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# Multi-criticality and long-range effects in non-Hermitian topological models

Y R Kartik  Theoretical Sciences Division, Poornaprajna Institute of
Scientific Research, Bidalur, Bengaluru 562164, India.  Graduate Studies,
Manipal Academy of Higher Education, Madhava Nagar, Manipal-576104, India.
Sujit Sarkar  Theoretical Sciences Division, Poornaprajna Institute of
Scientific Research, Bidalur, Bengaluru 562164, India.

###### Abstract

Long-range effects induce some interesting behavior and considered as a
gateway to understand the non-local behavior in the quantum systems.
Especially, the long-range topological models became a platform for the
realization of new quasi-particles, which are believed to be potential
candidates for the topological qubits. In this work, we consider non-Hermitian
Su-Schriffer-Heeger (SSH) model and discuss the interplay of non-Hermiticity
and long-range effects. We use the approach of momentum space
characterization, critical exponents and curvature renormalization group (CRG)
method to understand the aspects of interplay. The longer-range (finite
neighbors) effect produces higher winding numbers, where we observe a
staircase of transitions among the even-even and odd-odd winding numbers which
depends on the number of interacting neighbors. Here we also highlight the
effect of multi-criticality in the system and show that they belong to a
different universality class. The interplay of long-range (infinite neighbors)
effect and non-Hermiticity produces fractional topological invariants, and we
analyze them from the behavior of pseudo-spin vectors. We also determine the
long-range and short-range limit of the model through universality class of
critical exponents. Our work mainly showcases that how the study of
criticality in topological system is interesting in exploring the interplay of
non-Hermiticity and long-range effects.

##  I Introduction

Topological state of matter is the milestone in the area of condensed matter,
both from the perspective of theory and experimentsHasan and Kane (2010);
Moore (2010); Stanescu (2016); Bernevig and Hughes (2013); Ortmann  _et al._
(2015); Haldane (2017). The transitions among the topological phases can not
be apprehended by the conventional Landau theory of symmetry breaking, as they
do not possess an order parameterLandau (1965); Berry (1985); Wilczek and
Shapere (1989); Wen (2016). The topological phases possess localized edge
modes which are protected by certain symmetries and are robust against the
external perturbationsNiu  _et al._ (2012); Sarma  _et al._ (2015); Kitaev
(2001). Topological invariant is the most accepted tool to differentiate the
index of the topological phases, where topological invariant is equal to the
number of localized edge modesAlecce and Dell’Anna (2017). Even though
topological invariant can not be considered as local order parameter, it has a
well defined form in gapped phases and ill-defined at criticality. The edge
mode contains a topological localization length, which diverges as the system
drives towards criticalityContinentino  _et al._ (2020); Chen and Sigrist
(2019); Molignini  _et al._ (2020a, 2018); Chen (2016); Chen  _et al._ (2017);
Malard  _et al._ (2020a); Chen  _et al._ (2016); Chen (2018); Chen and
Schnyder (2019); Molignini  _et al._ (2020b); Abdulla  _et al._ (2020);
Molignini  _et al._ (2020a); Kumar  _et al._ (2022). This diverging nature
gives the idea of explaining the criticality through set of critical exponents
and renormalization group methodsKempkes  _et al._ (2016); Rufo  _et al._
(2019); Kartik  _et al._ (2021); Kumar  _et al._ (2021a). Especially
renormalization group (RG) method can be very helpful during the topological
transitions where there are actually no order parametersSachdev (2007).
Introducing non-Hermiticity in topological systems opened a new doorway
towards the unexplored parts of novel phases of matter Ashida  _et al._
(2020). The works in non-Hermitian topological systems from the perspective of
bulk-boundary correspondenceJin and Song (2019), loss-gainAgarwal and Joglekar
(2021), dissipationRahul (2022), symmetryShiozaki and Ono (2021); Wang  _et
al._ (2015); Kawabata  _et al._ (2019, 2018), skin effectYao and Wang (2018)
and critical scalingArouca  _et al._ (2020) gained a lot of attention in
recent days. The realization of non-Hermitian models in experimental setups
provided new understandings in optical systemsSchomerus (2010); El-Ganainy
_et al._ (2019), electric circuitsHofmann  _et al._ (2020); Zhang and Franz
(2020) and ultra-cold atomic gasesJotzu  _et al._ (2014); Bloch  _et al._
(2012). The longer-range effects in non-Hermitian topological system created
some fractional topological phases including non metallic phasesBao  _et al._
(2021); Yin  _et al._ (2018), which gained attention both theoretically and
experimentally Xu  _et al._ (2021); Li and Miroshnichenko (2018); Bao  _et
al._ (2021).
The longer-range models are important milestones in topological systems due to
the formation of more than one edge modes and formation of multi-critical
pointsKartik  _et al._ (2021). The increase in the number of coupling neighbor
generates higher winding numbers (WNs). Furthermore, the increase in the decay
parameter reduces the system towards short-range through staircase of
topological transitionKartik  _et al._ (2021). Multi-criticality is an
interesting phenomenon which can exhibit a combined behavior of two or more
different criticalitiesKumar  _et al._ (2021a, b); Kartik  _et al._ (2021). In
Hermitian systems, multi-critical point witnesses transition between different
(gapped-gapped, gapless-gapless, gapped-gapless) phasesKartik  _et al._
(2021). In addition, under some cases, multi-critical points have been
recognized as the point where Lorentz invariance violatesKumar  _et al._
(2021a). Also, there are observations, where the introduction of non-
Hermiticity violates multi-critical effectsRahul (2022).
On the other hand, long-range effects in condensed matter has a long history
and had been an area of curiosity over the decadesDefenu  _et al._ (2021).
Especially, long-range effects in topological systems been used to understand
the non-local behavior of the fermionsVodola (2015). Long-range topological
models exhibit the emergence of massive edge modesVodola  _et al._ (2014),
topological transition without gap closingVodola  _et al._ (2014); Kartik  _et
al._ (2021), area law violation for Von Neumann entropy and breaking of
Lorentz invarianceVodola  _et al._ (2015). The long-range models are also
important as they can be realized in the experiment setups of trapped ionsDeng
_et al._ (2005); Britton  _et al._ (2012); Hauke  _et al._ (2010); Roy  _et
al._ (2019), multi-mode cavitiesDouglas  _et al._ (2015), magnetic
impuritiesZhang  _et al._ (2019); Ménard  _et al._ (2015) and simulated
circuitsAmin  _et al._ (2019). Even in two dimensional models, long-range
effects has enhanced the topological behaviorsViyuela  _et al._ (2018). The
massive edge modes has been predicted to be a potential qubit for topological
computationViyuela  _et al._ (2016). The extension of long-range effects
towards the non-Hermitian systems naturally triggers the curiosity to explore
the interplay among them. Here our motivation is as follows.

  * •

To understand the behavior of non-Hermitian system with increased neighboring
coupling.

  * •

To explore the multi-critical effects in non-Hermitian systems and to
differentiate the multi-criticality from normal criticalities through
universality class and curvature renormalization group (CRG).

  * •

To understand the interplay of non-Hermiticity and long-range effect in a
topological system, thus to understand the non-local behaviors of the fermions
in these systems.

  * •

The interface of short-range and long-range limit is an interesting topic of
discussion. There are many arguments to decide the short-range limit based on
entanglement entropyVodola  _et al._ (2014), conformal invarianceVodola  _et
al._ (2015) and bulk-edge correspondenceLepori and Dell’Anna (2017). Here we
give an approach of universality class of critical exponents to decide the
short-range limit in a non-Hermitian long-range system.

With these motivations, we present the manuscript in following pattern. In
Sec.II we introduce the model Hamiltonian and explain the momentum space
properties. This section includes the study of universality class of critical
exponents, CRG method and behavior of multi-critical points in longer-range
model. In Sec.III, we study on long-range models with winding vector analysis,
universality class, higher order quantum transitions and interplay among non-
Hermiticity and long-range effect. In Sec.IV we give outlook and experimental
possibility of our work and conclude in Sec.V.

##  II Model Hamiltonian and properties

We consider chiral non-Hermitian 1D Su-Schriffer-Heeger (SSH) chain with long-
range interaction. Here, the non-Hermiticity is due to the imbalanced
intracell hoppings, i.e., the hopping within the lattice $i$ (between the sub-
lattice $A_{i},B_{i}$) is a real non-reciprocal quantity, which violates the
Hermiticity in the HamiltonianYin  _et al._ (2018).

$\displaystyle H$ $\displaystyle=$
$\displaystyle\sum_{j=1}^{L-l}(t-\delta)c^{\dagger}_{j,a}c_{j,b}+(t+\delta)c^{\dagger}_{j,b}c_{j,a}$
(1) $\displaystyle+$
$\displaystyle\sum_{j=1}^{L-l}\sum_{l=1}^{r}\frac{t^{\prime}}{l^{\alpha}}(c^{\dagger}_{j+l,a}c_{j,b}+c^{\dagger}_{j,b}c_{j+l,a}),$

where $\frac{t^{\prime}}{l^{\alpha}}$ is the intercell long-range hopping with
$l$ as site index and $\alpha$ as decay parameter respectively. The imbalance
in the intracell coupling is introduced through the non-zero term $\delta$,
which breaks the Hermiticity and leads to non-Hermitian skin effectYao and
Wang (2018); Yin  _et al._ (2018). The term $L$ is the system size where the
intercell coupling occurs between sites $i$ and $i+l$. The term $l$ can take
the upper limit $r$, where $r$ can be infinite. In such cases, it is called
long-range systems with infinite number of coupling neighbors. On the other
hand, if $r$ is finite, then it is longer-range with finite number of coupling
neighbors. In addition to these factors, as $\alpha\rightarrow\infty$, the
model reduces to short-range version irrespective of number of interacting
neighbors.

Figure 1: (Color online) Schematic representation of non-Hermitian long-range
SSH chain with open boundary condition. Colored rectangles represent the unit
cell, $t+\delta$ and $t-\delta$ are the forward and backward hopping
(intracell hopping) between sub-lattice sites $A$ and $B$ respectively. Here
$t^{\prime}$ is the intercell hopping (short-range limit) and
$\frac{t^{\prime}}{r^{\alpha}}$ is the longer-range hopping parameter.

After the Fourier transformation, BdG Hamiltonian can be written as

$H_{BdG}(k)=\left(\begin{matrix}0&&\chi_{x}(k)+i\chi_{y}(k)\\\
\chi_{x}(k)-i\chi_{y}(k)&&0\\\ \end{matrix}\right).$ (2)

With the use of Anderson pseudo-spin approach Anderson (1958); Sarkar (2018),
we write the BdG Hamiltonian in the pseudo-spin basis as

$H_{BdG}(k)=\chi_{x}(k)\vec{\sigma_{x}}+\chi_{y}(k)\vec{\sigma_{y}},$ (3)

where $\sigma_{x},\sigma_{y}$ are the Pauli matrices and the corresponding
coefficients,

$\displaystyle\chi_{x}(k)$ $\displaystyle=$ $\displaystyle
t+t^{\prime}\sum_{l=1}^{r}\frac{\cos(kl)}{l^{\alpha}},$
$\displaystyle\chi_{y}(k)$ $\displaystyle=$ $\displaystyle
t^{\prime}\sum_{l=1}^{r}\frac{\sin(kl)}{l^{\alpha}}-i\delta.$ (4)

It is to be noted that for $r\rightarrow\infty$ the series involving
$\frac{\cos(kl)}{l^{\alpha}}$ and $\frac{\sin(kl)}{l^{\beta}}$ terms give rise
to polylogarithmic functions Alecce and Dell’Anna (2017); Vodola  _et al._
(2014). Energy dispersion is given byRufo  _et al._ (2019); Chen and Sigrist
(2019)

$\displaystyle E(k,\mathbf{M})$ $\displaystyle=$
$\displaystyle\pm\sqrt{(\chi_{z}(k))^{2}+(\chi_{y}(k))^{2}},$ $\displaystyle=$
$\displaystyle\sqrt{(t^{\prime})^{2}+t^{2}-\delta^{2}+2tt^{\prime}\frac{\cos(lk)}{l^{\alpha}}+i2\delta
t^{\prime}\frac{\sin(lk)}{l^{\alpha}}}.$

Due to the non-Hermiticity, the Eq II gives the complex energy spectrum for
finite $\delta$Yin  _et al._ (2018). At the phase boundary the imaginary part
vanishes, such that the criticality condition is monitored by the real part.
For a non-Hermitian system, the Fermi sea level is contributed by the real
part, while the imaginary part signals the decoherence (information leak to
the environment)Rahul (2022); Yin  _et al._ (2018).

###  II.1 Winding number in non-Hermitian systems

For topological conditions, the winding vectors form a closed loop in the
parameter space, due to the periodicity in the boundary condition. In
Hermitian systems, when the parameter $k$ runs from $-\pi$ to $\pi$, which
gives topological invariantWilczek and Shapere (1989),

$W=\left(\frac{1}{2\pi}\right)\int_{-\pi}^{\pi}\frac{\chi_{x}\partial_{k}\chi_{y}-\chi_{y}\partial_{k}\chi_{x}}{\chi_{x}^{2}+\chi_{y}^{2}}dk.$
(6)

The imbalance in the intercell hopping induces non-Hermiticity in the
Hamiltonian which results in the complex topological angle ($\phi$), i.e.,

$\phi=\phi^{re}+i\phi^{im},$ (7)

where the imaginary parts do not contributes to the argument of the
topological angleYin  _et al._ (2018). But the imaginary part creates two
exceptional points (with topological angles $\phi_{1},\phi_{2}$) instead of
single Dirac (gap closing) point. The angles $\phi_{1}$ and $\phi_{2}$ are
real and they produce the winding number asYin  _et al._ (2018)

$W=\frac{1}{2}\left(\frac{1}{2\pi}\right)(\oint\partial_{k}\phi_{1}dk+\oint\partial_{k}\phi_{2}dk),$
(8)

where
$\phi_{1}=\tan^{-1}\left(\frac{\chi_{y}^{re}(k)+\chi_{x}^{im}(k)}{\chi_{x}^{re}(k)-\chi_{y}^{im}(k)}\right)$
and
$\phi_{2}=\tan^{-1}\left(\frac{\chi_{y}^{re}(k)-\chi_{x}^{im}(k)}{\chi_{x}^{re}(k)+\chi_{y}^{im}(k)}\right)$.(For
detailed derivation, refer Appendix A)
Here, we initially consider the simplest longer-range model with two neighbor
couplings. We gradually increase the number of neighbor couplings to get
further longer-range models and to analyze the transitions among them. At the
end, we consider infinite couplings to understand the interplay of non-
Hermiticity and long-range behavior.

###  II.2 Critical exponents and universality class

In the vicinity of phase transition point, the physical quantities shows the
diverging nature, which can be quantified by the critical
exponentsContinentino (2017); Continentino  _et al._ (2020). The set of
critical exponents constitutes the universality class, which effectively
categorizes the phase transitions based on their behavior. Here we analyze a
few critical exponents (relevant to zero temperature TQPTs) and analyze their
behavior.

The dynamical critical exponent $(z)$ defines the nature of dispersion and can
be calculated by expanding Eq. II around gap closing point $k_{0}$Rufo  _et
al._ (2019). i.e.,

$E(k,\mathbf{M})=\sqrt{|\delta g|^{2}+A_{1}k+A_{2}k^{2}+A_{4}k^{4}+...},$ (9)

where the dominating terms among $A_{1},A_{2},A_{4}$ are real numbers which
decides the nature of dispersion and corresponding $z$.
For the simplest longer-range model with two neighbors, the winding vectors
are given by

$\displaystyle\chi_{x}$ $\displaystyle=$ $\displaystyle
t+\cos(k)+\frac{\cos(2k)}{2^{\alpha}},$ $\displaystyle\chi_{y}$
$\displaystyle=$ $\displaystyle\sin(k)+\frac{\sin(2k)}{2^{\alpha}}-i\delta,$
(10)

with $t^{\prime}=1$. The criticality occurs at three values of $k$, i.e.,
$k=0,\pi$ and $\cos^{-1}(-2^{\alpha-1})$ which results in six exceptional
points respectively. Thus the model possesses six TQCLs

$\displaystyle k$ $\displaystyle=$ $\displaystyle 0\implies
t=-\frac{1}{2^{\alpha}}+\delta-1,\hskip
5.69046ptt=-\frac{1}{2^{\alpha}}-\delta-1,$ $\displaystyle k$ $\displaystyle=$
$\displaystyle\pi\implies t=-\frac{1}{2^{\alpha}}+\delta+1,\hskip
5.69046ptt=-\frac{1}{2^{\alpha}}-\delta+1,$ $\displaystyle k$ $\displaystyle=$
$\displaystyle\cos^{-1}(-2^{\alpha-1})\implies
t=\frac{1}{2^{\alpha}}+\delta,\hskip 5.69046ptt=\frac{1}{2^{\alpha}}-\delta.$

with red, blue, green, black, magenta and gray respectively (Fig 2 a).
Here we observe two different spectrum of real and absolute energy dispersion
with different critical exponents. Due to the non Hermiticity, the energy
dispersion need not to be symmetric around the gap closing values, which may
result as different exponents around a same pointArouca  _et al._ (2020). In
our study, both real and absolute spectrum yield $z=1/2$ for all TQCLs except
at multi-criticality (Fig 3 a). At multi-criticality the real spectrum yield
$z=2,1$ and 0.5 while (Fig 3 b)absolute spectrum yield $z=1,1$ and 0.5 for
MC1, MC2 and MC3 respectively (absolute spectrum is not showed here).
The longer-range non-Hermitian SSH model contains both fractional and integer
WNs, where the WN is given by Eq. 8, which physically means the encircling of
the winding vectors around the origin of the pseudo-spin parameter space. Due
to non-Hermiticity, there occurs two exceptional points instead of single
Dirac point which act as center of pseudo-spin space. If the winding vectors
encircle both the exceptional points equal times, then the WN is integer. If
the winding vectors encircle either one of the exceptional point, or unequal
number of time, then there occur fractional WN.
In Fig. 2 b, when winding vectors encircle both the EPs once (twice), we get
$W=1(W=2)$ topological phase which is equivalent to Hermitian version of
topological phases. If the winding vectors encircle either one of the
exceptional point, it gives $W=1/2$ and encircling first (second) EP twice
(once) while other only once (twice) give rise to $W=3/2$, which do not have
any Hermitian counterpart. Under some special cases, winding vectors encircle
one of the exceptional point even number of times and do not encircles the
other, which also results in the integer WN. With the increasing decay
parameter, the longer-range reduces to its short-range version at $\alpha=1$
through the intersection of TQCLs i.e., multi-critical points.

Localization critical exponent $(\nu)$: In a topological system, the zero
energy modes are localized at the edges and protected by the bulk Bloch
states.

Figure 2: (Color online) (a) Phase diagram for non-Hermitian longer-range SSH
chain with two interacting neighbors. Here the red, blue, green, black,
magenta and brown lines represent six TQCLs respectively. The points MC1, MC2
and MC3 correspond to first, second and third multi-critical points
respectively. The positions $a,b,c,d$ and $e$ represent the topological
regions $W=0,1/2,1,3/2$ and $W=2$ respectively. (b) Variation of winding
number against parameter $t$. (c) Energy dispersion (real part) for TQCLs. (d)
Energy dispersion (real part) for multi-critical points.(e) Energy dispersion
(imaginary part) for TQCLs. (f) Energy dispersion (imaginary part) for multi-
critical points.

As a simple example, we consider Eq.1 with two neighbor couplings ($r=2$). For
a semi-infinite system with $j=1,2,3...$, with wave function $\psi_{n}$, the
energy relation yieldsYin  _et al._ (2018)

$\displaystyle(t+\delta)\psi_{a,j}+t^{\prime}\psi_{a,j+1}+\frac{t^{\prime}}{2^{\alpha}}\psi_{a,j+2}=E_{n}\psi_{b,j}$
$\displaystyle(t-\delta)\psi_{b,j}+t^{\prime}\psi_{b,j-1}+\frac{t^{\prime}}{2^{\alpha}}\psi_{b,j-2}=E_{n}\psi_{a,j}.$
(11)

We construct the boundary condition with zero energy state ($E=0$) as

$\displaystyle(t+\delta)\psi_{a,j}+t^{\prime}\psi_{a,j+1}+\frac{t^{\prime}}{2^{\alpha}}\psi_{a,j+2}=0$
$\displaystyle(t-\delta)\psi_{b,j}+t^{\prime}\psi_{b,j-1}+\frac{t^{\prime}}{2^{\alpha}}\psi_{b,j-2}=0.$
(12)

Here the boundary condition $\psi_{b,j}=0$ is true for every $n$. The ratio of
nth localized zero eigenstate to that of first is given byContinentino (2017);
Continentino  _et al._ (2020)

$\delta\psi_{n}=\frac{\psi_{n}(E=0)}{\psi_{1}(E=0)}=\left|\frac{\delta
g}{A_{1,2,4}}\right|^{n-1}.$ (13)

These states are ensured by the condition $e^{ik_{0}}=-\left(\frac{\delta
g}{A_{1,2,4}}\right)$. Thus the $k_{0}$ is a complex number with
$k_{0}=i\left(\frac{\delta g}{A_{1,2,4}}\right)$, where $A_{1,2,4}$ are the
real numbers and dominating term among them decides the value of $k_{0}$. With
the system size $n$, the localization of the eigenstates is given by,

$\delta\psi_{n}=e^{\frac{-(n-1)}{\xi}},$ (14)

where $\xi$ is the localization length with $\xi=\frac{A_{n}}{|\delta
g|^{\nu}}\implies\xi=\xi_{0}|\delta g|^{-\nu}$, with $\xi_{0}$ as the natural
length of the system. Here $A_{n}$ is the dominating term in Eq. 9 and $\nu$
is the localization critical exponent. In non-Hermitian systems, the edge
localization is highly influenced by the skin-effect which decides the bulk-
edge correspondence. In our model, we get $\nu=1$ for both TQCLs and multi-
critical points (Fig.3 e,f).
To understand the number of localized edge modes at each end, we write Eq.12
as

$\displaystyle(t+\delta)+t^{\prime}\lambda+\frac{t^{\prime}}{2^{\alpha}}\lambda^{2}=0,$
$\displaystyle(t-\delta)+t^{\prime}\lambda^{\prime}+\frac{t^{\prime}}{2^{\alpha}}\lambda^{\prime
2}=0.$ (15)

with $\lambda=\frac{\psi_{a,j+1}}{\psi_{a,j}}$ and
$\lambda^{\prime}=\frac{\psi_{b,j+1}}{\psi_{b,j}}$, corresponding to the left
and right localized modes respectively. The quadratic form gives two roots

$\lambda_{\pm}=\frac{-t^{\prime}\pm\sqrt{(t^{\prime})^{2}-4(\frac{t^{\prime}}{2^{\alpha}})(t\pm\delta)}}{2^{1-\alpha}(t^{\prime})}.$
(16)

The topological limit forms a unit circle, where the number of solutions less
than unity represent the number of localized modes. The roots of $\lambda$ and
$\lambda^{\prime}$ represent the left and right localized modes respectively.
The combination of these two solution constitutes the phase diagram as
mentioned in Ref. Yin  _et al._ (2018).

Crossover critical exponent $(y)$: The scaling behavior near the TQCL is given
byArouca  _et al._ (2020),

$\displaystyle E(k\rightarrow k_{0},\mathbf{M}=\mathbf{M}_{c})$
$\displaystyle\propto$ $\displaystyle k^{z},$ $\displaystyle
E(k=k_{0},\mathbf{M}\rightarrow\mathbf{M}_{c})$ $\displaystyle\propto$
$\displaystyle\left|\partial g\right|^{y},$ (17)

where $y$ is called crossover or gap critical exponent with $y=z\nu$. This
exponent gives the relation between $z$ and $\nu$ and can be calculated in the
limit $E(k_{0},\mathbf{M}\rightarrow\mathbf{M}_{c})$. The energy gap at
criticality behaves like $\Delta\propto|\delta g|^{z\nu}$, where $\delta g$ is
the distance from the criticality. In our case, both real and absolute
spectrum give $y=0.5$ for TQCLs as well as for multi-critical points (Fig. 3
c,d).

Figure 3: (Color online) Critical exponents through non-linear curve fitting
method with the parameter space $t^{\prime}=1$ and $\delta=0.2$ respectively.
The red, blue, green and black colors in left panel correspond to first
($\alpha=1.5,t_{c}=-1.15$), second ($1.5,-1.55$), third ($1.5,1.84$) and
fourth ($1.5,0.44$) TQCLs respectively. The red, blue and green colors in
right panel correspond to first (1,0.7), second (1,0.3) and third (0.53,0.5)
multi-critical points respectively.(a-b) Dynamical critical exponent ($z$)
(c-d) Crossover critical exponent ($y$) (e-f) Localization critical exponent
($\nu$). (g-h) Susceptibility critical exponent ($\gamma$)

Canonical critical exponent $(\alpha^{*})$: The concept of thermodynamic grand
potential ($\Omega$) helps to understand the order of phase transition and
corresponding critical exponentArouca  _et al._ (2020); Kempkes  _et al._
(2016). The divergence or non-analyticity in the derivative of the grand
potential signals the order of the corresponding transition. For a system with
periodic boundary condition at zero temperature, the grand potential density
is given by,

$\omega=\frac{\Omega}{N}=-\int_{-\pi}^{\pi}E(k,\mathbf{M})dk,$ (18)

where $N$ is the system size. The above integral is convergent for all
positive $\alpha$, which makes the grand potential as independent of system
size in this limit. The order of the transition can be calculated through the
derivative of the grand potential. The transition can be nth order if the nth
derivative of grand potential shows a cusp or spike against the parameter and
the (n+1)th derivative shows a discontinuityMalard  _et al._ (2020b); Chen
_et al._ (2008).

Figure 4: (Color online) Derivatives of ground state energy for $k=\pi$
critical line at different values of $\alpha$. (a1-a3) Normal critical lines
showing cusp for second order derivatives. (b1-b3) Behavior of multi-critical
points, where MC1 and MC2 show cusp for second and first order derivatives
respectively.

In Fig. 4, we study $k=\pi$ critical line
(t=1-$\frac{1}{2^{\alpha}}\pm\delta$) under two different configurations. For
$\alpha=0.1$ the second derivative with respect to $t$ shows a cusp at
criticality followed by a discontinuous third order derivative. At multi-
criticality ($\alpha=1$), the first order derivative shows cusp at MC2 and a
continuous curve at MC2. The second order derivative shows discontinuity at
MC2 and a cusp at MC1 respectively. This observation signals that, the multi-
criticalities MC1 and MC2 exhibit different nature of transitions. To
understand this in detail we study the hyperscaling relation.
Here, the grand potential ($\omega$) contains two parts, i.e., regular and
singular parts. The singular part of the grand potential density scales
asArouca  _et al._ (2020)

$\omega_{singular}\propto|g|^{2-\alpha^{*}},$ (19)

where $\alpha^{*}$ is the canonical critical exponent and $2-\alpha^{*}$
signals the order of transition. These critical exponents are connected by the
relations called scaling laws which governs the phase transitions. One such
relation is the Josephson’s hyperscaling relation given byContinentino (2017)

$2-\alpha^{*}=\nu(z+d),$ (20)

where $z+d$ is the effective dimension. This law connects the dimensionality,
dynamical and correlation critical exponents with order of transition through
hyper-scaling relation. Thus, through hyper-scaling relation (Eq.20),

$\displaystyle 2-\alpha^{*}$ $\displaystyle=$ $\displaystyle
1(1/2+1)=3/2\hskip 14.22636pt\text{for TQCLs},$ $\displaystyle 2-\alpha^{*}$
$\displaystyle=$ $\displaystyle 1(2+1)=3\hskip 14.22636pt\text{for MC1},$
$\displaystyle 2-\alpha^{*}$ $\displaystyle=$ $\displaystyle 1(1+1)=2\hskip
14.22636pt\text{for MC2},$ $\displaystyle 2-\alpha^{*}$ $\displaystyle=$
$\displaystyle 1(1/2+1)=3/2\hskip 14.22636pt\text{for MC3}.$ (21)

which clearly signals the difference between the order of transitions among
normal criticality and multi-criticality. The fraction order of transitions
are the interesting observations in non-Hermitian systems, which are different
than their corresponding Hermitian counterparts. In Ref. Arouca  _et al._
(2020), the authors have tried to analyze this issue by scaling the singular
part with the factor $\xi\propto|\delta g|^{-\nu}$. As the grand potential has
a contribution from system size $L$, every distance is divided by the
correlation length. This has showed that, for the lower range of $\delta g$
the singular part of grand potential scales as $\delta g^{1/2}$, whereas for
the higher range it scales as $\delta g^{1}$. This is the reason behind the
difficulty to decide whether the transitions are first order or second order.
However, in our model also we find similar observation at normal criticality,
but multi-criticality remains interesting and merits more attention in this
regard.

###  II.3 Criticality and renormalization group approach

Due to the non-Hermitian effects, the TQPT occurs through the exceptional
points at $(0,\chi_{y}^{im})$ and $(0,-\chi_{y}^{im})$ instead of the Dirac
points. The WN is given by Eq. 8, with

$\displaystyle F_{1}(k,\mathbf{M})$ $\displaystyle=$
$\displaystyle\partial_{k}\phi_{1}=\frac{(\chi_{x}^{re}-\chi_{y}^{im})\partial_{k}\chi_{y}^{re}-\chi_{y}^{re}\partial_{k}(\chi_{x}^{re}-\chi_{y}^{im})}{(\chi_{x}^{re}-\chi_{y}^{im})^{2}+(\chi_{y}^{re})^{2}},$
$\displaystyle F_{2}(k,\mathbf{M})$ $\displaystyle=$
$\displaystyle\partial_{k}\phi_{2}=\frac{(\chi_{x}^{re}+\chi_{y}^{im})\partial_{k}\chi_{y}^{re}-\chi_{y}^{re}\partial_{k}(\chi_{x}^{re}+\chi_{y}^{im})}{(\chi_{x}^{re}+\chi_{y}^{im})^{2}+(\chi_{y}^{re})^{2}}.$

Here we deal with an interesting quantity called curvature function (CF). CF
is the quantity whose integral over the Brillouin zone gives the topological
invariant, like Berry connection, Berry curvature and Pfaffian wave
functionKumar  _et al._ (2021a); Chen and Sigrist (2019). In 1D models, we
consider Berry connection as our CF which is an gauge dependent quantity. The
CF exhibits non-analyticity at the criticality and yields ill-defined
topological invariant at these points. In case of non-Hermitian systems,
exceptional points are the points of non-analyticity. Based on the behavior in
the vicinity of exceptional points, CF can be classified as following.

  1. 1.

High symmetry points (HSP): When the CF exhibits a symmetric nature around the
gap closing point $k_{0}$, i.e.,$F(k_{0}+\delta k,\mathbf{M})=F(k_{0}-\delta
k,\mathbf{M})$ is called a HSP. Here the Lorentzian peak is is symmetric under
condition $k_{0}=-k_{0}$. The CF diverges as one approach $k_{0}$ from one
direction $\mathbf{M}\rightarrow\mathbf{M}_{c}^{+}$ and flips the direction as
one moves away $\mathbf{M}\rightarrow\mathbf{M}_{c}^{-}$ from $k_{0}$ with
same diverging curve. This can be represented as fixed peak with varying
height (Fig. 5 a1,b1).

  2. 2.

Non-High Symmetry points (non-HSP): In this case, as one approaches
criticality, there occurs diverging peak with different $k_{0}$ values. Here
one can observe the flipping of peaks but the Lorentzian peak keep moving
along the BZ (Fig. 5 a2,b2).

  3. 3.

Fixed point (FP): Generally this kind of behavior is observed at the multi-
criticality where HSP and non-HSP meet togetherKartik  _et al._ (2021); Kumar
_et al._ (2021a, b). Here the CF shows a diverging peak as
$\mathbf{M}\rightarrow\mathbf{M}_{c}^{+}$ from one side, and at $k_{0}$ we
observe a peak with diminished height. As we go away
$\mathbf{M}\rightarrow\mathbf{M}_{c}^{-}$ we observe a peak with constant
height with increasing width. The multi-critical points MC1 and MC2 show the
FP nature (Fig. 5 a3,b3).

Figure 5: (Color online) Behavior of curvature function around HSP, non-HSP
and fixed points respectively. The left panels represent criticality
$t=1-\frac{1}{2^{\alpha}}-\delta$ ($F1(k,\mathbf{M})$ parameter space) and the
right represent $t=1-\frac{1}{2^{\alpha}}+\delta$ ($F2(k,\mathbf{M})$
parameter space) respectively.

Susceptibility critical exponent $(\gamma)$: Here the CF exhibits non-analytic
property at the exceptional points, which gives the idea of critical scaling
near these pointsChen and Sigrist (2019). In present case, we consider Berry
connection as our CF, which is a gauge dependent quantity. Without loss of
generality, we write CF in Ornstein-Zernike form as,

$F(k_{0}+\delta k,\mathbf{M})=\frac{F(k_{0},\mathbf{M})}{1\pm\xi^{2}\delta
k^{2}+\xi^{4}\delta k^{4}},$ (23)

where $k_{0}$ is a gap closing point, $\delta k$ is a small deviation from gap
closing and $\xi$ is the characteristic scale respectively. At $k=k_{0}$ and
$\mathbf{M}\rightarrow\mathbf{M}$, the CF diverges signaling a TQPT where the
characteristic length vanishes.

$lim_{\mathbf{M}\rightarrow\mathbf{M}_{c}^{+}}F(k_{0},\mathbf{M})=-lim_{\mathbf{M}\rightarrow\mathbf{M}_{c}^{-}}F(k_{0},\mathbf{M})=\pm\infty.$
(24)

and CF exhibiting an interesting nature

$F(k_{0},\mathbf{M}^{\prime})=F(k_{0}+\delta k,\mathbf{M}),$ (25)

which gives the idea of renormalization group study near criticality.

Figure 6: (Color online) CRG flow diagram for non-Hermitian longer-range SSH
chain with two interacting neighbors. (a) Flow for $F1(k,\mathbf{M})$ at
$k=0$. (a) Flow for $F1(k,\mathbf{M})$ at $k=\pi$. (a) Flow for
$F2(k,\mathbf{M})$ at $k=0$. (a) Flow for $F2(k,\mathbf{M})$ at $k=\pi$. Here
the colored lines represent critical and fixed configurations. For detailed
derivation, refer AppendixB.

The divergence of CF and characteristic length gives the critical exponent
$F(k_{0},\mathbf{M})\propto|\mathbf{M}-\mathbf{M}_{c}|^{-\gamma}$ and
$\xi(k_{0},\mathbf{M})\propto|\mathbf{M}-\mathbf{M}_{c}|^{-\nu}$, where
$\gamma$ and $\nu$ are the susceptibility and characteristic critical
exponents respectively, where the susceptibility exponent explains the quality
of the transition to be topological. Here we get $\gamma=1$ for both TQCLs and
multi-critical points.
Expanding Eq 25 up to a leading order, we get CRG equation asChen and Sigrist
(2019),

$\frac{d\mathbf{M}}{dl}=\frac{(\frac{1}{2})\frac{d^{2}F(k,\mathbf{M})}{dk^{2}}|_{k=k_{0}}}{\frac{dF(k_{0},\mathbf{M})}{d\mathbf{M}}},$
(26)

where $\left|\frac{d\mathbf{M}}{dl}\right|\rightarrow\infty$ defines
criticality and $\left|\frac{d\mathbf{M}}{dl}\right|\rightarrow 0$ defines
fixed point configuration respectively. In our model, at $k=\pi$ (both
$F1(k,\mathbf{M})$ and $F2(k,\mathbf{M})$), there occurs a superposition of
all critical and fixed lines at $\alpha=1$. Here we find a whirlpool kind of
behavior where the flow lines encircle a single point (Fig. 6). At this point,
$\frac{d^{2}F}{dk^{2}},\frac{dF}{d\alpha},\frac{dF}{d\mu}\rightarrow 0$ and
$\frac{d\alpha}{dl},\frac{d\mu}{dl}\rightarrow\frac{0}{0}$. Interestingly at
this point, we observe a different class of UCCE through multi-critical
points. On the other hand, we do not observe any such behavior for $k=0$
criticality. (For detailed derivation, refer AppendixB)

Model | $\hskip 5.69046ptk\hskip 5.69046pt$ | $\hskip 5.69046ptz\hskip 5.69046pt$ | $\hskip 5.69046pt\nu\hskip 5.69046pt$ | $\hskip 5.69046pt\gamma_{1,2}\hskip 5.69046pt$ | $\hskip 5.69046pty\hskip 5.69046pt$ | $2-\alpha^{*}$
---|---|---|---|---|---|---
Short-range model | 0 | 0.5 | 1 | 1 | 0.5 | 1.5
| $\pi$ | 0.5 | 1 | 1 | 0.5 | 1.5
Longer-range model | 0 | 0.5 | 1 | 1 | 0.5 | 1.5
($r=$ Even and $\alpha=1$) | $\pi$ | (2,1) | 1 | 1 | 0.5 | (3,2)
Longer-range model | 0 | 0.5 | 1 | 1 | 0.5 | 1.5
($r=$ Odd, $\forall\alpha$) | $\pi$ | 0.5 | 1 | 1 | 0.5 | 1.5
($r=$ Even, $\alpha\neq 1$) |  |  |  |  |  |
Table 1: A comparison of universality class of critical exponents between the
original Kitaev chain and the reduced long-range Kitaev chain.

###  II.4 Significance of multi-criticality

Multi-criticalities are the intersection of (at least) two TQCLs and
distinguish (at least) three different topological regimes. So far, multi-
critical points acted as unique points, where we witnessed the breaking of
Lorentz invariance, topological transition without bulk gap and as a point
with zero entanglement entropyKumar  _et al._ (2021b). There are category of
multi-critical points based on the conformal field theory where we get
different central charge. However, in case of non-Hermitian systems, multi-
critical points may behave differently based on the nature of non-Hermitian
factor. For example, recently there has been the observation on the vanishing
of multi-criticality when the non-Hermiticity is introduced into an extended
Kitaev chain through chemical potentialRahul (2022). In our case, we have
introduced non-Hermiticity through local imbalance in the hopping, which
splits the Hermitian multi-critical points into two. We can observe that
multi-critical points as the superposition of HSP and non-HSP with a fixed
point behavior and belong to separate UCCE. This part is in similarity with
Hermitian counterpart.

Figure 7: (Color online) Localization of edge modes at criticality calculated
through Eq. 15. The unit circle represent the topological limit. The solution
less than, equal to and greater than unity represent the edge mode,
criticality and no edge mode condition respectively. (a1-a3) Edge mode
behavior at left end. (b1-b3) Edge mode behavior at right end.

The expansion of Eq. II around exceptional points $k_{0}$ gives the better
understanding towards the underlying physics of multi-criticality. At normal
criticalities, the dispersion behaves as $E(k\rightarrow
k_{0},\mathbf{M}_{c})\propto k^{1/2}$. At multi-criticalities, the dispersion
takes first ($E(k\rightarrow k_{0},\mathbf{M}_{c})\propto k^{1}$ for MC1) and
second order ($E(k\rightarrow k_{0},\mathbf{M}_{c})\propto k^{2}$ for MC2)
corrections as shown in Eq.9 respectively. On the other hand, the multi-
critical point MC3 is no different than the normal criticalities, as it shows
similar behaviors of normal criticality. However, we observe similar cases in
Hermitian counterpart, where the the instance is recognized as breaking of
Lorentz invarianceKumar  _et al._ (2021a).
Exponent $z$ gives information about the massless condition of the system. In
condensed matter, the energy dispersion at Fermi surface resembles the
relativeistic energy form $E^{2}=c^{2}p^{2}+m^{2}c^{4}$. For gap closing
points, the equation gives masless condition with $E^{2}=c^{2}p^{2}$. Under
some cases, the equation takes correction form as,
$E^{2}=c^{2}p^{2}+m^{2}c^{4}+Ap^{4}$ with $A$ as constantSadhukhan  _et al._
(2020). Similar effects we observe at exceptional points. For normal
criticalities, we get $E^{2}=\sqrt{AK}$ and for multi-criticalities, we obtain
$E^{2}=\sqrt{Ak+Bk^{2}}$ and $E^{2}=\sqrt{Ak+Bk^{4}}$ respectively. This
creates the difference in the nature of energy dispersion and thereby
dynamical critical exponent. Transition among gapless phases: Localization at
criticality is an important phenomenon, which is important both from the
perspective of theory and experimental aspects Jones and Verresen (2019);
Thorngren  _et al._ (2020); Kestner  _et al._ (2011); Cheng and Tu (2011);
Fidkowski  _et al._ (2012); Sau  _et al._ (2011); Kraus  _et al._ (2013);
Scaffidi  _et al._ (2017); Jiang  _et al._ (2018), especially in longer-range
modelsRahul  _et al._ (2021); Kumar  _et al._ (2021a); Verresen  _et al._
(2019, 2018); Verresen (2020); Kumar  _et al._ (2021b). In our model, the
multi-critical points act as an unique point, which witnesses the topological
transition among critical phases. Here MC1 and MC2 are present on the
$k=\pi\rightarrow t=1\pm\delta+\frac{t^{\prime}}{2^{\alpha}}$ respectively.
Through Eq.15, we find the solutions of quadratic equation as shown in Fig. 7.
Throughout the critical line, one of the solution remains unity, which
confirms the gapless condition. For $\alpha<1$, one of the solution is less
than unity ($\lambda=0.4$), indicating a localization of edge mode at
criticality. At $\alpha=1$, the both the solutions are unity, indicating a
transition among gapless phases through multi-criticality. For $\alpha>1$, one
of the solution is greater than unity ($\lambda=1.8$), indicating the absence
of localized modes. Hence, both MC1 and MC2 act as transition points among
gapless phases. A similar phenomenon has been observed in the Hermitian
counterpart of the similar modelRahul  _et al._ (2021); Kumar  _et al._
(2021a).

###  II.5 Staircase of topological transitions:

Due to the feature of long-range coupling in Eq.1, it is possible to achieve
higher winding number with the increase of interacting neighbors, i.e., finite
$r$. For the limit $r<L/2$, we can obtain higher WNs where the uppermost WN is
$W=r$. This is an effective method to obtain more localized modes through
static method while similar can be achieved through periodic driving which is
dynamical in natureKartik  _et al._ (2021); Tong  _et al._ (2013). However the
higher WNs are comparatively less stable and reduce to consecutive lower WNs
with the increasing of $\alpha$. Here we find an interesting patter of
transitions , i.e., a staircase of TQPTs.

  * •

Even-Even transition: When the number of neighbor is an even number (for
$r>2$), the uppermost WN will be an even number and the transition occurs
among only even WNs.

  * •

Odd-Odd transition: When the number of neighbor is an odd number (for $r>2$),
the uppermost WN will be an odd number and the transition occurs among only
odd WNs.

  * •

Even-Odd transition: When the number of neighbor is an even number (for
$r>2$), the uppermost WN is an even number and with the increasing $\alpha$,
the longer-range model reduces to short-range version. At the interface of
short-range and longer-range, there occurs a transition between
$W:2\rightarrow 1$. For very small $\delta$, this can be clearly observed and
with the increase in $\alpha$, the transition occurs like
$W:2\rightarrow\frac{3}{2}\rightarrow 1$. This transition occurs through the
multi-critical points.

Figure 8: (Color online) Staircase of TQPTs with different number of
interacting neighbors. For even (odd) ’$r$’, transitions occurs among only
even (odd) winding numbers. Near short-range limit, we do not observe this
behavior. The star mark indicates the numerical limitation of converging for
higher WNs.

Here we choose parameter space ($t=0.5,\delta=0.1$), where the increase in
$\delta$ suppresses the uppermost WN and creates fractional WNs. For the even
$r$, the fractional WNs are observable with small $\delta$ and large $\alpha$,
whereas the similar effect is not visible for odd $r$. The interface of short-
range and longer-range occurs through multi-critical points as shown in Fig.
8(d). These multi-critical points can also be recognized by the CRG method
also (Not shown here).

##  III Long-range

Figure 9: (Color online) (a) Phase diagram of long-range model in a 3D
representaion. (b) 2D representation of phase diagram. (c1-c6) Behavior of
winding vectors in long-range non-Hermitian SSH chain. The blue and green
winding vectors represent the F1(k,$\mathbf{M}$) and F2(k,$\mathbf{M}$)
parameter spaces respectively. The combined effect determines the topological
index of the system.

With the introduction of infinite number of neighbors ($r\rightarrow\infty$),
Eq.1 represents long-range non-Hermitian SSH chain, where the pseudo-spin
parameters exhibit the polylogarithmic natureVodola  _et al._ (2014), i.e.,

$\displaystyle\chi_{x}(k)$ $\displaystyle=$ $\displaystyle
t+t^{\prime}\left(\frac{Li_{\alpha}[e^{ik}]+Li_{\alpha}[e^{-ik}]}{2}\right),$
$\displaystyle\chi_{y}(k)$ $\displaystyle=$ $\displaystyle
t^{\prime}\left(\frac{Li_{\alpha}[e^{ik}]-Li_{\alpha}[e^{-ik}]}{2i}\right)-i\delta.$
(27)

Expansions of polylogarithmic function around a gap closing point $k_{0}$ is
given byOlver  _et al._ (2010),

$Li_{\alpha}[e^{ik}]=\Gamma[1-\alpha](-ik)^{\alpha-1}+\sum_{n=0}^{\infty}\frac{\zeta[\alpha-n]}{n!}(ik)^{n}.$
(28)

For the detailed study, we can expand the polylogarithmic function around
$k=0$ as

$\displaystyle\chi_{x}$ $\displaystyle=$ $\displaystyle
t+t^{\prime}(\Gamma[1-\alpha](k)^{\alpha-1}\sin\left(\frac{\pi\alpha}{2}\right)$
$\displaystyle+$
$\displaystyle\sum_{n=0}^{\infty}\frac{\zeta[\alpha-n]}{n!}(k)^{n}\cos\left(\frac{\pi
n}{2}\right)),$ $\displaystyle\chi_{y}$ $\displaystyle=$ $\displaystyle
t^{\prime}(\Gamma[1-\alpha](k)^{\alpha-1}\cos\left(\frac{\pi\alpha}{2}\right)$
(29) $\displaystyle+$
$\displaystyle\sum_{n=0}^{\infty}\frac{\zeta[\alpha-n]}{n!}(k)^{n}\sin\left(\frac{\pi
n}{2}\right))-i\delta.$

We analyze the properties of momentum space to characterize the non-Hermitian
long-range SSH chain.

###  III.1 Energy dispersion, Fermi velocity, GSE density, critical exponents:

The quasi-particle energy dispersion is given by Eq. II, where the band gap
vanishes for $k=0$ and $\pi$ respectively. Due to the polylogarithmic nature,
the band gap closes for all values of $\alpha$ at $k=\pi$ and only $\alpha>1$
at $k=0$. The parameter space possesses two exceptional points and four TQCLs
corresponding to two values of $k_{0}$. i.e.,

  1. 1.

$t=\delta-\text{Li}_{\alpha}(1)$ for $k=0$ for $\alpha>1$,

  2. 2.

$t=-\delta-\text{Li}_{\alpha}(1)$ for $k=0$ for $\alpha>1$,

  3. 3.

$t=\delta-\text{Li}_{\alpha}(-1)$ for $k=\pi$ and $\forall\alpha$,

  4. 4.

$t=-\delta-\text{Li}_{\alpha}(-1)$ for $k=\pi$ and $\forall\alpha$.

Here the pseudo-spin parameters go as $\sin(k)$ instead of $\sin(lk)$ which
result in the single time encircling of the origin. Thus the phase diagram is
given by Fig. 9(a,b).

Behavior of winding vectors: Due to the long-range effect, the winding vectors
exhibit a discontinuity at $k=0$ for $\alpha<1$ which give rise to fractional
winding number. Thus the region
$\pm\delta-\text{Li}_{\alpha}(1)<\mu<\pm\delta-\text{Li}_{\alpha}(-1)$ for
$\alpha>1$ gives topological (Hermitian and non-Hermitian) phase where the
winding vectors encircle the origin of the parameter space without any
discontinuity. Due to the polylogarithmic behavior, we find a less population
of winding vectors in the region $1<\alpha<2$, technically which do not
affects the winding number. In the region
$\delta-\text{Li}_{\alpha}(1)<\mu<-\delta-\text{Li}_{\alpha}(-1)$ for
$\alpha>1$, we find that both set of winding vectors encircle the origin of
each parameter space which yield $W=1$. The region
$\mu<\pm\delta-\text{Li}_{\alpha}(-1)$ for $\alpha<1$ gives topologically ill-
defined (fractional) phase where winding vectors corresponding to both the
parameter space shows a discontinuity at $k=0$. If both set of vectors exhibit
half encirclement around their corresponding origin, then it yield $W=1/2$. If
any one exhibits this behavior, it yields $W=\pm 1/4$ where the sign indicates
the direction of encircling. On the other hand, due to the imbalance in
intracell hopping, there arise fractional WN for
$-\delta-\text{Li}_{\alpha}(1)<\mu<\delta-\text{Li}_{\alpha}(1)$ and
$-\delta-\text{Li}_{\alpha}(-1)<\mu<\delta-\text{Li}_{\alpha}(-1)$ for
$\alpha>1$. These are the non-Hermitian topological phases and do not posses
any Hermitian counterpart. Here any one set of the winding vector encircle the
origin by yielding $W=1/2$. Extended winding vectors are given by

$\displaystyle\chi_{x}(ext)$ $\displaystyle=$
$\displaystyle\pm\delta+\frac{\text{Li}_{\alpha}(\exp(-ik))+\text{Li}_{\alpha}(\exp(ik))}{2}+t,$
$\displaystyle\chi_{y}(ext)$ $\displaystyle=$
$\displaystyle\frac{\text{Li}_{\alpha}(\exp(ik))-\text{Li}_{\alpha}(\exp(-ik))}{2i}.$
(30)

The behavior of extended winding vectors are presented in Fig. 9(c1-c6).

Momentum space behavior: Due to the polylogarithmic nature, in the limit
$k\rightarrow 0$, the $\chi_{y}$ term shows a divergence for $\alpha<1$ and
converges towards zero for $\alpha>1$ and a discontinuity at $\alpha=1$
respectively(Fig. 10c). This reflects in the behavior of CF and energy
dispersion. For $\alpha<1$ the CF shows a divergence (Fig. 10d) at $k=0$ which
leads to the emergence of the fractional WN. The corresponding energy spectrum
shows a gapped region with diverging bands(Fig. 10a).The transition from
$\alpha<1$ to $\alpha>1$ occurs at $\alpha=1$, where we observe topological
transition without gap closing $(W:-1/4\rightarrow 0,1/4\rightarrow
1/2,W:-1/2\rightarrow 1)$. This is an interesting behavior which can be
observed in long-range models.

Figure 10: (Color online) Properties of non-Hermitian long-range SSH chain.
(a) Real part of energy spectrum where the spectrum is divergent for
$\alpha<1$ and representation of topological transition at $\alpha=1$ without
gap closing. (b) A sharp variation of imaginary part of spectrum at $\alpha=1$
interface. (c) Divergence of $\chi_{y}$ term for $\alpha<1$ region due to the
polylogarithmic behavior. (d) Behavior of curvature function for $\alpha<1$
region. The divergences at $k=\pi$ and 0 is due to the criticality and
removable singularity respectively. (e) Decay of Wannier state correlation
function around $k=\pi$. The nature is similar $\forall\alpha$. (f) Decay of
correlation function around $k=0$, only for $\alpha>2$.

Curvature function and Wannier state correlation function: Substituting the
values of Eq. 27 into Eq. LABEL:cf we get the CF in Ornstein-Zernike form as

$\displaystyle
F(k,\mathbf{M})\propto\frac{\frac{Ak^{\alpha-2}}{Bk^{\alpha-1}}-\frac{Ck^{2\alpha-3}}{Dk^{2\alpha-2}}}{1+(\frac{Ek^{\alpha-1}}{Fk^{\alpha-1}})^{2}}.$
(31)

There are three possible cases,

  * •

When $\alpha<1$, the term $k^{\alpha-2}$ dominates and
$F(k,\mathbf{M})\rightarrow\infty$ as $k\rightarrow 0$, irrespective of
$\mathbf{M}\rightarrow\mathbf{M}_{c}$.

  * •

When $1<\alpha<2$, the term $k^{\alpha-2}$ dominates and
$F(k,\mathbf{M})\rightarrow\infty$ as $k\rightarrow 0$, irrespective of
$\mathbf{M}\rightarrow\mathbf{M}_{c}$.

  * •

When $\alpha>2$, again the term $k^{\alpha-2}$ dominates and
$F(k,\mathbf{M})\rightarrow\infty$ as $k\rightarrow 0$ with
$\mathbf{M}\rightarrow\mathbf{M}_{c}$.

Thus we can define the CF in the Ornstein-Zernike form and corresponding
critical exponents only for $\alpha>2$ around $k=0$ regionSadhukhan and
Dziarmaga (2021). For $k=\pi$, the CF can expressed Ornstein-Zernike form for
all values of $\alpha$.
Wannier state correlation function is the quantity, which discusses the phase
transition among different phasesKumar  _et al._ (2021a); Chen and Sigrist
(2019); Kumar  _et al._ (2021b). Conceptually, it is the overlap between the
Wannier state centering at the origin and that of distance $R$. i.e.,

$\displaystyle\langle r|R\rangle=W_{n}(r-R)$ (32)

with

$\displaystyle|R_{n}\rangle=\frac{1}{N}\sum_{k}e^{ik.(r-R)}|u_{k}\rangle$

(the form of $|u_{k}\rangle$ is expressed in Appendix A.)
Mathematically, Wannier states can be expressed as Fourier transform of the
CF, i.e.,

$\lambda_{R}=\int\frac{dk}{2\pi}e^{ik.R}F(k,\mathbf{M}).$ (33)

In our case, we observe the decay of Wannier state correlation function around
$k=\pi$ for all values of $\alpha$, with a critical exponent $\nu=1$ (Fig. 10
e). For $k=0$, there occurs no correlation decay in the region $1<\alpha<2$,
which results in the undefined critical exponent. For the region $\alpha>2$,
we observe the decay of correlation function, which is very sharp near the
criticality with an exponent $\nu=1$ (Fig. 10 f).

Energy dispersion and Fermi velocity: By substituting Eq. 27 into Eq. II, we
get energy dispersion. The first order derivative of the energy dispersion
gives the Fermi velocity as

$\displaystyle E(k,\mathbf{M})$ $\displaystyle=$
$\displaystyle\sqrt{Ak^{2\alpha-2}+Bk^{\alpha-1}+C},$
$\displaystyle\frac{dE_{k}}{dk}$ $\displaystyle=$
$\displaystyle\frac{Ak^{2\alpha-3}+Bk^{\alpha-1}+Ck}{\sqrt{A^{2}k^{2\alpha-2}+Bk^{\alpha}+C^{2}k^{2}}}.$
(34)

where $A,B$ and $C$ are constants. For the values $\alpha<1$, the Fermi
velocity is dependent on $\alpha$ as shown in Table 2

Condition | Relation | $k=0$ | $k=\pi$
---|---|---|---
$\alpha<1$ | $\frac{Ak^{2\alpha-3}}{Bk^{\alpha-1}}$ | $v_{F}\rightarrow\infty$ | $v_{F}(\mathbf{M}_{c})\rightarrow 0$
$1<\alpha<1.5$ | $Ak^{2\alpha-3}$ | $v_{F}\rightarrow\infty$ | $v_{F}(\mathbf{M}_{c})\rightarrow 0$
$\alpha>1.5$ | $Ck$ | $v_{F}(\mathbf{M}_{c})\rightarrow 0$ | $v_{F}(\mathbf{M}_{c})\rightarrow 0$
Table 2: Behavior of Fermi velocity with respect to decay parameter $\alpha$.
At $k=0$, the Fermi velocity vanishes as the system drives towards criticality
only for $\alpha>1.5$

Thus, as $k\rightarrow 0$ we obtain a diverging energy spectrum and Fermi
velocity for $\alpha<1$. For $1<\alpha<1.5$ we obtain gapless energy spectrum
and a diverging Fermi-velocity as $k\rightarrow 0$. For $\alpha>1.5$, we
obtain vanishing Fermi velocity as the system drives towards criticality.

Figure 11: (Color online) Critical exponents through non-linear curve fitting
method. The left and right panels represent the $k=0$ and $k=\pi$
criticalities respectively. The parameter space $\delta=0.2,t^{\prime}=1$ is
considered. The red, blue, green and black colors represent the first
($\alpha=2.5,t_{c}=-1.14$), second (2.5,-1.54), third (2.5,1.06) and fourth
(2.5,0.66) critical lines respectively. (a-b) Dynamical critical exponent. The
inset represent the region $1<\alpha<2$ with parameter space
($\alpha=1.1,t_{c}=-10.38$) and (1.1,-10.78), where $z<0.5$. (c-d) Crossover
critical exponent. (e-f) Localization critical exponent. (g-h) Susceptibility
critical exponent.

Here we analyze the UCCE through nonlinear curve fitting method (Fig. 11).
Around $k=0$, we obtain $z<0.5$ for the region $1<\alpha<2$, whereas $z=0.5$
for $\alpha>2$. In the region $1<\alpha<2$, we obtain undefined $\nu,\gamma$
and $y$ critical exponents, and $\nu=1,\gamma=1,y=0.5$ for $\alpha>2$. On the
other hand, we get $z=0.5,y=0.5,\nu=1$ and $\gamma=1$ for all values of
$\alpha$ around $k=\pi$. Table 3 summaries the critical exponents for
different regions of non-Hermitian long-range SSH chain.

###  III.2 Higher order bulk transitions:

To understand the order of phase transition, we check the derivatives of grand
potential density for different orders. The phase transitions act as
singularities in the density parameter space and the derivatives of density
signals the order of transition with non-analytic curves. The polylogarithmic
nature creates a singularity in the vicinity of $k\rightarrow 0$, while the
entire BZ contributes to the geometric phase. i.e., if $\omega_{singular}$
represents the grand potential density of the bulk, then it can be expressed
asCats  _et al._ (2018)

$\omega_{singular}=\omega_{x}+\omega_{y},$ (35)

where $\omega_{x}$ integral around $k=0$ and $\omega_{y}$ is the integral over
the rest of the BZ respectively.

Figure 12: (Color online) Different order derivatives of ground state energy
density. The singularities in the ground state represents distinction between
the topological phases. The order of the transition represents the order of
transition.

Through Eq. 28, we write an expanded term around $k=0$ as

$\omega_{x}(k=0,\mathbf{M}(\alpha>2))=\int_{0-\epsilon}^{0+\epsilon}\sqrt{\mathbf{M}^{2}+k^{2}}dk\propto\mathbf{M}^{2}\ln|\mathbf{M}|.$
(36)

In this case we can observe a divergence in the second order derivative of the
above equation, signaling a second order bulk transition. For the region
$1<\alpha<2$,

$\omega_{x}(k=0,\mathbf{M}(1<\alpha<2))\approx\int_{0-\epsilon}^{0+\epsilon}\sqrt{\mathbf{M}^{2}+k^{2(\alpha-1)}}dk.$
(37)

Through the Eq. 28, we observe the above term is proportional to
$\Gamma(-\frac{\alpha}{2(\alpha-1)})\Gamma(\frac{2\alpha-1}{2(\alpha-1)})\mathbf{M}^{\frac{\alpha}{\alpha-1}}$,
where $\Gamma$ function diverges for $\alpha=\frac{n}{n-1}$ where $n\in N$.
Thus as one approaches $\alpha=1$ from the side of $\alpha=2$, the integral
shows divergence for higher orders signaling a higher order bulk transitions.
Thus as we take the higher order derivatives, the criticalities corresponding
to $k=\pi$ show the peaks while $k=0$ fails to produce the same. This signals
a possible higher order phase transitions as one approaches $\alpha=1$. The
region with integral WN (Fig. 9) has similar Hermitian counterpart as
explained in Ref. , while the fractional WN do not have Hermitian counterpart.
Interestingly, the fractional WN corresponding to $k=0$ shows a very narrow
phase boundary, and as $\alpha\rightarrow 1$, the non-Hermitian phase
gradually vanishes. Thus at $\alpha=1$, the Hermitian phase exhibits an
infinite order of bulk transition, while non-Hermitian phases shows almost
none. This is also in good agreement with the hyper-scaling relation (Eq. 20)
also.

$\displaystyle 2-\alpha^{*}$ $\displaystyle=$ $\displaystyle
1(1/2+1)=3/2\hskip 14.22636pt\text{for $k=\pi$},$ $\displaystyle 2-\alpha^{*}$
$\displaystyle=$ $\displaystyle 1(1/2+1)=3/2\hskip 14.22636pt\text{for
$k=0,\alpha>2$},$ $\displaystyle 2-\alpha^{*}$ $\displaystyle=$
$\displaystyle\text{Higher as $\alpha\rightarrow 1$}\hskip 14.22636pt\text{for
$k=0,1<\alpha<2$}.$

$\hskip 5.69046pt\alpha\hskip 5.69046pt$ | $\hskip 5.69046ptk\hskip 5.69046pt$ | $\hskip 5.69046ptz\hskip 5.69046pt$ | $\hskip 5.69046pt\nu\hskip 5.69046pt$ | $\hskip 5.69046pt\gamma_{1,2}\hskip 5.69046pt$ | $\hskip 5.69046pty\hskip 5.69046pt$ | $2-\alpha^{*}$
---|---|---|---|---|---|---
$\alpha<1$ | 0 | - | - | - | - | -
$1<\alpha<2$ |  | $z<0.5$ | - | - | 0.5 | higher
$\alpha>2$ |  | 0.5 | 1 | 1 | 0.5 | 3/2
$\forall$$\alpha$ | $\pi$ | 0.5 | 1 | 1 | 0.5 | 3/2
Table 3: A comparison of universality class of critical exponents of long-
range Kitaev chain for different parameter spaces.

##  IV Outlook and experimental aspects

Non-Hermitian topological systems are sensitive to boundary conditions, where
the phase diagram may vary based on the choice of the boundary conditions.
Here, we have worked on the periodic boundary condition, while the appearance
of multi-criticality may be different in other conditions. We have observed
the signature of localized modes at the gapless phases, which is an
interesting phenomenon and merits further analysis in that regard.
Hermitian long-range models are the platform to realize the massive edge
modes, which also can act as an effective topological qubit. Such possibility
to explore massive edge modes in non-Hermitian counterparts can be interesting
both from the perspective of theory and experiment. The field theoretical
analysis of the Hermitian long-rang models reveals the breakdown of Lorentz
invariance for the region $1<\alpha<2$. Our current work also signals such
behavior through dynamical critical exponent, however, a detailed field
theoretical analysis may answer this.
Long-range models are the field of interest from the experimental aspects
also. The nature of shrinking characteristic length with the longer-range
interaction helps to suppress the finite-size effects even by using a
relatively small number of ionsGong  _et al._ (2016). Long-range models have
been realized in trapped ions Deng  _et al._ (2005); Britton  _et al._ (2012);
Hauke  _et al._ (2010); Roy  _et al._ (2019), atom coupled to multi-mode
cavities Douglas  _et al._ (2015), magnetic impurities Zhang  _et al._ (2019);
Ménard  _et al._ (2015) and quantum computation Amin  _et al._ (2019). The
effects of non-Hermiticity may give a different dimension towards the
understanding of long-range systems.

##  V Discussion and Conclusion

Understanding the behavior of topological system with different range of
coupling has attracted the attention of scientific community. The UCCE is an
efficient tool to characterize the criticalities based on their behavior. The
uniqueness of the multi-criticalities can be observed through UCCE. Here CRG
is an useful tool to obtain the fixed and critical line configurations. But
the CRG limits its coverage only to certain parameter spaces like HSP and non-
HSPs corresponding to lower WNs. The symmetry plays a major role in defining
the topological invariant. The current methodology may vary for the systems
which lack chiral symmetry.
To conclude, we have provided a detailed analysis of criticality in longer and
long-range non-Hermitian SSH chain. We have adopted the approach of separating
the real and imaginary parts of the complex angle and defined the topological
invariant. With the construction of staircase of transitions, we have observed
the transition among even-even and odd-odd winding numbers, which actually has
the Hermitian counterparts. To analyze the critical exponents, we have
considered the $r=2$ case and calculated the universality class of critical
exponents along with order of transition. The multi-critical point have shown
some interesting feature, where the universality class is different than the
rest of the parameter space. We adopt the CRG method to non-Hermitian systems
and observe the fixed point configuration which actually depicts the
difference in the universality class.
The long-range models are well known in Hermitian systems, due to the
emergence of exotic particles like massive edge modes and physics of short-
range,long-range inter-phase. Here we analyze model from momentum space
characterization, and observe the interplay of polylogarithmic and non-
Hermitian effect in winding vectors. Here we preferred the universality class
of critical exponents to find the short-range limit rather than the
topological invariant. CRG analysis gives the understanding of fixed/critical
point configurations in long-range models. We observe the in the
$\alpha\rightarrow 1$ limit, it is possible to observe the higher order
topological transitions.
The studies of long-range effects in non-hermitian models are less in
literature. We believe, our work can be useful in understanding the interplay
of long-range and non-Hermitian effects in topological state of matter.

Acknowledgments: SS would like to acknowledge DST (CRG/2021/000996) for the
support and RRI library for the books and journals. YRK is grateful to AMEF
for providing a PhD fellowship. The authors would like to acknowledge B S
Ramachandra, C Sivaram, Rahul S and Ranjith Kumar R for useful discussions.

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##  Appendix A Derivation of winding number for non-Hermitian system

The non-Hermiticity of the Hamiltonian reflects in the eigenvectors of the
Hamiltonian asYin  _et al._ (2018)

$W=\left(\frac{1}{2\pi}\right)\int_{-\pi}^{\pi}\frac{\langle
u_{k}^{L}|i\partial_{k}|u_{k}^{R}\rangle}{\langle
u_{k}^{L}|u_{k}^{R}\rangle}dk,$ (38)

where

$\langle
u_{k}^{L}|=\frac{1}{\sqrt{2}}\left(\begin{matrix}\frac{\chi_{x}+i\chi_{y}}{\sqrt{\chi_{x}^{2}+\chi_{y}^{2}}}\\\
-1\end{matrix}\right),\hskip
14.22636pt|u_{k}^{R}\rangle=\frac{1}{\sqrt{2}}\left(\begin{matrix}\frac{\chi_{x}-i\chi_{y}}{\sqrt{\chi_{x}^{2}+\chi_{y}^{2}}}\\\
-1\end{matrix}\right).$ (39)

Due to the non-Hermiticity, the winding vectors become complex (at least one
of them) and the criticality condition is given by $h_{x}^{2}+h_{y}^{2}=0$. In
Hermitian case, criticality occurs through single gap closing Dirac cone
whereas in non-Hermitian case it is through (at least two) exceptional points.
Due to the complex nature, the position of the exceptional point is given by,

$\displaystyle\chi_{x}^{re}(k)=-\chi_{y}^{im}(k)$ and
$\displaystyle\chi_{y}^{re}(k)=\chi_{x}^{im}(k),$ or
$\displaystyle\chi_{x}^{re}(k)=\chi_{y}^{im}(k)$ and
$\displaystyle\chi_{y}^{re}(k)=-\chi_{x}^{im}(k).$ (40)

In this model, we observe only two exceptional points. Thus we define a
complex topological angle as,

$\phi=\phi^{re}+i\phi^{im}.$ (41)

Here the real and imaginary parts contribute to the argument and amplitude
respectively. Thus the winding number is not affected by the imaginary part.
Thus

$\displaystyle e^{i2\phi}$ $\displaystyle=$
$\displaystyle\frac{1+i\tan(\phi)}{1-i\tan(\phi)}=\frac{\chi_{x}+i\chi_{y}}{\chi_{x}-i\chi_{y}},$
$\displaystyle e^{-i2\phi}$ $\displaystyle=$
$\displaystyle\left|\frac{\chi_{x}+i\chi_{y}}{\chi_{x}-i\chi_{y}}\right|,\hskip
14.22636pte^{-i2\phi_{r}}=\frac{\frac{\chi_{x}+i\chi_{y}}{\chi_{x}-i\chi_{y}}}{\left|\frac{\chi_{x}+i\chi_{y}}{\chi_{x}-i\chi_{y}}\right|},$
$\displaystyle\tan(2\phi_{r})$ $\displaystyle=$
$\displaystyle\frac{Im(\frac{\chi_{x}+i\chi_{y}}{\chi_{x}-i\chi_{y}})}{Re(\frac{\chi_{x}+i\chi_{y}}{\chi_{x}-i\chi_{y}})}.$
(42)

which can be represented in a single equation as,

$\tan(2\phi_{r})=\frac{\tan(\phi_{1})+\tan(\phi_{2})}{1-\tan(\phi_{1})\tan(\phi_{2})}=\tan(\phi_{1}+\phi_{2}),$
(43)

where
$\tan(\phi_{1})=\frac{\chi_{y}^{re}(k)+\chi_{x}^{im}(k)}{\chi_{x}^{re}(k)+\chi_{y}^{im}(k)}$
and
$\tan(\phi_{2})=\frac{\chi_{y}^{re}(k)-\chi_{x}^{im}(k)}{\chi_{x}^{re}(k)+\chi_{y}^{im}(k)}$.
The angles $\phi_{1}$ and $\phi_{2}$ are real and they produce the winding
number as

$W=\frac{1}{2}\left(\frac{1}{2\pi}\right)(\oint\partial_{k}\phi_{1}dk+\oint\partial_{k}\phi_{2}dk).$
(44)

For our model only $\chi_{y}$ contains the imaginary part. Thus the
exceptional points are located at $(0,\chi_{y}^{im})$ and
$(0,-\chi_{y}^{im})$. The wrapping of winding vectors
$(\chi_{x}^{re},\chi_{y}^{re})$ around first (second) exceptional point gives
complex angle $\phi_{1}(\phi_{2})$ with corresponding winding number
$W_{1}(W_{2})$ respectively. Due to the non-Hermiticity, each exceptional
point induces its own origin of pseudo-spin space and corresponding winding
vectors. Thus we work on extended winding vectors to understand the non-
Hermitian effect in the parameter space. i.e.,

$\displaystyle\chi_{x}(extended)=\chi_{x}^{re}(k)\pm\chi_{y}^{im}(k),$
$\displaystyle\chi_{y}(extended)=\chi_{y}^{re}(k)\pm\chi_{x}^{im}(k),$ (45)

which correspond to parameter space $F_{1}(k,\mathbf{M})$ and
$F_{2}(k,\mathbf{M})$ respectively.

##  Appendix B Detailed derivation of CRG equations

Considering Eq. 26, we calculate the CRG flow equations to the longer-range
model with $r=2$. Here $x1,x2$ and $x3,x4$ are critical and fixed lines
respectively.The terms $X_{1,2},Y_{1,2}$ and $Z_{1,2}$ are given by

$\displaystyle X_{1,2}$ $\displaystyle=$
$\displaystyle-\frac{2\left(2^{1-\alpha}\mp
1\right)^{3}}{\left(2^{-\alpha}-\delta+t\mp 1\right)^{3}}-\frac{3\left(\pm
1-2^{2-\alpha}\right)\left(2^{1-\alpha}\mp
1\right)}{\left(2^{-\alpha}-\delta+t\mp 1\right)^{2}},$ $\displaystyle+$
$\displaystyle\frac{\pm 1-2^{3-\alpha}}{2^{-\alpha}-\delta+t\mp 1}$
$\displaystyle Y_{1,2}$ $\displaystyle=$ $\displaystyle\frac{2^{1-\alpha}\mp
1}{\left(2^{-\alpha}-\delta+t\mp 1\right)^{2}},$ $\displaystyle Z_{1,2}$
$\displaystyle=$ $\displaystyle\frac{2^{-\alpha}\left(2^{1-\alpha}\mp
1\right)\log(2)}{\left(2^{-\alpha}-\delta+t\mp
1\right)^{2}}-\frac{2^{1-\alpha}\log(2)}{2^{-\alpha}-\delta+t\mp 1}.$

1) In the vicinity of first exceptional point (parameter space corresponding
to $F1(k,\mathbf{M})$), The CRG equations corresponding to $k=0$ are given by,

$\displaystyle\frac{dt}{dl}$ $\displaystyle=$
$\displaystyle\left(\frac{1}{2}\right)\left(-\frac{X_{2}}{Y_{2}}\right),\frac{d\alpha}{dl}=\left(\frac{1}{2}\right)\left(\frac{X_{2}}{Z_{2}}\right).$
(46)

with
$x1=\frac{1}{2}(-2\delta-1)\text{(red)},x2=-\frac{1}{2^{\alpha}}-\delta-1\text{(blue)},x3=2^{-\alpha}-\delta\text{(green)},x4=-\frac{2^{\alpha}\delta-2^{\alpha}+8\delta+1}{2^{\alpha}+8}\text{(black)}$.

2) In the vicinity of first exceptional point (parameter space corresponding
to $F1(k,\mathbf{M})$), The CRG equations corresponding to $k=\pi$ are given
by,

$\displaystyle\frac{dt}{dl}$ $\displaystyle=$
$\displaystyle\left(\frac{1}{2}\right)\left(-\frac{X_{1}}{Y_{1}}\right),\frac{d\alpha}{dl}=\left(\frac{1}{2}\right)\left(\frac{X_{1}}{Z_{1}}\right),$
(47)

with
$x1=\frac{1}{2}(1-2\delta)\text{(red)},x2=-\frac{1}{2^{\alpha}}-\delta+1\text{(blue)},x3=\frac{1}{2}\left(2^{1-\alpha}-\delta\right)\text{(green)},x4=-\frac{2^{\alpha}\delta+2^{\alpha}-8\delta+1}{2^{\alpha}-8}\text{(black)}.$

3) In the vicinity of second exceptional point (parameter space corresponding
to $F2(k,\mathbf{M})$), The CRG equations corresponding to $k=0$ are given by,

$\displaystyle\frac{dt}{dl}$ $\displaystyle=$
$\displaystyle\left(\frac{1}{2}\right)\left(-\frac{X_{2}}{Y_{2}}\right),\frac{d\alpha}{dl}=\left(\frac{1}{2}\right)\left(\frac{X_{2}}{Z_{2}}\right).$
(48)

Here
$x1=\frac{1}{2}(2\delta-1)\text{(red)},x2=-\frac{1}{2^{\alpha}}+\delta-1\text{(blue)},x3=2^{-\alpha}+\delta\text{(green)},x4=\frac{2^{\alpha}\delta+2^{\alpha}+8\delta-1}{2^{\alpha}+8}\text{(black)}$.

4) In the vicinity of second exceptional point (parameter space corresponding
to $F2(k,\mathbf{M})$), The CRG equations corresponding to $k=\pi$ are given
by,

$\displaystyle\frac{dt}{dl}$ $\displaystyle=$
$\displaystyle\left(\frac{1}{2}\right)\left(-\frac{X_{1}}{Y_{1}}\right),\frac{d\alpha}{dl}=\left(\frac{1}{2}\right)\left(\frac{X_{1}}{Z_{1}}\right).$
(49)

Here
$x1=\frac{1}{2}(2\delta+1)\text{(red)},x2=-\frac{1}{2^{\alpha}}-\delta+1\text{(blue)},x3=2^{-\alpha}+\delta\text{(green)},x4=\frac{2^{\alpha}\delta-2^{\alpha}-8\delta-1}{2^{\alpha}-8}\text{(black)}$.