# Cosmic evolution of Kantowski-Sachs universe in the context of bulk viscous
string in teleparallel gravity

S.R.Bhoyar Email<EMAIL_ADDRESS>0000-0001-8427-4540 Yash B. Ingole
Email<EMAIL_ADDRESS>0009-0006-7208-1999

###### Abstract

In the present work, we have analysed the Kantowski-Sachs cosmological model
with teleparallel gravity, where a bulk viscous fluid containing one-
dimensional cosmic strings serves as the source for the energy-momentum
tensor. To obtain the deterministic solution of the field equations, we
employed the proportionality condition linking the shear scalar $(\sigma)$ and
the expansion scalar $(\theta)$, which establishes a relationship between
metric potentials. Another approach employed was the utilisation of the Hybrid
Expansion Law (HEL). The discussion focuses on the behaviour of the
accelerating universe concerning the specific choice of a highly non-linear
form of teleparallel gravity $f(T)=\alpha T+\beta T^{m}$, where $T$ is torsion
scalar, $\alpha$, $\beta$ are model parameter and $m$ is a positive integer.
The equation of state parameter $(\omega)$ in this model supports the
acceleration behaviour of the universe. We observed that the null energy
condition and weak energy condition are obeyed while violating the strong
energy condition as per the present accelerating expansion. It has been
observed that, under specific model parameter constraints, the universe
transitions from a decelerating to an accelerated phase.

Keywords: Kantowski-Sachs, universe, bulk viscous string, $f(T)$ gravity,
(HEL).

##  1 Introduction

Albert Einstein presented the General Theory of Relativity (GR) in $1915$.
Even though GR has performed incredibly well in experiments conducted on the
solar system, it cannot explain every occurrence of gravity that has been seen
yet. Numerous cosmological studies over the last 20 years have come together
to suggest that our universe is presently going through an accelerated
expansion phase. The current universe is speeding rapidly, as evidenced by
data gathered from Type Ia supernovae [1, 2]. Strong observational evidence
for this rapid expansion is provided by Baryon Acoustic Oscillations (BAO) [3,
4], large-scale structure [5, 6], galaxy redshift survey [7], and Cosmic
Microwave Background Radiations (CMBR) [8, 9]. They have presented remarkable
proof that two unknown types of matter and energy, which we term dark matter
(DM) and dark energy (DE) with negative pressure, contribute to 95 % of the
universe’s contents [10]. The vacuum energy (also known as the cosmological
constant, $\Lambda$) with a constant equation of state parameter (EoS)
$\omega_{\Lambda}=-1$ is the simplest and most appealing candidate for dark
energy (DE) [11]. Although this model is generally consistent with current
astronomical observations, it encounters challenges in reconciling the
observed small value of DE density with that predicted by quantum field
theories. This is known as the cosmological constant problem [12]. The late-
time acceleration of the universe, as detected, continues to be one of the
fundamental mysteries in theoretical physics, and the mechanism accountable
for this expansion remains an open question.

Recently, it was demonstrated that the $\Lambda$CDM model may also encounter
an age problem [13]. Consequently, it is therefore logical to pursue alternate
possibilities to explain the current acceleration of the universe. Examining
the slight deviations for the EoS parameter from $-1$ requires a depiction of
the DE that permits the EoS to evolve across the phantom divide line $-1$
conceivably multiple times. The current data appear to marginally prefer an
evolving DE with the EoS parameter crossing $-1$ from above to below in the
recent past [14]. One may interpret the observed accelerating expansion as an
indication of the breakdown of our understanding of the laws of gravitation.
Although this model aligns reasonably well with observational data, it is
afflicted by several significant challenges. Among these, the two primary
obstacles are the cosmic coincidence problem and the fine-tuning problem [15].

Several models have been put forward in the literature to account for this
current acceleration. Fundamentally, two approaches have been adopted to
interpret this recent acceleration of the universe. The first approach
involves hypothesising the presence of a mysterious force with a raised
negative pressure, commonly referred to as dark energy (DE), which is
accountable for the current acceleration of the universe. To overcome this
problem, time-varying dark energy models have been proposed in the literature,
such as quintessence [16, 17], k-essence [18, 19], and perfect fluid models
(such as the Chaplygin gas model) [20, 21]. The second approach to clarifying
the current acceleration of the universe is to alter the geometry of
spacetime. This can be accomplished by modifying the left-hand side of the
Einstein equation. Modified theories of gravity represent the geometric
extensions of Einstein’s general theory of relativity in which universe
acceleration can be accomplished through modifications to the Einstein-Hilbert
action of GR. Recently, cosmologists have shown keen interest in modified
theories of gravity for understanding the role of dark energy. Within modified
gravity, the source of dark energy is acknowledged as a modification of
gravity [22]. A considerable amount of research indicates that modified
theories of gravity can account for both the early and late-time acceleration
of the universe. Consequently, there are abundant motivations to discover
theories surpassing the standard formulation of GR thus, modification of the
gravity theory is needed. Numerous modified theories have been proposed in the
literature.

Some of the modified theories include the $f(R)$ theory of gravity, the
modification of GR, introduced in [23], the $f(R,T)$ theory, an extension of
$f(R)$ gravity coupled with the trace of energy-momentum tensor $T$ [24, 25,
26], the $f(G)$ gravity [27, 28, 29], $f(R,G)$ theory [30, 31], $f(T)$ theory
[32, 33, 34], $f(T,B)$ theory [35], $f(Q,T)$ theory [36, 37, 38], $f(G)$
theory [39]. etc. We are going to use teleparallel gravity for this work, The
literature survey of $f(T)$ gravity is as follows: It was first introduced by
Einstein himself as an equivalent alternative to GR [40]. Due to the features
of translations, gauge theories including these transformations differ from
the standard internal gauge models in several ways. S. Bhoyar et al. studied
Bianchi type-I space-time with the linear equation of state filled with a
perfect fluid within the framework of $f(T)$ gravity [41]. D.D. Pawar et al.
have investigated two-fluid cosmological models in $f(T)$ framework [42]. R.
Myrzakulov et al. reconstructed some explicit examples of $f(T)$ from the
background FRW expansion history [43]. Baojiu Li et al. gave a numerical
solution that shows that the extra degree of freedom of such $f(T)$ gravity
models generally decays as one goes to smaller scales, and consequently, its
effects on scales such as galaxies and galaxy clusters are small [44]. M. E.
Rodrigues et al. explore different cosmological models, such as Bianchi
type-I, Bianchi type-III, and Kantowski-Sachs, in the context of $f(T)$
gravity. This can contribute to the development of alternative theories of
gravity and expand our understanding of the fundamental laws governing the
universe[45].

It has been demonstrated that viscosity is one of the most important aspects
of exploring the evolution of the universe. According to current
understanding, the expansion of the universe is an outcome of negative
pressure in the cosmic fluid. This has led to refreshed curiosity regarding
the fluid as a two-component mixture: a regular fluid component and a dark
energy component [46]. A significant role is played by dissipative processes
such as heat transport, bulk viscosity, and shear viscosity in cosmic
expansion [10]. In the early universe, the viscosity may originate due to
various processes, for example, the decoupling of matter from radiation during
the recombination era, particle collisions involving gravitons, and the
formation of galaxies [47]. It is evident that the bulk viscosity contributes
unfavourably to the total pressure, and thus a dissipative fluid appears to be
a potential dark energy nominee [48].

Several researchers have explored the impact of bulk viscous content in
different cosmological situations such as viscous dark energy [49, 48, 14],
viscous dark matter [50, 51], inflation in a dense fluid model [52], late-time
cosmic acceleration [53, 54], and so on. Brevik et al. [55, 56] established
the crucial role of viscosity in explaining the era of inflation and the
current epoch of the universe. Subsequently, Padmanabhan and Chitre [57]
employed the notion of bulk viscosity to characterise inflation in the early
universe. Furthermore, the connection between the modified equation of state
of viscous cosmological models, scalar fields, and extended theories of
gravity is being examined [54, 58].  Deng et al. [59] introduced innovative
models of viscous dark energy that were constrained by current cosmic
observations. Mark and Harko [60] closely examined the impact of bulk
viscosity in the Brans-Dicke gravitational theory. Bali and Pradhan [61]
investigated string cosmological models of Bianchi type III with a bulk
viscous fluid for massive strings. Cosmic strings are topologically stable
entities that may manifest during a phase transition in the early universe
[62]. Cosmic strings play a pivotal role in the study of the early universe.
These emerge during the phase transition after the explosion of the Big Bang
as the temperature decreases below a certain critical temperature, as
predicted by grand unified theories [63, 64, 65, 66]. These cosmic strings
possess stress energy and are coupled with the gravitational field. Thus, it
is interesting to examine the gravitational effects arising from strings. The
general relativistic treatment of strings was initiated by Letelier [67, 68]
and Stachel [69]. Letelier [67] has derived the solution to Einstein’s field
equations for a cloud of strings with spherical, planar, and cylindrical
symmetry. Subsequently, in 1983, he resolved Einstein’s field equations for a
cloud of massive strings and acquired cosmological models in Bianchi I and
Kantowski-Sachs space-times. [70] examined an axially symmetric Bianchi type I
string dust cosmological model in the presence and absence of a magnetic
field. The effect of a magnetic field on string cosmological models was also
explained by [71]. Carminati and McIntosh unveiled new exact solutions for
string cosmology with and without a source-free magnetic field for a Bianchi
type I spacetime, constructing upon the work [72]. [73] examined charged
strange quark matter connected to the string cloud in a spherically symmetric
spacetime admitting a one-parameter group of conformal motion.

In this work, we have focused on studying the cosmic acceleration of the
universe in $f(T)$ gravity in the presence of bulk viscous fluid. The
motivation for working in the $f(T)$ gravity is that in this framework, the
motion equations are in the high-order nonlinear form. In this article, we
investigate the cosmological model with bulk viscosity in the framework of the
highly non-linear form of teleparallel gravity. Our motivation for this study
stems from the above-mentioned discussions and analyses. The structure of this
paper is as follows: Firstly, we derive the field equations employing the
hybrid expansion law, which leads to a deceleration parameter that varies with
time, from the observed data. Moreover, it is noteworthy to mention that this
law is a broader formulation encompassing both the exponential and power law
cosmologies. We dedicate the subsequent sections to the comprehensive
examination of Physical Parameters and Kinematical Parameters. The final
section contains the energy conditions of the model, and at last, we conclude
the paper.

##  2 f(T) gravity

The dynamical object employed in teleparallelism is a Vierbein field
$e_{i}(x^{\mu}),\quad i=0,1,2,3$ representing an orthonormal basis for the
tangent space at each point $x^{\mu}$ of the manifold. This Vierbein field
satisfies the condition, $e_{i}.e_{j}=\eta_{ij}=diag(1,-1,-1,-1)$. The
components of each vector $e_{i}$ can be expressed as $e_{i}^{\mu}$,
$\mu=0,1,2,3$ in a coordinate basis, denoted as
$e_{i}=e_{i}^{\mu}\partial_{\mu}$. It is important to note that Latin indices
relate to the tangent space, while Greek indices indicate coordinates on the
manifold. The metric tensor, $g_{\mu\nu}(x)$ is derived from the dual Vierbein
by $g_{\mu\nu}(x)=\eta_{ij}e{{}^{i}_{\mu}}(x)e{{}^{j}_{\nu}}(x)$. In contrast
to general relativity, which employs the torsionless Levi-Civita connection,
teleparallelism utilises the curvature-less Weitzenböck connection, which
possesses a non-zero torsion.

$T{{}^{\alpha}{}_{\mu\nu}}=\Gamma{{}^{\alpha}{}_{\mu\nu}}-\Gamma{{}^{\alpha}{}_{\nu\mu}}=e{{}^{\alpha}_{i}}(\partial_{\mu}e{{}^{i}_{\nu}}-\partial_{\nu}e{{}^{i}_{\mu}})\
$ (1)

This tensor encompasses all the information about the gravitational field. The
e teleparallel equivalent of general relativity (TEGR) Lagrangian is built
with the torsion equation (1) and its dynamical equations for the vierbein
imply the Einstein equations for the metric. The torsion scalar $(T)$ is given
as

$T=T{{}^{\alpha}{}_{\mu\nu}}S{{}_{\alpha}{}^{\mu\nu}}$ (2)

where,

$S{{}_{\alpha}{}^{\mu\nu}}=\frac{1}{2}(K{{}^{\mu\nu}{}_{\alpha}}+\delta{{}^{\mu}_{\alpha}}T{{}^{\beta\nu}{}_{\beta}}-\delta{{}^{\nu}_{\alpha}}T{{}^{\beta\mu}{}_{\beta}})$
(3)

and $K{{}^{\mu\nu}{}_{\alpha}}$ is contorsion tensor

$K{{}^{\mu\nu}{}_{\alpha}}=-\frac{1}{2}(T{{}^{\mu\nu}{}_{\alpha}}-T{{}^{\nu\mu}{}_{\alpha}}-T{{}_{\alpha}{}^{\mu\nu}})$
(4)

which equals the difference between Weitzenböck and Levi-Civita connections.

The action in $f(T)$ gravity is defined as

$S=\int[f(T)+L_{M}]d^{4}x\sqrt{-g}$ (5)

where $\ f(T)$ denotes an algebraic function of the torsion scalar $T$,
$L_{M}$ represents the matter Lagrangian, and
$\sqrt{-g}=det[e{{}^{i}_{\mu}}]=e$. The equation of motion is obtained by the
functional variation of action equation $\eqref{5}$ concerning tetrad as

$S{{}_{\mu}{}^{\nu\rho}}(\partial_{\rho}T)f_{TT}+[e^{-1}e{{}^{i}_{\mu}}\partial_{\rho}(ee{{}^{\alpha}_{i}}S{{}_{\alpha}{}^{\nu\rho}})T{{}^{\alpha}{}_{\lambda\mu}}S{{}_{\alpha}{}^{\nu\lambda}}]f_{T}+\frac{1}{4}\delta{{}^{\nu}_{\mu}}f=4\pi
T{{}^{\nu}_{\mu}}$ (6)

Where the energy-momentum tensor $T{{}^{\nu}_{\mu}}$ is considered as bulk
viscous fluid coupled with a one-dimensional cosmic string. $f_{T}$ and
$f_{TT}$ represents the first and second order derivative of $f(T)$ with
respect to torsion scalar $T$.

The source is bulk viscous fluid coupled with a one-dimensional cosmic string
given by

$T{{}^{\nu}_{\mu}}=(\rho+\overline{p})u_{\mu}u^{\nu}-\overline{p}g{{}^{\nu}_{\mu}}-\lambda
x_{\mu}x^{\nu}$ (7) $\overline{p}=p-3\xi H$ (8)

where $\rho=\rho_{p}+\lambda$ is the proper string energy density with
particles attached to them and $\rho_{p}$ is the particle energy density,
$\lambda$ is the string tension density, $p-3\xi H$ is the bulk viscous
pressure, $\xi(t)$ is the coefficient of bulk viscosity, $H$ is Hubble’s
parameter, $x^{\nu}$ is a unit space-like vector for cloud string and
$u^{\nu}$ denotes the four-velocity satisfying the conditions
$u^{\nu}u_{\nu}=1=-x^{\nu}x_{\nu}$ and $u_{\nu}x^{\nu}=0$.

In the co-moving coordinate system, we have

$u^{\nu}=(0,0,0,1),\quad x^{\nu}=(A^{-1},0,0,0)$ (9)

##  3 Kantowski-Sachs universe

The Kantowski-Sachs geometry can be viewed as the anisotropic extension of the
closed Friedmann Lemaître–Robertson–Walker (FLRW) geometry. The metric of this
geometry is dependent on two crucial scale factors in the spacelike
hypersurface. By employing Misner-like variables [74, 75], one scale factor
represents the radius of the spacelike hypersurface, while the second scale
factor characterises the anisotropy. When the anisotropy remains constant, the
Kantowski-Sachs spacetime exhibits a six-dimensional Lie algebra as killing
vector fields, and it reaches the limit of the closed FLRW geometry [76]. The
Kantowski-Sachs space is closely linked to the Bianchi spacetime [74]. The
Kantowski-Sachs geometry arises from a Lie contraction in the locally
rotational spacetime (LRS) Bianchi type-$IX$.

Various studies have been conducted on the Kantowski-Sachs geometries due to
their importance. The cosmological constant’s effects in general relativity
were studied [77, 78], while the case of a perfect fluid obeying the
barotropic equation of state was examined by C.B. Collins et al. [79]. The
presence of the cosmological constant leads the Kantowski-Sachs universe to
become a de Sitter universe in the future [77], making it suitable for
describing the pre-inflationary era [78]. In [79], it was discovered that
models with a fluid source exhibit a past asymptotic behaviour resulting in a
big-bang singularity and a future attractor leading to a big crunch. The
effects of the electromagnetic field were explored in [80, 81, 82]. The
significance of the Kantowski-Sachs geometry can be observed in inhomogeneous
models. For silent universes, this importance is clear [83]. In the case of
Szekeres spacetimes, [84] the field equations of the Kantowski-Sachs restrict
the dynamical variables of the field equations system. This means that there
are silent inhomogeneous and anisotropic universes that reduce to the
anisotropic Kantowski-Sachs geometry when homogenised [85].

In this case, we consider the spatially homogeneous and anisotropic
Kantowski–Sachs spacetime of the form

$ds^{2}=dt^{2}-A^{2}(t)dr^{2}-B^{2}(t)(d\theta^{2}-\sin^{2}{\theta}d\phi^{2})$
(10)

where the metric potentials $A$ and $\ B$ are functions of cosmic time $t$
only. Now the set of the diagonal tetrads is related to the metric equation
$\eqref{10}$ and is given as

$[e{{}^{i}_{\mu}}]=diag[1,A,B,B\sin{\theta}]$ (11)

The determinant of equation (10) is

$e=AB^{2}\sin{\theta}$ (12)

The torsion scalar $T$ is derived from equation (2)

$T=-2\biggl{(}\frac{2\dot{A}\dot{B}}{AB}+\frac{\dot{B}^{2}}{B^{2}}\biggr{)}$
(13)

The field equation for Kantowski-Sachs Model equation(10) from equations
(5)-(9) in $f(T)$ gravity is given as

$\begin{split}f-4\biggl{(}\frac{\dot{B}}{B}\biggr{)}\dot{T}f_{TT}+4\biggl{(}\frac{\dot{A}\dot{B}}{AB}-\frac{\ddot{B}}{B}-\frac{\dot{B}^{2}}{B^{2}}\biggr{)}f_{T}=-16\pi(\bar{p}-\lambda)\end{split}$
(14)

$\begin{split}f-2\biggl{(}\frac{\dot{A}}{A}+\frac{\dot{B}}{B}\biggr{)}\dot{T}f_{TT}-2\biggl{(}\frac{\dot{A}\dot{B}}{AB}+\frac{\ddot{A}}{A}+\frac{\ddot{B}}{B}-\frac{\dot{B}^{2}}{B^{2}}\biggr{)}f_{T}=-16\pi\bar{p}\end{split}$
(15)

$f+4\biggl{(}2\frac{\dot{A}\dot{B}}{AB}+\frac{\dot{B}^{2}}{B^{2}}\biggr{)}f_{T}=16\pi\rho$
(16)

Where the overhead dot $(.)$ represents the derivative concerning cosmic time
$t$. $f_{T}$ and $f_{TT}$ are the first and second order derivative of $f(T)$
with respect to $t$ respectively. We have three non-linear differential
equations and six unknowns, namely $f$, $A$, $B$, $\bar{p}$, $\rho$ and
$\lambda$. The Volume $V$ and average scale factor $a(t)$ is defined as

$V=[a(t)]^{3}=AB^{2}$ (17)

The Anisotropic parameter $(A_{m})$ is given by

$A_{m}=\frac{1}{3}\sum_{i=1}^{3}\biggl{(}\frac{H_{i}-H}{H}\biggr{)}^{2}$ (18)

Where $H_{1}=\frac{\dot{A}}{A}$, $H_{2}=\frac{\dot{B}}{B}=H_{3}$ are
directional Hubble’s parameters and mean Hubble’s parameter is defined as

$H=\frac{\dot{a}}{a}=\frac{1}{3}\biggl{(}\frac{\dot{A}}{A}+2\frac{\dot{B}}{B}\biggr{)}$
(19)

The expansion scalar $(\theta)$ and shear scalar $(\sigma)$ are defined as

$\theta=u^{i}_{;i}=\frac{\dot{A}}{A}+2\frac{\dot{B}}{B}$ (20)
$\sigma^{2}=\frac{3}{2}H^{2}A_{m}$ (21)

The deceleration parameter $(q)$ is given as

$q=-\frac{\ddot{a}a}{\dot{a}^{2}}=-1+\frac{d}{dt}\biggl{(}\frac{1}{H}\biggr{)}$
(22)

##  4 Solution of field equations

To solve highly non-linear differential field equations (14)-(16) The
following physically conceivable scenarios have been taken into consideration.

  * •

We consider the Hybrid Expansion Law (HEL) proposed by [86] as

$a(t)=a_{0}\biggl{(}\frac{t}{t_{0}}\biggr{)}^{n}e^{b(\frac{t}{t_{0}}-1)}$ (23)

Where $a_{0}$ and $t_{0}$ respectively denote the scale factor and age of the
universe today and $n>0$, $b>0$ are the free parameters.

The HEL, also known as the generalised form of scale factor, gives rise to the
power-law cosmology for $b=0$ and exponential-law for $n=0$, while the
combination of $n>0$ and $b>0$ results in a new cosmology arising from the
HEL. Therefore, the HEL cosmology encompasses both the power-law and
exponential law cosmologies as special cases. By choosing this scale factor,
we obtain a time-dependent deceleration parameter [equation (36) and figure
(1(b))]that exhibits inflation and a radiation/matter-dominated era before the
dark energy era, followed by a transition from deceleration to acceleration.
Consequently, our choice of scale factor is deemed physically acceptable.
Recently, Bhoyar et al. [87], explored certain characteristics of a nonstatic
plane-symmetric universe filled with magnetised anisotropic dark energy using
HEL.

  * •

$A=B^{k},\quad k>1$ (24)

As observed velocity of the redshift relation for beyond galactic sources
suggests that the Hubble expansion of the universe is isotropic within a range
of 30$\%$. In a more precise manner, studies on redshifts establish a limit of
$(\frac{\sigma}{H})\leq 0.3$ on the ratio of shear $(\sigma)$ to the Hubble
constant $H$ in the vicinity of our galaxy. It has been highlighted by Collin
et al. [88] and Y. Aditya et al. [89] that for a spatially homogeneous metric,
the normal congruence to the homogeneous expansion fulfils the condition that
$(\frac{\sigma}{\theta})$ remains constant. In other words, the expansion
scalar is directly proportional to the shear scalar. This provides the
relationship between the metric potentials (24).

  * •

As pointed out by [90] and [91] the combined effect of bulk viscous pressure
and proper pressure is expressed as

$\bar{p}=p-3\xi H=\epsilon\rho,$ (25)

where

$\epsilon=\varepsilon_{0}-\gamma\quad(0\leq\varepsilon_{0}\leq 1),\quad
p=\varepsilon_{0}\rho$ (26)

$\epsilon_{0}$, $\gamma$ are constant. The highly non-linear form of $f(T)$
[92] gravity as

$f(T)=\alpha T+\beta T^{m},\quad m>2$ (27)

where $\alpha\geq 0$ and $\beta\geq 0$ are model parameters.

Now utilizing equation (24) in equations (14)$-$(16) we obtained

$f-4\biggl{(}\frac{\dot{B}}{B}\biggr{)}\dot{T}f_{TT}+4\biggl{(}-\frac{B\ddot{B}}{B}+(k-1)\frac{\dot{B^{2}}}{B^{2}}\biggr{)}f_{T}=-16\pi(\bar{p}-\lambda)$
(28)

$f-2\biggl{(}\frac{(k+1)\dot{B}}{B}\biggr{)}\dot{T}f_{TT}-2\biggl{(}\frac{B\ddot{B}(k+1)+\dot{B}^{2}(k^{2}-1)}{B^{2}}\biggr{)}f_{T}=-16\pi\bar{p}$
(29)

$f+4\biggl{(}(2k+1)\frac{\dot{B^{2}}}{B^{2}}\biggr{)}f_{T}=16\pi\rho$ (30)

The Values of metric potentials $A$ and $\ B$ are obtained from equations
(17), (23) and (24) are obtained as

$A=\biggl{[}a_{0}\biggl{(}\frac{t}{t_{0}}\biggr{)}^{n}e^{b(\frac{t}{t_{0}}-1)}\biggr{]}^{\frac{3k}{k+2}}$
(31)
$B=\biggl{[}a_{0}\biggl{(}\frac{t}{t_{0}}\biggr{)}^{n}e^{b(\frac{t}{t_{0}}-1)}\biggr{]}^{\frac{3}{k+2}}$
(32)

Now the metric equation (10) is written with the help of equations (31) and
(32) as

$\begin{split}ds^{2}=dt^{2}-\biggl{[}a_{0}\biggl{(}\frac{t}{t_{0}}\biggr{)}^{n}e^{b(\frac{t}{t_{0}}-1)}\biggr{]}^{\frac{3k}{k+2}}dr^{2}-\biggl{[}a_{0}\biggl{(}\frac{t}{t_{0}}\biggr{)}^{n}e^{b(\frac{t}{t_{0}}-1)}\biggr{]}^{\frac{3}{k+2}}\biggl{(}d\theta^{2}-\sin^{2}{\theta}d\phi^{2}\biggr{)}\end{split}$
(33)

The Volume $(V)$ of the model is given by equations (17) and (23) as

$V=\biggl{[}a_{0}\biggl{(}\frac{t}{t_{0}}\biggr{)}^{n}e^{b(\frac{t}{t_{0}}-1)}\biggr{]}^{3}$
(34)

The mean Hubble’s parameter $(H)$ and deceleration $(q)$ are given by
equations (19), (22) and (23) as

$H=\frac{n}{t}+\frac{b}{t_{0}}$ (35)

$q=-\frac{b^{2}t^{2}+2bntt_{0}+n(n-1)t^{2}_{0}}{(bt+nt_{0})^{2}}$ (36)

The graphical depictions in figure (1(a)) (left panel) showcase the behaviour
of Hubble’s parameter $H$. The Hubble parameter exhibits a discernible trend,
its values diminish as cosmic time $t$ progresses. The consistently positive
values of $H$ throughout the cosmic evolution unequivocally signify the
ongoing expansion of the universe. The Hubble parameter gradually converges
towards zero.

The deceleration parameter $q$. shown by figure (1(b)) (right panel) changes
over time, showing a transition from the early decelerating phase to the
current acceleration phase. This pattern nicely matches what we observe in
Type-I supernovae, we observe that as $q<0$ for
$\frac{t_{0}(\sqrt{n}-n)}{b}<t$. The paper showed that our current universe
entered an accelerating phase with a deceleration parameter ranging between
$-1\leq q\leq 0$. These results agree well with [93, 94, 95, 96].

(a) $n=1.25$, $b=0.20$, $t_{0}=0.57$

(b) $n=1.25,b=0.20,t_{0}=0.57$

Figure 1: The plot of Hubble’s parameter $(H)$ and Deceleration parameter
$(q)$ Vs cosmic time $(t)$

##  5 Geometrical and physical significance of the model

From equations (23), (27), (30) and (32) we have obtained the energy density
as

$\rho=\frac{-1}{16\pi}\biggl{\\{}-18\alpha E-18^{m}(2m-1)\beta
E^{m}\biggr{\\}}$ (37)

where,

$E=\biggl{[}\frac{-(1+2k)(bt+nt_{0})^{2}}{(k+2)^{2}t^{2}t_{0}^{2}}\biggr{]}$

The coefficient of bulk viscosity parameter $(\xi)$ is obtained from equations
(25) and(26) as

$\xi=\frac{\rho\gamma}{3H}$ (38)

The bulk viscosity coefficient $\xi$ consistently decreases with time,
maintaining a positive value throughout cosmic evolution as per figure (4(a))
(left panel). They are ultimately leading to the emergence of inflationary
models. Additionally, the models do not exhibit an initial singularity,
supporting Murphy’s [97] conclusion that the introduction of bulk viscosity
successfully avoids such singularities. This observed behaviour aligns well
with the anticipated physical characteristics of $\xi$, and these reports are
in good agreement with [98, 99].

The bulk viscous pressure $(\bar{p})$ is obtained from equations (27), (29)
and (32) as

$\begin{split}\bar{p}=&\frac{1}{16\pi(2+k)^{2}t^{2}t_{0}^{2}}\Bigg{\\{}18\alpha(1+2k)(bt+nt_{0})^{2}+mn\beta
t_{0}(2+k)(1+k)(1-m)^{2}2^{(1+m)}3^{(-1+2m)}\\\ &\ E^{(-1+m)}-18^{m}t^{2}\beta
t_{0}^{2}(2+k)^{2}E^{m}+6(1+k)\biggl{(}3b^{2}kt^{2}+6bkntt_{0}+n(-2+k(-1+3n))t_{0}^{2}\biggr{)}\\\
&\ \biggl{(}\alpha+18^{(-1+m)}m\beta
E^{(-1+m)}\biggr{)}\Biggr{\\}}\end{split}$ (39)

The string tension density ($\lambda$) is obtained from equations (27), (28),
(32) and (39) as

$\begin{split}\lambda=&\frac{(-1+k)}{8\pi(2+k)^{2}t^{2}t_{0}^{2}}\Biggl{\\{}mn\beta
t_{0}^{2}(2+k)(1-m)2^{m}3^{(-1+2m)}E^{(-1+m)}+3\biggl{(}3b^{2}(4+k)t^{2}+6b(4+k)ntt_{0}\\\
&\ +n(-2-k+3(4+k)n)t_{0}^{2}\biggr{)}\biggl{(}\alpha+18^{(-1+m)}m\beta
E^{(-1+m)}\biggr{)}\Biggr{\\}}\end{split}$ (40)

We have noted that the string tension density $\lambda$ exhibits a positive
correlation with time, always maintaining a negative value until it eventually
converges to zero at a subsequent point in time. Letelier [68] highlighted the
potential occurrence of both positive and negative values for $\lambda$. In
the case where $\lambda<0$, the string phase of the universe ceases to exist,
resulting in the presence of an anisotropic fluid comprised of particles. This
characteristic behaviour of $\lambda$ is also depicted in figure (6(a)) (left
panel). Our findings show comparable outcomes with [100, 101].

The particle energy density $(\rho_{p})$ is calculated from equations (39) and
(40) as

$\begin{split}\rho_{p}=&\frac{1}{16\pi(2+k)^{2}t^{2}t_{0}^{2}}\Biggl{\\{}18\alpha(1+2k)(bt+nt_{0})^{2}+18^{m}\beta
t^{2}t_{0}^{2}(2+k)^{2}(1-2m)E^{m}+2(1-k)\Biggl{[}mn\beta\\\ &\
t_{0}^{2}(2+k)(1-m)2^{m}3^{(-1+2m)}E^{(-1+m)}+3\biggl{(}3b^{2}(4+k)t^{2}+6b(4+k)ntt_{0}+n(-2-k+3(4\\\
&\ +k)n)t_{0}^{2}\biggr{)}\biggl{(}\alpha+18^{(-1+m)}m\beta
E^{(-1+m)}\biggr{)}\Biggr{]}\Biggr{\\}}\end{split}$ (41)

The proper energy density $\rho$ and particle density $\rho_{p}$ exhibit a
swift decline from positive values to the same constant as shown in figures
(2(a)) (left panel) and (2(b)) (right panel). The substantial initial values
of $\rho$ and $\rho_{p}$ figure (2) signify the domination of densities during
the early stages, diminishing significantly as time progresses. Importantly,
$\rho$ and $\rho_{p}$ maintain positive values throughout the entire evolution
of the universe. This trend underscores the expansive nature of the universe.
According to [41] $\rho\rightarrow 0$, $\rho_{p}\rightarrow 0$ for
$t\rightarrow\infty$ thus the universe approaches a flat universe at late
time. As a result, our model is in good agreement with the recent observation,
these respondents have strong agreement with [102, 103, 104].

(a) $\alpha=0.01,\beta=0.056,k=1.2,t_{0}=0.57$

(b) $\alpha=0.01,\beta=0.056,k=1.2,t_{0}=0.57$

Figure 2: Plot of energy density $(\rho)$ and particle energy density
$(\rho_{p})$ conditions Vs cosmic time $(t)$

From equations (25), (35), (38) and (39) The proper pressure $(p)$ is
calculated as

$\begin{split}p=&\frac{1}{16\pi(2+k)^{2}t^{2}t_{0}^{2}}\Bigg{\\{}18\alpha(1+2k)(bt+nt_{0})^{2}+mnt_{0}^{2}(2+k)(1+k)(1-m)2^{(1+m)}3^{(-1+2m)}\\\
&\
E^{(-1+m)}\beta-18^{m}\beta(2+k)^{2}t^{2}t_{0}^{2}E^{m}+6(1+k)\biggl{(}3b^{2}kt^{2}+6bkntt_{0}+n(-2+k(-1+3n))t_{0}^{2}\biggr{)}\\\
&\ \biggl{(}\alpha+18^{(-1+m)}m\beta
E^{(-1+m)}\biggr{)}+\beta\Biggl{(}18\alpha(1+2k)(bt+nt_{0})^{2}\alpha+18^{m}\beta(2+k)^{2}(1-2m)t^{2}t_{0}^{2}\\\
&\ E^{m}\Biggr{)}\Biggr{\\}}\end{split}$ (42)

(a) $\alpha=0.01,\beta=0.056,k=1.2,t_{0}=0.57$

(b) $\alpha=0.01,\beta=0.056,k=1.2,t_{0}=0.57$

Figure 3: Plot of pressure $(\bar{p})$ and bulk viscous pressure particle
energy density $(p)$ conditions Vs cosmic time $(t)$

The examination of figures (3(a)) (left panel) and (3(b)) (right panel)
reveals a noteworthy trend in the behaviour of bulk viscous ${\bar{p}}$
pressure and pressure $p$ across cosmic evolution. Both parameters ${\bar{p}}$
and $p$ figure (3) consistently exhibit a negative trajectory throughout this
evolution, beginning with substantial negative values and at the present epoch
$\bar{p}\rightarrow 0$ and $p\rightarrow 0$ this characteristic behaviour
implies a phase of accelerated expansion in the universe. The persistent
negativity in pressure and bulk viscous pressure signifies a sustained
repulsive effect, contributing to the observed cosmic acceleration, these
arguments hold similar results with [102, 103, 104, 100, 101].

The equation of state parameter (EoS) is given by

$\omega=\frac{p}{\rho}$ (43)

Currently one of the largest endeavours in observational cosmology is the
measurement of the EoS parameter for dark energy (DE). The traditional
description of the DE model is provided by the EoS parameter
$\omega=\frac{p}{\rho}$, which not necessarily be constant. Figure (4(b))
(right panel) portrays the behaviour of the EoS parameter along with cosmic
time using hybrid expansion law for some fixed values of $n$, $b$ and $m$. It
is clear that at the present epoch $-1\leq\omega<0$ and $\omega\leq-1$ in the
recent past thus the model shows a phase transition from the phantom era to
the quintessence era and this holds a good agreement with [104, 105, 106].

(a) $\alpha=0.01,\beta=0.056,k=1.2,t_{0}=0.57$

(b) $\alpha=0.01,\beta=0.056,k=1.2,t_{0}=0.57$

Figure 4: Plot of coefficient of bulk viscosity $(\xi)$ and EoS parameter
$(\omega)$ Vs cosmic time $(t)$

The anisotropic parameter $(A_{m})$, expansion scalar $(\theta)$, shear scalar
$(\sigma^{2})$ are given as

$A_{m}=\frac{2(k-1)^{2}}{(k+2)^{2}}$ (44) $\theta=\frac{3(bt+nt_{0})}{tt_{0}}$
(45) $\sigma^{2}=\frac{3(k-1)(bt+nt_{0})^{2}}{(k+2)^{2}t^{2}t_{0}^{2}}$ (46)

From equations (45) and (46), it is seen that $\theta\rightarrow 0$ and
$\sigma^{2}\rightarrow 0$ as $t\rightarrow\infty$ and they both get diverges
as $t\rightarrow 0$ as shown in figure (5(a)) (left panel) and (5(b)) (right
panel). It is observed from equation (44) that for $k\neq 1$ we got $A_{m}\neq
0$ this infers that our constructed model of the universe remains anisotropic
throughout cosmic evolution.

(a) $\alpha=0.01,\beta=0.056,k=1.2,t_{0}=0.57$

(b) $\alpha=0.01,\beta=0.056,k=1.2,t_{0}=0.57$

Figure 5: Plot of expansion scalar $(\theta)$ and shear scalar $(\sigma^{2})$
conditions Vs cosmic time $(t)$

###  5.1 Energy conditions

Numerous methods are employed to figure out the universe’s history and are
used in many approaches to understanding the evolution of the universe. Energy
conditions play a crucial role in confirming hypotheses about the singularity
of space-time and black holes [107], including those put forth by Roger
Penrose and Stephen Hawking  [108]. Verifying the universe’s expansion is the
primary goal of these energy conditions in this context. Energy conditions can
take many different forms, including dominant energy condition (DEC), weak
energy condition (WEC), null energy condition (NEC), and strong energy
condition these conditions are given as [109].

(a) $\alpha=0.01,\beta=0.056,k=1.2,t_{0}=0.57$

(b) $\alpha=0.01,\beta=0.056,k=1.2,t_{0}=0.57$

Figure 6: Plot of string energy density $(\lambda)$ and energy conditions
conditions Vs cosmic time $(t)$

1). $\rho\geq 0,$ $\rho\pm\bar{p}\geq 0$, (DEC) 2). $\rho\geq 0$,
$\rho-\bar{p}\geq 0$ (WEC) 3). $\rho+\bar{p}\geq 0$, $\rho+3\bar{p}\geq 0$
(SEC)

From figure (6(b)) (right panel) the weak energy condition (WEC) and dominant
energy condition (DEC) are satisfied for the constructed model. However, the
strong energy condition (SEC) is violated. There is a reverse gravitational
impact when the (SEC) is violated. The cosmos jerks as a result, and the
current acceleration phase gives way to the previous deceleration phase [110].
As a result, the model proposed in this research appears to be a good fit for
explaining the universe’s late temporal acceleration.

##  6 Conclusion

Here we have discussed spatially homogeneous and anisotropic Kantowski-Sachs
space-time in the presence of bulk viscous fluid with one-dimensional cosmic
strings in $f(T)$ gravity by modifying general relativity to explain the
challenging problem of late time acceleration of the universe. To obtain a
definite solution to the highly non-linear field equations (14)$-$(16) of this
theory, we have taken the help of the special law of variation (23)$-$(25). It
is interesting to observe that,

  * •

The Hubble parameter $(H)$ figure (1(a)) (left panel) diminishes and
ultimately approaches a small value with the passage of cosmic time. The
deceleration parameter $(q)$ figure (1(b)) (right panel) shows phase
transition as its value changes from positive to negative. It shows early
deceleration to late time acceleration, hence for the late time our model
shows $H>0$, $q<0$ implies the model shows the expanding and accelerating
nature of the universe.

  * •

Figure (2) represent the behaviour of proper energy density $(\rho)$ and
particle energy density $(\rho_{p})$. It is observed that both are positive
decreasing functions of time which shows the expansion nature of the universe.

  * •

Eventually, we arrive at an inflationary model since the bulk viscosity
$(\xi)$ diminishes with time (4(a)).

  * •

The negative pressure $(p)$ and $(\bar{p})$ figure (3) corresponds to cosmic
acceleration according to standard cosmology. Therefore, constructed models of
the universe show an accelerated expansion phase at present as well as in
future.

  * •

The string energy density $(\lambda)$ is negative throughout cosmic evolution
figure (6(a)) (left panel) and at the late time it approaches zero this means
at the present epoch we get string free model. When $\lambda<0$, the string
phase of the universe disappears i.e. we have an anisotropic fluid of
particles.

  * •

The EoS parameter $(\omega)$ figure (4(b)) (right panel) resembles the dark
energy model and behaves like the quintessence model $-1\leq\omega<0$ at
present.

  * •

The Energy Conditions figure (6(b)) (right panel) namely WEC, DEC and SEC, we
confirmed that WEC and DEC are valid but violate SEC. The violation of SEC
gives an anti-gravitational effect for which the universe gets jerked and thus
our Universe exhibits a transition from early decelerating to the present
accelerating Universe.

## References

  * Riess _et al._ [(1998] A. G. Riess, A. V. Filippenko, P. Challis, A. Clocchiatti, A. Diercks, P. M. Garnavich, R. L. Gilliland, C. J. Hogan, S. Jha, R. P. Kirshner _et al._ , _The astronomical journal_ , (1998), 116, 1009.
  * Perlmutter _et al._ [(1999] S. Perlmutter, G. Aldering, G. Goldhaber, R. Knop, P. Nugent, P. G. Castro, S. Deustua, S. Fabbro, A. Goobar, D. E. Groom _et al._ , _The Astrophysical Journal_ , (1999), 517, 565.
  * Eisenstein _et al._ [(2005] D. J. Eisenstein, I. Zehavi, D. W. Hogg, R. Scoccimarro, M. R. Blanton, R. C. Nichol, R. Scranton, H.-J. Seo, M. Tegmark, Z. Zheng _et al._ , _The Astrophysical Journal_ , (2005), 633, 560.
  * Percival _et al._ [(2010] W. J. Percival, B. A. Reid, D. J. Eisenstein, N. A. Bahcall, T. Budavari, J. A. Frieman, M. Fukugita, J. E. Gunn, Ž. Ivezić, G. R. Knapp _et al._ , _Monthly Notices of the Royal Astronomical Society_ , (2010), 401, 2148–2168.
  * Koivisto and Mota [(2006] T. Koivisto and D. F. Mota, _Physical Review D_ , (2006), 73, 083502.
  * Daniel _et al._ [(2008] S. F. Daniel, R. R. Caldwell, A. Cooray and A. Melchiorri, _Physical Review D_ , (2008), 77, 103513.
  * Fedeli _et al._ [(2009] C. Fedeli, L. Moscardini and M. Bartelmann, _Astronomy & Astrophysics_, (2009), 500, 667–679.
  * Caldwell and Doran [(2004] R. R. Caldwell and M. Doran, _Physical Review D_ , (2004), 69, 103517.
  * Huang _et al._ [(2006] Z.-Y. Huang, B. Wang, E. Abdalla and R.-K. Su, _Journal of Cosmology and Astroparticle Physics_ , (2006), 2006, 013.
  * Gadbail _et al._ [(2021] G. N. Gadbail, S. Arora and P. Sahoo, _The European Physical Journal C_ , (2021), 81, 1088.
  * Capozziello _et al._ [(2011] S. Capozziello, V. Cardone, H. Farajollahi and A. Ravanpak, _Physical Review D_ , (2011), 84, 043527.
  * Tsujikawa [(2011] S. Tsujikawa, in _Dark Matter and Dark Energy: A Challenge for Modern Cosmology_ , Springer, (2011), pp. 331–402.
  * Yang and Zhang [(2010] R.-J. Yang and S. N. Zhang, _Monthly Notices of the Royal Astronomical Society_ , (2010), 407, 1835–1841.
  * Feng _et al._ [(2005] B. Feng, X. Wang and X. Zhang, _Physics Letters B_ , (2005), 607, 35–41.
  * Copeland _et al._ [(2006] E. J. Copeland, M. Sami and S. Tsujikawa, _International Journal of Modern Physics D_ , (2006), 15, 1753–1935.
  * Carroll [(1998] S. M. Carroll, _Physical Review Letters_ , (1998), 81, 3067.
  * Fujii [(1982] Y. Fujii, _Physical Review D_ , (1982), 26, 2580.
  * Chiba _et al._ [(2000] T. Chiba, T. Okabe and M. Yamaguchi, _Physical Review D_ , (2000), 62, 023511.
  * Armendariz-Picon _et al._ [(2000] C. Armendariz-Picon, V. Mukhanov and P. J. Steinhardt, _Physical Review Letters_ , (2000), 85, 4438.
  * Bento _et al._ [(2002] M. Bento, O. Bertolami and A. A. Sen, _Physical Review D_ , (2002), 66, 043507.
  * Kamenshchik _et al._ [(2001] A. Kamenshchik, U. Moschella and V. Pasquier, _Physics Letters B_ , (2001), 511, 265–268.
  * Solanki _et al._ [(2021] R. Solanki, S. Pacif, A. Parida and P. Sahoo, _Physics of the Dark Universe_ , (2021), 32, 100820.
  * Buchdahl [(1970] H. A. Buchdahl, _Monthly Notices of the Royal Astronomical Society_ , (1970), 150, 1–8.
  * Harko _et al._ [(2011] T. Harko, F. S. Lobo, S. Nojiri and S. D. Odintsov, _Physical Review D_ , (2011), 84, 024020.
  * Shabani and Farhoudi [(2013] H. Shabani and M. Farhoudi, _Phys. Rev. D_ , (2013), 88, 044048.
  * Sahoo _et al._ [(2018] P. Sahoo, P. Moraes and P. Sahoo, _The European Physical Journal C_ , (2018), 78, 1–7.
  * De Felice and Tsujikawa [(2009] A. De Felice and S. Tsujikawa, _Physics Letters B_ , (2009), 675, 1–8.
  * Goheer _et al._ [(2009] N. Goheer, R. Goswami, P. K. Dunsby and K. Ananda, _Physical Review D_ , (2009), 79, 121301.
  * Bamba _et al._ [(2017] K. Bamba, M. Ilyas, M. Bhatti and Z. Yousaf, _General Relativity and Gravitation_ , (2017), 49, 1–17.
  * Elizalde _et al._ [(2010] E. Elizalde, R. Myrzakulov, V. V. Obukhov and D. Sáez-Gómez, _Classical and Quantum Gravity_ , (2010), 27, 095007.
  * Bamba _et al._ [(2010] K. Bamba, S. D. Odintsov, L. Sebastiani and S. Zerbini, _The European Physical Journal C_ , (2010), 67, 295–310.
  * Ferraro and Fiorini [(2007] R. Ferraro and F. Fiorini, _Physical Review D_ , (2007), 75, 084031.
  * Linder [(2010] E. V. Linder, _Physical Review D_ , (2010), 81, 127301.
  * Bamba _et al._ [2011] K. Bamba, C.-Q. Geng, C.-C. Lee and L.-W. Luo, _Journal of Cosmology and Astroparticle Physics_ , 2011, (2011), 021.
  * Bahamonde _et al._ [(2017] S. Bahamonde, S. Capozziello, M. Faizal and R. C. Nunes, _The European Physical Journal C_ , (2017), 77, 1–8.
  * Arora _et al._ [(2021] S. Arora, J. Santos and P. Sahoo, _Physics of the Dark Universe_ , (2021), 31, 100790.
  * Xu _et al._ [(2019] Y. Xu, G. Li, T. Harko and S.-D. Liang, _The European Physical Journal C_ , (2019), 79, 1–19.
  * Solanke _et al._ [(2023] Y. Solanke, A. Kale, D. Pawar and V. Dagwal, _Canadian Journal of Physics_ , (2023), 102, 85–95.
  * Nojiri _et al._ [(2008] S. Nojiri, S. D. Odintsov and P. V. Tretyakov, _Progress of Theoretical Physics Supplement_ , (2008), 172, 81–89.
  * Unzicker and Case [(2005] A. Unzicker and T. Case, _arXiv preprint physics/0503046_ , (2005).
  * Bhoyar _et al._ [(2017] S. Bhoyar, V. Chirde and S. Shekh, _Astrophysics_ , (2017), 60, 259–272.
  * Dagwal and Pawar [(2020] V. Dagwal and D. Pawar, _Modern Physics Letters A_ , (2020), 35, 1950357.
  * Myrzakulov [(2011] R. Myrzakulov, _The European Physical Journal C_ , (2011), 71, 1752.
  * Li _et al._ [(2011] B. Li, T. P. Sotiriou and J. D. Barrow, _Physical Review D_ , (2011), 83, 104017.
  * Rodrigues _et al._ [(2012] M. Rodrigues, M. Houndjo, D. Saez-Gomez and F. Rahaman, _Physical Review D_ , (2012), 86, 104059.
  * Brevik _et al._ [(2012] I. Brevik, R. Myrzakulov, S. Nojiri and S. Odintsov, _Physical Review D_ , (2012), 86, 063007.
  * Barrow and Matzner [(1977] J. D. Barrow and R. A. Matzner, _Monthly Notices of the Royal Astronomical Society_ , (1977), 181, 719–727.
  * Colistete Jr _et al._ [(2007] R. Colistete Jr, J. Fabris, J. Tossa and W. Zimdahl, _Physical Review D_ , (2007), 76, 103516.
  * Capozziello _et al._ [(2006] S. Capozziello, V. F. Cardone, E. Elizalde, S. Nojiri and S. D. Odintsov, _Physical Review D_ , (2006), 73, 043512.
  * Zimdahl _et al._ [(2001] W. Zimdahl, D. J. Schwarz, A. B. Balakin and D. Pavon, _Physical Review D_ , (2001), 64, 063501.
  * Velten and Schwarz [(2012] H. Velten and D. J. Schwarz, _Physical Review D_ , (2012), 86, 083501.
  * Bamba and Odintsov [(2016] K. Bamba and S. D. Odintsov, _The European Physical Journal C_ , (2016), 76, 1–12.
  * Arora _et al._ [(2022] S. Arora, S. Pacif, A. Parida and P. Sahoo, _Journal of High Energy Astrophysics_ , (2022), 33, 1–9.
  * Ren and Meng [(2006] J. Ren and X.-H. Meng, _Physics Letters B_ , (2006), 636, 5–12.
  * Brevik and Normann [(2020] I. Brevik and B. D. Normann, _Symmetry_ , (2020), 12, 1085.
  * Brevik and Grøn [(2013] I. Brevik and Ø. Grøn, _Astrophysics and Space Science_ , (2013), 347, 399–404.
  * Padmanabhan and Chitre [(1987] T. Padmanabhan and S. Chitre, _Physics Letters A_ , (1987), 120, 433–436.
  * Capozziello _et al._ [(2006] S. Capozziello, S. Nojiri and S. Odintsov, _Physics Letters B_ , (2006), 632, 597–604.
  * Wang _et al._ [(2017] D. Wang, Y.-J. Yan and X.-H. Meng, _The European Physical Journal C_ , (2017), 77, 660.
  * Mak and Harko [(2003] M. Mak and T. Harko, _International Journal of Modern Physics D_ , (2003), 12, 925–939.
  * Bali and Pradhan [(2007] R. Bali and A. Pradhan, _Chinese Physics Letters_ , (2007), 24, 585.
  * Kibble [(1976] T. W. Kibble, _Journal of Physics A: Mathematical and General_ , (1976), 9, 1387.
  * Zel’dovich _et al._ [(1974] Y. B. Zel’dovich, I. Y. Kobzarev and L. B. Okun, _Zh. Eksp. Teor. Fiz._ , (1974), 40, 3–11.
  * Kibble [(1980] T. W. Kibble, _Physics Reports_ , (1980), 67, 183–199.
  * Everett [(1981] A. E. Everett, _Physical Review D_ , (1981), 24, 858.
  * Vilenkin [(1981] A. Vilenkin, _Physical Review D_ , (1981), 24, 2082.
  * Letelier [(1979] P. S. Letelier, _Physical Review D_ , (1979), 20, 1294.
  * [68] P. S. Letelier, _Physical review D_ , 28, 2414.
  * Stachel [(1980] J. Stachel, _Physical Review D_ , (1980), 21, 2171.
  * Banerjee _et al._ [(1990] A. Banerjee, A. K. Sanyal and S. Chakraborty, _Pramana_ , (1990), 34, 1–11.
  * Chakraborty [(1991] S. Chakraborty, _Indian Journal of Pure and Applied Physics_ , (1991), 29, 31–33.
  * Ram and Singh [(1995] S. Ram and J. Singh, _General Relativity and Gravitation_ , (1995), 27, 1207–1213.
  * Yilmaz [(2006] İ. Yilmaz, _General Relativity and Gravitation_ , (2006), 38, 1397–1406.
  * Ryan and Shepley [(2015] M. P. Ryan and L. C. Shepley, _Homogeneous relativistic cosmologies_ , Princeton University Press, (2015), vol. 65.
  * Misner [(1968] C. W. Misner, _The Astrophysical Journal_ , (1968), 151, 431.
  * Weber [(1984] E. Weber, _Journal of mathematical physics_ , (1984), 25, 3279–3285.
  * Weber [(1985] E. Weber, _Journal of mathematical physics_ , (1985), 26, 1308–1310.
  * Grøn and Eriksen [(1987] Ø. Grøn and E. Eriksen, _Physics Letters A_ , (1987), 121, 217–220.
  * Collins [(1977] C. Collins, _Journal of Mathematical Physics_ , (1977), 18, 2116–2124.
  * Dhurandhar _et al._ [(1980] S. Dhurandhar, C. Vishveshwara and J. M. Cohen, _Physical Review D_ , (1980), 21, 2794.
  * Banerjee and Ghosh [(1999] A. Banerjee and T. Ghosh, _Classical and Quantum Gravity_ , (1999), 16, 3981.
  * Garcia-Salcedo and Bretón [(2005] R. Garcia-Salcedo and N. Bretón, _Classical and Quantum Gravity_ , (2005), 22, 4783.
  * Bruni _et al._ [(1994] M. Bruni, S. Matarrese and O. Pantano, _arXiv preprint astro-ph/9406068_ , (1994).
  * Szekeres [(1975] P. Szekeres, _Communications in Mathematical Physics_ , (1975), 41, 55–64.
  * Bonnor and Tomimura [(1976] W. B. Bonnor and N. Tomimura, _Monthly Notices of the Royal Astronomical Society_ , (1976), 175, 85–93.
  * Akarsu _et al._ [(2014] Ö. Akarsu, S. Kumar, R. Myrzakulov, M. Sami and L. Xu, _Journal of Cosmology and Astroparticle Physics_ , (2014), 2014, 022.
  * Shekh [(2015] S. B. V. C. S. Shekh, _International Journal_ , (2015), 3, 492–500.
  * Collins _et al._ [(1980] C. Collins, E. Glass and D. Wilkinson, _General Relativity and Gravitation_ , (1980), 12, 805–823.
  * Prasanthi and Aditya [(2020] U. D. Prasanthi and Y. Aditya, _Results in Physics_ , (2020), 17, 103101.
  * Reddy _et al._ [2013] D. Reddy, P. Rao, T. Vidyasagar, R. Bhuvana Vijaya _et al._ , _Advances in High Energy Physics_ , 2013, (2013), .
  * Naidu _et al._ [(2013] R. Naidu, D. Reddy, T. Ramprasad and K. Ramana, _Astrophysics and Space Science_ , (2013), 348, 247–252.
  * Bhoyar _et al._ [(2019] S. Bhoyar, V. Chirde and S. Shekh, _Journal of Scientific Research_ , (2019), 11, .
  * Vidyasagar _et al._ [(2014] T. Vidyasagar, R. Naidu, R. Bhuvana Vijaya and D. Reddy, _The European Physical Journal Plus_ , (2014), 129, 1–7.
  * Mishra and Dua [(2019] R. Mishra and H. Dua, _Astrophysics and Space Science_ , (2019), 364, 195.
  * Yadav [(2012] A. K. Yadav, _Research in Astronomy and Astrophysics_ , (2012), 12, 1467.
  * Bharali and Das [(2023] J. Bharali and K. Das, _Bulgarian Journal of Physics_ , (2023), 50, .
  * Murphy [(1973] G. L. Murphy, _Physical Review D_ , (1973), 8, 4231.
  * Reddy _et al._ [(2013] D. Reddy, P. Rao, T. Vidyasagar, R. Bhuvana Vijaya _et al._ , _Advances in High Energy Physics_ , (2013), 2013, .
  * Pawar and Dabre [(2022] K. Pawar and A. Dabre, _Int. J. Sci. Res. in Physics and Applied Sciences Vol_ , (2022), 10, .
  * Amirhashchi _et al._ [(2011] H. Amirhashchi, H. Zainuddin and A. Pradhan, _International Journal of Theoretical Physics_ , (2011), 50, 2531–2545.
  * Amirhashchi [(2013] H. Amirhashchi, _Pramana_ , (2013), 80, 723–738.
  * Reddy _et al._ [(2013] D. Reddy, R. Naidu, K. Dasu Naidu and T. Ram Prasad, _Astrophysics and Space Science_ , (2013), 346, 261–265.
  * Moreira and Almeida [(2023] A. Moreira and C. Almeida, _General Relativity and Gravitation_ , (2023), 55, 89.
  * Sahoo _et al._ [(2016] P. Sahoo, B. Mishra, P. Sahoo and S. Pacif, _The European Physical Journal Plus_ , (2016), 131, 1–12.
  * Santhi _et al._ [(2022] M. V. Santhi, A. S. Rao, T. Chinnappalanaidu and S. S. Madhu, _Mathematical Statistician and Engineering Applications_ , (2022), 71, 1056–1072.
  * Rodrigues _et al._ [(2015] M. Rodrigues, A. Kpadonou, F. Rahaman, P. Oliveira and M. Houndjo, _Astrophysics and Space Science_ , (2015), 357, 1–7.
  * Wald [(2010] R. M. Wald, _General relativity_ , University of Chicago press, (2010).
  * Ellis [(2014] G. F. Ellis, _The European Physical Journal H_ , (2014), 39, 403–411.
  * Koussour and Bennai [(2022] M. Koussour and M. Bennai, _International Journal of Geometric Methods in Modern Physics_ , (2022), 19, 2250038.
  * Caldwell _et al._ [(2006] R. R. Caldwell, W. Komp, L. Parker and D. A. Vanzella, _Physical Review D_ , (2006), 73, 023513.


# Supplemental Material for:
Negative superinflating bipartite fluctuations near exceptional points in
$\cal PT$-symmetric models

Wei Pan  Beijing Computational Science Research Center, Beijing 100084, China
Xiaoqun Wang<EMAIL_ADDRESS>School of Physics, Zhejiang University,
Hangzhou, 310058, China  Haiqing Lin<EMAIL_ADDRESS>Beijing
Computational Science Research Center, Beijing 100084, China  School of
Physics, Zhejiang University, Hangzhou, 310058, China  Shijie Hu
<EMAIL_ADDRESS>Beijing Computational Science Research Center, Beijing
100084, China  Department of Physics, Beijing Normal University, Beijing,
100875, China

###### Abstract

The Supplemental Material provides comprehensive information on $\cal
PT$-symmetric Su-Schrieffer-Heeger models. Firstly, we briefly derive the
correlation matrix, partial particle number fluctuations, leading-order
bipartite particle number fluctuations, and correlation length. Secondly, we
discuss the intriguing commensurate-to-incommensurate transition at
exceptional points, which may belong to a universality class of non-Hermitian
Pokrovsky-Talapov. Thirdly, we demonstrate a novel spatial profile of the von
Neumann entanglement entropy resulting from the ${\cal P}_{s}{\cal T}_{s}$
symmetry breaking. Finally, we present additional data for the open boundary
condition.

##  I I. Correlation matrix

This section explains how the correlation matrix $\cal C$ is derived. The
quadratic Hamiltonian of the Su-Schrieffer-Heeger (SSH) model presented in the
main text sums the partial Hamiltonian ${\hat{H}}_{k}$ over all momenta $k$ in
the first Brillouin zone (FBZ). Unless otherwise stated, we typically use the
periodic boundary condition (PBC), so $k=2m\pi/N$ and the integer number $m$
ranges from $0$ to $N-1$, where the chain is composed of $N$ unit cells. For
any $k$,

$\displaystyle{\hat{H}}_{k}=\left(\begin{array}[]{cc}{\hat{c}}^{\dagger}_{k,A}&{\hat{c}}^{\dagger}_{k,B}\end{array}\right)\left(\begin{array}[]{cc}iu&-f_{k}\\\
-f^{*}_{k}&-iu\end{array}\right)\left(\begin{array}[]{c}{\hat{c}}_{k,A}\\\
{\hat{c}}_{k,B}\end{array}\right)$ (6)

with the creation and annihilation operators

$\displaystyle{\hat{c}}_{k,\sigma}=\frac{1}{\sqrt{N}}\sum^{N}_{l=1}{\hat{c}}_{l,\sigma}e^{-ikl},\quad{\hat{c}}^{\dagger}_{k,\sigma}=\frac{1}{\sqrt{N}}\sum^{N}_{l=1}{\hat{c}}^{\dagger}_{l,\sigma}e^{ikl},$
(7)

and the structure factor $f_{k}=t_{1}+t_{2}e^{-ik}$ defined as usual.

The model has high ($+$) and low ($-$) energy bands
$\epsilon^{\pm}_{k}=\pm\sqrt{2t_{1}t_{2}\cos k+t^{2}_{1}+t_{2}^{2}-u^{2}}$,
which open a gap at the center momentum $k_{c}=\pi$ in the gapped $\cal
PT$-unbroken region $0\leq u<u_{c1}=t_{1}-t_{2}$. The gap closes at
$u_{c}=u_{c1}$, corresponding to an ordinary critical point (CP) $u_{c}=0$ or
a set of exceptional points (EPs) $0<u_{c}<1$. Two-dimensional vectors $|{\cal
L}^{-}_{k}\rangle=({\cal L}^{-}_{k,A},{\cal L}^{-}_{k,B})^{T}$ and $|{\cal
R}^{-}_{k}\rangle=({\cal R}^{-}_{k,A},{\cal R}^{-}_{k,B})^{T}$ represent the
corresponding left and right eigenvectors in the low energy band
$\epsilon^{-}_{k}$, respectively. Here, ${\cal L}^{-}_{k,A}={\cal
R}^{-*}_{k,B}=f_{k}$ and ${\cal L}^{-}_{k,B}={\cal
R}^{-}_{k,A}=p_{k}=-iu-\epsilon^{-}_{k}$.

In the correlation matrix

$\displaystyle{\cal
C}_{(l,\sigma),(l^{\prime},\sigma^{\prime})}=\frac{1}{N}\sum_{k}e^{ikD}{\cal
Q}^{\sigma\sigma^{\prime}}_{k},$ (8)

with a displacement $D=l^{\prime}-l$ in unit cells, the momentum distribution
function is

$\displaystyle{\cal Q}^{\sigma\sigma^{\prime}}_{k}={\cal
L}^{-*}_{k,\sigma}{\cal R}^{-}_{k,\sigma^{\prime}}/({\cal L}^{-*}_{k,A}{\cal
R}^{-}_{k,A}+{\cal L}^{-*}_{k,B}{\cal R}^{-}_{k,B}).$ (9)

Particularly,

$\displaystyle\begin{split}{\cal
Q}^{AA}_{k}&=\frac{1}{2}+i\frac{u}{2\epsilon^{-}_{k}},\quad{\cal
Q}^{AB}_{k}=-\frac{f^{*}_{k}}{2\epsilon^{-}_{k}},\\\ {\cal
Q}^{BB}_{k}&=\frac{1}{2}-i\frac{u}{2\epsilon^{-}_{k}},\quad{\cal
Q}^{BA}_{k}=-\frac{f_{k}}{2\epsilon^{-}_{k}}.\end{split}$ (10)

For $k=k_{c}+\delta k$ very close to $k_{c}$ at $u_{c}$, we get
$f_{k}=t_{1}-t_{2}e^{-i\delta k}\approx u_{c}+it_{2}\delta k$, $p_{k}\approx-
iu_{c}+v|\delta k|$ and a wave velocity $v=\sqrt{t_{1}t_{2}}$, up to the first
order of $\delta k$.

Firstly, the momentum distribution functions for $AA$ and $BB$ become

$\displaystyle{\cal Q}^{AA}_{k}=\frac{1}{2}+i{\cal Q}^{(nH)}_{k},\quad{\cal
Q}^{BB}_{k}=\frac{1}{2}-i{\cal Q}^{(nH)}_{k},$ (11)

with a function ${\cal Q}^{(nH)}_{k}=\pi\beta/|\delta k|$ and
$\beta=u_{c}/(2\pi v)=(\sqrt{t_{2}/t_{1}}-\sqrt{t_{1}/t_{2}})/2\pi$. ${\cal
Q}^{(nH)}_{k}$ becomes singular at $\delta k=0$. The function of $1/|\delta
k|$ originates from the loss of the biorthogonalization relation between the
left and right eigenvectors at EPs $u_{c}>0$ and $\beta\neq 0$, also called
the defectiveness of the Hilbert space in the main text. Meanwhile, we employ
a trick of removing the ill-defined contributions at or near $k_{c}$ in the
physical sum in Eq. (8). In Sec. V, we also examine the negligible effects of
the trick in the gapless $\cal PT$-broken region.

Secondly, the term for $AB$ is

$\displaystyle\begin{split}{\cal Q}^{AB}_{k}&=\frac{1}{2v|\delta
k|}\left[u_{c}+t_{2}e^{i\delta k/2}(e^{-i\delta k/2}-e^{i\delta
k/2})\right]\\\ &\approx\frac{1}{2v|\delta k|}\left[u_{c}-it_{2}\delta
ke^{i\delta k/2}\right]=-{\cal Q}^{(nH)}_{k}+{\cal
Q}^{(H)}_{k},\quad\end{split}$ (12)

and similarly ${\cal Q}^{BA}_{k}\approx-{\cal Q}^{(nH)}_{k}+{\cal
Q}^{(H)\dagger}_{k}$, where we define a step function ${\cal
Q}^{(H)}_{k}=-i\pi\alpha e^{i\delta k/2}\text{Sgn}(\delta k)/2$,
$\alpha=t_{2}/(\pi v)=\sqrt{t_{2}/t_{1}}/\pi$ and a conventional sign function
“Sgn”. It is noticed that the non-singular term $\text{Sgn}(\delta k)$ also
exists for massless fermions at the Hermitian CP $u_{c}=0$. Besides, ${\cal
Q}^{AA}_{k}=({\cal Q}^{BB}_{k})^{*}$ and ${\cal Q}^{BA}_{k}=({\cal
Q}^{AB}_{k})^{*}$ obviously due to the $\cal PT$-symmetric eigenvectors
$|{\cal L}^{-}_{k}\rangle$ and $|{\cal R}^{-}_{k}\rangle$ in the gapped $\cal
PT$-unbroken region.

In the continuum limit $1\ll D\ll(N/2)$, we get

$\displaystyle\begin{split}{\cal
C}_{(l,A),(l^{\prime},A)}&=\frac{1}{2\pi}\int_{-\pi}^{\pi}e^{ikD}{\cal
Q}^{AA}_{k}d(\delta k)\\\
&=\frac{1}{2}\delta_{l,l^{\prime}}-i\beta\gamma\int_{\pi/N}^{\pi}\frac{e^{i\delta
kD}d(\delta k)}{|\delta k|}\\\
&\approx\frac{1}{2}\delta_{l,l^{\prime}}-i\beta\gamma\int_{\pi/N}^{\pi}\frac{e^{i\delta
kD}d(\delta k)}{\sqrt{2(1-\cos\delta k)}}\\\
&=\frac{1}{2}\delta_{l,l^{\prime}}-i\gamma{\cal Z}_{l,l^{\prime}}\end{split}$
(13)

with a uniform gauge gradient $\gamma=(-1)^{D}$, a new quasi-long-range
correlation ${\cal Z}_{l,l^{\prime}}=\beta{\cal L}_{D}$ and a logarithmic
function ${\cal L}_{z}=\ln|2z/N|$ rescaled by $N/2$. Here we get rid of the
momentum $\delta k=0$ and use a formula Lee (2022)

$\displaystyle\int_{\pi/N}^{\pi}\frac{e^{izD}}{\sqrt{2(1-\cos
z)}}\text{d}z\approx{\cal L}_{D}.$ (14)

Moreover, the term $\delta_{l,l^{\prime}}/2$ in Eq. (13) gives the fermion
density at a site, consistent with the prior half-filling. Likewise, ${\cal
C}_{(l,B),(l^{\prime},B)}\approx\delta_{l,l^{\prime}}/2+i\gamma{\cal
Z}_{l,l^{\prime}}$.

The correlation matrix for $AB$ is obtained by integrating all momenta in FBZ
and takes the form

$\displaystyle\begin{split}&\quad\ {\cal
C}_{(l,A),(l^{\prime},B)}=\frac{1}{2\pi}\int_{-\pi}^{\pi}e^{ikD}{\cal
Q}^{AB}_{k}d(\delta k)\\\ &=\gamma{\cal
Z}_{l,l^{\prime}}+\frac{1}{2\pi}\int_{-\pi}^{\pi}e^{ikD}{\cal
Q}^{(H)}_{k}d(\delta k)\\\ &=\gamma\left({\cal
Z}_{l,l^{\prime}}+\frac{\alpha}{2}\int_{0}^{\pi}\sin[\delta k(2D+1)/2]d(\delta
k)\right)\\\ &=\gamma\left({\cal Z}_{l,l^{\prime}}+{\cal
W}_{(l,A),(l^{\prime},B)}\right).\end{split}$ (15)

We quickly discover that ${\cal
W}_{(l,\sigma),(l^{\prime},\sigma^{\prime})}=\alpha{\cal H}_{x}$ is
proportional to a harmonic function ${\cal H}_{z}=1/z$, which is a well-known
correlation for massless fermions. And, the prefactor $\alpha$ is subject to
renormalization by non-Hermitian effects. The displacement (in sites) of the
site-$(l^{\prime},\sigma^{\prime})$ with respect to the site-$(l,\sigma)$ is
given by $x=2D+y(\sigma^{\prime})-y(\sigma)$, depending of $\sigma$ and
$\sigma^{\prime}=A$ or $B$. In particular, $x=2D+1$ for the $AB$-term, while
$x=2D-1$ for the $BA$-term. In a similar vein, ${\cal
C}_{(l,B),(l^{\prime},A)}$ can be expressed as $\gamma\left({\cal
Z}_{l,l^{\prime}}+{\cal W}_{(l,B),(l^{\prime},A)}\right)$. Only a purely
logarithmic term survives as the wave velocity approaches zero at $t_{2}=0$.

##  II II. Partial particle number fluctuations

It is convenient to specify the range of summation prior to conducting
analytical calculations. In the gapped $\cal PT$-unbroken region, the
correlation matrix preserves a symmetry where ${\cal
C}_{(l,\sigma),(l^{\prime},\sigma^{\prime})}={\cal
C}^{*}_{(N-l+1,{\bar{\sigma}}),(N-l^{\prime}+1,{\bar{\sigma}^{\prime}})}$
under PBC, as mentioned in the main text. Therefore, for fixed $l$, the
correlation matrix ${\cal C}_{(l,\sigma),(l^{\prime},\sigma^{\prime})}$ decays
as a function of the distance $D$ and has a zero-value node at
$l^{\prime}=l+N/2$. The unit cells in subsystem-$e$ are naturally regrouped
into two sets. The first set, set-$[s|e]$, includes unit cells from
$l^{\prime}=N/2+1$ to $N/2+l-1$, which are closer to interface-$[s|e]$. The
second set, set-$[e|s]$, comprises the left unit cells in subsystem-$e$ near
interface-$[e|s]$. Since both sets have equal contributions to partial
particle number fluctuations, we only need to consider the contribution from
interface-$[s|e]$. By doubling this value, we can determine the total amount.

As previously discussed, the unit-cell-$l^{\prime}$ of subsystem-$e$
contributes $({\cal Z}_{l,l^{\prime}})^{2}-({\cal
W}_{(l,A),(l^{\prime},B)}+{\cal Z}_{l,l^{\prime}})^{2}$ to ${\cal
F}^{(e)}_{(l,A)}$ for site-$(l,A)$ in subsystem-$s$, which is decomposed into
a purely power-law term $-({\cal W}_{(l,A),(l^{\prime},B)})^{2}$ and a mixed
term $-2{\cal W}_{(l,A),(l^{\prime},B)}{\cal Z}_{l,l^{\prime}}$. Thus, the sum
of the purely power-law terms from set-$[s|e]$ is equal to

$\displaystyle\begin{split}{\cal
P}_{(l,A)}&=-\sum^{N/2+l-1}_{l^{\prime}=N/2+1}({\cal
W}_{(l,A),(l^{\prime},B)})^{2}\\\ &=-\alpha^{2}\sum^{N/2-1}_{D=D_{I}+1}{\cal
H}^{2}_{2D+1}\quad\end{split}$ (16)

with a displacement $D_{I}=N/2-l$ in unit cells with respect to
interface-$[s|e]$.

According to the properties of the Riemann Zeta function $\zeta(x)$, we get

$\displaystyle\sum^{p}_{n=1}{\cal
H}^{2}_{2n+1}=\frac{3}{4}\zeta(2)-1-\int^{\infty}_{0}\frac{ze^{-(2p+1)z}}{e^{2z}-1}dz.$
(17)

Once $p\gg 1$, Eq. (17) approximates to $(3/4)\zeta(2)-1-(1/2){\cal
H}_{2p+1}$. Therefore,

$\displaystyle\begin{split}{\cal P}_{(l,A)}&=\alpha^{2}\sum^{D_{I}}_{D=1}{\cal
H}^{2}_{2D+1}-\alpha^{2}\sum^{N/2-1}_{D=1}{\cal H}^{2}_{2D+1}\\\
&\approx\frac{\alpha^{2}}{2}\left({\cal H}_{N-1}-{\cal
H}_{2D_{I}+1}\right).\end{split}$ (18)

Meanwhile, if $N\gg 1$, ${\cal P}_{(l,A)}\approx-(\alpha^{2}/2){\cal
H}_{2D_{I}+1}$ shown in the main text. For site-$(l,B)$, similarly if $N\gg
1$,

$\displaystyle{\cal P}_{(l,B)}=-\alpha^{2}\sum^{N/2-1}_{D=D_{I}+1}{\cal
H}^{2}_{2D-1}\approx-\frac{\alpha^{2}}{2}{\cal H}_{2D_{I}-1}.$ (19)

Secondly, if $D_{I}\ll N/2$, the mixed term

$\displaystyle\begin{split}{\cal
M}_{(l,A)}&=-2\sum^{N/2+l-1}_{l^{\prime}=N/2+1}{\cal
W}_{(l,A),(l^{\prime},B)}{\cal Z}_{l,l^{\prime}}\\\
&=-2\alpha\beta\sum^{N/2-1}_{D=D_{I}+1}{\cal H}_{2D+1}{\cal L}_{D}\\\
&\approx-\alpha\beta\int^{1-2/N}_{2D_{I}/N}\frac{\ln
z}{z}dz=\frac{\alpha\beta}{2}{\cal L}^{2}_{D_{I}}.\end{split}$ (20)

Combined with ${\cal M}_{(l,B)}=2\sum^{N/2+l-1}_{l^{\prime}=N/2+1}{\cal
W}_{(l,B),(l^{\prime},A)}{\cal Z}_{l,l^{\prime}}$, the sum is given by

$\displaystyle\begin{split}&\quad\ {\cal M}_{(l,A)}\\!+\\!{\cal M}_{(l,B)}\\\
&=2\alpha\beta\sum^{N/2-1}_{D=D_{I}+1}\left({\cal H}_{2D-1}\\!-\\!{\cal
H}_{2D+1}\right){\cal L}_{2D}\\\
&\approx\frac{2\alpha\beta}{N}\int^{(N-2)/N}_{2D_{I}/N}\frac{\ln
z}{z^{2}}dz\approx 2\alpha\beta{\cal H}_{2D_{I}}({\cal
L}_{D_{I}}+1).\quad\end{split}$ (21)

So for unit-cell-$l$,

$\displaystyle\begin{split}&\quad{\cal F}^{(s)}_{l}={\cal
F}^{(s)}_{(l,A)}+{\cal F}^{(s)}_{(l,B)}=-{\cal F}^{(e)}_{(l,A)}-{\cal
F}^{(e)}_{(l,B)}\\\ &\approx(\alpha^{2}/2)({\cal H}_{2D_{I}+1}+{\cal
H}_{2D_{I}-1})-2\alpha\beta{\cal H}_{2D_{I}}({\cal
L}_{D_{I}}+1).\quad\quad\end{split}$ (22)

##  III III. Leading-order bipartite particle number fluctuations

The bipartite particle number fluctuations (BPNFs) at the ordinary CP
$u_{c}=0$ with $\alpha=1/\pi$ and $\beta=0$ are given by

$\displaystyle\begin{split}{\cal
F}^{(s)}&=\alpha^{2}\sum^{N/4}_{D_{I}=1}({\cal H}_{2D_{I}+1}+{\cal
H}_{2D_{I}-1})\\\ &\approx\alpha^{2}\int^{N/4}_{1}\frac{dz}{z}\approx\frac{\ln
N}{\pi^{2}}.\end{split}$ (23)

At EPs $u_{c}=u_{c1}>0$, the leading-order BPNFs arise from the mixed term,
namely

$\displaystyle\begin{split}{\cal
F}^{(s)}&=-4\alpha\beta\sum^{N/4}_{D_{I}=1}{\cal H}_{2D_{I}}{\cal
L}_{D_{I}}\\\ &\approx-2\alpha\beta\int^{1/2}_{2/N}\frac{\ln
z}{z}dz\approx\alpha\beta(\ln N)^{2}.\end{split}$ (24)

##  IV IV. Correlation length

In this section, we derive the analytical form of the correlation length in
the gapped $\cal PT$-unbroken region. For the momentum near $k_{c}$, denoted
as $k=k_{c}+\delta k$ , the high energy band $\epsilon^{+}_{k}$ can be
approximated by a Lorentz form $\sqrt{\Delta^{2}+v^{2}(\delta k)^{2}}$ with a
gap $\Delta=\sqrt{(t_{1}-t_{2})^{2}-u^{2}}$ and a wave velocity
$v=\sqrt{t_{1}t_{2}}$. The correlation matrix has an alternative form as
follows:

$\displaystyle{\cal
C}_{(l,\sigma),(l^{\prime},\sigma^{\prime})}=\sum_{m}\frac{\langle{\cal
L}^{-}|{\hat{c}}_{l,\sigma}^{\dagger}|{\cal R}^{m}\rangle\langle{\cal
L}^{m}|{\hat{c}}_{l^{\prime},\sigma^{\prime}}|{\cal
R}^{-}\rangle}{\langle{\cal L}^{-}|{\cal R}^{-}\rangle}.$ (25)

Here, $|{\cal L}^{m}\rangle$ and $|{\cal R}^{m}\rangle$ refer to the left and
right eigenvectors of an intermediate excited state with one less fermion
regarding the ground state $|{\cal L}^{-}\rangle=\prod_{k\in\text{FBZ}}|{\cal
L}^{-}_{k}\rangle$ and $|{\cal R}^{-}\rangle=\prod_{k\in\text{FBZ}}|{\cal
R}^{-}_{k}\rangle$, respectively.

Due to a finite gap $\Delta$ or equivalently a finite correlation length $\xi$
at $k_{c}$, we suppose that ${\cal
C}_{(l,\sigma),(l^{\prime},\sigma^{\prime})}\sim\exp(-D/\xi)$ for a long
distance, which means that the probability of creating a fermion at unit-
cell-$l$ after removing a fermion at unit-cell-$l^{\prime}$ is an
exponentially-vanishing function of the distance $D$. The above statement
yields a scattering equation in the continuum limit, that is,

$\displaystyle\sqrt{\Delta^{2}+v^{2}(\delta k)^{2}}\exp(-D/\xi)=0$ (26)

with $\delta k=-i\partial/\partial D$.

Solving the above equation gives the reciprocal relation $\xi=v/\Delta$
between the gap $\Delta$ and correlation length $\xi$. To consider higher-
order corrections, the scattering equation becomes

$\displaystyle\sqrt{t_{1}^{2}+t_{2}^{2}-u^{2}+2t_{1}t_{2}\cos(\delta
k)}\exp(-D/\xi)=0.$ (27)

And we obtain a solution, i.e.,

$\displaystyle\frac{1}{\xi}=\text{arccosh}\left(\frac{t^{2}_{1}+t^{2}_{2}-u^{2}}{2t_{1}t_{2}}\right).$
(28)

Subsequently, the effective wave velocity is

$\displaystyle v_{\text{eff}}=\Delta\times\xi.$ (29)

Moreover, we numerically fit the correlation function $\ln|{\cal
C}_{(l,A),(l^{\prime},A)}|$ with higher-order corrections Zauner  _et al._
(2015)

$\displaystyle\frac{-D}{\xi}+b_{1}\ln
D+\ln\left(1+\frac{b_{2}}{D}+\frac{b_{3}}{D^{2}}\right)+b_{4}.\quad\quad$ (30)

In Fig. IVS1, we find that values of the effective wave velocity
$v_{\text{eff}}$ keep consistent with the above-mentioned analytical
prediction in Eq. (29).

Figure IVS1: The effective wave velocity $v_{\text{eff}}$ (
$\mathbin{\vbox{\hbox{\scalebox{0.8}{$\Circle$}}}}$) as a function of $u$ for
different values of $t_{2}$ in the gapped $\cal PT$-unbroken region:
$t_{2}=0.9$ (a), $0.8$ (b), $0.7$ (c), and $0.6$ (d) The correlation length
$\xi$ is obtained from the numerical fitting by Eq. (30). For instance, when
$t_{2}=0.6$ and $u=0.2$, the fitting yields $\xi=2.25448$, $b_{1}=-0.50048$,
$b_{2}=-1.00738$, $b_{3}=0.50088$, and $b_{4}=0.28158$, which is in close
agreement with the analytical prediction $\xi=2.25444$ based on Eq. (29)
(black lines). We choose $t_{1}=1$ and $N=2048$ with PBC.

##  V V. Commensurate-to-incommensurate transition

Figure VS2:  (a) and (b) Correlation matrix $|{\cal
C}_{(l,A),(l^{\prime},A)}|$ (
$\mathbin{\vbox{\hbox{\scalebox{0.8}{$\Circle$}}}}$) as a function of the
distance $D$ for $t_{2}=0.6$. (a) At $u=0.35$ in the gapped $\cal PT$-unbroken
region, $|{\cal C}_{(l,A),(l^{\prime},A)}|$ for $N=512$ decays exponentially
for a long distance. (b) At $u=0.45$ in the gapless $\cal PT$-broken region,
$|{\cal C}_{(l,A),(l^{\prime},A)}|$ behaves like a power-law function of $D$
for $N=65536$. Insets show zoomed-in plots of the real part of $\cal C$ as a
function of $x$, indicating a period of four sites in (a) and a non-integer
period in (b), respectively. Scaling behaviors of the correlation length $\xi$
(c) and the pitch angle discrepancy $\delta q$ (d) for $t_{2}=0.8$ (
$\mathbin{\vbox{\hbox{\scalebox{0.8}{$+$}}}}$) and $0.6$ (
$\mathbin{\vbox{\hbox{\scalebox{0.9}{$\times$}}}}$) with $N=16384$ match the
analytical prediction in Eq. (28) and (31) (solid black lines) very well. (e)
The real (solid lines) and imaginary part (dashed lines) of the correlation
matrix $\cal C$ at $u=0.41$ as a function of $D$ for $N=8192$ (blue), and
$10808$ (orange). We choose $t_{1}=1$ with PBC.

In the case of the Hermitian model with $u=0$, two insulators can be
distinguished by a berry phase $\pi$ detected in the filled low energy band,
as well as a pair of zero-energy Majorana modes at edges Su  _et al._ (1979).
These insulators are connected at an ordinary CP $u_{c}=0$ with a CFT of
massless fermions. According to Eq. (28), the correlation length $\xi$
approximates to $\sqrt{2}t_{1}(t_{1}-t_{2})^{-\nu}$ with $\nu=1$ as
$0<1-t_{2}/t_{1}\ll 1$.

Starting from a Hermitian insulator as $t_{1}>t_{2}$, a gap at $k_{c}$ remains
open in the presence of a tiny imaginary staggered potential $u$. The ground
state retains the $\cal PT$ symmetry, and the correlation matrix $\cal C$
exhibits an oscillation with a commensurate period of two unit cells (or four
lattice sites), as shown in the inset of Fig. VS2(a). As $u$ approaches the EP
at $u_{c}>0$, $1/\xi^{2}\approx(t_{1}-t_{2})(u_{c}-u)/(2t_{1}t_{2})$ or
equivalently $\xi\approx\sqrt{2t_{1}t_{2}/(t_{1}-t_{2})}(u_{c}-u)^{-\nu}$ with
a critical exponent $\nu=1/2$. This result is consistent with the closure of
the gap $\Delta\approx(u_{c}-u)^{\nu}$ near the rank-$2$ EP. The validity of
the above conclusion is also supported by the large-scale numerical data in
Fig. VS2(c).

Once $u>u_{c}$, the presence of filled imaginary energy levels results in the
breaking of $\cal PT$ symmetry. In TDL, the closure of both real and imaginary
gaps occurs at

$\displaystyle
k^{I}_{c}=\arccos\left(\frac{u^{2}-t_{1}^{2}-t_{2}^{2}}{2t_{1}t_{2}}\right),$
(31)

which can be reached through subtly-tuned boundary conditions and the
appropriate number of unit cells. On the one hand, because of the gapless
excitations at $k^{I}_{c}$, the up envelope of $\cal C$ gradually collapses to
a power-law curve in double logarithmic scales. On the other hand, a composite
oscillation comprising a short period of four sites and a longer period
structure suggests an incommensurate pitch angle $q=k^{I}_{c}/2=k_{c}/2+\delta
q$. In TDL, the value of $\delta q$ is approximately given by
$(u-u_{c})^{\nu^{\prime}}$, where another critical exponent
$\nu^{\prime}=1/2$, as depicted in Fig. VS2(d).

In two-dimensional (2D) crystal surfaces, a transition from a gapped
commensurate phase to a gapless incommensurate phase may exhibit the
Pokrovsky-Talapov (PT) universality, characterized by critical exponents
$\nu=\nu^{\prime}=1/2$ on both sides of CP Huse and Fisher (1982). However, in
Hermitian systems, robust evidence of the quantum counterpart of PT
universality is limited due to the rare presence of quantum floating phases
Chepiga and Mila (2021), which are incommensurate, gapless, and disordered.
Interestingly, in our model, EPs show a similar criticality of the PT
universality. The critical exponent $\nu=1/2$ governs the divergence of the
correlation length near EPs in the gapped $\cal PT$-unbroken region. While the
other critical exponent $\nu^{\prime}=1/2$ gives the scaling behavior of the
pitch angle discrepancy $\delta q$ in the gapless $\cal PT$-broken region,
which may be a non-Hermitian floating phase.

To avoid the numerical instability caused by the loss of the
biorthogonalization relation between the left and right eigenvectors, we
remove the contribution from the momentum $k^{I}_{c}$ and those nearby, which
has a biorthogonalization error exceeding a specified criterion, such as
$10^{-2}$ in practice. Despite this adjustment, we found that the revised data
still support consistent conclusions in TDL, similar to artificially adding a
small momentum offset Chang  _et al._ (2020). For instance, we take
$t_{2}=0.6$ and $u=0.41$ with PBC as an example in Fig. VS2(e) and compare the
incommensurate pitch angle for different system sizes $N=8192$ and $10808$.
According to Eq. (31), we have $k^{I}_{c}=0.96300\pi$. For $N=10808$, the mode
with momentum $k=(10410/10808)\pi$ has a biorthogonalization error larger than
the criterion, and we erase its contribution in the sum of Eq. (8). The best
fitting of the correlation function suggests $q=0.48156\pi$. In contrast, for
$N=8192$, we consider all the low-energy levels, and the data fitting gives
$q=0.48148\pi$. Therefore, the relative error $0.0166\%$ caused by the trick
is negligible.

##  VI VI. ${\cal P}_{s}{\cal T}_{s}$ symmetry and the von Neumann
entanglement entropy

Figure VIS3:  (a) The phase diagram of the model (1) in the main text. The
gapped $\cal PT$-broken (blank) and gapless $\cal PT$-unbroken regions are
separated by a critical line $u_{c1}$ (black line). For a finite $N=4$, the
$\cal PT$-unbroken region is further divided into ${\cal P}_{s}{\cal
T}_{s}$-broken (cyan) and ${\cal P}_{s}{\cal T}_{s}$-unbroken (green) regions
by a quasi-critical line $u^{\cal C}_{c1}(N)$ (green dashed line). (a1, a2)
$2$D radar maps with a radius $r=|z|$ and an angle $\phi=\arg(z)$, for
visualizing the spectrum $\\{w^{\cal C}_{q}\\}$ of the correlation matrix
$\cal C$. The set-$0$ $z=w^{\cal C}_{q}$ (
$\scalerel*{\stackinset{c}{-.125pt}{c}{}{\scalebox{0.7}{\color[rgb]{1,1,1}{$\bigstar$}}}{$\bigstar$}}{\bigstar}$
and
$\scalerel*{\stackinset{c}{-.125pt}{c}{}{\scalebox{0.7}{\color[rgb]{1,1,1}{$\bigstar$}}}{$\bigstar$}}{\bigstar}$)
comprises all modes with $\text{Re}(w^{\cal C}_{q})<1/2$. For the set-$1$ (
$\varhexagon$ and  $\varhexagon$) that includes the remaining modes, we assign
$z=w^{\cal C}_{q}-1$. The display area is limited by a high bound $r=1$ (black
outer circle border) and a low bound $r=10^{-2}$ (origin) in the log-$10$
scale. (b1-b4) Von Neumann entanglement entropy $\cal S$ as a function of
$\ln\xi$ for $t_{2}=0.6$ (b1), $t_{2}=4/9$ (b2), $t_{2}=0.2$ (b3) for EPs
$u_{c1}$ and $t_{2}=0.5$ for $u_{c2}$ (b4). $N=512$ (
$\mathbin{\vbox{\hbox{\scalebox{1.0}{$\bullet$}}}}$) and $8192$ (
$\mathbin{\vbox{\hbox{\scalebox{1.0}{$\Circle$}}}}$) are compared. Here,
$t_{1}=1$ is chosen with PBC.

Due to the ${\cal P}_{s}{\cal T}_{s}$ symmetry of the partial correlation
matrix for subsystem-$s$, its spectrum levels are either real numbers or
complex conjugated pairs El-Ganainy  _et al._ (2018). Under PBC, the ${\cal
P}_{s}{\cal T}_{s}$ symmetry breaks for an infinitesimal $u$, resulting in the
emergency of complex conjugate pairs in the radar map Fig. VIS3(a1). However,
the ${\cal P}_{s}{\cal T}_{s}$ symmetry is restored once $u>u^{\cal
C}_{c1}(N)$, characterized by a signal of two fresh real pairs in Fig.
VIS3(a2), despite the stronger non-Hermitian $u$ near EPs. In TDL, the ${\cal
P}_{s}{\cal T}_{s}$ symmetry never survives in the whole $\cal PT$-unbroken
region, i.e., $u^{\cal C}_{c1}(\infty)=u_{c1}$. Consequently, for a fixed $u$
close to EPs, the bipartite von Neumann entanglement entropy (BvNEE) $\cal S$
follows an area law of ${\cal S}\sim(c_{e}/3)\ln\xi$ as $(N/2)\gg\xi$ in Fig.
VIS3(b1-b4).

##  VII VII. Novel spatial profiles of the von Neumann entanglement entropy

Figure VIIS4:  (a-c) Spatial profiles of the von Neumann entanglement entropy
${\cal S}$ ( $\mathbin{\vbox{\hbox{\scalebox{0.8}{$\Circle$}}}}$,
$\mathbin{\vbox{\hbox{\scalebox{0.8}{$\Circle$}}}}$ and
$\mathbin{\vbox{\hbox{\scalebox{0.8}{$\Circle$}}}}$) as a function of the size
$N_{s}$ of subsystem-$s$. (a1-c1) and (a2-c2) $2$D radar maps with a radius
$r=|z|$ and an angle $\phi=\arg(z)$, for visualizing the spectrum $\\{w^{\cal
C}_{q}\\}$ of the correlation matrix $\cal C$. The set-$0$ $z=w^{\cal C}_{q}$
(
$\scalerel*{\stackinset{c}{-.125pt}{c}{}{\scalebox{0.7}{\color[rgb]{1,1,1}{$\bigstar$}}}{$\bigstar$}}{\bigstar}$,
$\scalerel*{\stackinset{c}{-.125pt}{c}{}{\scalebox{0.7}{\color[rgb]{1,1,1}{$\bigstar$}}}{$\bigstar$}}{\bigstar}$
and
$\scalerel*{\stackinset{c}{-.125pt}{c}{}{\scalebox{0.7}{\color[rgb]{1,1,1}{$\bigstar$}}}{$\bigstar$}}{\bigstar}$)
comprises all modes with $\text{Re}(w^{\cal C}_{q})<1/2$. For the set-$1$ (
$\varhexagon$,  $\varhexagon$ and  $\varhexagon$) that includes the remaining
modes, we assign $z=w^{\cal C}_{q}-1$. The display area is limited by a high
bound $1$ (grey outer circle border) and a low bound $10^{-9}$ (origin) in the
log-$10$ scale. We investigate three system size: $N=128$ (a, a1, a2), $256$
(b, b1, b2) and $1024$ (c, c1, c2), with three typical parameters $u=0.19998$
(blue), $0.199995$ (orange) and $0.199999$ (green) while fixing $t_{2}=0.8$.
Moreover, in radar maps, we zoom in on two cases of distinct cutting lengths
$N_{s}=5$ (a1-c1) and $10$ (a2-c2). Here, we choose $t_{1}=1$ with PBC.

In Sec. VI, we demonstrated that the ${\cal P}_{s}{\cal T}_{s}$ symmetry might
break when the subsystem-$s$ size $N_{s}$ is larger than a certain healing
size $N_{h}$. We now estimate the change in the von Neumann entanglement
entropy (vNEE) for a pair of eigenstates in the spectrum $\\{w^{\cal
C}_{q}\\}$ during the ${\cal P}_{s}{\cal T}_{s}$ symmetry breaking process. In
the quadratic model of fermions, like our model (1) in the main text, vNEE
${\cal S}=-\sum_{q}[w^{\cal C}_{q}\ln w^{\cal C}_{q}+(1-w^{\cal
C}_{q})\ln(1-w^{\cal C}_{q})]$ Peschel and Eisler (2009). In the ${\cal
P}_{s}{\cal T}_{s}$ unbroken region $N<N_{h}$, two real eigenvalues $r_{1}$
and $r_{2}$ give ${\cal S}_{12}=-r_{1}\ln r_{1}-r_{2}\ln
r_{2}-(1-r_{1})\ln(1-r_{1})-(1-r_{2})\ln(1-r_{2})$. However, once
$N_{s}>N_{h}$, these eigenvalues become a complex conjugate pair $w^{\cal
C}_{q}=r_{3}\exp(\pm i\phi_{3})$. Assuming that $1-w^{\cal
C}_{q}=r_{4}\exp(\pm i\phi_{4})$ with
$r_{4}=\sqrt{1+r^{2}_{3}-2r_{3}\cos\phi_{3}}$,
$\cos\phi_{4}=(1-r_{3}\cos\phi_{3})/r_{4}$ and
$\sin\phi_{4}=-r_{3}\sin\phi_{3}/r_{4}$, the total contribution is given by

$\displaystyle{\cal S}_{34}$ $\displaystyle=$ $\displaystyle-2(r_{3}\ln
r_{3})\cos\phi_{3}+2r_{3}\phi_{3}\sin\phi_{3}$ (32) $\displaystyle-2(r_{4}\ln
r_{4})\cos\phi_{4}+2r_{4}\phi_{4}\sin\phi_{4}.$

In the vicinity of EP $N_{s}=N_{h}$, the smoothly connecting condition implies
$2r_{3}\cos\phi_{3}=r_{1}+r_{2}$ and $2r_{4}\cos\phi_{4}=2-r_{1}-r_{2}$.
Therefore, we have $-2(r_{3}\ln r_{3})\phi_{3}>-r_{1}\ln r_{1}-r_{2}\ln r_{2}$
and $-2(r_{4}\ln r_{4})\phi_{4}>-(1-r_{1})\ln(1-r_{1})-(1-r_{2})\ln(1-r_{2})$,
which yields the conclusion that ${\cal S}_{12}<{\cal S}_{34}$. Consequently,
vNEE increases when a real pair becomes a complex conjugate one.

Figure VIIS5:  (a) The bipartite particle number fluctuations ${\cal F}^{(s)}$
as a function of $(\ln N)^{2}$ (blue) or $(\ln\xi)^{2}$ (green). (b) The
coefficients $a_{n}$ ( $\times$,  $\times$ and  $\times$) in ${\cal
F}^{(s)}=a_{2}(\ln N)^{2}+a_{1}(\ln N)+a_{0}$ as a function of $u_{c}$. (c)
The bipartite von Neumann entanglement entropy as a function of $\ln N$ (blue)
or $\ln\xi$ (green). (d) The effective central charge $c_{e}$ ( $+$) in the
CFT prediction ${\cal S}\propto(c_{e}/6)\ln N$ as a function of $u_{c}$. In
both (a) and (c), we choose two typical parameters $u_{c}=0.4$ (
$\mathbin{\vbox{\hbox{\scalebox{0.8}{$\blacksquare$}}}}$ and
$\mathbin{\vbox{\hbox{\scalebox{1.1}{$\Square$}}}}$) and $0.7$ (
$\mathbin{\vbox{\hbox{\scalebox{1.4}{$\bullet$}}}}$ and
$\mathbin{\vbox{\hbox{\scalebox{0.8}{$\Circle$}}}}$). Meanwhile, black lines
give the best fittings of ${\cal F}^{(s)}$ and ${\cal S}$, respectively. We
choose $t_{1}=1$ with OBC.

In Fig. VIIS4, we examine the spatial profile of vNEE near EP $u_{c}=0.2$ at
$t_{2}=0.8$. At $u=0.19998$ relatively far away from EP with a distance
$\delta u=(u_{c}-u)=2\times 10^{-5}$, the spectrum of the correlation matrix
$\cal C$ displayed in radar maps exhibits a complex conjugate pair, as
depicted in Fig. VIIS4(a1-c1, a2-c2). The resulting spatial profile of vNEE is
convex, as shown in Fig. VIIS4(a-c). Closer to EP, specifically at
$u=0.199999$ or $\delta u=10^{-6}$ with a larger correlation length, we
observe that the spectrum is purely real for small $N=128$, but a complex
conjugate pair emerges for larger $N=1024$. Accordingly, the spatial profile
of vNEE transforms from a concave to a convex function. For an intermediate
size $N=256$, a novel kink in Fig. VIIS4(b) indicates the appearance of a
complex conjugate pair.

##  VIII VIII. Open boundary condition

For the open boundary condition (OBC) lacking an interface, we observe that
BvNEE and BPNFs still behave as a polynomial function of either $\ln N$ or
$\ln\xi$ in Fig. VIIS5(a, c). Moreover, coefficients $a_{1}$ and $a_{2}$ are
half of the counterparts for PBC, as shown in Fig. VIIS5(b). This is
consistent with the CFT prediction for Hermitian quantum criticality. But, the
curve of $a_{0}$ has an exceptional pace near $u_{c}=0$, lacking a theoretical
explanation yet.

## References

  * Lee (2022) Ching Hua Lee, “Exceptional bound states and negative entanglement entropy,” Physical Review Letters 128, 010402 (2022).
  * Zauner  _et al._ (2015) V Zauner, D Draxler, L Vanderstraeten, M Degroote, J Haegeman, M M Rams, V Stojevic, N Schuch, and F Verstraete, “Transfer matrices and excitations with matrix product states,” New Journal of Physics 17, 053002 (2015).
  * Su  _et al._ (1979) W. P. Su, J. R. Schrieffer, and A. J. Heeger, “Solitons in polyacetylene,” Physical Review Letters 42, 1698 (1979).
  * Huse and Fisher (1982) David A. Huse and Michael E. Fisher, “Domain walls and the melting of commensurate surface phases,” Physical Review Letters 49, 793 (1982).
  * Chepiga and Mila (2021) Natalia Chepiga and Frédéric Mila, “Lifshitz point at commensurate melting of chains of rydberg atoms,” Physical Review Research 3, 023049 (2021).
  * Chang  _et al._ (2020) Po-Yao Chang, Jhih-Shih You, Xueda Wen, and Shinsei Ryu, “Entanglement spectrum and entropy in topological non-hermitian systems and nonunitary conformal field theory,” Physical Review Research 2, 033069 (2020).
  * El-Ganainy  _et al._ (2018) Ramy El-Ganainy, Konstantinos G. Makris, Mercedeh Khajavikhan, Ziad H. Musslimani, Stefan Rotter, and Demetrios N. Christodoulides, “Non-hermitian physics and PT symmetry,” Nature Physics 14, 11 (2018).
  * Peschel and Eisler (2009) Ingo Peschel and Viktor Eisler, “Reduced density matrices and entanglement entropy in free lattice models,” Journal of Physics A: Mathematical and Theoretical 42, 504003 (2009).