††thanks: Corresponding author

# General Exceptional Points

Xiao-Ran Wang  Center for Advanced Quantum Studies, Department of Physics,
Beijing Normal University, Beijing 100875, China  Fei Yang  Center for
Advanced Quantum Studies, Department of Physics, Beijing Normal University,
Beijing 100875, China  Xiao-Jie Yu  Center for Advanced Quantum Studies,
Department of Physics, Beijing Normal University, Beijing 100875, China  Xian-
Qi Tong  Center for Advanced Quantum Studies, Department of Physics, Beijing
Normal University, Beijing 100875, China  Su-Peng Kou<EMAIL_ADDRESS>Center
for Advanced Quantum Studies, Department of Physics, Beijing Normal
University, Beijing 100875, China

###### Abstract

Exceptional points are interesting physical phenomena in non-Hermitian physics
at which the eigenvalues are degenerate and the eigenvectors coalesce. In this
paper, we find that the universal feature of arbitrary non-Hermitian two-level
systems with singularities is basis defectiveness rather than energy
degeneracy or state coalescence. This leads to the discovery of general
exceptional points (GEPs). For GEPs, more subtle structures (e.g., the so-
called Bloch peach), additional classification, and “hidden” quantum phase
transitions are explored. By using the topologically protected sub-space from
two edge states in the non-Hermitian Su–Schrieffer–Heeger model as an example,
we illustrate the physical properties of different types of GEPs.

##  I Introduction

“ _Exceptional point_ ” (EP) is a mathematical term introduced by T. Kato over
half a century agoKato . In mathematics, EPs are branch point singularities of
a spectrum and eigenfunctions for non-Hermitian matricesHeiss1 ; Moiseyev . In
the form of Jordan block matrix, at EPs the algebra multiplier of a matrix
becomes larger than its geometric multiplier. Since the publication of the
related paper by Bender and BoettcherBender1 , EPs have become one of the most
interesting phenomena in non-Hermitian physicsMiri ; Minganti . As a
particular example, for non-Hermitian Hamiltonians with parity–time
($\mathcal{PT}$) symmetryBender2 ; Heiss2 ; Mostafazadeh ; Lee ; Kawabata ;
El-Ganainy ; Yao ; Rotter ; Jin ; Deng ; Liu ; Ran ; Ashida ; Guo ; Guo1 ;
wang ; wang1 , spontaneous $\mathcal{PT}$-symmetry breaking corresponds to a
typical EP, at which the energy levels become degenerate and the eigenvectors
coalesce. In experiments, the phenomenon of EPs has been realized and
simulated using various approachesFeng ; Peng ; Hodaei ; AGuo ; Chen ; Hodaei1
; El-Ganainy2 ; Schindler ; Feng2 ; Regensburger ; Bittner ; MBender ; Hang ;
Zhu ; Brandstetter ; Popa ; Fleury ; Zhen ; Xu ; Peng1 ; Zhang ;
Assawaworrarit ; Choi ; Scheel ; Wu ; L. Xiao1 ; Naghiloo ; L. Xiao2 ; Liu1 ;
ChenPX ; Ding .

On the other hand, in some quantum many-body models, due to special conditions
of symmetry/topology, there may exist protected sub-systems. For example, for
topological insulators, there exist topologically protected edge states with
gapless energy spectra (or zero modes for one dimensional cases)Kane2010 ;
Qi2011 ; for the many-body systems with spontaneously symmetry breaking there
exist symmetry-protected degenerate ground states; for the topological orders
with long range entanglement, there exist topologically protected degenerate
ground states (on a torus) that make up topological qubits and may be possible
to be applied to incorporate intrinsic fault tolerance into a quantum
computerKitaev ; wen ; wen1 ; S. P. Kou .

For these topologically/symmetry protected sub-systems in different quantum
many-body models, the quantum properties will be changed under non-Hermitian
perturbations. _How the non-Hermitian perturbations affect the
topologically/symmetry protected sub-systems? Do there exist new types of
EPs?_ In this paper, to answer these questions, we investigate the properties
of EPs and find that basis defectiveness plays a key role in EPs. Furthermore,
we find that in certain non-Hermitian systems (subsystems of certain non-
Hermitian models) there may exist EPs without eigenvalue degeneracy, EPs
without the coalescence of different eigenvectors, or those without both
features (see the following discussions). This leads to the discovery of
general EPs.

The remainder of the paper is organized as follows. In Sec. II, we review the
theory of singularity for usual EPs in a simple two-level systems and show the
reason why eigenstates coalesce at an EP. In Sec. III, we discuss a general
theory for protected non-Hermitian sub-spaces (or topologically/symmetry
protected sub-systems in different quantum many-body models) and show how to
derive the effective Hamiltonian and the corresponding (initial) basis. In
Sec. IV, we develop a theory of singularity for the general EPs and show their
complete classification. In Sec. V, an example of a one-dimensional (1D) non-
Hermitian topological insulator – 1D nonreciprocal Su–Schrieffer–Heeger (SSH)
model is focused on. According to the global phase diagram of the
topologically protected sub-systems from the two edge states, we show the
occurrence of different types of general EPs. Finally, the conclusions are
given in Sec. VI.

##  II Singularity at an EP in a non-Hermitian two-level $\mathcal{PT}$ system

###  II.1 EP in a non-Hermitian two-level $\mathcal{PT}$ system

To learn the nature of the singularity at an EP, we study a two-level
$\mathcal{PT}$ system described by the following Hamiltonian:

$\hat{H}_{\mathrm{NH}}=h_{x}\sigma_{x}+ih_{z}\sigma_{z}.$ (1)

For this two-level non-Hermitian system,
$\\{|\psi_{0}^{\mathrm{R}}\rangle\\}=\\{|\psi_{0}\rangle\\}=\\{\left|\uparrow\right\rangle,\left|\downarrow\right\rangle\\}$
denotes an _initial basis_ obeying orthogonal and normalization conditions,
i.e., $\langle\uparrow|\downarrow\rangle=0$ and
$\langle\uparrow|\uparrow\rangle=\langle\downarrow|\downarrow\rangle=1$.

At $h_{x}=h_{z}$, a typical spontaneous $\mathcal{PT}$-symmetry breaking
occurs: for the case of $h_{x}>h_{z},$ the energy levels
$\left|+\right\rangle$ and $\left|-\right\rangle$ are
$E_{\pm}=\pm\sqrt{h_{x}^{2}-h_{z}^{2}};$ For the case $h_{x}<h_{z},$ these two
energy levels are $E_{\pm}=\pm i\sqrt{h_{z}^{2}-h_{x}^{2}};$ For the case of
$h_{x}=h_{z}$, the system is at an EP with eigenstate coalescence and energy
degeneracy. To strictly characterize the coalescence of the eigenstates, we
define their state similarity,
$\Lambda=|\langle\tilde{\psi}_{+}^{\mathrm{R}}|\tilde{\psi}_{-}^{\mathrm{R}}\rangle|$,
where $|\tilde{\psi}_{+}^{\mathrm{R}}\rangle$ and
$|\tilde{\psi}_{-}^{\mathrm{R}}\rangle$ are eigenstates satisfying the self-
normalization conditions
$|\langle\tilde{\psi}_{+}^{\mathrm{R}}|\tilde{\psi}_{+}^{\mathrm{R}}\rangle|=1$
and
$|\langle\tilde{\psi}_{-}^{\mathrm{R}}|\tilde{\psi}_{-}^{\mathrm{R}}\rangle|=1$.
Fig. 1(a) and Fig. 1(b) show the two energy levels and the state similarity
$\Lambda$, respectively. It is obvious that both $E_{+}=E_{-}$ and $\Lambda=1$
hold at the EP ($h_{x}=h_{z}$).

Figure 1: (Color online) Physical properties of the EP ($h_{x}=h_{z}$) for the
non-Hermitian two-level model
$\hat{H}_{\mathrm{NH}}=h_{x}\sigma_{x}+ih_{z}\sigma_{z}$. $h_{x}$ is set to
unity. (a) The two energy levels, $E_{\pm}.$ At the EP ($h_{x}=h_{z}$), these
two energy levels become degenerate; (b) The state similarity
$\Lambda=|\langle\tilde{\psi}_{+}^{\mathrm{R}}|\tilde{\psi}_{-}^{\mathrm{R}}\rangle|$
for two eigenstates $|\tilde{\psi}_{+}^{\mathrm{R}}\rangle$ and
$|\tilde{\psi}_{-}^{\mathrm{R}}\rangle$ satisfying the self-normalization
conditions
$|\langle\tilde{\psi}_{+}^{\mathrm{R}}|\tilde{\psi}_{+}^{\mathrm{R}}\rangle|=1$
and
$|\langle\tilde{\psi}_{-}^{\mathrm{R}}|\tilde{\psi}_{-}^{\mathrm{R}}\rangle|=1$.
At the EP ($h_{x}=h_{z}$), we have $\Lambda=1$; (c) The self-normalization of
the matrix basis, $\frac{1}{\mathcal{N}}^{\mathrm{R}}\langle
1|1\rangle^{\mathrm{R}}$ and $\frac{1}{\mathcal{N}}^{\mathrm{R}}\langle
2|2\rangle^{\mathrm{R}}$, where $\mathcal{N}=^{\mathrm{R}}\langle
1|1\rangle^{\mathrm{R}}+^{\mathrm{R}}\langle 2|2\rangle^{\mathrm{R}}.$ At the
EP ($h_{x}=h_{z}$), we have $\frac{1}{\mathcal{N}}^{\mathrm{R}}\langle
1|1\rangle^{\mathrm{R}}=1$ and $\frac{1}{\mathcal{N}}^{\mathrm{R}}\langle
2|2\rangle^{\mathrm{R}}=0$; (d) The Bloch peach for the quantum states
$|\psi^{\mathrm{R}}\rangle=\cos\frac{\theta}{2}\left|1\right\rangle^{\mathrm{R}}+e^{i\phi}\sin\frac{\theta}{2}\left|2\right\rangle^{\mathrm{R}}$
at the EP ($h_{x}=h_{z}$). Here, we have ${}^{\mathrm{R}}\langle
1|2\rangle^{\mathrm{R}}=0$, ${}^{\mathrm{R}}\langle
1|1\rangle^{\mathrm{R}}=1$, and ${}^{\mathrm{R}}\langle
2|2\rangle^{\mathrm{R}}=e^{-2\beta^{M}}$.

###  II.2 Singular non-Hermitian similar transformation

Let us understand _why eigenvectors coalesce at EPs._

By performing a non-Hermitian similarity transformation
$\mathcal{\hat{S}}_{M}=e^{-\beta^{M}\sigma_{y}}$, we transform the original
non-Hermitian Hamiltonian $\hat{H}_{\mathrm{NH}}$ into a Hermitian/anti-
Hermitian one, $\hat{H}_{0}$ ($\hat{H}_{0}^{\dagger}=\hat{H}_{0}$ or
$\hat{H}_{0}^{\dagger}=-\hat{H}_{0}$), i.e.,

$\hat{H}_{\mathrm{NH}}\rightarrow\mathcal{\hat{S}}_{M}^{-1}\hat{H}_{\mathrm{NH}}\mathcal{\hat{S}}_{M}=\hat{H}_{0},$
(2)

where
$\beta^{M}=\left|\frac{1}{2}\ln\left|\frac{h_{x}+h_{z}}{h_{x}-h_{z}}\right|\right|$.
The eigenvalue of $\hat{H}_{0}$ is the same as that of
$\hat{H}_{\mathrm{NH}}.$ Under the non-Hermitian similarity transformation
$\mathcal{\hat{S}}_{M}$, the basis of $\hat{H}_{\mathrm{NH}}$ is
correspondingly changed, i.e.,

$\\{|\psi_{0}^{\mathrm{R}}\rangle\\}\rightarrow\\{|\psi^{\mathrm{R}}\rangle\\}=\\{\mathcal{\hat{S}}_{M}(\sigma_{y},\beta^{M})|\psi_{0}\rangle\\}.$
(3)

In the following, we refer to $\\{|\psi^{\mathrm{R}}\rangle\\}$ as the _matrix
basis_. If we select the initial basis $\\{|\psi_{0}^{\mathrm{R}}\rangle\\}$
to be the eigenstates of $\sigma_{y},$, i.e.,

$\displaystyle\\{|\psi_{0}^{\mathrm{R}}\rangle\\}$
$\displaystyle=\\{\left|1\right\rangle,\left|2\right\rangle\\}$
$\displaystyle=\\{\frac{1}{\sqrt{2}}(\left|\uparrow\right\rangle+i\left|\downarrow\right\rangle),\frac{1}{\sqrt{2}}(\left|\uparrow\right\rangle-i\left|\downarrow\right\rangle)\\},$
(4)

then the matrix basis $\\{|\psi^{\mathrm{R}}\rangle\\}$ becomes

$\\{|\psi^{\mathrm{R}}\rangle\\}=\\{e^{-\beta^{M}\sigma_{y}}|\psi_{0}\rangle\\}=\\{\left|1\right\rangle^{\mathrm{R}},\left|2\right\rangle^{\mathrm{R}}\\},$
(5)

where $\left|1\right\rangle^{\mathrm{R}}=\left|1\right\rangle$ and
$\left|2\right\rangle^{\mathrm{R}}=e^{-\beta^{M}}\left|2\right\rangle$. The
matrix basis
$\\{\left|1\right\rangle^{\mathrm{R}},\left|2\right\rangle^{\mathrm{R}}\\}$
obeys the orthogonal condition ${}^{\mathrm{R}}\langle
1|2\rangle^{\mathrm{R}}=0$ but does not obey the normalization conditions,
i.e., ${}^{\mathrm{R}}\langle 1|1\rangle^{\mathrm{R}}=1$ and
${}^{\mathrm{R}}\langle 2|2\rangle^{\mathrm{R}}=e^{-2\beta^{M}}$.

Approaching the EP, the non-Hermitian similarity transformation becomes
singular, i.e.,

$\mathcal{\hat{S}}_{M}(\sigma_{y},\beta^{M})=e^{-\beta^{M}\sigma_{y}}$ (6)

with
$\beta^{M}=\left|\frac{1}{2}\ln\left|\frac{h_{x}+h_{z}}{h_{x}-h_{z}}\right|\right|\rightarrow\infty$.
As a result, the matrix basis becomes defective, i.e.,

$\displaystyle\\{|\psi^{\mathrm{R}}\rangle\\}$
$\displaystyle=\\{\mathcal{\hat{S}}_{M}(\sigma_{y},\beta^{M}\rightarrow\infty)|\psi_{0}\rangle\\}$
$\displaystyle=\\{\left|1\right\rangle^{\mathrm{R}},\left|2\right\rangle^{\mathrm{R}}\\}\rightarrow\\{\left|1\right\rangle^{\mathrm{R}},0\\}.$
(7)

Here, the base $\left|2\right\rangle^{\mathrm{R}}$ _disappears_! To illustrate
the defectiveness of the matrix basis, in Fig. 1(c), we plot
$\frac{1}{\mathcal{N}}^{\mathrm{R}}\langle 1|1\rangle^{\mathrm{R}}$ and
$\frac{1}{\mathcal{N}}^{\mathrm{R}}\langle 2|2\rangle^{\mathrm{R}}$, where
$\mathcal{N}=^{\mathrm{R}}\langle 1|1\rangle^{\mathrm{R}}+^{\mathrm{R}}\langle
2|2\rangle^{\mathrm{R}}$ is a normalization factor. Near the EP,
$\frac{1}{\mathcal{N}}^{\mathrm{R}}\langle 2|2\rangle^{\mathrm{R}}$ becomes
zero (${}^{\mathrm{R}}\langle 2|2\rangle^{\mathrm{R}}\rightarrow 0$).

As a result, according to the defective matrix basis without
$\left|2\right\rangle^{\mathrm{R}},$ the two eigenstates
$|\tilde{\psi}_{+}^{\mathrm{R}}\rangle$ and
$|\tilde{\psi}_{-}^{\mathrm{R}}\rangle$ must coalesce into
$\left|1\right\rangle$, i.e.,

$\displaystyle|\tilde{\psi}_{+}^{\mathrm{R}}\rangle$
$\displaystyle=\frac{1}{\sqrt{1+e^{-2\beta^{M}}}}(\left|1\right\rangle^{\mathrm{R}}+\left|2\right\rangle^{\mathrm{R}})$
$\displaystyle\rightarrow\frac{1}{\sqrt{1+e^{-2\beta^{M}}}}\left|1\right\rangle$
(8)

and

$\displaystyle|\tilde{\psi}_{-}^{\mathrm{R}}\rangle$
$\displaystyle=\frac{1}{\sqrt{1+e^{-2\beta^{M}}}}(\left|1\right\rangle^{\mathrm{R}}-\left|2\right\rangle^{\mathrm{R}})$
$\displaystyle\rightarrow\frac{1}{\sqrt{1+e^{-2\beta^{M}}}}\left|1\right\rangle.$
(9)

Now, near the EP, where $\beta^{M}\rightarrow\infty$, the state similarity
$\Lambda=|\langle\tilde{\psi}_{+}^{\mathrm{R}}|\tilde{\psi}_{-}^{\mathrm{R}}\rangle|=\frac{1}{1+e^{-2\beta^{M}}}$
obviously approaches $1$. Thus, one can see that the singularity of the EP
arises from the defective matrix basis
$\\{\left|1\right\rangle^{\mathrm{R}},0\\}$ due to __ the singular non-
Hermitian similarity transformation
$\mathcal{\hat{S}}_{M}(\sigma_{y},\beta^{M}\rightarrow\infty)$.

###  II.3 Bloch peach for two-level states under the non-Hermitian similarity
transformation

To further illustrate the singularity of EPs, we use a geometric approach to
illustrate the deformation of the Bloch sphere under a singular non-Hermitian
similarity transformation
$\mathcal{\hat{S}}_{M}(\sigma_{y},\beta^{M}\rightarrow\infty)$.

For the Hermitian case, one may use a point on the Bloch sphere with SU(2)
rotation symmetry to represent an arbitrary quantum state of the two-level
system, i.e.,

$|\psi\rangle=\cos\frac{\theta}{2}\left|1\right\rangle+e^{i\varphi}\sin\frac{\theta}{2}\left|2\right\rangle.$
(10)

Here, $\theta\in[0,\pi]$ and $\varphi\in[0,2\pi]$ are real numbers, and the
radius $r$ of the Bloch sphere can be obtained as
$r=\langle\psi|\psi\rangle=1$.

For a two-level $\mathcal{PT}$ system, a quantum state under the non-Hermitian
similarity transformation $\mathcal{\hat{S}}_{M}(\sigma_{y},\beta^{M})$
becomes

$|\psi^{\mathrm{R}}\rangle=\mathcal{\hat{S}}_{M}|\psi\rangle=\cos\frac{\theta}{2}\left|1\right\rangle+e^{i\varphi}e^{-\beta^{M}}\sin\frac{\theta}{2}\left|2\right\rangle$
(11)

(with $\theta\in[0,\pi]$ and $\varphi\in[0,2\pi]$), and the radius $R$ of the
Bloch sphere can be obtained as

$R=\langle\psi^{\mathrm{R}}|\psi^{\mathrm{R}}\rangle=\cos^{2}\frac{\theta}{2}+e^{-2\beta^{M}}\sin^{2}\frac{\theta}{2}.$
(12)

Therefore, the original Bloch sphere changes into a peach-like closed surface
with residue U(1) symmetry along the y-axis (we call this the _Bloch peach_).
With increasing $\beta$, one pole of the Bloch peach moves upward. At the EP,
this pole touches the origin of the coordinate system. See the illustration of
the Bloch peach in the limit of $\beta^{M}\rightarrow\infty$ in Fig. 1(d).

##  III Protected non-Hermitian sub-systems

Before giving the definition of new type of EPs, we define _protected non-
Hermitian sub-systems_.

Due to special conditions of symmetry/topology, there may exist protected sub-
systems. To completely characterize the many-body model and its protected sub-
systems, we give their definitions, $\\{\hat{H}_{\mathrm{MB}},$
$B_{\mathrm{MB}}\\}$ and $\\{\hat{H}_{\mathrm{S}},$ $B_{\mathrm{S}}\\},$
respectively. Here, both $\hat{H}_{\mathrm{MB}}$ and $\hat{H}_{\mathrm{S}}$
are Hermitian Hamiltonians, and both
$B_{\mathrm{MB}}=\\{|\Psi_{j}\rangle,j=1,2,...,N\\}$ and
$B_{\mathrm{S}}=\\{\mathcal{P}|\Psi_{j}\rangle,j=1,2,...,N\\}=\\{|\mathrm{S}_{j}\rangle,j=1,2,...,K\\}$
are normal basis with $B_{\mathrm{S}}=\mathcal{P}B_{\mathrm{MB}}\in
B_{\mathrm{MB}}$ where $\mathcal{P}$ is a projected operator on the basis of
the many-body system. It is obvious that $K<N$.

When one adds a non-Hermitian perturbation $i\delta\hat{H}$ on the many-body
model, the total Hamiltonian becomes non-Hermitian, i.e.,

$\hat{H}_{\mathrm{MB}}\rightarrow\hat{H}_{\mathrm{NH-
MB}}=\hat{H}_{\mathrm{MB}}+i\delta\hat{H}.$ (13)

Because the basis has no changing, the many-body model is denoted by
$\\{\hat{H}_{\mathrm{NH-MB}},$ $B_{\mathrm{MB}}\\}$. In general, for a non-
Hermitian sub-system, the biorthogonal set for the basis is defined by
$|\Psi_{j}^{\mathrm{R}}\rangle$ and $|\Psi_{j}^{\mathrm{L}}\rangle$
($j=1,2,...,K$), i.e.,

$\hat{H}_{\mathrm{NH-
MB}}|\Psi_{j}^{\mathrm{R}}\rangle=E_{j}|\Psi_{j}^{\mathrm{R}}\rangle,$ (14)

and

$\hat{H}_{\mathrm{NH-
MB}}^{\dagger}|\Psi_{j}^{\mathrm{L}}\rangle=(E_{j})^{\ast}|\Psi_{j}^{\mathrm{L}}\rangle,$
(15)

and $\langle\Psi_{j}^{\mathrm{L}}|\Psi_{j}^{\mathrm{R}}\rangle=1$ where $j$ is
state index.

We then assume the non-Hermitian terms are perturbation and don’t change the
existence of the protected sub-system and use
$\\{\hat{H}_{\mathrm{NH-S}},B_{\mathrm{S}}\\}=\\{\mathcal{P}\hat{H}_{\mathrm{MB}}\mathcal{P}^{-1},\mathcal{P}B_{\mathrm{MB}}\\}$
to describe the protected (non-Hermitian) sub-system. Here,
$\hat{H}_{\mathrm{NH-S}}=\mathcal{P}\hat{H}_{\mathrm{MB}}\mathcal{P}^{-1}$ is
the effective Hamiltonian of the protected sub-system. Under the projected
operation $\mathcal{P},$ the basis of the protected sub-system is obtained as

$\displaystyle B_{\mathrm{MB}}$
$\displaystyle=\\{|\Psi_{j}^{\mathrm{R}}\rangle\\}$ (16)
$\displaystyle\rightarrow
B_{\mathrm{S}}=\mathcal{P}B_{\mathrm{MB}}=\\{\mathcal{P}|\Psi_{j}^{\mathrm{R}}\rangle\\}=\\{|\psi_{j}^{\mathrm{R}}\rangle\\}.$

In addition, we show the method to calculate
$\mathcal{\hat{H}}_{\mathrm{NH}}.$ Based on the basis
$S_{\mathrm{NH}}=\\{|\psi_{j}^{\mathrm{R}}\rangle\\}$, an effective
Hamiltonian of protected sub-system is derived as

$\hat{H}_{\mathrm{NH-S}}={\displaystyle\sum\limits_{ij}}h_{IJ}$ (17)

where

$h_{IJ}=\left\langle\psi_{j}^{\mathrm{L}}\right|\hat{H}_{\mathrm{NH}}\left|\psi_{j}^{\mathrm{R}}\right\rangle,\text{
}I,J=1,2,...,K.$ (18)

In particular, the basis of protected sub-sysrem
$B_{\mathrm{S}}=\\{|\psi_{j}^{\mathrm{R}}\rangle\\}$ is always _abnormal._ For
example, it does’t follow usual normalization
$\langle\psi_{j}^{\mathrm{R}}|\psi_{j}^{\mathrm{R}}\rangle\neq 1$. In this
paper, we only focus on the case of $K=2$.

##  IV General EPs – universal feature and classification

We next develop a universal theory for an arbitrary non-Hermitian protected
two-level sub-system with a singularity and introduce the concept of _general
EPs_. This phenomenon always occurs in subsystems of certain non-Hermitian
models, for example, the defective edge states of a non-Hermitian topological
insulator, the defective degenerate ground states in non-Hermitian systems
with spontaneous symmetry breaking, or the topologically protected degenerate
ground states in intrinsic topological orders.

A general non-Hermitian protected two-level sub-system is described by
$\\{\hat{H}_{\mathrm{NH}},\\{|\psi_{0}^{\mathrm{R}}\rangle\\}\\}$. The
Hamiltonian is

$\hat{H}_{\mathrm{NH}}=h_{0}+\vec{h}\cdot\vec{\sigma},$ (19)

where $h_{0}$ is a complex number, $\vec{h}=(h^{x},$ $h^{y},$ $h^{z})$ is a
complex vector and
$\vec{\sigma}=(\hat{\sigma}_{x},\hat{\sigma}_{y},\hat{\sigma}_{z})$ is the
vector of Pauli matrices.
$\\{|\psi_{0}^{\mathrm{R}}\rangle\\}=\\{\mathcal{\hat{S}}_{B}|\psi_{0}\rangle\\}$
denotes the initial basis where
$\mathcal{\hat{S}}_{B}=e^{-\beta^{B}\cdot\vec{\sigma}^{B}}$ is a non-Hermitian
similarity transformation with a real positive $\beta^{B}$ and
$|\psi_{0}\rangle$ is a normal basis consisting of eigenstates of
$\vec{\sigma}^{B}$ (or
$\vec{\sigma}^{B}|\psi_{0}\rangle=\pm|\psi_{0}\rangle$).

###  IV.1 Representation under matrix basis

To identify the universal feature of general EPs, we transform the original
non-Hermitian Hamiltonian $\hat{H}_{\mathrm{NH}}$ into a Hermitian/anti-
Hermitian one and derive its representation under matrix basis,
$\\{\hat{H}_{0},\\{|\psi^{\mathrm{R}}\rangle\\}\\}$.

First, we transform the non-Hermitian Hamiltonian
$\hat{H}_{\mathrm{NH}}=h_{0}+\vec{h}\cdot\vec{\sigma}$ into

$\hat{H}_{\mathrm{NH}}=h_{0}+(\operatorname{Re}\vec{h})\vec{\sigma}_{\operatorname{Re}}+i(\operatorname{Im}\vec{h})\vec{\sigma}_{\operatorname{Im}},$
(20)

where $\vec{\sigma}_{\operatorname{Re}}$ and
$\vec{\sigma}_{\operatorname{Im}}$
($\vec{\sigma}_{\operatorname{Re}}\cdot\vec{\sigma}_{\operatorname{Re}}=1$ and
$\vec{\sigma}_{\operatorname{Im}}\cdot\vec{\sigma}_{\operatorname{Im}}=1$) are
the Pauli matrices corresponding to the real and imaginary parts of $\vec{h}$,
respectively.

Second, we divide
$i(\operatorname{Im}\vec{h})\vec{\sigma}_{\operatorname{Im}}$ into two parts,
$i(\operatorname{Im}\vec{h})^{A}\vec{\sigma}_{\operatorname{Im}}^{A}$ and
$i(\operatorname{Im}\vec{h})^{C}\vec{\sigma}_{\operatorname{Im}}^{C}$, with
$\left[\vec{\sigma}_{\operatorname{Im}}^{C},\vec{\sigma}_{\operatorname{Re}}\right]=0$
and
$\\{\vec{\sigma}_{\operatorname{Im}}^{A},\vec{\sigma}_{\operatorname{Re}}\\}=0.$
Then, the non-Hermitian Hamiltonian becomes

$\hat{H}_{\mathrm{NH}}=h_{0}+\vec{h}_{\operatorname{Re}}^{\prime}\cdot\vec{\sigma}_{\operatorname{Re}}+i\vec{h}_{\operatorname{Im}}^{\prime}\cdot\vec{\sigma}_{\operatorname{Im}}^{A},$
(21)

where
$\vec{h}_{\operatorname{Re}}^{\prime}=(\operatorname{Re}\vec{h})+i(\operatorname{Im}\vec{h})^{C}$
and $\vec{h}_{\operatorname{Im}}^{\prime}=(\operatorname{Im}\vec{h})^{A}$.

Thirdly, under a non-Hermitian similarity transformation
$\mathcal{\hat{S}}_{M}(\vec{\sigma}^{M},\beta^{M})$, we transform the original
NH Hamiltonian $\hat{H}_{\mathrm{NH}}$ into a Hermitian/anti-Hermitian one,
i.e.,

$\displaystyle\hat{H}_{\mathrm{NH}}$
$\displaystyle\rightarrow\mathcal{\hat{S}}_{M}^{-1}(\vec{\sigma}^{M},\beta^{M})\hat{H}_{\mathrm{NH}}\mathcal{\hat{S}}_{M}(\vec{\sigma}^{M},\beta^{M})$
$\displaystyle=\hat{H}_{0}=h_{0}+\sqrt{\vec{h}^{2}}\vec{\sigma}_{\operatorname{Re}},$
(22)

where

$\mathcal{\hat{S}}_{M}(\vec{\sigma}^{M},\beta^{M})=e^{-\beta^{M}\cdot\vec{\sigma}^{M}}$
(23)

with
$\vec{\sigma}^{M}=\frac{1}{2i}[\vec{\sigma}_{\operatorname{Im}}^{A},\vec{\sigma}_{\operatorname{Re}}]$
and
$\beta^{M}=\left|\frac{1}{2}\ln\frac{\left|\vec{h}_{\operatorname{Re}}^{\prime}+\vec{h}_{\operatorname{Im}}^{\prime}\right|}{\left|\vec{h}_{\operatorname{Re}}^{\prime}-\vec{h}_{\operatorname{Im}}^{\prime}\right|}\right|$.
As a result, the eigenvalue of $\hat{H}_{0}$ is the same as that of
$\hat{H}_{\mathrm{NH}}$, i.e.,

$E_{\pm}=h_{0}\pm\sqrt{\vec{h}^{2}}.$ (24)

For the case of $\vec{h}^{2}>0,$ $\hat{H}_{0}$ is a Hermitian Hamiltonian,
$\hat{H}_{0}=\hat{H}_{0}^{\dagger}$, whose energy levels
$E_{\pm}=h_{0}\pm\left|\vec{h}\right|$ are real; for the case of
$\vec{h}^{2}<0,$ $\hat{H}_{0}$ is an anti-Hermitian Hamiltonian,
$\hat{H}_{0}=-\hat{H}_{0}^{\dagger}$, whose energy levels $E_{\pm}=h_{0}\pm
i\left|\vec{h}\right|$ are an imaginary pair.

Finally, one can see that under the non-Hermitian similarity transformation,
the initial basis $\\{|\psi_{0}^{\mathrm{R}}\rangle\\}$ is correspondingly
changed into the unique matrix basis $\\{|\psi^{\mathrm{R}}\rangle\\}$, i.e.,

$\displaystyle\\{|\psi_{0}^{\mathrm{R}}\rangle\\}$
$\displaystyle\rightarrow\\{|\psi^{\mathrm{R}}\rangle\\}=\\{\left|1\right\rangle^{\mathrm{R}},\left|2\right\rangle^{\mathrm{R}}\\}$
$\displaystyle=\\{\mathcal{\hat{S}}_{M}(\vec{\sigma}^{B},\beta^{B})|\psi_{0}^{\mathrm{R}}\rangle\\}$
(25)
$\displaystyle=\\{\mathcal{\hat{S}}_{M}(\vec{\sigma}^{M},\beta^{M})\mathcal{\hat{S}}_{B}(\vec{\sigma}^{B},\beta^{B})|\psi_{0}\rangle\\}.$

In addition, to describe the same non-Hermitian two-level system, one can also
use another representation based on the normal basis,
$\\{\hat{H}_{\mathrm{NH}}^{\beta^{B}},\\{|\psi_{0}\rangle\\}\\}$, i.e.,

$\hat{H}_{\mathrm{NH}}^{\beta^{B}}=\mathcal{\hat{S}}_{B}^{-1}\hat{H}_{\mathrm{NH}}\mathcal{\hat{S}}_{B}$
(26)

and
$\\{|\psi_{0}\rangle\\}=\\{\mathcal{\hat{S}}_{B}^{-1}|\psi_{0}^{\mathrm{R}}\rangle\\}$.
To clearly show the relationship among the three representations for the
Hamiltonians of the same non-Hermitian two-level system,
$\\{\hat{H}_{\mathrm{NH}},\\{|\psi_{0}^{\mathrm{R}}\rangle\\}\\}$ under the
initial basis, $\\{\hat{H}_{0},\\{|\psi^{R}\rangle\\}\\}\\}$ under the matrix
basis, and $\\{\hat{H}_{\mathrm{NH}}^{\beta^{B}},\\{|\psi_{0}\rangle\\}\\}$
under the normal basis, we plot Fig. 2. $\mathcal{\hat{S}}_{M}$,
$\mathcal{\hat{S}}_{B}$, and $\mathcal{\hat{S}}_{M}\mathcal{\hat{S}}_{B}$ are
different non-Hermitian similarity transformations relating these
representations.

Figure 2: (Color online) Schematic diagram of the relationship among three
equivalent representations for a two-level system:
$\\{\hat{H}_{\mathrm{NH}},\\{|\psi_{0}^{\mathrm{R}}\rangle\\}\\}$, under the
initial basis; $\\{\hat{H}_{0},\\{|\psi^{R}\rangle\\}\\}$, under the matrix
basis; and $\\{\hat{H}_{\mathrm{NH}}^{\beta^{B}},\\{|\psi_{0}\rangle\\}\\}$,
under the normal basis. Here, $\hat{H}_{\mathrm{NH}}$ is the original non-
Hermitian Hamiltonian with initial basis
$\\{|\psi_{0}^{\mathrm{R}}\rangle\\}$.
$\\{|\psi^{R}\rangle\\}=\\{\mathcal{\hat{S}}_{M}|\psi_{0}^{\mathrm{R}}\rangle\\}=\\{\mathcal{\hat{S}}_{M}\mathcal{\hat{S}}_{B}|\psi_{0}\rangle\\}$
is a matrix basis that classifies different types of GEPs (M-GEPs, B-GEPs, and
H-GEPs).
$\hat{H}_{\mathrm{NH}}^{\beta^{B}}=\mathcal{\hat{S}}_{B}\hat{H}_{\mathrm{NH}}\mathcal{\hat{S}}_{B}^{-1}$
classifies the subclasses of B-GEPs.

###  IV.2 Definition of general EPs

We point out that the universal feature of arbitrary non-Hermitian two-level
systems with singularities is _defectiveness of the matrix basis_
$\\{|\psi^{\mathrm{R}}\rangle\\}$__ as $\beta^{M}\rightarrow\infty$ and/or
$\beta^{B}\rightarrow\infty$_rather than energy degeneracy or state
coalescence_. Thus, we define _general EPs_ as follows:

Definition – general EPs (GEPs): For an arbitrary non-Hermitian two-level
system, GEPs exist if and only if the matrix basis
$\\{|\psi^{\mathrm{R}}\rangle\\}=\\{\left|1\right\rangle^{\mathrm{R}},\left|2\right\rangle^{\mathrm{R}}\\}$
becomes defective, i.e., $\frac{1}{\mathcal{N}}^{\mathrm{R}}\langle
1|1\rangle^{\mathrm{R}}=1$ and ${}^{\mathrm{R}}\langle
2|2\rangle^{\mathrm{R}}=0$ or $\frac{1}{\mathcal{N}}^{\mathrm{R}}\langle
2|2\rangle^{\mathrm{R}}=1$ and ${}^{\mathrm{R}}\langle
1|1\rangle^{\mathrm{R}}=0$. Here, $\mathcal{N}=^{\mathrm{R}}\langle
1|1\rangle^{\mathrm{R}}+^{\mathrm{R}}\langle 2|2\rangle^{\mathrm{R}}$ is a
normalization factor.

As a result, at GEPs, the two energy levels may not necessarily be degenerate,
i.e., $E_{+}=E_{-}$ and $E_{+}\neq E_{-}$ are both allowed; the two
eigenstates $\left|\tilde{\psi}_{+}^{\mathrm{R}}\right\rangle$ and
$\left|\tilde{\psi}_{-}^{\mathrm{R}}\right\rangle$ may not necessarily
coalesce, i.e., $\Lambda\equiv 1$ and $\Lambda\equiv 0$ are both allowed. In
addition, in the form of Jordan block matrix, at GEPs the algebra multiplier
of a matrix may be same to its geometric multiplier.

###  IV.3 Classification of general EPs

By considering different behaviors of defective bases, we can classify GEPs
for an arbitrary non-Hermitian two-level system. Depending on the behavior of
the defective matrix basis
$\\{|\psi^{\mathrm{R}}\rangle\\}=\\{\mathcal{\hat{S}}_{M}(\vec{\sigma}^{M},\beta^{M})\mathcal{\hat{S}}_{B}(\vec{\sigma}^{B},\beta^{B})|\psi_{0}\rangle\\},$
there are _three_ types of GEPs: _matrix-type GEPs_ (or M-GEPs), with
$\beta^{M}\rightarrow\infty$ and $\beta^{B}\nrightarrow\infty$; _basis-type
GEPs_ (or B-GEPs), with $\beta^{M}\nrightarrow\infty$ and
$\beta^{B}\rightarrow\infty$; and _hybrid-type GEPs_ (or H-GEPs), with
$\beta^{M}\rightarrow\infty$ and $\beta^{B}\rightarrow\infty$.

In addition, there are _two_ different classes of B-GEPs, i.e., IB-GEPs and
IIB-GEPs. For IB-GEPs, the two eigenstates will never coalesce with each
other, i.e., $\Lambda\equiv 0$; for IIB-GEPs, the two eigenstates will always
coalesce each other, i.e., $\Lambda\equiv 1$.

To show the physical properties of these two classes of B-GEPs, we perform a
non-Hermitian similarity transformation $\mathcal{\hat{S}}_{B}^{-1}$ on the
initial basis $\\{|\psi_{0}^{\mathrm{R}}\rangle$ and obtain a representation
under the normal basis, i.e.,

$\\{|\psi_{0}^{\mathrm{R}}\rangle\\}\rightarrow\\{\mathcal{\hat{S}}_{B}^{-1}|\psi_{0}^{\mathrm{R}}\rangle\\}=\\{|\psi_{0}\rangle\\}.$
(27)

Correspondingly, the original Hamiltonian $\hat{H}_{\mathrm{NH}}$ is
transformed into

$\hat{H}_{\mathrm{NH}}^{\beta^{B}}=\mathcal{\hat{S}}_{B}\hat{H}_{\mathrm{NH}}\mathcal{\hat{S}}_{B}^{-1}.$
(28)

There are _two_ possibilities for $\hat{H}_{\mathrm{NH}}^{\beta^{B}}$, which
correspond to the two classes of B-GEPs: one possibility is that all elements
of $\hat{H}_{\mathrm{NH}}^{\beta^{B}}$ are finite, in which case the
eigenstates do not coalesce (or $\Lambda\equiv 0$), and the other is one or
more elements diverge, i.e.,
$(\hat{H}_{\mathrm{NH}}^{\beta^{B}})_{ij}\rightarrow\infty$, in which case the
eigenstates coalesce (or $\Lambda\equiv 1$). It is obvious that at IB-GEPs the
algebra multiplier of the Hamiltonian $\hat{H}_{\mathrm{NH}}$ is equal to its
geometric multiplier.

Therefore, there exists a quantum phase transition between IB-GEPs (the region
without eigenstate coalescence) and IIB-GEPs (the region with eigenstate
coalescence). Let us give a simple explanation of this fact.

For IB-GEPs, the Hamiltonian $\hat{H}_{\mathrm{NH}}$ commutes with
$\vec{\sigma}^{B}$, i.e., $[\hat{H}_{\mathrm{NH}},\vec{\sigma}^{B}]=0$. As a
result, the Hamiltonian $\hat{H}_{\mathrm{NH}}$ must be written as
$\lambda\vec{\sigma}^{B}$ with $\lambda\neq 0$. Under the non-Hermitian
similarity transformation
$\mathcal{\hat{S}}_{B}^{-1}=e^{\beta^{B}\vec{\sigma}^{B}}$, we have

$\hat{H}_{\mathrm{NH}}^{\beta^{B}}=\mathcal{\hat{S}}_{B}^{-1}\hat{H}_{\mathrm{NH}}\mathcal{\hat{S}}_{B}=\lambda\vec{\sigma}^{B}.$
(29)

Now, the matrix basis becomes normal, i.e.,

$\\{|\psi_{0}^{\mathrm{R}}\rangle\\}\rightarrow\\{\mathcal{\hat{S}}_{B}^{-1}|\psi_{0}^{\mathrm{R}}\rangle\\}=\\{|\psi_{0}\rangle\\}.$
(30)

Here, the $|\psi_{0}\rangle$ are eigenstates of $\vec{\sigma}^{B}$, or
$\vec{\sigma}^{B}|\psi_{0}\rangle=\pm|\psi_{0}\rangle$. Because the
eigenstates $|\psi_{\pm}^{\mathrm{R}}\rangle$ of
$\hat{H}_{\mathrm{NH}}=\lambda\vec{\sigma}^{B}$ are also those of
$|\psi_{0}\rangle$, the state similarity of $|\psi_{\pm}^{\mathrm{R}}\rangle$
must be zero, i.e.,

$\Lambda=|\langle\tilde{\psi}_{+}^{\mathrm{R}}|\tilde{\psi}_{-}^{\mathrm{R}}\rangle|=0.$
(31)

On the other hand, for IIB-GEPs, the Hamiltonian $\hat{H}_{\mathrm{NH}}$ does
not commute with $\vec{\sigma}^{B},$ i.e.,
$[\hat{H}_{\mathrm{NH}},\vec{\sigma}^{B}]\neq 0$. The Hamiltonian
$\hat{H}_{\mathrm{NH}}$ must be written as

$\hat{H}_{\mathrm{NH}}=\lambda\vec{\sigma}^{B}+\eta(\vec{\sigma}^{B})^{\perp}$
(32)

with $\lambda^{2}+\eta^{2}\neq 0$ and
$\\{(\vec{\sigma}^{B})^{\perp},\vec{\sigma}^{B}\\}=0$. Now, one element of
$\hat{H}_{\mathrm{NH}}^{\beta^{B}}=\mathcal{\hat{S}}_{B}^{-1}\hat{H}_{\mathrm{NH}}\mathcal{\hat{S}}_{B}$
diverges. On the basis of $\\{|\psi_{0}\rangle\\}$, the divergent term is
proportional to $(\begin{array}[c]{cc}0&1\\\ 0&0\end{array})$ or
$(\begin{array}[c]{cc}0&0\\\ 1&0\end{array})$. As a result, the Hamiltonian
$\hat{H}_{\mathrm{NH}}^{\beta^{B}}=\mathcal{\hat{S}}_{B}^{-1}\hat{H}_{\mathrm{NH}}\mathcal{\hat{S}}_{B}$
is dominated by this divergent term, and we can ignore other terms. In this
case, the state similarity of $|\psi_{\pm}^{\mathrm{R}}\rangle$ must be $1$,
i.e.,

$\Lambda=|\langle\tilde{\psi}_{+}^{\mathrm{R}}|\tilde{\psi}_{-}^{\mathrm{R}}\rangle|=1.$
(33)

Without sudden changing the energy levels and the defectiveness of matrix
basis, this quantum phase transition is always “ _hidden_ ”.

##  V Example: 1D nonreciprocal Su–Schrieffer–Heeger model

###  V.1 The model

In this section, we take the 1D nonreciprocal Su–Schrieffer–Heeger (SSH) model
as an example to illustrate the different types of GEPs for its topologically
protected sub-space of two edge states.

The Bloch Hamiltonian for the nonreciprocal SSH model under periodic boundary
conditions (PBC) is given by

$\displaystyle\hat{H}_{\mathrm{PBC}}(k)$
$\displaystyle=\sum_{k}c_{k}^{\dagger}\tau_{x}\left(t_{1}+t_{2}\cos
k\right)c_{k}$ (34)
$\displaystyle+\sum_{k}c_{k}^{\dagger}\tau_{y}\left(t_{2}\sin
k+i\gamma\right)c_{k}+i\varepsilon\sum_{k}c_{k}^{\dagger}\tau_{z}c_{k},$

where
$c_{k}^{\dagger}=(c_{k,\mathrm{A}}^{\dagger},c_{k,\mathrm{B}}^{\dagger})$; the
$\tau_{i}$ are the Pauli matrices acting on the (A or B) sublattice subspaces;
$t_{1}$ and $t_{2}$ describe the intracell and intercell hopping strengths,
respectively; $\gamma$ describes unequal intracell hopping; and $\varepsilon$
denotes the strength of an imaginary staggered potential on the two
sublattices. $t_{1}$, $t_{2}$, $\gamma$, and $\varepsilon$ are all real. In
this paper, we set $t_{2}=1$.

Under a non-Hermitian similarity transformation
$\mathcal{\hat{S}}_{\mathrm{NHP}}$, the physics properties of the 1D
nonreciprocal SSH model under open boundary conditions (OBC) are characterized
by $\hat{H}_{\mathrm{OBC}}(k)$ rather than $\hat{H}_{\mathrm{PBC}}(k)$Yao .
Here, the non-Hermitian similarity transformation
$\mathcal{\hat{S}}_{\mathrm{NHP}}$ given by

$c_{k}^{\dagger}\rightarrow\tilde{c}_{k}^{\dagger}=c_{k-iq_{0}}^{\dagger}=\mathcal{\hat{S}}_{\mathrm{NHP}}c_{k}^{\dagger}$
(35)

or

$c_{n}^{\dagger}\rightarrow\tilde{c}_{n}^{\dagger}=e^{-q_{0}(n-1)}c_{n}^{\dagger}=\mathcal{\hat{S}}_{\mathrm{NHP}}c_{n}^{\dagger}$
(36)

where $e^{q_{0}}=\sqrt{\frac{t_{1}-\gamma}{t_{1}+\gamma}}.$ Consequently, the
effective hopping parameters become
$\bar{t_{1}}=\sqrt{(t_{1}+\gamma)(t_{1}-\gamma)}$ and $\bar{t_{2}}=t_{2}$.

To characterize the topological properties of the non-Hermitian topological
system, the non-Bloch topological invariant $\bar{w}$ of
$\hat{H}_{\mathrm{OBC}}(k)$ is introduced, i.e.,

$\bar{w}=\frac{1}{2\pi}\int_{-\pi}^{\pi}\partial\bar{\phi}(k)dk$ (37)

where $\bar{\phi}(k)=\tan^{-1}(\bar{h}_{y}/\bar{h}_{x})$ and
$\bar{h}_{x}=\bar{t}_{1}+\bar{t}_{2}\cos k$, $\bar{h}_{y}=\bar{t}_{2}\sin k$.
In the region of $\left|\bar{t}_{1}\right|<\left|\bar{t}_{2}\right|$ and
$\bar{w}=1$, the system is a topological insulator (the gray region in Fig.
3(a)); in the region of $\left|\bar{t}_{1}\right|>\left|\bar{t}_{2}\right|$
and $\bar{w}=0$, the system is a normal insulator (the white region in Fig.
3(a)). A quantum phase transition occurs at
$\left|\bar{t}_{1}\right|=\left|\bar{t}_{2}\right|$, where the bulk energy gap
under OBC is closed.

###  V.2 Two-level systems from two edge states in topological phase

In the topological phase with $\bar{w}=1$, there exist two edge states
$\left|\psi_{1}^{\mathrm{R}}\right\rangle$ and
$\left|\psi_{2}^{\mathrm{R}}\right\rangle$. These two edge states make up a
topologically protected subspace denoted by
$\\{\hat{H}_{\mathrm{NH}},\\{|\psi_{0}^{\mathrm{R}}\rangle\\}\\}$. Under the
biorthogonal set, the initial basis $\\{|\psi_{0}^{\mathrm{R}}\rangle\\}$ is
$\\{\left|\psi_{1}^{\mathrm{R}}\right\rangle,\
\left|\psi_{2}^{\mathrm{R}}\right\rangle\\}$. The effective Hamiltonian
$\hat{H}_{\mathrm{NH}}$ is written as

$\hat{H}_{\mathrm{NH}}=\left(\begin{array}[c]{cc}h_{11}&h_{12}\\\
h_{21}&h_{22}\end{array}\right),$ (38)

where
$h_{ij}=\left\langle\psi_{i}^{\mathrm{L}}\right|\hat{H}\left|\psi_{j}^{\mathrm{R}}\right\rangle$,
$i,j=1,2$. Through straightforward calculationsRan , we can analytically
obtain the effective Hamiltonian of the two edge states as

$\hat{H}_{\mathrm{NH}}=\bar{\Delta}\sigma^{x}+i\varepsilon\sigma^{z},$ (39)

where
$\bar{\Delta}=\frac{(\bar{t}_{2}^{2}-\bar{t}_{1}^{2})}{\bar{t}_{2}}(\frac{\bar{t}_{1}}{\bar{t}_{2}})^{N}$.
The two energy levels for the two eigenstates $|\psi_{+}^{\mathrm{R}}\rangle$
and $|\psi_{-}^{\mathrm{R}}\rangle$ are
$E=\pm\sqrt{\bar{\Delta}^{2}-\varepsilon^{2}}$.

Figure 3: (Color online) (a) is the global phase diagram for the 1D
nonreciprocal SSH model under open boundary conditions ($N=20$ and
$\varepsilon=0.001$): four red dots $\rightarrow$ M-GEPs; solid black lines
$\rightarrow$ H-GEPs; yellow region $\rightarrow$ IB-GEPs; gray region
$\rightarrow$ IIB-GEPs; dashed lines $\rightarrow$ hidden quantum phase
transition between IB-GEPs and IIB-GEPs. (b1), (b2) and (b3) show examples of
the Bloch peaches for M-GEPs, B-GEPs, and H-GEPs, respectively.

###  V.3 General EPs

Then, based on
$\\{\hat{H}_{\mathrm{NH}},\\{|\psi_{0}^{\mathrm{R}}\rangle\\}\\}$, we study
the GEPs for the two edge states in the 1D nonreciprocal SSH model.

In topological phase with $\bar{w}=1$ (or
$\left|\bar{t}_{1}\right|<\left|\bar{t}_{2}\right|$), we have GEPs for the two
edge states except for the Hermitian/anti-Hermitian cases at $\gamma=0$ or
$t_{1}=0$. Fig. 3(a) is an illustration of the global phase diagram for GEPs:
the four red dots correspond to M-GEPs, the solid black lines correspond to
H-GEPs, the yellow region corresponds to IB-GEPs, and the gray region
corresponds to IIB-GEPs. The Bloch peaches for different types of GEPs are
illustrated in Fig. 3(b1) (an example for M-GEPs), Fig. 3(b2) (an example for
B-GEPs), and Fig. 3(b3) (an example for H-GEPs).

Let us separately discuss the different types of GEPs one-by-one.

Figure 4: (Color online) (a) is the energy levels of two edge states; (b) is
the state similarity of two edge states,
$\Lambda=|\langle\tilde{\psi}_{+}^{\mathrm{R}}|\tilde{\psi}_{-}^{\mathrm{R}}\rangle|$.
(a) and (b) are all for the case of $N=50$ , $t_{2}=1$, $\varepsilon=0.001$
and $t_{1}=0$. The two energy levels become degenerate and the state
similarity becomes $1$ at the gray dotted line ($\gamma=\pm 0.86$), where the
M-GEP occurs.

####  V.3.1 M-GEPs

Firstly, we consider the case of M-GEPs by setting $\varepsilon$ to a purely
real value and $\gamma=0$. Now, the initial basis becomes normal, i.e.,

$\\{|\psi_{0}^{\mathrm{R}}\rangle\\}=\\{|\psi_{0}\rangle\\}$ (40)

with $\mathcal{\hat{S}}_{B}(\vec{\sigma}^{B},\beta^{B})=1$. The Hamiltonian of
the effective two-level model is reduced to

$\hat{H}_{\mathrm{NH}}=i\varepsilon\sigma^{z}+\Delta_{0}\sigma^{x},$ (41)

where
$\Delta_{0}=\frac{\left(t_{2}^{2}-t_{1}^{2}\right)}{t_{2}}(-\frac{t_{1}}{t_{2}})^{N}$.

A spontaneous $\mathcal{PT}$-symmetry-breaking transition occurs at
$\left|\varepsilon\right|=\Delta_{0}.$ The energy levels become degenerate,
shown as Fig. 4(a) i.e.,

$E_{\pm}=\pm\sqrt{\varepsilon^{2}-\Delta_{0}^{2}}\rightarrow 0,$ (42)

and the non-Hermitian similarity transformation becomes singular, i.e.,

$\mathcal{\hat{S}}_{M}=e^{-\beta^{M}\cdot\sigma^{y}}$ (43)

with
$\beta^{M}=\left|\frac{1}{2}\ln\left|\frac{\Delta_{0}+\varepsilon}{\Delta_{0}-\varepsilon}\right|\right|\rightarrow\infty.$
We have an M-GEP with the following defective matrix basis:

$\\{|\psi^{\mathrm{R}}\rangle\\}=\\{\mathcal{\hat{S}}_{M}(\sigma^{y},\beta^{M}\rightarrow\infty)|\psi_{0}\rangle\\}.$
(44)

The state similarity
$\Lambda=|\langle\tilde{\psi}_{+}^{\mathrm{R}}|\tilde{\psi}_{-}^{\mathrm{R}}\rangle|=1$
indicates the coalescence of the two edge states, shown as the Fig. 4(b). From
the illustration of Fig. 4, we can obtain that at M-GEPs two energy levels
become degenerate and two energy states coalesce ($E_{+}\neq E_{-}$,
$\Lambda=1$), shown as the gray dotted line at $\gamma=\pm 0.86$. In addition,
as shown in Fig. 3(b1), the Bloch peach has a symmetric axis along the
y-direction.

Figure 5: (Color online) (a) is the energy levels of two edge states; (b) is
the state similarity of two edge states,
$\Lambda=|\langle\tilde{\psi}_{+}^{\mathrm{R}}|\tilde{\psi}_{-}^{\mathrm{R}}\rangle|$.
(a) and (b) are all for the case of $N=50$, $t_{2}=1$, $\varepsilon=0.001$ and
$\gamma=0.5$. The two energy levels become degenerate ($E_{+}=E_{-}$) at the
gray dotted line ($t_{1}=1.03$), where the H-GEP occurs with $\Lambda=1$; The
orange dotted line ($t_{1}=0.38$) denoted the phase transition between IB-GEPs
($E_{+}\neq E_{-}$, $\Lambda=0$) and IIB-GEPs ($E_{+}\neq E_{-}$,
$\Lambda=1$).

####  V.3.2 B-GEPs

Secondly, we consider the case of B-GEPs by setting
$\left|\bar{\Delta}\right|\neq\left|\varepsilon\right|$ with $t_{1}\neq 0$ and
$\gamma\neq 0$. Now, the initial basis becomes defective, i.e.,

$\\{|\psi_{0}^{\mathrm{R}}\rangle\\}=\\{\mathcal{\hat{S}}_{B}|\psi_{0}\rangle\\}$

with $\mathcal{\hat{S}}_{B}=e^{-\beta^{B}\cdot\sigma_{z}}$ and
$\beta^{B}=Nq_{0}.$ Here,
$q_{0}=\frac{1}{2}\ln(\frac{t_{1}-\gamma}{t_{1}+\gamma})$ is an imaginary wave
vector that characterizes the non-Hermitian skin effect. In the thermodynamic
limit $N\rightarrow\infty$, we have a B-GEP with the following defective
matrix basis:

$\\{|\psi^{\mathrm{R}}\rangle\\}=\\{|\psi_{0}^{\mathrm{R}}\rangle\\}=\\{\mathcal{\hat{S}}_{B}(\sigma^{z},\beta^{B}\rightarrow\infty)|\psi_{0}\rangle\\}.$
(45)

In Fig. 3(a), except for the black lines and red dots, B-GEPs exist throughout
the whole topological insulator region. An interesting fact is that the energy
levels are not degenerate, shown as Fig. 5(a), i.e.,

$E_{\pm}=\pm\sqrt{\bar{\Delta}^{2}-\varepsilon^{2}}\neq 0.$ (46)

As a result, _this is an example of GEPs without energy degeneracy._ Now, the
Bloch peach has a symmetric axis along the z-direction; see the illustration
in Fig. 3(b2).

Let us discuss the classes of B-GEPs in detail.

After the application of a non-Hermitian similarity transformation
$\mathcal{\hat{S}}_{B}$, the effective two-level model $\hat{H}_{\mathrm{NH}}$
is transformed into another non-Hermitian model, i.e.,

$\displaystyle\hat{H}_{\mathrm{NH}}^{\beta^{B}}$
$\displaystyle=(\mathcal{\hat{S}}_{B})^{-1}\hat{H}_{\mathrm{NH}}(\mathcal{\hat{S}}_{B})$
$\displaystyle=\bar{\Delta}^{+}\sigma^{+}+\bar{\Delta}^{-}\sigma^{-}+i\varepsilon\sigma^{z},$
(47)

where $\bar{\Delta}^{+}=\bar{\Delta}\exp{(-Nq_{0})}$ and
$\bar{\Delta}^{-}=\bar{\Delta}\exp{(Nq_{0})}$. In the thermodynamic limit
$N\rightarrow\infty$, there exist three phases: a phase with
$\left|\bar{\Delta}^{+}\right|\rightarrow\infty$ and
$\left|\bar{\Delta}^{-}\right|\rightarrow 0$, a phase with
$\left|\bar{\Delta}^{+}\right|\rightarrow 0$ and
$\left|\bar{\Delta}^{-}\right|\rightarrow\infty$, and a phase with
$\left|\bar{\Delta}^{+}\right|\rightarrow 0$ and
$\left|\bar{\Delta}^{-}\right|\rightarrow 0$. At
$\left|\bar{\Delta}^{\pm}\right|=1$ or $t_{1}\pm\gamma=\pm 1$, the “hidden”
quantum phase transition occurs, with a sudden change between IB-GEPs
($\Lambda=0$) and IIB-GEPs ($\Lambda=1$). Shown as Fig. 5, the orange dotted
line at $t_{1}=0.38$ identifies the phase transition between IB-GEPs
($E_{+}\neq E_{-}$, $\Lambda=0$) and IIB-GEPs ($E_{+}\neq E_{-}$,
$\Lambda=1$). We find that the situation changes for the state similarity
$\Lambda=|\langle\tilde{\psi}_{+}^{\mathrm{R}}|\tilde{\psi}_{-}^{\mathrm{R}}\rangle|$
of two edge states. In Fig. 6, one can see the “hidden” quantum phase
transition from the numerical results for the case of $N=20$ and
$\varepsilon=0.001$ that $\Lambda$ suddenly changes from $0$ to $1$ at
$\left|\bar{\Delta}^{\pm}\right|=1.$

Figure 6: (Color online) The state similarity of two edge states,
$\Lambda=|\langle\tilde{\psi}_{+}^{\mathrm{R}}|\tilde{\psi}_{-}^{\mathrm{R}}\rangle|$,
for the case of $N=20$ and $\varepsilon=0.001$ as a function of $\gamma$ and
$t_{1}$. A hidden phase transition occurs with a sudden change between IB-GEPs
($\Lambda=0$, green regions) and IIB-GEPs ($\Lambda=1$, red regions).

####  V.3.3 H-GEPs

Thirdly, we consider the case of H-GEPs by setting $\varepsilon$ to a purely
real value, $\gamma\neq 0,$ and
$\left|\bar{\Delta}\right|=\left|\varepsilon\right|$. On the one hand, the
initial basis is defective, i.e.,

$\\{|\psi_{0}^{\mathrm{R}}\rangle\\}=\\{\mathcal{\hat{S}}_{B}|\psi_{0}\rangle\\}=\\{e^{-\beta^{B}\cdot\sigma_{z}}|\psi_{0}\rangle\\}$
(48)

with $\beta^{B}=Nq_{0}\rightarrow\infty$. In the thermodynamic limit
$N\rightarrow\infty$, the two-level system may be regarded as a B-GEP. On the
other hand, for the case of $\left|\frac{\bar{\Delta}}{\varepsilon}\right|=1$,
the same system can be regarded as an M-GEP with another singular non-
Hermitian similarity transformation
$\mathcal{\hat{S}}_{B}(\beta^{M}\rightarrow\infty)$.

Therefore, in thermodynamic limit $N\rightarrow\infty$, we have an H-GEP with
the following defective matrix basis:

$\displaystyle\\{|\psi^{\mathrm{R}}\rangle\\}$
$\displaystyle=\\{\mathcal{\hat{S}}_{M}(\sigma_{y},\beta^{M}=\left|\frac{1}{2}\ln\left|\frac{\bar{\Delta}+\varepsilon}{\bar{\Delta}-\varepsilon}\right|\right|\rightarrow\infty)$
$\displaystyle\cdot\mathcal{\hat{S}}_{B}(\sigma_{z},\beta^{B}$
$\displaystyle=Nq_{0}\rightarrow\infty)|\psi_{0}\rangle\\}.$

Now, the quantum state under the non-Hermitian similarity transformations
$\mathcal{\hat{S}}_{M}(\sigma_{y},\beta^{M})$ and
$\mathcal{\hat{S}}_{B}(\sigma_{z},\beta^{B})$ becomes

$\displaystyle|\psi^{\mathrm{R}}\rangle$
$\displaystyle=\frac{1}{\sqrt{2}}(\cos\frac{\theta}{2}+e^{-\beta^{B}}\sin\frac{\theta}{2})\left|1\right\rangle$
(49)
$\displaystyle+\frac{i}{\sqrt{2}}e^{i\varphi}e^{-\beta^{M}}(\cos\frac{\theta}{2}-e^{-\beta^{B}}\sin\frac{\theta}{2})\left|2\right\rangle$

(with $\theta\in[0,\pi]$ and $\varphi\in[0,2\pi]$), and the radius of the
Bloch sphere $R$ can be obtained as

$\displaystyle R$
$\displaystyle=\langle\psi^{\mathrm{R}}|\psi^{\mathrm{R}}\rangle=\frac{1}{\sqrt{2}}[(\cos\frac{\theta}{2}+e^{-\beta^{B}}\sin\frac{\theta}{2})^{2}$
(50)
$\displaystyle+e^{-2\beta^{M}}(\cos\frac{\theta}{2}-e^{-\beta^{B}}\sin\frac{\theta}{2})^{2}]^{1/2}.$

As shown in Fig. 3(b3), the Bloch peach, with a symmetric axis along the
z-direction, shrinks.

In addition, for the non-Hermitian SSH model, the H-GEP is an unusual
spontaneous $\mathcal{PT}$-symmetry breaking accompanied by a transition from
real to complex spectra. From Fig. 6, the two energy levels become degenerate
($E_{+}=E_{-}$) at the gray dotted line with $t_{1}=1.03$, where the H-GEP
occurs, and we can see that the state similarity of the two edge states is
constant, i.e., $\Lambda\equiv 1$.

##  VI Conclusion and discussion

In this paper, we illustrate that the phenomenon of EPs in non-Hermitian
quantum systems is much more interesting than expected. For general non-
Hermitian two-level systems, there may exist three types of GEPs — M-GEPs
(with a defective matrix basis and a normal initial basis), B-GEPs (with a
normal matrix basis and a defective initial basis), and H-GEPs (with a
defective matrix basis and a defective initial basis). In addition, there are
two classes of B-GEPs — IB-GEPs, without eigenstate coalescence, and IIB-GEPs,
with eigenstate coalescence. By taking the topologically protected edge states
in the non-Hermitian SSH model as an example, we explore the physical
properties of GEPs. In the future, we will study higher-order GEPs in non-
Hermitian systems and attempt to develop a complete theory of GEPs.

###### Acknowledgements.

This work was supported by NSFC Grant Nos. 11974053 and 12174030.

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