# Magnetization in a non-equilibrium quantum spin system

X. Z. Zhang<EMAIL_ADDRESS>College of Physics and Materials Science,
Tianjin Normal University, Tianjin 300387, China

###### Abstract

The dynamics described by the non-Hermitian Hamiltonian typically capture the
short-term behavior of open quantum systems before quantum jumps occur. In
contrast, the long-term dynamics, characterized by the Lindblad master
equation (LME), drive the system towards a non-equilibrium steady state
(NESS), which is an eigenstate with zero energy of the Liouvillian
superoperator, denoted as $\mathcal{L}$. Conventionally, these two types of
evolutions exhibit distinct dynamical behaviors. However, in this study, we
challenge this common belief and demonstrate that the effective non-Hermitian
Hamiltonian can accurately represent the long-term dynamics of a critical two-
level open quantum system. The criticality of the system arises from the
exceptional point (EP) of the effective non-Hermitian Hamiltonian.
Additionally, the NESS is identical to the coalescent state of the effective
non-Hermitian Hamiltonian. We apply this finding to a series of critical open
quantum systems and show that a local dissipation channel can induce
collective alignment of all spins in the same direction. This direction can be
well controlled by modulating the quantum jump operator. The corresponding
NESS is a product state and maintains long-time coherence, facilitating
quantum control in open many-body systems. This discovery paves the way for a
better understanding of the long-term dynamics of critical open quantum
systems.

##  I Introduction

Open quantum many-body systems have emerged as a captivating research field at
the intersection of theoretical and experimental physics Breuer and
Petruccione (2002); Weiss (2012); Rivas and Huelga (2012). Comprising numerous
interacting quantum particles, these systems exhibit intricate and captivating
dynamics that elude traditional closed quantum systems. The interaction of
these systems with an external environment leads to dissipation and
decoherence, presenting new challenges and opportunities for exploring quantum
phenomena Pellizzari  _et al._ (1995); Balasubramanian  _et al._ (2009);
Lanyon  _et al._ (2011); Paik  _et al._ (2011). Recent advancements have been
made in the realization and manipulation of open quantum many-body systems in
atomic, molecular, and optical (AMO) systems Kasprzak  _et al._ (2006); Bloch
_et al._ (2008); Bloch (2008); Diehl  _et al._ (2008); Syassen  _et al._
(2008); Baumann  _et al._ (2010); Barreiro  _et al._ (2011); Schauß  _et al._
(2012); Ritsch  _et al._ (2013); Carusotto and Ciuti (2013); Daley (2014),
which offer precise control over individual quantum particles and enable the
engineering of complex interactions and dissipation mechanisms. In addition,
state-of-the-art measurement techniques, such as quantum state tomography and
quantum non-demolition measurements, provide unprecedented opportunities to
investigate the dynamics of these systems with high precision Nelson  _et al._
(2007); Gericke  _et al._ (2008); Hofferberth  _et al._ (2008); Bakr  _et al._
(2009); Sherson  _et al._ (2010); Miranda  _et al._ (2015); Cheuk  _et al._
(2015); Parsons  _et al._ (2015); Haller  _et al._ (2015); Omran  _et al._
(2015); Edge  _et al._ (2015); Yamamoto  _et al._ (2016); Alberti  _et al._
(2016).

The dynamics of an open quantum system are typically described by a quantum
master equation, specifically the Lindblad master equation (LME). This is
attributed to the weak coupling and separation of timescales between the
system and its environment. The Liouvillian superoperator $\mathcal{L}$
governs the time evolution of the density matrix, fully characterizing the
relaxation dynamics of an open quantum system through its complex spectrum and
eigenmodes Rivas and Huelga (2012). A notable feature of open quantum systems
is the presence of long-lived states that emerge far from equilibrium, known
as non-equilibrium steady states (NESS). These NESS can exhibit novel
properties, such as the presence of quantum correlations and the breakdown of
conventional statistical mechanics Eisert  _et al._ (2015). Investigating the
conditions and properties of NESS is currently an active area of research.

Figure 1: Schematic illustration of the magnetization in a non-equilibrium
quantum spin system. The system comprises six spins, with the third spin
subjected to a local external field represented as $\lambda s_{3}^{x}$, and a
dissipation channel characterized by the quantum jump operator $L=e^{-i\pi
s^{x}/2}s_{3}^{-}$. The green-shaded region in the illustration depicts the
local external field. The long-lived NESS of the system converges to a
coherent product state, with all spins aligning in the same direction, when a
critical field is activated. This behavior results from the identical dynamic
effects induced by two types of probability of the the SSE in each quantum
trajectory. Importantly, this finding remains robust irrespective of the
system’s structure and the initial spin configuration.

The non-Hermitian Hamiltonian is an extension of standard quantum mechanics
that allows for the description of dissipative systems in a minimalistic
manner. In recent years, there has been a growing interest in using non-
Hermitian descriptions to study condensed matter systems Lee (2016); Kunst
_et al._ (2018); Yao  _et al._ (2018); Gong  _et al._ (2018); El-Ganainy  _et
al._ (2018); Nakagawa  _et al._ (2018); Shen and Fu (2018); Wu  _et al._
(2019); Yamamoto  _et al._ (2019); Song  _et al._ (2019); Yang and Hu (2019);
Hamazaki  _et al._ (2019); Li  _et al._ (2019); Kawabata  _et al._ (2019a, b);
Lee  _et al._ (2019); Yokomizo and Murakami (2019); Jin  _et al._ (2020) .
These descriptions have not only expanded the realm of condensed-matter
physics, providing insightful perspectives, but also offered a fruitful
framework for understanding inelastic collisions Xu  _et al._ (2017), disorder
effects Shen and Fu (2018); Hamazaki  _et al._ (2019), and system-environment
couplings Nakagawa  _et al._ (2018); Yang and Hu (2019); Song  _et al._
(2019). The interplay between non-Hermiticity and interactions can lead to
exotic quantum many-body effects, such as non-Hermitian extensions of the
Kondo effect Nakagawa  _et al._ (2018); Lourenço  _et al._ (2018), many-body
localization Hamazaki  _et al._ (2019), and fermionic superfluidity Yamamoto
_et al._ (2019); Okuma and Sato (2019). One intriguing feature of non-
Hermitian systems is the presence of exceptional points (EPs), which are
degeneracies of non-Hermitian operators where the eigenvalues and
corresponding eigenstates merge into a single state Berry (2004); Heiss
(2012); Lee (2016); Miri and Alù (2019); Zhang and Gong (2020). These EPs give
rise to fascinating dynamical phenomena, including asymmetric mode switching
Doppler  _et al._ (2016), topological energy transfer Xu  _et al._ (2016),
robust wireless power transfer Assawaworrarit  _et al._ (2017), and enhanced
sensitivity Wiersig (2014, 2016); Hodaei  _et al._ (2017); Chen  _et al._
(2017), depending on the nature of their EP degeneracies. High-order EPs,
where more than two eigenstates coalesce, have attracted significant attention
due to their topological and distinct dynamical properties Zhang  _et al._
(2012, 2020a, 2020b); Zhang and Song (2020, 2021); Yang and Song (2021); Xu
_et al._ (2023); Xu and Jin (2023).

In the context of open quantum systems, the evolved density matrix driven by
the LME can be obtained by averaging an ensemble of quantum trajectories. Each
trajectory is determined by the stochastic Schröinger equation (SSE). The SSE
involves two types of probability evolution: a non-unitary evolution
determined by the effective non-Hermitian Hamiltonian and a state collapse
induced by the quantum jump operator. Generally, the non-Hermitian Hamiltonian
captures the short-term dynamics of the open quantum system before a quantum
jump occurs or describes a post-selected trajectory that necessitates
substantial experimental resources. The dynamical consequences of the
effective non-Hermitian system are irrelevant to the NESS of the open quantum
system. The objective of this paper is to establish the connection between the
dynamics determined by the non-Hermitian Hamiltonian and the LME. First, we
review the connection between the stochastic SSE and the LME. We demonstrate
how to modulate the quantum jump operator to make the evolved state converge
to the coalescent state determined by the critical non-Hermitian Hamiltonian
in each quantum trajectory. Essentially, the evolution direction dictated by
the critical non-Hermitian Hamiltonian coincides with that determined by the
quantum jump operator. This is a unique characteristic of the critical non-
Hermitian Hamiltonian that lacks in non-Hermitian Hamiltonians without
exceptional points (EPs) or imaginary energy levels. Furthermore, we
generalize this mechanism to the critical quantum spin system with high-order
EPs. We demonstrate that a single local dissipation can cause the collective
rotation of spins in a specific direction, which is shown in Fig. 1. The
achieved NESS is equivalent to the coalescent state of such a critical non-
Hermitian Hamiltonian. Remarkably, this non-equilibrium behavior remains
unaffected by the system’s geometry, initial spin configuration, and weak
disorder, thus highlighting its robustness. These analytical findings possess
independent interest and hold the potential to inspire future analytical
studies on critical open quantum systems.

The remainder of the paper is organized as follows: Sec. II provides a review
of the LME and the SSE, demonstrating the underlying mechanism using a two-
level open quantum system. Sec. III applies the obtained mechanism to an open
quantum spin system. We showcase the coincidence between EP dynamics and the
magnetization of the open quantum spin system. Furthermore, we analyze the
proposed scheme across various system parameters. We conclude the paper in
Sec. IV. Supplementary details of our calculation are provided in the
Appendix.

Figure 2: Plot of the purity Tr$\left(\rho_{\mathrm{NESS}}^{2}\right)$ as the
function of $\lambda/\gamma$. The blue and red lines correspond to the quantum
jump operators $L=s^{-}$, and $\widetilde{L}=e^{-i\frac{\pi}{2}s^{x}}s^{-}$,
respectively. The blue line monotonically decreases to $0.5$, indicating a
completely mixed state. The red line initially decreases and then returns to
$1$. There is a range around $0.5$ where the evolved state can be approximated
as a pure state. When $\lambda/\gamma=0.5$, the NESS is $\rho_{\mathrm{NESS}}$
$|y,+\rangle\langle y,+|$, which is also the coalescent of $\mathcal{H}$.

##  II Heuristic derivation

The dynamics of open quantum systems coupled to a Markovian environment are
commonly described by the LME. The equation describing the time evolution of
the density matrix $\rho$ is given by

$\frac{\text{{d}}\rho}{\text{{d}}t}=-i(\mathcal{H}\rho-\rho\mathcal{H}^{\dagger})+\sum_{\mu}\Gamma_{\mu}L_{\mu}\rho
L_{\mu}^{\dagger}\equiv\mathcal{L}\rho.$ (1)

In this equation, $\rho$ represents the density matrix. The non-Hermitian
Hamiltonian $\mathcal{H}$ is given by $\mathcal{H}$ $\mathcal{=}$
$H-\frac{i}{2}\sum_{\mu}\Gamma_{\mu}L_{\mu}^{\dagger}L_{\mu}$, where $H$ is a
Hermitian operator representing the system Hamiltonian. The non-Hermitian
nature of $\mathcal{H}$ accounts for the non-unitary dynamics observed in open
quantum systems. The jump operators $L_{\mu}$ describe the dissipative quantum
channels with a strength of $\Gamma_{\mu}$. $\mathcal{L}$ is the Liouvillian
superoperator. Alternatively, one can track the trajectory of a pure state
using a SSE, such as

$\displaystyle\text{{d}}|\Psi\rangle$ $\displaystyle=$
$\displaystyle-i\mathcal{H}|\Psi\rangle\text{{d}}t+\frac{1}{2}\sum_{\mu}\Gamma_{\mu}[\langle\Psi|L_{\mu}^{\dagger}L_{\mu}|\Psi\rangle]|\Psi\rangle\text{{d}}t$
(2)
$\displaystyle+\sum_{\mu}(\frac{L_{\mu}|\Psi\rangle}{\sqrt{\langle\Psi|L_{\mu}^{\dagger}L_{\mu}|\Psi\rangle}}-|\Psi\rangle)\text{{d}}N_{\mu},$

where the Poisson increment d$N_{\mu}$ satisfies
d$N_{\mu}$d$N_{\nu}=\delta_{\mu\nu}$, taking the value $0$ or $1$. The jump
operators in the LME correspond to the stochastic jumps in the SSE. If
d$N_{\mu}=0$, the evolution is solely described by the non-Hermitian
Hamiltonian $\mathcal{H}$, which is referred to as the no-click limit Daley
(2014). However, this limit is rarely achieved in experiments since its
realization requires exponentially many experiments to be carried out before a
desired trajectory is obtained. The connection between the SSE and the LME
lies in the relationship between the individual wave function trajectories and
the ensemble-averaged density matrix. By averaging over the different
realizations of the stochastic trajectories generated by the SSE, one can
recover the ensemble-averaged dynamics described by the LME. In this way, the
SSE provides a more detailed and microscopic description of the dynamics,
while the LME provides a coarse-grained description that captures the averaged
behavior of the system Daley (2014).

To fully grasp the essence of this paper, we begin by considering a simple
quantum system comprising two levels with orthonormal states. This model has
diverse applications and can describe phenomena such as the spin degree of
freedom of an electron, a simplified representation of an atom with only two
atomic levels, the lowest eigenstates of a superconducting circuit, or the
discrete charge states of a quantum dot. In this model, the system Hamiltonian
is given by $H=\lambda s^{x}$, where $\lambda$ represents the energy
difference between the two states and $\sigma^{x}=2s^{x}$ is the Pauli matrix
corresponding to the $x$-direction. The quantum jump operator is denoted as
$L_{\mu}=s^{-}$ with a strength $\Gamma_{\mu}=\gamma$, where $s^{-}$
represents the lowering operator responsible for the spin flip from the spin-
up state to the spin-down state. The initial state
$|\Psi\left(0\right)\rangle$ is assumed to be an arbitrary pure state
applicable to various many-body examples. In this context, the initial state
is represented by the density matrix
$\rho\left(t=0\right)=|\Psi\left(0\right)\rangle\langle\Psi\left(0\right)|$.
Referring to the Eq. (2), the evolution of $|\Psi\left(t+\delta
t\right)\rangle$ is determined by either $\frac{\left(1-i\mathcal{H}\delta
t\right)}{\sqrt{1-\delta p}}|\Psi\left(t\right)\rangle$ with a probability of
$1-\delta p$ or $\frac{L}{\sqrt{\delta p/\delta t}}|\Psi\left(t\right)\rangle$
with a probability of $\delta p$. Here $\delta p$ is red defined as

$\delta p=\langle\Psi\left(t\right)|s^{+}s^{-}|\Psi\left(t\right)\rangle\delta
t.$ (3)

Notice that when $\lambda>\gamma/2$, the state $|\Psi\left(t\right)\rangle$
oscillates between the two eigenstates of the non-Hermitian Hamiltonian
$\mathcal{H}$ , which possesses a full real spectrum except for a common
imaginary part eliminated by the amplitude $1/\sqrt{1-\delta p}$. On the other
hand, if $\lambda<\gamma/2$, $|\Psi\left(t\right)\rangle$ relaxes to the
eigenstate with the maximum imaginary part, as $\mathcal{H}$ has two complex
eigenvalues. It is worth noting that when $\lambda=\gamma/2$, an exceptional
point (EP) exists in the spectrum of $\mathcal{H}$, where there is only one
coalescent eigenstate
$|\psi_{\mathrm{c}}\rangle=\frac{1}{\sqrt{2}}\left(1,i\right)^{T}$. For an
arbitrary initial state $|\Psi\left(0\right)\rangle$, it evolves towards the
coalescent state $|\psi_{\mathrm{c}}\rangle$ due to the nilpotent matrix
property of $\mathcal{H}$, i.e., $\mathcal{H}^{2}=0$ (see Appendix V.1 for
more details). Alternatively, if the final state is a steady pure state, it
can be projected onto the Bloch sphere, revealing a definite spin direction.
However, the presence of the quantum jump operator $s^{-}$ disrupts the
evolution driven by $\mathcal{H}$ and consequently affects the direction. The
steady state must strike a balance between these two probabilistic evolutions.
To gain further insight into the NESS $\rho_{\mathrm{NESS}}$ defined by
d$\left(\rho_{\mathrm{NESS}}\right)/$d$t=0$, we employ a spin bi-base mapping,
also known as the Choi-Jamiołkowski isomorphism, to map a density matrix to a
vector in the computational bases (see Appendix V.2 for more details). The
NESS corresponds to the eigenstate of $\mathcal{L}$ with zero eigenvalue,
which can be expressed as

$\rho_{\mathrm{NESS}}=\left(\begin{array}[]{cc}\frac{\lambda^{2}}{2\lambda^{2}+\gamma^{2}}&-i\frac{\lambda\gamma}{2\lambda^{2}+\gamma^{2}}\\\
i\frac{\lambda\gamma}{2\lambda^{2}+\gamma^{2}}&\frac{\lambda^{2}+\gamma^{2}}{2\lambda^{2}+\gamma^{2}}\end{array}\right).$
(4)

The coherence of a system can be measured by its purity, which is quantified
by the function Tr$(\rho_{\mathrm{NESS}}^{2})$. In Fig. 2, we depict the
behavior of Tr$(\rho_{\mathrm{NESS}}^{2})$ as a function of $\lambda$, while
keeping $\gamma$ fixed at 1. Let us first consider two limiting cases: When
$\lambda=0$, the non-Hermitian Hamiltonian $\mathcal{H}$ drives the initial
state to $|\psi_{\mathrm{f}}\rangle=(0,1)$. Simultaneously, the quantum jump
operator projects the spin to the down state. Consequently,
$|\psi_{\mathrm{f}}\rangle$ becomes the non-equilibrium steady state (NESS).
On the other hand, when $\lambda\gg\gamma$, the density matrix
$\rho_{\mathrm{NESS}}$ simplifies to

$\rho_{\mathrm{NESS}}=\left(\begin{array}[]{cc}1/2&0\\\
0&1/2\end{array}\right),$ (5)

which corresponds to a completely mixed state. This can be understood as
follows: Under the influence of the non-Hermitian Hamiltonian $\mathcal{H}$,
the evolved state does not have a definite direction. Instead, it oscillates
between the two eigenvectors along the $x$-direction, i.e.,
$|\psi_{\mathrm{1}}\rangle=\left(1,\text{ }1\right)^{T}/\sqrt{2}$ and
$|\psi_{\mathrm{2}}\rangle=\left(1,\text{ }-1\right)^{T}/\sqrt{2}$. However,
the quantum jump operator forces the spin to align parallel to the
$-z$-direction. The consequences of the two effects are completely independent
and cannot be reconciled, leading to a thermal state with infinite
temperature. Generally, the NESS is not a pure state, except for a few
limiting cases. Therefore the evolution of the state cannot be mapped onto the
Bloch sphere, making it impossible to analyze its trajectory on the sphere.
However, by applying a rotation to the quantum jump operator, i.e.,
$\widetilde{L}=Us^{-}$, where $U=e^{-i\frac{\pi}{2}s^{x}}$ corresponds to a
unitary feedback operator Lloyd (2000); Nelson  _et al._ (2000), the new
quantum jump operator $\widetilde{L}$ drives the evolved state
$|\Psi\left(t\right)\rangle$ towards $|y,+\rangle=|\psi_{\mathrm{c}}\rangle$,
which is the eigenstate of the operator $s^{y}$ with eigenenergy $1/2$.
Importantly, the probability $\delta p$ and the non-Hermitian Hamiltonian
$\mathcal{H}$ remain unchanged since
$\widetilde{L}^{\dagger}\widetilde{L}=s^{+}s^{-}=L^{\dagger}L$. When
$\lambda=\gamma/2$, $|y,+\rangle$ also represents the coalescent state of
$\mathcal{H}$. The EP dynamics guides the evolved state
$|\Psi\left(t\right)\rangle$ towards $|y,+\rangle$. These two probabilistic
evolutions tend to drive the arbitrary initial state
$|\Psi\left(0\right)\rangle$ to $|y,+\rangle$, resulting in the final steady
state being a pure state $|y,+\rangle$. This can be demonstrated in Fig. 2.
The purity of $\rho_{\mathrm{NESS}}$, represented by the red line, initially
decays and then recovers to $1$ when $\lambda/\gamma=1/2$. At this point,
$\rho_{\mathrm{NESS}}=\left(I_{2}+\sigma^{y}\right)/2=|y,+\rangle\langle
y,+|$, which validates our previous analysis. Based on the above calculations,
one can appropriately choose the quantum jump operator to achieve the desired
spin polarization.

##  III Magnetization in open quantum spin systems induced by a local
dissipation channel

Figure 3: The time evolutions of the Uhlmann fidelity
$F(\rho(t),\rho_{\mathrm{c}})$ and quantum mutual information $I_{AB}(t)$. The
blue and red lines represent the driven systems under the quantum jump
operators $L_{1}$ and $\widetilde{L}_{1}$, respectively. The system parameters
are chosen as $\lambda/\gamma=0.5$ and $J_{ij}/\gamma=2$. The system is
initially prepared in the state $|\Downarrow\rangle$ such that
$F(\rho(0),\rho_{\mathrm{c}})=1/N$, where $N=4$ for simplicity. In the absence
of modulation, the Uhlmann fidelity $F(\rho(t),\rho_{\mathrm{c}})$ (blue
curve) saturates at approximately 0.75. On the other hand, the local
dissipation channel $\widetilde{L}_{1}$ drives the system towards the state
$\rho_{\mathrm{c}}$ with zero quantum mutual information
$I_{AB}(t\rightarrow\infty)=0$. It is noteworthy that these conclusions hold
irrespective of the system’s configuration and size.

In this section, we extend our main conclusion to a many-body quantum spin
system based on the mechanism described above. In this case, the system is
assumed to be described by a Heisenberg Hamiltonian under the influence of an
external field. The Hamiltonian $H$ is defined as follows:

$\displaystyle H$ $\displaystyle=$ $\displaystyle
H_{\text{{spin}}}+H_{\mathrm{e}},$ (6) $\displaystyle H_{\text{{spin}}}$
$\displaystyle=$ $\displaystyle-\sum_{i,j\neq
i}(J_{ij}/2)\left(s_{i}^{+}s_{j}^{-}+s_{i}^{-}s_{j}^{+}\right)+\sum_{i,j\neq
i}\Delta_{ij}s_{i}^{z}s_{j}^{z},$ $\displaystyle H_{\mathrm{e}}$
$\displaystyle=$
$\displaystyle\sum_{i}\lambda_{i}\mathbf{h}\cdot\mathbf{s}_{i}.$ (7)

Here the operators $s_{i}^{\pm}=s_{i}^{x}\pm is_{i}^{y}$ and $s_{i}^{z}$
represent spin-$1/2$ operators at the $i$-th site, which obey the standard
SU(2) symmetry relations: $[s_{i}^{z},s_{j}^{\pm}]=\pm s_{i}^{\pm}\delta_{ij}$
and $[s_{i}^{+},s_{j}^{-}]=2s_{i}^{z}\delta_{ij}$, where $\delta_{ij}$ is the
Dirac delta function. The summation $\sum_{i,j\neq i}$ implies the summation
over of possible pair interactions within an arbitrary range. The parameter
$J_{ij}$ represents the inhomogeneous spin-spin interaction, while
$\Delta_{ij}$ characterizes the anisotropy of the spin system
$H_{\text{{spin}}}$. The local external field $\mathbf{h}=(1,0,0)$ can be
interpreted as a magnetic field along the $x$-direction and is experimentally
accessible in ultracold atom experiments Lee and Chan (2014); Ashida  _et al._
(2017); Pan  _et al._ (2019). The strength experienced by each spin is denoted
as $\lambda_{i}$. When $\Delta_{ij}=J_{ij}/2$, the system $H_{\text{{spin}}}$
corresponds to a ferromagnetic Heisenberg Hamiltonian that respects the SU(2)
symmetry, i.e., $[\sum_{i}s_{i}^{\sigma},H_{\text{{spin}}}]=0$ with
$\sigma=\pm,z$. Thus, the eigenstates of $H_{\text{{spin}}}$ can be classified
based on the total spin number $s$. Among these states, a fully polarized
ferromagnetic state, denoted as
$\left|\Uparrow\right\rangle=\prod\limits_{i=1}^{N}\left|\uparrow\right\rangle_{i}$
belongs to the ground states multiplet, where
$|\uparrow\rangle_{i}\left(|\downarrow\rangle_{i}\right)$ is the eigenstate of
$s_{i}^{z}$ with eigenenergy $\frac{1}{2}(-\frac{1}{2})$ Heisenberg (1928);
Yang and Yang (1966). The degenerate ground states
$\left\\{\left|G_{n}\right\rangle\right\\}$ belonging to the subspace $s=N/2$
are given by
$\left|G_{n}\right\rangle=(\sum_{i}s_{i}^{-})^{n-1}\left|\Uparrow\right\rangle$,
where $n$ ranges from $1$ to $N+1$. Clearly,
$\left\\{\left|G_{n}\right\rangle\right\\}$ are the degenerate groundstates of
$H_{\text{{spin}}}$ with an $(N+1)$-fold degeneracy, where all the spins are
aligned in the same direction. However, the presence of the external field
$H_{\mathrm{e}}$ breaks the SU(2) symmetry of the system, consequently
splitting the degeneracy of these states. From this point onward, we will
assume $\Delta_{ij}=J_{ij}/2$ for clarity.

Figure 4: Plots of Uhlmann fidelity
$F(\rho_{\text{{nh}}}\left(t\right),\rho_{\text{{l}}}\left(t\right))$ as a
function of time $t$. Both $\rho_{\text{{nh}}}\left(t\right)$ and
$\rho_{\text{{l}}}\left(t\right)$ are initialized in the state
$\left|\Downarrow\right\rangle$ and driven by the effective non-Hermitian
Hamiltonian $\mathcal{H}_{\mathrm{spin}}$ and $\mathcal{L}$. The blue, red,
and yellow lines correspond to values of $\lambda/\gamma$ equal to $2/3$,
$1/2$, and $1/4$, respectively. At $\lambda/\gamma=1/2$, which corresponds to
the EP of $\mathcal{H}{\text{spin}}$,
$F(\rho_{\text{{nh}}}\left(t\right),\rho_{\text{{l}}}\left(t\right))$
approaches $1$ since the coalescent state $\rho_{\mathrm{c}}$ is the final
steady state for $\mathcal{L}$. For $\lambda/\gamma>1/2$, a full real spectrum
of $\mathcal{H}_{\mathrm{spin}}$ emerges leading to the periodic evolution of
$\rho_{\text{{nh}}}\left(t\right)$ without a definite direction in the Bloch
sphere. This oscillatory behavior is represented by the blue line. When
$\lambda/\gamma<1/2$, $\mathcal{H}{\text{spin}}$ exhibits imaginary energy
levels, and the final steady state is determined by the maximum value among
them. However, this state does not coincide with
$\rho_{\text{l}}(t\rightarrow\infty)$ due to the different effects of the jump
operator $L_{1}$ and $\mathcal{H}{\text{spin}}$. Consequently,
$F(\rho_{\text{nh}}(t\rightarrow\infty),\rho_{\text{l}}(t\rightarrow\infty))<1$,
indicating a deviation from unity. This can be observed in the plot
represented by the yellow line.

The dissipation channels $L_{i}=s_{i}^{-}$ are now applied to all the local
sites under the influence of a magnetic field. For simplicity, we will focus
on the case where only a single lattice site is affected by the external field
and dissipation channel. Specifically, we set
$\lambda_{i}=\lambda\delta_{i,1}$ and $\Gamma_{\mu}=\gamma\delta_{\mu,1}$. The
extension to multiple lattice sites is straightforward. Following Eq. (2), we
can divide the dynamics into two parts: the first part involves the quantum
jump operator that flips the spin on the first site from the up state
$|\uparrow\rangle_{1}$ to the down state $|\downarrow\rangle_{1}$. Considering
the external field as a perturbation, the low-energy excitation of the
ferromagnetic Heisenberg model can be described by magnons. Intuitively, the
collective behavior of spins leads to the spreading of the effect of
$s_{1}^{-}$ across the entire system, ultimately resulting in the attainment
of the final steady state $\left|\Downarrow\right\rangle$. The second part
characterizes the non-unitary dynamics driven by the non-Hermitian spin
Hamiltonian

$\mathcal{H}_{\mathrm{spin}}=H-i\gamma s_{1}^{+}s_{1}^{-}/2.$ (8)

Clearly, the local external field $H_{\mathrm{e}}=\lambda s_{1}^{x}$ and on-
site dissipation channel $-i\gamma s_{1}^{+}s_{1}^{-}/2$ can be combined into
a complex filed $H_{\mathrm{ec}}=\lambda s_{1}^{x}-i\gamma
s_{1}^{+}s_{1}^{-}/2$ applied to the ferromagnetic Heisenberg Hamiltonian
$H_{\text{{spin}}}$. In general, the commutation relation
$[H_{\text{{spin}}},$ $H_{\mathrm{ec}}]\neq 0$ leads to a splitting of the
ground state of $H_{\text{{spin}}}$ under the influence of $H_{\mathrm{ec}}$.
However, when $\lambda\rightarrow\gamma/2$, the spliting approaches $0$,
allowing us to treat $H_{\mathrm{ec}}$ as a non-Hermitian perturbation. To
facilitate this treatment, we introduce the unitary transformation
$U=\prod\nolimits_{j}U^{j}$ with $U^{j}=e^{-i\pi s_{j}^{x}/2}$, which
represents a collective spin rotation along the $s^{x}$ direction by an angle
$\pi/2$. The matrix form of $H_{\mathrm{ec}}$ in the degenerate subspace
spanned by $\\{|\widetilde{G}_{n}\rangle\\}=\\{U|G_{n}\rangle\\}$ can be given
as

$\displaystyle W_{m,n}$ $\displaystyle=$
$\displaystyle\sqrt{\left(N+1-m\right)m}[\left(\lambda-\gamma/2\right)\delta_{m+1,n}$
(9) $\displaystyle+\left(\lambda+\gamma/2\right)\delta_{m,n+1}]/2N.$

Here,
$W_{m,n}=\langle\widetilde{G}_{m}|UH_{\mathrm{ec}}U^{-1}|\widetilde{G}_{n}\rangle$.
When $\lambda=\gamma/2$, it reduces to a Jordan block matrix with an EP of
($N+1$) order. The corresponding coalescent state with geometric multiplicity
of $1$ is given as

$|\widetilde{G}_{1}\rangle=\prod\nolimits_{j}|y,+\rangle_{j},$ (10)

which represents all the spins aligning parallel to the $+y$ direction. These
results are detailed and exemplified in the Appendix V.3. For an arbitrary
initial state $\sum_{n}c_{n}\left(0\right)|\widetilde{G}_{n}\rangle$ within
the subspace $s=N/2$, the coefficient $c_{m}\left(t\right)$ is given by the EP
dynamics as

$\displaystyle c_{m}\left(t\right)$ $\displaystyle=$ $\displaystyle
c_{m}\left(0\right)+\sum_{n\neq
m}\left(\frac{-it\lambda}{N}\right)^{m-n}\frac{h\left(m-n\right)}{\left(m-n\right)!}$
(11)
$\displaystyle\times[\prod\limits_{p=n+1}^{m}p\left(N+1-p\right)]^{1/2}c_{n}\left(0\right),$

where $h\left(m-n\right)$ is the Heaviside step function (refer to Appendix
V.3 for more details). The expression shows that the coefficient
$c_{N+1}\left(t\right)$ of the evolved state always contains the highest power
of time $t$. As a result, the component $c_{N+1}\left(t\right)$ dominates over
the other components, leading to the final steady state being the coalescent
state
$|\psi_{\mathrm{c}}\rangle=e^{-i\frac{\pi}{2}s^{x}}\left|\Downarrow\right\rangle$
with $s^{x}=\sum_{i}s_{i}^{x}$. This implies that all spins align in parallel
to the $y$-direction. We would like to emphasize that while our primary focus
lies on the subspace indexed by $s=N/2$, the critical complex magnetic field
resulting from local dissipation can also lead to the coalescence of
eigenstates in each degenerate subspace with different quantum number $s$.
Consequently, the coalescent states in each subspace have a geometric
multiplicity of $1$. By following the EP$(N+1)$ dynamics in the $s=N/2$
subspace, an arbitrary initial state in a $s\neq N/2$ subspace evolves towards
the corresponding coalescent state. If the initial state consists of multiple
different types of coalescent states, then the final state is determined by
the coalescent state whose time-dependent coefficient has the highest power of
$t$.

Figure 5: Time evolutions of the correlator $\mathcal{C}\left(t\right)$ for
the evolved states $\rho_{\text{{l}}}\left(t\right)$ and
$\rho_{\text{{nh}}}\left(t\right)$. The system parameters are set as follows:
(a) $\lambda/\gamma=2/3$, (b) $\lambda/\gamma=1/2$, and (c)
$\lambda/\gamma=1/4$. In each panel, the blue and red lines represent the
$\mathcal{C}(t)$ driven by the Liouvillian superoperator $\mathcal{L}$ and the
effective non-Hermitian Hamiltonian $\mathcal{H}_{\text{spin}}$, respectively.
The correlator $\mathcal{C}\left(t\right)$ driven by $\mathcal{L}$
characterizes the spin-spin correlation of the NESS and asymptotically
approaches a steady value over time. However, in the presence of an EP or
imaginary energy levels in the Hamiltonian $\mathcal{H}_{\mathrm{spin}}$, the
correlator $\mathcal{C}\left(t\right)$ converges to a specific asymptotic
value instead of exhibiting indefinite oscillation. At the EP
($\lambda/\gamma=1/2$), both lines converge to the same value,
$\mathcal{C}(t\rightarrow\infty)=1/4$, as shown in Fig. 4(b).

It can be imagined that the system will not approach the coalescent state
$|\psi_{\mathrm{c}}\rangle$ under the effect of the Liouvillian superoperator
$\mathcal{L}$ due to the distinct operations of two types of evolutions. To
confirm this conjecture, we introduce Uhlmann fidelity Jozsa (1994) which
measures the distance between density operators, defined by

$F\left(\rho\left(t\right),\rho_{\mathrm{c}}\right)=(\text{{Tr}}\sqrt{\sqrt{\rho\left(t\right)}\rho_{\mathrm{c}}\sqrt{\rho\left(t\right)}})^{2}\text{,}$
(12)

where $\rho_{\mathrm{c}}=|\psi_{\mathrm{c}}\rangle\langle\psi_{\mathrm{c}}|$
and $\rho\left(t\right)$ denotes the evolved density matrix. The system is
initialized in the state $\left|\Downarrow\right\rangle$. The second physical
quantity of interest is the quantum mutual information of the bipartite state
$\rho_{AB}\left(t\right)$, which is defined as

$I_{AB}=S\left(\rho_{A}\right)+S\left(\rho_{B}\right)-S\left(\rho_{AB}\right),$
(13)

where $S\left(\rho_{\sigma}\right)=-$Tr$\rho_{\sigma}$ln$\rho_{\sigma}$
($\sigma=A,B$) represents the Von Neumann entropy of
$\rho_{\sigma}\left(t\right)$. This entropy is obtained by tracing out system
$B$ or $A$ from the joint density matrix $\rho_{AB}$. More specifically,
$\rho_{A}\left(t\right)=$Tr${}_{B}[\rho_{AB}\left(t\right)]$ or
$\rho_{B}\left(t\right)=$Tr${}_{A}[\rho_{AB}\left(t\right)]$.
$S\left(\rho_{AB}\right)$ denotes the Von Neumann entropy of the total state.
The quantity $I_{AB}$ is formally equivalent to the classical mutual
information, with the Shannon entropy replaced by its quantum counterpart.
Utilizing $I_{AB}$, we can effectively capture the separability of the evolved
state. If $I_{AB}=0$, the evolved state $\rho_{AB}\left(t\right)$ is
considered simply separable or a product state. In our system, we divide it
into two parts: part $A$ consists of a single local spin, while part $B$
represents its complement. When the NESS assumes a product form, $I_{AB}$ will
be $0$. In Fig. 3, we conduct a numerical simulation on these two quantities.
The results indicate that $F\left(\rho\left(t\right),\rho_{\mathrm{c}}\right)$
initially increases during the short-time evolution, as it is determined by
the non-Hermitian Hamiltonian $\mathcal{H}_{\mathrm{spin}}$, which drives
$\rho\left(t\right)$ towards $\rho_{\mathrm{c}}$. However, as the long-time
evolution progresses, a compromise between two distinct types of probabilistic
evolution emerges, leading to a deviation of the NESS from
$\rho_{\mathrm{c}}$. Additionally, we observe that the minimum value of
$I_{AB}$ is approximately $0.136$, implying that the spin at the first site
remains correlated with the other component. Consequently, the evolved state
$\rho(t)$ is not a product state.

To recover the final state $|\psi_{\mathrm{c}}\rangle$, one should introduce a
unitary operator $U=e^{-i\frac{\pi}{2}s^{x}}$ to the quantum jump operator

$\widetilde{L}_{1}=Us_{1}^{-}=\left(s_{1}^{x}-is_{1}^{z}\right)U.$ (14)

The operator $\widetilde{L}$ performs two operations: firstly, it rotates the
spin by an angle of $\pi/2$ along the $x$-direction, and secondly, it projects
the spin at the first site onto the $y$-direction, resulting in the state
$|y_{1},+\rangle$. The unitary operation $U=e^{-i\frac{\pi}{2}s^{x}}$ does not
affect non-Hermitian Hamiltonian $\mathcal{H}_{\mathrm{spin}}$ since
$\widetilde{L}_{1}^{\dagger}\widetilde{L}_{1}=L_{1}^{\dagger}L_{1}$. Its
effect is limited to the quantum trajectories that deviate from the post-
selected no-click trajectory. However, the effect of $\widetilde{L}_{1}$ on
the first spin is equivalent to that of $\mathcal{H}_{\mathrm{spin}}$ which
tends to freeze each spin along $y$-direction. As a result, regardless of the
type of probabilistic evolution in each quantum trajectory, the long-term
tendency leads to the same consequence, suggesting that
$\rho_{\mathrm{NESS}}=|\psi_{\mathrm{c}}\rangle\langle\psi_{\mathrm{c}}|$
represents the NESS of the open quantum spin system. In the Appendix V.4, we
verify that $\rho_{\mathrm{NESS}}$ is indeed the eigenfunction of the
Liouvillian superoperator $\mathcal{L}$ with zero energy. Consequently,
$\mathcal{H}_{\mathrm{spin}}$ and $\mathcal{L}$ share the same steady state
within the subspace $\left\\{\left|G_{n}\right\rangle\right\\}$. This is
further confirmed in Fig. 3, where the Uhlmann fidelity
$F\left(\rho\left(t\right),\rho_{\mathrm{c}}\right)\rightarrow 1$ and
$I_{AB}\left(t\right)\rightarrow 0$ correspond to the final product state of
$|\psi_{\mathrm{c}}\rangle$. Furthermore, we compare the evolution of two
density matrices driven by $\mathcal{H}_{\mathrm{spin}}$ and $\mathcal{L}$, at
$\lambda=\gamma/2$, respectively. The initial state is
$\left|\Downarrow\right\rangle$, prepared within the subspace
$\left\\{\left|G_{n}\right\rangle\right\\}$. We examine the Uhlmann fidelity
between the two evolved states $\rho_{\text{{nh}}}\left(t\right)$ and
$\rho_{\text{{l}}}\left(t\right)$ as depicted in Fig. 4, where
$\rho_{\text{{nh}}}\left(t\right)=e^{-i\mathcal{H}_{\mathrm{spin}}t}\rho_{\text{{nh}}}\left(0\right)e^{i\mathcal{H}_{\mathrm{spin}}^{\dagger}t}$
and
$\rho_{\text{{l}}}\left(t\right)=e^{\mathcal{L}t}\rho_{\text{{l}}}\left(0\right)$.
The two states $\rho_{\text{{nh}}}\left(0\right)$ and
$\rho_{\text{{l}}}\left(0\right)$ are initialized in the state
$\left|\Downarrow\right\rangle$. The fidelity initially decreases and then
rapidly increases to $1$, indicating that the long-time dynamics of
$\rho\left(t\right)$ driven by $\mathcal{L}$ can be effectively described by
$\mathcal{H}_{\mathrm{spin}}$. To gain further insight into the two types of
the evolution, we also investigate the time evolution of the correlator
$\mathcal{C}\left(t\right)=\mathrm{Tr}[\rho\left(t\right)s_{1}^{+}s_{N}^{-}]$
for two such evolved states. In Fig. 5(b), the two curves exhibit the same
long-time tendency and finally approaches $\mathcal{C}\left(t\right)=0.25$
when $\lambda/\gamma=1/2$, which can be also captured by the Uhlmann fidelity.
This result is quite astonishing as it challenges the common belief that
$\mathcal{H}_{\mathrm{spin}}$ captures the short-time dynamics before a
quantum jump occurs, while $\mathcal{L}$ characterizes the long-time dynamics.
Before ending this discussion, it is worth noting that when the initial state
is prepared in a different degenerate subspace ($s\neq N/2$), the final
evolved state becomes an entangled state rather than a separable state where
all the spins align in the same direction, as observed in the $s=N/2$
subspace. Achieving collective magnetization requires careful modulation of
the quantum jump operator $\mathcal{L}$ to align its action with the effect of
$\mathcal{H}_{\mathrm{spin}}$. This process may involve multiple dissipation
channels and present significant challenges in both theoretical and
experimental aspects. Consequently, our proposal is specifically applicable to
the $s=N/2$ subspace.

Figure 6: Plot of $F\left(\rho_{\mathrm{NESS}},\rho_{\mathrm{c}}\right)$ as a
function of $\gamma$. The strength of the external field is fixed at
$\lambda=0.5$. The configuration of the system is shown in Fig. 1. Notably, a
peak is observed at $\gamma=2\lambda$, which corresponds to the EP of the
effective non-Hermitian Hamiltonian $\mathcal{H}_{\mathrm{spin}}$. It is worth
mentioning that slight deviations from $1$ do not significantly impact the
NESS. This observation indicates the existence of a parameter window that
allows for magnetization induced by local dissipation.

Now let us further investigate whether this conclusion holds when the system
parameters are not finely tuned. First, we consider the case where $\gamma$
deviates from $\gamma_{c}=\lambda/2$. We plot Fig. 6, which shows
$F\left(\rho\left(t\rightarrow\infty\right),\rho_{\mathrm{c}}\right)$ as a
function of $\gamma$. It can be observed that the final steady state is almost
unaffected as $\gamma$ deviates slightly from $\gamma_{c}$. However, when
$\gamma\ll\gamma_{c}$, there are no EP and complex energy in
$\mathcal{H}_{\mathrm{spin}}$. In this case, the state initialized in the
subspace $\left\\{\left|G_{n}\right\rangle\right\\}$ will not tend to a
definite state but instead oscillates between different eigenenergies,
resulting in a periodic oscillation of the physical observables. This can be
seen from the Fig. 5(a). Consequently, the density matrix driven by
$\mathcal{H}_{\mathrm{spin}}$ will exhibit distinct dynamics from the quantum
jump operator which forces the spin along the $y$-direction. Combining both
effects, the NESS deviates from $\rho_{\mathrm{c}}$. On the other hand, when
$\gamma\gg\gamma_{c}$, $\mathcal{H}_{\mathrm{spin}}$ drives all the spins to
the down states since the eigenstate $\left|\Downarrow\right\rangle$ has the
largest imaginary part. However, this contradicts the action of the quantum
jump operator. As a consequence of the parameter deviation, the final state is
no longer a product state with a definite direction but a mixed state that
loses some of its coherence.

Figure 7: Numerical simulations of
$\overline{F}\left(\rho_{\mathrm{NESS}},\rho_{\mathrm{c}}\right)$,
$\overline{\mathcal{C}}\left(t\rightarrow\infty\right)$, and
$\overline{I}_{AB}\left(t\rightarrow\infty\right)$ as functions of the
disorder strength $h$. The system parameters are fixed at
$\lambda/\gamma=0.5$, and $J_{ij}/\gamma=1$. The structure of the system is
depicted in Fig. 1. The average Uhlmann fidelity
$\overline{F}\left(\rho_{\mathrm{NESS}},\rho_{\mathrm{c}}\right)$, average
correlator $\overline{\mathcal{C}}\left(t\rightarrow\infty\right)$, and
average quantum mutual information
$\overline{I}_{AB}\left(t\rightarrow\infty\right)$ are calculated by averaging
over 1000 disorder configurations. The simulations demonstrated that the
realization of $\rho_{\mathrm{c}}$ remains unaffected by the specific system
configuration and is immune to weak disorder, as indicated by the gray shaded
region. This property is advantageous for observing the magnetization induced
by single local dissipation in experimental setups.

Besides the deviation from the EP, another factor influencing the success of
the scheme is the presence of disorder. In the experiment, our proposal can be
realized in a cold atom system, particularly in the Rydberg atom quantum
simulator. Smith  _et al._ (2016); Zhang  _et al._ (2017); Marcuzzi  _et al._
(2017); Shibata  _et al._ (2020); Mondragon-Shem  _et al._ (2021) The system
can be subjected to disorder through external fields, such as electric or
magnetic fields. Fluctuations or variations in the strength and direction of
these fields can impact the energy levels and dynamics. It is crucial to
examine the system’s robustness to disorder. To achieve this, we introduce
disorder by considering a random magnetic field in the $z$ direction. The
modified system Hamiltonian is given as

$H_{\text{{spin}}}^{\mathrm{d}}=H_{\text{{spin}}}+H^{\mathrm{d}},$ (15)

with

$H^{\mathrm{d}}=\sum_{i}h_{i}s_{i}^{z},$ (16)

where $h_{i}$ represents a random number within the range $(-h,$ $h)$.
Clearly, $H^{\mathrm{d}}$ breaks the SU(2) symmetry of
$H_{\text{{spin}}}^{\mathrm{d}}$ and hence prevents the formation of the
subpace of $\left\\{\left|G_{n}\right\rangle\right\\}$. Although exact SU(2)
symmetry is spoiled, it can be inferred that the directed evolution in the
$\left\\{\left|G_{n}\right\rangle\right\\}$ subspace may still exist under
weak disorder. In Fig. 7, we perform the numerical simulation to examine the
average Uhlmann fidelity
$\overline{F}\left(\rho_{\mathrm{NESS}},\rho_{\mathrm{c}}\right)$, average
correlator $\overline{\mathcal{C}}\left(t\rightarrow\infty\right)$ and average
quantum mutual information $\overline{I}_{AB}\left(t\rightarrow\infty\right)$.
The results demonstrate that a small distribution of $h$ does not induce a
transition in the final state as the degenerate subspace
$\left\\{\left|G_{n}\right\rangle\right\\}$ is approximately preserved, as
manifested by the behavior of
$\overline{\mathcal{C}}\left(t\rightarrow\infty\right)$ and
$\overline{I}_{AB}\left(t\rightarrow\infty\right)$ in the grey shaded region.
However, when $h$ is large enough to completely destroy the SU(2) symmetry,
the dynamics of EP cannot be maintained as the degenerate subspace ceases to
exist. In such a scenario, the action of $\mathcal{H}_{\mathrm{spin}}$ and the
quantum jump operator $\widetilde{L}$ exhibit distinct dynamics, leading to
the collapse of the final ferromagnetic state.

##  IV Summary

In conclusion, we have demonstrated that the critical non-Hermitian system
accurately captures the long-term dynamics of the open quantum system.
Specifically, the master equation of the open quantum system can be rephrased
as a stochastic average over individual trajectories, which can be numerically
evolved as pure states over time. Each trajectory’s evolution is determined by
the SSE. There are two types of probabilistic evolution: a non-unitary
evolution driven by the effective non-Hermitian Hamiltonian and a state
projection determined by the quantum jump operator. The trade-off between
these two evolutions determines the final NESS. For the non-Hermitian
Hamiltonian, a definite final evolved state can be achieved if the system
possesses the EP or an imaginary energy level. In the former case, the evolved
state is forced towards the coalescent state, while in the latter case, it
approaches the eigenstate with the maximum value of the imaginary energy
level. If the final evolved state coincides with the state under the quantum
jump operation, then the NESS of the open quantum system is identical to the
coalescent state of the effective non-Hermitian Hamiltonian. Furthermore, we
apply this mechanism to the open quantum spin system and find that local
critical dissipation can induce a high-order EP in the effective non-Hermitian
ferromagnetic Heisenberg system. The dimension of the degenerate subspace
determines the order of the EP. The corresponding coalescent state represents
all the spins aligned in parallel to the $y$-direction. From a dynamical
perspective, when the initial state is prepared within the degenerate
subspace, the EP dynamics force all the spins to align in the y-direction
regardless of the initial spin configuration. On the other hand, the quantum
jump operator rotates the spin that passes the first site to align with the
direction of the coalescent state. Both actions of the two probabilistic
propagations are identical, leading to the NESS being the coalescent state.
The realization of this type of NESS is immune to weak disorder and holds
within a certain range of system parameters. These findings serve as the
building blocks for understanding critical open quantum systems from both
theoretical and experimental perspectives.

###### Acknowledgements.

X.Z.Z. gratefully acknowledge Y.L. for hosting and generously providing the
necessary resources for this study. We acknowledge the support of the National
Natural Science Foundation of China (Grants No. 12275193, No. 12225507, and
No. 12088101), and NSAF (Grant No. U1930403).

##  V Appendix

###  V.1 EP dynamics of two-level system

In this subsection, we analyze the EP dynamics in a non-Hermitian two-level
system. The Hamiltonian, given by

$\mathcal{H}=\lambda s^{x}-\frac{i\gamma}{2}s^{+}s^{-},$ (17)

is non-Hermitian due to the dissipation channel. In the basis of
$\\{|\uparrow\rangle,$ $|\downarrow\rangle\\}$, the matrix form of
$\mathcal{H}$ is expressed as

$\mathcal{H}=\frac{1}{2}\left(\begin{array}[]{cc}-i\gamma/2&\lambda\\\
\lambda&i\gamma/2\end{array}\right)-\frac{i\gamma}{4}I,$ (18)

where $\frac{i\gamma}{4}I$ is a constant term that does not affect the
relative probability of populating the two different energy states. The two
eigenstates of $\mathcal{H}$ coalesce at the EP when $\lambda=\gamma/2$. The
corresponding coalescent state is
$|\psi_{\mathrm{c}}\rangle=|y,+\rangle=\frac{1}{\sqrt{2}}\left(1,\text{
}i\right)^{T}$ which also represents the eigenstate of $s^{y}$ with
eigenenergy $1/2$. Now, let us turn focus on the system propagator
$\mathcal{U}_{2}$. Due to the nilpotent matrix property of $\mathcal{H}$,
i.e., $\mathcal{H}^{2}=0$, $\mathcal{U}_{2}$ simplifies to

$\mathcal{U}_{2}=e^{-i\mathcal{H}t}=e^{-\gamma t/4}\left[1-\frac{i\lambda
t}{2}\left(\begin{array}[]{cc}-i&1\\\ 1&i\end{array}\right)\right].$ (19)

For an arbitrary initial state $|\psi\left(0\right)\rangle=\left(a,\text{
}b\right)^{T}$, the evolved state can be given as

$|\psi\left(t\right)\rangle=e^{-\gamma t/4}\left(\begin{array}[]{c}a-i\lambda
t/2\\\ b+\lambda t/2\end{array}\right),$ (20)

As time tends to infinity, $|\psi\left(t\right)\rangle$ normalized in terms of
Dirac probability approaches
$|\psi\left(\infty\right)\rangle=|\psi_{\mathrm{c}}\rangle$.

###  V.2 NESS of an open two-level system

In this subsection, we derive the NESS of the two-level open system under
consideration. The LME describing the open system dynamics is given by

$\frac{\text{{d}}\rho}{\text{{d}}t}=-i\lambda[s^{x},\rho]-\frac{\gamma}{2}\left(s^{+}s^{-}\rho+\rho
s^{+}s^{-}\right)+\gamma s^{-}\rho s^{+}.$ (21)

By applying the Choi-Jamiołkowski isomorphism, the LME can be written as an
equivalent form:

$\frac{\text{{d}}\rho}{\text{{d}}t}\equiv\widetilde{\mathcal{L}}\rho,$ (22)

where the vectorized density matrix
$|\rho\rangle=\sum_{m,n}\rho_{m,n}|m\rangle\otimes|n\rangle$ represents the
density matrix in the double space. The Liouvillian superoperator is given by

$\displaystyle\widetilde{\mathcal{L}}$ $\displaystyle=$
$\displaystyle-i\lambda\left(s^{x}\otimes I-I\otimes s^{x}\right)+\gamma
s^{-}\otimes s^{-}$ (23)
$\displaystyle-\frac{\gamma}{2}\left(s^{+}s^{-}\otimes I+I\otimes
s^{+}s^{-}\right).$

The matrix representation of $\widetilde{\mathcal{L}}$ can be expressed as

$\widetilde{\mathcal{L}}=\left(\begin{array}[]{cccc}-\gamma&i\frac{\lambda}{2}&-i\frac{\lambda}{2}&0\\\
i\frac{\lambda}{2}&-\frac{1}{2}\gamma&0&-i\frac{\lambda}{2}\\\
-i\frac{\lambda}{2}&0&-\frac{1}{2}\gamma&i\frac{\lambda}{2}\\\
\gamma&-i\frac{\lambda}{2}&i\frac{\lambda}{2}&0\end{array}\right).$ (24)

The complete spectrum of the Liouvillian superoperator $\mathcal{L}$ can be
obtained by solving the eigen-equation:
$\widetilde{\mathcal{L}}|\rho_{k}\rangle=\varepsilon_{k}|\rho_{k}\rangle$,
where $k$ represents the eigenvalue and $|\rho_{k}\rangle$ denotes its
corresponding eigenmatrix. The NESS is unique and corresponds to the
eigenvalue $\varepsilon_{k}=0$. Straightforward algebra reveals that the
corresponding eigenmatrix is given by

$\rho_{\mathrm{NESS}}=\left(\begin{array}[]{cc}\frac{\lambda^{2}}{2\lambda^{2}+\gamma^{2}}&-i\frac{\lambda\gamma}{2\lambda^{2}+\gamma^{2}}\\\
i\frac{\lambda\gamma}{2\lambda^{2}+\gamma^{2}}&\frac{\lambda^{2}+\gamma^{2}}{2\lambda^{2}+\gamma^{2}}\end{array}\right).$
(25)

###  V.3 Non-Hermitian Heisenberg model and EP dynamics

####  V.3.1 model and EP

In this subsection, we analyze the non-Hermitian Heisenberg model with a local
dissipation channel and identify the EP. According to the main text, the
Hamiltonian of the effective non-Hermitian Heisenberg model in the LME under
an external field is given by

$\mathcal{H}_{\mathrm{spin}}=H_{\mathrm{spin}}+H_{\mathrm{ec}},$ (26)

where

$H_{\mathrm{spin}}=-\frac{1}{2}\sum_{i,j\neq
i}J_{ij}(s_{i}^{+}s_{j}^{-}+s_{i}^{-}s_{j}^{+}+2s_{i}^{z}s_{j}^{z}),$ (27)

and

$H_{\mathrm{ec}}=\sum_{\left\\{i\right\\}}\lambda_{i}\mathbf{h}\cdot\mathbf{s}_{i}-\frac{i}{2}\sum_{i}\Gamma_{i}s_{i}^{+}s_{i}^{-}.$
(28)

$H_{\mathrm{ec}}$ can be deemed as the external complex magnetic filed. Here,
$\left\\{i\right\\}$ represents a set of multiple local sites that are
subjected to the local complex fields. The presence of inhomogeneous magnetic
fields breaks the $SU(2)$ symmetry, i.e.,
$\left[s^{\pm},\mathcal{H}_{\mathrm{spin}}\right]\neq 0$. However,
$H_{\mathrm{spin}}$ and $H_{\mathrm{ec}}$, commute with each other when the
homogeneous magnetic field and dissipation are applied, i.e.,
$\lambda_{i}=\lambda$ and $\Gamma_{i}=\gamma$. Although these Hamiltonians
share common eigenstates, the properties of the ground states are unclear due
to the non-Hermitian nature of $H_{\mathrm{ec}}$. This poses a challenge to
perturbation theory in Hermitian quantum mechanics since the omission of high-
order corrections cannot be guaranteed in the complex regime. To simplify the
analysis, we consider $\lambda_{i}=\lambda\delta_{i,1}$, and
$\Gamma_{i}=\gamma\delta_{i,1}$ from this point onwards. To proceed, we
introduce a similarity transformation
$\mathcal{S}_{1}=\prod\nolimits_{j}\mathcal{S}_{1}^{j}$, where
$\mathcal{S}_{1}^{j}=e^{-i\theta s_{j}^{y}}$ represents a counter-clockwise
spin rotation in the $s_{x}$-$s_{z}$ plane around the $s_{y}$-axis by an angle
$\theta$. Here $\theta$ is a complex number dependent on the strength of the
complex field, given by $\theta=\tan^{-1}\left(2\lambda/i\gamma\right)$. It is
important to note that the spin rotation $\mathcal{S}_{1}^{j}$ is valid at
arbitrary $\gamma$ except at EP of $H_{\mathrm{ec}}$ ($\lambda=\gamma/2$),
where $H_{\mathrm{ec}}$ takes a non-diagonalizable Jordan block form. Under
the spin-rotation, the transformed Hamiltonian is as follows:

$\displaystyle\overline{\mathcal{H}}_{\mathrm{spin}}$ $\displaystyle=$
$\displaystyle\overline{H}_{\mathrm{spin}}+\overline{H}_{\mathrm{ec}},$ (29)
$\displaystyle\overline{H}_{\mathrm{spin}}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\sum_{i,j\neq
i}J_{ij}(\tau_{i}^{+}\tau_{j}^{-}+\tau_{i}^{-}\tau_{j}^{+}+2\tau_{i}^{z}\tau_{j}^{z}),$
(30) $\displaystyle\overline{H}_{\mathrm{ec}}$ $\displaystyle=$
$\displaystyle\sqrt{\lambda^{2}-\gamma^{2}/4}\tau_{1}^{z}-\frac{i\gamma}{4},$
(31)

where the new set of operators
$\tau_{j}^{\pm}=(\mathcal{S}_{1}^{j})^{-1}s_{j}^{\pm}\mathcal{S}_{1}^{j}$ and
$\tau_{j}^{z}=(\mathcal{S}_{1}^{j})^{-1}s_{j}^{z}\mathcal{S}_{1}^{j}$ also
satisfies the Lie algebra. We omit the overall decay factor $-i\gamma/4$ which
served as the energy base has no effects on the subsequent evolution.
Specifically, they obey the commutation relations
$[\tau_{i}^{z},\tau_{j}^{\pm}]=\pm\tau_{i}^{\pm}\delta_{ij}$ and
$[\tau_{i}^{+},\tau_{j}^{-}]=2\tau_{i}^{z}\delta_{ij}$. It is important to
note that $\tau_{j}^{\pm}\neq(\tau_{j}^{\mp})^{\dagger}$ due to the complex
rotation angle $\theta$. We consider the eigenstates of the operator
$\sum_{i}s_{i}^{z}$, denoted as
$\left\\{\left|\psi_{n}\right\rangle\right\\}$, which represent possible spin
configurations along the $+z$ direction. Under the biorthogonal basis of
$\\{\mathcal{S}^{-1}\left|\psi_{n}\right\rangle\\}$ and
$\\{\mathcal{S}^{\dagger}\left|\psi_{n}\right\rangle\\}$, the matrix form of
$\overline{\mathcal{H}}_{\mathrm{spin}}$ is Hermitian for $\lambda>\gamma/2$
except a complex energy base. Although the presence of the local complex field
breaks the SU(2) symmetry of the system, as indicated by
$[\overline{H}_{\mathrm{ec}},$ $\overline{\mathcal{H}}_{\mathrm{spin}}]\neq
0$, the entirely real spectrum remains without symmetry protection. When
$\lambda=\gamma/2$, the transformation of $\mathcal{S}_{1}$ is ill-defined,
indicating that $H_{\mathrm{ec}}$ is non-diagonalizable, which corresponds to
the presence of an EP. In principle, the EP of $\mathcal{H}_{\mathrm{spin}}$
or $\overline{\mathcal{H}}_{\mathrm{spin}}$ may not coincide with the EP of
$H_{\mathrm{ec}}$ at $\lambda=\gamma/2$. In the following, we will demonstrate
that $\mathcal{H}_{\mathrm{spin}}$ and $H_{\mathrm{ec}}$ exhibit the same EP
behavior within the framework of perturbation theory.

The Hermiticity of the matrix representation of
$\overline{\mathcal{H}}_{\mathrm{spin}}$ allows us to apply various
approximation methods in quantum mechanics. As$\ \gamma$ approaches
$2\lambda$, the value of $\sqrt{\lambda^{2}-\gamma^{2}/4}$ becomes small,
allowing $\overline{H}_{\mathrm{ec}}$ in the new frame to be treated as a weak
perturbation. Our focus is on the influence of $\overline{H}_{\mathrm{ec}}$ on
the ground states $\left\\{\left|G_{n}^{\prime}\right\rangle\right\\}$ of
$\overline{H}_{\mathrm{spin}}$. Due to the properties of the ferromagnetic
spin system $\overline{H}_{\mathrm{spin}}$, the ground states
$\left\\{\left|G_{n}^{\prime}\right\rangle\right\\}$ exhibit $(N+1)$ fold-
degeneracy and can be expressed as

$\left|G_{n}^{\prime}\right\rangle=(\sum_{i}\tau_{i}^{-})^{n-1}\left|\Uparrow\right\rangle^{\prime}\text{
}\left(n=1,\text{ }2\text{ }...\text{ }N+1\right)\text{, }$ (32)

where

$\left|\Uparrow\right\rangle^{\prime}=(\mathcal{S}_{1})^{-1}\left|\Uparrow\right\rangle\text{,
and
}\left|\Uparrow\right\rangle=\prod\limits_{i=1}^{N}\left|\uparrow\right\rangle_{i}\text{.}$
(33)

$\left|G_{n}^{\prime}\right\rangle$ is also the eigenstate of
$\mathbf{\tau}^{2}=\sum_{i}\mathbf{\tau}_{i}^{2}$ with $\tau=N/2$, where $N$
denotes the total number of spins. Noticeably, the presence of degenerate
ground states is irrelevant to the system’s structure. This property can be
observed in other types of systems as well Heisenberg (1928); Yang and Yang
(1966). Following the principles of degenerate perturbation theory, the
eigenvalues up to first order can be determined by the matrix representation
of $\overline{H}_{\mathrm{ec}}$ within the subspace spanned by
$\left\\{\left|G_{n}^{\prime}\right\rangle\right\\}$. For simplicity, we refer
to the corresponding perturbed matrix as $W^{\prime}$, with elements given by
$W_{m,n}^{\prime}=\langle\overline{G}_{m}^{\prime}|\overline{H}_{\mathrm{ec}}|G_{n}^{\prime}\rangle$.
The biorthogonal left eigenvectors are denoted as
$\\{\langle\overline{G}_{m}^{\prime}|\\}$ and can be expressed as

$\langle\overline{G}_{m}^{\prime}|=\left\langle\Uparrow\right|\mathcal{S}_{1}(\sum_{i}\tau_{i}^{+})^{m-1}\text{
}\left(m=1,\text{ }2\text{ }...\text{ }N+1\right).$ (34)

Two important points are highlighted: (i) Owing to the Hermiticity of the
matrix $W^{\prime}$, higher-order corrections can be safely disregarded as$\
\gamma$ approaches $2\lambda$. (ii) When a homogeneous magnetic field is
applied, $\left[\overline{H}_{\mathrm{ec}},\text{
}\overline{\mathcal{H}}_{\mathrm{spin}}\right]=0$, enabling the decomposition
of $\overline{\mathcal{H}}_{\mathrm{spin}}$ into block matrices based on the
eigenvectors of $\mathbf{\tau}^{2}$. Consequently, the eigenvalues of
$W^{\prime}$ comprise the energies of the ground state and $N$ excited states
of $\overline{\mathcal{H}}_{\mathrm{spin}}$. After straightforward algebras,
the entry of the matrix can be obtained as

$\
W_{m,n}^{\prime}=\sqrt{\lambda^{2}-\gamma^{2}/4}[\left(N/2-m+1\right)\delta_{m,n}]/N,$
(35)

where the factor $1/N$ arises from the translation symmetry of the ground
state $\\{\left|G_{n}\right\rangle^{\prime}\\}$. By performing the
transformation $W=U\mathcal{S}_{1}W^{\prime}(\mathcal{S}_{1})^{-1}U^{-1}$
($W_{m,n}=\langle\widetilde{G}_{m}|UH_{\mathrm{ec}}U^{-1}|\widetilde{G}_{n}\rangle$
with $|\widetilde{G}_{n}\rangle=U\left|G_{n}\right\rangle$), the matrix
element of $W$ can be expressed as

$\displaystyle W_{m,n}$ $\displaystyle=$
$\displaystyle\sqrt{\left(N+1-m\right)m}[\left(\lambda-\gamma/2\right)\delta_{m+1,n}$
(36) $\displaystyle+\left(\lambda+\gamma/2\right)\delta_{m,n+1}]/2N.$

When $\lambda=\gamma/2$, it reduces to a Jordan block form, and an EP of order
$N+1$ occurs. The corresponding coalsecent is
$|\psi_{\mathrm{c}}\rangle=\prod\nolimits_{j}e^{-i\frac{\pi}{2}s_{j}^{x}}\left|\Downarrow\right\rangle$.
It is worth mentioning that if we express $H_{\mathrm{ec}}$ in the basis of
$\left\\{\left|G_{n}\right\rangle\right\\}$, it describes a
$\mathcal{PT}$-symmetric hypercube graph of $N+1$ dimension Zhang  _et al._
(2012). The EP also emerges when $\lambda=\gamma/2$.

####  V.3.2 high order EP dynamics

In this subsection, our objective is to generate a saturated ferromagnetic
state where all local spins (or conduction electron spins) are aligned
parallel to the $y$-direction. The non-Hermitian Heisenberg Hamiltonian is
represented by Eq. (8) in the main text. Considering the EP $\lambda=\gamma/2$
within the subspace $\\{|\widetilde{G}_{n}\rangle\\}$, the matrix form of $W$
can be expressed as

$W_{m,n}=\lambda\sqrt{\left(N+1-m\right)m}\delta_{m,n+1}/N,$ (37)

which corresponds to a Jordan block of dimension $N+1$. The coalescent
eigenstate is $|\widetilde{G}_{N+1}\rangle$. It is important to note that $W$
is a nilpotent matrix with order $\left(N+1\right)$ meaning that
$\left(W\right)^{N+1}=0$. The element of matrix $W^{k}$ can be given as

$\left(W^{k}\right)_{mn}=[\prod\limits_{p=m+1-k}^{m}p\left(N+1-p\right)]^{1/2}(\frac{\lambda}{N})^{k}\delta_{m,n+k},$
(38)

where $k<m+1$. Our attention now shifts to the dynamics of the critical matrix
$W$, and the evolution of states within this subspace is governed by the
propagator $\mathcal{U=}e^{-iWt}$. Utilizing Eq. (38), we can derive the
elements of the propagator $\mathcal{U}$ as follows:

$\displaystyle\mathcal{U}_{m,n}$ $\displaystyle=$
$\displaystyle\delta_{mn}+\left(\frac{-it\lambda}{N}\right)^{m-n}\frac{h\left(m-n\right)}{\left(m-n\right)!}$
(39) $\displaystyle\times[\prod\limits_{p=n+1}^{m}p\left(N+1-p\right)]^{1/2},$

where $h\left(x\right)$ is a step function defined as $h\left(x\right)=1$
$\left(x>0\right),$ and $h\left(x\right)=0$ $\left(x<0\right)$. Considering an
arbitrary initial state
$\sum_{n}c_{n}\left(0\right)|\widetilde{G}_{n}\rangle$, the coefficient
$c_{m}\left(t\right)$ of the evolved state is given by

$\displaystyle c_{m}\left(t\right)$ $\displaystyle=$ $\displaystyle
c_{m}\left(0\right)+\sum_{n\neq
m}\left(\frac{-it\lambda}{N}\right)^{m-n}\frac{h\left(m-n\right)}{\left(m-n\right)!}$
(40)
$\displaystyle\times[\prod\limits_{p=n+1}^{m}p\left(N+1-p\right)]^{1/2}c_{n}\left(0\right).$

It is evident that regardless of the initial state chosen, the coefficient
$c_{N+1}\left(t\right)$ of evolved state always contains the highest power of
time $t$. As time progresses, the component $c_{N+1}\left(t\right)$ of the
evolved state overwhelms the other components, ensuring the final state is
coalescent state
$|\psi_{\mathrm{c}}\rangle=\prod\nolimits_{j}e^{-i\frac{\pi}{2}s_{j}^{x}}\left|\Downarrow\right\rangle$
under the Dirac normalization. The different types of initial states only
determine how the total probability of the evolved state increases over time
and the relaxation time for it to evolve towards the coalescent state.

###  V.4 NESS of the open quantum spin system subjected to a local magnetic
field

In this subsection, we demonstrate that the critical density matrix
$\rho_{\mathrm{c}}=|\psi_{\mathrm{c}}\rangle\langle\psi_{\mathrm{c}}|$ is also
the NESS of the open quantum spin system. The dynamics of the open quantum
spin system under consideration is governed by LME, expressed as:

$\displaystyle\frac{\text{{d}}\rho}{\text{{d}}t}$ $\displaystyle=$
$\displaystyle-i(\mathcal{H}_{\mathrm{spin}}\rho-\rho\mathcal{H}_{\mathrm{spin}}^{\dagger})$
(41) $\displaystyle+\gamma\left(s_{1}^{x}-is_{1}^{z}\right)U\rho
U^{-1}\left(s_{1}^{x}+is_{1}^{z}\right)$ $\displaystyle\equiv$
$\displaystyle\mathcal{L}\rho,$

where $U$ is defined as the product of operators $U=\prod\nolimits_{j}e^{-i\pi
s_{j}^{x}/2}$ and

$\displaystyle\mathcal{H}_{\mathrm{spin}}$ $\displaystyle=$ $\displaystyle
H_{\mathrm{spin}}+H_{\mathrm{ec}},$ (42) $\displaystyle H_{\mathrm{spin}}$
$\displaystyle=$ $\displaystyle-\frac{1}{2}\sum_{i,j\neq
i}J_{ij}(s_{i}^{+}s_{j}^{-}+s_{i}^{-}s_{j}^{+}+2s_{i}^{z}s_{j}^{z}),$ (43)
$\displaystyle H_{\mathrm{ec}}$ $\displaystyle=$ $\displaystyle\lambda
s_{1}^{x}-\frac{i\gamma}{2}s_{1}^{+}s_{1}^{-},$ (44)

Here $\lambda=\gamma/2$ is assumed when $\mathcal{H}_{\mathrm{spin}}$ is at
EP. Next, we substitute
$\rho_{\mathrm{c}}=|\psi_{\mathrm{c}}\rangle\langle\psi_{\mathrm{c}}|$ into
the above equation. Recalling that the
$|\psi_{\mathrm{c}}\rangle=\prod\nolimits_{j}e^{-i\frac{\pi}{2}s_{j}^{x}}\left|\Downarrow\right\rangle$,
we can readily deduce that
$\mathcal{H}_{\mathrm{spin}}|\psi_{\mathrm{c}}\rangle=-i\gamma/4|\psi_{\mathrm{c}}\rangle$,
resulting in

$-i(\mathcal{H}_{\mathrm{spin}}\rho_{\mathrm{c}}-\rho_{\mathrm{c}}\mathcal{H}_{\mathrm{spin}}^{\dagger})=-\frac{\gamma}{2}\rho_{\mathrm{c}}.$
(45)

Applying $\left(s_{1}^{x}-is_{1}^{z}\right)U$ to $|\psi_{\mathrm{c}}\rangle$
yields $\frac{1}{\sqrt{2}}|\psi_{\mathrm{c}}\rangle$. Thus, we can conclude
that $\mathcal{L}\rho_{\mathrm{c}}=0$, demonstrating that $\rho_{\mathrm{c}}$
is indeed the NESS $\rho_{\mathrm{NESS}}$ of the open quantum spin system.

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