# Emergent phase transition in Cluster Ising model with dissipation

Zheng-Xin Guo  These authors contributed equally.  Guangdong-Hong Kong Joint
Laboratory of Quantum Matter, Frontier Research Institute for Physics, South
China Normal University, Guangzhou 510006, China  Wilczek Quantum Center and
Key Laboratory of Artificial Structures and Quantum Control, School of Physics
and Astronomy, Shanghai Jiao Tong University, Shanghai 200240, China  Xue-Jia
Yu  These authors contributed equally.  International Center for Quantum
Materials, School of Physics, Peking University, Beijing 100871, China  Xi-Dan
Hu  Guangdong Provincial Key Laboratory of Quantum Engineering and Quantum
Materials, SPTE, South China Normal University, Guangzhou 510006, China  GPETR
Center for Quantum Precision Measurement, South China Normal University,
Guangzhou 510006, China  Zhi Li<EMAIL_ADDRESS>Guangdong-Hong Kong
Joint Laboratory of Quantum Matter, Frontier Research Institute for Physics,
South China Normal University, Guangzhou 510006, China

###### Abstract

We study a cluster Ising model with non-Hermitian external field which can be
exactly solved in the language of free fermions. By investigating the second
derivative of energy density and fidelity, the possible new critical points
are tentatively located. String order parameter and staggered magnetization
are then detected to reveal emergent phases of brand new characteristics. To
categorize the exotic phases and phase transitions induced by non-Hermiticity,
we calculate the variation mode of spin correlation function as well as string
correlation function, which characterize the emergent phases and critical
points with different patterns of decay and critical exponents. With the help
of string order parameter and staggered magnetization, we find that there are
four phases after introducing the non-Hermiticity—the cluster phase, the
gapless phase, the paramagnetic (PM) phase and the antiferromagnetic (AF)
phase. A phase diagram is then presented to graphically illustrate, based on
two ‘KT-like’ phase transitions and an Ising phase transition, respectively,
the generation of three critical lines as non-Hermitian strength increases.
Our theoretical work is expected to be realized in the experiment of ultra-
cold atoms, pushing for progress in exploring novel phases and phase
transitions.

###### pacs:

03.65.Ud, 05.30.Rt, 03.67.-a, 42.50.-p

##  I Introduction

According to conventional Landau-Ginzberg-Wilson paradigm LDLandau1980 , an
ordered phase is characterized by certain symmetry breaking of the system,
whereas a disordered phase features the preserve of symmetry. The phase
transition between ordered phases and disordered phases can be detected by
certain local parameters. However, Kosterlitz and Thouless discovered a
continuous phase transition without symmetry breaking in systems involving
classical vortex topology, which is then named as Kosterlitz-Thouless (KT)
phase transition JKosterlitz1973 . The key characteristic of KT phase
transition is that it occurs between a disordered phase and a gapless phase
with quasi-long-range order which can be detected by the power-law decay of
correlation function.

With the rapid development of quantum simulation and computation, the marriage
of traditional condensed matter physics and cutting-edge experimental
techniques gives birth to various new topics. In highly pure and controllable
ultra-cold atom platforms, a triangle optical lattice can be set up with atoms
loaded in a unique way CBecker2010 , so as to create an equivalent three-spin
ring-exchange interaction in spin system that can be mapped to a “zig-zag
chain”. Notably, the ground state of the system is cluster state JKPachos2004
, a disordered state whose spin spacial rotational symmetry is protected.
However, research papers also have it that except for three-spin ring-
interaction, two-spin interaction relating to symmetry breaking state is also
observed in such a system, causing competition between the two kinds of
interaction. From the view of quantum information, researchers regard the two-
spin interaction as a perturbation for it will damage the symmetry protected
topological (SPT) cluster state, and they are curious about SPT’s threshold of
robustness ACDoherty2009 ; SOSkrovseth2009 ; YCLi2011 ; XGWen2017 ;
FPollmann2012 ; ZCGu2019 . From the perspective of condensed matter physics,
the exotic continuous quantum phase transitions (QPT) between SPT phases and
symmetry breaking phases are fascinating for their own sake SSachdev1999 ;
PSmacchia2011 ; CXDing2019 . Therefore, cluster Ising model is put forward as
a good toy model to investigate the phase transitions in such quantum many-
body systems. Generally, SPT cluster phase is protected by a
$\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ symmetry WSon2011  and can be
characterized by non-local string order MPopp2005 ; LCampos2005  while the
symmetry breaking antiferromagnetic (AFM) state can be identified by local
staggered magnetization CCChiang2019 ; VGrigorev2021 .

The above discussion is based on traditional quantum mechanics, which requires
the Hermiticity of observables to ensure that their eigenvalues are real
numbers. However, non-Hermitian physics has currently attracted extensive
research interest as the non-Hermitian experimental techniques grow mature in
a wide range of table-top experimental platforms ZGong2018 ; KKawabata2019 ,
inclusive of ultra-cold atom system JLi2019 ; LLi2020 , optical system
WChen2017 ; BPeng2014 ; ACerjan2019 ; BZhen2015 , nitrogen-vacancy center
YWu2019 , etc. Many novel non-Hermitian phenomena have been detected, such as
parity-time ($\mathcal{PT}$) symmetry CMBender1998 ; CMBender2002 ;
CMBender2007 , non-Hermitian skin effect SYao2018 ; SYao20182 , new
topological behaviors associated with exceptional points DWZhang2020 ; PHe2020
; HShen2018 ; EJBergholtz2021 ; AGhatak2019 ; KKawabata2019 ; YHe20202 ,
disorder induced by non-Hermiticity DWZhang20202 ; XWLuo2020 ; HJiang2019 ,
etc. Recently, much attention has been paid on how non-Hermiticity influences
quantum phase transitions in systems with rotation-time-reversal
($\mathcal{RT}$) symmetry TELee2014 ; XZZhang2013 ; CWang2020 ; CLi2014 ;
KLZhang2020 , spin model with imaginary external field SB2021 ; YNishiyama2020
; YNishiyama20202 ; YGLiu2021 ; NShibata2019  and so on. It is also worth
mentioning that some researchers have developed a set of non-Hermitian linear
response theory, which uses dissipation as a means to reveal the properties of
Hermitian equilibrium system LPan2020 .

In this work, we try to uncover the connection between novel emergent phases
and non-Hermiticity without $\mathcal{PT}$ or $\mathcal{RT}$ symmetry. The
main body of this paper is organized as follows. In Sec. II, we introduce
cluster Ising model with on-site dissipations and analytically transform it
into free fermion expression. In Sec. III, the observables and the methods we
adopted to characterize quantum phase transitions are presented. In Sec. IV,
we give the theoretical results of observables and discuss the exotic phase
diagram and phase transitions. We summarize our work in Sec. V.

##  II Model and Exact solution

In this section, we establish the non-Hermitian cluster Ising model and
conduct the diagonalization procedure to obtain exact solution of the ground
state. Based on the conventional cluster Ising model, we build up our
Hamiltonian by inserting dissipation, which is equivalent to an external
imaginary field. The expression reads

$H=-J\sum_{l=1}^{N}\sigma_{l-1}^{x}\sigma_{l}^{z}\sigma_{l+1}^{x}+\lambda\sum_{l=1}^{N}\sigma_{l}^{y}\sigma_{l+1}^{y}+\frac{i\Gamma}{2}\sum_{l=1}^{N}\sigma_{l}^{u},$
(1)

where $\sigma^{x}_{l},\sigma^{y}_{l}$ and $\sigma^{z}_{l}$ are pauli matrices
of the $l^{th}$ spin and $\sigma^{u}$ denotes the matrix
$\begin{bmatrix}1&0\\\ 0&0\end{bmatrix}$ corresponding to the loss or gain,
which is a simple way to involve non-Hermiticity in optical or atomic
experiments BPeng2014 ; XWLuo2020 ; Muller2012 ; JMZeuner2015 . $N$ is large
enough for us to view $N/2$ as a ‘decent half’ regardless of the parity of
$N$. There are three control parameters $\it J$, $\lambda$ and $\Gamma$ in our
model, where the former two indicate the competition between SPT phase and
symmetry breaking phase while the latter determines the strength of complex
field (see Fig. 1). Notably, in this work, we set $J=1$ and take it as energy
unit in the following calculations.

Figure 1: (Color online). Graphic demonstration of cluster Ising model with
dissipation, where $\lambda/J$ is the ratio of Ising exchange strength to
cluster exchange strength and $\it\Gamma$ is the strength of loss or gain. We
take $J$=1 in the following discussion.

The diagonalization process can be divided into three steps. First of all, we
rewrite the Hamiltonian in free fermionic language using the standard Jordan-
Wigner transformation, which is defined as

$\sigma_{l}^{z}=1-2c_{l}^{\dagger}c_{l},$ (2)
$\sigma_{l}^{+}=\prod_{j<l}(1-2c_{j}^{\dagger}c_{j})c_{l},$ (3)

where $c_{l}^{\dagger}$ and $c_{l}$ are the creation and annihilation
operators at site $l$, respectively. Then we can obtain the Hamiltonian in
spinless fermion expression as

$\displaystyle H=$
$\displaystyle\sum_{l}^{N}\left[-\left(c_{l}^{\dagger}-c_{l}\right)\left(c_{l+2}^{\dagger}+c_{l+2}\right)\right.$
(4)
$\displaystyle\left.+\lambda\left(c_{l}^{\dagger}+c_{l}\right)\left(c_{l+1}^{\dagger}-c_{l+1}\right)\right]$
$\displaystyle-\frac{i\Gamma}{2}\sum_{l}^{N}\left(1-c_{l}^{\dagger}c_{l}\right).$

Secondly, a Fourier transformation
$c_{l}=\frac{1}{\sqrt{N}}\sum_{k=-\pi/2}^{\pi/2}$
$e^{2\pi ikl/N}b_{k}$ is conducted and we have

$\displaystyle H=$ $\displaystyle
2\sum_{k}\left[iy_{k}\left(c_{k}^{\dagger}c_{-k}^{\dagger}+c_{k}c_{-k}\right)\right.$
(5)
$\displaystyle\left.+z_{k}\left(c_{k}^{\dagger}c_{k}+c_{-k}^{\dagger}c_{-k}-1\right)\right].$

Here, $y_{k}=-\rm{sin}(2k)-\lambda\rm{sin}(k)$ and $z_{k}=-\rm{cos}(2\it
k)+\lambda\rm{cos}(\it k)+\frac{i\Gamma}{4}$. Thirdly, a Bogoliubov
transformation helps diagonalize the above equation, which reads

$b_{k}=\cos(\frac{\theta_{k}}{2})\gamma_{k}+i\sin(\frac{\theta_{k}}{2})\gamma_{-k}^{\dagger},$
(6)

$b_{-k}=\cos(\frac{\theta_{k}}{2})\gamma_{-k}-i\sin(\frac{\theta_{k}}{2})\gamma_{k}^{\dagger}.$
(7)

Eventually, a diagonalized solution is acquired as

$H=\sum_{k>0}\Lambda_{k}(\gamma_{k}^{\dagger}\gamma_{k}-\frac{1}{2}),$ (8)

where

$\Lambda_{k}=\sqrt{y_{k}^{2}+z_{k}^{2}},$ (9)
${\rm{tan}}(\theta_{k})=-\frac{y_{k}}{z_{k}},$ (10)

and the ground state of the model is

$\mathinner{|{G}\rangle}=\prod_{k>0}[{\rm{cos}}(\frac{\theta_{k}}{2})+i{\rm{sin}}(\frac{\theta_{k}}{2})c^{\dagger}_{k}c^{\dagger}_{-k}]\mathinner{|{{\rm{Vac}}}\rangle},$
(11)

where $\mathinner{|{{\rm{Vac}}}\rangle}$ denotes the vacuum state of the free
fermion.

##  III Observables and methods

###  III.1 Ground state energy density and its second derivative

The non-analyticity in the second derivative of ground state energy density
implies that continuous QPT occurs at zero temperature. According to the above
solution, the ground state energy can be calculated analytically and
numerically via the equation

$U_{g}=-\frac{2}{N}\sum_{k>0}\sqrt{y_{k}^{2}+z_{k}^{2}}=-\frac{1}{\pi}\int_{0}^{\pi}\sqrt{y_{k}^{2}+z_{k}^{2}}dk,$
(12)

and we can easily obtain the second derivative of $U_{g}$ with respect to
$\lambda$, i.e., $\frac{\partial^{2}U_{g}}{\partial\lambda^{2}}$.

Figure 2: (Color online). The second derivative of ground state energy density
with respect to $\it\lambda$. (a) When $\it\Gamma=$0, the singularity occurs
at $\it\lambda=$1, which is in good agreement with the critical point of
standard clusing-Ising model. (b) When $\it\Gamma=$0.6, the non-analytical
point in Hermitian case splits into 3 points, indicating new potential
critical points. (c) When $\it\Gamma=$1.6, distance between the three non-
analytical points increases with the growing non-Hermitian strength.

###  III.2 Fidelity

As the inner product of two wave functions with a tiny difference in
parameters, fidelity is also efficient in indicating the critical points of
QPT, whose expression reads

$F(\lambda,\lambda+\epsilon)=\langle\mathrm{G}(\lambda)\mid\mathrm{G}(\lambda+\epsilon)\rangle=\prod_{k>0}F_{k},$
(13)

with

$F_{k}=\cos\frac{\theta_{k}(\lambda)-\theta_{k}(\lambda+\epsilon)}{2}.$ (14)

It is noticeable that $\epsilon$ should take a small value. However, some
arbitrariness in choosing the value of $\epsilon$ can be allowed within a
reasonable range, and the selection rule is just as what we have discussed
before CXDing2019 . Here, we set $\epsilon=10^{-5}$, which can ensure the
stability of results under different parameters.

###  III.3 Order parameters

Two kinds of order parameters will be investigated in this section, i.e., non-
local string order parameter characterizing disordered phase, and local
staggered magnetization characterizing AFM phase. These order parameters will
be non-zero as long as the system is in the corresponding phase.

Figure 3: (Color online). The distribution of $1-F$ (fidelity) versus
$\lambda$. (a) When $\it\Gamma=$0, the singular point well indicates the
critical point of conventional SPT-AFM phase transition. (b) When
$\it\Gamma=$0.6, the fidelity also characterizes possible new phase transition
points. (c) As $\it\Gamma$ ascends to 1.6, the behavior of singularity is the
same as that of the second derivative of energy density.

Let us begin with the string order parameter, which is defined as

$\mathcal{O}^{x}=\lim_{N\rightarrow\infty}(-1)^{N}\left\langle\sigma_{1}^{x}\sigma_{2}^{y}\left(\prod_{k=3}^{N-2}\sigma_{k}^{z}\right)\sigma_{N-1}^{y}\sigma_{N}^{x}\right\rangle_{0}.$
(15)

Using the technique in Ref. GCWick1950 , we can express it by the product of
$A_{j}=c_{j}^{\dagger}+c_{j}$ and $B_{j}=c_{j}-c_{j}^{\dagger}$, i.e.,

$\mathcal{O}^{x}=\lim_{r\rightarrow\infty}\left\langle B_{2}A_{3}B_{3}\ldots
A_{r}B_{r}A_{r+1}B_{r+1}A_{r+2}\right\rangle.$ (16)

Then, with the help of Wick theorem GCWick1950 , we can go on expanding it by
the contractions $\langle A_{j}A_{l}\rangle$, $\langle B_{j}B_{l}\rangle$ and
$\langle B_{j}A_{l}\rangle$, whose expression can be acquired using the ground
state function. We have

$\langle A_{j}A_{l}\rangle=\delta_{jl},$ (17)

$\langle B_{j}B_{l}\rangle=-\delta_{jl},$ (18)

$\displaystyle\langle B_{j}A_{l}\rangle$ $\displaystyle=G_{j,l}=G_{r}$ (19)
$\displaystyle=\frac{1}{\pi}\int_{0}^{\pi}dk[\cos(kr)\cos\theta_{k}+\sin(kr)\sin\theta_{k}],$

where $r=j-l$. Since $\langle A_{j}A_{l}\rangle$ and $\langle
B_{j}B_{l}\rangle$ are always equal to zero and by considering the
characteristics of their expression, $\mathcal{O}^{x}$ can be transformed to a
Toeplitz determinant,

$\mathcal{O}^{x}=\lim_{r\rightarrow\infty}\left|\begin{array}[]{cccc}G_{-2}&G_{-3}&\cdots&G_{-r-1}\\\
G_{-1}&G_{-2}&\cdots&G_{-r}\\\ G_{0}&G_{-1}&\cdots&G_{-r+1}\\\
\vdots&\vdots&\ddots&\vdots\\\ G_{r-2}&G_{r-5}&\cdots&G_{-3}\\\
G_{r-3}&G_{r-4}&\cdots&G_{-2}\end{array}\right|.$ (20)

From the perspective of numerical calculation, we can take finite $r$ and
calculate finite number of integrals to obtain $\mathcal{O}^{x}$. Note that,
$r$ should be as large as possible in order to approach the thermodynamic
limit.

Next, we can acquire staggered magnetization at temperature $T$ via the
calculation of spin correlation function

$R_{jl}^{\alpha}(T)=\left\langle\sigma_{j}^{\alpha}\sigma_{l}^{\alpha}\right\rangle_{T},$
(21)

where $\alpha$ can be $x$, $y$ or $z$. Let us take $R_{jl}^{x}(T)$ as an
example. Similar to the precedure of calculating string correlation function,
$R_{jl}^{x}(T)$ can also be expanded by
$A_{j}=c_{j}^{\dagger}+c_{j},B_{j}=c_{j}-c_{j}^{\dagger}$, that is,

$\displaystyle R_{jl}^{x}(T)$
$\displaystyle=\left\langle\left(c_{j}-c_{j}^{\dagger}\right)\prod_{j<m<l}\left(1-2c_{m}c_{m}^{\dagger}\right)\left(c_{l}^{\dagger}+c_{l}\right)\right\rangle_{T}$
(22) $\displaystyle=\left\langle B_{j}A_{j+1}B_{j+1}\ldots
A_{l-1}B_{l-1}A_{l}\right\rangle_{T}.$

Figure 4: (Color online). The numerical results of string correlation function
$\it O_{x}$ and staggered magnetization $\it m_{y}$ for (a) Hermitian case
with $\it\Gamma$=0 and (b) non-Hermition case with $\it\Gamma$=2.0. The insets
of (b) are close-ups of the continuous change region around the two phase
transition points.

By using Wick theorem again we can also convert $R_{jl}^{x}(T)$ into a
Toeplitz determinant

$R_{r}^{x}(T)=\left|\begin{array}[]{cccc}D(-1,T)&D(-2,T)&\cdots&D(-r,T)\\\
D(0,T)&D(-1,T)&\cdots&D(-r+1,T)\\\ \vdots&\vdots&\ddots&\vdots\\\
D(r-2,T)&D(r-3,T)&\cdots&D(-1,T)\end{array}\right|,$ (23)

where the elements of the determinant reads

$\left\langle B_{j}A_{l}\right\rangle_{T}=D_{jl}(T)=D(j-l,T)=D(r,T).$ (24)

Similarly, $R_{r}^{y}(T)$ can be expressed by

$R_{r}^{y}(T)=\left|\begin{array}[]{cccc}D(1,T)&D(0,T)&\cdots&D(-r+2,T)\\\
D(2,T)&D(1,T)&\cdots&D(-r+3,T)\\\ \vdots&\vdots&\ddots&\vdots\\\
D(r,T)&D(r-1,T)&\cdots&D(1,T)\end{array}\right|.$ (25)

With $R_{r}^{y}$, we can calculate staggered magnetization $m_{y}$ with the
definition

$\lim_{r\rightarrow\infty}(-1)^{r}R_{r}^{y}(0)=m_{y}^{2}.$ (26)

Overall, with the above expression, we can investigate the influence of non-
Hermiticity on the CIM’s phase distribution at certain temperature.

Figure 5: (Color online). The variation modes of string parameter $|O_{x}(r)|$
and correlation function $|R_{y}(r)|$ at different $\lambda$ for (a) and (c)
Hermitian case with $\it\Gamma$=0 as well as (b) and (d) non-Hermitian case
with $\it\Gamma$=2. When $\it\Gamma$=0, the insets plot the curves at critical
point $\lambda=$1 in ln-ln coordinates and show the slope $K$ of the lines. As
$\it\Gamma$ increases to 2, the three critical points are at 0.551, 0.956 and
1.424. The insets also show the curves featuring power-law decay in ln-ln
coordinates (with subscript $e$).

###  III.4 Variation mode of correlation function

In phase transition theory, the spin correlation function $R_{y}(r)$ decreases
to a fixed non-zero value as $r$ increases in symmetry breaking phase, in
contrast with the exponential decay to zero in disordered phase. At the
critical point between these two phases, $R_{y}(r)$ exhibits a unique power-
law decay. Similarly, the variation mode of string order parameter $O_{x}(r)$
also carry the key information of the phases and phase transitions. $O_{x}(r)$
tends to be a constant in non-trivial cluster phase, rather than the
exponential decay in trivial phases. Moreover, it is worth mentioning that if
one plot the curve of power-law decay in the ln-ln coordinates, it will become
a straight line and the slope of the line will directly reveal one of the
critical exponents $\eta$.

##  IV Results and Discussions

In this section, we are going to illustrate the influence of non-Hermiticity
on different observables. Let us start with the second derivative of ground
state energy density. In Hermitian case, as it is shown in Fig. 2 (a), the
singularity emerges at $\lambda=1$, which corresponds to the critical point of
standard cluster Ising model. However, when non-Hermitian strength increases
[see Fig. 2 (b) and (c)], the singular point turns into three points, which
drift apart from one another with an increasing $\it\Gamma$. The emergence of
new non-analytical points indicates possible new phase transitions that
remains unknown. Notably, the continuous first derivative of $U_{g}$ is also
shown in Appendix A to prove that all these three phase transitions are
continuous phase transitions.

We also demonstrate the fidelity whose singularity characterizes phase
transition as well. As is shown in Fig. 3 (a), in Hermitian case, the singular
point also works well in characterizing the SPT-AFM phase transition at
$\lambda=1$. Then, we turn to the non-Hermitian cases as $\it\Gamma$ increases
to $0.6$ and $1.6$ [see Fig. 3 (b) and (c)]. The behavior of singularity is in
good agreement with that of $\partial^{2}U_{g}/\partial\lambda^{2}$. We can
observe two more emerging singular points and witness them moving away from
each other as non-Hermiticity increases, which denotes the emergence of the
unknown phase transition. The results of fidelity and the second derivative of
ground state energy nicely support each other.

When one investigate Hermitian cluster Ising model, it is a standard procedure
to calculate the string order parameter $\it O_{x}$ and staggered
magnetization $\it m_{y}$. One can directly recognize the domination of
disordered (ordered) phase with the help of non-local (local) order parameter
string order parameter (staggered magnetization) provided it is non-zero. We
also investigate the distribution of these two order parameters under
different non-Hermitian strengths. In Hermitian case [see Fig. 4 (a)], the
behaviors of $\it O_{x}$ and $\it m_{y}$ are the same as the previous
researches, where $\it O_{x}$ ($\it m_{y}$) is non-zero when $\lambda<1$
($\lambda>1$) and its decline is continuous. However, when we increase the
non-Hermitian strength $\it\Gamma$ from 0 to 2, the distribution of order
parameters changes greatly. At first, on the one hand, $\it O_{x}$ shows an
intense but continuous decline after the kink at the first critical point,
while on the other hand, the parameter $\it m_{y}$ illustrates a sharp but
continuous increase before the kink at the third critical point. The insets of
Fig. 4 (b) show the close-ups of continuous variation around the two kinks at
critical points. Note that, although the kinks at continuous critical points
are not to be found in Hermitian case, recent studies have proved that this
kind of kinks is actually a real phenomenon in the non-Hermition case.
TELee2014 ; BBWei2017 . Thus, after involving dissipation strength, one can
witness that the original two phase zones are divided into four. Then, one can
see that in zone I and II, $\it O_{x}$ is non-zero and $\it m_{y}$ is zero. In
zone IV, $\it m_{y}$ is non-zero and $\it O_{x}$ is zero. However, in zone
III, both $\it O_{x}$ and $\it m_{y}$ are zero, which is beyond the way of
phase classification in Hermitian case. Besides, it is noticeable that this is
a numerical result where $\it r$ in Eq. (20) and Eq. (26) is taken as $1000$
instead of infinite. Thus, tiny numerical error is involved, leading to some
continuous transition areas between different zones. In principle, there will
be some non-analytical slump as long as $\it r$ is large enough to approach
the thermodynamics limit, the position of which is also consistent with that
of singular points as we calculated above.

From the above discussions, it is obvious that the introduction of non-
Hermitian term leads to new phases and phase transitions that transcend the
framework of Hermitian case. After that, we want to classify the emergent
phases and phase transitions with the help of the variation modes of spin
correlation function $R_{y}(r)$ and string parameter $O_{x}(r)$. We
investigate the variation of $O_{x}(r)$ and $R_{y}(r)$ at different $\lambda$
and under different non-Hermitian strengths $\it\Gamma$. When $\it\Gamma$ = 0,
the critical point is at $\lambda$ = 1, which denotes the cluster-AFM phase
transition with symmetry breaking. From Fig. 5 (a), we can see that $O_{x}(r)$
remains a constant value before $\lambda=$ 1 and shows exponential decay to
zero after $\lambda=$ 1, indicating non-trivial cluster phase and trivial AFM
phase, respectively. Meanwhile, from Fig. 5 (c), we can see that the curves of
$R_{y}(r)$ before $\lambda=$ 1 exhibit the exponential decay and those after
$\lambda=$ 1 decrease to fixed non-zero values, denoting symmetry protected
cluster phase and symmetry breaking AFM phase, respectively. Notably, as is
shown in the insets of the two graphs, the curves at the critical point show
the power-law decay, indicating the phase transition. The critical exponent
$\eta$ at that critical point can also be obtained as 3/4 according to the
slope of line in the ln-ln coordinates (natural logarithm), which is different
from that of Ising universality class. Previous work PSmacchia2011  also
points out that although most of the critical exponents of Hermitian cluster
Ising model equal to that of Ising universality class, the central charge of
them are different, which means phase transition in Hermitian cluster Ising
model is beyond Ising universality class. From our calculation, the difference
in critical exponent $\eta$ also provides more evidence to support the above
claim.

Interestingly, as $\it\Gamma$ increases to 2 [see Fig. 5 (b) and (d)], exotic
phases and phase transitions are revealed by the variation mode of $O_{x}(r)$
and $R_{y}(r)$. Let us start with $O_{x}(r)$ [see Fig. 5 (b)]. In zone I
before the first critical point, $O_{x}(r)$ remains a constant as string
length $\it r$ increases, which indicates it is a non-trivial cluster phase.
In zone III and IV, $O_{x}(r)$ decreases exponentially to zero, which
indicates they are trivial phases. However, in the whole zone II, $O_{x}(r)$
exhibits a power-law decay and the slope of curves in ln-ln coordinate system
is close to -0.5, which indicates that zone II is a gapless phase and the
critical exponent $\eta$ of the phase transitions is 1/2. Then, let us move on
to the behavior of $R_{y}(r)$ [see Fig. 5 (d)]. One can only observe the
power-law decay at the third critical point, which means that the symmetry
breaking only occurs at the third phase transition. Analyzing the above two
clues with the distributions of $\it O_{x}$ and $\it m_{y}$ shown in Fig. 4
(b), we find that in zone IV, $\it O_{x}$ is zero and AFM order parameter $\it
m_{y}$ is non-zero, which means that zone IV is a trivial AFM phase with
symmetry breaking. However, both $\it O_{x}$ and $\it m_{y}$ are zero in zone
III, which means it is a trivial phase without symmetry breaking and
magnetism. Thus, it can only be a PM phase in one dimensional chain.
Therefore, it should be a PM-AFM phase transition of standard Ising
universality class between zone III and IV. This claim is also supported by
the critical exponent obtained by the slope of $R_{y}(r)$ at the critical
point between zone III and IV: $\eta=$1/8, which is consistent with that of
standard Ising universality class. As a result, the phase transition between
zone I and II as well as zone II and III are phase transitions between gapped
phase and gapless phase, so we call them ‘KT-like’ phase transition. To
conclude, the introducing of non-Hermitian term gives rise to novel phases and
phase transitions as well as the shift of critical points.

Figure 6: (Color online). The $\it\Gamma-\lambda$ phase diagram according to
$\it U_{g}$’s second derivative. In Hermitian case ($\it\Gamma=$ 0), there is
a conventional critical point of Ising universal class at $\lambda=$1\. With
the increase of $\it\Gamma$, the critical point splits into three critical
points, denoting the phase transitions among the four different phases.

To be more comprehensive and intuitive, we investigate the $\it\Gamma-\lambda$
phase diagram according to the non-anlytical points of $\it U_{g}$’s second
derivative (see Fig. 6). When $\it\Gamma=$ 0, there is only one SPT-AFM phase
transition that is beyond Ising universality class at $\lambda=$1\. As
$\it\Gamma$ grows, the critical point extends to three consecutive critical
lines. Blue, red and yellow lines correspond to two ‘KT-like’ phase transition
and a standard Ising phase transition, respectively. Notably, only phase
transition on yellow critical line still accompanies with the breaking of spin
rotational symmetry.

##  V Summary

There is an SPT phase and an AFM phase in Hermitian cluster Ising model and
the phase transition between them is beyond Ising universality class. The
influence of non-Hermiticity on cluster Ising model is investigated in this
work. We first detect the singular behaviors of second derivative of energy
density and fidelity, finding that new critical points may emerge and move
away from each other. Next, we numerically investigate the string order
parameter and staggered magnetization, and the results show that the
introduction of non-Hermiticity will give rise to a very rich phase diagram
featuring phase transitions between four phases. In order to characterize
them, we then investigate the variation modes of string parameter and spin
correlation function, which help us distinguish different phases and
characterize three phase transitions with the critical exponent. By combining
the result of string order parameter and staggered magnetization, the four
phases are identified as a cluster phase, a gapless phase, a paramagnetic
phase and an antiferromagnetic phase, respectively. The phase transitions
between them are two ‘KT-like’ phase transitions and one standard Ising phase
transition with the corresponding critical exponents $\eta=$1/2, 1/2 and 1/8,
respectively. Eventually, we give the $\it\Gamma-\lambda$ phase diagram to
visualize the emergence and extension of critical lines.

In the end, we briefly clarify the possible mechanism behind the novel
phenomena. In Hermitian case, when $\lambda<$1, SPT phase of the cluster Ising
model is well protected by a $\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ symmetry,
i.e., both the $Z2$ symmetry of $\sigma_{z}$’s product and the symmetry of
time-reversal are unbroken. However, after introducing the non-Hermitian
external field, though the $\mathbb{Z}_{2}$ symmetry of $\sigma_{z}$’s product
is intact, the time-reversal symmetry is broken, which leads to the emergence
of cluster phase in zone I, the closing of gap in zone II as well as the
emergence of paramagnetic phase in zone III. Deeper mechanism behind the
relation between non-Hermitian external field and gapless phase remains
unclear nowadays, which is related to the developing non-unitary conformal
field theory and we will discuss it in our future work. Our work can be
realized in ultra-cold atom experiment and will shed light on experimental
construction of novel phases and phase transitions in open quantum many body
systems.

## Acknowledgment

We thank C. X. Ding, Y. G. Liu, Q. Q. Su, M.Gong, S. Liu and D. C. Lu for
helpful discussions. This work was supported by the NSFC (Grant No. 11704132),
the NSAF (Grant No. U1830111), the Natural Science Foundation of Guangdong
Province (Grant No. 2021A1515012350), and the KPST of Guangzhou (Grant No.
201804020055).

##  Appendix A First derivative of ground state energy density

To prove that all the phase transitions in non-Hermitian case are continuous,
we present the first derivative of the ground state energy density $U_{g}$
(see Fig. 7), from which one can see that the first derivative of the order
parameters is continuous. Since the second derivative of $U_{g}$ are
discontinuous (see Fig. 2), all the phase transitions after introducing non-
Hermition dissipation are standard continuous phase transitions.

Figure 7: (Color online). The first derivative of ground state energy density
$U_{g}$ under different dissipation strength $\it\Gamma$. (a) When
$\it\Gamma=$ 0, the distribution of $\partial U_{g}/\partial\lambda$ with
$\lambda$. (b) When $\it\Gamma=$ 0.6, the distribution of $\partial
U_{g}/\partial\lambda$ with $\lambda$. (c) When $\it\Gamma=$ 1.6, the
distribution of $\partial U_{g}/\partial\lambda$ with $\lambda$. The insets:
the zoom in of the central area where the second critical point at. All of the
above distributions are continuous at the critical points.

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