# Proposal for observing Yang-Lee criticality in Rydberg atomic arrays

Ruizhe Shen<EMAIL_ADDRESS>Department of Physics, National University of
Singapore, Singapore 117542  Tianqi Chen<EMAIL_ADDRESS>School of
Physical and Mathematical Sciences, Nanyang Technological University,
Singapore 639798  Fang Qin  Department of Physics, National University of
Singapore, Singapore 117542  Yin Zhong  School of Physical Science and
Technology $\&$ Key Laboratory for Magnetism and Magnetic Materials of the
MoE, Lanzhou University, Lanzhou 730000, China  Lanzhou Center for Theoretical
Physics, Key Laboratory of Theoretical Physics of Gansu Province, Lanzhou
730000, China  Ching Hua Lee<EMAIL_ADDRESS>Department of Physics,
National University of Singapore, Singapore 117542

###### Abstract

Yang-Lee edge singularities (YLES) are the edges of the partition function
zeros of an interacting spin model in the space of complex control parameters.
They play an important role in understanding non-Hermitian phase transitions
in many-body physics, as well as characterizing the corresponding non-unitary
criticality. Even though such partition function zeroes have been measured in
dynamical experiments where time acts as the imaginary control field,
experimentally demonstrating such YLES criticality with a physical imaginary
field has remained elusive due to the difficulty of physically realizing non-
Hermitian many-body models. We provide a protocol for observing the YLES by
detecting kinked dynamical magnetization responses due to broken
$\mathcal{P}\mathcal{T}$ symmetry, thus enabling the physical probing of non-
unitary phase transitions in non-equilibrium settings. In particular, scaling
analyses based on our non-unitary time evolution circuit with matrix product
states (tMPS) accurately recover the exponents uniquely associated with the
corresponding non-unitary CFT. We provide an explicit proposal for observing
YLES criticality in Floquet quenched Rydberg atomic arrays with laser-induced
loss, which paves the way towards an universal platform for simulating non-
Hermitian many-body dynamical phenomena.

Introduction.– In 1952, Yang and Lee established a relationship between phase
transitions and special points where the partition function vanish, also known
as Yang-Lee zeros [1, 2]. For a spin model in the thermodynamic limit, non-
unitary critical points known as Yang-Lee edge singularities (YLES) [3, 4, 5,
6] lie at the ends of a dense line of partition function zeros in the space of
complex control parameters such as the magnetic field or inverse temperature
[1, 2]. To observe YLES, one can examine a non-Hermitian quantum ferromagnetic
many-body Hamiltonian involving complex magnetic fields [7, 8, 5, 6]. The YLES
of such non-Hermitian ferromagnetic models lead to anomalous critical scaling
behaviours associated with their governing non-unitary conformal field
theories (CFTs) [9, 8, 6].

For a long time, Yang-Lee edge singularities have been deemed as purely
theoretical constructs since it is experimentally challenging to realize the
requisite imaginary field. Recently, it was realized that the partition
function of a classical spin model can also be mathematically simulated by
real-time evolution, and Yang-Lee zeros were finally observed through probing
spin coherence in a series of landmark experiments involving externally
coupled local spins [10, 11, 12]. However, what these experiments achieved was
the measurement of partition function zeros through a dynamical process, not
the physical observation of the YLES and their associated non-unitary phase
transition. It is still difficult to realize such esoteric phase transitions
in a finite-size quantum ferromagnetic model with physical complex fields.
While various single-body non-Hermitian phenomena have already been
demonstrated [13, 14, 15], experimental demonstrations in interacting many-
body non-Hermitian models have just begun, primarily with cold atoms [16, 17,
18, 19, 20].

Indeed, ultracold atomic systems have lately proven to be ideal platforms for
simulating many-body physics due to their excellent tunability and high
controllability [21, 22, 23, 24], with demonstrated successes in topological
physics [25, 26], strongly correlated matter [27, 28, 29] and thermalization
phenomena [30, 31]. Rydberg atomic arrays are particularly promising, with
Rydberg interactions successfully deployed to simulate many-body Hamiltonians
[32, 33, 34, 35, 36, 37, 38]. Rydberg-dressing techniques enjoy great
tunability in shaping interactions, leading to new prospects in engineering
complicated many-body phases [39, 40, 41, 42, 43, 44, 45, 46, 47]. To
introduce non-Hermiticity in an ultracold atomic lattice, an increasingly
established approach [48, 19, 49, 50, 51, 52, 20] is laser-induced loss, i.e.,
the application of lasers onto trapped atoms such that they are excited to
“external” states [18, 53, 19]. Encouraged by these recent rapid advances in
ultracold atoms, we combine Rydberg-dressing techniques with laser-induced
loss to realize a ferromagnetic chain with an imaginary effective field, such
as to observe the YLES and its critical scaling in a genuinely physical
complex parameter space.

In this work, we first introduce a transverse-field Ising model with imaginary
field and discuss signatures of its non-unitary YLES criticality associated
with spontaneous $\mathcal{P}\mathcal{T}$ breaking [54, 55, 56]. We next
devise a Floquet quench for observing the YLES phase boundary by measuring
kinked dynamical responses in a non-equilibrium setting, before describing our
proposed experiment involving a dissipative Rydberg-dressed optical tweezer
array [41, 42, 46, 57, 6], where the imaginary field is implemented through
laser-induced atom loss [18, 53, 19].

Figure 1: Yang-Lee edge singularities (YLES) of $\hat{H}_{\text{TFI}}$ (Eq.
(1)) and their associated ground state discontinuities (which cannot be
directly measured yet): (a) Phase diagram of $\hat{H}_{\rm TFI}$ in the space
of real and complex fields $h_{x}$ and $\gamma$. The YLES (dashed curve) is
the phase boundary demarcating the paramagnetic (PM) ${\rm Im}E_{\rm g}=0$
phase and the ferromagnetic (FM) $|{\rm Im}E_{\rm g}|>0$ phase, where $E_{g}$
is the ground state eigenenergy with the minimal real part. (b) Plots of ${\rm
Im}E_{\rm g}$ vs. $\gamma$ at the four values of $h_{x}$ indicated in (a),
revealing that $E_{g}$ is non-analytic at YLES $\gamma_{YL}$. (c)
Identification of Yang-Lee phase transitions by the ground state magnetization
order parameter
$M_{x}=\left|\left\langle\sum_{j}\hat{\sigma}^{x}_{j}/L\right\rangle\right|$.
Magnetization kinks (dashed lines) occur at the same $\gamma_{YL}$ as in (b).
(d) The critical point (at $\gamma_{YL}=0.1837$ for $h_{x}=1.5$) can be
equivalently extracted from divergences in either the derivative of the
imaginary ground state energy $\frac{d|{\rm Im}(E_{\rm g}(\gamma))|}{d\gamma}$
(blue), or that of the ground-state x-magnetization $\frac{dM_{x}}{d\gamma}$
(red). All results are obtained from exact diagonalization (ED) with open
boundary conditions, with interaction strength $J=1$ and system size $L=8$.

Model for Yang-Lee edge singularities.– For a concrete platform for observing
Yang-Lee criticality, we consider the prototypical non-Hermitian ferromagnetic
transverse-field Ising chain [58, 59], which we will later show how to realize
with Rydberg atoms:

$\hat{H}_{\rm
TFI}=-\sum_{j}(h_{x}\hat{\sigma}_{j}^{x}+J\hat{\sigma}^{z}_{j}\hat{\sigma}^{z}_{j+1})+\sum_{j}i\gamma\hat{\sigma}^{z}_{j},$
(1)

where $J$ sets the strength of the interaction, $\gamma$ is the imaginary
field strength and $\hat{\sigma}_{j}^{\alpha}(\alpha=x,z)$ are Pauli
matrices.The YLES for this model form a curve in the plane of real and
imaginary magnetic fields $(h_{x},\gamma)$ in FIG. 1 (a) [6], which we denote
as $\gamma_{YL}$ at each value of $h_{x}$. When approaching $\gamma_{YL}$, the
ground states of our model experience spontaneous $\mathcal{P}\mathcal{T}$
symmetry breaking [6], with real ground-state eigenenergies $E_{g}$ splitting
into complex eigenenergies with equal and opposite $\text{Im}E_{g}$, which
demarcate the paramagnetic and ferromagnetic ground states in our model, as
shown in FIG. 1 (a) for $\hat{H}_{\rm TFI}$: the paramagnetic (ferromagnetic)
phases are characterized by vanishing (non-vanishing) $\text{Im}E_{g}$.

Even though YLES were originally defined for classical systems in the
thermodynamic limit, they equivalently exist in finite-size quantum systems
due to a quantum-classical mapping [60, 61, 9, 6], as shown in FIGs. 1
(a),(b), which were computed through exact diagonalization (ED) with $L=8$
sites. In FIGs. 1 (c),(d), the associated ground state magnetization
$M_{z}=\left|\left\langle\sum_{j}\hat{\sigma}^{z}_{j}/L\right\rangle\right|$
and
$M_{x}=\left|\left\langle\sum_{j}\hat{\sigma}^{x}_{j}/L\right\rangle\right|$
also exhibit kinks at these YLES locations. Indeed, comparing the derivatives
$\frac{d|({\rm Im}E_{\rm g}(\gamma))|}{d\gamma}$ against that of the
x-magnetization $|\frac{dM_{x}(\gamma)}{d\gamma}|$ (FIG. 1 (d)), we observe
divergences at the same $\gamma=\gamma_{YL}=0.1837$ where the YLES is located
(for $J=1,h_{x}=1.5$).

Dynamical response from Yang-Lee edge singularities.– Since Eq. 1 is non-
Hermitian, any physical realization, cold atom or otherwise, will undergo non-
equilibrium evolution. That makes it difficult to directly probe ground state
properties such as Yang-Lee criticality via a static ensemble measurement.
Detecting a specific critical ground state transition through dynamics
requires different dynamical behaviors across its two phases. In particular,
the ground state of one phase can dominate the dynamics and be isolated [62,
63, 64, 6].

Figure 2: (a) Flow of the complex spectrum of $\hat{H}_{\text{TFI}}$ (Eq. (1))
as $\gamma$ is tuned across $\gamma_{YL}\approx 0.1837$. The ground state
eigenenergies (boxed) undergo spontaneous $\mathcal{P}\mathcal{T}$ symmetry
breaking and rapidly acquires imaginary parts (green to yellow, red arrow) as
$\gamma$ increases slightly above $\gamma_{YL}$, much more so than other
eigenenergies. (b) and (c): Evolution of the overlap
$|\bra{\psi(T)}\ket{\psi_{i}}|$ between the dynamically evolved state
$\ket{\psi(T)}\\!=\\!e^{-iT\hat{H}_{\rm
TFI}}\ket{\psi(0)}/||e^{-iT\hat{H}_{\rm TFI}}\ket{\psi(0)}||$ and all
eigenstates $\ket{\psi_{i}}$ (blue) of $\hat{H}_{\rm TFI}$, with initial state
being $\ket{\psi(0)}=\ket{\downarrow\downarrow......}$. For $\gamma$ below the
YLES $\gamma_{YL}$ (b), the ground state overlap (red dashed) decreases
rapidly due to mixing. But for $\gamma>\gamma_{YL}$, the ground state
component becomes dominant beyond time $TJ\sim\mathcal{O}(10)$ due to the
large $\text{Im}E_{g}$ of the ground state, leading to kinked magnetization
responses detectable in our proposed Rydberg atom system of FIG. 4 below. All
results are obtained from ED with $J=1$, $h_{x}=1.5$ and $L=8$.

To design a dynamical means to probe the YLES, we turn to the spectral flow
across the critical transition. The under-appreciated but crucial observation
is that due to $\mathcal{PT}$-symmetry breaking, imaginary eigenenergies
appear and that leads to markedly different non-unitary dynamics across the
transition [6]. From FIG. 2 (a), the ground state energy $E_{g}$ (with
smallest $\text{Re}E_{g}$) is seen to rapidly acquire larger
$\pm\text{Im}E_{g}$ immediately after $\gamma$ is tuned to be greater than
$\gamma_{YL}$ (inset). By contrast, other non-ground state eigenenergies in
FIG. 2 (a) are largely stationary. This drastic real-to-complex ground state
eigenenergy transition does not just imply that ground state observables i.e.
$M_{x}$ exhibit a kink at the YLES - more importantly, the rapidly increasing
$\text{Im}E_{g}$ at $\gamma>\gamma_{YL}$ suggests that upon time evolution by
$\hat{H}_{\text{TFI}}$, any initial state with significant ground state
overlap will converge towards the ground state and dominate.

As such, our proposal to experimentally detect non-unitary critical YLES
involves measuring the dynamical $x$-magnetization order parameter
$M_{x}(T)\\!=\\!\bra{\psi(t)}\sum_{j}\hat{\sigma}^{x}_{j}/L\ket{\psi(t)}$,
which we henceforth expect to exhibit similar kinks as the ground state
$M_{x}$ (FIG. 1 (c)). The edge of such a non-unitary phase transition can be
mapped out in parameter space by plotting the kink locations of $M_{x}(T)$.
With our Rydberg array implementation in mind, we propose the following
protocol:

  1. 1.

Prepare a ferromagnetic initial state
$\ket{\psi(0)}=\ket{\downarrow\downarrow......}$, which in a Rydberg array,
can be achieved by first optically pumping all atoms into either
$\ket{\uparrow}$ or $\ket{\downarrow}$. Next, a microwave field which couples
the two ground states can pump all atoms to $\ket{\downarrow}$ [65].

  2. 2.

Apply a dynamical quench on this ferromagnetic initial state by evolving it
under $\hat{H}_{\text{TFI}}$: $\ket{\psi(t)}\\!=\\!e^{-it\hat{H}_{\rm
TFI}}\ket{\psi(0)}/||e^{-it\hat{H}_{\rm TFI}}\ket{\psi(0)}||$, where
$||\ket{\psi(t)}||=\sqrt{|\langle\psi(t)|\psi(t)\rangle|}$ gives the
normalization.

  3. 3.

Measure the x-magnetization order parameter
$M_{x}(T)\\!=\\!|\bra{\psi(T)}\sum_{j}\hat{\sigma}^{x}_{j}\ket{\psi(T)}|/L$
after a sufficiently long stipulated time $T$ for different $\gamma$, keeping
$J$ and $h_{x}$ fixed.

As shown in FIGs. 2 (b),(c), the initial ferromagnetic state $|\psi(0)\rangle$
already overlaps with the ground state of $\hat{H}_{\text{TFI}}$ more than
most other eigenstates. Due to PT-symmetry breaking, after evolving for
$TJ\sim\mathcal{O}(10^{1})$, it is dominated by the ground state for
$\gamma>\gamma_{YL}$, but not when $\gamma<\gamma_{YL}$. As such, we expect to
observe a kink in $M_{x}(T)$ across the critical YLES $\gamma=\gamma_{YL}$,
even though there is considerable ground state overlap at one side of the
transition. Note that even though the ground state $M_{x}(T)$ is extracted
through a dynamical quench, what we are measuring is not mathematically a
dynamical phase transition, as previously measured in Refs. [10, 11, 12, 66]
through partition function zeroes. Instead, our proposal allows for the
demonstration of non-unitary criticality by providing access to a non-
Hermitian Ising chain with complex fields from atom loss in an optical array.

Nonunitary criticality of Yang-Lee edge singularities.– Next, to show how our
protocol above can be used to extract the critical exponents of our YLES
through finite-size scaling. After fixing the quenching duration $T$, the
critical values $\gamma=\gamma^{L}_{YL}$ at different system sizes $L$ can be
extracted from the peak divergences of the plots of the derivative of
$M_{x}(T)$ with respect to $\gamma$. This is illustrated in FIG. 3 (a) with
data from our tMPS simulation (described shortly after), computed with
$T=20/J$ with $J=1$ [6]. Here, for different system sizes $L=8,10,...,16$, the
critical values of $\gamma=\gamma^{L}_{YL}$ are marked by dashed vertical
lines, where the derivatives peak. In an experiment, $\gamma^{L}_{YL}$ can be
extracted by adjusting $\gamma$ and measuring $M_{x}(T)$ in separate spin
chains of various lengths $L$.

Figure 3: Anomalous critical scaling of dynamically-determined Yang-Lee edge
singularities (YLES) of $\hat{H}_{\text{TFI}}$, with $J=1$ and $h_{x}=1.5$.
(a) Size-dependent YLES $\gamma_{YL}^{L}$ as kinks in the derivative
$\frac{dM_{x}(T)}{d\gamma}$. The dynamical magnetization
$M_{x}(T)=|\bra{\psi(T)}\sum_{j}\hat{\sigma}^{x}_{j}\ket{\psi(T)}|/L$ is
measured at time $T=20/J$ for different system sizes $L=8$ to $16$ using our
tMPS algorithm from (d). (b) Extrapolation of $\gamma_{YL}^{L}$ with $1/L$
gives the thermodynamic limit YLES value $\gamma_{YL}^{\infty}\approx 0.139$
through polynomial fitting. (c) Plotting our data for
$\log(\gamma_{YL}^{L}-\gamma_{YL}^{\infty})$ against $-\log L$ yields the
critical exponent $\alpha\approx 2.423$ as the gradient (Eq. (2)), which
agrees closely with the CFT result $\alpha=12/5$ [67, 68]. (d) One time step
($\delta t=0.01$) of the Hamiltonian time evolution in our tMPS-based second-
order Suzuki-Trotter decomposition algorithm [6]. Each step $U(\delta t)$ of
Eq.(3) is decomposed into odd-bond $U_{\text{odd}}(\delta t/2)$ (light green)
and even-bond $U_{\text{even}}(\delta t)$ (dark green) parts, with
normalization (red solid line) performed at the end.

The characteristic critical exponents for the Yang-Lee singularity can be
extracted via the following universal critical scaling law, which holds for
its non-unitary $c=-22/5$ conformal field theory [69, 9]:

$\displaystyle\gamma^{L}_{YL}-\gamma^{\infty}_{YL}\propto
L^{-\alpha}=L^{-\left(\beta_{1}\delta_{1}/\nu_{1}\right)},$ (2)

where $\gamma^{L}_{YL}$ is the location of the YLES obtained from our protocol
at finite size $L$. $\gamma^{\infty}_{YL}$ represents the YLES as
$L\rightarrow\infty$, which can be obtained from our finite-sized data by
extrapolating $\gamma^{L}_{YL}$ with respect to $1/L$, as performed by
polynomial fitting in FIG. 3 (b). Upon obtaining $\gamma_{YL}^{\infty}$, one
can further plot $\log(\gamma^{L}_{YL}-\gamma^{\infty}_{YL})$ against
$\log(1/L)$, such that the critical exponent $\alpha$ in Eq. 2 can be
extracted from the gradient of the fitted line in FIG. 3 (c). From our tMPS
calculations, we obtain $\alpha_{\rm MPS}\approx 2.423$, which is in excellent
agreement with its theoretical value from the non-unitary CFT with central
charge $c=-22/5$ for the YLES: $\alpha_{\rm
CFT}=\beta_{1}\delta_{1}/\nu_{1}=2.40$ with
$\beta_{1}=1,\delta_{1}=-6,\nu_{1}=-5/2$ [67, 68].

Before proposing how our protocol for finite-size systems can be
experimentally implemented, we briefly describe the numerical approach of time
evolution with matrix product states (tMPS) for computing the results in FIG.
3, which is a state-of-art tool for handling generic one-dimensional quantum
many-body systems with nearest-neighbor couplings [70, 71, 72]. We implement
the discretized non-unitary time evolution operator $e^{-i\delta t\hat{H}_{\rm
TFI}}$ for our non-Hermitian Hamiltonian through a non-unitary circuit [73,
74, 75, 6]. As sketched in FIG. 3 (d) [6], for each time step, we utilize a
second-order Suzuki-Trotter decomposition as

$U(\delta t)=U_{\text{odd}}(\delta t/2)U_{\text{even}}(\delta
t)U_{\text{odd}}(\delta t/2)+\mathcal{O}(\delta t^{3}),$ (3)

where the non-unitary components are embedded in every even and odd bond [6].
At the end of each time step, normalization is applied to suppress numerical
divergences: this gives a direct implementation of non-unitary dynamics more
efficient than ancilla-based approaches with significant information wastage
[73, 76]. In this work, we only consider open boundary conditions 111Due to
the absence of the non-Hermitian skin effect, we do not expect the results to
differ significantly between open and periodic boundary conditions., which is
less costly for MPS calculations.

Figure 4: Proposed Rydberg atomic array for realizing the Floquet drives
$U_{1}^{\text{Floq}}$, $U_{2}^{\text{Floq}}$ for the dynamical
$\hat{H}_{\text{TFI}}$ evolution (Eq. (4)). Optical tweezers (red, Left) trap
Cesium atoms (cyan), with their two hyperfine ground states defining
pseudospin-$1/2$ states
$\ket{\downarrow}=\left|6S_{1/2},F=3,m_{F}=0\right\rangle$ and
$\ket{\uparrow}=\left|6S_{1/2},F=4,m_{F}=0\right\rangle$ (Right). In
$U_{1}^{\text{Floq}}$ (Step 1), $\Omega$ driving (blue dashed arrow) couples
$\ket{\uparrow}$ and the Rydberg state
$|R\rangle=\left|43P_{3/2}\right\rangle$, resulting in a Rydberg-dressed
nearest-neighbor $\ket{\uparrow\uparrow}$ interaction $\hat{H}_{\text{int}}$
(purple arrow) that can be converted to a ferromagnetic interaction via
additional transverse field kicks (red loop). In $U_{2}^{\text{Floq}}$ (Step
2), a microwave field (brown loop) generates the transverse field
$F\hat{\sigma}^{x}$, while another depopulates $\ket{\uparrow}$, resulting in
laser-induced loss interpreted as the imaginary field as $ig\hat{\sigma}^{z}$
[6]. The quenching durations $\tau_{J}$, $\tau_{h_{x}}$ and $\tau_{\gamma}$
are chosen such that
$(\tau_{h_{x}}F,\tau_{\gamma}g,\tau_{J}J_{0})=\frac{T}{N}(h_{x},\gamma,J)$,
where $N\gg 1$.

Experimental proposal with Rydberg atoms.– To observe YLES in Rydberg atomic
arrays through our driving protocol, what needs to be implemented is the time
evolution operator $e^{-iT\hat{H}_{\rm TFI}}$. For this, we note that in a
Rydberg system [46], it is feasible to implement a transverse field
Hamiltonian $\hat{H}_{X}(F)=-\sum_{j}F\hat{\sigma}_{j}^{x}$, an atomic loss
$\hat{H}_{Z}(g)=i\sum_{j}g\hat{\sigma}_{j}^{z}$ and, as we shall show, a
ferromagnetic interaction $\hat{H}_{ZZ}$ that is proportional to a coupling
constant $J_{0}$. As such, we can Trotter-approximate $e^{-i{T}\hat{H}_{\rm
TFI}}$ by a two-step Floquet driving protocol (FIG. 4)

$\displaystyle e^{-i{T}\hat{H}_{\rm TFI}}$
$\displaystyle\approx[(e^{i\frac{T}{N}\sum_{j}(h_{x}\hat{\sigma}_{j}^{x}-i\gamma\hat{\sigma}^{z}_{j})})(e^{i\frac{T}{N}\sum_{j}J\hat{\sigma}^{z}_{j}\hat{\sigma}^{z}_{j+1}})]^{N}$
(4)
$\displaystyle=\left[e^{+i\tau_{h_{x}}\hat{H}_{X}(\\!F\\!)+i\tau_{\gamma}\hat{H}_{Z}(\\!g\\!)}e^{+i\tau_{J}\hat{H}_{ZZ}}\right]^{N}$
$\displaystyle=[U_{2}^{\text{Floq}}U_{1}^{\text{Floq}}]^{N},$

which consists of $N$ Floquet periods, each period being a two-step quench
governed by $U_{1}^{\text{Floq}}=e^{i\tau_{J}\hat{H}_{ZZ}}$ followed by
$U_{2}^{\text{Floq}}=e^{i\tau_{h_{x}}\hat{H}_{X}(\\!F\\!)+i\tau_{\gamma}\hat{H}_{Z}(\\!g\\!)}$.
As elaborated below, $F$, $g$ and $J_{0}$ are respectively the magnitudes of
the physical interactions, transverse fields and atom loss, and are related to
their corresponding quenching durations $\tau_{J}$, $\tau_{h_{x}}$ and
$\tau_{\gamma}$ via
$(\tau_{h_{x}}F,\tau_{\gamma}g,\tau_{J}J_{0})=\frac{T}{N}(h_{x},\gamma,J)$.
They should be chosen such that the number of Floquet cycles
$N=\frac{TJ}{J_{0}\tau_{J}}\gg 1$, so as to minimize the Trotterization error
222In this limit, the first order correction can sometimes already give rise
to interesting new phenomena [121, 122, 123, 124, 125, 126], particularly in
many-body settings [127, 128, 129].. We next detail how the Floquet quenches
$U_{1}^{\text{Floq}}$ and $U_{2}^{\text{Floq}}$ can be implemented in our
Rydberg setup. As shown in FIG. 4, we encode the two hyperfine ground states
of trapped Caesium atoms to form a pseudospin-1/2:
$\ket{\downarrow}=\left|6S_{1/2},F=3,m_{F}=0\right\rangle$ and
$\ket{\uparrow}=\left|6S_{1/2},F=4,m_{F}=0\right\rangle$ [46]. To engineer
interactions in the first step $U_{1}^{\text{Floq}}$, we employ Rydberg-
dressing [41, 42, 79, 46] by coupling the state $\ket{\uparrow}$ with the
Rydberg state $\ket{R}=\ket{43P_{3/2}}$. As derived in [6], in the limit of
large detuning $\Delta$ vs. Rabi frequency $\Omega$ (blue arrow in FIG. 4),
the interplay between the bare Rydberg interactions and $\ket{R}$ leads to an
energy shift of $J_{0}\approx\frac{\Omega^{4}}{8\Delta^{3}}$ [40]. In the
subspace $\ket{\uparrow\uparrow}_{i,i+1}$ of the $\ket{\uparrow}$ states of
nearest neighbor atoms, $J_{0}$ behaves as an effective interaction [40, 80,
6]

$\displaystyle\hat{H}_{\rm
int}(J_{0})=-\sum_{i}J_{0}\hat{P}_{i}\hat{P}_{i+1},$ (5)

where $\hat{P}_{i}=\frac{\hat{\sigma}_{i}^{z}+\hat{I}_{i}}{2}$ with
$\hat{\sigma}^{z}_{i}=\ket{\uparrow}_{i}\bra{\uparrow}_{i}-\ket{\downarrow}_{i}\bra{\downarrow}_{i}$
and
$\hat{I}_{i}=\ket{\uparrow}_{i}\bra{\uparrow}_{i}+\ket{\downarrow}_{i}\bra{\downarrow}_{i}$.
While this is still not the ferromagnetic interaction required in
$\hat{H}_{\text{TFI}}$ of Eq.(4), inspired by the operator identity
$e^{-i\frac{\pi}{2}\hat{\sigma}^{x}}e^{i\tau_{J}{J_{0}}\hat{\sigma}^{z}}e^{-i\frac{\pi}{2}\hat{\sigma}^{x}}e^{i\tau_{J}{J_{0}}\hat{\sigma}^{z}}=-\hat{I}$,
one can convert it into a clean ferromagnetic interaction
$\hat{H}_{ZZ}(J_{0})=-\sum_{j}J_{0}\hat{\sigma}_{i}^{z}\hat{\sigma}_{i+1}^{z}$
by applying two transverse $\hat{\sigma}^{x}$ field kicks:

$\displaystyle(e^{-i\frac{\pi}{2}\sum_{j}\hat{\sigma}_{j}^{x}}e^{+i2\tau_{J}\hat{H}_{\rm
int}(J_{0})})^{2}\approx e^{+i\tau_{J}\hat{H}_{ZZ}}=U_{1}^{\text{Floq}},$ (6)

as elaborated in the supplemental materials [6]. Such transverse-field kicks
can be generated by microwave fields, shown as the red circle in FIG. 4 [46].

To realize the next evolution step $U_{2}^{\text{Floq}}$, the Rydberg dressing
for $U_{1}^{\text{Floq}}$ is turned off immediately, and another microwave
field (brown circle in FIG. 4) is turned on to generate the transverse field
$\hat{H}_{X}(F)=-\sum_{j}F\hat{\sigma}_{j}^{x}$ [46]. At the same time, a
strong laser is shone on $\ket{\uparrow}$ such as to excite it to another
state $\ket{6P_{3/2},F=5}$, leading to effective laser-induced loss
$\hat{H}_{Z}(g)=\sum_{j}ig\hat{\sigma}_{j}^{z}$ with imaginary field/decay
rate $g$ [81, 48, 50, 51, 82, 6].

After repeatedly alternating between Floquet steps $U_{1}^{\text{Floq}}$ and
$U_{2}^{\text{Floq}}$ (FIG. 4) over $N=\frac{TJ}{J_{0}\tau_{J}}$ iterations,
the dynamically evolved magnetization $M_{x}(T)$ can be obtained by measuring
the normalized populations in the $\ket{\uparrow}$ and $\ket{\downarrow}$
levels [14, 6]. The YLES can be observed as kinks in $M_{x}(T)$ as
$\tau_{\gamma}$ or $g$ are tuned (Fig. 3). From that, the associated anomalous
scaling behavior and exponents (Eq. 2) can be simply be extracted by
controlling the number of trapped atoms $L$ [36, 83].

Discussions.– The ground state properties of non-Hermitian quantum systems are
often deemed experimentally inaccessible due to overwhelming decoherence or
the lack of thermal equilibrium. Yet, for the Yang-Lee phase transition in our
model, we found that the spontaneously broken $\mathcal{PT}$-symmetry can give
rise to a pronounced kink in the dynamical magnetization $M_{x}(T)$ without
the need for reaching thermal equilibrium. As such, we provide a realistic
Floquet evolution protocol for observing the YLES and its associated non-
unitary phase transition in a Rydberg chain, distinct from the observation of
partition function zeroes in previous experiments [10, 11, 12]. Our proposal
paves the way for future experimental observation of not just the YLES, but
also other non-unitary phase transitions [84, 85, 86, 87, 82].

The rapid development of hardware and algorithms in universal quantum
computation also opens up the possibility of implementing our YLES measurement
protocol in quantum computers via ancilla-based methods [88, 89, 90, 91].
Moreover, our MPS implementation, which is related to mid-circuit measurements
[92, 93, 94, 95], provides an approach to improve the current ancilla-based
methods for dynamically simulating various non-Hermitian many-body phenomena
[96, 97, 98, 99, 100, 101, 102, 103] and unconventional non-Hermitian topology
[104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117], on
quantum circuits.

###### Acknowledgements.

Acknowledgements.– T. C. thanks Bo Yang for fruitful discussions. The exact
diagonalization is numerically computed with QuSpin Python library [118, 119],
and the MPS results are calculated with ITensor [120]. T. C. acknowledges
support from the National Research Foundation, Singapore under the NRF
fellowship award (NRF-NRFF12-2020-005). F. Q. is supported by the National
Research Foundation, Singapore under its QEP2.0 programme
(NRF2021-QEP2-02-P09).

## References

  * Yang and Lee [1952] C.-N. Yang and T.-D. Lee, Statistical theory of equations of state and phase transitions. i. theory of condensation, Physical Review 87, 404 (1952).
  * Lee and Yang [1952] T.-D. Lee and C.-N. Yang, Statistical theory of equations of state and phase transitions. ii. lattice gas and ising model, Physical Review 87, 410 (1952).
  * Cardy [1985] J. L. Cardy, Conformal invariance and the yang-lee edge singularity in two dimensions, Physical review letters 54, 1354 (1985).
  * Von Gehlen [1991] G. Von Gehlen, Critical and off-critical conformal analysis of the ising quantum chain in an imaginary field, Journal of Physics A: Mathematical and General 24, 5371 (1991).
  * Jian  _et al._ [2021] S.-K. Jian, Z.-C. Yang, Z. Bi, and X. Chen, Yang-lee edge singularity triggered entanglement transition, Physical Review B 104, L161107 (2021).
  * [6] Supplemental Materials .
  * Fisher [1980] M. E. Fisher, Yang-lee edge behavior in one-dimensional systems, Progress of Theoretical Physics Supplement 69, 14 (1980).
  * Matsumoto  _et al._ [2020] N. Matsumoto, M. Nakagawa, and M. Ueda, Embedding the yang-lee quantum criticality in open quantum systems, arXiv preprint arXiv:2012.13144 (2020).
  * Yin  _et al._ [2017] S. Yin, G.-Y. Huang, C.-Y. Lo, and P. Chen, Kibble-zurek scaling in the yang-lee edge singularity, Physical Review Letters 118, 065701 (2017).
  * Peng  _et al._ [2015] X. Peng, H. Zhou, B.-B. Wei, J. Cui, J. Du, and R.-B. Liu, Experimental observation of lee-yang zeros, Physical review letters 114, 010601 (2015).
  * Francis  _et al._ [2021] A. Francis, D. Zhu, C. Huerta Alderete, S. Johri, X. Xiao, J. K. Freericks, C. Monroe, N. M. Linke, and A. F. Kemper, Many-body thermodynamics on quantum computers via partition function zeros, Science Advances 7, eabf2447 (2021).
  * Wei and Liu [2012] B.-B. Wei and R.-B. Liu, Lee-yang zeros and critical times in decoherence of a probe spin coupled to a bath, Physical review letters 109, 185701 (2012).
  * Ding  _et al._ [2021] L. Ding, K. Shi, Q. Zhang, D. Shen, X. Zhang, and W. Zhang, Experimental determination of p t-symmetric exceptional points in a single trapped ion, Physical Review Letters 126, 083604 (2021).
  * Zhou  _et al._ [2021] L. Zhou, H. Li, W. Yi, and X. Cui, Engineering non-hermitian skin effect with band topology in ultracold gases, arXiv preprint arXiv:2111.04196 (2021).
  * Liang  _et al._ [2022a] Q. Liang, D. Xie, Z. Dong, H. Li, H. Li, B. Gadway, W. Yi, and B. Yan, Dynamic signatures of non-hermitian skin effect and topology in ultracold atoms, Physical review letters 129, 070401 (2022a).
  * Malossi  _et al._ [2014] N. Malossi, M. Valado, S. Scotto, P. Huillery, P. Pillet, D. Ciampini, E. Arimondo, and O. Morsch, Full counting statistics and phase diagram of a dissipative rydberg gas, Physical Review Letters 113, 023006 (2014).
  * Goldschmidt  _et al._ [2016] E. A. Goldschmidt, T. Boulier, R. C. Brown, S. B. Koller, J. T. Young, A. V. Gorshkov, S. Rolston, and J. V. Porto, Anomalous broadening in driven dissipative rydberg systems, Physical review letters 116, 113001 (2016).
  * Li  _et al._ [2019] J. Li, A. K. Harter, J. Liu, L. de Melo, Y. N. Joglekar, and L. Luo, Observation of parity-time symmetry breaking transitions in a dissipative floquet system of ultracold atoms, Nature communications 10, 1 (2019).
  * Ren  _et al._ [2022] Z. Ren, D. Liu, E. Zhao, C. He, K. K. Pak, J. Li, and G.-B. Jo, Chiral control of quantum states in non-hermitian spin–orbit-coupled fermions, Nature Physics 18, 385 (2022).
  * Liang  _et al._ [2022b] Q. Liang, D. Xie, Z. Dong, H. Li, H. Li, B. Gadway, W. Yi, and B. Yan, Observation of non-hermitian skin effect and topology in ultracold atoms, arXiv preprint arXiv:2201.09478 (2022b).
  * Bloch [2005] I. Bloch, Ultracold quantum gases in optical lattices, Nature physics 1, 23 (2005).
  * Bloch  _et al._ [2008] I. Bloch, J. Dalibard, and W. Zwerger, Many-body physics with ultracold gases, Reviews of modern physics 80, 885 (2008).
  * Giorgini  _et al._ [2008] S. Giorgini, L. P. Pitaevskii, and S. Stringari, Theory of ultracold atomic fermi gases, Reviews of Modern Physics 80, 1215 (2008).
  * Bloch  _et al._ [2012] I. Bloch, J. Dalibard, and S. Nascimbene, Quantum simulations with ultracold quantum gases, Nature Physics 8, 267 (2012).
  * Miyake  _et al._ [2013] H. Miyake, G. A. Siviloglou, C. J. Kennedy, W. C. Burton, and W. Ketterle, Realizing the harper hamiltonian with laser-assisted tunneling in optical lattices, Physical review letters 111, 185302 (2013).
  * Jotzu  _et al._ [2014] G. Jotzu, M. Messer, R. Desbuquois, M. Lebrat, T. Uehlinger, D. Greif, and T. Esslinger, Experimental realization of the topological haldane model with ultracold fermions, Nature 515, 237 (2014).
  * Greiner  _et al._ [2002] M. Greiner, O. Mandel, T. Esslinger, T. W. Hänsch, and I. Bloch, Quantum phase transition from a superfluid to a mott insulator in a gas of ultracold atoms, nature 415, 39 (2002).
  * Mazurenko  _et al._ [2017] A. Mazurenko, C. S. Chiu, G. Ji, M. F. Parsons, M. Kanász-Nagy, R. Schmidt, F. Grusdt, E. Demler, D. Greif, and M. Greiner, A cold-atom fermi–hubbard antiferromagnet, Nature 545, 462 (2017).
  * Salomon  _et al._ [2019] G. Salomon, J. Koepsell, J. Vijayan, T. A. Hilker, J. Nespolo, L. Pollet, I. Bloch, and C. Gross, Direct observation of incommensurate magnetism in hubbard chains, Nature 565, 56 (2019).
  * Schreiber  _et al._ [2015] M. Schreiber, S. S. Hodgman, P. Bordia, H. P. Lüschen, M. H. Fischer, R. Vosk, E. Altman, U. Schneider, and I. Bloch, Observation of many-body localization of interacting fermions in a quasirandom optical lattice, Science 349, 842 (2015).
  * Kyprianidis  _et al._ [2021] A. Kyprianidis, F. Machado, W. Morong, P. Becker, K. S. Collins, D. V. Else, L. Feng, P. W. Hess, C. Nayak, G. Pagano, _et al._ , Observation of a prethermal discrete time crystal, Science 372, 1192 (2021).
  * Barredo  _et al._ [2015] D. Barredo, H. Labuhn, S. Ravets, T. Lahaye, A. Browaeys, and C. S. Adams, Coherent excitation transfer in a spin chain of three rydberg atoms, Physical review letters 114, 113002 (2015).
  * Labuhn  _et al._ [2016] H. Labuhn, D. Barredo, S. Ravets, S. De Léséleuc, T. Macrì, T. Lahaye, and A. Browaeys, Tunable two-dimensional arrays of single rydberg atoms for realizing quantum ising models, Nature 534, 667 (2016).
  * de Léséleuc  _et al._ [2017] S. de Léséleuc, D. Barredo, V. Lienhard, A. Browaeys, and T. Lahaye, Optical control of the resonant dipole-dipole interaction between rydberg atoms, Physical Review Letters 119, 053202 (2017).
  * Lin and Motrunich [2019] C.-J. Lin and O. I. Motrunich, Exact quantum many-body scar states in the rydberg-blockaded atom chain, Physical review letters 122, 173401 (2019).
  * Browaeys and Lahaye [2020] A. Browaeys and T. Lahaye, Many-body physics with individually controlled rydberg atoms, Nature Physics 16, 132 (2020).
  * Lin  _et al._ [2020] C.-J. Lin, V. Calvera, and T. H. Hsieh, Quantum many-body scar states in two-dimensional rydberg atom arrays, Physical Review B 101, 220304 (2020).
  * Bluvstein  _et al._ [2021] D. Bluvstein, A. Omran, H. Levine, A. Keesling, G. Semeghini, S. Ebadi, T. T. Wang, A. A. Michailidis, N. Maskara, W. W. Ho, _et al._ , Controlling quantum many-body dynamics in driven rydberg atom arrays, Science 371, 1355 (2021).
  * Bouchoule and Mølmer [2002] I. Bouchoule and K. Mølmer, Spin squeezing of atoms by the dipole interaction in virtually excited rydberg states, Physical Review A 65, 041803 (2002).
  * Johnson and Rolston [2010] J. Johnson and S. Rolston, Interactions between rydberg-dressed atoms, Physical Review A 82, 033412 (2010).
  * Henkel  _et al._ [2010] N. Henkel, R. Nath, and T. Pohl, Three-dimensional roton excitations and supersolid formation in rydberg-excited bose-einstein condensates, Physical review letters 104, 195302 (2010).
  * Pupillo  _et al._ [2010] G. Pupillo, A. Micheli, M. Boninsegni, I. Lesanovsky, and P. Zoller, Strongly correlated gases of rydberg-dressed atoms: quantum and classical dynamics, Physical review letters 104, 223002 (2010).
  * Schempp  _et al._ [2015] H. Schempp, G. Günter, S. Wüster, M. Weidemüller, and S. Whitlock, Correlated exciton transport in rydberg-dressed-atom spin chains, Physical review letters 115, 093002 (2015).
  * Zeiher  _et al._ [2016] J. Zeiher, R. Van Bijnen, P. Schauß, S. Hild, J.-y. Choi, T. Pohl, I. Bloch, and C. Gross, Many-body interferometry of a rydberg-dressed spin lattice, Nature Physics 12, 1095 (2016).
  * Jau  _et al._ [2016] Y.-Y. Jau, A. Hankin, T. Keating, I. H. Deutsch, and G. Biedermann, Entangling atomic spins with a rydberg-dressed spin-flip blockade, Nature Physics 12, 71 (2016).
  * Borish  _et al._ [2020] V. Borish, O. Marković, J. A. Hines, S. V. Rajagopal, and M. Schleier-Smith, Transverse-field ising dynamics in a rydberg-dressed atomic gas, Physical review letters 124, 063601 (2020).
  * Hollerith  _et al._ [2022] S. Hollerith, K. Srakaew, D. Wei, A. Rubio-Abadal, D. Adler, P. Weckesser, A. Kruckenhauser, V. Walther, R. van Bijnen, J. Rui, _et al._ , Realizing distance-selective interactions in a rydberg-dressed atom array, Physical Review Letters 128, 113602 (2022).
  * Lee  _et al._ [2019] W. Lee, M. Kim, H. Jo, Y. Song, and J. Ahn, Coherent and dissipative dynamics of entangled few-body systems of rydberg atoms, Physical Review A 99, 043404 (2019).
  * Li  _et al._ [2020] L. Li, C. H. Lee, and J. Gong, Topological switch for non-hermitian skin effect in cold-atom systems with loss, Physical review letters 124, 250402 (2020).
  * Bienias  _et al._ [2020] P. Bienias, J. Douglas, A. Paris-Mandoki, P. Titum, I. Mirgorodskiy, C. Tresp, E. Zeuthen, M. J. Gullans, M. Manzoni, S. Hofferberth, _et al._ , Photon propagation through dissipative rydberg media at large input rates, Physical review research 2, 033049 (2020).
  * Wintermantel  _et al._ [2020] T. Wintermantel, Y. Wang, G. Lochead, S. Shevate, G. Brennen, and S. Whitlock, Unitary and nonunitary quantum cellular automata with rydberg arrays, Physical Review Letters 124, 070503 (2020).
  * Qin  _et al._ [2023] F. Qin, R. Shen, C. H. Lee, _et al._ , Non-hermitian squeezed polarons, Physical Review A 107, L010202 (2023).
  * Lapp  _et al._ [2019] S. Lapp, F. A. An, B. Gadway, _et al._ , Engineering tunable local loss in a synthetic lattice of momentum states, New Journal of Physics 21, 045006 (2019).
  * Giorgi [2010] G. L. Giorgi, Spontaneous p t symmetry breaking and quantum phase transitions in dimerized spin chains, Physical Review B 82, 052404 (2010).
  * Galda and Vinokur [2018] A. Galda and V. M. Vinokur, Parity-time symmetry breaking in spin chains, Physical Review B 97, 201411 (2018).
  * Wang  _et al._ [2015] X. Wang, T. Liu, Y. Xiong, and P. Tong, Spontaneous pt-symmetry breaking in non-hermitian kitaev and extended kitaev models, Physical Review A 92, 012116 (2015).
  * Schäfer  _et al._ [2020] F. Schäfer, T. Fukuhara, S. Sugawa, Y. Takasu, and Y. Takahashi, Tools for quantum simulation with ultracold atoms in optical lattices, Nature Reviews Physics 2, 411 (2020).
  * Li  _et al._ [2014] C. Li, G. Zhang, X. Zhang, and Z. Song, Conventional quantum phase transition driven by a complex parameter in a non-hermitian pt- symmetric ising model, Physical Review A 90, 012103 (2014).
  * Pires and Macrì [2021] D. P. Pires and T. Macrì, Probing phase transitions in non-hermitian systems with multiple quantum coherences, Physical Review B 104, 155141 (2021).
  * Liu  _et al._ [2015] C.-W. Liu, A. Polkovnikov, A. W. Sandvik, and A. Young, Universal dynamic scaling in three-dimensional ising spin glasses, Physical Review E 92, 022128 (2015).
  * Silvi  _et al._ [2016] P. Silvi, G. Morigi, T. Calarco, and S. Montangero, Crossover from classical to quantum kibble-zurek scaling, Physical Review Letters 116, 225701 (2016).
  * Dóra  _et al._ [2019] B. Dóra, M. Heyl, and R. Moessner, The kibble-zurek mechanism at exceptional points, Nature communications 10, 1 (2019).
  * Xiao  _et al._ [2021] L. Xiao, D. Qu, K. Wang, H.-W. Li, J.-Y. Dai, B. Dóra, M. Heyl, R. Moessner, W. Yi, and P. Xue, Non-hermitian kibble-zurek mechanism with tunable complexity in single-photon interferometry, PRX Quantum 2, 020313 (2021).
  * Haldar  _et al._ [2021] A. Haldar, K. Mallayya, M. Heyl, F. Pollmann, M. Rigol, and A. Das, Signatures of quantum phase transitions after quenches in quantum chaotic one-dimensional systems, Physical Review X 11, 031062 (2021).
  * Borish [2020] V. F. Borish, Many-body spin dynamics with rydberg-dressed atoms, Stanford University PhD Thesis (2020).
  * Brandner  _et al._ [2017] K. Brandner, V. F. Maisi, J. P. Pekola, J. P. Garrahan, and C. Flindt, Experimental determination of dynamical lee-yang zeros, Physical review letters 118, 180601 (2017).
  * Fisher [1978] M. E. Fisher, Yang-lee edge singularity and $\phi$ 3 field theory, Physical Review Letters 40, 1610 (1978).
  * Uzelac  _et al._ [1979] K. Uzelac, P. Pfeuty, and R. Jullien, Yang-lee edge singularity from a real-space renormalization-group method, Physical Review Letters 43, 805 (1979).
  * von Gehlen and Honecker [1993] G. von Gehlen and A. Honecker, Multi-particle structure in the zn-chiral potts models, Journal of Physics A: Mathematical and General 26, 1275 (1993).
  * White [1992] S. R. White, Density matrix formulation for quantum renormalization groups, Phys. Rev. Lett. 69, 2863 (1992).
  * Vidal [2004] G. Vidal, Efficient simulation of one-dimensional quantum many-body systems, Physical review letters 93, 040502 (2004).
  * White and Feiguin [2004] S. R. White and A. E. Feiguin, Real-time evolution using the density matrix renormalization group, Phys. Rev. Lett. 93, 076401 (2004).
  * Gingrich and Williams [2004] R. M. Gingrich and C. P. Williams, Non-unitary probabilistic quantum computing, (2004).
  * Koczor and Benjamin [2019] B. Koczor and S. C. Benjamin, Quantum natural gradient generalised to non-unitary circuits, arXiv preprint arXiv:1912.08660 10 (2019).
  * Xie  _et al._ [2022] H.-N. Xie, S.-J. Wei, F. Yang, Z.-A. Wang, C.-T. Chen, H. Fan, and G.-L. Long, A probabilistic imaginary time evolution algorithm based on non-unitary quantum circuit, arXiv preprint arXiv:2210.05293 (2022).
  * Daskin and Kais [2017] A. Daskin and S. Kais, An ancilla-based quantum simulation framework for non-unitary matrices, Quantum Information Processing 16, 1 (2017).
  * Note [1] Due to the absence of the non-Hermitian skin effect, we do not expect the results to differ significantly between open and periodic boundary conditions.
  * Note [2] In this limit, the first order correction can sometimes already give rise to interesting new phenomena [121, 122, 123, 124, 125, 126], particularly in many-body settings [127, 128, 129].
  * Zeiher  _et al._ [2017] J. Zeiher, J.-y. Choi, A. Rubio-Abadal, T. Pohl, R. Van Bijnen, I. Bloch, and C. Gross, Coherent many-body spin dynamics in a long-range interacting ising chain, Physical review X 7, 041063 (2017).
  * Balewski  _et al._ [2014] J. B. Balewski, A. T. Krupp, A. Gaj, S. Hofferberth, R. Löw, and T. Pfau, Rydberg dressing: understanding of collective many-body effects and implications for experiments, New Journal of Physics 16, 063012 (2014).
  * Saffman  _et al._ [2010] M. Saffman, T. G. Walker, and K. Mølmer, Quantum information with rydberg atoms, Reviews of modern physics 82, 2313 (2010).
  * Lourenço  _et al._ [2022] J. A. Lourenço, G. Higgins, C. Zhang, M. Hennrich, and T. Macrì, Non-hermitian dynamics and pt-symmetry breaking in interacting mesoscopic rydberg platforms, Physical Review A 106, 023309 (2022).
  * Scholl  _et al._ [2021] P. Scholl, M. Schuler, H. J. Williams, A. A. Eberharter, D. Barredo, K.-N. Schymik, V. Lienhard, L.-P. Henry, T. C. Lang, T. Lahaye, _et al._ , Quantum simulation of 2d antiferromagnets with hundreds of rydberg atoms, Nature 595, 233 (2021).
  * Wang  _et al._ [2019] K. Wang, X. Qiu, L. Xiao, X. Zhan, Z. Bian, B. C. Sanders, W. Yi, and P. Xue, Observation of emergent momentum–time skyrmions in parity–time-symmetric non-unitary quench dynamics, Nature communications 10, 1 (2019).
  * Hamazaki [2021] R. Hamazaki, Exceptional dynamical quantum phase transitions in periodically driven systems, Nature communications 12, 1 (2021).
  * Zhang  _et al._ [2022] P. Zhang, C. Liu, S.-K. Jian, and X. Chen, Universal entanglement transitions of free fermions with long-range non-unitary dynamics, Quantum 6, 723 (2022).
  * Lin  _et al._ [2022] Q. Lin, T. Li, L. Xiao, K. Wang, W. Yi, and P. Xue, Topological phase transitions and mobility edges in non-hermitian quasicrystals, Physical Review Letters 129, 113601 (2022).
  * Lin  _et al._ [2021] S.-H. Lin, R. Dilip, A. G. Green, A. Smith, and F. Pollmann, Real-and imaginary-time evolution with compressed quantum circuits, PRX Quantum 2, 010342 (2021).
  * Koh  _et al._ [2022a] J. M. Koh, T. Tai, Y. H. Phee, W. E. Ng, and C. H. Lee, Stabilizing multiple topological fermions on a quantum computer, npj Quantum Information 8, 1 (2022a).
  * Koh  _et al._ [2022b] J. M. Koh, T. Tai, and C. H. Lee, Simulation of interaction-induced chiral topological dynamics on a digital quantum computer, Physical Review Letters 129, 140502 (2022b).
  * Chen  _et al._ [2022] T. Chen, R. Shen, C. H. Lee, and B. Yang, High-fidelity realization of the aklt state on a nisq-era quantum processor, arXiv preprint arXiv:2210.13840 (2022).
  * Pino  _et al._ [2021] J. M. Pino, J. M. Dreiling, C. Figgatt, J. P. Gaebler, S. A. Moses, M. Allman, C. Baldwin, M. Foss-Feig, D. Hayes, K. Mayer, _et al._ , Demonstration of the trapped-ion quantum ccd computer architecture, Nature 592, 209 (2021).
  * Egan  _et al._ [2021] L. Egan, D. M. Debroy, C. Noel, A. Risinger, D. Zhu, D. Biswas, M. Newman, M. Li, K. R. Brown, M. Cetina, _et al._ , Fault-tolerant control of an error-corrected qubit, Nature 598, 281 (2021).
  * Bharti  _et al._ [2022] K. Bharti, A. Cervera-Lierta, T. H. Kyaw, T. Haug, S. Alperin-Lea, A. Anand, M. Degroote, H. Heimonen, J. S. Kottmann, T. Menke, _et al._ , Noisy intermediate-scale quantum algorithms, Reviews of Modern Physics 94, 015004 (2022).
  * Hua  _et al._ [2022] F. Hua, Y. Jin, Y. Chen, J. Lapeyre, A. Javadi-Abhari, and E. Z. Zhang, Exploiting qubit reuse through mid-circuit measurement and reset, arXiv preprint arXiv:2211.01925 (2022).
  * Le Boité  _et al._ [2013] A. Le Boité, G. Orso, and C. Ciuti, Steady-state phases and tunneling-induced instabilities in the driven dissipative bose-hubbard model, Physical review letters 110, 233601 (2013).
  * Sieberer  _et al._ [2013] L. Sieberer, S. D. Huber, E. Altman, and S. Diehl, Dynamical critical phenomena in driven-dissipative systems, Physical review letters 110, 195301 (2013).
  * Joshi  _et al._ [2013] C. Joshi, F. Nissen, and J. Keeling, Quantum correlations in the one-dimensional driven dissipative x y model, Physical Review A 88, 063835 (2013).
  * Landa  _et al._ [2020] H. Landa, M. Schiró, and G. Misguich, Multistability of driven-dissipative quantum spins, Physical Review Letters 124, 043601 (2020).
  * Jian  _et al._ [2020] C.-M. Jian, B. Bauer, A. Keselman, and A. W. Ludwig, Criticality and entanglement in non-unitary quantum circuits and tensor networks of non-interacting fermions, arXiv preprint arXiv:2012.04666 (2020).
  * Pistorius  _et al._ [2020] T. Pistorius, J. Kazemi, and H. Weimer, Quantum many-body dynamics of driven-dissipative rydberg polaritons, Physical Review Letters 125, 263604 (2020).
  * Deuar  _et al._ [2021] P. Deuar, A. Ferrier, M. Matuszewski, G. Orso, and M. H. Szymańska, Fully quantum scalable description of driven-dissipative lattice models, PRX Quantum 2, 010319 (2021).
  * Naji  _et al._ [2022] J. Naji, M. Jafari, R. Jafari, and A. Akbari, Dissipative floquet dynamical quantum phase transition, Physical Review A 105, 022220 (2022).
  * Okuma  _et al._ [2020] N. Okuma, K. Kawabata, K. Shiozaki, and M. Sato, Topological origin of non-hermitian skin effects, Physical review letters 124, 086801 (2020).
  * Lee [2021] C. H. Lee, Many-body topological and skin states without open boundaries, Physical Review B 104, 195102 (2021).
  * Zhai  _et al._ [2022] L.-J. Zhai, G.-Y. Huang, and S. Yin, Nonequilibrium dynamics of the localization-delocalization transition in non-hermitian aubry-andr$\backslash$’$\\{$e$\\}$ model, arXiv preprint arXiv:2205.08271 (2022).
  * Li  _et al._ [2021] L. Li, S. Mu, C. H. Lee, and J. Gong, Quantized classical response from spectral winding topology, Nature communications 12, 5294 (2021).
  * Li and Lee [2022] L. Li and C. H. Lee, Non-hermitian pseudo-gaps, Science Bulletin 67, 685 (2022).
  * Shen and Lee [2022] R. Shen and C. H. Lee, Non-hermitian skin clusters from strong interactions, Communications Physics 5, 1 (2022).
  * Fu and Zhang [2022] Y. Fu and Y. Zhang, Anatomy of open-boundary bulk in multiband non-hermitian systems, arXiv preprint arXiv:2212.13753 (2022).
  * Lee [2022] C. H. Lee, Exceptional bound states and negative entanglement entropy, Physical Review Letters 128, 010402 (2022).
  * Kawabata  _et al._ [2022] K. Kawabata, K. Shiozaki, and S. Ryu, Many-body topology of non-hermitian systems, Physical Review B 105, 165137 (2022).
  * Alsallom  _et al._ [2022] F. Alsallom, L. Herviou, O. V. Yazyev, and M. Brzezińska, Fate of the non-hermitian skin effect in many-body fermionic systems, Physical Review Research 4, 033122 (2022).
  * Qin  _et al._ [2022a] F. Qin, Y. Ma, R. Shen, and C. H. Lee, Universal competitive spectral scaling from the critical non-hermitian skin effect, arXiv preprint arXiv:2212.13536 (2022a).
  * Yoshida and Hatsugai [2023] T. Yoshida and Y. Hatsugai, Fate of exceptional points under interactions: Reduction of topological classifications, Physical Review B 107, 075118 (2023).
  * Poddubny [2023] A. N. Poddubny, Interaction-induced analog of a non-hermitian skin effect in a lattice two-body problem, Physical Review B 107, 045131 (2023).
  * Guo  _et al._ [2023] T. Guo, K. Kawabata, R. Nakai, and S. Ryu, Non-hermitian boost deformation, arXiv preprint arXiv:2301.05973 (2023).
  * Weinberg and Bukov [2017] P. Weinberg and M. Bukov, QuSpin: a Python Package for Dynamics and Exact Diagonalisation of Quantum Many Body Systems part I: spin chains, SciPost Phys. 2, 003 (2017).
  * Weinberg and Bukov [2019] P. Weinberg and M. Bukov, QuSpin: a Python Package for Dynamics and Exact Diagonalisation of Quantum Many Body Systems. Part II: bosons, fermions and higher spins, SciPost Phys. 7, 20 (2019).
  * Fishman  _et al._ [2020] M. Fishman, S. R. White, and E. M. Stoudenmire, The ITensor software library for tensor network calculations (2020), arXiv:2007.14822 .
  * Čadež  _et al._ [2017] T. Čadež, R. Mondaini, and P. D. Sacramento, Dynamical localization and the effects of aperiodicity in floquet systems, Physical Review B 96, 144301 (2017).
  * Li  _et al._ [2018] L. Li, C. H. Lee, and J. Gong, Realistic floquet semimetal with exotic topological linkages between arbitrarily many nodal loops, Physical review letters 121, 036401 (2018).
  * Rudner and Lindner [2019] M. S. Rudner and N. H. Lindner, Floquet topological insulators: from band structure engineering to novel non-equilibrium quantum phenomena, arXiv preprint arXiv:1909.02008 (2019).
  * Nathan  _et al._ [2021] F. Nathan, R. Ge, S. Gazit, M. Rudner, and M. Kolodrubetz, Quasiperiodic floquet-thouless energy pump, Physical Review Letters 127, 166804 (2021).
  * Castro  _et al._ [2022] A. Castro, U. De Giovannini, S. A. Sato, H. Hübener, and A. Rubio, Floquet engineering the band structure of materials with optimal control theory, Physical Review Research 4, 033213 (2022).
  * Qin  _et al._ [2022b] F. Qin, C. H. Lee, R. Chen, _et al._ , Light-induced phase crossovers in a quantum spin hall system, Physical Review B 106, 235405 (2022b).
  * Kuwahara  _et al._ [2016] T. Kuwahara, T. Mori, and K. Saito, Floquet–magnus theory and generic transient dynamics in periodically driven many-body quantum systems, Annals of Physics 367, 96 (2016).
  * Lee  _et al._ [2018] C. H. Lee, W. W. Ho, B. Yang, J. Gong, and Z. Papić, Floquet mechanism for non-abelian fractional quantum hall states, Physical review letters 121, 237401 (2018).
  * Ye  _et al._ [2021] B. Ye, F. Machado, and N. Y. Yao, Floquet phases of matter via classical prethermalization, Physical Review Letters 127, 140603 (2021).
  * Sachdev [1999] S. Sachdev, Quantum phase transitions, Physics world 12, 33 (1999).
  * Sachdev [2011] S. Sachdev, The quantum ising model, in  _Quantum Phase Transitions_ (Cambridge University Press, 2011) p. 58–78, 2nd ed.
  * Wei [2018] B.-B. Wei, Probing conformal invariant of non-unitary two-dimensional systems by central spin decoherence, Scientific Reports 8, 1 (2018).
  * Suzuki [1990] M. Suzuki, Fractal decomposition of exponential operators with applications to many-body theories and monte carlo simulations, Physics Letters A 146, 319 (1990).

### Supplementary Materials for “Proposal for observing Yang-Lee criticality
in Rydberg atomic arrays”

In this supplement, we provide the following:

I. Elaboration on engineering a non-Hermitian transverse-field Ising model in
a Rydberg atomic array.

II. Phase transitions associated with Yang-Lee edge singularities.

III. Details of our tMPS numerical simulation approach and results.

##  Appendix A I. Rydberg atomic platforms

In the main text, we have suggested observing the Yang-Lee edge singularities
(YLES) in a non-Hermitian transverse-field Ising model, which is proposed in a
Rydberg atomic platform. In a Rydberg atomic platform, the Rydberg-dressing
technique can generate ferromagnetic interactions among ground-state atoms
[41, 42, 79, 46], and atom loss leads to imaginary fields [18, 53, 19]. In
this section, we further provide more details for our experimental proposal.

###  A.1 Atoms with Rydberg dressing

To engineer interactions among atoms in our setup by Rydberg-dressing
techniques [41, 42, 79, 46], we couple the ground state
$\ket{\uparrow}=\left|6S_{1/2},F=4,m_{F}=0\right\rangle$ with the Rydberg
state $|R\rangle=\left|43P_{3/2}\right\rangle$ as shown FIG 1 of the main
text. Note that the ground state here refers to the atomic energy level, which
is different from the eigenstate of our transverse-field Ising model in
Eq.(S10) discussed later. To illustrate how such a coupling process leads to
effective interactions, we consider a simple two-atom case, where we couple
the Rydberg state $\ket{R}$ with the ground state $\ket{\uparrow}$ by means of
a Rabi frequency $\Omega$ [40]. This setup can be described by a Hamiltonian
in the basis of $(|\uparrow\uparrow\rangle,1/\sqrt{2}(|\uparrow
R\rangle+|R\uparrow\rangle),|RR\rangle)$:

$\hat{H}_{\rm
dressing}=\left(\begin{array}[]{ccc}0&\frac{\Omega}{\sqrt{2}}&0\\\
\frac{\Omega}{\sqrt{2}}&\Delta&\frac{\Omega}{\sqrt{2}}\\\
0&\frac{\Omega}{\sqrt{2}}&2\Delta+U_{dd}\end{array}\right),$ (S1)

where $\Delta$ is laser detuning at the ground state $\ket{\uparrow}$, and
$U_{dd}$ is for dipole-dipole interaction between two nearest Rydberg states
[40]. We can approximate the ground-state energy of $\hat{H}_{\rm dressing}$
as

$E_{G}\approx\frac{1}{2}\left(\Delta-\sqrt{2\Omega^{2}+\Delta^{2}}\right)$
(S2)

in the strong interaction limit as $U_{dd}\gg\Delta$ [40]. Compared with the
ground-state energy without Rydberg dressing
$E_{G}|_{U_{dd}=0}=\Delta-\sqrt{\Omega^{2}+\Delta^{2}}$, we obtain an energy
shift on states $\ket{\uparrow\uparrow}$, which in the large detuning limit
$\Delta\gg\Omega$ is

$\delta E_{G}\approx-\frac{\Omega^{4}}{8\Delta^{3}},$ (S3)

Such an energy shift can be interpreted as the effective interaction between
two nearest ground-state atoms as $\ket{\uparrow\uparrow}$, which takes the
value of $-2\pi\times 0.18$ kHz under our suggested physical parameters
$\Omega=2\pi\times 1.9$ MHz and $\Delta=2\pi\times 21.0$ MHz [46].

###  A.2 Kicks from transverse fields

As discussed in the previous section, the Rydberg-dressing process produces
the effective interaction as follows

$\hat{H}_{\rm
int}(J_{0})=-J_{0}\sum_{i}\ket{\uparrow\uparrow}_{i,i+1}\bra{\uparrow\uparrow}_{i,i+1}=-\sum_{i}\frac{J_{0}}{4}\left(\hat{\sigma}_{i}^{z}+\hat{I}_{i}\right)\left(\hat{\sigma}_{i+1}^{z}+\hat{I}_{j}\right),$
(S4)

with $J_{0}=\Omega^{4}/\left|8\Delta^{3}\right|$. Here, the long-range
interactions are not significant [40]. To realize clean ferromagnetic
interactions as
$\hat{H}_{ZZ}(J_{0})=-\sum_{i}J_{0}\hat{\sigma}_{i}^{z}\hat{\sigma}_{i+1}^{z}$,
which is required for the transverse field Ising model which is used to
demonstrate YLES, we note the operator identity

$e^{-i\frac{\pi}{2}\hat{\sigma}^{x}}e^{+i\delta\tau{J_{0}}\hat{\sigma}^{z}}e^{-i\frac{\pi}{2}\hat{\sigma}^{x}}e^{+i\delta\tau{J_{0}}\hat{\sigma}^{z}}=\left(\begin{array}[]{cc}-1&0\\\
0&-1\end{array}\right).$ (S5)

Physically, the exponential terms involving $\hat{\sigma}^{x}$ can be
interpreted as two kicks from transverse $\hat{\sigma}^{x}$ fields. The fact
that the right-hand-side of Eq. S5 is proportional to the identity means that
the kicks can remove the $e^{+i\delta\tau{J_{0}}\hat{\sigma}^{z}}$ field.

When generalized to the whole array, such kicks can also eliminate the
intermediate $\hat{\sigma}^{z}$-field contributions in Eq. S4, such as to
convert $\hat{H}_{\text{int}}$ to a clean ferromagnetic interaction. To
describe this explicitly, we first expand the fast-evolving operation in Eq.6
of the main text, dropping errors of subleading order in $\delta\tau$ [46]:

$\displaystyle(e^{-i\frac{\pi}{2}\sum_{j}\hat{\sigma}_{j}^{x}}e^{+i\tau_{J}\sum_{u}\frac{J_{0}}{2}\left(\hat{\sigma}_{u}^{z}+\hat{I}_{u}\right)\left(\hat{\sigma}_{u+1}^{z}+\hat{I}_{u+1}\right)})^{2}$
(S6)
$\displaystyle=(e^{-i\frac{\pi}{2}\sum_{j}\hat{\sigma}_{j}^{x}}e^{+i\delta\tau(\sum_{u}\frac{J_{0}}{2}\sigma_{u}^{z}\hat{\sigma}_{u+1}^{z})})(e^{+i\sum_{j}\delta\tau{J_{0}}\hat{\sigma}_{j}^{z}}e^{-i\frac{\pi}{2}\sum_{j}\hat{\sigma}_{j}^{x}}e^{+i\sum_{j}\delta\tau{J_{0}}\hat{\sigma}_{j}^{z}})(e^{+i\delta\tau(\sum_{u}\frac{J_{0}}{2}\hat{\sigma}_{u}^{z}\hat{\sigma}_{u+1}^{z})})+\mathcal{O}(\delta\tau^{2}).$

Then, by mathematically inserting the $\hat{\sigma}^{z}$-field kicks as
$e^{+i\frac{\pi}{2}\sum_{j}\hat{\sigma}_{j}^{x}}e^{-i\frac{\pi}{2}\sum_{j}\hat{\sigma}_{j}^{x}}$,
we can write the above process as follows

$\displaystyle(e^{-i\frac{\pi}{2}\sum_{j}\hat{\sigma}_{j}^{x}}e^{+i\delta\tau(\sum_{u}\frac{J_{0}}{2}\sigma_{u}^{z}\hat{\sigma}_{u+1}^{z})}e^{+i\frac{\pi}{2}\sum_{j}\hat{\sigma}_{j}^{x}})(e^{-i\frac{\pi}{2}\sum_{j}\hat{\sigma}_{j}^{x}}e^{+i\sum_{j}\delta\tau{J_{0}}\hat{\sigma}_{j}^{z}}e^{-i\frac{\pi}{2}\sum_{j}\hat{\sigma}_{j}^{x}}e^{+i\sum_{j}\delta\tau{J_{0}}\hat{\sigma}_{j}^{z}})(e^{+i\delta\tau(\sum_{u}\frac{J_{0}}{2}\hat{\sigma}_{u}^{z}\hat{\sigma}_{u+1}^{z})}).$
(S7)

Since
$(e^{-i\frac{\pi}{2}\sum_{j}\hat{\sigma}_{j}^{x}}e^{+i\sum_{j}\delta\tau{J_{0}}\hat{\sigma}_{j}^{z}}e^{-i\frac{\pi}{2}\sum_{j}\hat{\sigma}_{j}^{x}}e^{+i\sum_{j}\delta\tau{J_{0}}\hat{\sigma}_{j}^{z}})$
can be removed due to the operator identity in Eq.(S5), the following sequence
of driving terms can recover the clean ferromagnetic interaction
$\hat{H}_{ZZ}(J_{0})=-\sum_{i}J_{0}\hat{\sigma}_{i}^{z}\hat{\sigma}_{i+1}^{z}$:

$\displaystyle(e^{-i\frac{\pi}{2}\sum_{j}\hat{\sigma}_{j}^{x}}e^{+i\delta\tau(\sum_{u}\frac{J_{0}}{2}\sigma_{u}^{z}\hat{\sigma}_{u+1}^{z})}e^{+i\frac{\pi}{2}\sum_{j}\hat{\sigma}_{j}^{x}})(e^{+i\delta\tau(\sum_{u}\frac{J_{0}}{2}\hat{\sigma}_{u}^{z}\hat{\sigma}_{u+1}^{z})})$
(S8) $\displaystyle\approx
e^{+i\delta\tau\sum_{u}{J_{0}}\hat{\sigma}_{u}^{z}\hat{\sigma}_{u+1}^{z}},$

with
$e^{-i\frac{\pi}{2}\sum_{j}\hat{\sigma}_{j}^{x}}e^{+i\delta\tau(\sum_{u}\frac{J_{0}}{2}\sigma_{u}^{z}\hat{\sigma}_{u+1}^{z})}e^{+i\frac{\pi}{2}\sum_{j}\hat{\sigma}_{j}^{x}}=e^{+i\delta\tau(\sum_{u}\frac{J_{0}}{2}\sigma_{u}^{z}\hat{\sigma}_{u+1}^{z})}.$

###  A.3 Atom loss as imaginary fields

Figure S1: Structure of the key energy levels of the Rydberg system. For the
two selected hyperfine ground states:
$\ket{\downarrow}=\left|6S_{1/2},F=3\right\rangle$ and
$\ket{\uparrow}=\left|6S_{1/2},F=4\right\rangle$, we couple the state
$\ket{\uparrow}=\left|6S_{1/2},F=4\right\rangle$ with an excited state
$\left|6P_{3/2},F=5\right\rangle$. The pink arrow represents laser-induced
loss which acts as the imaginary field in Eq.(S9).

In our proposed setup, we consider laser-induced atom loss which gives rise to
effective imaginary fields. Such a process is shown in FIG. S1. Here, the
tuneable laser-induced loss is added on the state
$\ket{\uparrow}=\left|6S_{1/2},F=4\right\rangle$, which is generated by
coupling a ground state to an intermediate excited state
$\left|6P_{3/2},F=5\right\rangle$ [18, 53, 19]. In this case, such imaginary
fields from atom loss can be presented as

$\displaystyle i2g\ket{\uparrow}\bra{\uparrow}$
$\displaystyle=ig(\ket{\uparrow}\bra{\uparrow}-\ket{\downarrow}\bra{\downarrow})+ig(\ket{\uparrow}\bra{\uparrow}+\ket{\downarrow}\bra{\downarrow})$
(S9) $\displaystyle=ig\hat{\sigma}^{z}+ig\hat{I},$

where $g$ is the tuneable decay rate controlled by laser. Here, we can drop
the global loss term $ig\hat{I}$ by post-selection on $\ket{\uparrow}$ and
$\ket{\downarrow}$ during experimental measurements [14].

##  Appendix B II. Phase transitions in the non-Hermitian transverse-field
Ising model

The Yang-Lee theorem indicates that phase transitions are associated with
zeros in partition functions called the Yang-Lee zeros [1, 2]. For a classical
spin model in the thermodynamic limit, the Yang-Lee edge singularities
correspond to edges of the Yang-Lee zeros [7]. For our proposed quantum model
used in the main text

$\hat{H}_{\rm
TFI}=-\sum_{j}(h_{x}\hat{\sigma}_{j}^{x}+{J}\hat{\sigma}^{z}_{j}\hat{\sigma}^{z}_{j+1})+\sum_{j}i{\gamma}\hat{\sigma}^{z}_{j},$
(S10)

the Yang-Lee theorem is also applicable in understanding its quantum phase
transitions [4, 5] via a quantum-classical mapping, as described below [130].

##  Appendix C Quantum-classical mapping

To understand the existence of the Yang-Lee edge singularities in our finite
size model, we consider its partition function
$\mathcal{Z}=\operatorname{Tr}\exp(-\beta H_{\rm TFI})$ with inverse
temperature $\beta$. Following the quantum-classical mapping in Ref. [131], we
decompose this partition function expanded into $N_{\beta}\rightarrow\infty$
slices ordered by imaginary time:

$\displaystyle\operatorname{Tr}\exp(-\beta\hat{H}_{\rm TFI})$
$\displaystyle=\sum_{\sigma_{i}=\uparrow,\downarrow}\bra{\left\\{\sigma_{i}\right\\}}\exp(\beta\sum_{j=1}^{L}(h_{x}\hat{\sigma}_{j}^{x}+{J}\hat{\sigma}^{z}_{j}\hat{\sigma}^{z}_{j+1})-\sum_{j=1}^{L}i{\gamma}\sigma^{z}_{j})\ket{\left\\{{\sigma}_{i}\right\\}}$
(S11) $\displaystyle=\lim_{N_{\beta}\rightarrow\infty}\sum_{\sigma_{i,j}=\pm
1}C\exp(\frac{\beta}{N_{\beta}}(\sum_{i=1}^{L}\sum_{j=1}^{N_{\beta}}({J}\sigma_{i+1,j}\sigma_{i,j}+h^{\prime}\sigma_{i,j}\sigma_{i,j+1})-\sum_{i=1}^{L}\sum_{j=1}^{N_{\beta}}i{\gamma}\sigma_{i,j})),$

with $\exp(-2h^{\prime})=\tanh(\frac{\beta}{N_{\beta}}h_{x})$. Here, we
express the partition function of our quantum Ising chain in the basis of
product state as
$\ket{\left\\{\sigma_{i}\right\\}}=\prod_{i=1}^{L}\ket{\sigma_{i}}$. For the
classical model, $\sigma_{i,j}$ denotes a classical spin, where the coordinate
$j$ runs along slices in imaginary time. Note that the anisotropic couplings
in the classical spin model do not impact phase transitons, and details of the
above mapping are discussed in Ref. [131]. In this case, the quantum phase
transition of our model will occur when the zeros of its partition function
appear under the condition $\beta\rightarrow\infty$. Thus, we have mapped the
partition function of a finite-size (1+1) D quantum spin chain to that of a
(2+1) D classical spin model in the thermodynamic limit, with the additional
dimension arising from the discretization of the imaginary time dimension.
With the above quantum-classical mapping, critical points of the Hamiltonian
in Eq. (S10) can be explained as the YLES in the dual classical spin model [4,
3, 132, 5].

###  C.1 Phase transitions of the finite quantum model

To determine the phase transitions of our model in Eq. (S10) (but not that of
the dynamical quench in the main text), we introduce the following order
parameters:

$\begin{aligned}
M_{x}=|\left\langle\hat{\sigma}^{x}\right\rangle|=\left|\left\langle\frac{\sum_{j}\hat{\sigma}^{x}_{j}}{L}\right\rangle\right|,\end{aligned}~{}\begin{aligned}
M_{z}=|\left\langle\hat{\sigma}^{z}\right\rangle|=\left|\left\langle\frac{\sum_{j}\hat{\sigma}^{z}_{j}}{L}\right\rangle\right|,\end{aligned}$
(S12)

which are computed in the ground state with the minimum real eigenvalue.

Figure S2: Order parameters $M_{x}$ and $M_{z}$ as defined in Eq. (S12), which
are computed by the ground state of the model from Eq. (S10) under open
boundary conditions (OBCs). We set $L=8$, $h_{x}=1.5$, and $J=1$. The YLES
appears at $\gamma_{YL}=0.184$ where order parameters are discontinuous. Both
order parameters are obtained using exact diagonalization (ED).

As plotted in FIG. S2, we present such two kinds of order parameters, and both
of them can capture critical points $\gamma_{YL}$ which appear at the kink
along $\gamma$. Since a singular value cut-off is required in the matrix
product states computation discussed later, we choose the order parameter
$M_{x}$ for identifying phase transition in this work. The divergence near
critical points is not intense, and the measurement in $M_{x}$ helps to
minimize the error in our MPS computation.

Figure S3: Order parameters $M_{x}$ and $M_{z}$ computed by the ground state
of the model in Eq. (S10) under OBCs, for different transverse fields $h_{x}$.
We set $L=8$, and $J=1$ for all plots. Order parameters are discontinuous at
critical points $\gamma_{YL}$.

Next, in FIG. S3, we plot order parameters computed in ground states under
different transverse fields $h_{x}$, and the critical point $\gamma_{YL}$
appears when $h_{x}>J$. Under different $h_{x}$, we find that critical
behaviors of $M_{x}$ and $M_{z}$ are qualitatively similar. Hence both
eligible to describe critical phase transitions in our model. To avoid overly
dramatic divergences near the critical points, we choose $M_{x}$ as our phase-
transition indicator.

###  C.2 Dynamical signature of phase transitions

Due to the atom loss in our proposed setup, it is difficult to access the
ground state. Below, we provide more detailed results supporting the use of a
dynamical signature for detecting the phase transitions, as we have used in
our main proposal.

Figure S4: (a) Dynamical overlap $|\bra{\psi(t)}\ket{\psi_{i}}|$ of the
evolved state $\ket{\psi(t)}\propto e^{-iT\hat{H}_{\text{TFI}}}\ket{\psi_{0}}$
with each of the eigenstates $\ket{\psi_{i}}$ of the Hamiltonian
$\hat{H}_{\text{TFI}}$ in Eq. (S10) under OBCs and $J=1$, $h_{x}=1.5$. The
initial state is $\ket{\psi_{0}}=\ket{\downarrow\downarrow......}$. The red
dashed curves refer to the overlap with the ground state. For (a) and (b), we
set $\gamma<\gamma_{YL}$, and $\gamma>\gamma_{YL}$ for (c) and (d). When
$\gamma>\gamma_{YL}$, the ground state with imaginary eigenvalue ${\rm
Im}E_{g}>0$ evolves to a high overlap which is a dominated state through
dynamics. The huge contrast in the dynamical behavior and overlap provides a
clear signature of the phase transition shown in FIG. S3.

According to the protocol introduced in the main text, we propose to measure
the YLES through responses under quench dynamics $\ket{\psi(t)}\propto
e^{-iT\hat{H}_{\text{TFI}}}\ket{\psi_{0}}$. To understand why such dynamics
can exhibit dramatic response near critical points, we plot the dynamical
overlap for every eigenstate in FIG. S4. Here, the ground-state overlaps (red
curves) exhibits significantly different behaviors across the phase
transition. When $\gamma>\gamma_{YL}$ for the ferromagnetic phase, the ground-
state overlap evolves to a high plateau value under the ferromagnetic initial
state as $\ket{\psi_{0}}=\ket{\downarrow\downarrow......}$, which indicates
that the ground state govern the dynamics. However, when $\gamma<\gamma_{YL}$,
the ground states host real eigenvalues, and accordingly, the dynamics will be
governed by other states with complex eigenvalues, as shown in FIG. S4 (a) and
(b).

Figure S5: Spectra of the Hamiltonian in Eq. (S10) under OBCs with different
colors indicating the overlap between the particular eigenstate $\ket{\psi}$
and the ferromagnetic initial state
$\ket{\downarrow}=\ket{\downarrow\downarrow\downarrow...}$. We set $L=8,J=1$
and $h_{x}=1.5$ which correspond to the critical point value
$\gamma_{YL}=0.184$. Due to the rapid repulsion of the ground state (with
smallest $\text{Re}(E)$) eigenenergies away from the real line when
$\gamma>\gamma_{YL}$, the ground states evolve to those with the maximum
imaginary spectra rapidly.

To understand the above dynamical behavior under $\gamma>\gamma_{YL}$, we plot
the spectra in FIG. S5 and check the eigenstate overlap with our prepared
initial state $\ket{\downarrow\downarrow......}$ before dynamical evolution.
As shown in FIG. S5, the ground states host relativily high overlap with
$\ket{\downarrow\downarrow......}$. In particular, one of the ground states
evolves to that with the maximum imaginary eigenvalue quickly when $\gamma$
exceeds $\gamma_{YL}=0.184$ under $J=1,h_{x}=1.5,L=8$. The above facts
indicate that such a ground state can be a dominating state in our designed
quenching dynamics when $\gamma>\gamma_{YL}$.

##  Appendix D III. Numerical method

In our work, we use ED and time evolution with matrix product states (tMPS) to
compute the dynamics for our model. Compared with ED, the method of tMPS is a
state-of-art tool to simulate larger system sizes for generic one-dimensional
quantum many-body systems [70, 71, 72]. As presented in the main text, we
simulate quenching dynamics for our model with tMPS to determine the YLES and
emulate the experimental process. Here, we provide more details about our tMPS
method. From our results from tMPS, we also discuss the scaling behavior of
Yang-Lee edge singularities.

###  D.1 Time evolution with matrix product states (tMPS)

Figure S6: Time evolution with MPS (tMPS) with a second-order Suzuki-Trotter
decomposition algorithm of the evolution operator: (a) Odd and even-bond time
evolution operator in the form of a rank-$4$ matrix product operator (MPO).
The MPO representation for the time evolution operator on the odd bond
$\exp[-i\delta t/2\,h_{2j-1,2j}]$ (light green) and the even bond
$\exp[-i\delta t/2\,h_{2j,2j+1}]$ (dark green) has four indices, of which each
has a dimension of two. The indices without prime are contracted with the MPS
in the time evolution. (b) Two adjacent steps of tMPS. The initial state is
prepared in the right-canonical form (red). The total time evolution operator
for each time step $U(\delta t)$ is decomposed into two odd-bond parts
$U_{\text{odd}}(\delta t/2)$ (light green) and one even-bond part
$U_{\text{even}}(\delta t)$ (dark green) by a symmetrical decomposition. At
the end of each time step $\delta t$, a normalization of the state (solid red
line) is performed.

Here, we treat the total Hamiltonian evolution as a hybrid real-time and
imaginary-time evolution and apply a second-order Suzuki-Trotter decomposition
[133] to each trotter step $U(\delta t)$, after which this time evolution
operator $U(\delta t)$ can be written as [see Fig. S6(b)]

$\displaystyle U(\delta t)=U_{\text{odd}}(\delta t/2)U_{\text{even}}(\delta
t)U_{\text{odd}}(\delta t/2)+\mathcal{O}(\delta t^{3}),$ (S13)

where

$\displaystyle U_{\nu}(\delta t)=\exp\left[-i\delta
t\left(H_{ZZ}^{\nu}+H_{X}^{\nu}\right)-i\left(i\delta
t\right)\tilde{H}_{Z}^{\nu}\right](\nu\text{ is either odd or even}).$ (S14)

The terms in the above equation are explicitly:

$\displaystyle
H_{ZZ}^{\text{odd}}=\sum_{j\,\,\text{odd}}h_{j,j+1}^{ZZ}=\sum_{j=1}^{N/2}-J\sigma_{2j-1}^{z}\otimes\sigma_{2j}^{z},$
(S15) $\displaystyle
H_{ZZ}^{\text{even}}=\sum_{j\,\,\text{even}}h_{j,j+1}^{ZZ}=\sum_{j=1}^{N/2-1}-J\sigma_{2j}^{z}\otimes\sigma_{2j+1}^{z},$
$\displaystyle
H_{X}^{\text{odd}}=\sum_{j\,\,\text{odd}}h_{j,j+1}^{X}=(-h_{x})\sigma_{1}^{x}\otimes\mathds{1}_{2}+\mathds{1}_{1}\otimes(-h_{x}/2)\sigma_{2}^{x}+\sum_{j=1}^{N/2-1}(-h_{x}/2)\sigma_{2j-1}^{x}\otimes\mathds{1}_{2j}+\mathds{1}_{2j-1}\otimes(-h_{x}/2)\sigma_{2j}^{x}$
$\displaystyle+(-h/2)\sigma_{N-1}^{x}\otimes\mathds{1}_{N}+\mathds{1}_{N-1}\otimes(-h)\sigma_{N}^{x},$
$\displaystyle
H_{X}^{\text{even}}=\sum_{j\,\,\text{even}}h_{j,j+1}^{X}=\sum_{j=1}^{N/2-1}(-h_{x}/2)\sigma_{2j}^{x}\otimes\mathds{1}_{2j+1}+\mathds{1}_{2j}\otimes(-h_{x}/2)\sigma_{2j+1}^{x}$
$\displaystyle\tilde{H}_{Z}^{\text{odd}}=\sum_{j\,\,\text{odd}}\tilde{h}_{j,j+1}^{Z}=\gamma\sigma_{1}^{z}\otimes\mathds{1}_{2}+\mathds{1}_{1}\otimes(\gamma/2)\sigma_{2}^{z}+\sum_{j=1}^{N/2-1}(\gamma/2)\sigma_{2j-1}^{z}\otimes\mathds{1}_{2j}+\mathds{1}_{2j-1}\otimes(\gamma/2)\sigma_{2j}^{z},$
$\displaystyle+(\gamma/2)\sigma_{N-1}^{z}\otimes\mathds{1}_{N}+\mathds{1}_{N-1}\otimes\gamma\sigma_{N}^{z},$
$\displaystyle\tilde{H}_{Z}^{\text{even}}=\sum_{j\,\,\text{even}}\tilde{h}_{j,j+1}^{Z}=\sum_{j=1}^{N/2-1}(\gamma/2)\sigma_{2j}^{z}\otimes\mathds{1}_{2j+1}+\mathds{1}_{2j}\otimes(\gamma/2)\sigma_{2j+1}^{z},$

and therefore $U_{\text{odd}}(\delta t/2)$ or $U_{\text{even}}(\delta t)$ can
be re-written as a product of local two-body terms [see Fig. S6(a)]:

$\displaystyle U_{\text{odd}}(\delta t/2)=\exp\left[-i\delta
t/2\left(H_{ZZ}^{\text{odd}}+H_{X}^{\text{odd}}\right)-i\left(i\delta
t/2\right)\tilde{H}_{Z}^{\text{odd}}\right]$ (S16)
$\displaystyle=\exp\left[-i\delta
t/2\left(\sum_{j\,\,\text{odd}}h_{j,j+1}^{ZZ}+\sum_{j\,\,\text{odd}}h_{j,j+1}^{X}\right)-i(i\delta
t/2)\sum_{j\,\,\text{odd}}\tilde{h}_{j}^{Z}\right]$
$\displaystyle=\prod_{j\,\,\text{odd}}\exp\left[-i\delta
t/2\left(h_{j,j+1}^{ZZ}+h_{j,j+1}^{X}\right)-i\left(i\delta
t/2\right)\tilde{h}_{j}^{Z}\right]$
$\displaystyle=\prod_{j\,\,\text{odd}}\exp\left[-i\delta
t/2\,h_{j,j+1}\right],$ $\displaystyle U_{\text{even}}(\delta
t)=\exp\left[-i\delta
t\left(H_{ZZ}^{\text{even}}+H_{X}^{\text{even}}\right)-i\left(i\delta
t\right)\tilde{H}_{Z}^{\text{even}}\right]$ $\displaystyle=\exp\left[-i\delta
t\left(\sum_{j\,\,\text{even}}h_{j,j+1}^{ZZ}+\sum_{j\,\,\text{even}}h_{j,j+1}^{X}\right)-i(i\delta
t)\sum_{j\,\,\text{even}}\tilde{h}_{j}^{Z}\right]$
$\displaystyle=\prod_{j\,\,\text{even}}\exp\left[-i\delta
t\left(h_{j,j+1}^{ZZ}+h_{j,j+1}^{X}\right)-i\left(i\delta
t\right)\tilde{h}_{j}^{Z}\right]$
$\displaystyle=\prod_{j\,\,\text{even}}\exp\left[-i\delta
t\,h_{j,j+1}\right].$

For our setup, the system size $L$ is even, and we have explicitly written
both odd-bond and even-bond Hamiltonians as two-body tensor products of local
spin operators. $\mathds{1}_{j}$ is the local identity operator in the
spin-$1/2$ basis. We have also introduced $\tilde{H}_{Z}^{\text{odd}}$ and
$\tilde{H}_{Z}^{\text{even}}$ so that the imaginary $i$ from the imaginary
field of the original $H_{Z}$ has been absorbed into the imaginary time
evolution step $i\delta t$. Considering a long time $T$ which is discretized
into $T/\delta t$ steps, if applying the $U(\delta t)$, the error accumulates
to $T/\delta t\mathcal{O}(\delta t^{3})\approx\mathcal{O}(\delta t^{2})$ for
the total time interval $T$, and therefore it renders a second-order time
evolution. Since the implementation of tMPS algorithm with PBC at this stage
is computationally expensive, we only consider the OBCs for the tMPS results
in this work.

###  D.2 Additional numerical results

As described in the main text, we propose to measure the dynamical order
parameter as
$M_{x}(T)=|\bra{\psi(T)}\frac{\sum_{j}\sigma^{x}_{j}}{L}\ket{\psi(T)}|$ at a
fixed time $T$ and fixed $h$ and $J$, such that our suggested order parameter
$M_{x}(T)$ diverges when $\gamma$ sweeps past the critical point
$\gamma_{YL}$. For this, we need to choose the duration of evolution $T$ for
$M_{x}(T)$ properly. Here, we suggest measuring the order parameter at time
$JT=20$ with $J=1$, and supporting insights are shown in FIG. S7 (b1) -(b5),
where we present the full picture of the dynamics under different sizes $L$. A
duration $T=20$ ensures that the curves spread out for $\gamma>\gamma_{YL}$
but not $\gamma<\gamma_{YL}$ for all sizes $L$. As such, measuring at $T=20$
can reveal the mentioned divergence in the dynamical order parameter at
$\gamma_{YL}$.

Figure S7: (a) Dynamical order parameter $M_{x}(T)$ measured at time $T=20$
under $J=1$ (black dashed lines marked in (b1)-(b5)) with different $\gamma$.
We set parameters as $J=1$, $h_{x}=1.5$, and $L=8,10,12,14,16$, and prepare
the initial state as
$\ket{\downarrow\downarrow\downarrow\downarrow\downarrow\downarrow\downarrow\downarrow}$
under OBCs. Order parameters exhibit intense growth near critical points
$\gamma_{YL}$ marked by dashed lines. (b1)-(b5) Dynamical order parameter
$M_{x}(T)$ with initial state
$\ket{\downarrow\downarrow\downarrow\downarrow\downarrow\downarrow\downarrow\downarrow}$
under different sizes $L$ and $\gamma$. We set $JT=20$ as the measuring time
(black dashed lines) for plot (a) and the main text such that we can
distinctively see the curves spreading out for $\gamma>\gamma_{YL}$ but not
$\gamma<\gamma_{YL}$.