. V. We couple the photonic crystal to its
environment with non-local jump operators $L_{x,n}=b_{2,n}-ib_{2,n+\hat{x}}$
and $L_{z,n}=b_{2,n}-ib_{2,n+\hat{z}}$, where $b_{2,n+\hat{e}}^{\dagger}$
($b_{2,n+\hat{e}}$) is a real-space creation (annihilation) operator at site
$2$ in unit cell $n+\hat{e}$. We then find that

$\sum_{n}(\kappa L^{\dagger}_{x,n}L_{x,n}+L^{\dagger}_{z,n}L_{z,n})=2\sum_{\bf
q}(2+\kappa\sin{q_{x}}+\sin{q_{z}})b^{\dagger}_{2,\bf q}b_{2,\bf q},$ (72)

where we have performed a Fourier transform $b_{n}=\sum_{\bf q}b_{\bf
q}e^{i{\bf q}\cdot{\bf r}_{n}}$. Expanding ${\bf q}$ around zero, this term
reads $2\sum_{\bf k}(2+\kappa k_{x}+k_{z})b^{\dagger}_{2,\bf k}b_{2,\bf k}$,
where ${\bf q}\approx{\bf k}$. We then find $H_{\textrm{eff}}=\sum_{\bf k}{\bf
b}^{\dagger}_{\bf k}\mathcal{H}_{\textrm{eff}}{\bf b}_{\bf k}$ with ${\bf
b}_{\bf k}=(b_{1,\bf k},b_{2,\bf k})^{T}$ and

$\mathcal{H}_{\textrm{eff}}=k_{x}\sigma^{x}+k_{y}\sigma^{y}-i(2+\kappa
k_{x}+k_{z})(\sigma_{0}-\sigma_{z}).$ (73)

This effective description has eigenvalues
$E_{\textrm{eff},\pm}=-i(2+k_{z}+\kappa
k_{x})\pm\sqrt{k_{x}^{2}+k_{y}^{2}-(2+k_{z}+\kappa k_{x})^{2}}$ from which we
see that the effective Hamiltonian thus indeed realizes ECs albeit at a
shifted $k_{z}$ value $(k_{z}\rightarrow k_{z}-2)$.