# Surface exceptional points in a topological Kondo insulator

Robert Peters<EMAIL_ADDRESS>Department of Physics, Kyoto
University, Kyoto 606-8502, Japan  Kazuhiro Kimura  Department of Physics,
Kyoto University, Kyoto 606-8502, Japan  Yoshihiro Michishita  Department of
Physics, Kyoto University, Kyoto 606-8502, Japan  Tsuneya Yoshida  Department
of Physics, University of Tsukuba, Ibaraki 305-8571, Japan  Norio Kawakami
Department of Physics, Kyoto University, Kyoto 606-8502, Japan

(August 27, 2024)

###### Abstract

Correlated materials have appeared as an arena to study non-Hermitian effects
as typically exemplified by the emergence of exceptional points. We show here
that topological Kondo insulators are an ideal platform for studying these
phenomena due to strong correlations and surface states exhibiting a
nontrivial spin texture. Using numerical simulations, we demonstrate the
emergence of exceptional points in the single-particle Green’s function on the
surface of the material while the bulk is still insulating. We reveal how
quasiparticle states with long lifetimes are created on the surface by non-
Hermitian effects while the Dirac cones are smeared, which explains the
surface Kondo breakdown at which heavy Dirac cones disappear from the single-
particle spectrum and are replaced by light states. We further show how the
non-Hermiticty changes the spin texture inherent in the surface states, which
might help identify exceptional points experimentally. Besides confirming the
existence of non-Hermitian effects on the surface of a topological Kondo
insulator, this paper demonstrates how the eigenstates and eigenvalues of the
effective non-Hermitian matrix describing the single-particle Green’s function
help understand the properties of correlated materials.

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##  I Introduction

Recently, it has become apparent that non-Hermitian phenomenaAshida  _et al._
(2020); Shen  _et al._ (2018); El-Ganainy  _et al._ (2018) can be observed in
single- or two-particle properties of correlated materials at equilibrium
Yoshida  _et al._ (2018, 2020); Rausch  _et al._ (2021); Shen and Fu (2018);
Papaj  _et al._ (2019); Michishita  _et al._ (2020); Nagai  _et al._ (2020);
Kozii and Fu (2017); Shen and Fu (2018). Ordinarily, non-Hermiticity has been
connected to systems out-of-equilibrium experiencing gain and loss, e.g.,
photonic systems, open quantum systemsFeng  _et al._ (2017); El-Ganainy  _et
al._ (2018); Miri and Alù (2019), mechanical systemsScheibner  _et al._
(2020); Yoshida and Hatsugai (2019), and active matterSone and Ashida (2019);
Sone  _et al._ (2020), leading to exciting phenomena, such as the emergence of
exceptional pointsHeiss (2012); Budich  _et al._ (2019). Exceptional points
are topologically protected band touchings, where the eigenvalues (energies)
and the corresponding eigenvectors of the non-Hermitian Hamiltonian are equal.
Furthermore, experiments have demonstrated the emergence of several
spectacular properties at or close to exceptional pointsZhou  _et al._ (2018);
Özdemir  _et al._ (2019), such as loss-induced transparency, unidirectional
invisibilityLin  _et al._ (2011); Regensburger  _et al._ (2012); Feng  _et
al._ (2013), topological chiralityDembowski  _et al._ (2001); Gao  _et al._
(2015); Doppler  _et al._ (2016); Xu  _et al._ (2016); Yoon  _et al._ (2018),
band mergingZhen  _et al._ (2015), and enhanced sensitivityHodaei  _et al._
(2017); Wiersig (2014); Liu  _et al._ (2016); Chen  _et al._ (2017).

In correlated materials at equilibrium, non-Hermitian properties emerge due to
the finite lifetime of quasiparticles in the single- or two-particle Green’s
functionsYoshida  _et al._ (2018, 2020); Rausch  _et al._ (2021); Shen and Fu
(2018); Papaj  _et al._ (2019); Nagai  _et al._ (2020); Yoshida (2021);
Yoshida  _et al._ (2019); Kimura  _et al._ (2019); Matsushita  _et al._ (2021,
2019); Okuma and Sato (2021). Because Green’s functions describe the
material’s response to an external perturbation and can be directly measured
in experiments, the emergence of exceptional points is expected to affect
experimental observations. For example, the single-particle Green’s function
can be observed in angle-resolved photoemission spectroscopy (ARPES) and
tunneling experiments. It has been shown that exceptional points induce peaks
in the observed spectral function. Furthermore, exceptional points have also
been found in two-particle Green’s functions, e.g., resulting in the
appearance of Fermi arcs in the charge-structure factor Rausch  _et al._
(2021). Notably, it has been demonstrated that the non-Hermitian Hamiltonian
describing an open quantum system with loss and gain and the effective non-
Hermitian Hamiltonian describing the single-particle Green’s function are
equivalentMichishita and Peters (2020). Thus, it should be possible to observe
similar spectacular phenomena in the single-particle Green’s function of
strongly correlated materials as in photonic and open quantum systems.

It has been shown that exceptional points can easily emerge in band structures
of correlated materials hosting Dirac cones, where two noninteracting bands
coalesce Yoshida  _et al._ (2018); Papaj  _et al._ (2019). In this situation,
a small difference in the lifetime of the quasiparticles of these two bands
will lead to a splitting of the Dirac cone and the emergence of two
exceptional points connected by a Fermi arc. Even small correlation effects
can induce this behavior. Thus, correlated systems hosting a Dirac point seem
to be particularly interesting when studying non-Hermitian properties and
exceptional points. One such type of material hosting Dirac cones and being
correlated is that of a topological Kondo insulator. Here, Dirac cones emerge
on the surface of the material. Furthermore, the Dirac cones are composed of
weakly interacting conduction ($c$) electrons and strongly correlated $f$
electrons. Thus, the lifetimes of the bands forming the Dirac cones are
different, satisfying the desired condition for exceptional points to appear.
It has been shown that correlation effects are often stronger on the surface
than in bulk due to the reduced coordination number Alexandrov  _et al._
(2015); Peters  _et al._ (2016). Thus, one might expect that exceptional
points emerge at the surface while the bulk is still insulating, which would
make the observation easy. Moreover, Dirac cones on the surfaces of a 3D Kondo
insulator exhibit a spin texture. This spin texture will be influenced by the
exceptional points and can be used to detect them in experiments. Topological
Kondo insulators thus seem to provide an exciting playground to study the
emergence of exceptional points and the interplay between non-Hermitian
properties induced by correlations and band topologyKawabata  _et al._ (2019);
Bergholtz  _et al._ (2021); Gong  _et al._ (2018); Bergholtz and Budich
(2019); Denner  _et al._ (2021).

In this paper, we study non-Hermitian properties in a three-dimensional
topological Kondo insulator. The noninteracting model hosts Dirac cones on
each surface at $\vec{k}=(0,0)$, $\vec{k}=(\pi,0)$, and $\vec{k}=(0,\pi)$ of
the surface Brillouin zone (BZ). Using dynamical mean-field theory (DMFT), we
calculate a self-consistent self-energy of this model in a wide range of
temperatures and study the emergence of exceptional points in the bulk and on
the surface. We confirm that correlation effects are enhanced on the surface
compared to the bulk, leading to the emergence of exceptional points pinned to
the surface while the bulk is still insulating. However, contrary to previous
studies Yoshida  _et al._ (2018); Papaj  _et al._ (2019), the Dirac cones
themselves do not split and do not form exceptional points. Instead, we find
that exceptional points are formed by hybridization between the states
comprising the Dirac cones and other surface states. Finally, we analyze the
impact of non-Hermiticity and exceptional points on the surface band structure
and the surface spin texture.

This paper is organized as follows: We introduce the model and the methods in
the next section. This is followed by Sec. III describing our analysis of the
bulk properties of this model. In Sec. IV, we show how non-Hermitian
properties affect the surface band structure of the topological Kondo
insulator, demonstrating the existence of exceptional points in the single-
particle Green’s function on the surface of the topological Kondo insulator.
Finally, we summarize and conclude the paper.

##  II Model and Methods

###  II.1 Model Hamiltonian

To study the impact of non-Hermiticity on the single-particle properties of a
topological Kondo insulator, we use the following model introduced in Refs.
Peters  _et al._ (2018, 2019)

$\displaystyle H$ $\displaystyle=$ $\displaystyle H_{0}+H_{\mathrm{int}}\;,$
(1) $\displaystyle H_{0}$ $\displaystyle=$
$\displaystyle\sum_{\vec{k}}\sum_{\tau=\\{\uparrow,\downarrow\\}}\sum_{o=\\{c,f\\}}\epsilon^{o}_{\vec{k}}c^{\dagger}_{\vec{k},\tau,o}c_{\vec{k},\tau,o}$
(2)
$\displaystyle+V\sum_{\vec{k},\tau_{1},\tau_{2}}c^{\dagger}_{\vec{k},\tau_{1},c}c_{\vec{k},\tau_{2},f}\sin
k_{x}\sigma^{x}_{\tau_{1}\tau_{2}}$
$\displaystyle+V\sum_{\vec{k},\tau_{1},\tau_{2}}c^{\dagger}_{\vec{k},\tau_{1},c}c_{\vec{k},\tau_{2},f}\sin
k_{y}\sigma^{y}_{\tau_{1}\tau_{2}}$
$\displaystyle+V\sum_{\vec{k},\tau_{1},\tau_{2}}c^{\dagger}_{\vec{k},\tau_{1},c}c_{\vec{k},\tau_{2},f}\sin
k_{z}\sigma^{z}_{\tau_{1}\tau_{2}}$
$\displaystyle+E_{c}\sum_{i,\tau}n_{i,\tau,c}\;,$
$\displaystyle\epsilon^{c}_{\vec{k}}$ $\displaystyle=$
$\displaystyle-2t\Bigl{(}\cos(k_{x})+\cos(k_{y})+\cos(k_{z})\Bigr{)}$ (3)
$\displaystyle+4t^{\prime}\cdot\cos(k_{x})\cos(k_{y})$
$\displaystyle+4t^{\prime}\cdot\cos(k_{y})\cos(k_{z})$
$\displaystyle+4t^{\prime}\cdot\cos(k_{x})\cos(k_{z})$
$\displaystyle+8t^{\prime\prime}\cdot\cos(k_{x})\cos(k_{y})\cos(k_{z})\;,$
$\displaystyle\epsilon^{f}_{\vec{k}}$ $\displaystyle=$
$\displaystyle-0.1\epsilon^{c}_{\vec{k}}\;,$ $\displaystyle H_{\mathrm{int}}$
$\displaystyle=$ $\displaystyle
U\sum_{i}n_{i,\uparrow,f}n_{i,\downarrow,f}\;.$ (4)

The operator $c^{\dagger}_{\vec{k},\sigma,o}$ creates an electron with
momentum $\vec{k}$, spin direction $\sigma$ in orbital $o\in\\{c,f\\}$.
$\epsilon^{o}_{\vec{k}}$ describes the energy depending on the momentum for
each orbital. $E_{c}$ is an energetical shift of the $c$ electrons. We include
nearest-neighbor, next-nearest neighbor, and next-next-nearest neighbor
hoppings on a cubic lattice and take the nearest-neighbor hopping $t$ as a
unit of energy throughout this paper. The model respects inversion and time-
reversal symmetry. We choose the parameters as
$t^{\prime}=t^{\prime\prime}=0.375t$ and $E_{c}=t$ resulting in a band
structure similar to SmB6. We find band inversions between the $c$ electrons
and $f$ electrons at $(k_{x},k_{y},k_{z})$=$(\pi,0,0)$, $(0,\pi,0)$, and
$(0,0,\pi)$ in the Brillouin zone, identical to the situation in SmB6. Due to
the hybridization, $V$, between the $c$-electron band and the $f$-electron
band, a gap opens in the bulk spectrum.Throughout this paper, we choose
$V=0.8t$ and $U=16t$, which results in a strongly correlated insulating state
at $T=0$. Thus, this model describes a strong topological Kondo insulator in
3D with inversion and time-reversal symmetry. The sign of the hopping of the
$c$ and the $f$ electrons should be opposite to open a topologically
nontrivial gap. We note that the results reported in this paper do not
strongly depend on the choice of parameters as long as the model remains in
the strong topological insulating phase and the interaction strength is strong
enough to form a crossover from an insulator to a metal due to the Kondo
effect at finite temperature. Furthermore, a more realistic treatment of SmB6
should also include the bands stemming from boron. The spectral function
visible on the surface will then depend on the surface termination of the
crystalKim  _et al._ (2014); Denlinger  _et al._ ; Thunström and Held (2021).
While the described phenomena will occur, as long as surface states are
present, exceptional points might be more difficult to detect experimentally
because of the additional spectral weight of the boron bands.

At each momentum, the noninteracting model reads

$\displaystyle H_{0}$ $\displaystyle=$
$\displaystyle\sum_{\tau_{1},\tau_{2}}\left(\epsilon^{f}c_{\tau_{1},f}^{\dagger}c_{\tau_{2},f}\delta_{\tau_{1}\tau_{2}}+\epsilon^{c}c_{\tau_{1},c}^{\dagger}c_{\tau_{2},c}\delta_{\tau_{1}\tau_{2}}\right.$
$\displaystyle\left.+c_{\tau_{1},f}^{\dagger}c_{\tau_{2},c}\vec{V}\cdot\vec{\sigma}_{\tau_{1}\tau_{2}}+\text{h.c.}\right),$

where $\vec{V}$ is a vector describing the hybridization between $c$ and $f$
electrons, and $\tau_{1}$, $\tau_{2}$ are spin indices. By rotating the spin
quantization axis of the $c$\- and the $f$-electron band in the direction of
$\vec{V}$, only $\sigma_{z}$ appears in the rotated Hamiltonian, and the
Hamiltonian is block-diagonal. Thus, regardless of the exact momentum
dependence of $\epsilon^{f}_{\vec{k}}$, $\epsilon^{c}_{\vec{k}}$, and
$\vec{V}_{\vec{k}}$, this Hamiltonian can be block diagonalized.

###  II.2 Dynamical Mean-Field Theory

To include correlation effects and calculate a self-consistent self-energy, we
use DMFTGeorges  _et al._ (1996). DMFT maps each lattice site onto a quantum
impurity model by calculating the local Green’s function. DMFT calculates a
frequency-dependent self-energy. Furthermore, DMFT can be easily extended to
real-space DMFT, where each atom of a finite cluster or a slab is mapped onto
its impurity model. This enables one to study models with open boundaries. In
Sec. III, we perform calculations using DMFT for the bulk, where the self-
energy of each atom is the same. The local Green’s function is then calculated
via integration over the whole BZ. In Sec. IV, we use real-space DMFT to
analyze a slab of $20$ layers with open boundary conditions Peters  _et al._
(2016, 2018). By integrating over the slab’s two-dimensional (2D) BZ, we
calculate a local Green’s function for each layer, map each layer onto its
impurity model, and calculate the layer-dependent self-energy. Our results
show that the self-energy is nearly layer-independent for layers $4$ to $16$.
Therefore, we believe that the number of $20$ layers is enough to study
correlation effects and the emergence of exceptional points on the surface of
a topological Kondo insulator.

To calculate the self-energy of a quantum impurity model, we use the numerical
renormalization group (NRG)Bulla  _et al._ (2008), which is well suited for
calculating real-frequency spectral functions and self-energies at low
temperatures with high resolution around the Fermi energy for arbitrary
interaction strengths. We note that we have used the same model and a
combination of methods to study the magnetic properties of topological Kondo
insulatorsPeters  _et al._ (2018).

###  II.3 Non-Hermitian properties in the single-particle Green’s function

Single-particle properties of a correlated system are given by the retarded
single-particle Green’s functions,

$G(\vec{k},\omega)=\Bigl{(}\omega-H_{0}(\vec{k})-\Sigma(\omega)\Bigr{)}^{-1},$
(6)

where $\vec{k}$ is the momentum, $\omega$ the frequency, $\eta\rightarrow 0$ a
convergence factor, $H_{0}$ the noninteracting tight-binding Hamiltonian, and
$\Sigma(\omega)$ the retarded self-energy. $H_{0}$ and $\Sigma(\omega)$ are
thereby in general matrices. Using DMFT, the self-energy is a momentum-
independent diagonal matrix, ignoring nonlocal fluctuations. Thus, the Green’s
function can be written as

$G(\vec{k},\omega)=\left(\omega+i\eta-
H_{\mathrm{eff}}(\vec{k},\omega)\right)^{-1},$ (7)

where $H_{\mathrm{eff}}(\vec{k},\omega)=H_{0}(\vec{k})+\Sigma(\omega)$ is an
effective non-Hermitian Hamiltonian. The effective Hamiltonian is a non-
Hermitian matrix because of the imaginary part of the self-energy
corresponding to the lifetime of the particles. It may happen that this matrix
is defective at certain points $(\vec{k},\omega)$, i.e., the effective
Hamiltonian cannot be diagonalized. At these points, $(\vec{k},\omega)$, at
least two eigenvalues and eigenstates of the effective Hamiltonian coalesce.
We call these points exceptional points.

The eigenvalues and eigenvectors of this effective Hamiltonian completely
describe the single-particle Green’s function. Away from the exceptional
points, we can diagonalize the effective Hamiltonian as

$\displaystyle H_{\mathrm{eff}}|n_{R}\rangle$ $\displaystyle=$ $\displaystyle
E_{n}|n_{R}\rangle,$ (8) $\displaystyle\langle n_{L}|H_{\mathrm{eff}}$
$\displaystyle=$ $\displaystyle E_{n}\langle n_{L}|,$ (9)
$\displaystyle\langle n_{L}|m_{R}\rangle$ $\displaystyle=$
$\displaystyle\langle n_{R}|m_{L}\rangle=\delta_{nm},$ (10)

where $|n_{L}\rangle$ and $|n_{R}\rangle$ are the left- and right-eigenvectors
of the non-Hermitian matrix $H_{\mathrm{eff}}$, respectively. $E_{n}$ is a
complex-valued eigenvalue of the matrix. Furthermore, $\langle
n_{L}|=\left(|n_{L}\rangle\right)^{H}$ and $\langle
n_{R}|=\left(|n_{R}\rangle\right)^{H}$ hold, where the superscript $H$ is the
Hermitian adjoint. Using these vectors, we can write the retarded Green’s
function as

$\displaystyle G_{ij}(\omega,\vec{k})$ $\displaystyle=$
$\displaystyle\sum_{n}\langle i|n_{R}\rangle\frac{1}{\omega-E_{n}}\langle
n_{L}|j\rangle,$ (11)

where $i$ and $j$ are orbital and spin indices of the original model. Note
that the eigenvectors and the eigenvalues depend on the momentum and the
frequency.

Besides being directly observable in ARPES and tunneling experiments, the
single-particle Green’s function also determines single-particle expectation
values, such as the spin expectation value. In this paper, we will analyze the
impact of non-Hermiticity in the spectral function and the impact on the spin
expectation values that can be observed in spin-resolved ARPES.

We can write the single-particle spectral function and expectation values
using the left- and right-eigenvectors of the non-Hermitian matrix
$H_{\mathrm{eff}}$,

$\displaystyle A_{ij}(\omega,\vec{k})$ $\displaystyle=$
$\displaystyle-\frac{1}{\pi}\text{Im}\left(\sum_{n}\langle
i|n_{R}\rangle\frac{1}{\omega-E_{n}}\langle n_{L}|j\rangle\right),$
$\displaystyle\langle\sigma^{x,y,z}\rangle(\omega,\vec{k})$ $\displaystyle=$
$\displaystyle-\frac{1}{\pi}\text{Im}\sum_{ij}\left(\sigma^{x,y,z}_{ij}G_{ji}(\omega)\right),$
(13) $\displaystyle=$
$\displaystyle-\frac{1}{\pi}\text{Im}\left(\sum_{ij}\sum_{n}\langle
j|\sigma^{x,y,z}|i\rangle\right.$
$\displaystyle\quad\quad\quad\left.\times\langle
i|n_{R}\rangle\frac{1}{\omega-E_{n}}\langle n_{L}|j\rangle\right),$
$\displaystyle=$ $\displaystyle-\frac{1}{\pi}\text{Im}\left(\sum_{n}\langle
n_{L}|\sigma^{x,y,z}|n_{R}\rangle\right.$
$\displaystyle\quad\quad\quad\quad\quad\quad\times\left.\frac{1}{\omega-
E_{n}}\right),$

where $A_{ij}(\omega,\vec{k})$ is the spectral function for orbitals $i$ and
$j$ at frequency $\omega$ and momentum $\vec{k}$, and
$\langle\sigma^{x,y,z}\rangle(\omega,\vec{k})$ is the spin expectation value
as observed in spin-resolved ARPES. $\sigma^{x,y,z}$ are the matrix
representations of the spin operators for the noninteracting Hamiltonian
$H_{0}$. We note that the eigenstates and eigenvalues depend via the effective
Hamiltonian on the frequency $\omega$. Because the eigenstates of the
effective Hamiltonian determine the spin direction of a photoelectron in ARPES
at a specific frequency, we will later show the spin expectation values of the
eigenstates.

##  III bulk properties

Figure 1:  Imaginary part of the bulk self-energy for different temperatures
(a). Comparison between the imaginary part of the bulk and surface self-energy
at $\omega=0$ (b). The self-energies in (a) are calculated at the following
temperatures $T/t=\\{0.02$, $0.04$, $0.08$, $0.12$, $0.16$, $0.2$, $0.28$,
$0.44$, $0.56\\}$.  Figure 2: Momentum-resolved spectral functions in the bulk
for three different temperatures, $T/t=0$ (a), $T/t=0.1$ (b), and $T/t=0.4$
(c).

Figure 3: Real part of the eigenvalues over the momentum of the effective
Hamiltonian, $H_{\mathrm{eff}}(k_{x},k_{y},k_{z}=0,\omega=0)$, in the bulk for
$T/t=0.0$ (a), $T/t=0.16$ (b), and $T/t=0.2$ (c). We show only a part of the
BZ in (b) and (c) for better visibility of the non-Hermitian effect.  Figure
4: Real and imaginary part of the eigenvalues of the effective Hamiltonian for
$\omega=0$ and $k_{y}=0.74$ and $k_{z}=0$ for different $k_{x}$ at $T/t=0.2$.
This line of momenta goes through two exceptional points in the BZ.

In this section, we analyze the bulk properties of this model at finite
temperatures. In particular, we demonstrate the appearance of exceptional
points in the bulk spectrum around the Kondo temperature at which the system
changes from insulating to metallic, which has also been shown for other
$f$-electron systems in Ref. Michishita  _et al._ (2020); Nagai  _et al._
(2020). We will show in Sec. IV that exceptional points emerge at the surface
of the material at a lower temperature than in the bulk.

The imaginary part of the bulk self-energy for different frequencies and a
comparison between the bulk and the surface self-energy at $\omega=0$ for
different temperatures are shown in Fig. 1. This imaginary part, which
corresponds to an inverse lifetime of the quasiparticles, leads to the
emergence of non-Hermitian properties in the single-particle Green’s function.
While the imaginary part vanishes at $\omega=0$ for $T=0$, it becomes large at
finite temperatures. Furthermore, we see that the imaginary part of the self-
energy is larger in magnitude on the surface than in the bulk, indicating that
the surface is more strongly correlated than the bulk. This has also been
found in a previous study Peters  _et al._ (2016).

The effective Hamiltonian at the Fermi energy completely describes the single-
particle properies of the material at the Fermi energy. We thus mainly focus
on the properties of the effective Hamiltonian at $\omega=0$. Because the
imaginary part of the self-energy vanishes at $T=0$ at the Fermi energy, the
effective Hamiltonian at $T=0$ is just given by the noninteracting
Hamiltonian, where energy levels are possibly shifted by the real part of the
self-energy. In the current model, the hybridization between the $c$ and $f$
electrons leads to a gap at $\omega=0$ in the noninteracting Hamiltonian in
the bulk. Thus, the effective Hamiltonian is gapped, and the system is
insulating, as shown in Fig. 2. At high temperatures, on the other hand, the
imaginary part of the self-energy becomes large. For a two-band model
consisting of a noninteracting $c$-electron band and a correlated $f$-electron
band, we can calculate the eigenvalues of the effective Hamiltonian in the
limit of a very large imaginary part as

$\displaystyle H_{\mathrm{eff}}(\vec{k},\omega=0)$ $\displaystyle=$
$\displaystyle\begin{pmatrix}\epsilon^{c}_{k}&V_{k}\\\
V_{k}^{\star}&\epsilon^{f}_{k}+\Sigma(\omega=0)\end{pmatrix}$ (14)
$\displaystyle\Rightarrow(E_{1},E_{2})$ $\displaystyle=$
$\displaystyle\left(\epsilon^{c}_{k},\epsilon^{f}_{k}+\Sigma(\omega=0)\right)\quad\text{for
Im}\Sigma\rightarrow\pm\infty.$

If the imaginary part of the self-energy becomes very large, the effect of the
hybridization becomes negligible, and we find a unperturbed $c$ electron and a
correlated $f$ electron. Thus, at high temperatures, when the imaginary part
of the self-energy is large, the system becomes metallic with a Fermi surface
given by the conduction electrons.

The single-particle spectrum of the bulk consisting of four bands is shown in
Fig. 2 for three different temperatures. At $T=0$, the single-particle
spectrum is gapped in the bulk. Because the imaginary part of the self-energy
vanishes, $f$ and $c$ electrons can hybridize at the Fermi energy, $\omega=0$,
and thus form a gap. With increasing temperature, the crossover from the
insulating state at $T=0$ to a metallic state at high temperatures occurs. At
high temperatures, $f$ electrons localize and thus cannot hybridize with the
$c$ electrons at the Fermi energy. Thus, the $c$ electron bands span the gap,
and the system becomes metallic.

Recently, it has become clear that exceptional points appear in the single-
particle spectrum close to the Fermi energy approximately around the Kondo
temperature Michishita  _et al._ (2020). Thus, we here analyze the energy
eigenvalues of the effective Hamiltonian at the Fermi energy, $\omega=0$.
Figure 3 shows the real part of the eigenvalues of the effective Hamiltonian
for $k_{z}=0$ over $(k_{x},k_{y}$). We see the existence of a gap at
$\omega=0$. Because these eigenvalues completely determine the spectral
function at $\omega=0$, we confirm that the system is insulating. Note that
the gap size in Fig. 3 appears larger than that in Fig. 2. The gap in Fig. 2
appears smaller because the effective Hamiltonian changes with the frequency.

Figure 3(b) shows the eigenvalues of the effective Hamiltonian for $T/t=0.16$
close to the Kondo temperature. While $H_{0}$ is the same as in Fig. 3(a), the
self-energy has changed. The imaginary part of the $f$-electron self-energy
results in the formation of ”pockets” where the eigenvalues from the band
below and above the Fermi energy approach each other. The momenta of these
pockets are given by the minima of the hybridization along the lines of
$\epsilon^{c}_{\vec{k}}=\epsilon^{f}_{\vec{k}}=0$ in the noninteracting
Hamiltonian. Further increasing the temperature and thus the imaginary part in
the self-energy leads to a merging of the pockets from below and above the
Fermi energy, as shown in Fig. 3(c). This merging leads to a line of momenta
where the real part of the eigenvalues of the effective Hamiltonian is
degenerate. Furthermore, at the edge of the pockets, the real and imaginary
parts of two eigenvalues are equal, corresponding to an exceptional point in
the effective Hamiltonian. The line inside the merged pockets, where the real
parts of two eigenvalues are the same, is called a Fermi arc.

Increasing the temperature above the Kondo temperature, the size of the
pockets increases, and finally, pockets developing at different momenta merge.
At this temperature, also the exceptional points merge and annihilate each
other Yoshida  _et al._ (2018).

To demonstrate the existence of these exceptional points, we show the
eigenvalues of the bulk Hamiltonian along a straight line through the
exceptional points at $k_{z}=0$ in Fig. 4. We see that exactly at the same
momentum, the real part and the imaginary part of the eigenvalues are the
same. Thus, all bands are degenerate in the effective Hamiltonian at this
point. We note that this cannot be an ordinary (Hermitian) degeneracy because
the hybridization between the bands does not vanish at this momentum. Thus,
this degeneracy originates in the non-Hermiticty induced by the imaginary part
of the self-energy.

##  IV Non-Hermitian effects on the surface states

Figure 5: Real part of the eigenvalues of the effective Hamiltonian
($T/t=0.106$) for an open system consisting of $20$ layers focusing on
$k_{x}\in[-1.5,1.5]$ and $k_{y}\in[-1.5,1.5]$ of the surface BZ. The bulk is
still gapped at this temperature. Pockets in the band structure are visible at
$(k_{x},k_{y})\approx(0,-0.9)$, $(-0.9,0)$, and $(0.9,0)$. The color is added
for better visibility and corresponds to the distance from the origin.  Figure
6:  Same as Fig. 5 but for $T/t=0.11$.

###  IV.1 Energy spectrum of $H_{\mathrm{eff}}$

We now turn to the main results of this paper, i. e., how non-Hermiticity
affects the surface states of a topological Kondo insulator. Previous
calculations have shown that even a small difference in the lifetime of the
particles forming a Dirac cone can result in the splitting of the Dirac cone
and the formation of exceptional points connected by Fermi arcs Yoshida  _et
al._ (2018). Thus, we next focus on the effect of non-Hermiticity on the
surface states of this topological Kondo insulator. Due to the topology of
this system, there are three protected Dirac cones on each surface with
surface momentum $(k_{x},k_{y})=(0,0)$, $(\pi,0)$, and $(0,\pi)$. Furthermore,
because the magnitude of the imaginary part of the self-energy at the surface
is enhanced compared to the bulk, as shown in Fig. 1, non-Hermitian effects
can be expected to be stronger on the surface than in the bulk. We here ask
whether the above scenario can apply for the surface Dirac states in the
present system and how else non-Hermiticity affects the surface spectrum.

In Fig. 5, we show the real part of the eigenvalues for a system with open
boundaries in the $z$-direction consisting of $20$ layers at $T/t=0.106$. We
here focus on the Dirac cone at the center of the surface BZ. We find a
similar behavior for the Dirac cones located at $(k_{x},k_{y})$=$(\pi,0)$ and
$(0,\pi)$.

The structure of the eigenvalues in Fig. 5 shows two notable features: a Dirac
cone at $(k_{x},k_{y})$=$(0,0)$ and pockets in the eigenvalue structure of the
effective Hamiltonian separated from the Dirac cone. The first observation is
that the Dirac cone on the surface of this topological Kondo insulator is not
strongly affected by the finite lifetime of the $f$ electrons at this
temperature. This is contrary to the results in Ref. Yoshida  _et al._ (2018),
where the Dirac cone in the bulk of a 2D system splits by forming exceptional
points. This difference can be explained in the following way: There is no
effective hybridization between the lower and the upper part of the Dirac cone
on the same surface. Analyzing the current model, we find that there would be
a hybridization between the Dirac cone on the bottom surface and that on the
top surface, but there is no hybridization between the two bands comprising
the Dirac cone on the same surface. The hybridization between states on two
different surfaces is too small to affect the results because of the distance
between the surfaces. Thus, exceptional points cannot emerge within the Dirac
cone because of the absence of hybridization between the surface states. The
second notable feature is the appearance of pockets in the eigenvalue
structure. These pockets, which are absent at $T=0$, are created by the non-
Hermiticity similar to Fig. 3(b) in the bulk at $T/t=0.16$. The pockets in the
eigenvalue spectrum with open boundaries appear at a much lower temperature
than in the bulk, which can be explained by larger correlation effects on the
surface.

With increasing the temperature to $T/t=0.11$, as shown in Fig. 6, we see that
the pockets from above and below the Fermi energy merge. The emergent
structure points to the existence of exceptional points at the edges of the
pockets, which can also be confirmed by showing the concrete level structure
(see Fig. 8) and analyzing the eigenvectors of the effective Hamiltonian.
Thus, exceptional points emerge on the surface of the topological Kondo
insulator for much smaller temperatures and do not split the Dirac cone.

Figure 7: Real part of the eigenvalues of the effective Hamiltonian at
$\omega=0$ and $T/t=0.106$. The color encodes the imaginary part of the
eigenvalues. Furthermore, we include indices ($1$ or $2$) corresponding to the
block of the block-diagonalized Hamiltonian, and mark degeneracies due to the
absence of any hybridization by green circles and an ’H’. We also include the
spin in the $y$-direction, $\langle\sigma^{x}\rangle$, of the surface states
located on the bottom of the slab by blue arrows. Figure 8: Same as Fig. 7 but
for $T/t=0.11$. We mark a degeneracy in the real part due to non-Hermiticity
by ’NH’.  Figure 9: Same as Fig. 7 but for $T/t=0.14$.

###  IV.2 Analysis of the degeneracies in the spectrum

A more detailed analysis of the eigenvalues including the spin expectation
values of the eigenstates is shown in Figs. 7-9. We show the real part of the
eigenvalues close to the Fermi energy of the effective Hamiltonian at
$\omega=0$ for different temperatures along the surface momentum from
$(k_{x},k_{y})$=$(-\pi,0)$ via $(0,0)$ to $(\pi,\pi)$. Besides showing the
real part of the eigenvalues, we also include the imaginary part as color. As
explained above, the Hamiltonian can always be block-diagonalized. To better
understand the effect of non-Hermiticity, we thus block-diagonalize the
Hamiltonian and include an index corresponding to the block in Figs. 7-9.
While bands with the same index lie in the same block, bands with different
indices lie in separate blocks and thus do not couple. When taking into
account energies of both blocks, all bands are degenerate. However, because of
inversion and time-reversal symmetry in this model, degenerate surface states
lie on opposite surfaces. Thus, by block-diagonalization, we can uniquely
identify states on a specific surface of the topological insulator. Finally,
we also include the spin expectation value in the $y$-direction,
$\langle\sigma^{y}\rangle$, of these states as blue arrows.

In Fig. 7 at $T/t=0.106$, which is the same temperature as in Fig. 5, we see
several band crossings. We see the Dirac cones at $(k_{x},k_{y})$=$(0,0)$ and
$(-\pi,0)$ due to the topological properties. Analyzing the bands forming the
Dirac cone at $(k_{x},k_{y})$=$(0,0)$ on the bottom surface of the slab, we
see that both bands lie in different blocks of the block-diagonalized
Hamiltonian. Thus, there is no hybridization between these states in the
noninteracting and the effective Hamiltonian. As a hybridization between
states is essential to form exceptional points, this explains why the Dirac
cones do not split and do not form exceptional points in the presence of
correlations, although they are made of particles with different lifetimes. We
note that the possibility to block-diagonalize the Hamiltonian is completely
general for a four-band model (consisting of two spinful bands) with inversion
and time-reversal symmetry, as explained in the model section.

Around $(k_{x},k_{y})$=$(-0.9,0)$ in Fig. 7, we see the appearance of the
pockets at this temperature. The pockets are created by the band forming the
lower part (block index $1$) of the Dirac cone at $(k_{x},k_{y})$=$(0,0)$, and
a band originating from the bulk spectrum at $(0,0)$. We see that these bands
have the same block index and thus hybridize with each other. Furthermore, we
see that the band, forming the pocket above the Fermi energy at
$(k_{x},k_{y})$=$(-0.9,0)$, crosses the band comprising the upper part of the
Dirac cone at $(k_{x},k_{y})$=$(0,0)$ without hybridization. By analyzing the
block indices of these bands, we see that these bands lie in different blocks
and thus do not hybridize. In Fig. 7, we have marked Hermitian degeneracies
due to the absence of hybridization by green circles and an ’H’.

In Fig. 8, at $T/t=0.11$, the pockets from above and below the Fermi energy
have merged and formed a point where the real parts of two eigenvalues are
degenerate, denoted as ’NH’ in this figure. Note that this non-Hermitian
degeneracy is absent on the line from $(k_{x},k_{y})$=$(0,0)$ to $(\pi,\pi)$.
Thus, when going around the center of the BZ, this line of real-part
degeneracies has to end at some momenta, as visible in Fig. 6. These endpoints
must be exceptional points. We have verified these exceptional points by
confirming that the real and imaginary parts of the eigenstates are degenerate
at these endpoints. We thus find exceptional points on the surface of the
material emerging due to correlation effects. Further analyzing the imaginary
part of the eigenstates forming the degeneracy at $(k_{x},k_{y})$=$(-0.9,0)$,
we see that the imaginary parts of these bands are smaller in magnitude than
that of the states comprising the Dirac cone at $(k_{x},k_{y})$=$(0,0)$, which
will become important further below.

At $T/t=0.11$ in Fig. 8, only the band contributing to the lower part of the
Dirac cone at $(k_{x},k_{y})$=$(0,0)$ forms a non-Hermitian degeneracy. With
increasing the temperature further to $T/t=0.14$, as shown in Fig. 9, the band
contributing to the upper part of the Dirac cone also forms a non-Hermitian
degeneracy at $(k_{x},k_{y})$=$(-1.74,0)$. While there is a degeneracy of the
real part on the line of momenta from $(k_{x},k_{y})$=$(-\pi,0)$ to $(0,0)$ in
the band with block index 2, it is absent on the line of momenta from
$(k_{x},k_{y})$=$(0,0)$ to $(\pi,\pi)$, which points again to the existence of
exceptional points. Besides these non-Hermitian degeneracies, there are
several new Hermitian degeneracies between bands that do not hybridize.

###  IV.3 Effect of the non-Hermiticity on the momentum-dependent spectral
weight

Figure 10: Surface spectral function for $\omega=0$ and the direction of the
spin expectation values on the bottom of the slab as calculated from the
Green’s function for $T/t=0.01$.  Figure 11: Same as Fig. 10 but for
$T/t=0.14$. We show the location of the exceptional points as red dots. We
stress the two ring-like structures in the spectrum, visible as dark blue
color, going around the origin of the BZ.

Finally, let us analyze the non-Hermitian effect directly on the single-
particle Green’s function as observable in ARPES. We show the spectral
function of the single-particle Green’s function at the Fermi energy for
different temperatures in Fig. 10 and 11. We also include the spin expectation
value as calculated from the Green’s function. However, we only take into
account electrons on the bottom surface of the slab by summing only over the
lower half of the slab

$\langle\vec{\sigma}\rangle=-\frac{1}{\pi}\text{Im}\sum_{ij<N/2}\vec{\sigma}_{ij}G_{ji}(\omega=0),$
(15)

where $i$ and $j$ are indices describing the orbital, spin, and layer of the
system. For a system consisting of $N$ layers, we only sum over layers in the
lower half of the system. We note that the spin expectation value of the full
system vanishes as the top surface and the bottom surface have opposite spin
expectation values.

In Fig. 10, we show the spectral function at $T/t=0.01$. Clearly visible is
the spectral weight created by the Dirac cones at $(k_{x},k_{y})$=$(0,0)$,
$(\pi,0)$, and $(0,\pi)$. Focusing on the Dirac cone at
$(k_{x},k_{y})$=$(0,0)$, we see that the spin rotates clockwise around the
origin. Comparing with Fig. 7, we see that the Dirac cone is located slightly
below the Fermi energy. The Fermi energy is located in the band forming the
upper part of the Dirac cone, as shown in Fig. 7. The spin expectation value
of this band (shown as an arrow in Fig. 7) exhibits a positive
$\langle\sigma_{y}\rangle$ value for $k_{x}<0$ and a negative value for
$k_{x}>0$, which explains the clockwise spin direction around
$(k_{x},k_{y})$=$(0,0)$. This band mainly contributes to the spin expectation
value seen in Fig. 10.

With increasing the temperature to $T/t=0.14$, as shown in Fig. 11, the
momenta with highest spectral weight (blue color) shift away from
$(k_{x},k_{y})$=$(0,0)$. This is an interesting feature as we see in Fig. 9
that the position of the Dirac cone does not change with increasing
temperature. However, Fig. 9 reveals the existence of bands crossing the Fermi
energy at $(k_{x},k_{y})$=$(-1.74,0)$, $(-1,0)$, and $(0.75,0.75)$ due to non-
Hermitian effects. Furthermore, the imaginary part of the bands forming the
Dirac cone is much larger in magnitude than the imaginary part of the bands
crossing the Fermi energy away from $(k_{x},k_{y})$=$(0,0)$. Thus, the
spectral weight of the Dirac cone is smaller than that of the bands crossing
the Fermi energy away from $(k_{x},k_{y})$=$(0,0)$. These bands create two
rings around the center of the BZ in the spectral function, as shown in Fig.
11. In Fig. 9, we see that these bands have a large velocity compared to the
bands comprising the Dirac cone. The situation described here is identical to
the phenomenon called surface Kondo breakdown in Refs. Alexandrov  _et al._
(2015); Peters  _et al._ (2016). The heavy Dirac cone (small velocity) becomes
invisible in the single-particle spectral function and is replaced by bands
with large velocity. By analyzing the effective non-Hermitian Hamiltonian, we
understand that the Dirac cone still exists at $(k_{x},k_{y})$=$(0,0)$ but is
smeared out due to a large imaginary part. The light states are created by a
non-Hermitian effect suppressing the hybridization between different surface
states.

Using the eigenstates of the effective non-Hermitian Hamiltonian describing
the Green’s function, as shown in Fig. 9, we can also understand the spin
expectation values in Fig. 11. The spin direction of the band with the smaller
imaginary part (longest lifetime) causing the degeneracy at
$(k_{x},k_{y})$=$(-1,0)$ rotates clockwise, and the spin direction of the band
with the smaller imaginary part creating the degeneracy at
$(k_{x},k_{y})$=$(-1.74,0)$ rotates anticlockwise, as shown in Fig. 9. We thus
understand the emergence of the two rings of high spectral weight in Fig. 11,
where the spin direction rotates clockwise for the inner ring and
anticlockwise for the outer ring.

Finally, as noted above, the degeneracy in the real part that exists in the
band with index $2$ between $(k_{x},k_{y})=(-\pi,0)$ and $(0,0)$ is absent on
the line between $(k_{x},k_{y})=(0,0)$ and $(\pi,\pi)$. Thus, there are line
segments in the BZ that end in exceptional points. In Fig. 11, we show these
exceptional points as dark red dots. These exceptional points lie on the
surface of the material. Furthermore, because the surface is more strongly
correlated than the bulk, the bulk is still insulating and does not form non-
Hermitian degeneracies. As expected from the analysis in Fig. 9, these points
lie in the outer circle going around the center of the BZ. This ring exhibits
variations of spectral weight depending on the momentum. These variations
originate in the changes of the imaginary part of the band and the sudden
change of the band structure at the exceptional points at which the band
degeneracy suddenly vanishes. Furthermore, close to these exceptional points,
the spin direction visible in the outer ring becomes very small and seems to
end. We believe that these variations of spectral weight and the spin
expectation values yield the necessary clues to identify exceptional points in
the single-particle Green’s function of correlated materials.

##  V Conclusions

We have analyzed non-Hermitian properties of a topological Kondo insulator in
3D. Using numerical calculations, we have confirmed that the crossover from
localized $f$ electrons at high temperatures to hybridized $f$ electrons at
low temperatures is a non-Hermitian effect accompanied by the appearance of
exceptional points in the single-particle Green’s function Michishita  _et
al._ (2020). Because correlation effects are stronger on the surface than in
the bulk, exceptional points emerge on the surface while the bulk is still
insulating. We found that the surface Dirac cones are stable against non-
Hermitian effects, although comprised of particles with different lifetimes.
This stability can be explained by the absence of hybridization between the
states comprising the Dirac cone in this time-reversal and inversion symmetric
system. Thus, exceptional points on the surface of the material form away from
the Dirac cone by hybridization between different surface states. Notably, the
emergence of exceptional points located on the surface is related to the
surface Kondo breakdown investigated in previous studies Alexandrov  _et al._
(2015); Peters  _et al._ (2016). While the states forming the Dirac cone
acquire a large imaginary part and are smeared out, two new bands with a small
imaginary part are created due to the non-Hermiticity of the effective
Hamiltonian. These bands become visible in the single-particle spectral
function. Furthermore, we have analyzed the spin texture arising due to the
topological surface states and non-Hermiticity and found that the surface
states inherited from the non-Hermiticity have opposite spin directions.

In summary, we believe that topological Kondo insulators host an ideal
platform to study non-Hermitian effects in correlated materials. These
materials provide exceptional points at the surface while the bulk is
insulating, and the states due to non-Hermiticity exhibit a spin texture that
can be used to detect those states. Finally, our analysis based on the
effective non-Hermitian Hamiltonian has been shown to be a powerful tool to
detect and understand changes in the single-particle spectral function, such
as the surface Kondo breakdown.

###### Acknowledgements.

RP would like to thank Youichi Yanase for fruitful discussions. This work is
supported by the WISE program, MEXT, and by the following JSPS KAKENHI grants
No. 20J12265, No. JP18K03511, No. JP19H01838, No. JP21K13850, and No.
JP20H04627. Computer simulations were done on the supercomputer of Tokyo
University at the ISSP.

##  Appendix A Frequency dependence of the exceptional points

Figure 12: Momentum-resolved spectral function and energy spectrum for
$T/t=0.106$. Panel (a) shows the spectral function and the spin expectation in
the $y$-direction on the bottom surface, $\langle S_{y}\rangle$, as calculated
by the Green’s function for different $\omega$ from $(-\pi,0)$ to $(0,0)$ in
the BZ. Panels (b)-(f) show the real part of the energies of the effective
Hamiltonian for different frequencies along the same path in the BZ.
Furthermore, we show the contribution of each state to the spin expectation
value, $\langle S_{y}\rangle$, of the bottom layer as an arrow attached to
each state. The sum of the contributions of all states corresponds to the
expectation value shown in panel (a). The frequencies are $\omega/t=-0.1$ (b),
$\omega/t=-0.04$ (c),$\omega/t=-0.02$ (d), $\omega/t=0$ (e), and
$\omega/t=0.02$ (f).The red lines denote the frequency of the effective
Hamiltonian.

Because the self-energy is frequency-dependent, the effective Hamiltonian also
possesses a frequency dependence. We show in Fig. 12 how the Green’s function,
the spin expectation value, and the energy spectrum of the effective
Hamiltonian change depending on the frequency. The temperature in this figure
is fixed to $T/t=0.106$, for which exceptional points at the Fermi energy are
absent, as shown in Fig. 7. Because the magnitude of the imaginary part of the
self-energy increases when changing the frequency away from the Fermi energy
(see Fig. 1), exceptional points may emerge in the spectrum away from
$\omega=0$. Besides this change in the imaginary part, we note that the real
part of the self-energy also varies with frequency. The real part of the self-
energy results in a shift of the eigenvalues of the effective Hamiltonian.
Figure 12(a) shows the spectral function and the spin expectation value in the
$y$-direction of the bottom layer along a path from $(k_{x},k_{y})=(-\pi,0)$
to $(0,0)$ in the BZ. In both functions, we see two peaks whose momenta are
nearly frequency independent. The spectral function shows only a slight
frequency dependence between the maxima. Figures 12(b)-(f) show the real part
of the eigenvalues of the effective Hamiltonians. We see that these
eigenvalues are strongly frequency-dependent and shift when we change the
frequency. Although the intersection between the surface states and the red
line (frequency of the Hamiltonian) occurs at different momenta, this
intersection does not lead to a peak in the spectral function, as shown in
Fig. 12(a). The reason for this behavior is the imaginary part of the
eigenstates present in all surface states at this temperature. Reducing the
temperature reduces the imaginary part and thus will lead to visible shifts in
the spectral function when we change the frequency. However, in that case,
exceptional points do not occur because the imaginary part in the self-energy
is necessary for their appearance. The same holds for the spin-expectation
value. Because of the imaginary part, all surface states contribute to the
spin expectation value.

Finally, let us analyze the emergence of exceptional points in the frequency-
dependent effective Hamiltonian. In Fig. 12(b), at $\omega/t=-0.1$, we see
that two levels cross at $(k_{x},k_{y})=(-1,0)$ at $\text{Re}(E)/t\approx
0.5$. This level crossing is due to the imaginary part of the self-energy, as
has been analyzed in detail in the main text. We see how these two bands
separate when increasing the frequency of the effective Hamiltonian from
$\omega/t=-0.02$ to $\omega/t=0.0$. Thus, there must be an exceptional point.
Again, we note that the changes in the spectral function and the spin
expectation value between these two frequencies are small, although an
exceptional point appears. Thus, changing the frequency does not provide
additional clues to the existence of exceptional points. As described in the
main text, the momentum dependence of the spectral function and the spin
expectation value at the Fermi energy displaying typical Fermi arc behavior
provide better information.

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