#  $\mathcal{PT}$-symmetric phase transition, hysteresis and resonances in the
complex Scarf II potential

Adipta Pal  Indian Institute of Science Education and Research Kolkata,
Mohanpur 741246, India  Subhrajit Modak<EMAIL_ADDRESS>Indian
Institute of Science Education and Research Kolkata, Mohanpur 741246, India
Chemical and Biological Physics Department, Weizmann Institute of Science,
76100, Rehovot, Israel.  Prasanta K. Panigrahi  Indian Institute of Science
Education and Research Kolkata, Mohanpur 741246, India

###### Abstract

We explicitly demonstrate the level dynamics and interferometric origin of
exceptional point and zero-width resonances in the $\mathcal{PT}$-symmetric
broken phase of the complex Scarf II potential. In the process, we show the
intimate connection of $\mathcal{PT}$-symmetry breaking and breaking of
supersymmetry in the unbroken regime and also point out the phenomenon of
hysteresis near exceptional point. We observe that eigenfunctions at spectral
singularities are highly sensitive to a change in parameters pertaining to
competing effects of the extent of $\mathcal{PT}$-symmetry breaking and modes
which coalesce at that point. This, in fact, results in wavepacket
localization at higher modes. Intriguingly, the $\mathcal{PT}$-symmetric
Hamiltonians related by SUSY are also found to be isospectrally deformed
counterparts for a specific parametric condition with the deformation
satisfying the Korteweg-deVries equation.

##  1 Introduction

The search for new materials with desired optical properties has been one of
the main driving force of research in optics and other related areas in recent
times. Emergence of Parity-Time ($\mathcal{PT}$)-symmetric [1, 2] optical
structures represent a new generation of artificial optical systems [3, 4, 5,
6] which utilize gain and loss in a balanced manner in order to achieve a
desired order of functionality. $\mathcal{PT}$-symmetry came into play after
the seminal work of Bender and Boettcher[1, 2] in which they suggested an
extension of quantum mechanics, where the notion of Hermiticity of the
underlying Hamiltonian is replaced by $\mathcal{PT}$ invariance. A one-
dimensional Hamiltonian is $\mathcal{PT}$-symmetric when the corresponding
potential fulfils the condition $V(x)=V^{*}(-x)$, where $x$ is the spatial
coordinate and the asterisk stands for a complex conjugation which requires
that the real part of the potential is even while its imaginary part is an odd
function of position. Such non-Hermitian systems exhibit many interesting
properties which are otherwise unattainable in conventional Hermitian systems.
One of the most striking $\mathcal{PT}$ properties is the appearance of a
sharp, symmetry-breaking transition [7], once the non-Hermitian parameter
crosses a certain threshold. This transition signifies the occurrence of a
spontaneous $\mathcal{PT}$-symmetry breaking from the exact phase where all of
the eigenvalues are real to a broken phase where eigenvalues occur in complex-
conjugate pairs. This symmetry breaking point is termed as exceptional point,
where not only the eigenvalues coalesce but the corresponding eigenvectors
collapse on each other. Exceptional point is typically associated with the
creation of bound and resonance states (along with its anti-resonance partner)
in the broken phase.

$\mathcal{PT}$ symmetric potentials have already appeared in quantum
mechanical structures as shown in [5, 8, 9, 17, 10]. In order to realize
complex $\mathcal{PT}$-symmetric structures in optical systems, the formal
equivalence of the quantum mechanical Schr$\ddot{o}$dinger equation to the
optical wave equation has been exploited by involving symmetric index guiding
and an antisymmetric gain/loss profile. The prospect of utilizing both optical
gain and loss has emerged as a new paradigm in shaping the flow of light.
These include effects like unidirectional invisibility [12], loss-induced
transparency [3], band merging [13], optically induced atomic lattices [16]
and new classes of lasers [14, 15] with improved lasing characteristics. Along
these lines the exceptional points (EPs) have also emerged as a new design
tool for engineering the response of optical systems. At exceptional point
both the eigenvalues coalesce with the merging of corresponding eigenvectors.
Collapse at the EP deeply changes light propagation, which becomes very
sensitive to small changes of initial conditions or system parameters,
similarly to what happens in models of classical or quantum catastrophes [18].
Recent studies have shown that controllable confinement of light can be made
possible at EPs [19]. For an account of the recent status of
$\mathcal{PT}$-symmetry, the interested reader is referred to [20].

Here, we demonstrate the consequence of optical wave propagation through a
medium having complexified dielectric distribution obeying
$\mathcal{PT}$-symmetry. For explicitness, we consider complex Scarf II
profile as refractive index, where all the aforementioned features of
$\mathcal{PT}$-symmetric system are analytically tractable. Recently, this
profile has been utilized for controlling the reflection of light, which at
certain parameter value can be completely reflectionless. Interestingly, these
potentials are characterized by supersymmetry which provides a new avenue for
designing partner index profiles with desired characters [25]. We show that
optics can provide a fertile ground where the ramifications of SUSY can be
explored and utilized to realize a new class of functional structures with
desired characteristics. Moreover, in the Helmholtz regime, SUSY endows two
very different scatterers with identical reflectivities and transmittivities
irrespective of the angle of incidence.

The paper is organized as follows. In section 3, the complex
$\mathcal{PT}$-symmetric potential is introduced and analysed in terms of SUSY
methods, while establishing their connection with optics. The energy
representations in the SUSY broken phases are also derived for the first time
for the complex Scarf II potential. The pertinent features of the
$\mathcal{PT}$ breaking transition in terms of exceptional points and spectral
singularities in the $\mathcal{PT}$-broken phase are also highlighted. Section
4 shows the high parameter sensitivity of the eigenfunctions at the spectral
singularities and in turn wave-packet localization as a consequence. Section 5
makes connection between energy eigenvalues in the -unbroken phase and KdV
equation through isospectral deformation. Section 6 shows yet another feature
of the $\mathcal{PT}$-phase transition as hysteresis-like phenomena in the
parameter space which enables jumping from one superpotential to another
possible. Section 7 concludes the paper with directions for future work. But
first we provide a basic introduction to the supersymmetric methods utilized
in this paper.

##  2 Supersymmetric Quantum Mechanics

Supersymmetry relates two Hamiltonians, which can be used in combination with
shape invariance to obtain the entire eigen-spectrum with their associated
eigenfunctions. These sibling Hamiltonians can be factorized as
$H_{-}=A^{\dagger}A$ and $H_{+}=AA^{\dagger}$. But we also have
$H_{\pm}=-\frac{\partial^{2}}{\partial x^{2}}+V_{\pm}$. A connection may be
established with the introduction of the superpotential,
$W(x)=-\frac{1}{\psi_{0}(x)}\frac{\partial\psi_{0}(x)}{\partial x}$ for the
given pair of potentials. It may then be trivially seen

$V_{\pm}=W^{2}(x)\pm\frac{\partial W(x)}{\partial x}.$

Here, $\psi_{0}(x)$ is the ground state of $H_{-}(x)$. Moreover, the ladder
operators are $A^{\dagger}=-\frac{\partial}{\partial x}+W(x)$,
$A=\frac{\partial}{\partial x}+W(x)$. One may notice that
$E_{n}^{+}=E_{n+1}^{-}$ and thus the spectra of partner potentials differ only
in the ground state of $H_{-}$.

This property can be utilized for our cause, in case of shape invariant
potentials [24]. Now, partner potentials are said to be shape invariant, if
they satisfy relations of the form,

$V_{+}(x;p_{0})=V_{-}(x;p_{1})+R(p_{1})$

where $p_{0}$ is a set of parameters for a given SUSY potential pair, $p_{1}$
is some function of $p_{0}$ and $R(p_{1})$ is not dependent on $x$. Repeated
application yields,

$H^{m}=-\frac{\partial^{2}}{\partial
x^{2}}+V_{+}(x;p_{m})+\sum_{k=1}^{m}R(p_{k})=-\frac{\partial^{2}}{\partial
x^{2}}+V_{+}(x;p_{m-1})+\sum_{k=1}^{m-1}R(p_{k})$

where the ground state energy is given as $E_{0}^{m}=\sum_{k=1}^{m}R(p_{k})$.
Putting $H^{0}=H_{-}$ and $H^{1}=H_{+}$, the spectrum of $H_{-}$ is easily
obtained as

$E_{0}^{n}=\sum_{k=1}^{n}R(p_{k})$

. From the previous paragraph, one may also deduce that
$\psi_{n+1}^{-}(x)\propto A^{\dagger}\psi_{n}^{+}$ and hence the eigenstates
may be derived as

$\psi_{n}^{-}(x;p_{0})\propto
A^{\dagger}(x;p_{0})A^{\dagger}(x;p_{1})...A^{\dagger}(x;p_{n-1})\psi^{-}_{0}(x,p_{n})$

.

Once we have defined the method to be used to determine the energy spectra, we
immediately proceed to the $\mathcal{PT}$ symmetric complex potential, where
we want to apply it. Before that, it is worthwhile to indulge in the physical
connection of $\mathcal{PT}$ symmetric configurations with optics. The results
discussed thereafter shall assume greater significance in an optical
framework.

##  3 Optical connection

Let a plane monochromatic optical wave be incident at an angle $\theta$ on a
medium with inhomogeneous dielectric distribution,
$\epsilon(x)=\epsilon_{b}+\alpha(x)$. The inhomogenity is assumed to vanish at
spatial infinity, so that $\epsilon(x)\rightarrow\epsilon_{b}$ as
$|x|\rightarrow\infty$. The problem can be made effectiveley one dimensional
due to the fact that dependence on $y$ is given by $e^{ik_{y}y}$,where
$k_{y}=k_{0}\sqrt{\epsilon_{b}}\text{sin}\theta$ . Note that $k_{y}$ is the
same for each layer (follows from the continuity of tangential fields) leading
to a propagation equation in $x$. The propagation equation for Transverse
Electric waves is found to satisfy Schr$\ddot{o}$dinger equation

$\frac{d^{2}\mathcal{E}}{dx^{2}}+(k_{0}^{2}\epsilon(x)-k_{y}^{2})\mathcal{E}=0,$
(1)

gives the relation between the energy and potential as,

$E=k_{0}^{2}\epsilon_{b}\text{cos}^{2}\theta.$ (2)
$V(x)=k_{0}^{2}\epsilon_{b}-k_{0}^{2}\epsilon(x).$ (3)

The potential $V(x)$ is said to be reflectionless if any wave with arbitrary
positive energy can pass through the potential completely. It is also clear
that Eq.(3) establishes the relation between the reflectionless potential and
the corresponding dielectric function profile $\epsilon(x)$. Since refractive
index is given by the square root of the dielectric function, Eq.(3) can be
rewritten to yield the corresponding reflectionless refractive index profile
$n(x)$ as

$n^{2}(x)=n_{b}^{2}-\frac{V(x)}{k_{0}^{2}}.$ (4)

In this framework, if $\mathcal{PT}$-symmetry is addressed, the refractive
index needs to get satisfied $n(x)=n^{*}(-x)$. Here we address the effects and
consequences of choosing complex Scarf profile as a substrate potential. One
must notice that the Scarf potential vanishes asymptotically and as already
pointed out, it is possible to regulate the permittivity and hence the
refractive index by controlling the real and imaginary parts of the potential.
We analyze this system from an algebraic point of view, applying the tools of
supersymmetric quantum mechanics (SUSY-QM). In order to see the role of SUSY
in the $\mathcal{PT}$-symmetric system, we start with the complex
superpotentials,

$W_{\mathcal{PT}}^{\pm}=\Big{(}A\pm iC^{\mathcal{PT}}\Big{)}tanh\alpha
x+\Big{(}\pm C^{\mathcal{PT}}+iB\Big{)}sech\alpha x,$ (5)

which leads to both real and complex spectra. Under certain parametrization,
when the superpotential is unique, the spectrum is real and shape-invariant,
leading to translational shift in a suitable parameter by real units. For a
different parametrization, two different superpotentials leading to same
potential, yields broken $\mathcal{PT}$-symmetry, the energy spectra in the
two phases being separated by a bifurcation. Interestingly, these two
superpotentials generate two disjoint sectors of the Hilbert space. In the
broken case, shape invariance produces complex parametric shifts. In order to
see the role of SUSY and $\mathcal{PT}$-symmetry, we take $V_{-}$
corresponding to Eq.5

$\displaystyle\begin{split}V(x)={}&-\big{[}(A\pm iC^{\mathcal{PT}})(A\pm
iC^{\mathcal{PT}}+\alpha)-(\pm C^{\mathcal{PT}}\\\ &+iB)^{2}]sech^{2}(\alpha
x)-i(\pm iC^{\mathcal{PT}}-B)\big{[}2(A\pm iC^{\mathcal{PT}})\\\
&+\alpha)\big{]}sech(\alpha x)tanh(\alpha x),\end{split}$ (6)

which in general may not be $\mathcal{PT}$-symmetric. To be
$\mathcal{PT}$-symmetric, the coefficient of the first term must be real and
that of the second term must be purely imaginary. Implementing these
conditions, one arrives at a unique relation [7]:

$C^{\mathcal{PT}}\big{[}2(A-B)+\alpha\big{]}=0.$ (7)

Choice of the condition decides whether the symmetry is maintained or not. If
$C^{\mathcal{PT}}=0$, the potential becomes $\mathcal{PT}$-symmetric,

$\displaystyle\begin{split}V_{\mathcal{PT}}(x)={}&-\big{[}A(A+\alpha)+B^{2}\big{]}sech^{2}(\alpha
x)\\\ &+iB(2A+\alpha)sech(\alpha x)tanh(\alpha x),\end{split}$ (8)

derived from the superpotential,

$W_{1}(x)=A\leavevmode\nobreak\ tanh(\alpha x)+iB\leavevmode\nobreak\
sech(\alpha x).$ (9)

This potential is invariant under the transformation $A\leftrightarrow
B-\frac{\alpha}{2}$ [22, 21], which provides another candidate for the
superpotential,

$W_{2}(x)=(B-\frac{\alpha}{2})tanh(\alpha x)+i(A+\frac{\alpha}{2})sech(\alpha
x).$ (10)

These superpotentials lead to two sets of real energy eigenvalues,

$E^{1}_{n}=-(A-n\alpha)^{2},$ (11)

and

$E^{2}_{n}=-(B-\frac{\alpha}{2}-n\alpha)^{2}.$ (12)

Here it is imperative to point out that the above expressions are valid only
if, $A>0$ and $B>\frac{\alpha}{2}$, respectively. For the other regions in the
parameter space, we notice that, at least one of the energy eigenvalue set
fails to give a normalizable ground state, which points to the case of broken
supersymmetry. Interestingly, for the broken SUSY cases also, exact energy
eigenvalues can be obtained by two set of shape invariance [28, 17]. The
convert first the potential to SUSY unbroken state and the next one finds the
energy eigenvalues. The regions of broken and unbroken SUSY have been depicted
in Fig. 1 explicitly. The energy eigenvalues in these regions are obtained as
follows:

Figure 1: The shaded regions I, II and III are broken SUSY areas in the
parameter space. The corrections of energy eigenvalues are carried out for the
broken SUSY domains by considering two shape invariances of the complex
$\mathcal{PT}$-symmetric Scarf II potential.

  * •

Domain I ($A<0,B>\alpha/2$): Consider first the shape invariance
$A\rightarrow-A$, $B\rightarrow-B$ and then apply the known shape invariance
$A\rightarrow A-\alpha$, $B\rightarrow B$, to get energy eigenvalues for (9)
as

$E^{1}_{n,I}=-(A+n\alpha)^{2},$ (13)

$E^{2}_{n}$ remains the same. Notice that the first shape invariance changes
the sign of A so that unbroken SUSY can be applied, and although sign of B is
changed it does not contribute to the energy eigenvalues for (9).

  * •

Domain II ($A>0,B<\alpha/2$)): Consider first the shape invariance
$B-\alpha/2\rightarrow-(B-\alpha/2)$, $A+\alpha/2\rightarrow-(A-\alpha/2)$ and
then apply the known shape invariance $B-\alpha/2\rightarrow
B-\alpha/2-\alpha$, $A-\alpha/2\rightarrow A-\alpha/2$, to get energy
eigenvalues for the superpotential (10) as

$E^{2}_{n,II}=-(B-\frac{\alpha}{2}+n\alpha)^{2},$ (14)

$E^{1}_{n}$ remains unchanged. The first shape invariance changes the sign and
takes it to the unbroken SUSY domain, sign change for the second parameter
does not affect as it does not contribute to the energy for (10).

  * •

Domain III ($A<0,B>\alpha/2$): Here one needs to consider both the techniques,
one for Domain I for the eigenvalues of (9) and one for Domain II for the
eigenvalues of (10), which yields

$E^{1}_{n,III}=-(A+n\alpha)^{2},\qquad
E^{2}_{n,III}=-(B-\frac{\alpha}{2}+n\alpha)^{2},$ (15)

###  3.1 Condition 2.

For $C^{\mathcal{PT}}\neq 0$, one gets $A=B-\frac{\alpha}{2}$, which when
substituted in Eqs. 5 and 6, yields

$W_{\mathcal{PT}}^{\pm}=\Big{(}A\pm iC^{\mathcal{PT}}\Big{)}tanh\alpha
x+\Big{(}\pm C^{\mathcal{PT}}+i(A+\frac{\alpha}{2})\Big{)}sech\alpha x.$ (16)

It needs to be emphasized that both the superpotentials leads to the same
$\mathcal{PT}$-symmetric complex potential, corresponding to broken
$\mathcal{PT}$-symmetry. Utilizing the shape invariance condition, one can
simply obtain the energy eigenvalues,

$E_{n}^{\pm}=-(n\alpha-A\pm iC^{\mathcal{PT}})^{2},$ (17)

which has complex conjugate pairs and each member of a complex conjugate pair
can be connected to one of the two superpotentials in Eq. 16. Same can be said
about the corresponding eigenfunctions,

$\psi_{n}^{\pm}(x)\propto\Big{(}sech(\alpha x)\Big{)}^{\frac{1}{\alpha}(A\pm
iC^{\mathcal{PT}})}\text{exp}\Big{[}\Big{(}-\frac{i}{\alpha}(A+\frac{\alpha}{2})\mp\frac{C^{\mathcal{PT}}}{\alpha}\Big{)}tan^{-1}(sinh(\alpha
x))\Big{]}P_{n}^{\mp\frac{2iC^{\mathcal{PT}}}{\alpha},-\frac{2A}{\alpha}-1}[i\leavevmode\nobreak\
sinh(\alpha x)].$ (18)

This pair of wavefunctions are related to each other with $\mathcal{PT}$
transformations. Here
$P_{n}^{\mp\frac{2iC^{\mathcal{PT}}}{\alpha},\frac{2A}{\alpha}+1}[i\sinh(\alpha
x)]$ is the $n^{\text{th}}$ order Jacobi polynomial. Notice that, inspite of
being in broken $\mathcal{PT}$ phase, complex eigenmodes coalease at
$A=n\alpha$, i.e., neither of them experiences a net gain or loss. Vanishing
of the damping term results in zero leakage, i.e., zero line width states with
infinite quality factor, in other words, zero-width resonances [14]. The
energy eigenvalue is trivially found there to be
$E_{SS}={C^{\mathcal{PT}}}^{2}$ and the conditions are essentially identical
to the one in [31]. Character of the modes change as soon as this condition
gets violated, i.e., $A\neq n\alpha$. In this regime, symmetry of each mode is
broken such that one of them enjoys amplification while the other one
experiences attenuation. These $\mathcal{PT}$-symmetric spectral singularities
are manifested at the transition between stable regions and region of
instabilities.

##  4 Parameter dependence of eigenfunction at spectral singularities:

The spectral singularity eigenfunctions show some interesting features as the
right parameters are varied. The eigenfunctions are naturally $A$ and
$C^{\mathcal{PT}}$ dependent. The $A$ dependence comes from the fact that for
spectral singularities $A$ must occupy integral multiples of $\alpha$, i.e.
$A=n\alpha$ and so higher values of A implies higher degree Jacobi
polynomials. This is clear from the expression of the degenerate
wavefunctions:

$\psi_{n,SS}^{\pm}(x)\propto\Big{(}sech(\alpha x)\Big{)}^{n\pm
i\frac{C^{\mathcal{PT}}}{\alpha}}\text{exp}\Big{[}(-i(n+\frac{1}{2})\mp\frac{C^{\mathcal{PT}}}{\alpha})tan^{-1}(sinh(\alpha
x)\Big{]}P_{n}^{\mp\frac{2iC^{\mathcal{PT}}}{\alpha},-2n-1}[i\leavevmode\nobreak\
sinh(\alpha x)].$ (19)

The nature of the wavefunction reveals competing effects between the value of
$A$ and $C^{\mathcal{PT}}$, for when $A$ is greater than $C^{\mathcal{PT}}$,
one starts to observe a localization at $x=0$ compared to the abrupt
transition from low amplitude constant function to high amplitude oscillations
at the same critical point, $x=0$ as shown in. The transition direction,
however, depends entirely on the sign of $C^{\mathcal{PT}}$. Varying the
parameters carefully, one notices that the transition from sudden high
frequency oscillations to the localized wavepacket is a process depending
entirely on how much greater $A$ is compared to $C^{\mathcal{PT}}$. The value
of $\alpha$ just monitors the rate of the transitions at a given value of $A$
and $C^{\mathcal{PT}}$. It may be further noticed that as the value of
$C^{\mathcal{PT}}$ is increased, varying the modes around $C^{\mathcal{PT}}$
leads to the same sequence of events as shown in Fig. 2 but oscillations have
a higher frequency.

(a) n=2, $C^{\mathcal{PT}}$=7, $\alpha=2$

(b) n=8, $C^{\mathcal{PT}}$=7, $\alpha=2$

(c) n=12, $C^{\mathcal{PT}}$=7, $\alpha=2$

(d) n=8, $C^{\mathcal{PT}}$=7, $\alpha=2$

(e) n=2, $C^{\mathcal{PT}}$=-7, $\alpha=2$

(f) n=2, $C^{\mathcal{PT}}$=-12, $\alpha=2$

Figure 2: The first five plots (unnormalized) represent the real (blue) and
imaginary (red) parts of $\psi^{+}_{n,SS}(x)$ for at $C^{\mathcal{PT}}=7$ and
$\alpha=2$ clearly showing oscillations at lower n values and localization for
higher n values. More generally, it is the difference in magnitude of n and
$C^{\mathcal{PT}}$ which results in such features.(f) shows the fifth plot for
higher $C^{\mathcal{PT}}$ values and is associated with an increase in
frequency. Actually at higher $C^{\mathcal{PT}}$ one may all the first five
plots just at higher frequency.

This behaviour of wavefunction is interesting in its own right when one
recollects the one to one relation between the wavefunction, $\psi(x)$ and the
electric field, $E(x)$. As noted here, a parameter variation on the potential,
or refractive index, acts as knob to control the release and storage of
electric field energy.

##  5 Isospectral deformation and connection with KdV

The two different superpotentials giving rise to the same potential is known
to appear in the case of the iso-spectral deformation in supersymmetric
quantum mechanics. In this, a given potential remains unchanged by an additive
deformation of the superpotential whereas its partner potential is changed
having the same energy levels. These deformed potentials satisfy the KdV
equation and the corresponding superpotentials obey the mKdV equation. Hence
its is natural to explore the physical nature of the additive deformation of
the superpotenial in the present case. This section is, therefore, concerned
with the possible application of KdV equation (and its modified version) to
the design of planar permittivity profiles and generating a family of complex
materials via isospectral deformation. If the permittivity profile is
subjected to evolve as KdV variable, it can be linked directly to mKdV
equation via Miura transformation. Use of Cole-Hopf transformation in one of
the Miura route yields Schr$\ddot{o}$dinger equation. Now, the Galilean
invariance of KdV equation allows a constant shift in KdV variable, leading to
a equivalent of Eq.1, where the KdV variable act as a potential. It can be
seen in the next steps how a deformation at the level of superpotential can
generate different permittvity profiles. In terms of superpotential $W(x)$ we
are assuming that there exist other superpotentials $\tilde{W}(x)$ which yield
the same $V_{-}(x)$. In general, if we set

$\tilde{W}(x)=W(x)+v(x),$ (20)

it leads to the following Bernoulli equation:

$\frac{dv(x)}{dx}-2W(x)v(x)-v^{2}(x)=0.$ (21)

It is to be noted that superpotential is dependent on the ground state that
encodes the boundary conditions which the potential by itself does not [32].
Thus isospectral deformation only keeps the value of $V_{\mathcal{PT}}$
unchanged, not the system boundaries. According to the formulation, difference
between the superpotentials are related by the function

$v(x)=(B-\frac{\alpha}{2}-A)tanh\alpha x-i(B-\frac{\alpha}{2}-A)sech\alpha x,$
(22)

satisfies Eq. 21 when $B-\frac{\alpha}{2}=-A$. Under this condition, Eq. 22
turns to,

$v(x)=-2Atanh\alpha x+i2Asech\alpha x.$ (23)

Interesting to note that, the functional form for the translation satisfies
exactly the mKdV equation [27]. Whereas two superpartners generated due to the
deformation satisfy KdV equation. Interestingly, the sum of the paired
solution is also a solution of the KdV equation, whereas the difference is
not. The existence of these solutions can be traced to the fact that, there
are two distinct mKdV equation whose solutions are related through
$v\rightarrow iv$, where $v$ is the mKdV variable. The Miura transformation
for the solution of KdV, $u=v^{2}\pm v_{x}$, then implies that, $u=-v^{2}\pm
iv_{x}$, is also a solution. This, in turn, generates more general complex
superposed solutions for the KdV equation with distinct permittivity profiles.

##  6 The Hysteresis phenomena

We now consider the energy eigenvalues (11) and (12) defined for the two
superpotentials (9) and (10) respectively. The lowest energy in both cases are
$-A^{2}$ and $-(B-\frac{\alpha}{2})^{2}$ respectively. Therefore depending on
the parameters $A$ and $B$, we can ascribe the ground state either to $-A^{2}$
or $-(B-\frac{\alpha}{2})^{2}$ based on which is lower in energy. Evidently,
the superpotential is dependent on the ground state wavefunction which are
different for the two cases and there can be only one ground state in the one
dimensional case. Therefore, only the superpotential corresponding to the
lower energy among the mentioned lowest energies should exist at a given time.
The condition for which ground state will be present at a given parametric
domain can be expressed as follows:

$A^{2}-(B-\alpha/2)^{2}=K^{2},\quad K=constant;\qquad
or,\frac{A^{2}}{K^{2}}-\frac{(B-\alpha/2)^{2}}{K^{2}}=1,$ (24)

which is a hyperbola in the parameter space of $A$ and $B$. The above
condition corresponds to $-A^{2}$ being the ground state and thus
superpotential (9) exists.

Figure 3: Regions I and II correspond to the specific ground states based on
the parameters involved. The asymptote with positive slope (a) corresponds to
the condition for exceptional points while that with negative slope (b)
corresponds to isospectral deformation. A small perturbation from the
condition for exceptional point or isospectral deformation may lead us to a
domain different from the domain of our original trajectory.

The contrary condition expressed as:

$(B-\alpha/2)^{2}-A^{2}=K_{1}^{2},\quad K_{1}=constant;\qquad
or,\frac{(B-\alpha/2)^{2}}{K_{1}^{2}}-\frac{A^{2}}{K_{1}^{2}}=1$ (25)

yields a conjugate hyperbola to the previous one, such that
$-(B-\frac{\alpha}{2})^{2}$ is the ground state and the superpotential (10)
holds. We then have the same asymptote to the two hyperbolas, where the
asymptotes $B=\pm A+\alpha/2$ point to conditons pertaining to the exceptional
point and isospectral deformation respectively as depicted in Fig. 2. The
different features of the above figure can be explicated as:

  * •

Region I contains all the hyperbola with $-A^{2}$ as the ground state with
superpotential (9).

  * •

Region II contains all the hyperbola with $-(B-\frac{\alpha}{2})^{2}$ as the
ground state with superpotential (10).

  * •

(a) corresponds to the asymptote given by condition $A=B-\alpha/2$ and thus
gives the exceptional points.

  * •

(b) is the other asymptote but it is given by the condition $A=-B+\alpha/2$
and is a possible region for real energy eigenvalues and it relates the
superpotentials (9) and (10) as isospectrally deformed counterparts.

  * •

The coordinates $(\alpha/2,0)$ and $(-\alpha/2,\alpha)$ are the parameters
under which the isospectral deformation and its associated potential satisfy
the mKdV equation and KdV respectively.

It is observed that if for a certain perturbation to the parameters, one
reaches the exceptional point then on the opposite path one can move either
through the hyperbola one had been originally along or the conjugate
hyperbola, making jumping from one superpotential to another possible. The two
superpotentials which characterize the complex Scarf II potential describe the
energy eigenvalues in disjoint regions of the $\mathcal{PT}$-unbroken phase
separated by the conditions of isospectral deformation and
$\mathcal{PT}$-symmetry breaking where the two energy eigenvalue sets merge.
However, the superpotentials do not merge at the isopectral deformation while
they merge for the $\mathcal{PT}$-symmetry breaking as shown in Fig. 2. The
hyperbolic natures of the difference between the two ground state energy
representations which asymptotically tends to the $\mathcal{PT}$-breaking show
that the $\mathcal{PT}$ phase transiition has an inbuilt hysteresis. SUSY
broken regions as shown in Fig. 1 can be considered to be arising due to
isospectral deformations which follows independently from the Bernoulli
equation. Together Fig. 1 and Fig. 3 depict four parameter domains for energy
representation. .

##  7 Conclusion

In conclusion, we have explicitly demonstrated the formation of exceptional
points and existence of both bound and resonance states in the broken
$\mathcal{PT}$ scenario for appropriate parametric conditions . Remarkably,
breaking of SUSY and $\mathcal{PT}$-symmetry are found to exist in mutually
exclusive domains with the $\mathcal{PT}$-symmetry boundaries showing
hysteresis. Moreover, the parametric conditions for isospectral deformation
connecting the two superpotentials describing the same potential in the
$\mathcal{PT}$-unbroken phase, derived here, has been enabled only due to the
corrections carried out for the broken SUSY phase. The intersecting lines as
depicted in Fig. 3 clearly demarcate the $\mathcal{PT}$-symmetry breaking
boundary as well as the isospectral deformation criteria along which lies
points connecting the deformation associated potential with KdV equation. The
unusual parameter sensitivity of the nature of eigenfunctions at spectral
singularities gives rise to wave packet localization for higher modes
corresponding to the magnitude of $C^{\mathcal{PT}}$. This is interesting in
the sense that it can be utilized to build novel devices to store electric
field energy where the zero-width resonance acts as an added bonus to
exponentially increase the efficiency of such devices. This direction is
pursued in later works.

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