#  Symmetry-protected exceptional and nodal points in non-Hermitian systems

Sharareh Sayyad<EMAIL_ADDRESS>Max Planck Institute for the
Science of Light, Staudtstraße 2, 91058 Erlangen, Germany  Marcus Stålhammar
Department of Physics, Stockholm University, AlbaNova University Center, 106
91 Stockholm, Sweden  Lukas Rødland  Department of Physics, Stockholm
University, AlbaNova University Center, 106 91 Stockholm, Sweden  Flore K.
Kunst  Max Planck Institute for the Science of Light, Staudtstraße 2, 91058
Erlangen, Germany

###### Abstract

One of the unique features of non-Hermitian (NH) systems is the appearance of
non-Hermitian degeneracies known as exceptional points (EPs). The extensively
studied defective EPs occur when the Hamiltonian becomes non-diagonalizable.
Aside from this degeneracy, we show that NH systems may host two further types
of degeneracies, namely, non-defective EPs and ordinary (Hermitian) nodal
points. The non-defective EPs manifest themselves by i) the diagonalizability
of the NH Hamiltonian at these points, ii) the non-diagonalizability of the
Hamiltonian along certain intersections of these points and iii) instabilities
in the Jordan decomposition when approaching the points from certain
directions. We demonstrate that certain discrete symmetries, namely parity-
time, parity-particle-hole, and pseudo-Hermitian symmetry, guarantee the
occurrence of both defective and non-defective EPs. We extend this list of
symmetries by including the NH time-reversal symmetry in two-band systems.
Two-band and four-band models exemplify our findings. Through an example, we
further reveal that ordinary nodal points may coexist with defective EPs in NH
models when the above symmetries are relaxed.

#### Introduction.

Despite violating the axioms of quantum mechanics, non-Hermitian (NH)
Hamiltonians offer compelling descriptions for numerous interacting/open
systems in various fields of physics Bender (2005); Ashida  _et al._ (2020);
Alexandre  _et al._ (2020); Sayyad  _et al._ (2021, 2022). The underlying
physics of these effective Hamiltonians goes beyond the realm of Hermitian
physics and has been immensely studied lately Bergholtz  _et al._ (2021); Zhou
_et al._ (2018); Berry (2004); Heiss (2012); Xu  _et al._ (2017a); Kozii and
Fu ; Carlström and Bergholtz (2018); Cerjan  _et al._ (2019); Carlström  _et
al._ (2019); Stålhammar  _et al._ (2019); Zhang  _et al._ (2021); Yang  _et
al._ (2020); Ghorashi  _et al._ (2021a, b). Aside from unraveling rich
physics, the properties of NH systems are well-reflected in abstract
mathematical frameworks, including homotopy theory Wojcik  _et al._ (2020); Hu
and Zhao (2021); Li and Mong (2021); Wojcik  _et al._ ; Hu  _et al._ and
K-theory Gong  _et al._ (2018); Zhou and Lee (2019); Kawabata  _et al._
(2019a). These frameworks provide a reliable toolbox to understand the exotic
properties of NH systems and distinguish their behavior from Hermitian
physics.

Noticeable distinctions of NH Hamiltonians from their Hermitian counterparts
include the appearance of exceptional points (EPs), as NH degeneracies. While
one generally should satisfy $n-1$ complex constraints to realize EPs of order
$n$ (EP$n$s), at which the Hamiltonian casts a Jordan block, recent studies
have shown that the presence of certain discrete symmetries, such as parity-
time ($\mathcal{PT}$), parity-particle-hole ($\cal CP$), or pseudo-Hermiticity
($\rm psH$) symmetry, reduces the total number of constraints Budich  _et al._
(2019); Yoshida  _et al._ (2019); Okugawa and Yokoyama (2019); Zhou  _et al._
(2019); Szameit  _et al._ (2011); Kimura  _et al._ (2019); Delplace  _et al._
(2021); Mandal and Bergholtz (2021); Stålhammar and Bergholtz (2021); Crippa
_et al._ (2021); Sayyad and Kunst (2022). Although the main focus of these
studies was on characterizing symmetry-induced restrictions on defective EPs,
drawing a link between discrete symmetries and non-defective degeneracies
associated with the NH Hamiltonian beyond case studies have received little
attention. Furthermore, while there is a near consensus on calling defective
degeneracies Xu  _et al._ (2017b); Kawabata  _et al._ (2019b) defective EPs
111 Except in Ref. Shen  _et al._ (2018) where ‘hybrid points’ have also been
introduced based on the asymptotic dispersion relations close to defective
degeneracies. Note that branch cuts do not terminate at these points Yang  _et
al._ (2021)., non-defective degeneracies in NH systems are referred to as
diabolic points Keck  _et al._ (2003); Xue  _et al._ (2020); Wiersig (2022);
Cui  _et al._ , Fermi points Yang  _et al._ (2021), Dirac points and vortex
points Shen  _et al._ (2018). These non-defective degeneracies are different
in nature, as both Dirac points and vertex points might appear in the vicinity
of defective EPs, while diabolic points and Fermi points are similar to
(Hermitian) ordinary nodal points (ONPs).

Recalling their mathematical origin, EPs were introduced by Kato as isolated
singularities of systems depending on one complex variable Kato (1995), and
they were recently classified into type I EPs and type II EPs Ashida  _et al._
(2020). EPs of type I are degeneracies with or without algebraic singularities
reminiscent of defective EPs and ONPs, while type II EPs are defined as points
in the complex plane where the Jordan normal form is unstable, i.e., the
eigenprojectors have a pole. While the appearance and existence of
degeneracies reminiscent of type I EPs have been studied extensively, type II
EPs have hitherto been overlooked in the literature.

In this letter, we introduce a natural extension of type II EPs to higher
dimensions, dubbed _non-defective EPs_. We present that the correct criterion
for detecting non-defective EPs is the form of the Hamiltonian matrix in the
_vicinity_ of these degenerate points: Non-defective EPs are surrounded by
defective EPs along certain intersections such that the Hamiltonian matrix
casts a Jordan block in certain directions away from the non-defective EPs.
Furthermore, we show that the Jordan decomposition is singular at these points
when approaching the points from certain directions, emphasizing that non-
defective EPs indeed are reminiscent of the type II EPs in Ref. Ashida  _et
al._ , 2020. To characterize the role of symmetries in witnessing NH
degeneracies, we study the coexistence of defective and non-defective EP$n$s
in two-, three- and four-band models in the presence of psH, ${\cal PT}$ or
${\cal CP}$ symmetry. Additionally, we show that symmetry-protected non-
defective EP2s may also appear in models with non-Hermitian time-reversal
symmetry. Finally, we find that defective EPs may coexist with ONPs instead of
non-defective EPs when lifting symmetry constraints. We illustrate this
finding with a fine-tuned example.

Finding possibilities to detect ONPs and (non-)defective EPs may pave the way
to advance applications of Hermitian and non-Hermitian topological properties
in various fields of research. For instance, topological lasers are one of the
platforms which owe their success to either Hermitian Wittek  _et al._ (2017);
Zeng  _et al._ (2020) or non-Hermitian St-Jean  _et al._ (2017); Parto  _et
al._ (2018); Harari  _et al._ (2018); Bandres  _et al._ (2018) topological
properties. Moreover, our systems, especially $\cal PT$-symmetric models, are
experimentally feasible as they can be implemented in experimental optical
setups with balanced gain and loss effectively Özdemir  _et al._ (2019).
Exploring the role of EPs in $\cal PT$-symmetric optical systems has already
unraveled many interesting phenomena, such as unidirectional invisibility
Regensburger  _et al._ (2012), induced mode-transition by encircling EPs
Doppler  _et al._ (2016), loss induced optical transparency Guo  _et al._
(2009), and stable single-mode lasing in multi-mode optical setups Feng  _et
al._ (2014); Hodaei  _et al._ (2015).

#### Symmetry-stabilized (non-)defective EPs.

A generic $n$-band Hamiltonian can be decomposed as
$\operatorname{\mathcal{H}}=d_{\mu}\Upsilon^{\mu}$, where
$\mu\in\\{0,\cdots,n^{2}-1\\}$, $d_{\mu}$ are continuously differentiable
complex-valued functions of the lattice momentum $\mathbf{k}$, $\Upsilon^{0}$
denotes the identity matrix of order $n$ and $\boldsymbol{\Upsilon}$ is the
basis of the $SU(n)$ group, which consists of three Pauli matrices when $n=2$,
eight Gell-Mann matrices when $n=3$, and fifteen generalized Gell-Mann
matrices when $n=4$ (see supplemental materials (SM) for the explicit
presentation of these matrices SuppMat ). The Hamiltonian
$\operatorname{\mathcal{H}}$ displays $\cal PT$ symmetry with generator ${\cal
PT}$, $\cal CP$ symmetry with generator ${\cal CP}$, or $\rm psH$ with
generator $\varsigma$, if it satisfies one of the following relations, namely,

$\displaystyle{\cal PT}:$
$\displaystyle\quad\operatorname{\mathcal{H}}(\mathbf{k})=({\cal
PT})\operatorname{\mathcal{H}}^{*}(\mathbf{k})({\cal PT})^{-1},$ (1)
$\displaystyle{\cal CP}:$
$\displaystyle\quad\operatorname{\mathcal{H}}(\mathbf{k})=-({\cal
CP})\operatorname{\mathcal{H}}^{*}(\mathbf{k})({\cal CP})^{-1},$ (2)
$\displaystyle{\rm psH}:$
$\displaystyle\quad\operatorname{\mathcal{H}}(\mathbf{k})=\varsigma\operatorname{\mathcal{H}}^{\dagger}(\mathbf{k})\varsigma^{-1}.$
(3)

These symmetry considerations reduce the number of nonzero $d_{\mu}$ values.
More precisely, for each basis matrix $\Upsilon^{\mu}$ merely a real or
imaginary part of $d_{\mu}$ remains nonzero, i.e., only one real-valued
function $d_{\mu}$ per each dimension of $\Upsilon^{\mu}$ survives Sayyad and
Kunst (2022). Trivial band touching points occur when the traceless part of
$\operatorname{\mathcal{H}}$ becomes a Null matrix ($[0]_{n\times n}$), i.e.,
all of the nonzero $d_{\mu}$ values for $\mu>0$ must vanish, which means that
one needs to satisfy $n^{2}-1$ real constraints. For $n=2,3,4$, we have
collected these $d_{\mu}$’s for each symmetry operation along side a choice
for its generator in Tables 1, 2, and 3, respectively.

The $n-1$ complex constraints to find defective EP$n$s can be expressed in
terms of the traces and the determinant of $\operatorname{\mathcal{H}}$, which
for two-, three-, and four-band models, respectively, read Sayyad and Kunst
(2022)

$\displaystyle n=2:$
$\displaystyle\quad\eta^{2b}=\mathop{\mathrm{tr}}[\operatorname{\mathcal{H}}]^{2}-4\det[\operatorname{\mathcal{H}}],$
(4) $\displaystyle n=3:$
$\displaystyle\begin{cases}\eta^{3b}&=\frac{1}{2}\left(\mathop{\mathrm{tr}}[{\cal
H}]^{2}-3\mathop{\mathrm{tr}}[{\cal H}^{2}]\right),\\\
\nu^{3b}&=\frac{1}{2}\left(54\det[{\cal H}]-5\mathop{\mathrm{tr}}[{\cal
H}]^{3}+9\mathop{\mathrm{tr}}[{\cal H}]\mathop{\mathrm{tr}}[{\cal
H}^{2}]\right),\end{cases}$ (5) $\displaystyle n=4:$
$\displaystyle\begin{cases}\eta^{4b}&=-3ac+b^{2}+12d,\\\
\nu^{4b}&=27a^{2}d-9abc+2b^{3}-72bd+27c^{2},\\\
\kappa^{4b}&=a^{3}-4ab+8c,\end{cases}$ (6)

where $a=\mathop{\mathrm{tr}}[{\cal H}]$,
$b=\big{(}(\mathop{\mathrm{tr}}[{\cal H}])^{2}-\mathop{\mathrm{tr}}[{\cal
H}^{2}]\big{)}/2$, $d=\det[{\cal H}]$ and $c=\left(\mathop{\mathrm{tr}}[{\cal
H}]^{3}-3\mathop{\mathrm{tr}}[{\cal H}]\mathop{\mathrm{tr}}[{\cal
H}^{2}]+2\mathop{\mathrm{tr}}[{\cal H}^{3}]\right)/6$. We refer to the SM for
details on how to derive these expressions SuppMat .

In the presence of ${\cal PT}$, ${\cal CP}$ and psH symmetry, some of these
constraints are automatically satisfied leaving us with exactly $n-1$ real
constraints, cf. Tables 1, 2 and 3 Sayyad and Kunst (2022). It is notable that
at trivial solutions (${\bf d}=0$), all traces and the determinant of
$\operatorname{\mathcal{H}}$ acquire zero values and subsequently, the
constraints in Eqs. (4)-(6) are also satisfied. As a result, the trivial
solutions mark non-defective EP$n$s with the binding signature that
$\operatorname{\mathcal{H}}$ is diagonalizable at these points. Perturbing the
system away from these non-defective points along intersections at which the
constraints vanish brings the Hamiltonian into a non-diagonalizable structure.
We set this behavior as a criterion to detect non-defective EPs. We note that
this is opposed to the situation in which trivial solutions are isolated, and
thus band touching points behave similarly to Hermitian degeneracies, i.e.,
ONPs.

The diagonalizability of the Hamiltonian at non-defective EPs enables us to
map our NH Hamiltonians into their Hermitian counterparts with nodal points.
In addition, having $n^{2}-1$ nonzero $d_{\mu}$’s as in Hermitian systems
enforces non-defective EPs to always appear in pairs. This statement
originates from the Poincaré-Hopf theorem Mathai and Thiang (2017), as the
number of nonzero $d_{\mu}$ functions equals the dimension of the vector space
($n^{2}-1$), cf. last columns in Tables 1, 2, and 3 Poincare .

Table 1: Summarized symmetries, their generators and their associated
constraints to find defective and non-defective EP2s in two-band systems.

Symm. |  | Generator |  | Constr. def. EP2s |  | Constr. non-def. EP2s
---|---|---|---|---|---|---
${\cal PT}$ |  | ${\mathbbm{1}}$ |  | $\eta^{2b}_{R}=0$ |  | $d_{xR}=d_{yI}=d_{zR}=0$
${\cal CP}$ |  | ${\mathbbm{1}}$ |  | $\eta^{2b}_{R}=0$ |  | $d_{xI}=d_{yR}=d_{zI}=0$
psH |  | ${\rm adiag}[1,1]$ |  | $\eta^{2b}_{R}=0$ |  | $d_{xR}=d_{yI}=d_{zI}=0$
TRS† |  | ${\rm adiag}[1,-1]$ |  | $\eta^{2b}_{R}=\eta^{2b}_{I}=0$ |  | $d_{xa}=d_{ya}=d_{za}=0$

Here we use either $d_{j}=d_{jR}+\mathrm{i}d_{jI}$ or $d_{j}=d_{js}+d_{ja}$,
where $d_{js}(d_{ja})$ is (anti-)symmetric with respect to
$\mathbf{k}\to-\mathbf{k}$. $\eta^{2b}$ is given in Eq. (4) with
$\eta^{2b}=\eta^{2b}_{R}+\mathrm{i}\eta^{2b}_{I}$.

Aside from $\cal PT$, $\cal CP$ and $\rm psH$ symmetries, a particular non-
Hermitian time-reversal symmetry, known as TRS†, in two-band systems may also
give rise to realizing non-defective EPs. To evidently see this behavior, we
recall that respecting TRS† symmetry imposes
$\operatorname{\mathcal{H}}(-\mathbf{k})={\cal
C}_{+}\operatorname{\mathcal{H}}^{\dagger}(\mathbf{k}){\cal C}^{\dagger}_{+}$.
This non-(momentum)-local transformation does not reduce the number of non-
vanishing (real/imaginary) parts of $d_{\mu}$. However, when ${\cal
C}_{+}=\mathrm{i}\sigma_{y}$, it enforces all symmetric parts of $d_{\mu}$ to
become zero. We further know that right at time-reversal invariant momenta
(TRIM), e.g., $\mathbf{k}_{\rm TRIM}=\\{(0,0),(0,\pi),(\pi,0),(\pi,\pi)\\}$ on
the square Brillouin zone, anti-symmetric functions vanish. Therefore, at
$\mathbf{k}_{\rm TRIM}$ both real and imaginary parts of anti-symmetric
$d_{\mu}$ functions become zero, which gives rise to observing non-defective
EPs in the spectra of two-band TRS†-symmetric $\operatorname{\mathcal{H}}$,
cf. Table 1.

Table 2: Summarized symmetries, their generators and their associated
constraints to find defective and non-defective EP3s in three-band systems.

Symm. |  | Generator |  | Constr. def. EP3s |  | Constr. non-def. EP3s
---|---|---|---|---|---|---
${\cal PT}$ |  | ${\rm diag}[1,-1,1]$ |  | $\eta^{3b}_{R}=0\,\&\,\nu^{3b}_{R}=0$ |  | $d_{1R}=d_{4I}=d_{2I}=d_{5R}=d_{3R}=d_{6I}=d_{7R}=d_{8R}=0$
${\cal CP}$ |  | ${\rm diag}[1,-1,1]$ |  | $\eta^{3b}_{R}=0\,\&\,\nu^{3b}_{I}=0$ |  | $d_{1I}=d_{4R}=d_{2R}=d_{5I}=d_{3I}=d_{6R}=d_{7I}=d_{8I}=0$
psH |  | ${\rm diag}[1,-1,1]$ |  | $\eta^{3b}_{R}=0\,\&\,\nu^{3b}_{R}=0$ |  | $d_{1I}=d_{4I}=d_{2R}=d_{5R}=d_{3I}=d_{6I}=d_{7R}=d_{8R}=0$

Here we use $d_{j}=d_{jR}+\mathrm{i}d_{jI}$. Complex valued $\eta^{3b}$ and
$\nu^{3b}$ constraints are given in Eq. (5) with
$\alpha^{3b}=\alpha^{3b}_{R}+\mathrm{i}\alpha^{3b}_{I}$ for
$\alpha\in\\{\eta,\nu\\}$.

Table 3: Summarized symmetries, their generators and their associated
constraints to get defective and non-defective EP4s in four-band systems.

Symm. |  | Generator |  | Constr. def. EP4s |  | Constr. non-def. EP4s
---|---|---|---|---|---|---
${\cal PT}$ |  | ${\rm diag}[1,-1,1,-1]$ |  | $\eta^{4b}_{R}=0\,\&\,\nu^{4b}_{R}=0\,\&\,\kappa^{4b}_{R}=0$ |  |  $d_{1R}=d_{2I}=d_{3R}=d_{4R}=d_{5I}=d_{6R}=d_{7I}=d_{8R}=0$ $d_{9I}=d_{10I}=d_{11R}=d_{12I}=d_{13R}=d_{14R}=d_{15R}=0$
${\cal CP}$ |  | ${\rm diag}[1,-1,1,-1]$ |  | $\eta^{4b}_{R}=0\,\&\,\nu^{4b}_{R}=0\,\&\,\kappa^{4b}_{I}=0$ |  |  $d_{1I}=d_{2R}=d_{3I}=d_{4I}=d_{5R}=d_{6I}=d_{7R}=d_{8I}=0$ $d_{9R}=d_{10R}=d_{11I}=d_{12R}=d_{13I}=d_{14I}=d_{15I}=0$
psH |  | ${\rm diag}[1,-1,1,-1]$ |  | $\eta^{4b}_{R}=0\,\&\,\nu^{4b}_{R}=0\,\&\,\kappa^{4b}_{R}=0$ |  |  $d_{1I}=d_{2R}=d_{3I}=d_{4I}=d_{5R}=d_{6I}=d_{7I}=d_{8R}=0$ $d_{9I}=d_{10I}=d_{11R}=d_{12I}=d_{13R}=d_{14R}=d_{15R}=0$

Here we use $d_{j}=d_{jR}+\mathrm{i}d_{jI}$. Complex valued $\eta^{4b}$,
$\nu^{4b}$ and $\kappa^{4b}$ constraints are given in Eq. (6) with
$\alpha^{4b}=\alpha^{4b}_{R}+\mathrm{i}\alpha^{4b}_{I}$ for
$\alpha\in\\{\eta,\nu,\kappa\\}$.

Figure 1:  Real part (a) and imaginary part (b) of spectra for Eq. (7) along
$k_{x}=k_{y}$ with $k_{z}=\pi/2$. Red points indicate non-defective EPs. Here
we set $t=V=\lambda_{0}=1.0$.

Before moving on to examples, we note that the different number of constraints
that need to be satisfied to find symmetry-protected defective and non-
defective EP$n$s also result in a different codimension of these EPs. Here,
the codimension is given by the difference between the total dimension of the
system and the dimension of the exceptional feature. Equivalently, the
codimension corresponds precisely to the number of nonvanishing constraints.
In particular, while the presence of ${\cal PT}$, ${\cal CP}$ and psH
symmetries reduces the number of real constraints for finding defective EP$n$s
to $n-1$, the number of real constraints to detect non-defective EP$n$s is
$n^{2}-1$. As a consequence, in the case of $n=2$, defective EP2s have
codimension _one_ , whereas non-defective EP2s have codimension _three_.
Therefore, the latter appear as points in three-dimensional systems, whereas
defective EP2s appear as two-dimensional surfaces. For the TRS† invariant two-
band model, the codimension is _two_ , and hence the defective EP2s are curves
connected at the TRIMs.

#### Examples for the coexistence of defective and non-defective EPs.

We start with introducing a two-band ${\cal PT}$-symmetric Weyl-like tight-
binding model described by

$\displaystyle\operatorname{\mathcal{H}}_{\cal PT}^{2b}$
$\displaystyle=d_{0}\Upsilon^{0}+d_{xR}\Upsilon^{1}+\mathrm{i}d_{yI}\Upsilon^{2}+d_{zR}\Upsilon^{3}$
$\displaystyle=2\lambda_{0}\sin{\left(k_{x}\right)}\Upsilon^{0}+2t\sin{\left(k_{x}\right)}\Upsilon^{1}+2t\sin{\left(k_{y}\right)}\Upsilon^{3}$
$\displaystyle+\mathrm{i}\left\\{2t\cos{\left(k_{z}\right)}+2V\left[2-\cos{\left(k_{x}\right)}-\cos{\left(k_{y}\right)}\right]\right\\}\Upsilon^{2}.$
(7)

Here $\lambda_{0}$, $t$ and $V$ are real-valued parameters. The real and
imaginary parts of the band structure are shown in Figs. 1(a) and (b),
respectively.

Non-defective EPs appear when all components of the Hamiltonian, except
$d_{0}$, vanish. More specifically, these degeneracies emerge when the
solutions of $d_{xR}=0$ (at $k_{x}=n\pi$ with $n\in\mathbb{Z}$), $d_{yI}=0$
(orange curves in Fig. 2), and $d_{zR}=0$ (grey line) intersect. Red points at
$\mathbf{k}=\left(0,0,\pm\pi/2\right)$ in Fig. 2 exemplify such solutions.
Note that the criterion for detecting non-defective EPs is satisfied for the
red points in Fig. 2 as they are surrounded by defective EPs (dashed blue
curves), residing on $\eta^{2b}_{R}=0$, where $\eta^{2b}_{R}$ is the real part
of $\eta^{2b}$ in Eq. (4).

Figure 2:  Solutions of $d_{yI}$ (orange), $d_{zR}$ (gray), and
$\eta_{R}^{2b}$ (dashed blue) at $k_{x}=0$, which is a solution to $d_{xR}=0$.
Red points at $\mathbf{k}=(0,0,\pm\pi/2)$ indicate the intersection between
solutions to $\eta_{R}^{2b}=d_{yI}=d_{zR}=0$, and are non-defective EPs. Here
we set $t=V=\lambda_{0}=1.0$.

Along the defective EPs, defined by $d_{xR}^{2}+d_{zR}^{2}=d_{yI}^{2}$, the
Jordan decomposition reads
$\operatorname{\mathcal{H}}_{\mathcal{PT}}^{2b}=SJS^{-1}$ with
$S=(s_{1},s_{2})$, where

$\displaystyle s_{1}$
$\displaystyle=\begin{pmatrix}\frac{\sqrt{t^{4}\left[2-\cos{\left(2k_{x}\right)}-\cos{\left(2k_{y}\right)}\right]}-\sqrt{2}t^{2}\sin{\left(k_{x}\right)}}{\sqrt{2}t^{2}\sin{\left(k_{y}\right)}}\\\
1\end{pmatrix},$ (8) $\displaystyle s_{2}$
$\displaystyle=\begin{pmatrix}\frac{\sqrt{2}\sqrt{t^{4}\left[2-\cos{\left(2k_{x}\right)}-\cos{\left(2k_{y}\right)}\right]}-2t^{2}\sin{\left(k_{x}\right)}}{4t^{3}\sin{\left(k_{y}\right)}^{2}}\\\
0\end{pmatrix},$ (9) $\displaystyle J$
$\displaystyle=\begin{pmatrix}2\lambda_{0}\sin{\left(k_{x}\right)}&1\\\
0&2\lambda_{0}\sin{\left(k_{x}\right)}\end{pmatrix}.$ (10)

The second column in the transformation matrix, $s_{2}$, has a singularity
when $k_{x}$ and $k_{y}$ goes to $n\pi$ simultaneously, corresponding exactly
to the non-defective EP. Hence, the Jordan decomposition is unstable at the
non-defective EPs.

We further note that the defective EPs separate two regions in the real part
of spectrum, where $\text{Re}[\Delta\epsilon]=0$ and
$\text{Re}[\Delta\epsilon]\neq 0$ with $\Delta\epsilon$ the difference between
the two energy bands as shown in Fig. 1(a). Regions where
$\text{Re}[\Delta\epsilon]=0$ are sometimes referred to as NH bulk real-Fermi
states, which merely appear in NH systems Bergholtz  _et al._ (2021). In the
SM, we show that besides these bulk Fermi states, this model also hosts states
on the boundary SuppMat . Therefore, there is a coexistence between defective
and non-defective EPs as well as between bulk Fermi states and boundary
states.

Figure 3:  Real (a) and imaginary (b) parts of band structure for
$\operatorname{\mathcal{H}}_{\rm psH}^{4b}$ in Eq. (11) along $k_{x}=k_{y}$
and $k_{z}$. Note that both bands are doubly degenerate. Red points indicate
non-defective EPs at $\mathbf{k}=(0,0,\pm k_{0})$. Here we set $t=t_{z}=1.0$,
$\lambda_{Ixx}=\lambda_{Ixy}=0.15$, $m^{\prime}_{I}=-0.27$ and $k_{0}=\pi/2$.

Let us now turn to a four-band model. We consider a Dirac-like $\rm
psH$-symmetric model described by

$\displaystyle\operatorname{\mathcal{H}}_{\rm psH}^{4b}=$ $\displaystyle
t\left[\cos{\left(k_{x}\right)}+\cos{\left(k_{y}\right)}-2\right]\left(\frac{2}{\sqrt{3}}\Upsilon^{14}+\sqrt{\frac{2}{3}}\Upsilon^{15}\right)$
$\displaystyle+t_{z}\left[\cos{\left(k_{z}\right)}-\cos{\left(k_{0}\right)}\right]\left[\frac{2}{\sqrt{3}}\Upsilon^{14}+\sqrt{\frac{2}{3}}\Upsilon^{15}\right]$
$\displaystyle+\mathrm{i}\lambda_{Ixy}\sin{\left(k_{y}\right)}(\Upsilon^{3}+\Upsilon^{4})$
$\displaystyle+\mathrm{i}\lambda_{Ixx}\sin{\left(k_{x}\right)}(\Upsilon^{9}+\Upsilon^{10})$
$\displaystyle+\mathrm{i}m^{\prime}_{I}\sin{\left(k_{z}\right)}\left[\cos{\left(k_{x}\right)}-\cos{\left(k_{y}\right)}\right]\left(\Upsilon^{7}-\Upsilon^{12}\right),$
(11)

where $t$, $t_{z}$, $\lambda_{Ixy}$, $\lambda_{Ixx}$, and
$m_{I}^{{}^{\prime}}$ are real-valued parameters. This model is a $\rm psH$
generalization of the tight-binding model studied in Ref. Kargarian  _et al._
, 2016. The trivial band touching points for obtaining the null form of the
traceless part of $\operatorname{\mathcal{H}}_{\rm psH}^{4b}$ in Eq. (11) are
located at $\mathbf{k}=\mathbf{k}_{\rm psH}=(0,0,\pm k_{0})$. Right at these
points and on lines connecting these points, constraints for realizing EP4s,
summarized in Table 3 and given in Eq. (6), are also satisfied. Hence, points
at $\mathbf{k}=\mathbf{k}_{\rm psH}$ in our $\rm psH$-symmetric model are non-
defective EP4s. We present these nodal points (red spheres) in the band
structure of $\operatorname{\mathcal{H}}_{\rm psH}^{4b}$ in Fig. 3. The real
part of the spectra (a) shows that two arc-shaped surfaces with (non)zero real
(imaginary) parts are terminated by the non-defective EPs as well as the
defective exceptional lines. These surfaces are the aforementioned NH bulk
real-Fermi surfaces. While our model hosts bulk Fermi surfaces, boundary
states are not stable, similar to its Hermitian counterpart Kargarian  _et
al._ (2016).

Figure 4: Real (a) and imaginary (b) parts of band structure for
$\operatorname{\mathcal{H}}_{\rm ONP}^{\rm 2b}$ in Eq. (12) along $k_{z}=\pi$.
(c) Solutions of the real (blue spheres) and imaginary (green manifold) parts
of $\eta^{2b}$ in Eq. (4). Red closed lines in (c) present the intersection of
these manifolds and correspond to defective EPs. Black points in panels (a),
(b) indicate ONPs. For visibility purposes, merely a corner of Brillouin zone
is shown in panels (a), (b).

#### Searching for the coexistence of ONPs and defective EPs.

So far, we have explored how the presence of one of the ${\cal PT}$, ${\cal
CP}$ or $\rm psH$ symmetries leads to the general coexistence of defective and
non-defective EPs. In the remaining, we address whether ONPs may also exist in
NH spectra. For this purpose, we lift (discrete) symmetry restrictions such
that some (or all) $d_{\mu}$’s have real and imaginary parts. We emphasize
that in this situation, in contrast to the EPs in the previous sections, band
degeneracies are generally unstable to small perturbations 222More precisely,
twofold degeneracies are protected by composite symmetries consisting of
multiple symmetry operations Sayyad2022d. Respecting all these symmetry
operations might be easily violated upon introducing perturbations.. This is
because finding solutions to ${\bf d}=0$ generally requires satisfying
$n^{2}-1$ complex constraints. Consequently, the appearance of ONPs in NH
models is vulnerable to the fine-tuning of parameters. Nevertheless, we show
in the following that this setting provides a platform to observe ONPs.

To illustrate our idea, we introduce a two-band model given by

$\displaystyle\operatorname{\mathcal{H}}_{\rm ONP}^{2b}$
$\displaystyle=\sin(k_{x})\left[1/2+\mathrm{i}\cos(k_{y})\right]\Upsilon^{1}$
$\displaystyle+\sin(k_{y})\left[1/2+\mathrm{i}\cos(k_{z})\right]\Upsilon^{2}$
$\displaystyle+\sin(k_{z})\left[1/2+\mathrm{i}\cos(k_{x})\right]\Upsilon^{3}.$
(12)

Fig. 4 exhibits real (a) and imaginary (b) parts of the energy dispersion of
this system along $k_{z}=\pi$. The black points in these panels mark real and
imaginary band touching points, located at $k_{x,y,z}^{\rm ONP}=\pm n\pi$.
Even though at these momenta the traceless part of
$\operatorname{\mathcal{H}}_{\rm ONP}^{2b}$ becomes a null matrix, we
emphasize that ${\bf k}={\bf k}^{\rm ONP}$ indicates the location of ONPs and
not non-defective EPs. The reason for this statement lies in the fact that the
criterion for the emergence of non-defective EPs is not satisfied, i.e., no
defective EPs reside close to ${\bf k}^{\rm ONP}$. This can be seen from Fig.
4 (c) in which we present defective EPs, red curves, as the intersection
between $\text{Re}[\eta^{2b}]=0$, blue spheres, and $\text{Im}[\eta^{2b}]=0$,
green manifold. Fig. 4 (c) reveals that defective EPs do not cross ${\bf
k}={\bf k}^{\rm ONP}$ and thus the black points at ${\bf k}^{\rm ONP}$ indeed
correspond to ONPs. Consequently, $H^{2b}_{\text{ONP}}$ is diagonalizable at
${\bf k}={\bf k}^{\rm ONP}$ and in any small neighborhood surrounding the
point.

Aside from these ONPs in the momentum-space, introducing models that host
boundary states connecting ONPs in NH systems is theoretically feasible. We
present an example of such a model in the SM SuppMat .

#### Conclusion.

Despite the intense focus on NH systems in recent studies, the possibility of
realizing different types of EPs has hitherto been overlooked. The present
work shows that two different types of EPs, dubbed defective and non-defective
EPs, may coexist in various setups of physical importance. Concretely, we show
that non-defective EPs are stabilized by certain symmetries, including $\cal
PT$, $\cal CP$, $\rm psH$, and time-reversal symmetry. To resolve the
confusion in the current literature, where non-defective EPs are mixed up with
ONPs, we have in this work introduced a clear criterion to distinguish these
concepts. We also highlight this difference in example models.

Even though the models we discuss in this work are experimentally tractable,
we do not expect a difference in the experimental signatures of non-defective
EPs and ONPs when single-particle, non-interacting Hamiltonians are explored.
Investigating interacting/coupled systems may allow identifying distinct
footprints of non-defective EPs and ONPs in experiments.

#### Acknowledgements.

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.
acknowledges the support from the Knut and Alice Wallenberg foundation under
grant no. 2017.0157.

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##  Appendix A Bases matrices for two-, three-, and four-band systems

###  A.1 Basis matrices for two-band systems

The basis matrices for two-band systems are Pauli matrices which read

$\displaystyle\Upsilon^{1}=\begin{pmatrix}0&1\\\
1&0\end{pmatrix},\,\,\Upsilon^{2}=\begin{pmatrix}0&-\mathrm{i}\\\
\mathrm{i}&0\end{pmatrix},\,\,\Upsilon^{3}=\begin{pmatrix}1&0\\\
0&-1\end{pmatrix}.$ (13)

###  A.2 Basis matrices for three-band systems

The basis matrices for three-band systems are the Gell-Mann matrices, that
span the Lie algebra of the SU(3) group,

$\displaystyle\Upsilon^{1}$ $\displaystyle=\begin{pmatrix}0&-\mathrm{i}&0\\\
\mathrm{i}&0&0\\\
0&0&0\end{pmatrix},\quad\Upsilon^{2}=\begin{pmatrix}0&0&-\mathrm{i}\\\
0&0&0\\\ \mathrm{i}&0&0\end{pmatrix},$ (14) $\displaystyle\Upsilon^{3}$
$\displaystyle=\begin{pmatrix}0&0&0\\\ 0&0&-\mathrm{i}\\\
0&\mathrm{i}&0\end{pmatrix},\quad\Upsilon^{4}=\begin{pmatrix}0&1&0\\\ 1&0&0\\\
0&0&0\end{pmatrix},$ (15) $\displaystyle\Upsilon^{5}$
$\displaystyle=\begin{pmatrix}0&0&1\\\ 0&0&0\\\
1&0&0\end{pmatrix},\quad\Upsilon^{6}=\begin{pmatrix}0&0&0\\\ 0&0&1\\\
0&1&0\end{pmatrix},$ (16) $\displaystyle\Upsilon^{7}$
$\displaystyle=\begin{pmatrix}1&0&0\\\ 0&-1&0\\\
0&0&0\end{pmatrix},\quad\Upsilon^{8}=\begin{pmatrix}\frac{1}{\sqrt{3}}&0&0\\\
0&\frac{1}{\sqrt{3}}&0\\\ 0&0&-\frac{2}{\sqrt{3}}\end{pmatrix}.$ (17)

###  A.3 Basis matrices for four-band systems

The basis matrices for four-band systems are the generalized Gell-Mann
matrices, that span the Lie algebra of the SU(4) group,

$\displaystyle\Upsilon^{1}$ $\displaystyle=\begin{pmatrix}0&-\mathrm{i}&0&0\\\
\mathrm{i}&0&0&0\\\ 0&0&0&0\\\ 0&0&0&0\\\
\end{pmatrix},\quad\Upsilon^{2}=\begin{pmatrix}0&0&-\mathrm{i}&0\\\ 0&0&0&0\\\
\mathrm{i}&0&0&0\\\ 0&0&0&0\\\ \end{pmatrix},$ (18)
$\displaystyle\Upsilon^{3}$ $\displaystyle=\begin{pmatrix}0&0&0&-\mathrm{i}\\\
0&0&0&0\\\ 0&0&0&0\\\ \mathrm{i}&0&0&0\\\
\end{pmatrix},\quad\Upsilon^{4}=\begin{pmatrix}0&0&0&0\\\ 0&0&-\mathrm{i}&0\\\
0&\mathrm{i}&0&0\\\ 0&0&0&0\\\ \end{pmatrix},$ (19)
$\displaystyle\Upsilon^{5}$ $\displaystyle=\begin{pmatrix}0&0&0&0\\\
0&0&0&-\mathrm{i}\\\ 0&0&0&0\\\ 0&\mathrm{i}&0&0\\\
\end{pmatrix},\quad\Upsilon^{6}=\begin{pmatrix}0&0&0&0\\\ 0&0&0&0\\\
0&0&0&-\mathrm{i}\\\ 0&0&\mathrm{i}&0\\\ \end{pmatrix},$ (20)
$\displaystyle\Upsilon^{7}$ $\displaystyle=\begin{pmatrix}0&1&0&0\\\
1&0&0&0\\\ 0&0&0&0\\\ 0&0&0&0\\\
\end{pmatrix},\quad\Upsilon^{8}=\begin{pmatrix}0&0&1&0\\\ 0&0&0&0\\\
1&0&0&0\\\ 0&0&0&0\\\ \end{pmatrix},$ (21) $\displaystyle\Upsilon^{9}$
$\displaystyle=\begin{pmatrix}0&0&0&1\\\ 0&0&0&0\\\ 0&0&0&0\\\ 1&0&0&0\\\
\end{pmatrix},\quad\Upsilon^{10}=\begin{pmatrix}0&0&0&0\\\ 0&0&1&0\\\
0&1&0&0\\\ 0&0&0&0\\\ \end{pmatrix},$ (22) $\displaystyle\Upsilon^{11}$
$\displaystyle=\begin{pmatrix}0&0&0&0\\\ 0&0&0&1\\\ 0&0&0&0\\\ 0&1&0&0\\\
\end{pmatrix},\quad\Upsilon^{12}=\begin{pmatrix}0&0&0&0\\\ 0&0&0&0\\\
0&0&0&1\\\ 0&0&1&0\\\ \end{pmatrix},$ (23) $\displaystyle\Upsilon^{13}$
$\displaystyle=\begin{pmatrix}1&0&0&0\\\ 0&-1&0&0\\\ 0&0&0&0\\\ 0&0&0&0\\\
\end{pmatrix},\,\Upsilon^{14}=\begin{pmatrix}\frac{1}{\sqrt{3}}&0&0&0\\\
0&\frac{1}{\sqrt{3}}&0&0\\\ 0&0&-\frac{2}{\sqrt{3}}&0\\\ 0&0&0&0\\\
\end{pmatrix},$ (24) $\displaystyle\Upsilon^{15}$
$\displaystyle=\begin{pmatrix}\frac{1}{\sqrt{6}}&0&0&0\\\
0&\frac{1}{\sqrt{6}}&0&0\\\ 0&0&\frac{1}{\sqrt{6}}&0\\\
0&0&0&-\sqrt{\frac{3}{2}}\\\ \end{pmatrix}.$ (25)

Figure S1:  Real parts (a) and imaginary parts (b) of spectra for Eq. (7) with
open boundary condition in the $y$ direction and $k_{x}=0$. Red lines present
boundary states. Here we set $t=V=\lambda_{0}=1.0$.

##  Appendix B Derivation of constraints to find EP$n$s

Here we briefly summarize how to derive Eqs. (4)-(6) in the main text, where
these constraints were originally derived in Ref. Sayyad and Kunst, 2022.
There it was shown that the characteristic polynomial of an $n$-band NH matrix
can be expressed in terms of its determinant and traces. Indeed, for two-,
three- and four-band matrices, these polynomials read

$\displaystyle F^{2b}(\lambda)$
$\displaystyle=\lambda^{2}-\mathop{\mathrm{tr}}[\mathcal{H}]\lambda+\det[\mathcal{H}]=0,$
(26) $\displaystyle F^{3b}(\lambda)$
$\displaystyle=\lambda^{3}-\mathop{\mathrm{tr}}[\mathcal{H}]\lambda^{2}+\frac{\mathop{\mathrm{tr}}[\mathcal{H}]^{2}-\mathop{\mathrm{tr}}[\mathcal{H}^{2}]}{2}\lambda-\det[\mathcal{H}]=0,$
(27) $\displaystyle F^{4b}(\lambda)$
$\displaystyle=\lambda^{4}-a\lambda^{3}+b\lambda^{2}-c\lambda+d=0,$ (28)

where

$\displaystyle a$ $\displaystyle=\mathop{\mathrm{tr}}[\mathcal{H}],\quad
b=\frac{\mathop{\mathrm{tr}}[\mathcal{H}]^{2}-\mathop{\mathrm{tr}}[\mathcal{H}^{2}]}{2},\quad
d=\det[\mathcal{H}],$ (29) $\displaystyle c$
$\displaystyle=\frac{\mathop{\mathrm{tr}}[\mathcal{H}]^{3}-3\mathop{\mathrm{tr}}[\mathcal{H}]\mathop{\mathrm{tr}}[\mathcal{H}^{2}]+2\mathop{\mathrm{tr}}[\mathcal{H}^{3}]}{6},$
(30)

and $\lambda$ are the eigenvalues.

To find degeneracies, the discriminant $D[\mathcal{H}]$ of these
characteristic polynomials need to be set to zero. The discriminants read

$\displaystyle D[\mathcal{H}^{2b}]$
$\displaystyle=\mathop{\mathrm{tr}}[\mathcal{H}]^{2}-4\det[\mathcal{H}],$ (31)
$\displaystyle D[\mathcal{H}^{3b}]$
$\displaystyle=-\frac{1}{27}[4(\eta^{3b})^{3}+(\nu^{3b})^{2}],$ (32)
$\displaystyle D[\mathcal{H}^{4b}]$
$\displaystyle=\frac{1}{27}[4(\eta^{4b})^{3}-(\nu^{4b})^{2}],$ (33)

where $\eta^{3b}$ and $\nu^{3b}$, and $\eta^{4b}$ and $\nu^{4b}$ are given in
Eqs. (5) and (6) in the main text, respectively. From here, we immediately see
that setting the discriminants in Eqs. (31) and (32) to zero gives us the
constraints in Eqs. (4) and (5) in the main text, respectively. In the case of
the four-band model, we note that for all roots of the discriminant to
coincide, not only $\eta^{4b}=0$ and $\nu^{4b}=0$ need to be satisfied, but
also $\kappa^{4b}=0$, where $\kappa^{4b}$ is defined in Eq. (6) in the main
text. We refer to Ref. Sayyad and Kunst, 2022 for a more detailed discussion
on this point.

##  Appendix C Spectra of the two-band model with open boundary condition

In addition to the properties of momentum-dependent spectra for ${\cal
H}_{\cal PT}^{\rm 2b}$ in Fig. 1, we present real (a) and imaginary (b) parts
of the energy dispersion with open boundary condition in the $y$ direction
plotted in Fig. S1. The figure exhibits boundary states, red lines, well-
separated from the bulk states (blue lines) when $|k_{z}|>\pi/2$.

Figure S2:  Real parts (a) and imaginary parts (b) of spectra for Eq. (34)
with open boundary condition in the $y$ direction and at $k_{z}=0$. Red lines
present boundary states. Black points mark ONPs. Here we set
$t=V=1.0,\lambda_{0}=2.3$.

##  Appendix D Realizing Hermitian boundary states in non-Hermitian systems

Here we present a non-Hermitian tight-binding model hosting Hermitian boundary
states, which connects ONPs with zero imaginary parts.

Our model Hamiltonian reads

$\displaystyle\operatorname{\mathcal{H}}_{\rm edg}^{\rm 2b}=$
$\displaystyle\lambda_{0}\cos(k_{x})\Upsilon^{0}-\mathrm{i}V\left[1-\cos(k_{z})\right]\Upsilon^{1}$
$\displaystyle+\left[2V\cos(k_{y})-2t\cos(k_{x})\right]\Upsilon^{1}$
$\displaystyle-2t\sin(k_{y})\Upsilon^{2}-2t\sin(k_{z})\Upsilon^{3},$ (34)

where $\lambda_{0}$, $t$ and $V$ are real-valued coupling constants. Along
$k_{z}=0$, the above Hamiltonian is fully Hermitian. As a result, nodal
points, which live on the $(k_{x},k_{y})$ plane, are band touching points with
zero imaginary parts. For instance, these ordinary nodal points appear at
$\mathbf{k}_{\rm ONPs}=(\pm n\pi,\pm n\pi,0)$ with
$n\in\operatorname{\mathbb{Z}}$ when $t=V$. Considering the open boundary
condition along the $y$ axis and at $k_{z}=0$ results in mid-gap boundary
states, red lines in Fig. S2(a), which connects $\mathbf{k}_{\rm ONPs}$.