Biorthogonal Majorana zero modes, extended waves in continuum of bound states
and non-Hermitian Toda soliton-fermion duality

Harold Blas

Instituto de Física

Universidade Federal de Mato Grosso

Av. Fernando Correa, $N^{0}$ 2367

Bairro Boa Esperança, Cep 78060-900, Cuiabá - MT - Brazil

<EMAIL_ADDRESS>
We study the non-Hermitian (NH) Toda model coupled to fermions through soliton
theory techniques and the realizations of the pseudo-chiral and pseudo-
Hermitian symmetries. The interplay of non-Hermiticity, integrability,
nonlinearity, and topology significantly influence the formation and behavior
of a continuum of bound state modes (CBM) and extended waves in the localized
continuum (ELC). The non-Hermitian soliton-fermion duality, the complex scalar
field topological charges and winding numbers in the spectral topology are
uncovered. The Hermitian bound states/solitons lie on the unit circle $|z|=1$
defined by the uniformization parameter $z\in\leavevmode\hbox{\,$\vrule
height=6.45831pt,width=0.4pt,depth=0.0pt\kern-3.00003pt{\rm
C}$}\backslash\\{0\\}$ related to the complex energy eigenvalue, whereas the
non-Hermitian bound states/solitons lie on the complex plane such that
$|z|\neq 1$. The biorthogonal Majorana zero modes, dual to the NH Toda
solitons with topological charges $\pm 1$, appear at the complex-energy point
gap and are pinned at zero energy. Our findings improve the understanding of
exotic quantum states, but also paves the way for future research in
harnessing non-Hermitian phenomena for topological quantum computation, as
well as the exploration of integrability and NH solitons in the theory of
topological phases of matter.

##  1 Introduction

A common assumption underlying the physics of isolated systems is the
Hermiticity property of the Hamiltonian. However, non-Hermitian (NH)
Hamiltonians have attracted considerable interest in recent years since they
are quite ubiquitous in nature and remarkable theoretical and experimental
results have been uncovered (see e.g. [1, 2, 3]). They have been studied in
three different approaches, a generic NH Hamiltonians with complex eigenvalues
[4, 5], (anti-)pseudo-Hermitian Hamiltonians [6, 7, 8], and ${\cal P}{\cal T}$
symmetric Hamiltonians with real eigenvalues [9]. Theoretical aspects and some
experimental validations of nonlinear wave dynamics in ${\cal P}{\cal T}$
symmetric systems have been reviewed in [10].

In physical systems with Hermitian and NH Hamiltonians a considerable amount
of work on topological phases has emerged as a rapidly growing research area
relevant to a variety of fields. An important concept in topological physics
is the bulk-boundary correspondence in topological phases, such that the
topological invariant related to the bulk states counts the number of gapless
boundary states. So, topological phenomena can be related to boundary
phenomena. For a review on non-Hermitian lattice systems in condensed matter
physics, focusing on the boundary phenomena, see e.g. [11]. Originally found
in condensed matter physics, robust edge states are currently found in
practically all wave dynamics originating from a variety of physical models.
For example, remarkable similarities have recently been found between the
shallow water equations in their linearized form and topological phases of
matter [12]. A duality relationship between the bound state waves and the
topological solitons has been uncovered for non-linear shallow water waves
[13].

The study of topological edge modes has initially been related to Hermitian
linear systems concepts, such as orthogonal eigenvectors and real energy
spectra. Actually, many physical systems exhibiting these phenomena naturally
present a non-linear behavior. This motivated recently a growing interest in
the confluence of nonlinearity and topology with applications to a wide range
of phenomena (see e.g. [14] and references therein). From the quantum field
theory perspective, it has been shown the presence of edge and bulk solitons
related to the problem of fractionally charged solitons and electron
fractionalization in $(1+1)$-dimensional systems [15, 16, 17]. In various ways
the concepts involved in the $(1+1)$-dimensional systems can be extended to
higher dimensions.

In this context, the exploration of the interplay of NH solitons and fermionic
systems emerges as an intriguing subject in theoretical and mathematical
physics. In this paper, we delve into the interplay between non-Hermiticity,
nonlinearity, integrability and topology in the framework of the non-Hermitian
Toda soliton coupled to Dirac fermions; so, extending the Hermitian soliton-
fermion duality reported in [18, 19, 20]. The exploration of non-Hermitian
systems has emerged as a fascinating and rapidly growing field, yielding
intriguing insights into the fundamental nature of quantum mechanics. Non-
Hermitian matrices exhibit unconventional properties compared with Hermitian
ones: eigenstates are, in general, nonorthogonal, instead it has been
introduced the concept of biorthogonality and a complex-energy spectrum can
possess exceptional points (EPs) giving rise to spectral topology. Non-
Hermiticity drastically changes the properties of topological boundary states.
Non-Hermiticity amplifies the edge states, which enables novel topologically
protected states. Recently a modified biorthogonal bulk-boundary
correspondence has been established that works in the presence of non-
Hermiticity [21].

The non-Hermitian Toda model coupled to matter exhibits an extraordinary
confluence of non-Hermiticity, nonlinearity, integrability and topological
solitons. Central to our investigations are the concepts of pseudo-chiral and
pseudo-Hermitian symmetries, which shed lights to disentangle the intricate
interplay between the mentioned properties. Remarkably, a relationship between
the complex energy, number wave and mass parameters in the NH soliton sector
of the model can be defined on the whole complex plane through a
uniformization parameter $z$, which allows us to discover that while Hermitian
bound state/soliton states lie on the unit circle $|z|=1$, the non-Hermitian
generalizations of that correspondence can be established on the continuous
region $|z|\neq 1$ of the complex plane. In this scenario we show the
emergence of biorthogonal Majorana zero modes as dual to the non-Hermitian
topological solitons with topological charges $\pm 1$, which manifest at the
complex-energy point gap and pinned at zero energy. These zero modes, we find,
hold the key to unlocking the topologically protected bulk-boundary
correspondence from the integrable field theory perspective.

An essential distinction between Hermitian and non-Hermitian systems is the
degrees of freedom that we have access to; nonunitary operations forbidden in
Hermitian systems can be performed in non-Hermitian systems. In other words, a
change in the spectrum from real to complex increases the number of parameters
that describe the system. Since topology crucially depends on the underlying
manifold, non-Hermiticity is expected to alter the topological classification
of the physical systems such as insulators and superconductors. The Hermitian
Toda model coupled to matter has been found in a variety of models, such as
the continuum limit of $s-$wave superconductor [22], effective theory
describing Majorana bound states [23], a model for high $T_{c}$
superconductivity [24], low-energy effective Lagrangian in QCD2 [25], etc; so,
our study not only becomes of academic interest, but physically motivated. As
pointed out above, pseudo-Hermiticity is a constraint unique to non-Hermitian
systems and may provide novel topological features. In view of the rapid
theoretical and experimental advances in non-Hermitian physics, there has been
a great interest and an urgent need for comprehensive topological
classifications that provide reference points for experiments and predict
novel non-Hermitian topological phases.

Our exploration has led us to unveil remarkable phenomena: the existence of
continuum spectra of bound states and extended waves embedded within this
localized continuum (ELC). These phenomena are profoundly shaped by the
interplay of four key characteristics: non-Hermiticity, integrability,
nonlinearity, and topology. The continuum of biorthogonal bound states emerge
coupled to a continuous family of topological complex solitons, such that
their topological charges depend on the energy eigenvalues. This type of
continuum of bound states in a non-Hermitian Hamiltonian have recently been
uncovered also in [26]. In the Hermitian Toda system coupled to fermion, which
has recently been presented in [18], the in-gap and the bound states in the
continuum (BIC) have quantized energies, whereas the free states form a
continuum. On the other hand, the NH extended wave states embedded into the NH
localized continuum can be regarded as some kind of inverted BIC states due to
the non-Hermicity of the model. Recently, these novel types of states in a NH
system have been dubbed as extended states in a localized continuum (ELC)
[27].

So, the NH models exhibit behaviors that challenge conventional understanding.
Within the NH Toda theory coupled to matter integrable field theory we clarify
the concept of the non-Hermitian soliton-fermion duality, revealing a profound
connection between these entities which survive non-Hermiticity effects, and
shed lights on the appearance of a pair of topological charges, such that the
second charge arises purely as the result of the interplay between the non-
Hermiticity, topology, and nonlinear effects. We believe that beyond the pure
theoretical inquiry, our discoveries might have important implications for the
realm of practical quantum technologies. The harnessing of non-Hermitian
phenomena in the context of topological quantum computation promise new
frontiers in the development of quantum computing paradigms. Furthermore, our
work establishes a bridge between quantum field theory, condensed matter
physics and integrable systems, inviting further exploration at the crossroads
of these three fields.

The paper is organized as follows. In the next section we present the
integrable NH Toda model coupled to fermions and the relevant eigenvalue
equations for the right and left eigenvectors. In the section 3 the pseudo-
chirality symmetry is discussed, as well as the concept of biorthogonality of
the left and right eigenvectors. The section 4 discusses the anti-pseudo-
Hermiticity symmetry. Some theorems and properties of pseudo-Hermitian
Hamiltonians are discussed in the presence of ${\cal P}$, ${\cal T}$ and
${\cal P}{\cal T}$ transformations. In the section 5 we uncover the
topological complex solitons and the biorthogonal spinor bound states through
the tau function formalism. The complex energy manifold is defined. Moreover,
the spectral topology and the associated winding number are discussed. In
section 6 a classification of these solutions are discussed. The continuum of
bound states/solitons and the extended waves embedded into this localized
continuum are presented. The biorthogonality relationships between the left
and right eigenvectors are computed. In section 7 the biorthogonal Majorana
zero mode-topological soliton duality is studied by providing a mapping
between the spinor and the scalar fields. Section 8 presents the conclusions
and discussions. In the appendix A we present the static equations of motion
in components.

##  2 The model

We consider two fermion fields chirally coupled to a complex scalar field
defined by the following non-Hermitian Lagrangian[28, 19, 20]111Notation:
$\partial^{2}=\partial^{2}_{t}-\partial^{2}_{x}$.
$\gamma_{0}=\left(\begin{array}[]{cc}0&i\\\ -i&0\end{array}\right)$,
$\gamma_{1}=\left(\begin{array}[]{cc}0&-i\\\ -i&0\end{array}\right)$,
$\gamma_{5}=\gamma_{0}\gamma_{1}=\left(\begin{array}[]{cc}1&0\\\
0&-1\end{array}\right)$, and $\bar{\psi}=\widetilde{\psi}^{T}\gamma_{0}$;
$\bar{\chi}=\widetilde{\chi}^{T}\gamma_{0}$.

$\displaystyle\mathcal{L}$ $\displaystyle=$
$\displaystyle\;\frac{1}{2}\partial_{\mu}\Phi\partial^{\mu}\Phi+i\widetilde{\psi}\gamma^{\mu}\partial_{\mu}\psi-M(\widetilde{\psi}\psi\cos(\beta\Phi)+i\widetilde{\psi}\gamma_{5}\psi\sin(\beta\Phi)),$
(2.1)

with $\Phi$ being a complex field and $\widetilde{\psi}$ and $\psi$ two Dirac
spinors. Below we will consider a complex mass parameter
$M\in\leavevmode\hbox{\,$\vrule
height=6.45831pt,width=0.4pt,depth=0.0pt\kern-3.00003pt{\rm C}$}$ and a real
$\beta$ as the coupling strength between the fermions and the complex scalar.
The Hermitian version of the model (2.1) has been studied in the literature
and it has been taken a real scalar field with the fermions reality condition
$\widetilde{\psi}=\psi^{\star}$ and with $M$ being a real mass parameter. In
those cases the model has been defined by a real Lagrangian and associated
Hermitian hamiltonian [19, 29, 20, 25, 18]. The Hermintian model has been
shown to describe the low-energy effective Lagrangian of QCD2 with one flavor
and $N$ colors [25] and the BCS coupling in spinless fermions in a two
dimensional model of high T superconductivity in which the solitons play the
role of the Cooper pairs [24]. Moreover, the model has earlier been studied as
a model for fermion confinement in a chiral invariant theory [30] and the
mechanism of fermion mass generation without spontaneously chiral symmetry
breaking in two-dimensions [31]. Recently, through analytical and numerical
methods the Hermitian version of the model (2.1) plus a scalar potential has
been shown to exhibit Majorana zero mode-soliton duality and in-gap and BIC
bound states, such that the Noether, topological and a novel nonlocal charge
densities satisfy a formula of the Atiyah-Patodi-Singer-type.

The general model (2.1), with two Dirac spinors $\psi$ and $\widetilde{\psi}$
and a complex field $\Phi\in\leavevmode\hbox{\,$\vrule
height=6.45831pt,width=0.4pt,depth=0.0pt\kern-3.00003pt{\rm C}$}$, belongs to
a wide class of integrable soliton theories, the so-called $\hat{sl}(2)$
affine Toda systems coupled to matter fields [28]. The integrability
properties, the construction of the general solution including the solitonic
ones, the soliton-fermion duality properties, as well as its symplectic
structures were discussed in [19, 20, 29]. We will consider the non-Hermitian
model (2.1) and show that the dynamics of the system exhibits a continuum of
kink type states and its associated biorthogonal spinor bound states. In
addition, it will be shown the appearance of extended wave states embedded
into the continuum of bound states. In this context we will realize the
various symmetry properties of the model such as the pseudo-chirality and its
associated biorthogonal bound states, such that the generalized non-Hermitian
particle-hole (NHPH) symmetry holds.

The equations of motion become

$\displaystyle\partial^{2}\Phi-M\beta\widetilde{\psi}\psi\sin(\beta\Phi)+iM\beta\widetilde{\psi}\gamma_{5}\psi\cos(\beta\Phi)=0$
(2.2) $\displaystyle
i\gamma_{\mu}\partial^{\mu}\psi-M(\psi\cos(\beta\Phi)+i\gamma_{5}\psi\sin(\beta\Phi))=0$
(2.3) $\displaystyle
i\gamma_{\mu}\partial^{\mu}\widetilde{\psi}-M(\widetilde{\psi}\cos(\beta\Phi)+i\gamma_{5}\widetilde{\psi}\sin(\beta\Phi))=0$
(2.4)

Let us define

$\displaystyle\psi$ $\displaystyle\equiv$ $\displaystyle e^{-i\lambda
t}\xi,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\widetilde{\psi}\equiv e^{i\lambda
t}\eta$ (2.5) $\displaystyle\xi$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{c}\xi_{A}\\\
\xi_{B}\end{array}\right),\,\,\,\,\,\,\,\,\eta=\left(\begin{array}[]{c}\eta_{A}\\\
\eta_{B}\end{array}\right),$ (2.10)

with $\lambda$ being a complex parameter. The equations (2.3)-(2.4) can be
written in matrix form as

$\displaystyle H\xi$ $\displaystyle=$ $\displaystyle\lambda\,\xi,$ (2.11)
$\displaystyle\widetilde{H}\eta$ $\displaystyle=$ $\displaystyle\lambda\,\eta$
(2.12)

where

$\displaystyle H=\left(\begin{array}[]{cc}i\frac{d}{dx}&iMe^{-i\beta\Phi}\\\
-iMe^{i\beta\Phi}&-i\frac{d}{dx}\end{array}\right),\,\,\,\,\,\,\,\widetilde{H}=\left(\begin{array}[]{cc}-i\frac{d}{dx}&-iMe^{i\beta\Phi}\\\
iMe^{-i\beta\Phi}&i\frac{d}{dx}\end{array}\right).$ (2.17)

Notice that the equations above define a pair of eigenvalue equations for the
non-Hermitian operators $H$ and $\widetilde{H}$ with right eigenvectors $\xi$
and $\eta$, respectively. Recently, the Hermitian version of the model (2.11)
has been considered as the continuum limit of a one-dimensional system of
spin-orbit coupled Dirac Hamiltonian and a $s-$wave superconducting pairing,
such that the $s-$wave pairing is represented by
$-iMe^{i\beta\Phi}\,(\Phi\in\leavevmode\hbox{\rm I\kern-1.79993ptR})$ [22].
Moreover, the system (2.11) has been considered in [23] as an effective theory
describing Majorana bound states such that the function $iMe^{-i\beta\Phi}$ is
related to the superconducting order parameter.

One can show that the NH operators $H$ and $\widetilde{H}$ are related by

$\displaystyle\Gamma H\Gamma=\widetilde{H},\,\,\,\,\,\,\Gamma=i\gamma_{1}.$
(2.18)

Due to the last relationship (2.18), given an eigenvector $\eta$ of
$\widetilde{H}$ with eingenvalue $\lambda$ one can get an eigenvector
$\Gamma\eta$ of $H$ with the same eigenvalue $\lambda$. In fact, from (2.12)
and taking into account (2.18) one can get

$\displaystyle\Gamma
H\Gamma\eta=\lambda\eta\,\,\,\rightarrow\,\,\,H(\Gamma\eta)=\lambda(\Gamma\eta)$
(2.19)

So, comparing (2.19) with (2.11) one can write

$\displaystyle\xi=\Gamma\eta\,\,\,\,\,\Rightarrow\,\,\,\,\,\xi_{A}=\eta_{B},\,\,\,\,\,\,\xi_{B}=\eta_{A}.$
(2.20)

From (2.2) and taking into account the Ansatz (2.5) one can get a static
version of the complex scalar field equation of motion

$\displaystyle-\partial^{2}_{x}\Phi+M\beta[e^{i\beta\Phi}\,\eta_{B}\xi_{A}+e^{-i\beta\Phi}\,\eta_{A}\xi_{B}]=0.$
(2.21)

We will examine the static solutions satisfying the relationship

$\displaystyle\partial_{x}\Phi$ $\displaystyle=$
$\displaystyle-\frac{\beta}{2}[\eta_{A}\xi_{A}+\eta_{B}\xi_{B}],$ (2.22)
$\displaystyle=$
$\displaystyle-\frac{\beta}{2}[\eta_{3}\xi_{3}-\eta_{4}\xi_{4}+\eta_{1}\xi_{1}-\eta_{2}\xi_{2}+i(\eta_{3}\xi_{4}+\eta_{4}\xi_{3}+\eta_{1}\xi_{2}+\eta_{2}\xi_{1})],$
(2.23)

where in the last equation we have written the real and imaginary parts of the
bilinear current terms using the component of $\xi$ and $\eta$ defined in
(A.3) and (A.6).

Remarkably, the first order equation (2.22) together with the first order
equations of motion for the Dirac spinors (2.11)-(2.12) imply the second order
equation for $\Phi$ in (2.21). Therefore, a given set of solutions of the
system of static first order differential equations (2.11)-(2.12) and (2.22)
will also be solutions of the second order equation for $\Phi$. The equation
(2.22) suggests a certain kind of equivalence relationship between the
$x-$derivative of the scalar and the fermion bilinear in the NH model. In
fact, this is a NH analog of the equivalence between the $U(1)$ Noether and
topological soliton currents considered in the literature in the Hermitian
case of the model [18]. In both condensed matter (e.g. Bogoliubov-de Gennes
(BdG) and gap equations ) and high energy physics (e.g. Gross-Neveu (GN) and
Nambu-Jona-Lasinio models) there have been studied analogous relationships to
(2.22) by self-consistency requirements with (2.11)-(2.12) which amounts to
write e.g.
$iMe^{i\beta\Phi}\sim[\xi_{A}^{\star}\xi_{B}+\eta_{A}^{\star}\eta_{B}]$ [32].
Actually, for a real scalar and one Dirac fermion these relationships resemble
to the bosonization rules which map some fermion bilinears to a topological
current and exponentials of the scalar [33].

One can notice that the imaginary component of the bilinear current in (2.23)
vanishes provided that

$\displaystyle\eta=\pm\xi^{\star}\,\,\Rightarrow\,\,\Big{\\{}\begin{array}[]{c}Case\,I:\,\eta_{3}=\xi_{3},\,\,\,\eta_{4}=-\xi_{4},\,\,\eta_{1}=\xi_{1},\,\,\eta_{2}=-\xi_{2},\\\
Case\,II:\,\eta_{3}=-\xi_{3},\,\,\,\eta_{4}=\xi_{4},\,\,\eta_{1}=-\xi_{1},\,\,\eta_{2}=\xi_{2}.\end{array}\Big{\\}}$
(2.26)

Then, for the cases (2.26) the scalar field must be real, i.e.
$\Re{(\Phi)}=\Phi_{1}\,(\Im{(\Phi)}=0)$. So, this implies
$\widetilde{H}=H^{\star}$ (for real $M$) and one recovers a Hermitian model
with

$\displaystyle H^{\dagger}=H,\,\,\,\,\,\widetilde{H}^{\dagger}=\widetilde{H},$
(2.27)

and real eigenvalues $\lambda\in\leavevmode\hbox{\rm I\kern-1.79993ptR}$. In
the both cases (2.26) the imaginary part of the second order equation (2.21)
for the scalar field identically vanishes for $\Phi_{2}=0$ and $M_{2}=0$. The
bilinear terms of the spinor components are presented in (A.18).

However, for $M_{2}\neq 0$, $\eta\neq\pm\xi^{\star}$ and $\Phi_{2}=0$ the eq.
(2.23) provides the general constraint
$(\eta_{3}\xi_{4}+\eta_{4}\xi_{3}+\eta_{1}\xi_{2}+\eta_{2}\xi_{1})=0$ and the
second order eq. (A.18) will provide an additional constraint (A.21).

In the case of NH Hamiltonians it is convenient to define the so-called left
eigenvectors, since in general they are not the same as the relevant Hermitian
conjugated right eingenvectors. So, the left eigenvectors $\bar{\xi}_{j}^{T}$
and $\bar{\eta}_{j}^{T}$ are defined as

$\displaystyle\bar{\xi}_{j}^{T}H=\lambda_{j}\,\bar{\xi}^{T}_{j},\,\,\,\,\,\bar{\eta}_{j}^{T}\widetilde{H}=\lambda_{j}\,\bar{\eta}^{T}_{j}$
(2.28)

Next, we will consider certain solutions satisfying

$\displaystyle\partial_{x}\bar{\Phi}=-\frac{\beta}{2}[\bar{\eta}_{A}\bar{\xi}_{A}+\bar{\eta}_{B}\bar{\xi}_{B}],$
(2.29)

where we have introduced a complex scalar field $\bar{\Phi}$, and
$\bar{\eta}^{T}\equiv(\bar{\eta}_{A}\,,\,\bar{\eta}_{B})\,$ and
$\bar{\xi}^{T}\equiv(\bar{\xi}_{A}\,,\,\bar{\xi}_{B}).$ In general one has
that the left and right eigenvectors are different, i.e.

$\displaystyle\bar{\eta}\neq\eta,\,\,\,\bar{\xi}\neq\xi,$ (2.30)

and, so, the scalar field $\bar{\Phi}$ can be considered to be in general
different from $\Phi$ in (2.22). However, we will see below that the left and
right eigenvectors satisfy some symmetry relationships, which imply a
relationship between the scalars $\Phi$ and $\bar{\Phi}$ provided that the
equation (2.29) holds. So, the second order differential equation for the
scalar $\bar{\Phi}$ becomes

$\displaystyle\partial^{2}_{x}\bar{\Phi}+M\beta[e^{i\beta\bar{\Phi}}\,\bar{\eta}_{A}\bar{\xi}_{B}+e^{-i\beta\bar{\Phi}}\,\bar{\eta}_{B}\bar{\xi}_{A}]=0.$
(2.31)

Notice that the equations (2.21) and (2.31) have different origins since the
first one has been derived from the Lagrangian, whereas the second one is a
consequence of the equivalence relationship (2.29) which has been assumed to
hold between a topological current associated to $\bar{\Phi}$ and the left-
eigenvector current density, respectively. However, we will see below that for
$\lambda$ and $-\lambda$ eigenvalues the relevant right and left eigenvectors
will be related by symmetry arguments, and so, those equations will be related
by the so-called pseudo-chirality symmetry. So, we are establishing some
equivalences between topological currents and biorthogonal fermion currents
represented by the correspondence relationships (2.22) and (2.29),
respectively.

##  3 Pseudo-chirality and biorthogonality of the eigenvectors

For a generic complex scalar field $\Phi$ the operators $H$ and
$\widetilde{H}$ are non-Hermitian with a spectra of complex eingenvalues
$\lambda$. Recently, there have been various attempts to realize particle-hole
symmetry in non-Hermitian (NH) systems. However, due to the fact that these
effective particles and holes have complex energies in a NH system, they do
not need to fulfill the Fermi-Dirac statistics. Non-Hermitian Hamiltonian
matrices exhibit more types of symmetries than the Hermitian counterparts
[11]. So, in order to examine certain extensions of the Noether’s theorem, it
has been proposed distinct ways to define the inner product in NH systems. An
inner product, without the complex conjugation, has been proposed in [5] which
focuses on asymmetric non-Hermitian Hamiltonians. So, let us write the next
inner product

$\displaystyle(\xi_{i}|\xi_{j})\equiv\xi_{i}^{T}\xi_{j}.$ (3.1)

With this inner product and using an alternative form of the Ehrenfest’s
theorem for the time evolution of $(\psi|{\cal G}|\psi)$ one can show that the
operator ${\cal G}$ is a constant of motion provided that $H$ satisfies [5]

$\displaystyle{\cal G}H{\cal G}^{-1}=-H^{T},$ (3.2)

where $H^{T}$ stands for matrix transpose. For the NH Hamiltonian $H$ in
(2.17) one has that

$\displaystyle{\cal G}=-i\gamma_{0}.$ (3.3)

In [5] the symmetry (3.2) has been defined as pseudo-chirality, and it
guaranties, as we will see below, a symmetric spectrum about the origin of the
complex energy plane, i.e. for each $\lambda$ there exists $-\lambda$, similar
to the consequence of particle-hole symmetry in the hermitian counterpart of
the model (2.1) studied recently in [18].

Below we will follow the biorthogonal approach to describe the orthogonality
properties of eigenvectors and eigenstates [2, 5]. So, let us define the right
eigenvectors $\xi_{j}$ and the relevant left eigenvectors $\bar{\xi}_{j}^{T}$
as

$\displaystyle
H\xi_{i}=\lambda_{i}\,\xi_{i},\,\,\,\,\,\,\,\,\,\,\bar{\xi}_{i}^{T}H=\lambda_{i}\,\bar{\xi}^{T}_{i}.$
(3.4)

The eigenvectors $\xi_{i}$ and $\bar{\xi}_{i}$ are different in general but
share the same eigenvalue $\lambda_{i}$. Moreover, they satisfy the
biorthogonality relationship [5]

$\displaystyle(\bar{\xi}_{j}|\xi_{i})\equiv\bar{\xi}_{j}^{T}\xi_{i}=\delta_{ji},$
(3.5)

which holds away from an exceptional point ($\lambda_{j}=0$). In fact,
considering the both eigenvalue equations in (3.4) for $\xi_{i}$ and
$\bar{\xi}_{j}^{T}$, respectively, one can write the identity

$\displaystyle\bar{\xi}_{j}^{T}H\xi_{i}=\lambda_{i}\,\bar{\xi}_{j}^{T}\xi_{i},\,\,\,\,\,\bar{\xi}_{j}^{T}H\xi_{i}=\lambda_{j}\,\bar{\xi}^{T}_{j}\xi_{i}\,\,\,\,\,\Rightarrow\,\,\,\,\,0=(\lambda_{i}-\lambda_{j})\,\bar{\xi}_{j}^{T}\xi_{i}.$
(3.6)

Then, from this equation one can see that the biorthogonality relationship
(3.5) is satisfied for $\lambda_{i}\neq\lambda_{j}$. The normalization
$(\bar{\xi}_{i}|\xi_{i})=1\,(\lambda_{i}\neq 0)$ will be considered in the
sections below in the context of the soliton type solutions of the model.

Next, for the Hamiltonian $H$ in (2.17) which possesses the property (3.2),
one can show the following property

$\displaystyle H^{T}[{\cal G}\xi_{\lambda_{j}}]=-{\cal
G}H\xi_{\lambda_{j}}=-\lambda_{j}[{\cal G}\xi_{\lambda_{j}}].$ (3.7)

Since the second eq. in (3.4) can be written as
$H^{T}\bar{\xi}_{-\lambda_{j}}=-\lambda_{j}\bar{\xi}_{-\lambda_{j}}$, then
from (3.7) one can write

$\displaystyle\bar{\xi}_{-\lambda_{j}}={\cal G}\xi_{\lambda_{j}}.$ (3.8)

This relation suggests that for each right eigenstate $\xi_{\lambda_{j}}$ with
eigenvalue $\lambda_{j}$, one can find a left eigenstate
$\bar{\xi}_{-\lambda_{j}}$ for the relevant eigenvalue $-\lambda_{j}$, such
that in components one has the next relationships

$\displaystyle\bar{\xi}_{-\lambda_{j}A}=\xi_{\lambda_{j}B},\,\,\bar{\xi}_{-\lambda_{j}B}=-\xi_{\lambda_{j}A}.$
(3.9)

In terms of the Dirac field $\psi_{\lambda_{j}}$ defined in (2.5), and
defining $\bar{\psi}_{-\lambda_{j}}\equiv
e^{-i\lambda_{j}t}\bar{\xi}_{-\lambda_{j}}$, one can write

$\displaystyle\bar{\psi}_{-\lambda_{j}}={\cal G}\psi_{\lambda_{j}}.$ (3.10)

So, the operator ${\cal G}$ maps a right Dirac eigenstate with eigenvalue
$\lambda_{j}$ to one of the left eigenstates with corresponding eigenvalue
$-\lambda_{j}$. This relationship exhibits the so-called NH particle-hole
symmetry of the system and then one expects that the ‘particles’ and ‘holes’
will have identical spectra, since the spectra of $H$ is symmetric about the
origin of the complex eigenvalue plane. So, for certain zero modes one can
formulate the NH analog of the Majorana fermions for a system of this kind
since there exists a biorthogonal ‘particle-hole’ symmetry.

A Majorana fermion in a Hermitian model is defined by considering the particle
and hole with the same energy as a single particle. Here we generalize the
Majorana zero modes to NH model which exhibits topological solitons, and show
a general but quite different properties from their Hermitian counterparts.
Then, following analogous reasoning, from the relationship (3.10) and for
zero-mode solutions ($\lambda_{j}=0$), one can write

$\displaystyle\bar{\psi}_{0}={\cal G}\psi_{0}.$ (3.11)

The biorthogonal states obeying the condition (3.11) will be regarded as NH
Majorana zero modes. These states are well known to lead to interesting
phenomena in condensed matter and we will search for them as bound states for
the biorthogonal states.

Notice that the biorthogonal relationship (3.5) is different from the NH inner
product (3.1). For example, the product (3.5) vanishes at an exceptional point
of $\lambda_{j}=0$, since taking into account (3.11), one has
$\bar{\xi}_{0A}=\xi_{0B},\,\,\bar{\xi}_{0B}=-\xi_{0A}$, which implies

$\displaystyle(\bar{\xi}_{0}|\xi_{0})$ $\displaystyle\equiv$
$\displaystyle\bar{\xi}_{0}^{T}\xi_{0}$ (3.12) $\displaystyle=$ $\displaystyle
0.$

On the other hand, the product (3.1) does not vanish at the same EP,
$\lambda_{j}=0$, i.e. it becomes

$\displaystyle(\xi_{0}|\xi_{0})$ $\displaystyle\equiv$
$\displaystyle\xi_{0}^{T}\xi_{0}$ (3.13) $\displaystyle=$
$\displaystyle(\xi_{0A})^{2}+(\xi_{0B})^{2},$ (3.14)

which is a complex number in general. This is in contradistinction to the
Hermitian case in which an inner product has been defined to provide a real
value for the Hermitian Majorana zero modes [18].

Similar constructions as above can be performed for the second Dirac field
$\widetilde{\psi}$. Since the pseudochirality property (3.2) also holds for
the second operator $\widetilde{H}$, i.e.

$\displaystyle{\cal G}\widetilde{H}{\cal G}^{-1}=-\widetilde{H}^{T},$ (3.15)

then one can follow throughout all the discussions above for the right
eigenvectors $\eta$ and the relevant left eigenvectors $\bar{\eta}^{T}$

$\displaystyle\widetilde{H}\eta_{i}=\lambda_{i}\,\eta_{i},\,\,\,\,\,\,\,\,\,\,\bar{\eta}_{i}^{T}\widetilde{H}=\lambda_{i}\,\bar{\eta}^{T}_{i}.$
(3.16)

So one can show

$\displaystyle\bar{\eta}_{-\lambda_{j}}={\cal G}\eta_{\lambda_{j}}.$ (3.17)

So, for each right eigenstate $\eta_{\lambda_{j}}$, one can find a left
eigenstate $\bar{\eta}_{-\lambda_{j}}$, which in components become

$\displaystyle\bar{\eta}_{-\lambda_{j}A}=\eta_{\lambda_{j}B},\,\,\bar{\eta}_{-\lambda_{j}B}=-\eta_{\lambda_{j}A}.$
(3.18)

For the Dirac field $\widetilde{\psi}$ as defined in (2.5)
($\widetilde{\psi}=e^{i\lambda t}\eta$) and taking into account (3.17) one can
written

$\displaystyle\overline{\widetilde{\psi}}_{-\lambda_{j}}={\cal
G}\widetilde{\psi}_{\lambda_{j}}.$ (3.19)

So, one can argue that this relationship also exhibits the NH particle-hole
symmetry for the second Dirac field $\widetilde{\psi}$ and then allows us to
discuss the Majorana zero modes. Following analogous reasoning from the
relationship (3.19) and for zero-mode solutions ($\lambda_{j}=0$) one can
write

$\displaystyle\overline{\widetilde{\psi}}_{0}={\cal G}\widetilde{\psi}_{0}.$
(3.20)

The biorthogonal states obeying the condition (3.20) will be regarded as the
second NH Majorana zero mode of the system. The analog of the product (3.5)
becomes

$\displaystyle(\bar{\eta}_{i}|\eta_{j})=\bar{\eta}_{i}^{T}\eta_{j}=\delta_{ij}.$
(3.21)

This product vanishes at the exceptional point $\lambda_{j}=0$, since taking
into account (3.20), one has
$\bar{\eta}_{0A}=\eta_{0B},\,\,\bar{\eta}_{0B}=-\eta_{0A}$, which implies

$\displaystyle(\bar{\eta}_{0}|\eta_{0})$ $\displaystyle\equiv$
$\displaystyle\bar{\eta}_{0}^{T}\eta_{0}$ (3.22) $\displaystyle=$
$\displaystyle 0.$

The exceptional points (EP) are defined by the zeros of the complex energy
$\lambda=0$, where the bi-orthogonal conditions (3.5) and (3.21) break down,
and the set of states $\\{\bar{\xi}_{0}$, $\xi_{0}\\}$ and
$\\{\bar{\eta}_{0}$, $\eta_{0}\\}$ coalesce, i.e. they become orthogonal, and
the Hamiltonian becomes non-diagonalizable, see e.g. [34, 35]. The results
(3.12) and (3.22) will be verified below for spinor bound state eigenvectors.
Notice that at the exceptional point $\lambda=0$ (EP) we will have an energy
degeneracy for the both types of eigenvector spaces, left and right. In fact,
the set of right and left eigenvectors $\\{\xi_{0}\,,\,\eta_{0}\\}$ and
$\\{\bar{\xi}_{0}\,,\,\bar{\eta}_{0}\\}$, respectively, correspond to the
vanishing eigenvalue $\lambda=0$.

At this point, let us examine the relationships between the currents
equivalence equations (2.22) and (2.29) in view of the pseudo-chirality
symmetry discussed above. So, taking into account the relationships (3.9) and
(3.18) one can write

$\displaystyle\bar{\eta}_{-\lambda,A}\,\bar{\xi}_{-\lambda,A}+\bar{\eta}_{-\lambda,B}\,\bar{\xi}_{-\lambda,B}=\eta_{\lambda,A}\,\xi_{\lambda,A}+\eta_{\lambda,B}\,\xi_{\lambda,B}.$
(3.23)

Then, due to the identity (3.23) one can argue that the both scalars $\Phi$
and $\bar{\Phi}$ satisfy the identity

$\displaystyle\bar{\Phi}=\Phi,$ (3.24)

provided that they satisfy the equations (2.22) and (2.29) for the right-
eigenvectors with $\lambda$ eigenvalue and the left-eigenvectors with
$-\lambda$ eigenvalue, respectively. Likewise, the second order differential
equations (2.21) and (2.31) can be mapped to each other using the pseudo-
chiral symmetry relationships (3.9) and (3.18) of the right and left
eigenvectors for eigenvalues $\lambda$ and $-\lambda$, respectively, and the
relationship (3.24) for the scalars which holds for field configurations
satisfying the currents equivalence equations (2.22) and (2.29).

##  4 Anti-pseudo-Hermiticity, ${\cal P}$, ${\cal T}$ and ${\cal P}{\cal T}$
transformations

Breaking the Hermiticity property leads to a larger number of symmetries since
the non-Hermitian Hamiltonian (2.17) exhibits the properties $H\neq
H^{\dagger}$ and $H^{\star}\neq H^{T}$. In that scenario the time-reversal
(TR) and particle-hole symmetries constitute two inequivalent representations,
i.e. depending on whether the symmetry considers conjugation or transposition
of $H$, respectively (see e.g. [36]). Notice that the symmetry property
considering the transposition of $H$ has already been discussed as the pseudo-
chiral symmetry in the last section. Next, we consider the other inequivalent
representation by taking the conjugation of $H$.

###  4.1 Anti-pseudo-Hermiticity

Let us consider a TR operator ${\cal T}$ such that ${\cal T}z{\cal
T}^{-1}=z^{\star}$ and ${\cal T}^{2}=\pm 1$ and so, ${\cal T}$ is an anti-
linear operator. Next, apply this operator on the both hand sides of the
pseudo-chirality relationship (3.2) for a generic complex scalar fied $\Phi$
and complex parameter $M$. So, one can write

$\displaystyle\sigma_{-}H\sigma_{-}^{-1}$ $\displaystyle=$
$\displaystyle-H^{\dagger},$ (4.1) $\displaystyle\sigma_{-}$
$\displaystyle\equiv$ $\displaystyle{\cal G}{\cal T},$ (4.2)

where ${\cal G}$ is given in (3.3). Notice that $\sigma_{-}$ is an antilinear
operator. Following the notations and conventions in the current literature
one has that the equations (4.1)-(4.2) define $H$ to be an anti-
pseudo-(anti)-Hermitian Hamiltonian222We denote $\sigma_{\pm}$ for pseudo-,
and pseudo-(anti)-Hermitian cases, respectively, such that
$\sigma_{\pm}H\sigma_{\pm}^{-1}=\pm H^{\dagger}.$ Further generalizations are
possible, $\sigma_{\pm}$ operators can be either linear or anti-linear. In our
case $\sigma_{-}$ is an anti-linear operator. So, $H$ can be dubbed as an
anti-pseudo-(anti)-Hermitian Hamiltonian [35, 7]. A similar property can be
discussed for the companion Hamiltonian $\widetilde{H}$, since it also
satisfies the pseudo-chirality relationship (3.15); so, $\widetilde{H}$ is
also an anti-pseudo-(anti)-Hermitian Hamiltonian

$\displaystyle\sigma_{-}\widetilde{H}\sigma_{-}^{-1}$ $\displaystyle=$
$\displaystyle-\widetilde{H}^{\dagger}.$ (4.3)

Next, let us define the right and left eigenvector equations for the operator
$H^{\star}$

$\displaystyle
H^{\star}\chi_{j}=\lambda_{j}\,\chi_{j},\,\,\,\,\hat{\chi}^{T}_{i}H^{\star}=\lambda_{i}\,\hat{\chi}^{T}_{i}.$
(4.4)

Taking the transpose of the left eigenvalue equation in (4.4) and subsequently
considering the pseudo-anti-Hermiticity relationship (4.1) one can write

$\displaystyle H^{\dagger}\hat{\chi}_{j}$ $\displaystyle=$
$\displaystyle\lambda_{j}\,\hat{\chi}_{j},$ (4.5)
$\displaystyle-\sigma_{-}H\sigma_{-}^{-1}\hat{\chi}_{j}$ $\displaystyle=$
$\displaystyle\lambda_{j}\,\hat{\chi}_{j},$ (4.6) $\displaystyle
H^{\star}({\cal G}^{-1}\hat{\chi}_{j})$ $\displaystyle=$
$\displaystyle-\lambda_{j}\,({\cal G}^{-1}\hat{\chi}_{j}),$ (4.7)

where (4.6) follows upon using (4.1) into (4.5). In order to get (4.7) we have
used ${\cal T}H{\cal T}^{-1}=H^{\star}$ and multiplied on the left by ${\cal
G}^{-1}$ on the b.h.s. of (4.6). So, comparing the last relationship (4.7) and
the right eigenvalue equation in (4.4) one gets

$\displaystyle\chi_{\widetilde{j}}={\cal G}^{-1}\hat{\chi}_{j},$ (4.8)

where $\chi_{\widetilde{j}}$ denotes a right eigenvector with eigenvalue
$(-\lambda_{j})$.

Notice that one can also relate the above left eigenvector $\hat{\chi}^{T}$ to
the left eigenvector of type $\bar{\xi}^{T}$ in (3.4) for the same eigenvalue
$\lambda_{j}$. In fact, by taking the complex conjugate of (3.4) one has

$\displaystyle\bar{\xi}_{j}^{\dagger}H^{\star}$ $\displaystyle=$
$\displaystyle\lambda_{j}^{\star}\bar{\xi}_{j}^{\dagger},$ (4.9)

which upon comparing to the relevant equation in (4.4) provides the left
eigenvector of $H^{\star}$

$\displaystyle\hat{\chi}^{T}_{j}=\bar{\xi}_{j}^{\dagger}$ (4.10)

with eigenvalue $\lambda_{j}^{\star}$. Moreover, using the symmetry (4.1) and
the right eingenvector equation (3.4) one can write the following left
eingenvector equations

$\displaystyle(\xi^{\dagger}\sigma_{-})H=-\lambda^{\star}(\xi^{\dagger}\sigma_{-}),$
(4.11) $\displaystyle(\xi^{\dagger}{\cal
G})H^{\star}=-\lambda^{\star}(\xi^{\dagger}{\cal G}),$ (4.12)

where in order to get the last relationship we have used the form of
$\sigma_{-}$ in (4.2). So, by comparing (4.12) with the left-eigenvector
equation in (4.4) one can conclude that for each left eigenvector
$\hat{\chi}^{T}_{j}$ with eigenvalue $\lambda_{j}$ one must have another
eigenvector $(\xi^{\dagger}{\cal G})$ with eigenvalue $-\lambda^{\star}_{j}$,
i.e.

$\displaystyle\hat{\chi}^{T}_{-\lambda_{j}^{\star}}=(\xi^{\dagger}_{\lambda_{j}}{\cal
G}),$ (4.13)

with $\xi_{j}$ being the eigenvector of $H$ with eigenvalue $\lambda_{j}$
defined in (3.4).

Analogous construction can be performed for the companion Hamiltonian
$\widetilde{H}$. So, let us define the right and left eigenvector equations
for $\widetilde{H}^{\star}$

$\displaystyle\widetilde{H}^{\star}\phi_{j}=\lambda_{j}\,\phi_{j},\,\,\,\,\hat{\phi}^{T}_{j}\widetilde{H}^{\star}=\lambda_{j}\,\hat{\phi}^{T}_{j}.$
(4.14)

So, taking the transpose of the left eigenvalue equation in (4.14) and
subsequently considering the pseudo-anti-Hermiticity relationship (4.3) one
can show the next relationship

$\displaystyle\phi_{\widetilde{j}}={\cal G}^{-1}\hat{\phi}_{j},$ (4.15)

where $\phi_{\widetilde{j}}$ denotes a right eigenvector with eigenvalue
$(-\lambda_{j})$.

Similarly, one can relate the above left eigenvector $\hat{\phi}^{T}$ to the
left eigenvector of type $\bar{\eta}^{T}$ in (3.16) for the conjugate
eigenvalue $\lambda_{j}^{\star}$

$\displaystyle\hat{\phi}^{T}_{j}=\bar{\eta}_{j}^{\dagger}.$ (4.16)

Moreover, using the symmetry (4.3) and the right eingenvector equation (3.16)
one can write the following left eingenvector equations

$\displaystyle(\eta^{\dagger}\sigma_{-})\widetilde{H}=-\lambda^{\star}(\eta^{\dagger}\sigma_{-}),$
(4.17) $\displaystyle(\eta^{\dagger}{\cal
G})\widetilde{H}^{\star}=-\lambda^{\star}(\eta^{\dagger}{\cal G}).$ (4.18)

So, by comparing (4.18) with the left-eigenvector equation in (4.14) one can
conclude that for each left eigenvector $\hat{\phi}^{T}_{j}$ with eigenvalue
$\lambda_{j}$ one must have another eigenvector $(\eta^{\dagger}{\cal G})$
with eigenvalue $-\lambda^{\star}_{j}$, i.e.

$\displaystyle\hat{\phi}^{T}_{-\lambda_{j}^{\star}}=(\eta^{\dagger}_{\lambda_{j}}{\cal
G}),$ (4.19)

with $\eta_{j}$ being the right eigenvector of $\widetilde{H}$ with eigenvalue
$\lambda_{j}$ defined in (3.16).

So, in contradistinction to the pseudo-chiral symmetry (3.2) which ensures
that for each energy level $\lambda$ there exists the opposite $-\lambda$,
i.e. the pairs $(\lambda,-\lambda)$, the symmetries (4.1) and (4.3) ensure the
energy levels in pairs with identical imaginary part and opposite real part,
i.e. the pairs $(\lambda,-\lambda^{\star})$. Remarkably, in our constructions
of the next section, we will provide analytic realizations of these eigenvalue
and eigenvector symmetries for a continuum of spinor bound states and extended
wave states in the localized continuum.

We write the biorthogonality relationship in this case. So, from (4.4) one can
get

$\displaystyle\hat{\chi}^{T}_{i}H^{\star}\chi_{j}=\lambda_{j}\,\hat{\chi}^{T}_{i}\chi_{j},\,\,\,\,\hat{\chi}^{T}_{i}H^{\star}\chi_{j}=\lambda_{i}\,\hat{\chi}^{T}_{i}\chi_{j},$
(4.20)

which implies

$\displaystyle
0=(\lambda_{i}-\lambda_{j})\hat{\chi}^{T}_{i}\chi_{j}\,\,\,\,\Rightarrow\,\,\,\,\hat{\chi}^{T}_{i}\chi_{j}=\delta_{ij}.$
(4.21)

Similar construction from (4.14) allows one to write

$\displaystyle
0=(\lambda_{i}-\lambda_{j})\hat{\phi}^{T}_{i}\phi_{j}\,\,\,\,\Rightarrow\,\,\,\,\hat{\phi}^{T}_{i}\phi_{j}=\delta_{ij}.$
(4.22)

Table 1: Pseudo-chiral and anti-pseudo-(anti-)Hermitian symmetries and representations | Pseudo-chiral | Anti-pseudo-Hermitian
---|---|---
Symmetry |  ${\cal G}X{\cal G}^{-1}=-X^{T}$, $X\equiv\\{H,\widetilde{H}\\}$ |  $\sigma_{-}X\sigma_{-}^{-1}=-X^{\dagger}$, $X\equiv\\{H,\widetilde{H}\\}$
Right-eigenvectors | $H\xi=\lambda\,\xi,\,\,\,\,\widetilde{H}\eta=\lambda\,\eta$ | $H^{\star}\chi=\lambda\,\chi,\,\,\,\,\widetilde{H}^{\star}\phi=\lambda\,\phi$
|  $\Gamma H\Gamma^{-1}=\widetilde{H}$ $\rightarrow$ $\eta=\Gamma\xi$, $\bar{\eta}=\Gamma\bar{\xi}$ |  $\Gamma H^{\star}\Gamma^{-1}=\widetilde{H}^{\star}$ $\rightarrow$ $\phi=\Gamma\chi$, $\hat{\phi}=\Gamma\hat{\chi}$
Left-eigenvectors | $\bar{\xi}^{T}H=\lambda\,\bar{\xi}^{T},\,\,\,\,\bar{\eta}^{T}\widetilde{H}=\lambda\,\bar{\eta}^{T}$ | $\hat{\chi}^{T}H^{\star}=\lambda\,\hat{\chi}^{T},\,\,\,\,\hat{\phi}^{T}\widetilde{H}^{\star}=\lambda\,\hat{\phi}^{T}$
|  $\Gamma H\Gamma^{-1}=\widetilde{H}$ $\rightarrow$ $\bar{\eta}=\Gamma\bar{\xi}$ |  $\Gamma H^{\star}\Gamma^{-1}=\widetilde{H}^{\star}$ $\rightarrow$ $\hat{\phi}=\Gamma\hat{\chi}$
Eigenvalues | $\\{\lambda,\,\,\,\,-\lambda\\}$ | $\\{\lambda,\,\lambda^{\star},\,-\lambda^{\star}\\}$
R/L-eigenvectors, $(\lambda,-\lambda)$ | $\bar{\xi}_{-\lambda}={\cal G}\xi_{\lambda},\,\,\,\,\bar{\eta}_{-\lambda}={\cal G}\eta_{\lambda}$ |  $\chi_{-\lambda}={\cal G}^{-1}\hat{\chi}_{\lambda}$, $\phi_{-\lambda}={\cal G}^{-1}\hat{\phi}_{\lambda}$
R/L-eigenvectors, $(\lambda,\pm\lambda^{\star})$ | - |  $\hat{\chi}_{-\lambda^{\star}}=-{\cal G}\xi^{\star}_{\lambda}$, $\hat{\phi}_{-\lambda^{\star}}=-{\cal G}\eta^{\star}_{\lambda}$
| - |  $\hat{\chi}_{\lambda^{\star}}=-\bar{\xi}^{\star}_{\lambda}$, $\hat{\phi}_{\lambda^{\star}}=-\bar{\eta}^{\star}_{\lambda}$
Biorthogonality | $\bar{\xi}^{T}_{i}\xi_{j}=\delta_{ij},\,\,\,\,\bar{\eta}^{T}_{i}\eta_{j}=\delta_{ij}$ | $\hat{\chi}_{i}^{T}\chi_{j}=\delta_{ij},\,\,\,\,\hat{\phi}_{i}^{T}\phi_{j}=\delta_{ij}$
Majorana modes ($\lambda=0$) | $\bar{\xi}^{T}_{0}\xi_{0}=0,\,\,\,\,\bar{\eta}^{T}_{0}\eta_{0}=0$ | $\hat{\chi}_{0}^{T}\chi_{0}=0,\,\,\,\,\hat{\phi}_{0}^{T}\phi_{0}=0$

The Table 1 summarizes the main results related to the pseudo-chiral and
$(\sigma_{-})$-anti-pseudo-Hermitian symmetries and their relevant
representations, respectively, for the set of Hamiltonians
$\\{H,\widetilde{H}\\}$ and $\\{H^{\star},\widetilde{H}^{\star}\\}$. So, we
have organized for each type of symmetry, the corresponding eigenvectors and
their mutual relationships, as well as the allowed values of the spectral
parameter $\lambda$ permitted by the relevant symmetry. The corresponding
biorthogonality relationships and the appearance of the Majorana zero modes at
the EP $\lambda=0$ and the coalescence of the eigenvectors are also included.

###  4.2 Some properties of pseudo-Hermitian Hamiltonians

Let us discuss some theorems and properties of pseudo-Hermiticity and anti-
pseudo-Hermiticity for NH Hamiltonians following the approach in [6, 7, 8,
37], which are relevant to our model. A NH Hamiltonian $H$ is said to be a
$\zeta$-pseudo-Hermitian Hamiltonian if one has

$\displaystyle\zeta H\zeta^{-1}=H^{\dagger},$ (4.23)

for $\zeta$ being a linear invertible Hermitian operator. Notice that for
$\zeta={\mathbb{1}}$ one recovers a Hermitian Hamiltonian, i.e.
$H^{\dagger}=H$. On the other hand, a $\sigma-$anti-pseudo-Hermitian
Hamiltonian is defined as [7, 37]

$\displaystyle\sigma H\sigma^{-1}=H^{\dagger},$ (4.24)

for $\sigma$ being an anti-linear Hermitian automorphism. These types of anti-
linear operators appearing in the definition of pseudo-Hermicity have further
been classified into $\sigma_{\pm}$, Hermitian [37] and anti-Hermitian ones
[7], respectively. The anti-linear operator $\sigma_{-}$ in (4.1) belongs to
the anti-Hermitian type in this classification; so, $H$ in (2.17) which
satisfies (4.1), can be dubbed as $(\sigma_{-})-$anti-pseudo-(anti-)Hermitian
Hamiltonian. An equivalence between a $\zeta-$pseudo-Hermicity and a
$\sigma-$anti-pseudo-Hermiticity has been established using spectral
decomposition techniques in [7, 37].

Next, we summarize the main properties of pseudo-Hermitian operators [6, 8] to
be useful for our discussions of the spectra of the pseudo-Hermitian
Hamiltonian (2.17). Let $H:{\cal H}\rightarrow{\cal H}$ be a diagonalizable
linear operator acting on a Hilbert space ${\cal H}$ and $\zeta:{\cal
H}\rightarrow{\cal H}$ be a linear Hermitian automorphism. Then

1). $H$ is pseudo-Hermitian if and only if its eigenvalues are real or come in
complex-conjugate pairs.

2). if H is pseudo-Hermitian with respect to two linear Hermitian
automorphisms $\zeta_{1}$ and $\zeta_{2}$, then $\zeta_{1}^{-1}\zeta_{2}$
generates a symmetry of H, i.e.

$\displaystyle[\zeta_{1}^{-1}\zeta_{2},H]=0.$ (4.25)

Moreover, in [7] it has been shown that every diagonalizable Hamiltonian
possessing an antilinear symmetry is necessarily pseudo-hermitian. This
applies, in particular, to diagonalizable PT-symmetric Hamiltonians. The proof
of the equivalence of pseudo-hermiticity and the presence of antilinear
symmetries has been performed provided that the Hamiltonian is diagonalizable
and has a discrete spectrum. It used the fact that such Hamiltonian $H$ is
anti-pseudo-hermitian, i.e. there is an antilinear, Hermitian, invertible
operator as $\sigma$ in (4.24). The analogous properties relevant to field
theory models, to our knowledge, have not yet been proved; however, it is
expected that some of them must hold, with relevant modifications, for the NH
Hamiltonians in the field theory context. In fact, we will see below that the
eigenvalues of the field theory NH model described by the $(\sigma_{-})-$anti-
pseudo-Hermitian Hamiltonian $H$ in (2.17) come in complex-conjugate pairs and
some of the Hamiltonian symmetries (4.25) can be realized provided that the
complex scalar $\Phi$ components satisfy some definite parities under space
inversion.

###  4.3 ${\cal P}$, ${\cal T}$ and ${\cal PT}$ symmetries

So, in order to examine additional symmetries we consider the ${\cal P}$,
${\cal T}$ and ${\cal PT}$ symmetry transformations of the NH Hamiltonian
(2.17). Let us define the next symmetry transformations for the scalar field
components $\Phi_{a}$, the complex parameter $z$ and the space coordinate $x$

$\displaystyle\Phi$ $\displaystyle\equiv$ $\displaystyle\Phi_{1}+i\Phi_{2},$
(4.26) $\displaystyle{\cal P}\Phi_{a}{\cal P}$ $\displaystyle=$
$\displaystyle\nu_{a}\Phi_{a},\,\,\,\,\,\,\,\nu_{a}=\pm 1,\,\,\,a=1,2.$ (4.27)
$\displaystyle{\cal P}x{\cal P}$ $\displaystyle=$
$\displaystyle-x,\,\,\,\,\,\,\,{\cal T}z{\cal T}=z^{\star}.$ (4.28)

Next, we will assume the scalar field components $\Phi_{a}\,(a=1,2)$ to
possess definite parities $\nu_{a}=\pm 1$ as in (4.27). So, let us consider
the following cases: Case A. $\nu_{1}=+1,\,\,\nu_{2}=+1$. Case B.
$\nu_{1}=+1,\,\,\nu_{2}=-1$. Case C. $\nu_{1}=-1,\,\,\nu_{2}=+1$. And Case D.
$\nu_{1}=-1,\,\,\nu_{2}=-1$.

One can show the next relationships for the Hamiltonian $H$ in (2.17) upon
making the relevant transformations in its matrix representation.

Case A. $\nu_{1}=+1,\,\,\nu_{2}=+1$. In this case one has

$\displaystyle{\cal P}\gamma_{5}H\gamma_{5}{\cal P}^{-1}=-H.$ (4.29)

This implies the anti-symmetry $\\{{\cal P}\gamma_{5},H\\}=0$.

Case B. $\nu_{1}=+1,\,\,\nu_{2}=-1$. In this case one has for $M$ real

$\displaystyle{\cal P}H{\cal P}^{-1}=H^{\dagger}.$ (4.30)

Making use of the anti-pseudo-Hermicity relationship (4.1) together with
(4.30) one can get the anti-symmetry $\\{\sigma_{-}^{-1}{\cal P}\,,\,H\\}=0.$

Case C. $\nu_{1}=-1,\,\,\nu_{2}=+1$.

$\displaystyle{\cal P}{\cal T}\gamma_{5}H[{\cal P}{\cal T}\gamma_{5}]^{-1}=H.$
(4.31)

So, in this case the NH Hamiltonian $H$ possesses the antilinear symmetry
${\cal P}{\cal T}\gamma_{5}$, i.e. $[{\cal P}{\cal T}\gamma_{5}\,,\,H]=0$. So,
the antilinear ${\cal P}{\cal T}$-like symmetry (4.31) must be taken into
account together with the pseudo-chirality and anti-pseudo-Hermiticity of the
model in order to examine its spectral properties in this case.

Case D. $\nu_{1}=-1,\,\,\nu_{2}=-1$.

In this case one has the relationship

$\displaystyle\bar{\sigma}_{-}H\bar{\sigma}_{-}^{-1}$ $\displaystyle=$
$\displaystyle-H^{\dagger},$ (4.32) $\displaystyle\bar{\sigma}_{-}$
$\displaystyle\equiv$ $\displaystyle{\cal P}{\cal T}.$ (4.33)

Notice that $\bar{\sigma}_{-}$ is an antilinear operator. So, this Hamiltonian
is a (anti)-pseudo-anti-Hermitian Hamiltonian as in (4.1), however in this
particular case under the ${\cal P}{\cal T}$ operator $\bar{\sigma}_{-}$.
Considering the both pseudo-anti-Hermiticity relationships (4.32) and (4.1)
one can show that

$\displaystyle\sigma^{-1}_{-}\bar{\sigma}_{-}H(\sigma^{-1}_{-}\bar{\sigma}_{-})^{-1}$
$\displaystyle=$ $\displaystyle H$ (4.34) $\displaystyle({\cal P}{\cal
G})H({\cal P}{\cal G})^{-1}$ $\displaystyle=$ $\displaystyle H,$ (4.35)

where the product $\sigma^{-1}_{-}\bar{\sigma}_{-}={\cal P}{\cal G}$ has been
inserted in the eq. (4.35). So, (4.35) implies the symmetry

$\displaystyle[H,{\cal P}{\cal G}]=0.$ (4.36)

Then, (4.36) establishes a symmetry of type $\sigma^{-1}_{-}\bar{\sigma}_{-}$
for a Hamitonian which is pseudo-anti-Hermitian with respect to two different
anti-linear operators $\sigma_{-}$ and $\bar{\sigma}_{-}$. So, it is an
analogous symmetry to the one in (4.25) which establishes a type of symmetry
($\zeta_{1}^{-1}\zeta_{2}$) for pseudo-Hermitian Hamiltonians, i.e. with
pseudo-Hermiticity property with respect to two different linear Hermitian
operators $\zeta_{1}$ and $\zeta_{2}$.

Below, in the context of the tau-function approach, we will perform the
construction of the right and left eigenvectors satisfying the pseudo-
chirality and pseudo-Hermitian properties discussed above, and for a special
solution we will show a NH Majorana zero mode-soliton duality description of
the model, such that the static scalar field is a self-consistent solution of
the equation (2.22) and (2.29). In fact, we will realize the theorem 1) above
by showing that the both NH Hamiltonians in (2.17) possess complex-conjugate
pairs of eigenvalues.

Table 2:  ${\cal P}$, ${\cal T}$ and ${\cal PT}$ symmetries Case | Parity:$(\nu_{1},\,\nu_{2})$ |  $H^{\prime}s$ property | (Anti-)symmetry
---|---|---|---
A |  $\left(+\,\,\,\,+\right)$ | Pseudo-chiral+Anti-pseudo-Hermitian | $\\{\gamma_{5}{\cal P}\,,\,H\\}=0$
B |  $\left(+\,\,\,\,-\right)$ | Anti-pseudo-Hermitian |  $\\{\sigma_{-}^{-1}{\cal P}\,,\,H\\}=0$ $(M\in\leavevmode\hbox{\rm I\kern-1.79993ptR}$)
C |  $\left(-\,\,\,\,\,+\right)$ | Pseudo-chiral+Anti-pseudo-Hermitian | $[\gamma_{5}{\cal P}{\cal T}\,,\,H]=0$
D |  $\left(-\,\,\,\,\,-\right)$ | Anti-pseudo-Hermitian | $[{\cal P}{\cal G}\,,\,H]=0$

In the Table 2 we summarize the main results of the cases $A,B,C$ and $D$
discussed above. We present the interplay of the ${\cal P}$, ${\cal T}$ and
${\cal PT}$ symmetry transformations together with the pseudo-chiral and anti-
pseudo-Hermiticity properties, which imply a (anti-)symmetry of the
Hamiltonian (last column of the Table 2) for each set of parities
($\nu_{1},\,\nu_{2}$) of the scalar fields $\Phi_{1}$ and $\Phi_{2}$,
respectively.

##  5 NH topological solitons coupled to the fermions

In this section we find exact analytical solutions of the system of eqs.
(2.11)-(2.12) and (2.21) associated to the NH model (2.1) and search for
nontrivial topological configurations for the complex field $\Phi$ interacting
with the bound states formed by the two fermion fields $\xi$ and $\eta$. The
Hermitian case has been considered before by defining a submodel for $\Phi$
being a real scalar and by imposing the reality condition
$\widetilde{\psi}\equiv\psi^{\star}$ to the model (2.1); so, the reduced model
defined a Dirac field interacting with real Toda solitons [28, 31, 29, 20, 25,
19, 18].

###  5.1 Topological solitons and spinor bound states

The soliton solutions, as well as the bound states of the Hermitian version of
this model has been considered in [18] through the tau function formalism.
Here we perform the construction of the general complex 1-solitons and spinor
bound state configurations of the non-Hermitian model (2.1). In this context,
for a suitable subspace in parameter space, it will emerge also some extended
wave states in the localized continuum.

Since the pair of first order differential equations (2.11)-(2.12) for the
spinors plus the first order equation (2.22) for $\Phi$ imply the second order
differential eq. for the complex scalar (2.21), each set of solutions of the
first order system of equations will be solutions of the second order
differential equation (2.21). So, it is sufficient to consider the system of
equations (2.11)-(2.12) and (2.22) in order to find the complex solitons and
related bound state solutions.

In the next steps we rewrite the equations of motion of the model in terms of
the tau functions $\tau_{0,1}$, $\tau_{A,B},\widetilde{\tau}_{A,B}$ as

$\displaystyle e^{i\beta\Phi}$ $\displaystyle=$
$\displaystyle\rho\,(\frac{\tau_{0}}{\tau_{1}})^{2},$ (5.1)
$\displaystyle\xi_{A}$ $\displaystyle=$
$\displaystyle\frac{\tau_{A}}{\tau_{0}},\,\,\,\xi_{B}=\frac{\tau_{B}}{\tau_{1}},\,\,\,\eta_{A}=\frac{\widetilde{\tau}_{A}}{\tau_{1}},\,\,\,\eta_{B}=\frac{\widetilde{\tau}_{B}}{\tau_{0}},$
(5.2)

with $\rho$ being a constant complex parameter. Next, substituting the field
components (5.1)-(5.2) into the system of first order equations (2.11)-(2.12)
and (2.22) one has, respectively

$\displaystyle
i\lambda\tau_{1}\tau_{B}-M\rho\,\tau_{0}\tau_{A}+\tau_{B}\tau^{\prime}_{1}-\tau_{1}\tau^{\prime}_{B}$
$\displaystyle=$ $\displaystyle 0\,$ (5.3) $\displaystyle
i\lambda\tau_{0}\tau_{A}+M\rho^{-1}\tau_{1}\tau_{B}-\tau_{A}\tau^{\prime}_{0}+\tau_{0}\tau^{\prime}_{A}$
$\displaystyle=$ $\displaystyle 0\,$ (5.4) $\displaystyle
i\lambda\tau_{0}\widetilde{\tau}_{B}+M\rho^{-1}\tau_{1}\widetilde{\tau}_{A}-\widetilde{\tau}_{B}\tau^{\prime}_{0}+\tau_{0}\widetilde{\tau}^{\prime}_{B}$
$\displaystyle=$ $\displaystyle 0\,$ (5.5) $\displaystyle
i\lambda\tau_{1}\widetilde{\tau}_{A}-M\rho\,\tau_{0}\widetilde{\tau}_{B}+\widetilde{\tau}_{A}\tau^{\prime}_{1}-\tau_{1}\widetilde{\tau}^{\prime}_{A}$
$\displaystyle=$ $\displaystyle 0.$ (5.6)

and

$\displaystyle
2i(\tau^{\prime}_{0}\tau_{1}-\tau_{0}\tau^{\prime}_{1})+M\beta(\tau_{A}\widetilde{\tau}_{A}+\tau_{B}\widetilde{\tau}_{B})=0.$
(5.7)

Next, let us assume the following tau functions

$\displaystyle\tau_{1}$ $\displaystyle=$ $\displaystyle
1+a\,e^{2kx},\,\,\,\,\tau_{0}=1+b\,e^{2kx},$ (5.8) $\displaystyle\tau_{A}$
$\displaystyle=$ $\displaystyle c\,e^{kx},\,\,\,\tau_{B}=d\,e^{kx},$ (5.9)
$\displaystyle\widetilde{\tau}_{A}$ $\displaystyle=$
$\displaystyle\widetilde{c}\,e^{kx},\,\,\,\widetilde{\tau}_{B}=\widetilde{d}\,e^{kx},$
(5.10)

with the complex parameters $a,b,c,d,\widetilde{c},\widetilde{d}$ and $k$. So,
the above tau functions satisfy the system of equations (5.3)-(5.6) and (5.7)
provided that

$\displaystyle a$ $\displaystyle=$
$\displaystyle\frac{\widetilde{c}c}{2M}\frac{\beta^{2}z^{3}}{(z^{2}-1)^{2}},\,\,\,\,\,\,\,\widetilde{d}=\frac{\widetilde{c}\,c}{d},\,\,\,\,\,\,\,\rho=-i\frac{d}{c}z,$
(5.11) $\displaystyle b$ $\displaystyle=$
$\displaystyle\frac{\widetilde{c}c}{2M}\frac{\beta^{2}\,z}{(z^{2}-1)^{2}},\,\,\,\,\,z\equiv-\frac{\lambda}{M}\pm\sqrt{\frac{\lambda^{2}}{M^{2}}-1}$
(5.12) $\displaystyle\lambda$ $\displaystyle=$
$\displaystyle-\frac{M}{2}(z+z^{-1}),\,\,\,\,\,\,\,k=\frac{iM}{2}(z-z^{-1}).$
(5.13)

From the above equations one can write

$\displaystyle\lambda^{2}+k^{2}=M^{2}.$ (5.14)

The equation (5.14) can be regarded as a dispersion relationship between the
complex energy $\lambda$ and the complex wave number $k$ of the quasi-
particles. Notice that this equation is invariant under the transformations

$\displaystyle\lambda$ $\displaystyle\rightarrow$ $\displaystyle-\lambda$
(5.15) $\displaystyle k$ $\displaystyle\rightarrow$ $\displaystyle-k.$ (5.16)

So, the eigenvalues come in pairs of type $(\lambda,-\lambda)$. This is in
accordance to the pseudo-chirality symmetry property discussed in section 3.
Then, one can argue that the eigenvectors corresponding to this pair of
eigenvalues realize the pseudo-chirality symmetry (3.2) and the left
$\bar{\xi}_{-\lambda}$ and right eigenvectors $\xi_{\lambda}$ can be related
by (3.8).

Next, one can take the complex conjugate of the dispersion relationship (5.14)
and get

$\displaystyle(\lambda^{\star})^{2}+(k^{\star})^{2}=(M^{\star})^{2}.$ (5.17)

Since this equation defines a new eigenvalue $\lambda^{\star}$, one has that
the eigenvalues come in complex-conjugate pairs. Notice that (5.17) is
invariant under
$\lambda^{\star}\rightarrow-\lambda^{\star},\,\,\,k^{\star}\rightarrow-k^{\star}$.
So, from (5.14) and (5.17) and their relevant symmetries under inversion
$\lambda\rightarrow-\lambda$ and $\lambda^{\star}\rightarrow-\lambda^{\star}$,
one can argue that the solutions above describe the corresponding left and
right eigenvectors with the set of eigenvalues
$\\{\lambda,-\lambda,\lambda^{\star},-\lambda^{\star}\\}$.

One can see that the complex dispersion relationship (5.17) arises from the
tau function representation of the eigenvalue equation in (4.4) for the right
eigenvector $\chi$ and the operator $H^{\star}$. In fact, by taking the
complex conjugate of $H\xi=\lambda\xi$ in (3.4) and the relevant tau function
components in (5.1)-(5.2) and subsequent substitution into the corresponding
tau function equations analogous to (5.3)-(5.6) one can get (5.17).

In fact, these solutions become the realizations of the pseudo-chiral and
pseudo-Hermitian symmetries of the model discussed in the sections 3 and 4,
respectively. Remarkably, the relevant spectral properties associated to the
symmetries (3.2) and (4.1) are realized above by providing the set of
eigenvectors with eingenvalues $\\{\lambda,-\lambda\\}$ and
$\\{\lambda,-\lambda^{\star}\\}$, respectively. Then, the eigenvalues come in
complex conjugate pairs and their reversions. This is in accordance to the
first theorem 1) which must hold for pseudo-anti-Hermitian Hamiltonians, once
the equivalence between the pseudo-Hermicity and the anti-pseudo-Hermicity
properties is assumed [7, 37].

Next, we define the complex parameters such that

$\displaystyle M$ $\displaystyle\equiv$ $\displaystyle M_{1}+i\,M_{2},$ (5.18)
$\displaystyle\lambda$ $\displaystyle\equiv$
$\displaystyle\lambda_{1}+i\,\lambda_{2},$ (5.19) $\displaystyle k$
$\displaystyle\equiv$ $\displaystyle k_{1}+i\,k_{2},$ (5.20)

and

$\displaystyle
a=|a|e^{i\alpha_{1}},\,b=|b|e^{i\alpha_{2}},\,\rho=|\rho|e^{i\delta},\,\,c=|c|e^{i\zeta_{1}},\,\,d=|d|e^{i\zeta_{2}}.$
(5.21)

So, from the eqs. (5.11)-(5.12) the complex variable $z$ can be written as

$\displaystyle
z=\sqrt{|a|/|b|}\,e^{i\sigma},\,\,\,\,\sigma\equiv\frac{\alpha_{1}-\alpha_{2}}{2}.$
(5.22)

Next, taking into account (5.18)-(5.20) and (5.11)-(5.13) one can write

$\displaystyle
z=\sqrt{\frac{(\lambda_{1}-k_{2})^{2}+(\lambda_{2}+k_{1})^{2}}{|M|^{2}}}\,e^{i\sigma},\,\,\,\,\sigma\equiv\arctan{\Big{[}\frac{M_{2}(k_{2}-\lambda_{1})+M_{1}(k_{1}+\lambda_{2})}{M_{2}(k_{1}+\lambda_{2})-M_{1}(k_{2}-\lambda_{1})}\Big{]}}.$
(5.23)

Notice that from (5.14) one can consider a complex function
$\lambda=\pm\sqrt{M^{2}-k^{2}}$ defined on a Riemann surface consisting of two
sheets, with $\pm M$ being the branch points, which are necessary to make the
square-root function single valued. However, the convenient parametrization in
the complex $z$ variable defined in (5.13) avoids the introduction of this
two-sheet surface, see below more about this construction.

The scalar field component solutions from (5.1) can be written as

$\displaystyle\Phi_{1}$ $\displaystyle=$
$\displaystyle\frac{2}{\beta}\arctan{\Big{[}i\,\frac{e^{-i\frac{\delta}{2}}\tau_{0}^{\star}\tau_{1}-e^{i\frac{\delta}{2}}\tau_{0}\tau_{1}^{\star}}{e^{i\frac{\delta}{2}}\tau_{0}\tau^{\star}_{1}+e^{-i\frac{\delta}{2}}\tau_{0}^{\star}\tau_{1}}\Big{]}}$
(5.24) $\displaystyle\Phi_{2}$ $\displaystyle=$
$\displaystyle-\frac{1}{\beta}\log{\Big{[}|\rho|\frac{\tau_{0}^{\star}\tau_{0}}{\tau_{1}^{\star}\tau_{1}}\Big{]}}.$
(5.25)

Next, taking into account the tau functions (5.8)-(5.10) one can write

$\displaystyle\Phi_{1}$ $\displaystyle=$
$\displaystyle\frac{2}{\beta}\arctan{\Big{[}\frac{\sin{\frac{\delta}{2}}+(|b|\sin{(\Xi_{2}(x))}-|a|\sin{(\Xi_{1}(x))}e^{2k_{1}x}+|a||b|\sin{(\Theta)}e^{4k_{1}x}}{\cos{\frac{\delta}{2}}+(|b|\cos{(\Xi_{2}(x))}+|a|\cos{(\Xi_{1}(x))}e^{2k_{1}x}+|a||b|\cos{(\Theta)}e^{4k_{1}x}}\Big{]}}$
(5.26) $\displaystyle\Xi_{1}(x)$ $\displaystyle\equiv$ $\displaystyle
2k_{2}x+\alpha_{1}-\frac{\delta}{2},\,\,\,\,\Xi_{2}(x)\equiv
2k_{2}x+\alpha_{2}-\delta/2,\,\,\,\,\,\,\Theta\equiv\alpha_{2}-\alpha_{1}+\frac{\delta}{2}$

and

$\displaystyle\Phi_{2}$ $\displaystyle=$
$\displaystyle-\frac{1}{\beta}\log{\Big{[}|\rho|\,\,\frac{1+2|b|\cos{(2k_{2}x+\alpha_{2})}e^{2k_{1}x}+|b|^{2}e^{4k_{1}x}}{1+2|a|\cos{(2k_{2}x+\alpha_{1})}e^{2k_{1}x}+|a|^{2}e^{4k_{1}x}}\big{]}}.$
(5.27)

For the wave number with real component different from zero, i.e. $k_{1}\neq
0$, one can define the next topological invariants for each component of the
complex scalar field as

$\displaystyle\Phi_{1}(+\infty)-\Phi_{1}(-\infty)$ $\displaystyle=$
$\displaystyle\pm\frac{2}{\beta}(\alpha_{1}-\alpha_{2}),$ (5.28)
$\displaystyle=$ $\displaystyle\pm\frac{4}{\beta}\,\sigma,$ (5.29)
$\displaystyle=$ $\displaystyle\pm\frac{4}{\beta}\,\arg{z}$ (5.30)
$\displaystyle\Phi_{2}(+\infty)-\Phi_{2}(-\infty)$ $\displaystyle=$
$\displaystyle\pm\frac{2}{\beta}\log{\frac{|b|}{|a|}},$ (5.31)
$\displaystyle=$ $\displaystyle\pm\frac{4}{\beta}\log{|z|}$ (5.32)

where $\sigma$ ($\sigma=\arg{z}$) and $|z|$ are the argument and modulus of
the complex parameter $z$, respectively, presented in (5.22)-(5.23).
Remarkably, from the results (5.30) and (5.32) one can write the next
suggestive formula

$\displaystyle e^{i\beta(\Phi(+\infty)-\Phi(-\infty))}$ $\displaystyle=$
$\displaystyle
e^{i\beta(\Phi_{1}(+\infty)-\Phi_{1}(-\infty))}e^{-\beta(\Phi_{2}(+\infty)-\Phi_{2}(-\infty))}$
(5.33) $\displaystyle=$ $\displaystyle z^{\pm 4}.$ (5.34)

This implies that

$\displaystyle\Phi(+\infty)-\Phi(-\infty)=\pm\frac{4}{i\beta}\log{z}.$ (5.35)

Next, we will regard the fist order differential equation (2.22) as an
equivalence relationship between the topological charge density associated to
the complex scalar and the spinor current density defined by a sum of the
spinor bilinear terms. So, integrating (2.22) in the whole real line, one can
define the next topological charges

$\displaystyle[\Phi_{1}+i\Phi_{2}]\Big{|}_{x=-\infty}^{x=+\infty}$
$\displaystyle=$
$\displaystyle-\frac{\beta}{2}\int^{+\infty}_{-\infty}(\eta_{A}\xi_{A}+\eta_{B}\xi_{B})\,dx,$
(5.36) $\displaystyle{\cal Q}_{1}$ $\displaystyle\equiv$
$\displaystyle\frac{\beta}{2\pi}[\Phi_{1}(+\infty)-\Phi_{1}(-\infty)]$ (5.37)
$\displaystyle{\cal Q}_{2}$ $\displaystyle\equiv$
$\displaystyle\frac{\beta}{2\pi}[\Phi_{2}(+\infty)-\Phi_{2}(-\infty)].$ (5.38)

Then, one can write

$\displaystyle{\cal Q}_{1}$ $\displaystyle\equiv$
$\displaystyle\mbox{Re}\Big{[}-\beta^{2}\int^{+\infty}_{-\infty}\xi_{A}\xi_{B}\,dx\Big{]}$
(5.39) $\displaystyle{\cal Q}_{2}$ $\displaystyle\equiv$
$\displaystyle\mbox{Im}\Big{[}-\beta^{2}\int^{+\infty}_{-\infty}\xi_{A}\xi_{B}\,dx\Big{]}$
(5.40)

where the relationship (2.19) between the $\xi$ and $\eta$ fields has been
used.

In some cases the solutions $\Phi_{1,2}$ exhibit vanishing topological
charges. In fact, for $\alpha_{2}=\alpha_{1}$ ($\arg{z}=0$) one has
$\Phi_{1}(+\infty)=\Phi_{1}(-\infty)$, while for $|b|=|a|$ ($|z|=1$) one gets
$\Phi_{2}(+\infty)=\Phi_{2}(-\infty)$. These would imply ${\cal Q}_{1}=0$ or
${\cal Q}_{2}=0$ in (5.37)-(5.38), respectively.

Next, let us discuss the analytical properties of the multi-valued function
$\lambda$ associated to the complex plane defined by the variable $z$. The so-
called uniformization variable $z$ has been introduced for the energy
$\lambda$ and wave number $k$ in (5.13). This parametrization avoids the
introduction of the Riemann surface consisting of two sheets with relevant
branching points as mentioned above. Notice that the relationship (5.14) will
hold for any arbitrary complex number $z$ in the entire complex plane
excluding the origin, i.e. $z\in\leavevmode\hbox{\,$\vrule
height=6.45831pt,width=0.4pt,depth=0.0pt\kern-3.00003pt{\rm
C}$}\backslash\\{0\\}$.

From (5.13) one can show the next useful identity

$\displaystyle\lambda$ $\displaystyle=$
$\displaystyle-\frac{M}{2}(z+z^{-1})\,\,\,\mbox{and}\,\,\,\,k=\frac{iM}{2}(z-z^{-1})\,\,\,\
\Rightarrow\,\,\,z^{2}=\frac{\lambda+ik}{\lambda-ik},\,\,\,\,(M\neq 0).$
(5.41)

Notice that the phase and modulus of the uniformization complex variable $z$
encode the topological charges of the two types of solitons as shown in (5.30)
and (5.32). So, we write the complex variable $z$ in polar form such that the
modulus $|z|$ and $\arg{z}$ depend on the real and complex parts of $\lambda$
and $k$ as follow

$\displaystyle z=\left(\frac{|\lambda+ik|}{|\lambda-
ik|}\right)^{1/2}\,e^{i\sigma},\,\,\,\,\,\,\sigma=\arctan\Big{\\{}\frac{|k|^{2}-|\lambda|^{2}+|\lambda-
ik||\lambda+ik|}{\lambda\,k^{\star}+\lambda^{\star}k}\Big{\\}}.$ (5.42)

Then, using the results (5.30) and (5.32) together with the polar
decomposition (5.42) into the formulas (5.37) and (5.38) for the topological
charges one has

$\displaystyle{\cal Q}_{1}$ $\displaystyle=$
$\displaystyle\pm\frac{2}{\beta\pi}\,\arctan\Big{\\{}\frac{|k|^{2}-|\lambda|^{2}+|\lambda-
ik|\,|\lambda+ik|}{\lambda\,k^{\star}+\lambda^{\star}k}\Big{\\}},$ (5.43)
$\displaystyle{\cal Q}_{2}$ $\displaystyle=$
$\displaystyle\pm\frac{1}{\beta\pi}\log{\frac{|\lambda+ik|}{|\lambda-ik|}}.$
(5.44)

Therefore, one has a family of localized continuum states with complex energy
$\lambda$ and wave number $k$ defining two types of solitons with topological
charges provided in (5.43)-(5.44). The second topological soliton with charge
${\cal Q}_{2}$ owns its existence purely to the non-Hermicity and non-
linearity of the model. In fact for $\lambda$ and $k$ real one has a trivial
topological charge, i.e. from (5.44) one has that ${\cal Q}_{2}=0$. So, the
appearance of the second soliton shows an interesting phenomena in which the
interplay between non-linearity and non-Hermicity of the model gives rise to a
new state provided by the composite topological soliton $\Phi$ with components
(5.26) and (5.27) and the next spinor bound states

$\displaystyle\xi_{A}$ $\displaystyle=$
$\displaystyle|c|e^{-i(k_{2}x+\alpha_{2}-\zeta_{1})}\,\frac{e^{k_{1}x}\left(|b|e^{2k_{1}x}+e^{i(2k_{2}x+\alpha_{2})}\right)}{(1+|b|e^{2k_{1}x})^{2}-2|b|e^{2k_{1}x}(1-\cos{(2k_{2}x+\alpha_{2})})},$
(5.45) $\displaystyle\xi_{B}$ $\displaystyle=$
$\displaystyle|d|e^{-i(k_{2}x+\alpha_{1}-\zeta_{2})}\,\frac{e^{k_{1}x}\left(|a|e^{2k_{1}x}+e^{i(2k_{2}x+\alpha_{1})}\right)}{(1+|a|e^{2k_{1}x})^{2}-2|a|e^{2k_{1}x}(1-\cos{(2k_{2}x+\alpha_{1})})},$
(5.46)

where the tau functions (5.8)-(5.10) have been inserted into (5.2) in order to
write the spinor solutions (5.45)-(5.46). The components $\eta_{A}$ and
$\eta_{B}$ can be shown to satisfy the relationships (2.20), so from
(5.45)-(5.46) one can get $\eta_{A}=\xi_{B}$ and $\eta_{B}=\xi_{A}$. The Fig.
2 shows a representative plot of this type of localized states.

Moreover, the $x-$integrated current density in (2.22) can be written as

$\displaystyle\int_{-\infty}^{+\infty}[\eta_{A}\xi_{A}+\eta_{B}\xi_{B}]\,dx$
$\displaystyle=$
$\displaystyle\frac{|c||d|}{|b|}\Big{\\{}\frac{(\log{\frac{|a|}{|b|}})^{2}+(\alpha_{1}-\alpha_{2})^{2}}{(k_{1}^{2}+k_{2}^{2})\Big{[}(\frac{|a|}{|b|})^{2}+1-2\frac{|a|}{|b|}\cos{(\alpha_{1}-\alpha_{2})}\Big{]}}\Big{\\}}^{1/2}\,\,e^{i\Delta}$
(5.47) $\displaystyle\Delta$ $\displaystyle\equiv$
$\displaystyle\zeta_{1}+\zeta_{2}+\delta_{o}+\alpha_{o}+\nu,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\nu=-\arctan{[\frac{k_{2}}{k_{1}}]},$
(5.48) $\displaystyle\delta_{o}$ $\displaystyle=$
$\displaystyle\arctan{\Big{\\{}\frac{\alpha_{1}-\alpha_{2}}{\log{|a|}-\log{|b|}}}\Big{\\}},\,\,\alpha_{o}=\arctan{\Big{\\{}\frac{|b|\sin{\alpha_{2}}-|a|\sin{\alpha_{1}}}{|a|\cos{\alpha_{1}}-|b|\cos{\alpha_{2}}}}\Big{\\}}.$
(5.49)

So, the above integration (5.47) provides a complex number such that the real
part will be proportional to ${\cal Q}_{1}$ and the imaginary part to ${\cal
Q}_{2}$ as in (5.39)-(5.40), respectively. The equations (5.43)-(5.44)
together with (5.47) will provide two relationships for the parameters.
Additional relationships which must be satisfied by the parameters come from
the biorthogonality relationship $(\bar{\xi}_{\lambda}|\xi_{\lambda})$ in
(3.5) which will be computed below for any complex eigenvalue $\lambda$.

Notice that for the Hermitian case one has $|a|=|b|,\,k_{2}=0$ and since
$\tau_{0}=\tau_{1}^{\star}$ in (5.8) one can take
$\alpha_{1}=\frac{\pi}{2},\,\alpha_{2}=\frac{3\pi}{2}$, and then
$\nu=0,\,\delta_{o}=\frac{\pi}{2},\,\alpha_{o}=\frac{\pi}{2}$. So, it must be
$\zeta_{1}+\zeta_{2}=n\pi$ in order to fix $e^{i\Delta}=\pm 1$, such that the
charge ${\cal Q}_{2}$ vanishes.

So, the right-eigenvector fermion and the complex topological soliton $\Phi$
currents, which satisfy the equivalence relationship (2.22), as well as the
relevant equivalence of the left-eigenvector fermion current to the scalar
field $\bar{\Phi}$ topological current in (2.29), can be regarded as the
analogs of the biorthogonal generalized bulk-boundary correspondence in non-
Hermitian condensed matter systems [21].

One can see that an essential distinction between the Hermitian Toda model
coupled to fermion and its non-Hermitian counterpart is the degrees of freedom
that one can access to; i.e. nonunitary transformations forbidden in the
Hermitian system can be realized in its non-Hermitian version. In fact, the
change in the spectrum from real parameters $(\lambda_{1},k_{1},M_{1})$ to the
complex ones ($\lambda,k,M$) increases the number of parameters that describe
the system. Since topology mainly depends on the underlying manifold, non-
Hermiticity is expected to alter the topological classification of the
solitons related to the scalar field components $\Phi_{1}$ and $\Phi_{2}$. So,
in order to classify the topological sectors of the NH model above the pseudo-
chiral and anti-pseudo-Hermitian symmetries will play an important role.

Next let us discuss the physical origin of the non-Hermiticity. Some physical
effects which introduce non-Hermiticity into Hermitian systems are, gain and
loss, dissipation of energy in open quantum systems, finite life-time of
resonances, etc. In this context, it is usual to ascribe some physical meaning
to the real and imaginary parts of a complex eigenvalue, i.e. the real part
codifies the excitation energy, whereas the imaginary part encrypts the decay
or amplification rate. However, in our constructions above, one can see that
the modulus and phase of the uniformization complex parameter $z$ in (5.41)
acquire some physical meaning. In fact, as mentioned above, the phase,
$\arg{z}$, and modulus, $|z|$, encode the topological charges of the scalar
fields $\Phi_{1}$ and $\Phi_{2}$ in (5.30) and (5.32), respectively.
Remarkably, the circle $|z|=1$ will correspond to the trivial topological
charge associated to the field $\Phi_{2}$. As we will discuss below, the bound
states in the Hermitian system (for $\Phi_{2}=0$) lie on the unit circle
$|z|=1$. So, one can argue that the topological soliton associated to
$\Phi_{2}$ is purely of NH origin. An alternative interpretation into the
realm of non-Hermitian physics, which involves neither gain nor loss, has
recently been introduced in [38], in which the complex eigenvalues emerge from
the amplitudes and phase differences of waves backscattered from the boundary
of insulators. So, we believe that the physical origin of the nonhermiticity
of our model can be understood in the framework of the inverse scattering
transform formalism associated to the model, which deserves a careful
consideration in future.

###  5.2 Spectral topology and winding number

One can analize the spectral properties of the model by examining the behavior
of the curves on the complex plane defined by the uniformization variable $z$
and the energy $\lambda$ and wave number $k$ expressions in (5.41)-(5.42).
Notice that for the bound states in the Hermitian case one has the real
parameters $M=M_{1}$, $\lambda=\lambda_{1}$. So, when
$k=k_{1}(k_{1}\in\leavevmode\hbox{\rm I\kern-1.79993ptR})$ one has the unit
circle $|z|=1$ (see Fig.1). This implies that the bound states lie on this
unit circle ($\lambda_{1}=\pm\sqrt{M_{1}^{2}-k_{1}^{2}}$). However, for purely
imaginary $k=ik_{2}$ one notices that the scattering states are allowed for
$z\in\leavevmode\hbox{\rm I\kern-1.79993ptR}$
($\lambda_{1}=\pm\sqrt{M_{1}^{2}+k_{2}^{2}}$). The so called half-bound states
at the edges of the continuous spectrum are given at $\lambda_{1}=\pm
M_{1}\rightarrow z=\pm 1$. So, the thresholds of the energy gap of the free
massive fermions are at $\lambda_{1}=\pm M_{1}$.

Figure 1: The complex plane defined by the uniformization variable $z$. The
complex energy $\lambda(z)$ (5.13) exhibits a pole at the origin $z=0$ and the
zeros at $z=\pm i$. The points $z=\pm 1$ correspond to the edge states
$k=0\,(\lambda=\pm M)$.

In the non-Hermitian case one has a complex eigenvalue spectrum, which must be
studied on the complex plane, and so leads to the emergence of eigenvalue
topology, or spectral topology. So, one has a continuum of bound states, such
that the energy $\lambda$ is defined as an analytical function of the complex
variable $z$. This additional input of spectral topological concepts is a
unique feature in non-Hermitian systems. In this context, the zero-energy
points at $z=\pm i$ in (5.41) are dubbed as exceptional points (EP) and they
can arise as branch points in the $k-$space eigenvalue manifold as can be
written from (5.14)

$\displaystyle\lambda=\pm\sqrt{M^{2}-k^{2}}.$ (5.50)

As we will see below, the branch points ($k=\pm M$) exhibit exotic topological
phenomena associated with the winding of eigenvalues and eigenvectors. The
complex plane defined by the uniformization variable $z$ turns out to be
convenient for performing the relevant calculations in order to study the
analytical properties of the energy $\lambda(z)$ and wave number $k(z)$
functions, i.e. in order to examine the zero modes and the points
corresponding to the edges of the model, which we will discuss below. The
analytic function $\lambda(z)$ in (5.41) has two zeros and one pole, located
on the imaginary axis at the points $z=\pm i\rightarrow k=\pm M$, each one of
first order, and the pole at the origin $z=0$. Next, let us define the winding
number

$\displaystyle W$ $\displaystyle=$ $\displaystyle\frac{1}{2\pi
i}\oint_{\Gamma}\,\frac{d\lambda}{\lambda}$ (5.51) $\displaystyle=$
$\displaystyle Z-P.$ (5.52)

In the second line above we have used the Cauchy’s argument principle, such
that $Z$ is the number of zeros and $P$ is the number of poles of the function
$\lambda$ inside the contour $\Gamma$ (see Fig. 1). The $Z$ and $P$ must be
counted with their multiplicities and order, respectively. It is interesting
to compute the winding number of $\lambda(z)$ in (5.13) as it traverses the
contour $\Gamma$ encircling the zero modes $\lambda=0$ located at the points
$z=\pm i$ and the pole at $z=0$ in the complex plane $z$. So, for the function
$\lambda(z)$ in (5.41) and using (5.51) one has

$\displaystyle W$ $\displaystyle=$ $\displaystyle\frac{1}{2\pi
i}\oint_{\Gamma}\,\frac{z^{2}-1}{z(z^{2}+1)}\,dz$ (5.53) $\displaystyle=$
$\displaystyle 1.$

Then, one can see that the eq. (5.52) is also verified, i.e. $W=1$, since
$Z=2$ and $P=1$ are the number of zeros and poles of the function $\lambda(z)$
inside $\Gamma$.

##  6 Continuum of bound states/solitons and extended ELC modes

In this section we discuss the general solution presented above and classify
it into two types of solutions: The dual continuum of bound states/solitons
and the extended modes embedded into the localized states (ELC). Moreover, we
also compute analytically the biorthogonality relationships between the
relevant left and right eigenvectors.

###  6.1 Continuum of bound state modes/solitons

In the Hermitian Toda system coupled to fermion, which has recently been
presented in [18], the in-gap and BIC bound states have quantized energies,
whereas the free states form a continuum. We have shown above how this
principle fails for the non-Hermitian Toda system coupled to fermions, by
analyzing the non-Hermitian Hamiltonian with a complex Toda field and the
properties of pseudo-chirality and pseudo-Hermicity. In fact, the biorthogonal
eigenstates, which we may call “continuum bound state modes” (CBM), have
spatial envelopes and form a continuum filling the complex energy plane. We
have shown that their strong coupling sector host a continuum complex Toda
field solitons, such that the fermion eigenstates are localized inside the
topological solitons with topological invariant quantities (5.47)-(5.48). In
the strong coupling sector the family of complex solitons can be dubbed as
continuum of topological solitons (CTS). These type of novel features of a
non-Hermitian Hamiltonian, i.e. the spatially localized eigenstates at every
value of the energy in the complex energy plane $\lambda$, have recently been
uncovered also in [26].

The spinor bound states requires the condition $k_{1}\neq 0$ in the next
relationships

$\displaystyle k_{1}$ $\displaystyle=$
$\displaystyle\pm\frac{\sqrt{2}}{2}\sqrt{\Delta+(\widetilde{M}-\widetilde{\lambda})}$
(6.1) $\displaystyle k_{2}$ $\displaystyle=$
$\displaystyle\pm\frac{\sqrt{2}}{2}\sqrt{\Delta-(\widetilde{M}-\widetilde{\lambda})}$
(6.2) $\displaystyle\Delta$ $\displaystyle\equiv$
$\displaystyle\sqrt{|M-\lambda|^{2}|M+\lambda|^{2}}$ (6.3)
$\displaystyle\widetilde{M}$ $\displaystyle\equiv$ $\displaystyle
M_{1}^{2}-M_{2}^{2},$ (6.4) $\displaystyle\widetilde{\lambda}$
$\displaystyle\equiv$ $\displaystyle\lambda_{1}^{2}-\lambda_{2}^{2},$ (6.5)

Notice that the parameters defined above satisfy
$\Delta>|\widetilde{M}-\widetilde{\lambda}|$ without any further restriction
on the complex plane $k=k_{1}+ik_{2}$ for a fixed value of the complex mass
parameter $M=M_{1}+iM_{2}$. So, for each complex eigenvalue
$\lambda\in\leavevmode\hbox{\,$\vrule
height=6.45831pt,width=0.4pt,depth=0.0pt\kern-3.00003pt{\rm C}$}$ one has a
continuum of complex scalar solutions with wave numbers $k_{1}$ and $k_{2}$
given in (6.1)-(6.2). As discussed above, a continuum of bound states emerge
coupled to a continuous family of topological solitons (see Fig 2.) such that
their topological charges depend on the energy eigenvalues as in
(5.28)-(5.32). In addition, the zero mode $\lambda=0$ implies $k=\pm M$ in
(6.1)-(6.2).

Figure 2: A representative of the continuum of bound states. Topological
complex soliton $\Phi=\Phi_{1}+i\Phi_{2}$ and bound state
$\xi^{T}=(\xi_{3}+i\xi_{4},\,\xi_{1}+i\xi_{2})$ for
$|a|=2,|b|=1.5,|c|=1,|d|=1.3,\zeta_{1}=1.2,\zeta_{2}=1.5,\alpha_{1}=1.1,\alpha_{2}=1.7,k_{1}=1,k_{2}=1,\beta=1,|\rho|=1.5,M_{1}=4.04,\,M_{2}=1.49$.
Figure 3: A representative of the extended wave states in the continuum of
bound states. The components of $\Phi=\Psi_{1}+i\Phi_{2}$ and
$\xi^{T}=(\xi_{3}+i\xi_{4},\,,\xi_{1}+i\xi_{2})$ for
$|a|=1.5,|b|=1.5,|c|=1,|d|=1.3,\zeta_{1}=1.2,\zeta_{2}=1.5,\alpha_{1}=\pi/3,\alpha_{2}=\pi/2,k_{1}=0,k_{2}=1,\beta=1,|\rho|=1.5,M_{1}=-0.05,\,M_{2}=3.9$.

On the other hand, the complex energy components become

$\displaystyle\lambda_{1}$ $\displaystyle=$
$\displaystyle\pm\frac{\sqrt{2}}{2}\sqrt{\Delta+(\widetilde{M}-\widetilde{k})}$
(6.6) $\displaystyle\lambda_{2}$ $\displaystyle=$
$\displaystyle\pm\frac{\sqrt{2}}{2}\sqrt{\Delta-(\widetilde{M}-\widetilde{k})}$
(6.7) $\displaystyle\Delta$ $\displaystyle\equiv$
$\displaystyle\sqrt{|M-k|^{2}|M+k|^{2}}$ (6.8) $\displaystyle=$
$\displaystyle\sqrt{[(k_{1}-M_{1})^{2}+(k_{2}-M_{2})^{2}][(k_{1}+M_{1})^{2}+(k_{2}+M_{2})^{2}]}$
(6.9) $\displaystyle\widetilde{M}$ $\displaystyle\equiv$ $\displaystyle
M_{1}^{2}-M_{2}^{2},$ (6.10) $\displaystyle\widetilde{k}$
$\displaystyle\equiv$ $\displaystyle k_{1}^{2}-k_{2}^{2}.$ (6.11)

From the above equations (6.6)-(6.11) one has the following $M=\pm
k\Rightarrow\Delta=0$ and $\widetilde{M}-\widetilde{k}=0$, which imply
$\lambda=0$. This is just the zero mode $\lambda=0$.

Figure 4: The left and right plots show the energy surfaces
$\lambda(k_{1},k_{2})$ for $M_{1}=2,\,M_{2}=-2$. The middle Fig. shows the
curves $\lambda(k_{1}=0,k_{2})$ (line colors distiguish the left/righ surfaces
each one belongs to) which correspond to the extended waves in the localized
continuum (ELC) states (LABEL:phi1e)-(6.19). The points of the left and right
manifolds (excluding the curves $\lambda(k_{1}=0,k_{2})$ of the middle plot)
represent the localized continuum states (5.26)-(5.27) and (5.45)-(5.46).
Notice the presence of the exceptional points (EP) at $k=\pm M$ (left and
right plots) which correspond to the zero modes $\lambda=0$.

Moreover, from the complex dispersion relationship (5.14) one can get the next
two equations, considering the real and imaginary parts, respectively

$\displaystyle
k_{1}^{2}-k_{2}^{2}-M_{1}^{2}+M_{2}^{2}+\lambda_{1}^{2}-\lambda_{2}^{2}$
$\displaystyle=$ $\displaystyle 0$ (6.12) $\displaystyle
k_{1}k_{2}-M_{1}M_{2}+\lambda_{1}\lambda_{2}$ $\displaystyle=$ $\displaystyle
0.$ (6.13)

These two relationships are equivalent to the next system of equations

$\displaystyle
k_{1}^{2}(k_{1}^{2}-M_{1}^{2}+M_{2}^{2}+\lambda_{1}^{2}-\lambda_{2}^{2})-(M_{1}M_{2}-\lambda_{1}\lambda_{2})^{2}$
$\displaystyle=$ $\displaystyle 0,\,\,\,k_{1}\neq 0,$ (6.14) $\displaystyle
k_{2}^{2}(k_{2}^{2}+M_{1}^{2}-M_{2}^{2}-\lambda_{1}^{2}+\lambda_{2}^{2})-(M_{1}M_{2}-\lambda_{1}\lambda_{2})^{2}$
$\displaystyle=$ $\displaystyle 0,\,\,\,k_{2}\neq 0.$ (6.15)

Once the values of $M_{1}$ and $M_{2}$ are fixed one has that the equations
(6.14)-(6.15) define a couple of families of curves in the pane
$(\lambda_{1},\lambda_{2})$ parametrized by the variables $k_{1}$ and $k_{2}$,
respectively. The Fig. 5 shows the spectral curves filling some regions of the
complex plane corresponding to the family of curves (6.14)-(6.15). In the Fig.
6 we plot the family of curves $(\lambda_{1},\lambda_{2})$ which intersect to
each other at the origin. Notice that the coalescence poit is at the EP
($\lambda=0$).

Figure 5: Spectral curves on the plane $\lambda=(\lambda_{1},\lambda_{2})$ for
$(M_{1},M_{2})=(\pm 2,\pm 2)$ (left) and $(M_{1},M_{2})=(\pm 2,\mp 2)$
(right). The $k_{1}$ and $k_{2}$ families correspond to (6.14) and (6.15),
respectively. The figures show the curves for the parameter ranges
$|k_{a}|<1.5\,(a=1,2)$. The exceptional points (EP) are at
$\lambda=0\,(k_{1}=M_{1},k_{2}=M_{2})$. The points $\lambda=(M_{1},M_{2})$
correspond to the threshold points $k=0$.

The parabolas $\lambda_{1}\lambda_{2}=M_{1}M_{2}=const.$ in the plane
$(\lambda_{1},\lambda_{2})$ can be obtained when one of the components $k_{1}$
or $k_{2}$ vanishes, and correspond to the extended wave states in the
localized continuum for $k_{1}=0$, or to a subset of the continuum bound
states for $k_{2}=0$.

So, the confluence of non-Hermiticity, nonlinearity and spectral topology
generate new phenomena such as the non- Hermitian currents equivalence (2.22),
the complex valued topological charges (${\cal Q}\equiv{\cal Q}_{1}+i{\cal
Q}_{2}$) and the presence of the winding numbers (5.53) related to the
spectral topology. The model exhibits the three phenomena unique to non-
Hermitian systems: the non-vanishing fermion current, nonzero winding number
of energy and the presence of the pair of topological solitons in the model.
One can argue that the validity of any one of the three properties is the
sufficient and necessary condition for the validity of the other two in this
non-Hermitian system.

In summary, we have shown that the non-Hermitian Hamiltonian $H$ can allow an
uncountably infinite set of bound states, contrary to the results in [18] in
which the in-gap and BIC bound states appear with quantized energies. The
biorthogonal continuum bound states possess localized envelopes and are
related to a continuum of topological complex Toda solitons. Regarding the
zero modes, one can also argue that the biorthogonal Majorana zero modes are
embedded into a family of continuum of bound states, with properties different
from those studied in their Hermitian counterparts [18].

Figure 6: The family of curves $(\lambda_{1},\lambda_{2})$ for
$M_{1}=-M_{2}=1$ (left) and $M_{1}=M_{2}=1$ (right) in the parameter ranges
$|k_{1,\,2}|<1.5$. Notice the coalescence of some states at the EP
($\lambda=0$).

###  6.2 Extended wave states embedded in the continuum of bound states

Next we construct the extended wave states localized in the continuum of bound
states (ELC). These states can be obtained by setting $k_{1}=0$ into the tau
functions (5.8)-(5.10). So, from (5.1) the scalar components become

$\displaystyle\Phi_{1}$ $\displaystyle=$
$\displaystyle-\frac{2}{\beta}\arctan{\Big{[}\frac{|a|\sin{(2k_{2}x+\alpha_{1}-\frac{\delta}{2})}-|b|\sin{(2k_{2}x+\alpha_{2}+\frac{\delta}{2})}+|a||b|\sin{(\alpha_{1}-\alpha_{2}-\frac{\delta}{2})}-\sin{\frac{\delta}{2}}}{|a|\cos{(2k_{2}x+\alpha_{1}-\frac{\delta}{2})}+|b|\cos{(2k_{2}x+\alpha_{2}+\frac{\delta}{2})}+|a||b|\cos{(\alpha_{1}-\alpha_{2}-\frac{\delta}{2})}+\cos{\frac{\delta}{2}}}\Big{]}},$
$\displaystyle\Phi_{2}$ $\displaystyle=$
$\displaystyle-\frac{1}{\beta}\log{\Big{[}|\rho|\frac{1+|b|^{2}+2|b|\cos{(2k_{2}x+\alpha_{2})}}{1+|a|^{2}+2|a|\cos{(2k_{2}x+\alpha_{1})}}\Big{]}}.$
(6.17)

Similarly, inserting (5.8)-(5.10) into (5.2) for $k_{1}=0$ we have the spinor
components

$\displaystyle\xi_{A}$ $\displaystyle=$
$\displaystyle|c|\,\frac{|b|\cos{(k_{2}x+\alpha_{2}-\zeta_{1})}+\cos{(k_{2}x+\zeta_{1})}}{1+|b|^{2}+2|b|\cos{(2k_{2}x+\alpha_{2})}}-$
(6.18) $\displaystyle
i|c|\,\frac{|b|\sin{(k_{2}x+\alpha_{2}-\zeta_{1})}-\sin{(k_{2}x+\zeta_{1})}}{1+|b|^{2}+2|b|\cos{(2k_{2}x+\alpha_{2})}},$
$\displaystyle\xi_{B}$ $\displaystyle=$
$\displaystyle|d|\,\frac{|a|\cos{(k_{2}x+\alpha_{1}-\zeta_{2})}+\cos{(k_{2}x+\zeta_{2})}}{1+|a|^{2}+2|a|\cos{(2k_{2}x+\alpha_{1})}}-$
(6.19) $\displaystyle
i|d|\,\frac{|a|\sin{(k_{2}x+\alpha_{1}-\zeta_{2})}-\sin{(k_{2}x+\zeta_{2})}}{1+|a|^{2}+2|a|\cos{(2k_{2}x+\alpha_{1})}}.$

Using the relationships (2.20) from (6.18)-(6.19) one can get
$\eta_{A}=\xi_{B}$ and $\eta_{B}=\xi_{A}$ with the $\xi_{A,B}$’s given above.
The Fig. 3 shows a representative plot of this type of extended states. The
middle plot in Fig. 4 shows the curves $\lambda(k_{1}=0,k_{2})$ which
correspond to the ELC extended wave states (LABEL:phi1e)-(6.19). The family of
spectral curves for $k_{1}=0$ in (6.14)-(6.15) can be written as

$\displaystyle
k_{2}^{2}M_{1}^{2}-(\lambda_{1}^{2}-M_{1}^{2})(\lambda_{2}^{2}+M_{2}^{2})=0.$
(6.20)

The Fig. 7 shows this family of curves parametrized by $k_{2}$ for
$M_{1}^{2}=M_{2}^{2}=2$. Notice that these curves lie inside the continuum of
bound states in the Fig. 4. So, these states constitute the extended waves in
the localized continuum (ELC), in contradistinction to the well known
Hermitian bound states in the continuum (BIC), which appear embedded into the
continuum of scattering states.

Figure 7: The family of curves $(\lambda_{1},\lambda_{2})$ for $k_{1}=0$ and
$M_{1}^{2}=M_{2}^{2}=2$ in the parameter range $|k_{2}|<1.5$ which correspond
to the extended waves in the localized continuum (ELC) states parametrized by
$k_{2}$ (6.20). Notice the ELC zero modes $\lambda=0\,(k_{2}=\pm M_{2})$.

The bound states in the continuum (BIC) are localized states such that
spatially and spectrally overlap with a continuum of free states. However,
those states do not hybridize with any of the extended or radiating states. It
has recently been established that those bound states remain localized even
though they coexist with the continuous spectrum of radiating waves that can
carry energy away, i.e. for energies above or below the threshold of the
Hermitian Toda coupled to fermion [18] . On the other hand, the extended wave
states on the localized continuum defined in (6.18)-(6.19) can be regarded as
some kind of inverted BIC states due to the non-Hermicity of the model (2.1).
Recently, these novel type of states in a NH system have been dubbed as
extended states in a localized continuum (ELC) [27].

###  6.3 Left and right eigenvectors and biorthogonality relationships

In the NH case there are some types of time-independent inner products [34,
35]. Thus, for the model above one can write: $\eta^{T}\xi$,
$\xi^{T}\Gamma\xi$, $\eta^{T}\Gamma\eta$, $\bar{\xi}^{T}\xi$,
$\xi^{T}\cal{G}\xi$, $\bar{\eta}^{T}\eta$, $\eta^{T}\cal{G}\eta$,
$\bar{\xi}^{T}{\cal G}^{-1}\bar{\xi}$ , $\bar{\eta}^{T}{\cal
G}^{-1}\bar{\eta}$, and the analog products associated to the
$\chi,\phi,\hat{\chi},\hat{\phi}$ eigenvectors. In the following, before
verifying the biorthogonality property (3.5), let us compute the NH inner
product (3.1) between two right spinors of types $\eta$ and $\xi$ at the
exceptional point $\lambda=0$. So, let us consider the inner product

$\displaystyle(\eta_{-\lambda}|\xi_{\lambda})=\int^{+\infty}_{-\infty}[\eta_{-\lambda,A}\xi_{\lambda,A}+\eta_{-\lambda,B}\xi_{\lambda,B}]\,dx.$
(6.21)

The soliton parameters in (5.11)-(5.13) can be mapped as
$\lambda\rightarrow-\lambda,\,\,k\rightarrow k$ by making the complex variable
transformation $z\rightarrow-\frac{1}{z}$; in addition, through this
transformation one can also relate the parameters of $\eta_{-\lambda}$ to that
of $\xi_{\lambda}$. So, from (6.21) one can write

$\displaystyle(\eta_{-\lambda}|\xi_{\lambda})$ $\displaystyle=$
$\displaystyle\frac{1}{2k}\left(\frac{\log{(a^{\prime}/b)}}{a^{\prime}-b}+\frac{\log{(a/b^{\prime})}}{a-b^{\prime}}\right),$
(6.22) $\displaystyle=$
$\displaystyle\pm\frac{M}{k\beta^{2}}\frac{(1-z^{2})^{2}}{\widetilde{c}cz^{2}+\widetilde{c}^{\prime}c^{\prime}}\frac{1+z^{2}}{z}\log{\left(\frac{\widetilde{c}c}{\widetilde{c}^{\prime}c^{\prime}}z^{2}\right)}.$
(6.23)

where the parameters $a^{\prime},b^{\prime},c^{\prime},\widetilde{c}^{\prime}$
and $a,b,c,\widetilde{c}$ correspond to the spinors $\eta_{-\lambda}$ and
$\xi_{\lambda}$, respectively, according to the notation in (5.11)-(5.13). For
the zero modes, i.e. $\lambda=0$ ($z=\pm i$), from (6.23) one has

$\displaystyle(\eta_{0}|\xi_{0})$ $\displaystyle=$ $\displaystyle 0.$ (6.24)

In the next steps we will compute the biorthogonality relationship (3.5) for
some special cases. So, let us find the left eigenvectors satisfying the
equations (2.28)-(2.29) for the relevant eigenvalue $\bar{\lambda}$. Following
similar steps as above we consider the tau functions $\bar{\tau}_{0,1}$,
$\bar{\tau}_{A,B},\bar{\tau}^{\prime}_{A,B}$ such that

$\displaystyle e^{i\beta\Phi}$ $\displaystyle=$
$\displaystyle\bar{\rho}\,(\frac{\bar{\tau}_{0}}{\bar{\tau}_{1}})^{2},$ (6.25)
$\displaystyle\bar{\xi}_{A}$ $\displaystyle=$
$\displaystyle\frac{\bar{\tau}_{A}}{\bar{\tau}_{1}},\,\,\,\bar{\xi}_{B}=\frac{\bar{\tau}_{B}}{\bar{\tau}_{0}},\,\,\,\bar{\eta}_{A}=\frac{\bar{\tau}^{\prime}_{A}}{\bar{\tau}_{0}},\,\,\,\bar{\eta}_{B}=\frac{\bar{\tau}^{\prime}_{B}}{\bar{\tau}_{1}},$
(6.26)

with $\bar{\rho}$ being a constant complex parameter.

So, let us assume the following tau functions

$\displaystyle\bar{\tau}_{1}$ $\displaystyle=$ $\displaystyle
1+\bar{a}\,e^{2\bar{k}x},\,\,\,\,\bar{\tau}_{0}=1+\bar{b}\,e^{2\bar{k}x},$
(6.27) $\displaystyle\bar{\tau}_{A}$ $\displaystyle=$
$\displaystyle\bar{c}\,e^{\bar{k}x},\,\,\,\bar{\tau}_{B}=\bar{d}\,e^{\bar{k}x},$
(6.28) $\displaystyle\bar{\tau}^{\prime}_{A}$ $\displaystyle=$
$\displaystyle\bar{c}^{\prime}\,e^{\bar{k}x},\,\,\,\bar{\tau}^{\prime}_{B}=\bar{d}^{\prime}\,e^{\bar{k}x},$
(6.29)

with the complex parameters
$\bar{a},\bar{b},\bar{c},\bar{d},\bar{c}^{\prime},\bar{d}^{\prime}$ and
$\bar{k}$. Next, making use of these tau functions and substituting the
expressions (6.25)-(6.26) into the eigenvalue equations for the left
eigenvectors (2.28) with eigenvalue $\lambda$, one can show that the equations
are satisfied provided that the following parameter relationships hold

$\displaystyle\bar{a}$ $\displaystyle=$
$\displaystyle\frac{\bar{c}^{\prime}\bar{c}}{2M}\frac{\beta^{2}w^{3}}{(w^{2}-1)^{2}},\,\,\,\,\,\,\,\bar{d}^{\prime}=\frac{\bar{c}^{\prime}\,\bar{c}}{\bar{d}},\,\,\,\,\,\,\,\bar{\rho}=i\frac{\bar{c}}{\bar{d}}\,w,$
(6.30) $\displaystyle\bar{b}$ $\displaystyle=$
$\displaystyle\frac{\bar{c}^{\prime}\bar{c}}{2M}\frac{\beta^{2}\,w}{(w^{2}-1)^{2}},\,\,\,\,\,w\equiv-\frac{\bar{\lambda}}{M}\pm\sqrt{\frac{\bar{\lambda}^{2}}{M^{2}}-1}$
(6.31) $\displaystyle\bar{\lambda}$ $\displaystyle=$
$\displaystyle-\frac{M}{2}(w+w^{-1}),\,\,\,\,\,\,\,\bar{k}=\frac{iM}{2}(w-w^{-1}).$
(6.32)

Notice that the form of the relationship (5.14) also holds in this case, i.e.
$\bar{\lambda}^{2}+\bar{k}=M^{2}$, and the $\bar{\lambda}$ and $\bar{k}$
parameters depend on the uniformization parameter $w$ in the same form as in
the right eigenvalue case (5.13).

Next, let us compute $(\bar{\xi}_{\bar{\lambda}}|\xi_{\lambda})$ for
$\bar{\lambda}=\pm\lambda\,$ by using the expressions (5.2) and (6.26) and the
relevant tau functions (5.8)-(5.10) and (6.27)-(6.29), respectively. First,
let us consider $\bar{\lambda}=\lambda\,(\lambda\neq 0)$. The transformation
$w=\frac{1}{z}$ maps $\bar{\lambda}(w)\rightarrow\lambda(z)$ and
$\bar{k}\rightarrow-k$. So, let us compute the next integral in $x$ for
$\bar{k}=-k$

$\displaystyle(\bar{\xi}_{\lambda}|\xi_{\lambda})$ $\displaystyle=$
$\displaystyle\int_{-\infty}^{\infty}[\bar{\xi}_{\lambda,A}\xi_{\lambda,A}+\bar{\xi}_{\lambda,B}\xi_{\lambda,B}]\,dx$
(6.33) $\displaystyle=$
$\displaystyle\int_{-\infty}^{\infty}[\eta_{-\lambda,A}\xi_{\lambda,A}-\eta_{-\lambda,B}\xi_{\lambda,B}]\,dx$
$\displaystyle=$
$\displaystyle\frac{1}{2k}\left(\frac{\bar{c}c\log{\left(\bar{a}b\right)}}{\bar{a}b-1}+\frac{\bar{d}d\log{\left(\bar{b}a\right)}}{a\bar{b}-1}\right)$
$\displaystyle=$ $\displaystyle 1.$

Then, in order to satisfy the biorthogonality relationship (3.5) the
parameters must satisfy the normalization condition (6.33).

Next, let us compute (3.5) for the special case
$(\bar{\xi}_{-\lambda}|\xi_{\lambda})=0$. Notice that the transformation
$w=-\frac{1}{z}$ maps $\bar{\lambda}(w)\rightarrow-\lambda(z)$ and
$\bar{k}\rightarrow k$. So, one has $\bar{\lambda}=-\lambda$. Let us compute
the integral in $x$ for $\bar{k}=k$

$\displaystyle(\bar{\xi}_{-\lambda}|\xi_{\lambda})$ $\displaystyle=$
$\displaystyle\int_{-\infty}^{\infty}[\bar{\xi}_{-\lambda,A}\xi_{\lambda,A}+\bar{\xi}_{-\lambda,B}\xi_{\lambda,B}]\,dx$
(6.34) $\displaystyle=$
$\displaystyle\frac{1}{2k}\left(\frac{\bar{c}c\log{\left(\bar{a}/b\right)}}{\bar{a}-b}+\frac{\bar{d}d\log{\left(\bar{b}/a\right)}}{\bar{b}-a}\right)$
(6.35)

In order to show the vanishing of the expression above let us impose the
pseudo-chirality relationship (3.9) to the left and right eigenvectors, i.e.
$\bar{\xi}_{-\lambda,\,A}=\xi_{\lambda,\,B},\,\,\bar{\xi}_{-\lambda,\,B}=-\xi_{\lambda,\,A}$.
So, from the the expressions (5.2) and (6.26) and the relevant tau functions
(5.8)-(5.10) and (6.27)-(6.29), respectively, one has the relationships

$\displaystyle\bar{c}=d,\,\,\bar{a}=a,\,\,\bar{d}=-c,\,\,\bar{b}=b.$ (6.36)

Therefore, making use of the relationships (6.36) into (6.35) one can show
that

$\displaystyle(\bar{\xi}_{-\lambda}|\xi_{\lambda})=0.$ (6.37)

The zero mode biorthogonality equation (3.12) follows by setting $\lambda=0$
in the last equation (6.37), i.e. $(\bar{\xi}_{0}|\xi_{0})=0$ . Notice that
for the general case of arbitrary states with $\lambda_{i}$ and $\lambda_{j}$
($\lambda_{i}\neq\lambda_{j}\,\rightarrow\,k_{i}\neq k_{j}$ ), the
$x-$integration
$(\bar{\xi}_{\lambda_{j}}|\xi_{\lambda_{j}})=\int^{+\infty}_{-\infty}\bar{\xi}_{\lambda_{j}}^{T}\xi_{\lambda_{j}}\,dx$,
in general, will not provide an exact integral, and one could perform the
$x-$integration using numerical methods. A similar procedure as above can be
performed in the case of $\eta$ and $\bar{\eta}$ eigenvectors and show
$(\bar{\eta}_{\lambda_{i}}|\eta_{\lambda_{j}})=\delta_{ij}$, except at the EP
$\lambda=0$.

##  7 Biorthogonal Majorana zero mode-topological soliton duality

Firstly, we find the zero mode right eigenvectors. So, let us set $\lambda=0$
into the system of equations (5.3)-(5.6) or equivalently one can set $z=\pm i$
into the set of parameter equations (5.11)-(5.12). So, the tau functions
(5.8)-(5.10) satisfy the equations provided that

$\displaystyle k$ $\displaystyle=$ $\displaystyle\pm
M,\,\,\,\,\,b=-a,\,\,\,\,\,\rho=\pm\frac{\widetilde{c}}{\widetilde{d}}\,,\,\,\,\,\,\widetilde{d}=\frac{c\widetilde{c}}{d}.$
(7.1)

and the charge densities equivalence equation (2.22) for

$\displaystyle a=\pm i\frac{c\widetilde{c}\beta^{2}}{8M},$ (7.2)

without any further restriction. Therefore, the one-soliton solution satisfy
the condition (2.22) for the equivalence between the spinor bilinear
$\eta^{T}\xi$ and the $x-$derivative of the complex field $\Phi$. An analog
relationship in the one-soliton sector has been defined as the equivalence
between Noether and topological currents in the Hermitian version of the model
[19, 29, 18].

So, from (5.37)-(5.38) and using (5.30) and (5.32), one can compute the
relevant topological charges associated to the complex scalar by setting
$z=\pm i$ in the next equations

$\displaystyle{\cal Q}_{1}$ $\displaystyle\equiv$
$\displaystyle\frac{\beta}{2\pi}[\Phi_{1}(+\infty)-\Phi_{1}(-\infty)]$ (7.3)
$\displaystyle=$ $\displaystyle\frac{2}{\pi}\arg{z}$ (7.4) $\displaystyle=$
$\displaystyle\pm 1$ (7.5) $\displaystyle{\cal Q}_{2}$ $\displaystyle\equiv$
$\displaystyle\frac{1}{2\pi}[\Phi_{2}(+\infty)-\Phi_{2}(-\infty)].$ (7.6)
$\displaystyle=$ $\displaystyle\frac{2}{\pi}\log{|z|}$ (7.7) $\displaystyle=$
$\displaystyle 0.$ (7.8)

The above soliton and bound state solutions of the model, provided the
parameters (7.1)-(7.2) relationships are satisfied, allow us to construct the
following mappings between the spinor components and the complex scalar field

$\displaystyle\xi_{0A}$ $\displaystyle=$
$\displaystyle\pm\frac{c}{2\sqrt{a}}e^{-i\beta\Phi/2}(\rho-e^{i\beta\Phi})^{1/2}$
(7.9) $\displaystyle\xi_{0B}$ $\displaystyle=$
$\displaystyle\frac{d}{2\sqrt{a\rho}}(\rho-e^{i\beta\Phi})^{1/2}$ (7.10)
$\displaystyle\eta_{0A}$ $\displaystyle=$
$\displaystyle\frac{d}{2\sqrt{a\rho}}(\rho-e^{i\beta\Phi})^{1/2}$ (7.11)
$\displaystyle\eta_{0B}$ $\displaystyle=$
$\displaystyle\pm\frac{c}{2\sqrt{a}}e^{-i\beta\Phi/2}(\rho-e^{i\beta\Phi})^{1/2}.$
(7.12)

Notice that these fields satisfy the relationships (2.20). In the Fig. 8 we
show the plots of $\xi_{0}$ Majorana zero mode components and the associated
scalar fields. The field $\Phi_{1}$ is a kink-type soliton with topological
charge ${\cal Q}_{1}=1$, whereas the excitation $\Phi_{2}$ develops a trivial
topological charge ${\cal Q}_{2}=0$. The second spinor $\eta_{0}$ can be
plotted using $\eta_{0A}=\xi_{0B},\eta_{0B}=\xi_{0A}$. One can see that this
non-Hermitian Majorana zero mode real components $\xi_{j},\,j=1,2,3,4$ are
independent from each other as compared to its Hermitian counterpart reported
in [18], in which the four real components reduce to two independent fields
due to the particle-hole symmetry, or, a charge conjugation symmetry which
holds in the relativistic model, Hermitian Toda model coupled to one Dirac
fermion.

Figure 8: Majorana zero mode bound state $\xi_{0}$ and the associated scalar
fields for
$|a|=|b|=1,|c|=|d|=1,\zeta_{1}=-\zeta_{2}=1.2,\alpha_{1}=\pi+1,\alpha_{2}=1,k_{1}=1,k_{2}=-1,\beta=1$.
The field $\Phi_{1}$ is a soliton with ${\cal Q}_{1}=1$, whereas $\Phi_{2}$
possesses zero topological charge.

Next, using the mapping (7.9)-(7.12) into the equations of motion (2.21) and
defining $\rho\equiv e^{\rho_{o}}e^{i\alpha_{o}}$ with
$\rho_{o}\,\mbox{and}\,\alpha_{o}$ being real parameters, one can write the
next equation for the complex scalar

$\displaystyle-\partial^{2}_{x}\Phi+\frac{4M^{2}}{\beta}\sin{(\beta\Phi-\alpha_{o}+i\rho_{o})}=0.$
(7.13)

Likewise, using the mapping (7.9)-(7.12) into the system of equations of
motion (2.11)-(2.12) with $\lambda=0$ one can get the following equations for
the zero mode components

$\displaystyle
i\xi^{\prime}_{0A}+i\frac{M}{\rho}\,\xi_{0B}+\frac{1}{2}\beta^{2}\,\xi_{0A}\,\xi_{0B}\,\eta_{0B}$
$\displaystyle=$ $\displaystyle 0,$ (7.14)
$\displaystyle-i\xi^{\prime}_{0B}-iM\rho\,\,\xi_{0A}+\frac{1}{2}\beta^{2}\,\xi_{0A}\,\xi_{0B}\,\eta_{0A}$
$\displaystyle=$ $\displaystyle 0,$ (7.15)
$\displaystyle-i\eta^{\prime}_{0A}-iM\rho\,\,\eta_{0B}+\frac{1}{2}\beta^{2}\,\eta_{0A}\,\eta_{0B}\,\xi_{0B}$
$\displaystyle=$ $\displaystyle 0,$ (7.16) $\displaystyle
i\eta^{\prime}_{0B}+i\frac{M}{\rho}\,\eta_{0A}+\frac{1}{2}\beta^{2}\,\eta_{0A}\,\eta_{0B}\,\xi_{0A}$
$\displaystyle=$ $\displaystyle 0.$ (7.17)

The model (7.13) can be considered as a NH sine-Gordon model describing the
dynamics of the complex scalar field $\Phi$, whereas the system (7.14)-(7.17)
as the NH Thirring model describing the biorthogonal Majorana fermions. Notice
that the coupling parameter $\beta^{2}$ appears in the three-fermion coupling
(or four-fermion term in the Lagrangian description) of the system
(7.14)-(7.17), whereas in the Lagrangian description of the NHSG model (7.13)
it would appear as $\frac{1}{\beta^{2}}$ factor of the
$\cos{(\beta\Phi-\alpha_{o}+i\rho_{o})}$ term of the relevant potential. So,
following the ideas of [18], the mapping (7.9)-(7.12) can be regarded as the
strong $\rightleftarrows$ weak coupling mappings of the dual sectors of the NH
Toda model coupled to Majorana fermions. The Hermitian version of the model
(7.14)-(7.17) has recently been considered for two independent spinors and
shown to be integrable in the bosonic and Grassmanian versions, respectively
[39, 40]. A Hermitian (generalized) massive Thirring/sine-Gordon duality has
also been discussed in [41, 42]. Non-Hermitian ${\cal PT}$-symmetric versions
of the massive Thirring model and the sine-Gordon model have recently been
considered, and some aspects of their duality relationships in the NH context
were examined [43, 44].

Next, for completeness we provide the zero mode left eigenvectors. So, one can
set $\lambda=0$ into the system of equations (2.28)-(2.29). Since we have
already found above the left eigenvectors for the general $\lambda$ eigenvalue
following the tau functions formalism (6.25)-(6.26) and (6.27)-(6.29), we
simply set $w=\pm i\,(\lambda=0)$ for the zero modes into the relevant
parameter relationships (6.30)-(6.32). So, one has

$\displaystyle\bar{k}$ $\displaystyle=$ $\displaystyle\pm
M,\,\,\,\,\,\bar{b}=-\bar{a},\,\,\,\,\,\bar{\rho}=\pm\frac{\bar{c}^{\prime}}{\bar{d}^{\prime}},\,\,\,\,\,\bar{d}^{\prime}=\frac{\bar{c}\bar{c}^{\prime}}{\bar{d}},$
(7.18) $\displaystyle\bar{a}^{\prime}$ $\displaystyle=$ $\displaystyle\pm
i\frac{\bar{c}^{\prime}\bar{c}\beta^{2}}{8M}.$ (7.19)

Topological superconductors can host Majorana zero modes and are currently
intensively pursued as a promising candidate for topological quantum
computation, but their experimental realization has so far been inconclusive.
It has recently been shown, using non-Hermitian exceptional points, that non-
Hermiticity can strongly enhance topological superconductivity, implying more
robust Majorana zero modes [45].

##  8 Discussions and conclusions

We have studied the NH Toda model coupled to Dirac fields (2.1) by examining
the pseudo-chiral and (anti)-pseudo-Hermitian symmetries of the model and
their relevant representations. The pseudo-chiral symmetry has recently been
introduced as a manifestation of Noether’s theorem in NH systems, such that a
new inner product was defined different from the one in standard quantum
mechanics [5]. We have discussed the properties of the right and left
eigenvectors and the related eigenvalues, as well as the role played by the
complex scalar field components. Among the key characteristics of the model
are the biorthogonality relationship between the left and right eigenvectos,
and the fact that the complex eigenvalues come in pairs of type
$\\{\lambda,-\lambda\\}$ and $\\{\lambda,-\lambda^{\star}\\}$. The Table 1
summarizes the main results related to these symmetries and their relevant
representations, respectively, for the set of Hamiltonians
$\\{H,\widetilde{H}\\}$ and $\\{H^{\star},\widetilde{H}^{\star}\\}$. This
Table describes each type of symmetry and the corresponding eigenvectors and
their mutual relationships, as well as the spectral parameter $\lambda$
properties allowed by the relevant symmetry. The biorthogonality relationships
and the presence of the Majorana zero modes at the EP $\lambda=0$ and the
coalescence of the eigenvectors are also indicated. Moreover, we present the
interplay of the ${\cal P}$, ${\cal T}$ and ${\cal PT}$ symmetry
transformations with the pseudo-chiral and anti-pseudo-Hermiticity properties,
which imply a symmetry of the Hamiltonian for each set of parities
($\nu_{1},\,\nu_{2}$) which can take the scalar fields $\Phi_{1}$ and
$\Phi_{2}$, respectively. In Table 2 we summarize the main results of the
cases $A,B,C$ and $D$ discussed such that the last column of this Table
indicates their relevant symmetries.

Some relationships between the complex scalar fields $\Phi$ and $\bar{\Phi}$
and the biorthogonal spinor sectors $\\{\xi,\eta\\}$ and
$\\{\bar{\xi},\bar{\eta}\\}$ of the model are furnished by the correspondences
between the spinors and topological currents densities in (2.22) and (2.29),
respectively, together with the equations (3.23) and (3.24) provided by the
pseudo-chirality symmetry. The emergence of the fields $\bar{\Phi}$ and
$\\{\bar{\xi},\bar{\eta}\\}$ is a feature which characterizes the non-
Hermitian Toda model coupled to Dirac spinors. We have shown that the
solutions (5.1) and (5.2) with the tau functions in (5.8)-(5.10) satisfy the
system of equations (2.11)-(2.12) which coupled the spinors and the scalar
field and the currents equivalence equation (2.22). In the soliton sector, the
$x-$integral in the whole real line of the current density (5.47) provides a
complex number, which are proportional to the topological charges ${\cal
Q}_{1}$ and ${\cal Q}_{2}$, respectively. In fact, the real part of (5.47) is
proportional to ${\cal Q}_{1}$, whereas the imaginary part to ${\cal Q}_{2}$
as in (5.39)-(5.40), respectively. So, the above NH feature is the analog of
the equivalence between the $U(1)$ Noether and real topological currents in
the Hermitian version of the model [18].

The construction of the analytical biorthogonal spinor bound states and their
associated complex scalar topological solitons were performed. Analytical
solutions of the static version of the system were obtained by using the tau
function approach. The complex eigenvalue $\lambda$, in each topological
sector, satisfies a second order algebraic equation as the dispersion relation
(5.14), $\lambda^{2}=M^{2}-k^{2}$, plus its complex conjugate. So, a related
uniformization parameter $z$ defined in (5.13) has played a key role in
understanding the asymptotic behavior of the scalar field. The solutions
exhibit some topological charges associated to the asymptotic values of the
complex scalar field (5.35), i.e.
$\Phi(+\infty)-\Phi(-\infty)=\pm\frac{4}{i\beta}\log{z}$, with
$z\in\leavevmode\hbox{\,$\vrule
height=6.45831pt,width=0.4pt,depth=0.0pt\kern-3.00003pt{\rm
C}$}\backslash\\{0\\}$ being the uniformization parameter. The biorthogonal
spinor bound states are found for each topological configurations of the
background scalar fields. In these developments the solitons shapes depend
crucially on the spinor bound state complex energy and wave number parameters,
$\lambda$ and $k$, i.e. the soliton shape depends on the fermionic state to
which it is coupled. So, it is worth mentioning that the back-reactions of the
spinors on the solitons are described nonperturbatively and exactly through
our analytical methods.

The energy $\lambda$ has been defined as an analytical function of the complex
variable $z$ (5.13). The spectral topology concept is a unique feature in non-
Hermitian systems. In this scenario, the zero-energy points at $z=\pm i$ in
(5.41) were dubbed as exceptional points (EP) and they arise as branch points
in the $k-$space eigenvalue manifold (see the points at $k=\pm M$ in the left
and right plots of the Fig. 4). The winding number of $\lambda(z)$ has been
computed as it traverses the contour $\Gamma$ encircling the zero modes
$\lambda=0$ located at the points $z=\pm i$ and the pole at $z=0$ in the
complex plane $z$ (see Fig. 1). The contour integral of the function
$\lambda(z)$ using (5.51) provides the winding number $W=1$ in (5.53), and so,
it is in accordance with the Cauchy’s argument principle (5.52).

The general solution (5.1) and (5.2) with the tau functions in (5.8)-(5.10)
can be classified into two types of solutions:

1\. The continuum of bound states/solitons (5.45)-(5.46) and (5.26)-(5.27),
respectively. For the wave number with non-vanishing real component, i.e.
$k_{1}\neq 0$, we have defined the topological invariants for each component
of the scalar field (5.28)-(5.31). So, the spinor solutions (5.45)-(5.46)
become spatially localized eigenstates at every value of the energy in the
complex plane $\lambda$ which satisfies the dispersion relation (5.14)
provided that $\Re{(k)}\neq 0$. So, this feature violates the general wisdom
that bound states should be quantized. In section 6 we have discussed in more
detail this important class of solutions of the non-Hermitian Toda model
coupled to fermions, and presented the Fig. 2 as a representative plot.
Recently, this phenomenon has also been reported in the context of NH lattice
models [26]. The charges $\Re{(Q_{topol})}=\pm 1,\,\Im{(Q_{topol})}=0$ are
special in the sense that the relevant solitons host the Majorana zero-modes,
as we will discuss below.

2\. Another class of solutions are the extended waves embedded into the
localized states (ELC). These states hold provided that one sets $k_{1}=0$
into the tau functions (5.8)-(5.10). The scalar fields take the form
(LABEL:phi1e)-(6.17) and the extended wave spinors become (6.18)-(6.19). The
Fig. 3 shows a representative plot of this type of state. The middle plot in
Fig. 4 shows the curves $\lambda(k_{1}=0,k_{2})$ corresponding to the ELC
states. These states can be regarded as a generic NH phenomenon representing
some kind of inverted BIC states due to the non-Hermicity of the model (2.1).
Recently, these novel type of ELC states in a NH system have been reported in
the context of an active mechanical lattice [27].

The above type of solutions may further be classified into subsets according
to the ${\cal P}$, ${\cal T}$ and ${\cal PT}$ symmetries of the Hamiltonians
$H$ and $\widetilde{H}$ for each set of parities ($\nu_{1},\,\nu_{2}$) of the
scalar field components $\Phi_{1}$ and $\Phi_{2}$, respectively. In this
context, the representations of the symmetries in the last column of the Table
2 presented in section 4.3 deserve to be performed. We are currently
investigating these and other important issues through analytical and
numerical methods.

In order to understand the NH soliton-particle duality we have examined the
zero mode bound states and corresponding scalar field solutions in section 7.
In contradistinction to the Hermitian case in [18] the NH Majorana zero mode
components $\xi_{j},\,j=1,2,3,4$ are independent from each other, and they are
hosted by the complex scalar field supporting a topological soliton in its
real component and a wave with vanishing topological charge in its imaginary
component (see Fig. 8). Then, one can argue that these non-Hermitian zero
modes are stable even in the presence of the imaginary component scalar wave
with vanishing topological charge. We uncovered the mapping (7.9)-(7.12) which
can be regarded as a strong $\rightleftarrows$ weak duality mappings of the
dual sectors of the NH Toda model coupled to Majorana fermions. In this
picture the strong coupling sector is described by a NH sine-Gordon model
(7.13), whereas the weak coupling sector by the NH Thirring model
(7.14)-(7.17) of biorthogonal Majorana fermions. In future works it would be
interesting to consider the NH (generalized) massive Thirring/sine-Gordon
duality extension of the relevant Hermitian models [41, 42].

Some analytical and numerical extensions of this work deserve to be
considered, such as the NH kink-antikink scattering in order to understand the
behavior of the NH fermions and hosted bound states, as well as the extensions
for more than one species of chiral fermions and scalars in the scalar-fermion
interaction term. In particular, a scalar potential term can be introduced
into the Lagrangian in order to study the NH version of the model studied in
[18]. So, our work presented the pseudo-chiral and (anti)-pseudo-Hermitian
symmetries and their representations from the field theory perspective which
were mainly reported before in the context of NH lattice models. We hope our
approach will be relevant to the understanding of the physical origins of non-
Hermiticity, the finite lifetimes of resonances, back reaction of fermions on
solitons and the interplay of nonlinearity, topology, and integrability in the
context of NH integrable field theory models.

Finally, it would be interesting to study the dynamics of kinks and bound
state solutions of the NH model (2.1) in the context of quasi-integrability,
and the relevant quasi-conserved quantities in the so-called
(quasi-)integrable approach [41, 46, 48, 47]. So far, to our knowledge, a NH
model possessing a strong-weak duality property had not been studied in the
quasi-integrability context. Besides, our findings could be relevant in the
context of one-dimensional topological superconductors in order to understand
how the non-hermiticity influence the shape and the lifetime of the bound
states, in particular the Majorana zero modes. It will be an interesting issue
to study the robustness against decoherence and disorder of our non-Hermitian
Majorana zero modes, as compared to their Hermitian counterparts. These type
of non-Hermitian modes, being more stable in the presence of imperfections,
make them appealing for quantum information processing and fault-tolerant
quantum computing. These and other extensions will be reported elsewhere.

Acknowledgements

The author is grateful to J.J. Monsalve and J. R. V. Pereira for useful
discussions.

##  Appendix A Equations in components

Taking
$\lambda=\lambda_{1}+i\lambda_{2},\,\,\Phi=\Phi_{1}+i\Phi_{2},\,\,M=M_{1}+iM_{2}$
and

$\displaystyle\psi$ $\displaystyle=$ $\displaystyle e^{-i\lambda
t}\xi,\,\,\,\,\,\,\,\,\,\xi=\left(\begin{array}[]{c}\xi_{3}+i\xi_{4}\\\
\xi_{1}+i\xi_{2}\end{array}\right),$ (A.3) $\displaystyle\chi$
$\displaystyle=$ $\displaystyle e^{i\lambda
t}\eta,\,\,\,\,\,\,\,\,\,\eta=\left(\begin{array}[]{c}\eta_{3}+i\eta_{4}\\\
\eta_{1}+i\eta_{2}\end{array}\right),$ (A.6)

one has the next equations for the spinor components

$\xi^{\prime}_{1}+(\lambda_{1}\xi_{2}+\lambda_{2}\xi_{1})+e^{-\beta\Phi_{2}}[(M_{1}\xi_{3}-M_{2}\xi_{4})\cos{(\beta\Phi_{1})}-(M_{2}\xi_{3}+M_{1}\xi_{4})\sin{(\beta\Phi_{1})}]=0,\\\
$ (A.7)
$\xi^{\prime}_{2}-(\lambda_{1}\xi_{1}-\lambda_{2}\xi_{2})+e^{-\beta\Phi_{2}}[(M_{2}\xi_{3}+M_{1}\xi_{4})\cos{(\beta\Phi_{1})}+(M_{1}\xi_{3}-M_{2}\xi_{4})\sin{(\beta\Phi_{1})}]=0,\\\
$ (A.8)
$\xi^{\prime}_{3}-(\lambda_{1}\xi_{4}+\lambda_{2}\xi_{3})+e^{\beta\Phi_{2}}[(M_{1}\xi_{1}-M_{2}\xi_{2})\cos{(\beta\Phi_{1})}+(M_{2}\xi_{1}+M_{1}\xi_{2})\sin{(\beta\Phi_{1})}]=0,\\\
$ (A.9)
$\xi^{\prime}_{4}+(\lambda_{1}\xi_{3}-\lambda_{2}\xi_{4})+e^{\beta\Phi_{2}}[(M_{2}\xi_{1}+M_{1}\xi_{2})\cos{(\beta\Phi_{1})}+(M_{2}\xi_{2}-M_{1}\xi_{1})\sin{(\beta\Phi_{1})}]=0,\\\
$ (A.10)

and the equations for the $\eta_{a}^{\prime}s\,(a=1,2,3,4)$ can be obtained
from (A.7)-(A.10) by replacing

$\displaystyle\xi_{3}$ $\displaystyle\rightarrow$
$\displaystyle\eta_{1},\,\,\,\,\xi_{4}\rightarrow\eta_{2},$ (A.11)
$\displaystyle\xi_{1}$ $\displaystyle\rightarrow$
$\displaystyle\eta_{3},\,\,\,\,\xi_{2}\rightarrow\eta_{4},$ (A.12)

in accordance to the relationship $\xi=\Gamma\eta$ in (2.19). So, let us write

$\eta^{\prime}_{3}+(\lambda_{1}\eta_{4}+\lambda_{2}\eta_{3})+e^{-\beta\Phi_{2}}[(M_{1}\eta_{1}-M_{2}\eta_{2})\cos{(\beta\Phi_{1})}-(M_{2}\eta_{1}+M_{1}\eta_{2})\sin{(\beta\Phi_{1})}]=0,\\\
$ (A.13)
$\eta^{\prime}_{4}-(\lambda_{1}\eta_{3}-\lambda_{2}\eta_{4})+e^{-\beta\Phi_{2}}[(M_{2}\eta_{1}+M_{1}\eta_{2})\cos{(\beta\Phi_{1})}+(M_{1}\eta_{1}-M_{2}\eta_{2})\sin{(\beta\Phi_{1})}]=0,\\\
$ (A.14)
$\eta^{\prime}_{1}-(\lambda_{1}\eta_{2}+\lambda_{2}\eta_{1})+e^{\beta\Phi_{2}}[(M_{1}\eta_{3}-M_{2}\eta_{4})\cos{(\beta\Phi_{1})}+(M_{2}\eta_{3}+M_{1}\eta_{4})\sin{(\beta\Phi_{1})}]=0,\\\
$ (A.15)
$\eta^{\prime}_{2}+(\lambda_{1}\eta_{1}-\lambda_{2}\eta_{2})+e^{\beta\Phi_{2}}[(M_{2}\eta_{3}+M_{1}\eta_{4})\cos{(\beta\Phi_{1})}+(M_{2}\eta_{4}-M_{1}\eta_{3})\sin{(\beta\Phi_{1})}]=0,\\\
$ (A.16)

For the scalar field components one has

$\displaystyle\partial^{2}\Phi_{1}+\beta\cos{(\beta\Phi_{1})}\Big{[}-e^{\beta\Phi_{2}}(\eta_{4}(M_{2}\xi_{1}+M_{1}\xi_{2})-\eta_{3}(M_{1}\xi_{1}-M_{2}\xi_{2}))+$
$\displaystyle
e^{-\beta\Phi_{2}}(\eta_{1}(M_{1}\xi_{3}-M_{2}\xi_{4})-\eta_{2}(M_{1}\xi_{4}+M_{2}\xi_{3}))\Big{]}+$
$\displaystyle\beta\sin{(\beta\Phi_{1})}\Big{[}e^{\beta\Phi_{2}}(\eta_{3}(M_{2}\xi_{1}+M_{1}\xi_{2})+\eta_{4}(M_{1}\xi_{1}-M_{2}\xi_{2}))-$
$\displaystyle
e^{-\beta\Phi_{2}}(\eta_{1}(M_{2}\xi_{3}-M_{1}\xi_{4})+\eta_{2}(M_{2}\xi_{4}-M_{1}\xi_{3}))\Big{]}$
(A.17)

and

$\displaystyle\partial^{2}\Phi_{2}+\beta\cos{(\beta\Phi_{1})}\Big{[}e^{\beta\Phi_{2}}(\eta_{3}(M_{2}\xi_{1}+M_{1}\xi_{2})+\eta_{4}(M_{1}\xi_{1}-M_{2}\xi_{2}))+$
$\displaystyle
e^{-\beta\Phi_{2}}(\eta_{1}(M_{2}\xi_{3}+M_{1}\xi_{4})-\eta_{2}(M_{2}\xi_{4}-M_{1}\xi_{3}))\Big{]}+$
$\displaystyle\beta\sin{(\beta\Phi_{1})}\Big{[}e^{\beta\Phi_{2}}(\eta_{4}(M_{2}\xi_{1}+M_{1}\xi_{2})-\eta_{3}(M_{1}\xi_{1}-M_{2}\xi_{2}))+$
$\displaystyle
e^{-\beta\Phi_{2}}(\eta_{1}(M_{1}\xi_{3}-M_{2}\xi_{4})-\eta_{2}(M_{2}\xi_{3}+M_{1}\xi_{4}))\Big{]}$
(A.18)

The relationship (2.22) becomes

$\displaystyle\Phi^{\prime}_{1}=-\frac{\beta}{2}[\eta_{1}\xi_{1}-\eta_{2}\xi_{2}+\eta_{3}\xi_{3}-\eta_{4}\xi_{4}],$
(A.19)
$\displaystyle\Phi^{\prime}_{2}=-\frac{\beta}{2}[\eta_{2}\xi_{1}+\eta_{1}\xi_{2}+\eta_{4}\xi_{3}+\eta_{3}\xi_{4}].$
(A.20)

For $M_{2}\neq 0$, $\Phi_{2}=0$ and $\eta\neq\pm\xi^{\star}$ one has that the
eq. (A.17) provides the constraint

$\displaystyle\tan{(\beta\Phi_{1})}=\frac{\eta_{3}(M_{2}\xi_{1}+M_{1}\xi_{2})+\eta_{4}(M_{1}\xi_{1}-M_{2}\xi_{2})+\eta_{1}(M_{2}\xi_{3}+M_{1}\xi_{4})-\eta_{2}(M_{2}\xi_{4}-M_{1}\xi_{3})}{\eta_{4}(M_{2}\xi_{1}+M_{1}\xi_{2})-\eta_{3}(M_{1}\xi_{1}-M_{2}\xi_{2})+\eta_{1}(M_{1}\xi_{3}-M_{2}\xi_{4})-\eta_{2}(M_{2}\xi_{3}+M_{1}\xi_{4})}.$
(A.21)

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