# Geometric phase and multipartite entanglement of Rydberg atom chains

Chang-Yan Wang<EMAIL_ADDRESS>Institute for Advanced Study,
Tsinghua University, Beijing 100084, China

###### Abstract

We investigate the behavior of geometric phase (GP) and geometric entanglement
(GE), a multipartite entanglement measure, across quantum phase transitions in
Rydberg atom chains. Using density matrix renormalization group calculations
and finite-size scaling analysis, we characterize the critical properties of
transitions between disordered and ordered phases. Both quantities exhibit
characteristic scaling near transition points, with the disorder to $Z_{2}$
ordered phase transition showing behavior consistent with the Ising
universality class, while the disorder to $Z_{3}$ phase transition displays
distinct critical properties. We demonstrate that GP and GE serve as sensitive
probes of quantum criticality, providing consistent critical parameters and
scaling behavior. A unifying description of these geometric quantities from a
quantum geometry perspective is explored, and an interferometric setup for
their potential measurement is discussed. Our results provide insights into
the interplay between geometric phase and multipartite entanglement near
quantum phase transitions in Rydberg atom systems, revealing how these
quantities reflect the underlying critical behavior in these complex quantum
many-body systems.

Rydberg atom arrays have emerged as a powerful platform for studying quantum
many-body physics [1, 2, 3], offering a high degree of controllability and the
ability to study a wide range of physical phenomena, such as quantum many-body
scars [4, 5, 6, 7, 8, 9, 10, 11, 12, 13], quantum spin liquid [14, 15, 16, 17,
18, 19, 20, 21, 22, 23, 24, 25, 26, 27] and Kibble-Zurek mechanism [28, 29].
These systems also have attracted significant attention due to their potential
applications in quantum information processing and quantum computation [30,
31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43]. Understanding the quantum
phases and phase transitions in Rydberg atom arrays is crucial for harnessing
their capabilities and exploring fundamental questions in quantum physics.

The geometric phase and quantum entanglement have proven to be powerful tools
for characterizing quantum critical behavior in many-body systems over the
past decades [44, 45, 46]. The geometric phase, originally introduced by Berry
[47], has proven invaluable in studying topological properties of quantum
systems, such as the quantum Hall effect [48]. It has also been instrumental
in investigating quantum critical phenomena in various systems, including the
$XY$ model [49, 50, 51, 52, 53, 54, 55] and the Dicke model [56, 57], etc.
Concurrently, geometric entanglement [58, 59, 60], a measure of multipartite
entanglement [46], has been successfully applied to study critical phenomena
and detect phase transitions [61, 62, 63, 64, 65, 66]. Mousolou et al. [67]
proposed a unifying description that combines these two quantities into a
complex-valued geometric entanglement, using the $XY$ model as an illustrative
example.

In this work, we investigate the behavior of GP and GE in Rydberg atom chains
across quantum phase transitions. We employ density matrix renormalization
group (DMRG) method [68, 69, 70, 71] to compute these quantities for various
system sizes. By applying finite-size scaling analysis, we extract quantum
phase transition points and critical exponents for the quantum phase
transitions from disorder to $Z_{2}$ ordered phase and from disorder to
$Z_{3}$ ordered phase, demonstrating the effectiveness of these geometric
quantities as probes of quantum criticality in Rydberg atom systems.

We also discuss the unifying description of GP and GE based on a quantum
geometry perspective adopted from previous work in the $XY$ model [67]. This
approach provides a deeper understanding of the connection between these
quantities in the context of quantum phase transitions. Additionally, we
explore the applicability of an interferometric setup, also proposed in the
context of the $XY$ model [67], for the potential experimental measurement of
these geometric quantities in Rydberg atom chains.

The rest of this paper is organized as follows: in Sec. I, we review the
Hamiltonian of the Rydberg atom chains, and their phases; In Sec. II, we
review the geometric phase and why it can be used to characterize quantum
phase transitions; in Sec. III, we discuss two different methods to calculate
the geometric entanglement; in Sec. IV, we present our numerical result for GP
and GE and extract the critical points and critical exponent using finite size
scaling; in Sec. V and Sec. VI, we discuss the relation of GP and GE based on
a quantum geometry perspective and an interferometer setup; we conclude in
Sec. VII.

##  I Hamiltonian of Rydberg atom chains

We consider the Hamiltonian of 1d Rydberg atom chain with length $L$ in open
boundary condition [1, 72]

$\displaystyle
H=\sum_{i}(\Omega\hat{\sigma}_{i}^{x}-\delta\hat{n}_{i})+\sum_{i<j}V(|i-j|)\hat{n}_{i}\hat{n}_{j},$
(1)

where $\hat{\sigma}_{i}=|r_{i}\rangle\langle g_{i}|+|g_{i}\rangle\langle
r_{i}|$ describes the couple between the internal atomic ground state
$|g_{i}\rangle$ and a Rydberg excited state $|r_{i}\rangle$ of the $i$th atom
and $\hat{n}_{i}=|r_{i}\rangle\langle r_{i}|$. $\Omega$ and $\delta$ denote
the Rabi frequency and detuning of the coherent laser driving, respectively.
The term $V(|i-j|)=C_{6}/|i-j|^{6}$ describes the van der Waals interaction
between the $i$th and $j$th atoms when both are excited to the Rydberg state.

The long-range van der Waals interaction can be parametrized by the Rydberg
blockade radius $R_{b}$, defined as $V(R_{b}/a)\equiv\Omega$, where $a$ is the
lattice spacing. When $R_{b}/a$ becomes sufficiently large, the long-range van
der Waals interaction disfavors the nearest-neighbor atoms being excited to
the Rydberg state, leading to the formation of the $Z_{2}$ ordered phase. As
$R_{b}/a$ further increases, the occupation of the Rydberg state in next-
nearest-neighbor atoms is also suppressed, resulting in the formation of the
$Z_{3}$ ordered phase [1], as shown in Fig. 1.

Figure 1: Schematics of the phase diagram of the 1d Rydberg atom chain. As the
blockade radius increases, the $Z_{2}$ order phase emerges as the occupation
of the Rydberg state in next-neighbor atoms is suppressed, then $Z_{3}$ order
phase emerges as occupation of the Rydberg state in next-next-neighbor atoms
is further suppressed.

##  II Geometric phase of the Rydberg array

The geometric phase, or Berry phase, is a phase factor acquired by a quantum
system undergoing an adiabatic cyclic evolution in parameter space. It arises
due to the non-trivial topology of the system’s Hilbert space and the
curvature associated with the parameter space. Geometric phase can profoundly
affect physical observables, such as energy levels, interference patterns, and
quantum transport properties, leading to phenomena like the Aharonov-Bohm
effect and the quantum Hall effect [48]. These effects have initiated
extensive studies of topological phases of matter [73, 74]. Moreover, the
geometric phase can be used to characterize quantum phase transitions, and
critical exponents can be extracted from the scaling behavior of geometric
phases.

When a quantum system’s parameters, denoted by $\mathbf{R}$, are varied slowly
and cyclically, its eigenstate $|\psi_{n}(\mathbf{R}(t))\rangle$, with
$t\in[0,1]$ parameterizing the adiabatic path, acquires a phase factor upon
returning to its initial state:
$|\psi_{n}(\mathbf{R}(1))\rangle=e^{i\Phi_{B}}|\psi_{n}(\mathbf{R}(0))\rangle$,
where $\Phi_{B}$ is the geometric phase. The geometric phase can be calculated
by integrating the Berry connection $A_{B}$ along the adiabatic path $C$ in
parameter space:

$\displaystyle\Phi_{B}=\int_{0}^{1}A_{B}dt,\
A_{B}=-i\langle\psi_{n}(\mathbf{R}(t))|\frac{d}{dt}|\psi_{n}(\mathbf{R}(t))\rangle.$
(2)

We define the GP density as $\phi_{B}=\frac{\Phi_{B}}{L}$ for later use.

To calculate the geometric phase, one must construct an adiabatic path. For
example, in the $XY$ model, a natural choice is spin rotation along the
$z$-axis. Similarly, we can construct such an adiabatic path for the Rydberg
atom chain by treating the ground state $|g_{i}\rangle$ and the Rydberg
excited state $|r_{i}\rangle$ of each atom as a pseudo-spin. Defining
$\hat{\sigma}_{i}^{z}=2|r_{i}\rangle\langle r_{i}|-1$, we have the adiabatic
Hamiltonian as:

$\displaystyle H(t)=U(t)HU^{\dagger}(t),\
U(t)=\prod_{i=1}^{L}e^{it\pi\hat{\sigma}_{i}^{z}}.$ (3)

With this adiabatic path, the adiabatic ground state is given by:

$\displaystyle|\psi(t)\rangle=U(t)|\psi_{0}\rangle,$ (4)

where $|\psi_{0}\rangle$ is the ground state of $H$ when the adiabatic
parameter $t=0$. One can then calculate the geometric phase according to
Eq.(2). Unlike the $XY$ model, which is exactly solvable using the Jordan-
Wigner transformation, the Rydberg atom array is not exactly solvable even in
1D in general. Thus, one needs to use sophisticated numerical methods such as
the DMRG method.

Before proceeding to the numerical results, we shall briefly explain why the
geometric phase can characterize quantum critical behavior. Quantum phase
transitions are associated with level-crossing, and the singularity of the
geometric phase indeed results from such level-crossing. The geometric phase
along a closed path $C$ in parameter space can also be given by the
integration of the Berry curvature over the region $S$ enclosed by this path:
$\Phi_{B}=-\int_{S}\mathbf{F}(\mathbf{R})\cdot d\mathbf{S}$, where
$d\mathbf{S}$ denotes the area element of region $S$, and the Berry curvature
for the ground state $|\psi_{0}\rangle$ is given by [47]:

$\displaystyle\mathbf{F}(\mathbf{R})=\mathrm{Im}\sum_{n\neq
0}\frac{\langle\psi_{0}|\nabla_{\mathbf{R}}H|\psi_{n}\rangle\times\langle\psi_{n}|\nabla_{\mathbf{R}}H|\psi_{0}\rangle}{(E_{n}-E_{0})^{2}},$
(5)

where $|\psi_{n}(\mathbf{R})\rangle$ is the $n$th eigenstate of the
Hamiltonian, i.e.,
$H(\mathbf{R})|\psi_{n}(\mathbf{R})\rangle=E_{n}(\mathbf{R})|\psi_{n}(\mathbf{R})\rangle$.
Hence, we can see that when there is level-crossing around the ground state
energy, the Berry curvature diverges. As the quantum phase transition occurs
at level crossings or avoided level crossings, thus, GP can capture such
quantum phase transition.

##  III Multipartite Entanglement of the Rydberg array

There are several multipartite entanglement measures; the geometric
entanglement (GE) is one of them. The idea of GE is based on the observation
that the multipartite entanglement of a quantum state $|\psi\rangle$ can be
captured by the overlap between this state and its closest product state, i.e.

$\displaystyle\Lambda_{\mathrm{max}}=\mathrm{max}_{\psi_{p}}|\langle\psi_{p}|\psi\rangle|.$
(6)

For the $XY$ model, the quantity $\Lambda_{\mathrm{max}}$ is actually the
entanglement eigenvalue of the $XY$ ground state. The GE of the state
$|\psi\rangle$ is defined as

$\displaystyle E(\psi)=-\log_{2}\Lambda_{\mathrm{max}}^{2}.$ (7)

And the GE density is defined as $\mathcal{E}(\psi)=E(\psi)/L$.

Figure 2: $\langle\hat{\sigma}_{i}^{z}\rangle$ for the exact ground state
calculated from the DMRG method and the product states calculated through the
L-BFGS and SVD truncation methods respectively for a Rydberg chain of length
$L=19$ in different phases. (a) $Z_{2}$ ordered state with
$\delta/\Omega=2.5,R_{b}/a=1.5$, (b) $Z_{3}$ ordered state with
$\delta/\Omega=2.50R_{b}/a=2.5$. (c) Disordered state with
$\delta/\Omega=1.5,R_{b}/a=2.5$.

It has been demonstrated that the GE of the ground state in the $XY$ model
exhibits features characteristic of quantum criticality. These features
include universality, critical exponents, and scaling behavior, which are
manifested through the singular behavior of the GE density. By performing
scaling analysis on the divergences observed at the singular points of the GE
density, critical exponents corresponding to different universality classes
have been extracted.

Due to the realness of the Hamiltonian of the Rydberg atom chains, we can
assume the closest product state to the ground state $|\psi\rangle$ of the
Hamiltonian takes the following form

$\displaystyle|\psi_{p}(\bm{\theta})\rangle=\otimes_{i=1}^{L}\Big{[}\cos\left(\theta_{i}\right)|r_{i}\rangle\pm\sin\left(\theta_{i}\right)|g_{i}\rangle\Big{]},$
(8)

where $\bm{\theta}=\\{\theta_{1},\theta_{2},...,\theta_{L}\\}$ are the
parameters to be determined to maximize the overlap i.e.
$\Lambda_{\mathrm{max}}=\mathrm{max}_{\bm{\theta}}|\langle\psi_{p}(\bm{\theta})|\psi\rangle|$.
In previous studies, such as for the $XY$ model, translational symmetry is
always adopted, hence, one can take $\theta_{i}=\theta$, which can simplify
the calculation. However, such an ansatz for the closest product state is not
valid for the $Z_{2}$ and $Z_{3}$ ordered states of the Rydberg atom chain, as
these states are no longer translationally invariant. Therefore, we cannot
simply set $\theta_{i}=\theta$, instead, we need to numerically minimize the
function
$\mathcal{F}(\bm{\theta})=-|\langle\psi_{p}(\bm{\theta})|\psi\rangle|$ over
possible $\bm{\theta}$, which includes as many as hundreds of parameters in
our case. Such a task can be achieved by the so-called limited memory Broyden-
Fletcher-Goldfarb-Shanno (L-BFGS) method [75], which is a large scale
optimization method that can minimize a function with many parameters. This
algorithm has been implemented in software packages such as Optim.jl [76].

An alternative approach to calculating the closest product state involves a
two-step process based on the DMRG method [77]. First, the exact matrix
product state (MPS) ground state is computed using DMRG calculations with a
high bond dimension. Then, a singular value decomposition (SVD) is performed
on the MPS, and it is truncated to a single dominant singular value, reducing
the MPS to a product state. More specifically, for the MPS ground state, the
SVD is given by

$\displaystyle|\Psi_{\mathrm{MPS}}\rangle=\sum_{i}\lambda_{i}|\psi_{i}^{L}\rangle\langle\psi_{i}^{R}|.$
(9)

Then we keep only the contribution from the largest singular value
$\lambda_{1}$, i.e.
$|\psi_{p}\rangle=\lambda_{1}|\psi_{1}^{L}\rangle\langle\psi_{1}^{R}|$. We can
repeat this process for each bond of the MPS state to obtain the product state
$|\psi\rangle$ that is closest to the MPS ground state
$|\Psi_{\mathrm{MPS}}\rangle$. Subsequently, this obtained product state
undergoes several variational optimization sweeps, where the bond dimension is
fixed at unity, until the overlap with the exact ground state converges.

To compare these two algorithms, in Fig. 2, we show the expected value of the
operator $\hat{\sigma}_{i}^{z}$ for the exact ground state calculated from the
DMRG method and the product states calculated through the L-BFGS and SVD
truncation methods respectively for a Rydberg chain of length $L=19$ in
different phases. In Fig. 2 (a) and (b), we show
$\langle\hat{\sigma}_{i}^{z}\rangle$ for the $Z_{2}$ and $Z_{3}$ ordered
states respectively. In these two phases, $\langle\hat{\sigma}_{i}^{z}\rangle$
of the product states obtained from the L-BFGS method and the SVD truncation
method almost coincide with that of the exact ground state. However, for the
disordered phase as shown in Fig. 2 (c), $\langle\hat{\sigma}_{i}^{z}\rangle$
of the product state calculated from L-BFGS method is closer to that of the
exact ground state that the SVD truncation method. Thus, we will adopt the
L-BFGS method in our numerical calculations.

##  IV Numerical results

Due to the lack of an analytical solution for the Hamiltonian Eq.(1), we
employ the state-of-the-art numerical method, specifically the density matrix
renormalization group method, to determine the ground state using the software
ITensors.jl. Although the Hamiltonian Eq.(1) contains long-range van der Waals
interaction, we adopt a truncation scheme for the parameter range
$R_{b}/a\lesssim 3$ considered here. We set the van der Waals interaction to
zero for atoms separated by more than $5a$, i.e., $V(|i-j|>5)=0$. In our
numerical calculations, we set both the truncation error and relative energy
tolerance error to $10^{-10}$.

Fig. 3 illustrates the geometric phase density $\phi_{B}$ and the geometric
entanglement density $\mathcal{E}$ as functions of the detuning
$\tilde{\delta}=\delta/\Omega$ and the Rydberg blockade radius $R_{b}/a$, for
a Rydberg chain of length $L=121$. Both quantities clearly exhibit lobe
shapes, corresponding to the $Z_{2}$ and $Z_{3}$ ordered states. This
demonstrates that the geometric phase and geometric entanglement can
characterize the phase of the Rydberg chain without prior knowledge of the
system’s symmetry. Furthermore, as we will demonstrate, analyzing the scaling
behavior of the derivatives of GP and GE density with respect to
$\tilde{\delta}$ allows us to extract the critical exponents of the phase
transition in the Rydberg chain.

Figure 3: (a) GP density of the 1d Rydberg atom chain. (b) GE density of the
1d Rydberg atom chain. The system size is $L=121$.

We define the derivative of the GP density with respect to the tuning
parameter $\tilde{\delta}=\delta/\Omega$ as:

$\displaystyle\phi_{B}^{\prime}=\frac{\partial\phi_{B}}{\partial\tilde{\delta}}$
(10)

and the derivative of the GE density as:

$\displaystyle\mathcal{E}^{\prime}=\frac{\partial\mathcal{E}}{\partial\tilde{\delta}}$
(11)

Fig. 4 presents $\phi_{B}^{\prime}$ and $-\mathcal{E}^{\prime}$ as functions
of the tuning parameter $\tilde{\delta}$ for different system sizes $L$, at
$R_{b}/a=1.4$. For finite system sizes, we observe no true divergence, but
rather a peak for each system size. The maximum of each peak defines a pseudo-
critical point $\delta_{m}$, which approaches the real critical point
$\delta_{c}$ as the system size increases. The insets show the values of
$\phi_{B}^{\prime}$ and $-\mathcal{E}^{\prime}$ at the pseudo-critical points
as functions of the logarithm of system size $L$. Both $\phi_{B,m}^{\prime}$
and $-\mathcal{E}_{m}^{\prime}$ increase linearly with $\ln L$. Specifically,
for $\phi_{B,m}^{\prime}$, we have:

$\displaystyle\phi_{B,m}^{\prime}=\kappa_{1,\phi^{\prime}}\ln
L+\mathrm{const.}$ (12)

By using least-square fitting, we have $\kappa_{1,\phi^{\prime}}=0.336$ for
the GP density derivative, and $\kappa_{1,\mathcal{E}^{\prime}}=0.025$ for the
GE density derivative when the tuning parameter $\tilde{\delta}=1.4$. This
logarithmic behavior at the pseudo-critical points suggests a logarithmic
behavior for the GP derivative around the critical point $\tilde{\delta}_{c}$
in the thermodynamic limit:

$\displaystyle\phi_{B}^{\prime}=\kappa_{2,\phi^{\prime}}\ln|\tilde{\delta}-\tilde{\delta}_{c}|+\mathrm{const.}$
(13)

The critical exponent $\nu$ can be obtained from
$\nu=|\kappa_{2,\phi^{\prime}}/\kappa_{1,\phi^{\prime}}|$. For systems like
the $XY$ model where $\kappa_{2}$ can be calculated analytically, this method
can directly extract the critical exponent.

However, for the Rydberg chain, $\kappa_{2}$ cannot be calculated
analytically. Therefore, we must employ the finite-size scaling method to
extract the critical exponent. For quantities with logarithmic scaling
behavior like $\phi_{B}^{\prime}$ (as in Eq. (13)), it can be shown that the
data for $F_{\phi_{B}^{\prime}}=1-\exp(\phi_{B}^{\prime}-\phi_{B,m}^{\prime})$
as a function of $L^{1/\nu}(\tilde{\delta}-\tilde{\delta}_{c})$ should
collapse onto a single curve, independent of system size $L$ [78]. The
challenge lies in not knowing the function of this curve a priori. To quantify
the quality of data collapse, we adopt the quantity
$S(\tilde{\delta}_{c},\nu)$ defined in Ref. [79] (See Appendix A for more
detail). This quantity measures the mean square distance between the data and
the fitted curve for various values of $\delta_{c}$ and $\nu$. By minimizing
$S(\tilde{\delta}_{c},\nu)$ over possible values of $\delta_{c}$ and $\nu$, we
can determine the critical point and critical exponent. Since the GE density
$\mathcal{E}^{\prime}$ also exhibits a logarithmic scaling behavior, we can
define the quantity
$F_{\mathcal{E}^{\prime}}=1-\exp(\mathcal{E}^{\prime}-\mathcal{E}_{m}^{\prime})$.
We can use the same finite size scaling method to find the best data collapse
of $F_{\mathcal{E}^{\prime}}$ as a function of
$L^{1/\nu}(\tilde{\delta}-\tilde{\delta}_{c})$ to extract the quantum critical
point and critical exponent.

Figure 4: (a) Derivative of the GP density, denoted as $\phi_{B}^{\prime}$
over the parameter $\delta/\Omega$, when $R_{b}/a=1.4$. Inset: max value of
$\phi_{B}^{\prime}$ for different system size L. (b) Derivative of the GE
density, denoted as $\mathcal{E}^{\prime}$ over the parameter $\delta/\Omega$,
when $R_{b}/a=1.4$. Inset: max value of $\mathcal{E}^{\prime}$ for different
system size L.

###  IV.1 Disorder to $Z_{2}$ ordered phase transition

Figure 5: Data collapse for the derivative of (a) GP and (b) GE, when
$R_{b}/a=1.4$.

Fig. 5 illustrates the data collapse for quantities $F_{\phi^{\prime}}$ and
$F_{\mathcal{E}^{\prime}}$ as functions of
$L^{1/\nu}(\tilde{\delta}-\tilde{\delta}_{c})$, for $R_{b}/a=1.4$. In this
regime, a phase transition from disorder to $Z_{2}$ ordered phase is expected.
The data collapse is robust across different system sizes for both GP and GE
derivatives. Analysis of the GP density data collapse yields a critical point
$\tilde{\delta}_{c,\phi^{\prime}}=1.020$ and a critical exponent
$\nu_{\phi^{\prime}}=1.058$. Similarly, for the GE density, we obtain
$\tilde{\delta}_{c,\mathcal{E}^{\prime}}=1.018$ and
$\nu_{c,\mathcal{E}^{\prime}}=1.067$. Notably, both GP and GE densities
produce critical exponents close to 1, consistent with the expectation that
the disorder to $Z_{2}$ ordered phase transition belongs to the Ising
universality class. These results are also compatible with that in Ref. [80].
These results demonstrate the effectiveness of both GP and GE in
characterizing this quantum phase transition.

###  IV.2 Disorder to $Z_{3}$ ordered phase transition

Figure 6: Derivative of (a) the GP density $\phi_{B}^{\prime}$ and (b) the GE
density over the tuning parameter $\tilde{\delta}=\delta/\Omega$, when
$R_{b}/a=2.3$. Inset: The values at pseudo-critical points of (a)
$\phi_{B}^{\prime}$ and (b) $\mathcal{E}^{\prime}$ varying with the logarithm
of system size $L$.

Figure 6 illustrates the numerical results for the GP and GE density
derivatives as functions of the tuning parameter $\tilde{\delta}$ for various
system sizes, when $R_{b}/a=2.3$. This regime corresponds to a potential phase
transition from a disordered to a $Z_{3}$ ordered phase. The insets show the
values of $\phi_{B}^{\prime}$ and $-\mathcal{E}^{\prime}$ at the pseudo-
critical points plotted against the logarithm of system size $L$. Both
quantities exhibit a clear linear increase with $\ln L$, consistent with the
expected logarithmic scaling behavior. Using least-squares fitting, we obtain
scaling coefficients $\kappa_{1,\phi^{\prime}}=0.663$ for $\phi_{B}^{\prime}$
and $\kappa_{1,\mathcal{E}^{\prime}}=0.027$ for $-\mathcal{E}^{\prime}$. This
logarithmic scaling provides a foundation for further finite-size scaling
analysis.

Figure 7 presents the data collapse for quantities $F_{\phi^{\prime}}$ and
$F_{\mathcal{E}^{\prime}}$ as functions of
$L^{1/\nu}(\tilde{\delta}-\tilde{\delta}_{c})$ for $R_{b}/a=2.3$. Employing
finite-size scaling analysis on the GP density derivative data yields a
critical point $\tilde{\delta}_{c,\phi^{\prime}}=1.797$ and a critical
exponent $\nu_{\phi^{\prime}}=0.665$. Similarly, analysis of the GE density
derivative data produces $\tilde{\delta}_{c,\mathcal{E}^{\prime}}=1.799$ and
$\nu_{\mathcal{E}^{\prime}}=0.828$. The close agreement between
$\tilde{\delta}_{c,\phi^{\prime}}$ and
$\tilde{\delta}_{c,\mathcal{E}^{\prime}}$ lends credibility to our results.
Moreover, these critical points are consistent with those obtained in Ref.
[80] using alternative methods, further validating our approach. And the
critical exponent $\nu_{\mathcal{E}^{\prime}}=0.828$ from the GE density
derivative is more close to the one in Ref. [80]. The discrepancy in critical
exponents between GP and GE analyses may warrant further investigation,
potentially revealing subtleties in how these quantities capture the $Z_{3}$
ordered phase transition.

Figure 7: Data collapse for the derivative of (a) GP and (b) GE, when
$R_{b}/a=2.3$.

##  V Unifying description of GP and GE through quantum geometry

At first glance, the GE and GP seem unrelated based on respective definitions.
However, they can be related to each other through the so-called quantum
geometric tensor [67]. In this section, we shall discuss the relationship
between them in the Rydberg atom chain case.

We consider the Hilbert space $\mathcal{H}$ formed by the normalized quantum
many-body states of the Rydberg atom chain of length $L$. Assuming this space
$\mathcal{H}$ can be parametrized by the coordinates
${\xi_{1},\xi_{2},...,\xi_{\dim\mathcal{H}}}$, We can define the quantum
metric tensor for a quantum state $|\psi\rangle\in\mathcal{H}$ as [81]

$\displaystyle\mathcal{T}_{\mu\nu}=\langle\partial_{\mu}\psi|\partial_{\nu}\psi\rangle-\langle\partial_{\mu}\psi|\psi\rangle\langle\psi|\partial_{\nu}\psi\rangle,$
(14)

where $\partial_{\mu}|\psi\rangle=\partial|\psi\rangle/\partial\xi_{\mu}$.
This quantum geometric tensor comprises a symmetric real part and an
antisymmetric part, i.e.

$\displaystyle\mathcal{T}_{\mu\nu}=g_{\mu\nu}+i\mathcal{F}_{\mu\nu}.$ (15)

The real part $g_{\mu\nu}$ of the quantum tensor is known as the Fubini-Study
metric in the Hilbert space, while the imaginary part $\mathcal{F}_{\mu\nu}$
corresponds to the Berry curvature.

For a state $|\psi\rangle$ in the Hilbert space $\mathcal{H}$, the overlap of
states and the Fubini-Study metric are related as follows [81]

$\displaystyle|\langle\psi(\xi)|\psi(\xi+d\xi)\rangle|=1-\frac{1}{2}ds^{2},$
(16)

where $ds^{2}=g_{\mu\nu}d\xi_{\mu}d\xi_{\nu}$. Consequently, the overlap
$\Lambda_{\mathrm{max}}$ between the state $|\psi\rangle$ and its closest
product state $|\psi_{p}\rangle$, can be represented by the integration of
$ds$ over the geodesics $\Gamma$ connecting $|\psi\rangle$ and
$|\psi_{p}\rangle$,

$\displaystyle\Lambda_{\mathrm{max}}=1-\frac{1}{2}\left[\int_{\Gamma}ds\right]^{2}.$
(17)

To obtain the geometric phase from the imaginary part of the geometric tensor,
we need to define a closed path in the space $\mathcal{H}$. Such a closed path
can be described by a unitary operator, $U(s)=\otimes_{i}U_{i}(s)$, where
$U_{i}(s)$ is a unitary operator acting on the $i$th Rydberg atom. Hence, the
geometric phase can be expressed as the integration over the surface $\Sigma$
enclosed by this closed path, i.e.

$\displaystyle\Phi_{B}=\int_{\Sigma}F_{\mu\nu}dS^{\mu\nu},$ (18)

where $dS^{\mu\nu}$ is the surface element of $\Sigma$.

To further understand why GP and GE can characterize quantum phase
transitions, we need to consider the space formed by the Hamiltonian $H(\xi)$,
whose unique ground state is given by $|\psi_{0}(\xi)\rangle$. Thus, the
geometric tensor $\mathcal{T}_{\mu\nu}$ can be written as [82, 83]

$\displaystyle\mathcal{T}_{\mu\nu}=\sum_{n\neq
0}\frac{\langle\psi_{0}|\partial_{\mu}H|\psi_{n}\rangle\langle\psi_{n}|\partial_{\nu}H|\psi_{0}\rangle}{(E_{0}-E_{n})^{2}},$
(19)

where $|\psi_{n}\rangle$ is the $n$th eigenstate of $H(\xi)$ with eigenvalue
$E_{n}(\xi)$. Quantum phase transitions occurs at level crossing or avoided
level crossing, and the geometric tensor $\mathcal{T}_{\mu\nu}$ in the form of
Eq.(19) can capture this level crossing, as it display a singular behavior
when level crossing happens. Since GP and GE are related to the geometric
tensor through Eqs.(17) and (18), they can also effectively capture quantum
phase transitions.

##  VI GP and GE in an interferometry setup

In addition to their unified description from the quantum geometry
perspective, the GE and GP can also be conceptualized through an
interferometry setup [67]. This approach not only provides a deeper
understanding but also offers a practical protocol for their measurement. The
setup requires an auxiliary qubit, which in our case can be another Rydberg
atom with states $|g_{0}\rangle,|r_{0}\rangle$. This auxiliary qubit serves as
the two arms of the interferometer. Utilizing this ancilla, we prepare the
initial state as:

$\displaystyle|\Psi_{i}\rangle=\frac{1}{\sqrt{2}}(|r_{0}\rangle+|g_{0}\rangle)\otimes|\psi\rangle,$
(20)

where $|\psi\rangle$ is the ground state of the Rydberg atom chain under
consideration.

This interferometer protocol consists of the following steps:

1\. In the $|r_{0}\rangle$ arm, project the state $|\psi\rangle$ onto its
closest product state $|\psi_{p}\rangle$. This is achieved by applying the
operator: $O_{1}=|g_{0}\rangle\langle g_{0}|\otimes I+|r_{0}\rangle\langle
r_{0}|\otimes|\psi_{p}\rangle\langle\psi_{p}|$ to the initial state
$|\Psi_{i}\rangle$.

2\. Apply the unitary operators:

$\displaystyle
U_{\psi}=\otimes_{i}e^{\frac{i}{2}\pi[\hat{\sigma}_{i}^{z}-\langle\psi|\hat{\sigma}_{i}^{z}|\psi\rangle]}$
(21)

and

$\displaystyle U_{\psi_{p}}=\otimes_{i}e^{\frac{i}{2}\pi\hat{\sigma}_{i}^{z}}$
(22)

to the state $|\psi\rangle$ and the product state $|\psi_{p}\rangle$
respectively. This is equivalent to applying the operator:
$O_{2}=|g_{0}\rangle\langle g_{0}|\otimes U\psi+|r_{0}\rangle\langle
r_{0}|\otimes U{\psi_{p}}$ to the state $O_{1}|\Psi_{i}\rangle$.

3\. Apply a $U(1)$ phase shift $e^{-if}$ to the $|g_{0}\rangle$ arm using the
operator: $O_{3}=|g_{0}\rangle\langle g_{0}|\otimes
e^{-if}I+|r_{0}\rangle\langle r_{0}|\otimes I$.

4\. Bring the two arms back to interference by applying the operator:
$O_{4}=|g_{0}\rangle\langle g_{0}|\otimes I+|g_{0}\rangle\langle r_{0}|\otimes
I$.

After this process, the final state is:

$\displaystyle|\Psi_{f}\rangle=\frac{1}{\sqrt{2}}|g_{0}\rangle\left[e^{-if}U_{\psi}|\psi\rangle+U_{\psi_{p}}|\psi_{p}\rangle\langle\psi_{p}|\psi\rangle\right]$
(23)

The resulting interference fringe is given by:

$\displaystyle
I=\frac{1}{2}\left[1+|\langle\psi_{p}|\psi\rangle|^{2}\right]+\mathrm{Re}(Ae^{-if})$
(24)

where

$\displaystyle A$ $\displaystyle=$
$\displaystyle\langle\psi|U_{\psi}^{\dagger}U_{\psi_{p}}|\psi_{p}\rangle\langle\psi_{p}|\psi\rangle$
(25) $\displaystyle=$
$\displaystyle|\langle\psi_{p}|\psi\rangle|^{2}e^{i\Phi_{B}}$

Based on these results, we can define a complex-valued GE as:

$\displaystyle E_{c}(\psi)=-\log_{2}A=E(\psi)-i\frac{\Phi_{B}}{\ln 2},$ (26)

where the real part represents the geometric entanglement $E(\psi)$ and the
imaginary part corresponds to the geometric phase. This formulation elegantly
combines both the GE and GP into a single complex-valued quantity, providing a
unified measure of these quantum properties.

##  VII Conclusions

In this study, we have demonstrated the effectiveness of GP and GE as
sensitive probes of quantum criticality in Rydberg atom chains. Through DMRG
calculations and finite-size scaling analysis, we have characterized the
critical properties of transitions from disordered to $Z_{2}$ and $Z_{3}$
ordered phases.

Our results show that GP and GE exhibit distinct signatures at quantum phase
transitions, providing complementary information about the system’s critical
properties. We have explored a unifying description of GP and GE from a
quantum geometric perspective, deepening our understanding of the connection
between these quantities and the critical behavior of Rydberg atom chains. We
have also discussed the potential adaptation of an interferometric setup for
the experimental measurement of GP and GE in these systems. While challenges
remain, this approach could enable future experimental verification of our
theoretical predictions.

A promising direction for future research is the investigation of the Uhlmann
phase [84, 85, 86, 87, 88, 89], which extends the concept of geometric phase
to finite temperatures and open quantum systems. This could provide insights
into the behavior of Rydberg atom systems under more realistic conditions.

In conclusion, our study highlights the power of geometric concepts in
characterizing quantum phase transitions in Rydberg atom chains. As quantum
simulation platforms advance, we anticipate that these approaches will play an
increasingly important role in our understanding of quantum many-body systems,
opening new avenues for exploring fundamental questions in quantum physics and
advancing quantum technologies.

###### Acknowledgements.

CYW is supported by the Shuimu Tsinghua scholar program at Tsinghua
University.

##  Appendix A Quality function used for the finite size scaling

In this appendix, we will briefly review the quality function that measures
how well data collapse in the finite size scaling. This quality function is
first introduced in Ref. [79], which refined the one introduced in Ref. [90].
In the following, we use the GP density derivative $\phi_{B}^{\prime}$ as an
example to define the quality function. One can also follow the same procedure
to define the quality function for the GE density derivative.

We let $i$ labels the different system sizes $L_{i}$, $i=1,2,...,k$, and $j$
labels the tuning parameters $\tilde{\delta}_{j}$, $j=1,2,...,n$, with
$\tilde{\delta}_{1}<\tilde{\delta}_{2}<\cdots<\tilde{\delta}_{n}$. Then, we
define the scaled data as

$\displaystyle x_{ij}$ $\displaystyle=$ $\displaystyle
L_{i}^{1/\nu}(\tilde{\delta}_{j}-\tilde{\delta}_{c}),$ (27) $\displaystyle
y_{ij}$ $\displaystyle=$ $\displaystyle
1-\exp[\phi_{B}^{\prime}(i,j)-\phi_{B,m}^{\prime}(i)].$ (28)

And $dy_{ij}$ is the standard error of $y_{ij}$. Then, with this set of data
$\\{x_{ij},y_{ij},dy_{ij}\\}$, we can define $Y_{ij}$ and $dY_{ij}$ which are
the estimated position and the standard error of the master curve, which is
the curve that all the data should collapse to, at $x_{ij}$.

$Y_{ij}$ and $dY_{ij}$ at $x_{ij}$ are defined as follows: For each
$i^{\prime}\neq i$, we select two points
$(x_{i^{\prime}j^{\prime}},y_{i^{\prime}j^{\prime}},dy_{i^{\prime}j^{\prime}})$
and
$(x_{i^{\prime},j^{\prime}+1},y_{i^{\prime},j^{\prime}+1},dy_{i^{\prime},j^{\prime}+1})$
that satisfy $x_{i^{\prime}j^{\prime}}\leq x_{ij}\leq
x_{i^{\prime},j^{\prime}+1}$. If there is no such points for this $i^{\prime}$
data set, we do not select any points from this set. After doing such
selections, suppose we have data $m$ points $(x_{l},y_{l},dy_{l})$,
$l=1,2,...,m$. With this data points, we can do a weighted least squares
linear fit. Then, $Y_{ij}$ is the value of that fitted straight line at
$x_{ij}$, and $dY_{ij}$ is the associated standard error, i.e.

$\displaystyle Y_{ij}$ $\displaystyle=$
$\displaystyle\frac{K_{xx}K_{y}-K_{x}K_{xy}}{\Delta}+x_{ij}\frac{KK_{xy}-K_{x}K_{y}}{\Delta},$
(29) $\displaystyle dY_{ij}$ $\displaystyle=$
$\displaystyle\frac{1}{\Delta}(K_{xx}-2x_{ij}K_{x}+x_{ij}^{2}K)$ (30)

where $w_{l}=\sum 1/dy_{l}^{2},\ K=\sum w_{l},\ K_{x}=\sum w_{l}x_{l},\
K_{y}=\sum w_{l}y_{l},\ K_{xx}=\sum w_{l}x_{l}^{2},\ K_{xy}=\sum
w_{l}x_{l}y_{l}$, and $\Delta=KK_{xx}-K_{x}^{2}$. Then we can define the
quality function is defined as the mean square distance of the data set to the
fitted master curve in units of standard errors, i.e.

$\displaystyle
S(\tilde{\delta}_{c},\nu)=\frac{1}{N}\sum_{ij}\frac{(y_{ij}-Y_{ij})^{2}}{dy_{ij}^{2}+dY_{ij}^{2}}.$
(31)

By minimizing this quality function over all possible $\tilde{\delta}_{c}$ and
$\nu$, we should get the correct critical point and critical exponent.

## References

  * Bernien  _et al._ [2017] H. Bernien, S. Schwartz, A. Keesling, H. Levine, A. Omran, H. Pichler, S. Choi, A. S. Zibrov, M. Endres, M. Greiner, V. Vuletić, and M. D. Lukin, Probing many-body dynamics on a 51-atom quantum simulator, Nature 551, 579 (2017).
  * Browaeys and Lahaye [2020] A. Browaeys and T. Lahaye, Many-body physics with individually controlled Rydberg atoms, Nat. Phys. 16, 132 (2020).
  * Ebadi  _et al._ [2021] S. Ebadi, T. T. Wang, H. Levine, A. Keesling, G. Semeghini, A. Omran, D. Bluvstein, R. Samajdar, H. Pichler, W. W. Ho, S. Choi, S. Sachdev, M. Greiner, V. Vuletić, and M. D. Lukin, Quantum phases of matter on a 256-atom programmable quantum simulator, Nature 595, 227 (2021).
  * Turner  _et al._ [2018a] C. J. Turner, A. A. Michailidis, D. A. Abanin, M. Serbyn, and Z. Papić, Weak ergodicity breaking from quantum many-body scars, Nature Phys 14, 745 (2018a).
  * Turner  _et al._ [2018b] C. J. Turner, A. A. Michailidis, D. A. Abanin, M. Serbyn, and Z. Papić, Quantum scarred eigenstates in a Rydberg atom chain: Entanglement, breakdown of thermalization, and stability to perturbations, Phys. Rev. B 98, 155134 (2018b).
  * Lin and Motrunich [2019] C.-J. Lin and O. I. Motrunich, Exact Quantum Many-Body Scar States in the Rydberg-Blockaded Atom Chain, Phys. Rev. Lett. 122, 173401 (2019).
  * Lin  _et al._ [2020] C.-J. Lin, V. Calvera, and T. H. Hsieh, Quantum many-body scar states in two-dimensional Rydberg atom arrays, Phys. Rev. B 101, 220304 (2020).
  * Serbyn  _et al._ [2021] M. Serbyn, D. A. Abanin, and Z. Papić, Quantum many-body scars and weak breaking of ergodicity, Nat. Phys. 17, 675 (2021).
  * Turner  _et al._ [2021] C. J. Turner, J.-Y. Desaules, K. Bull, and Z. Papić, Correspondence Principle for Many-Body Scars in Ultracold Rydberg Atoms, Phys. Rev. X 11, 021021 (2021).
  * Cheng  _et al._ [2022] Y. Cheng, S. Liu, W. Zheng, P. Zhang, and H. Zhai, Tunable Confinement-Deconfinement Transition in an Ultracold-Atom Quantum Simulator, PRX Quantum 3, 040317 (2022).
  * Pan and Zhai [2022] L. Pan and H. Zhai, Composite spin approach to the blockade effect in Rydberg atom arrays, Phys. Rev. Res. 4, L032037 (2022).
  * Yao  _et al._ [2022] Z. Yao, L. Pan, S. Liu, and H. Zhai, Quantum many-body scars and quantum criticality, Phys. Rev. B 105, 125123 (2022).
  * Chandran  _et al._ [2023] A. Chandran, T. Iadecola, V. Khemani, and R. Moessner, Quantum Many-Body Scars: A Quasiparticle Perspective, Annual Review of Condensed Matter Physics 14, 443 (2023).
  * Whitsitt  _et al._ [2018] S. Whitsitt, R. Samajdar, and S. Sachdev, Quantum field theory for the chiral clock transition in one spatial dimension, Phys. Rev. B 98, 205118 (2018).
  * Verresen  _et al._ [2021] R. Verresen, M. D. Lukin, and A. Vishwanath, Prediction of Toric Code Topological Order from Rydberg Blockade, Phys. Rev. X 11, 031005 (2021).
  * Semeghini  _et al._ [2021] G. Semeghini, H. Levine, A. Keesling, S. Ebadi, T. T. Wang, D. Bluvstein, R. Verresen, H. Pichler, M. Kalinowski, R. Samajdar, A. Omran, S. Sachdev, A. Vishwanath, M. Greiner, V. Vuletić, and M. D. Lukin, Probing topological spin liquids on a programmable quantum simulator, Science 374, 1242 (2021).
  * Giudice  _et al._ [2022] G. Giudice, F. M. Surace, H. Pichler, and G. Giudici, Trimer states with ${\mathbb{Z}}_{3}$ topological order in Rydberg atom arrays, Phys. Rev. B 106, 195155 (2022).
  * Giudici  _et al._ [2022] G. Giudici, M. D. Lukin, and H. Pichler, Dynamical Preparation of Quantum Spin Liquids in Rydberg Atom Arrays, Phys. Rev. Lett. 129, 090401 (2022).
  * Lee  _et al._ [2023] J. Y. Lee, J. Ramette, M. A. Metlitski, V. Vuletić, W. W. Ho, and S. Choi, Landau-Forbidden Quantum Criticality in Rydberg Quantum Simulators, Phys. Rev. Lett. 131, 083601 (2023).
  * Samajdar  _et al._ [2023] R. Samajdar, D. G. Joshi, Y. Teng, and S. Sachdev, Emergent ${\mathbb{Z}}_{2}$ Gauge Theories and Topological Excitations in Rydberg Atom Arrays, Phys. Rev. Lett. 130, 043601 (2023).
  * Slagle  _et al._ [2022] K. Slagle, Y. Liu, D. Aasen, H. Pichler, R. S. K. Mong, X. Chen, M. Endres, and J. Alicea, Quantum spin liquids bootstrapped from Ising criticality in Rydberg arrays, Phys. Rev. B 106, 115122 (2022).
  * Verresen and Vishwanath [2022] R. Verresen and A. Vishwanath, Unifying Kitaev Magnets, Kagome Dimer Models, and Ruby Rydberg Spin Liquids, Phys. Rev. X 12, 041029 (2022).
  * Yan  _et al._ [2022] Z. Yan, R. Samajdar, Y.-C. Wang, S. Sachdev, and Z. Y. Meng, Triangular lattice quantum dimer model with variable dimer density, Nat Commun 13, 5799 (2022).
  * Cheng  _et al._ [2023] Y. Cheng, C. Li, and H. Zhai, Variational approach to quantum spin liquid in a Rydberg atom simulator, New J. Phys. 25, 033010 (2023).
  * Kalinowski  _et al._ [2023] M. Kalinowski, N. Maskara, and M. D. Lukin, Non-Abelian Floquet Spin Liquids in a Digital Rydberg Simulator, Phys. Rev. X 13, 031008 (2023).
  * Kornjača  _et al._ [2023] M. Kornjača, R. Samajdar, T. Macrì, N. Gemelke, S.-T. Wang, and F. Liu, Trimer quantum spin liquid in a honeycomb array of Rydberg atoms, Commun Phys 6, 1 (2023).
  * Tarabunga  _et al._ [2022] P. S. Tarabunga, F. M. Surace, R. Andreoni, A. Angelone, and M. Dalmonte, Gauge-Theoretic Origin of Rydberg Quantum Spin Liquids, Phys. Rev. Lett. 129, 195301 (2022).
  * Keesling  _et al._ [2019] A. Keesling, A. Omran, H. Levine, H. Bernien, H. Pichler, S. Choi, R. Samajdar, S. Schwartz, P. Silvi, S. Sachdev, P. Zoller, M. Endres, M. Greiner, V. Vuletić, and M. D. Lukin, Quantum Kibble–Zurek mechanism and critical dynamics on a programmable Rydberg simulator, Nature 568, 207 (2019).
  * Chepiga and Mila [2021] N. Chepiga and F. Mila, Kibble-Zurek exponent and chiral transition of the period-4 phase of Rydberg chains, Nat Commun 12, 414 (2021).
  * Adams  _et al._ [2019] C. S. Adams, J. D. Pritchard, and J. P. Shaffer, Rydberg atom quantum technologies, J. Phys. B: At. Mol. Opt. Phys. 53, 012002 (2019).
  * Cohen and Thompson [2021] S. R. Cohen and J. D. Thompson, Quantum Computing with Circular Rydberg Atoms, PRX Quantum 2, 030322 (2021).
  * Cong  _et al._ [2022] I. Cong, H. Levine, A. Keesling, D. Bluvstein, S.-T. Wang, and M. D. Lukin, Hardware-Efficient, Fault-Tolerant Quantum Computation with Rydberg Atoms, Phys. Rev. X 12, 021049 (2022).
  * Evered  _et al._ [2023] S. J. Evered, D. Bluvstein, M. Kalinowski, S. Ebadi, T. Manovitz, H. Zhou, S. H. Li, A. A. Geim, T. T. Wang, N. Maskara, H. Levine, G. Semeghini, M. Greiner, V. Vuletić, and M. D. Lukin, High-fidelity parallel entangling gates on a neutral-atom quantum computer, Nature 622, 268 (2023).
  * Nguyen  _et al._ [2018] T. L. Nguyen, J. M. Raimond, C. Sayrin, R. Cortiñas, T. Cantat-Moltrecht, F. Assemat, I. Dotsenko, S. Gleyzes, S. Haroche, G. Roux, Th. Jolicoeur, and M. Brune, Towards Quantum Simulation with Circular Rydberg Atoms, Phys. Rev. X 8, 011032 (2018).
  * Paredes-Barato and Adams [2014] D. Paredes-Barato and C. S. Adams, All-Optical Quantum Information Processing Using Rydberg Gates, Phys. Rev. Lett. 112, 040501 (2014).
  * Ryabtsev  _et al._ [2005] I. I. Ryabtsev, D. B. Tretyakov, and I. I. Beterov, Applicability of Rydberg atoms to quantum computers, J. Phys. B: At. Mol. Opt. Phys. 38, S421 (2005).
  * Saffman  _et al._ [2010] M. Saffman, T. G. Walker, and K. Mølmer, Quantum information with Rydberg atoms, Rev. Mod. Phys. 82, 2313 (2010).
  * Saffman [2016] M. Saffman, Quantum computing with atomic qubits and Rydberg interactions: Progress and challenges, J. Phys. B: At. Mol. Opt. Phys. 49, 202001 (2016).
  * Wu  _et al._ [2021] X. Wu, X. Liang, Y. Tian, F. Yang, C. Chen, Y.-C. Liu, M. K. Tey, and L. You, A concise review of Rydberg atom based quantum computation and quantum simulation*, Chinese Phys. B 30, 020305 (2021).
  * Zhao  _et al._ [2017] P. Z. Zhao, X.-D. Cui, G. F. Xu, E. Sjöqvist, and D. M. Tong, Rydberg-atom-based scheme of nonadiabatic geometric quantum computation, Phys. Rev. A 96, 052316 (2017).
  * 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, S. Choi, M. Serbyn, M. Greiner, V. Vuletić, and M. D. Lukin, Controlling quantum many-body dynamics in driven Rydberg atom arrays, Science 371, 1355 (2021).
  * Ebadi  _et al._ [2022] S. Ebadi, A. Keesling, M. Cain, T. T. Wang, H. Levine, D. Bluvstein, G. Semeghini, A. Omran, J.-G. Liu, R. Samajdar, X.-Z. Luo, B. Nash, X. Gao, B. Barak, E. Farhi, S. Sachdev, N. Gemelke, L. Zhou, S. Choi, H. Pichler, S.-T. Wang, M. Greiner, V. Vuletić, and M. D. Lukin, Quantum optimization of maximum independent set using Rydberg atom arrays, Science 376, 1209 (2022).
  * Omran  _et al._ [2019] A. Omran, H. Levine, A. Keesling, G. Semeghini, T. T. Wang, S. Ebadi, H. Bernien, A. S. Zibrov, H. Pichler, S. Choi, J. Cui, M. Rossignolo, P. Rembold, S. Montangero, T. Calarco, M. Endres, M. Greiner, V. Vuletić, and M. D. Lukin, Generation and manipulation of Schrödinger cat states in Rydberg atom arrays, Science 365, 570 (2019).
  * Osterloh  _et al._ [2002] A. Osterloh, L. Amico, G. Falci, and R. Fazio, Scaling of entanglement close to a quantum phase transition, Nature 416, 608 (2002).
  * Carollo  _et al._ [2020] A. Carollo, D. Valenti, and B. Spagnolo, Geometry of quantum phase transitions, Physics Reports 838, 1 (2020).
  * Chiara and Sanpera [2018] G. D. Chiara and A. Sanpera, Genuine quantum correlations in quantum many-body systems: A review of recent progress, Rep. Prog. Phys. 81, 074002 (2018).
  * Berry [1984] M. V. Berry, Quantal phase factors accompanying adiabatic changes, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences 392, 45 (1984).
  * Thouless  _et al._ [1982] D. J. Thouless, M. Kohmoto, M. P. Nightingale, and M. den Nijs, Quantized Hall Conductance in a Two-Dimensional Periodic Potential, Phys. Rev. Lett. 49, 405 (1982).
  * Zhu [2006] S.-L. Zhu, Scaling of Geometric Phases Close to the Quantum Phase Transition in the $XY$ Spin Chain, Phys. Rev. Lett. 96, 077206 (2006).
  * Carollo and Pachos [2005] A. C. M. Carollo and J. K. Pachos, Geometric Phases and Criticality in Spin-Chain Systems, Phys. Rev. Lett. 95, 157203 (2005).
  * Hamma [2006] A. Hamma, Berry phases and quantum phase transitions (2006), arXiv:quant-ph/0602091 [quant-ph] .
  * Pachos and Carollo [2006] J. K. Pachos and A. C. Carollo, Geometric phases and criticality in spin systems, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 364, 3463 (2006).
  * Patra  _et al._ [2011] A. Patra, V. Mukherjee, and A. Dutta, Path-dependent scaling of geometric phase near a quantum multi-critical point, J. Stat. Mech. 2011, P03026 (2011).
  * Quan [2009] H. T. Quan, Finite-temperature scaling of magnetic susceptibility and the geometric phase in the XY spin chain, J. Phys. A: Math. Theor. 42, 395002 (2009).
  * Reuter  _et al._ [2007] M. E. Reuter, M. J. Hartmann, and M. B. Plenio, Geometric phases and critical phenomena in a chain of interacting spins, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 463, 1271 (2007).
  * Chen  _et al._ [2006] G. Chen, J. Li, and J.-Q. Liang, Critical property of the geometric phase in the Dicke model, Phys. Rev. A 74, 054101 (2006).
  * Plastina  _et al._ [2006] F. Plastina, G. Liberti, and A. Carollo, Scaling of Berry’s phase close to the Dicke quantum phase transition, EPL 76, 182 (2006).
  * Wei and Goldbart [2003] T.-C. Wei and P. M. Goldbart, Geometric measure of entanglement and applications to bipartite and multipartite quantum states, Phys. Rev. A 68, 042307 (2003).
  * Barnum and Linden [2001] H. Barnum and N. Linden, Monotones and invariants for multi-particle quantum states, J. Phys. A: Math. Gen. 34, 6787 (2001).
  * Shimony [1995] A. Shimony, Degree of Entanglementa, Annals of the New York Academy of Sciences 755, 675 (1995).
  * Wei  _et al._ [2005] T.-C. Wei, D. Das, S. Mukhopadyay, S. Vishveshwara, and P. M. Goldbart, Global entanglement and quantum criticality in spin chains, Phys. Rev. A 71, 060305 (2005).
  * Orús [2008] R. Orús, Universal Geometric Entanglement Close to Quantum Phase Transitions, Phys. Rev. Lett. 100, 130502 (2008).
  * Montakhab and Asadian [2010] A. Montakhab and A. Asadian, Multipartite entanglement and quantum phase transitions in the one-, two-, and three-dimensional transverse-field Ising model, Phys. Rev. A 82, 062313 (2010).
  * Hofmann  _et al._ [2014] M. Hofmann, A. Osterloh, and O. Gühne, Scaling of genuine multiparticle entanglement close to a quantum phase transition, Phys. Rev. B 89, 134101 (2014).
  * Orús and Wei [2010] R. Orús and T.-C. Wei, Visualizing elusive phase transitions with geometric entanglement, Phys. Rev. B 82, 155120 (2010).
  * Radgohar and Montakhab [2018] R. Radgohar and A. Montakhab, Global entanglement and quantum phase transitions in the transverse XY Heisenberg chain, Phys. Rev. B 97, 024434 (2018).
  * Azimi Mousolou  _et al._ [2013] V. Azimi Mousolou, C. M. Canali, and E. Sjöqvist, Unifying geometric entanglement and geometric phase in a quantum phase transition, Phys. Rev. A 88, 012310 (2013).
  * Perez-Garcia  _et al._ [2007] D. Perez-Garcia, F. Verstraete, M. M. Wolf, and J. I. Cirac, Matrix product state representations, Quantum Info. Comput. 7, 401 (2007).
  * Schollwöck [2005] U. Schollwöck, The density-matrix renormalization group, Rev. Mod. Phys. 77, 259 (2005).
  * Schollwöck [2011] U. Schollwöck, The density-matrix renormalization group in the age of matrix product states, Annals of Physics 326, 96 (2011).
  * White [1992] S. R. White, Density matrix formulation for quantum renormalization groups, Phys. Rev. Lett. 69, 2863 (1992).
  * 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, Phys. Rev. X 7, 041063 (2017).
  * Hasan and Kane [2010] M. Z. Hasan and C. L. Kane, Colloquium: Topological insulators, Rev. Mod. Phys. 82, 3045 (2010).
  * Qi and Zhang [2011] X.-L. Qi and S.-C. Zhang, Topological insulators and superconductors, Rev. Mod. Phys. 83, 1057 (2011).
  * Nocedal and Wright [2006] J. Nocedal and S. Wright,  _Numerical Optimization_ , 2nd ed. (Springer, New York, 2006).
  * Mogensen and Riseth [2018] P. K. Mogensen and A. N. Riseth, Optim: A mathematical optimization package for Julia, Journal of Open Source Software 3, 615 (2018).
  * Schneider  _et al._ [2022] J. T. Schneider, S. J. Thomson, and L. Sanchez-Palencia, Entanglement spectrum and quantum phase diagram of the long-range XXZ chain, Phys. Rev. B 106, 014306 (2022).
  * Barber [1983] M. N. Barber, in  _Phase Transitions and Critical Phenomena_ , Vol. 8 (Academic, New York, 1983) pp. 146–266.
  * Houdayer and Hartmann [2004] J. Houdayer and A. K. Hartmann, Low-temperature behavior of two-dimensional Gaussian Ising spin glasses, Phys. Rev. B 70, 014418 (2004).
  * Yu  _et al._ [2022] X.-J. Yu, S. Yang, J.-B. Xu, and L. Xu, Fidelity susceptibility as a diagnostic of the commensurate-incommensurate transition: A revisit of the programmable Rydberg chain, Phys. Rev. B 106, 165124 (2022).
  * Provost and Vallee [1980] J. P. Provost and G. Vallee, Riemannian structure on manifolds of quantum states, Commun.Math. Phys. 76, 289 (1980).
  * Zanardi  _et al._ [2007] P. Zanardi, P. Giorda, and M. Cozzini, Information-Theoretic Differential Geometry of Quantum Phase Transitions, Phys. Rev. Lett. 99, 100603 (2007).
  * Campos Venuti and Zanardi [2007] L. Campos Venuti and P. Zanardi, Quantum Critical Scaling of the Geometric Tensors, Phys. Rev. Lett. 99, 095701 (2007).
  * Uhlmann [1986] A. Uhlmann, Parallel transport and “quantum holonomy” along density operators, Reports on Mathematical Physics 24, 229 (1986).
  * Uhlmann [1991] A. Uhlmann, A gauge field governing parallel transport along mixed states, Lett Math Phys 21, 229 (1991).
  * Uhlmann [1993] A. Uhlmann, Density operators as an arena for differential geometry, Reports on Mathematical Physics 33, 253 (1993).
  * Ericsson  _et al._ [2003a] M. Ericsson, A. K. Pati, E. Sjöqvist, J. Brännlund, and D. K. L. Oi, Mixed State Geometric Phases, Entangled Systems, and Local Unitary Transformations, Phys. Rev. Lett. 91, 090405 (2003a).
  * Ericsson  _et al._ [2003b] M. Ericsson, E. Sjöqvist, J. Brännlund, D. K. L. Oi, and A. K. Pati, Generalization of the geometric phase to completely positive maps, Phys. Rev. A 67, 020101 (2003b).
  * Wang and He [2024] C.-Y. Wang and Y. He, The Uhlmann phase winding in Bose-Einstein condensates at finite temperature (2024), arXiv:2403.05127 [cond-mat.quant-gas] .
  * Kawashima and Ito [1993] N. Kawashima and N. Ito, Critical Behavior of the Three-Dimensional ±J Model in a Magnetic Field, J. Phys. Soc. Jpn. 62, 435 (1993).