# Dynamical phase transitions in quantum spin models with antiferromagnetic
long-range interactions

Jad C. Halimeh<EMAIL_ADDRESS>INO-CNR BEC Center and Department of
Physics, University of Trento, Via Sommarive 14, I-38123 Trento, Italy
Maarten Van Damme  Department of Physics and Astronomy, University of Ghent,
Krijgslaan 281, 9000 Gent, Belgium  Lingzhen Guo  Max Planck Institute for the
Science of Light, Staudtstrasse 2, 91058 Erlangen, Germany  Johannes Lang
<EMAIL_ADDRESS>Max Planck Institute for the Physics of Complex Systems,
01187 Dresden, Germany  Philipp Hauke<EMAIL_ADDRESS>INO-CNR BEC
Center and Department of Physics, University of Trento, Via Sommarive 14,
I-38123 Trento, Italy

(August 27, 2024)

###### Abstract

In recent years, dynamical phase transitions and out-of-equilibrium
criticality have been at the forefront of ultracold gases and condensed matter
research. Whereas universality and scaling are established topics in
equilibrium quantum many-body physics, out-of-equilibrium extensions of such
concepts still leave much to be desired. Using exact diagonalization and the
time-dependent variational principle in uniform martrix product states, we
calculate the time evolution of the local order parameter and Loschmidt return
rate in transverse-field Ising chains with antiferromagnetic power law-
decaying interactions, and map out the corresponding rich dynamical phase
diagram. Anomalous cusps in the return rate, which are ubiquitous at small
quenches within the ordered phase in the case of ferromagnetic long-range
interactions, are absent within the accessible timescales of our simulations.
We attribute this to much weaker domain-wall binding in the antiferromagnetic
case. For quenches across the quantum critical point, regular cusps appear in
the return rate and connect to the local order parameter changing sign,
indicating the concurrence of two major concepts of dynamical phase
transitions. Our results consolidate conclusions of previous works that a
necessary condition for the appearance of anomalous cusps in the return rate
after quenches within the ordered phase is for topologically trivial local
spin flips to be the energetically dominant excitations in the spectrum of the
quench Hamiltonian. Our findings are readily accessible in modern trapped-ion
setups, and we outline the associated experimental considerations.

###### Contents

  1. I Introduction
  2. II Brief review of dynamical quantum phase transitions
  3. III Model and its equilibrium physics
  4. IV Quench dynamics
    1. IV.1 Nonintegrable model at finite $\alpha>1$
    2. IV.2 Integrable case of $\alpha=0$
  5. V Experimental considerations
  6. VI Conclusions and outlook
  7. A Details regarding iMPS calculations
    1. A.1 Implementation of long-range interactions
    2. A.2 Backward time-evolution trick
    3. A.3 MPS transfer matrix and rate-function branches
  8. B Néel state with permutation invariance

##  I Introduction

Thermal Cardy (1996) and quantum Sachdev (2001) phase transitions in
equilibrium have mostly become very well-established disciplines in physics.
Not only are theoretical tools such as renormalization group Wilson (1971)
able to accurately predict equilibrium critical points and exponents, but also
modern ultracold-atom experiments can realize quantum phase transitions with
great reliability.Greiner  _et al._ (2002); Lewenstein  _et al._ (2007); Bloch
_et al._ (2008); Simon  _et al._ (2011) A foundational breakthrough in this
field has been the discovery of equilibrium universality classes, which
characterize a set of different models that share the same critical exponents
at a continuous phase transition.Cardy (1996); Sachdev (2001) In other words,
these models, though perhaps quite different away from the phase transition,
behave very similarly in its vicinity.

The situation is not as well developed in out-of-equilibrium physics, even
though impressive experimental advances in quantum simulators have allowed
unprecedented control in the observation of various intriguing out-of-
equilibrium phenomena such as prethermalization,Gring  _et al._ (2012); Langen
_et al._ (2015); Neyenhuis  _et al._ (2017) gauge-theory dynamics,Martinez
_et al._ (2016); Bernien  _et al._ (2017); Dai  _et al._ (2017); Klco  _et
al._ (2018); Kokail  _et al._ (2019); Schweizer  _et al._ (2019); Görg  _et
al._ (2019); Mil  _et al._ (2020); Yang  _et al._ (2020) time crystals,Choi
_et al._ (2017); Zhang  _et al._ (2017a); Rovny  _et al._ (2018); Smits  _et
al._ (2018) Kibble-Zurek mechanism,Xu  _et al._ (2014); Anquez  _et al._
(2016); Clark  _et al._ (2016); Cui  _et al._ (2016); Keesling  _et al._
(2019) dynamical phase transitions,Jurcevic  _et al._ (2017); Zhang  _et al._
(2017b); Fläschner  _et al._ (2018); Muniz  _et al._ (2020) many-body
localization,Choi  _et al._ (2016); Smith  _et al._ (2016) and many-body
dephasing.Kaplan  _et al._ (2020) Even though dynamical universality classes
have been discovered for classical systems, Hohenberg and Halperin (1977)
quantum many-body models still lack a clear classification of dynamical
universality. However, a connection has recently been made in out-of-
equilibrium quantum many-body systems to equilibrium criticality. Specfically,
the concept of dynamical quantum phase transitionsHeyl  _et al._ (2013); Heyl
(2014, 2015, 2018); Zvyagin (2016); Mori  _et al._ (2018) (DQPT; also known as
DPT-IIfoo (a)) has been introduced to offer a dynamical analog of the thermal
free energy. This dynamical analog, commonly called the Loschmidt return rate,
can exhibit nonanalytic cusps under certain conditions.Silva (2008) In the
seminal work of Ref. Heyl  _et al._ , 2013, it has been shown that quenches in
the nearest-neighbor transverse-field Ising chain (NN-TFIC) give rise to
nonanalyticities in the Loschmidt return rate only when carried out across the
equilibrium quantum critical point $h_{\mathrm{c}}^{\mathrm{e}}$. Early
assumptions that the appearance of such cusps in the return rate may be
specific to integrable models mappable onto two-band free fermionic systems
have been refuted through numerical studiesKarrasch and Schuricht (2013) in
the time-dependent density matrix renormalization group White (1992, 1993);
Schollwöck (2005); Schmitteckert (2004); Feiguin and White (2005); García-
Ripoll (2006); McCulloch (2007) based on matrix product states.Östlund and
Rommer (1995); Schollwöck (2011) Indeed, the next-nearest-neighbor transverse-
field and the nearest-neighbor tilted-field Ising chains, both nonintegrable
models, exhibit nonanalytic Loschmidt return rates in the wake of certain
quenches.Karrasch and Schuricht (2013) Additionally, quenching across the
equilibrium quantum critical point has been found to be neither a necessary
nor sufficient condition for cusps to appear in the Loschmidt return
rate.Vajna and Dóra (2014); Andraschko and Sirker (2014); Jafari (2019) Even
in two-band free fermionic models with long-range couplings, Dutta and Dutta
(2017); Defenu  _et al._ (2019); Uhrich  _et al._ (2020) a quench is found to
cause cusps in the Loschmidt return rate only when it crosses a dynamical
critical point $h_{\mathrm{c}}^{\mathrm{d}}\leq h_{\mathrm{c}}^{\mathrm{e}}$,
but it is not necessary for the quench to go across
$h_{\mathrm{c}}^{\mathrm{e}}$.

Naturally, the question then arises of how DPT-II is connected to the Landau-
type dynamical phase transition (DPT-I) characterized by the dynamics of a
local order parameter in the wake of a quench.Sciolla and Biroli (2010, 2011,
2013) Considering the case of an ordered initial state, the local order
parameter can either asymptotically decay to a constant value without changing
sign for small quenches to $h<h_{\mathrm{c}}^{\mathrm{d}}$, or it oscillates
around zero with an amplitude that vanishes in the long-time limit for large
quenches to $h>h_{\mathrm{c}}^{\mathrm{d}}$. Altman and Auerbach (2002);
Calabrese and Cardy (2006, 2007); Barmettler  _et al._ (2009); Halimeh  _et
al._ (2017); Calabrese  _et al._ (2011, 2012) In case the model has a finite-
temperature phase transition, the former (latter) case coincides with an
ordered (disordered) long-time steady state. Numerical studies have indicated
that $h_{\mathrm{c}}^{\mathrm{d}}$ is the same for DPT-I and DPT-II
particularly in the case of ferromagnetic (FM) long-range interactions,
Halimeh and Zauner-Stauber (2017); Zauner-Stauber and Halimeh (2017);
Homrighausen  _et al._ (2017); Lang  _et al._ (2018a, b); Žunkovič  _et al._
(2018); Piccitto  _et al._ (2019) although this depends on the specific
definition of the phases of DPT-II.foo (b) This picture is also found to
persist even at finite temperature. For example, quenches in the NN-TFIC at
finite temperature will always give rise to an analytic return rate,Abeling
and Kehrein (2016) which coincides with the absence of an equilibrium phase
transition in this model at finite temperature. On the other hand, the fully
connected transverse-field Ising model, also known as the Lipkin-Meshkov-Glick
(LMG) model, which supports a finite-temperature phase transition in
equilibrium, exhibits two different kinds of cusps depending on whether one
quenches below or above a dynamical critical point
$h_{\text{c}}^{\text{d}}(h_{\text{i}},T)$ from the ordered phase, where this
dynamical critical point depends on the initial value of the transverse-field
strength $h_{\text{i}}$ and the temperature $T$ at which the initial state is
prepared.Homrighausen  _et al._ (2017); Lang  _et al._ (2018a, b); Sehrawat
_et al._ (2020) This is also found to coincide with a long-time ferromagnetic
or paramagnetic steady state, respectively, and is directly related to the LMG
model hosting an equilibrium phase transition at a temperature
$T<T_{\mathrm{c}}$, where $T_{\mathrm{c}}$ is its thermal critical point.
Cusps arising in the return rate of the LMG model for quenches below
$h_{\mathrm{c}}^{\mathrm{d}}(h_{\text{i}},T)$ within the ordered phase, where
$T$ is the temperature at which the initial state is prepared, are called
anomalous and they do not correspond to any zeros in the dynamics of the order
parameter. Their regular counterparts occur when the quench goes from the
ordered phase to above $h_{\mathrm{c}}^{\mathrm{d}}(h_{\text{i}},T)$, and just
as in the traditional case of the NN-TFIC at $T=0$, these cusps correspond to
zeros in the dynamics of the order parameter. Anomalous cusps have been shown
to occur when the quench Hamiltonian’s spectrum in the ordered phase
energetically favors local spin excitations, such as in the case of the LMG
model,Homrighausen  _et al._ (2017); Lang  _et al._ (2018a, b) two-dimensional
quantum Ising models, Hashizume  _et al._ (2018, 2020) and transverse-field
Ising chains with sufficiently long-range FM interactions.Halimeh and Zauner-
Stauber (2017); Zauner-Stauber and Halimeh (2017); Halimeh  _et al._ (2020a)

In this work, we extend the connection between DPT-I and DPT-II to transverse-
field Ising chains with antiferromagnetic (AF) power-law interactions. Given
their experimental feasibility and more natural implementation in trapped ions
relative to their FM counterparts,Britton  _et al._ (2012); Islam  _et al._
(2013); Richerme  _et al._ (2014); Jurcevic  _et al._ (2014, 2015); Smith  _et
al._ (2016); Jurcevic  _et al._ (2017); Zhang  _et al._ (2017b) it is
important to understand the dynamical critical properties of these systems to
guide and complement modern ion-trap setups in their investigations. Moreover,
when interactions are long-range, frustration arises in the AF case, but not
in the FM case. It is therefore also worth studying the effect of frustration
on quench dynamics in the AF-TFIC.

The rest of the paper is organized as follows: In Sec. II, we provide a short
review of the theory of dynamical quantum phase transitions (DPT-II). In Sec.
III, we discuss the paradigmatic quantum Ising chain with power-law
interactions and its equilibrium features both in the AF and FM cases,
including analytic and numerical investigations of the underlying domain-wall
interactions. Our numerical results obtained from infinite matrix product
states (iMPS) and exact diagonalization (ED) are then presented in Sec. IV.
Experimental considerations of our findings are discussed in Sec. V. We
provide concluding remarks and an outlook in Sec. VI. Further details on our
iMPS framework are provided in Appendix A and a discussion of the ED
implementation can be found in Appendix B.

##  II Brief review of dynamical quantum phase transitions

For completeness, we recap here the theory of dynamical quantum phase
transitions, first described in Ref. Heyl  _et al._ , 2013. Its basic premise
lies in recognizing the connection between the thermal partition function
$\mathcal{Z}=\Tr\\{e^{-\beta H}\\}$ and the Loschmidt amplitude

$\displaystyle G(t)=\bra{\psi_{0}}e^{-iHt}\ket{\psi_{0}},$ (1)

where $\ket{\psi_{0}}$ is an initial state quenched by a Hamiltonian $H$. By
viewing complexified time $it$ as inverse temperature, one quickly determines
that $G(t)$ is nothing but a boundary partition function. Consequently, the
Loschmidt return rate

$\displaystyle r(t)=-\lim_{N\to\infty}\frac{1}{N}\ln\lvert G(t)\rvert^{2},$
(2)

is a dynamical analog of the thermal free energy, and one can extend the
Ehrenfest classification of thermal phase transitions to it. The time
$t_{\mathrm{c}}$ at which $r(t)$ exhibits a nonanalyticity is a critical time
at which a dynamical quantum phase transition occurs, much the same way as in
equilibrium where a thermal phase transition occurs at the critical
temperature $\beta_{\mathrm{c}}^{-1}$ at which the thermal free energy
$-\beta^{-1}\ln\mathcal{Z(\beta)}$ has a nonanalyticity. As such, the theory
of thermal phase transitions is extended into the far-from-equilibrium realm,
wherein DPT-II serves as a formal concept of dynamical phase transitions.
Further extensions of universal scaling behavior in the vicinity of cusps in
the return rate have also been recently explored.Halimeh and Gurarie (2019);
Wu (2019, 2020a, 2020b); Trapin  _et al._ (2020); Halimeh  _et al._ (2020b) A
rigorous treatment of DPT-II can be found in several recent reviews,Zvyagin
(2016); Heyl (2018); Mori  _et al._ (2018) along with experimental
observations of them.Jurcevic  _et al._ (2017); Zhang  _et al._ (2017b);
Fläschner  _et al._ (2018)

Figure 1: (Color online). (a) Initial states and single-domain excitations
with a domain of length $M$ spins. We call such excitations monopoles
(dipoles) if $M$ is odd (even). We note that even though we initially prepare
the system in a ground state of Eq. (3) at field strength $h_{\text{i}}=0$, we
have checked that our conclusions are independent of the choice of
$h_{\text{i}}<h_{\text{c}}^{\text{e}}$. (b) The domain potential
$V_{\alpha,\chi}(M)$, see Eq. (6), denotes the cost of a domain of $M$ sites
with respect to a ground state of Eq. (3) at zero transverse-field strength.
Shown here for $\alpha=1.6$, this potential is significantly greater in the
case of ferromagnetic (FM, $J>0$) compared to antiferromagnetic (AF, $J<0$)
interactions. Distinctly from the FM case, monopole excitations under AF
interactions are energetically more costly due to frustration, a feature
absent in the case of FM interactions. (c) Equilibrium phase diagram of the
power-law interacting ($\propto-J/r^{\alpha}$ with $r$ inter-spin distance)
transverse-field Ising chain (TFIC), given in Eq. (3) with FM or AF
interactions. The $\alpha$-dependent equilibrium quantum critical point
$h_{\text{c}}^{\text{e}}(\alpha)$ separates an ordered phase at
$h<h_{\text{c}}^{\text{e}}$ from a paramagnetic phase at
$h>h_{\text{c}}^{\text{e}}$. The crossover field strength
$h_{\text{cross}}(\alpha)$, also $\alpha$-dependent, is an upper bound with
regards to domain-wall binding. At sufficiently small $h<h_{\text{cross}}$, it
is always energetically favorable in the case of FM interactions for domain
walls to bind. When interactions are AF this applies only in case of dipole
single-domain states, whereas monopole domains are energetically favored to
either grow into larger monopole domains or decay into dipoles. Domain-wall
binding is particularly prominent in the TFIC with FM power-law interactions
where for small $\alpha$ (i.e., long-range interactions) we find that
$h_{\text{cross}}\approx h_{\text{c}}^{\text{e}}$. On the other hand, in the
TFIC with AF power-law interactions, domain-wall binding is very weak with a
much smaller value of $h_{\text{cross}}$, where we also find that
$h_{\text{cross}}\ll h_{\text{c}}^{\text{e}}$ for the range of $\alpha$
accessible in our matrix product state calculations.

##  III Model and its equilibrium physics

The power-law interacting TFIC is given by the Hamiltonian

$\displaystyle
H=-\frac{J}{\mathcal{K}_{\alpha}}\sum_{j<\ell}\frac{\sigma_{j}^{z}\sigma_{\ell}^{z}}{\lvert\ell-j\rvert^{\alpha}}-h\sum_{j}\sigma_{j}^{x},$
(3)

where $\sigma_{j}^{\\{x,y,z\\}}$ are the Pauli matrices on site $j$, $h$ is
the transverse-field strength, $\alpha$ is the interaction exponent, and $J$
is the spin coupling constant. We have also included the Kac normalizationKac
_et al._ (1963)

$\displaystyle\mathcal{K}_{\alpha}=\lim_{N\to\infty}\frac{1}{N-1}\sum_{m=1}^{N}\frac{N-m}{m^{\alpha}},$
(4)

which ensures energy extensivity for $0\leq\alpha\leq 1$. AF interactions are
realized for $J<0$, while interactions are FM when $J>0$. Generically, this
model is nonintegrable for finite $\alpha>0$, with two integrable points at
$\alpha=0$ giving rise to the LMG model in the limit of infinite-range
interactions, and $\alpha\to\infty$ leading to the NN-TFIC.Russomanno  _et
al._ (2020); Defenu (2020)

The excitations of the TFIC are either topological in the form of domain-wall
states, or topologically trivial in the case of local spin flips, and in the
FM case it has been shown that there is a crossover between these excitations
at sufficiently long-range interactions based on the quasiparticle
Ansatz.Vanderstraeten  _et al._ (2018) In Refs. Halimeh  _et al._ , 2020a; Liu
_et al._ , 2019, long-range interactions have been shown to strongly influence
quench dynamics and dynamical criticality when binding of domain walls becomes
energetically favorable in the spectrum of the quench Hamiltonian. Indeed, in
the presence of FM power-law interactions at small transverse-field strength,
single-domain states are energetically favored to have small domains, see Fig.
1. This domain-wall binding leads to constrained dynamics with very slow decay
of the order parameter in the wake of a quench, even when the model is in the
short-range equilibrium universality class ($\alpha\geq 3$).Halimeh  _et al._
(2017); Liu  _et al._ (2019) Domain-wall coupling has also been shown to be a
necessary condition for the appearance of anomalous cusps in the Loschmidt
return rate.Halimeh and Zauner-Stauber (2017); Defenu  _et al._ (2019) This
intriguing phenomenon also exists in quantum Ising chains with exponentially
decaying interactions.Halimeh  _et al._ (2020a) When bound domain walls
dominate the low-energy spectrum of the quench Hamiltonian, then even when the
model is of the short-range universality class in equilibrium, small quenches
from the ordered phase do not lead to an exponential decay in the order
parameter, as in the case of nearest-neighbor interactions where domain walls
propagate freely.Calabrese  _et al._ (2011, 2012) Instead, the order parameter
shows persistent long-time order and oscillations, as has been shown in recent
exact diagonalizationLiu  _et al._ (2019) and iMPS studies.Halimeh  _et al._
(2017, 2020a) Only at sufficiently large $h$ does the effect of domain-wall
binding vanish, as then domain walls are not attractive just as in the case of
the NN-TFIC, where domain walls freely propagate at any value of $h$ below the
equilibrium critical point.

This domain-wall binding effect is also present in the AF-TFIC, but there is
additionally domain-wall repulsion, depending on the type of single-domain
state considered. As depicted in Fig. 1(a), we can define in the cases of both
the FM-TFIC and the AF-TFIC two kinds of single-domain states: dipole
excitations, where the domain in between the two domain walls is of an even
length $M$, and monopole excitations when $M$ is odd. In order to better
understand how these two types of excitations behave under FM or AF
interactions, it is useful to consider the low-energy regime of the system in
Eq. (3) by mapping it onto the two-kink model.Coldea  _et al._ (2010);
Rutkevich (2010); Liu  _et al._ (2019) Inherently, the latter restricts the
Hilbert space to just single-domain states, which is a valid approximation
when $\lvert h\rvert\ll\lvert J\rvert/\mathcal{K}_{\alpha}$ in Eq. (3). These
states can be denoted $\ket{j,M}$, where $j$ indicates the spatial position of
the first site of the domain of length $M$. As such, the state $\ket{j,M}$ has
its spins on sites $j$ up to $j+M-1$ all flipped with respect to a ground
state of Eq. (3) at zero transverse-field strength. This ground state is
doubly degenerate for $\alpha>0$, and is thus one of two possible Néel states
in the case of AF interactions, or a fully $z$-up or $z$-down polarized state
when the interactions are FM; see Fig. 1(a). It is also evident that at zero
strength of the transverse field, the states $\ket{j,M}$ are eigenstates of
Eq. (3). The two-kink Hamiltonian can thus be written as

$\displaystyle H_{\mathrm{kink}}=$
$\displaystyle\sum_{j}\sum_{M}\Big{\\{}V_{\alpha,\chi}(M)\ket{j,M}\bra{j,M}$
$\displaystyle-h\big{[}\ket{j,M}\big{(}\bra{j,M-1}+\bra{j+1,M-1}$
$\displaystyle+\bra{j,M+1}+\bra{j-1,M+1}\big{)}+\text{H.c.}\big{]}\Big{\\}}.$
(5)

The terms $\propto h$ correspond to the growth or shrinking of the domain via
a magnetic-field induced spin-flip. The energetic cost of $\ket{j,M}$ with
respect to the corresponding ground state of Eq. (3) at zero transverse-field
strength is given by the domain potential

$\displaystyle
V_{\alpha,\chi}(M)=\frac{4J}{\mathcal{K}_{\alpha}}\sum_{\ell=1}^{M}\sum_{r=\ell}^{N\to\infty}\frac{\chi^{r}}{r^{\alpha}},$
(6)

with $\chi=\text{sgn}{(}J)$. The energetic cost of two domain walls separated
by $M$ sites is strongly influenced by whether the interactions are FM
($\chi=1$) or AF ($\chi=-1$), particularly when they are long-range. The
staggering due to $\chi=-1$ in the AF case leads to a much smaller
$V_{\alpha,\chi}(M)$ than in the FM case for the same $\alpha$ and $M$:
$V_{\alpha,-1}(M)\ll V_{\alpha,+1}(M)$. Indeed, this is demonstrated in Fig.
1(b) for $\alpha=1.6$. It will be interesting to see how the much weaker
domain-wall binding in the case of AF interactions will influence the quench
dynamics of the AF-TFIC compared to its FM counterpart. Another major
difference between the domain potentials of the FM and AF cases is that in the
former dipole and monopole excitations behave the same, insomuch that they
favor smaller domains at zero transverse-field strength. In the AF case, this
holds true for dipole domains, whereas monople domains are energetically
favored to be large; see inset of Fig. 1(b). In the regime of small quenches,
dipole excitations are expected to play a more dominant role in the case of AF
interactions, as monopole excitations are high-energy states.

We summarize the equilibrium physics of the AF-TFIC along with that of its FM
counterpart for comparison. The only difference from its FM counterpart is the
sign of $J$ in Eq. (3). As we have seen, this seemingly trivial difference
leads to vastly different equilibrium properties and phase diagrams,Koffel
_et al._ (2012); Jaschke  _et al._ (2017) as illustrated in Fig. 1(c). Whereas
the FM-TFIC hosts a finite-temperature phase transition for $\alpha<2$,Landau
and Lifshitz (2013); Dyson (1969); Thouless (1969); Dutta and Bhattacharjee
(2001) the AF-TFIC does not, regardless of the value of $\alpha$. Long-range
interactions are relevant for $\alpha<3$ in the FM case,Dutta and
Bhattacharjee (2001) while in the AF-TFIC they are relevant for $\alpha\leq
2.25$.Koffel  _et al._ (2012) Similarly to its FM counterpart, at zero
temperature the AF-TFIC has an $\alpha$-dependent equilibrium quantum critical
point $h_{\text{c}}^{\text{e}}(\alpha)$, which however increases from zero at
$\alpha=0$ to $|J|$ in the nearest-neighbor limit $\alpha\to\infty$, whereas
in the FM case it is $2J$ at $\alpha=0$ and decreases monotonically to $J$ in
the nearest-neighbor limit.Vidal  _et al._ (2004); Chen  _et al._ (2009);
Jaschke  _et al._ (2017) The difference between both models is indeed striking
at small $\alpha$, as shown by the MPS calculations in Fig. 1(c) of the
equilibrium quantum critical point $h_{\mathrm{c}}^{\mathrm{e}}$ and the
crossover field strength $h_{\mathrm{cross}}$. We define the latter as the
value of the transverse-field strength below which it is energetically more
favorable for two domain walls to be bound, rather than infinitely
separated.Halimeh  _et al._ (2020a) In the AF case, this applies only for
dipole excitations, because their monopole counterparts are repulsive. In this
sense, $h_{\mathrm{cross}}$ serves as an upper bound for domain-wall binding
in the FM-TFIC and for dipole excitations in the AF-TFIC.

The equilibrium quantum critical points of the FM-TFIC and AF-TFIC separate
further the smaller $\alpha$ is, whereas in the short-range limit $\alpha\geq
3$ they approach $|J|$, which is the equilibrium quantum critical point of the
NN-TFIC irrespective of whether interactions are FM or AF. Common to both
models, $h_{\mathrm{cross}}$ approaches $h_{\mathrm{c}}^{\mathrm{e}}$ at
longer interaction ranges. Within the accessible $\alpha$ values in our MPS
calculations, we find that $h_{\mathrm{cross}}\approx
h_{\mathrm{c}}^{\mathrm{e}}$ for $\alpha\lesssim 2$ in case of FM
interactions, whereas for the AF-TFIC $h_{\mathrm{cross}}\ll
h_{\mathrm{c}}^{\mathrm{e}}$ down to the smallest finite $\alpha$ we achieve.
As we will see in Sec. IV.1, this will lead to dynamical criticality in the
AF-TFIC that is fundamentally different from that of the FM-TFIC.

The spread of excitations, correlations, and entanglement in the wake of local
and global quenches has been studied in the AF-TFIC with long-range
interactions.Hauke and Tagliacozzo (2013); Schachenmayer  _et al._ (2013);
Richerme  _et al._ (2014); Jurcevic  _et al._ (2014); Foss-Feig  _et al._
(2015); Cevolani  _et al._ (2015); Neyenhuis  _et al._ (2017); Schneider  _et
al._ (2021) Ion-trap setups have proven to be an ideal platform for the
implementation of the AF-TFIC.Porras and Cirac (2004); Friedenauer  _et al._
(2008); Richerme  _et al._ (2013) To mediate interactions with $\alpha>0$,
ion-trap setups employ the radial phonons, which are less dispersive than the
axial ones. As the lowest-lying and best-separated mode of that branch is the
zig-zag mode,Lechner  _et al._ (2016) ion-trap setups naturally induce AF
power-law spin–spin interactions Islam  _et al._ (2013); Richerme  _et al._
(2014); Jurcevic  _et al._ (2014, 2015); Smith  _et al._ (2016); Jurcevic  _et
al._ (2017); Zhang  _et al._ (2017b) (see Sec. V). It is this relative ease of
experimental realization and the fundamental difference in its equilibrium
physics with respect to the FM-TFIC that motivate us to map out in this work
the dynamical phase diagram of the AF-TFIC.

##  IV Quench dynamics

We now present our numerical results for the time evolution of the local order
parameter

$\displaystyle\mathcal{M}(t)=\lim_{N\to\infty}\frac{1}{N}\sum_{j=1}^{N}(-1)^{j}\bra{\psi_{0}}e^{iHt}\sigma_{j}^{z}e^{-iHt}\ket{\psi_{0}},$
(7)

and return rate (2) in the AF-TFIC starting in a ground state of Eq. (3) at an
initial transverse-field strength $h_{\mathrm{i}}=0$, and then quenching to a
final value $h>0$. For $\alpha>0$, the ground-state manifold of the AF-TFIC
for $h_{\text{i}}<h_{\mathrm{c}}^{\mathrm{e}}$ is doubly degenerate,
consisting of the two Néel states
$\ket{\cdots\uparrow_{j}\downarrow_{j+1}\cdots}$ and
$\ket{\cdots\downarrow_{j}\uparrow_{j+1}\cdots}$ at $h_{\mathrm{i}}=0$. In the
fully connected limit of $\alpha=0$, the ground-state manifold is infinitely
degenerate, growing with system size $N$ as $\mathcal{O}(N!/[(N/2)!]^{2})$.
Even though in this work we present results only for $h_{\text{i}}=0$, we have
checked that our conclusions are valid also for other values of
$h_{\text{i}}<h_{\text{c}}^{\text{e}}$. We do not present results for
$h_{\text{i}}>h_{\text{c}}^{\text{e}}$, as the picture is qualitatively
identical to that of the FM-TFIC already presented in Ref. Halimeh and Zauner-
Stauber, 2017.

###  IV.1 Nonintegrable model at finite $\alpha>1$

Using iMPS in the framework of the time-dependent variational principle
(TDVP),Haegeman  _et al._ (2011, 2016); Vanderstraeten  _et al._ (2019) we
calculate the time evolution of the local order parameter (7) and the
Loschmidt-echo return rate (2). We restrict our calculations to $\alpha\geq
1.6$ due to numerical overhead; see Appendix A.1 for details. We use the trick
of backward time evolution in order to double the longest evolution time
accessible for the return rate, although this is not possible for the local
order parameter.Halimeh and Zauner-Stauber (2017) This is why in the following
iMPS results $r(t)$ achieves double the maximal evolution time reached in
$\mathcal{M}(t)$; see Appendix A.2 for details. Our most demanding
calculations achieve convergence at an evolution time-step $\tau=0.005/J$ with
a maximal bond dimension $\mathcal{D}=300$.

Figure 2: (Color online). Quench dynamics of the Loschmidt-echo return rate
(solid blue line), first rate-function branch (dotted gray line; see Appendix
A.3), and local order parameter (dot-dashed orange line) for the transverse-
field Ising chain with antiferromagnetic long-range interactions $\propto
1/r^{\alpha}$ with $\alpha=1.6$ after starting in a Néel; see Eq. (3) for
$J<0$. (a) Unlike the case of ferromagnetic interactionsHalimeh and Zauner-
Stauber (2017) ($J>0$), here anomalous cusps are absent for all evolution
times accessible in our iMPS calculations when $h<h_{\mathrm{cross}}\approx
0.051|J|$. Nevertheless, like in the FM case, the order parameter shows
persistent oscillations with little decayHalimeh  _et al._ (2017); Liu  _et
al._ (2019) at the same frequency as the return rate. (b) Similarly to the
ferromagnetic case,Halimeh  _et al._ (2020a) however, cusps are absent when
$h_{\mathrm{cross}}<h<h_{\mathrm{c}}^{\mathrm{d}}\approx 0.284|J|$, and (c)
dynamical criticality appears only in the form of regular cusps when
$h>h_{\mathrm{c}}^{\mathrm{d}}$, with these cusps coinciding with a zero
crossing of the order parameter. Note that within the precision of our iMPS
simulations, we have found in the case of antiferromagnetic interactions that
the dynamical and equilibrium critical points seem to coincide
($h_{\mathrm{c}}^{\mathrm{d}}\approx h_{\mathrm{c}}^{\mathrm{e}}$) regardless
of $\alpha$, which is different from the case of ferromagnetic
interactionsHalimeh and Zauner-Stauber (2017) where
$h_{\mathrm{c}}^{\mathrm{d}}\leq h_{\mathrm{c}}^{\mathrm{e}}$ with the two
critical points coinciding only in the nearest-neighbor limit
$\alpha\to\infty$.

Mean-field analysis shows that long-range interactions in the AF-TFIC are
relevant for $\alpha\leq 2.25$.Koffel  _et al._ (2012) As such, we show in
Fig. 2 the dynamics for $\alpha=1.6$. We consider three different ranges in
the values of the final transverse-field strength $h$. The first is
$h\in\big{(}0,h_{\mathrm{cross}}\big{)}$, where in the FM phase the return
rate can give rise to anomalous cusps without any zero crossings in the order
parameter.Halimeh and Zauner-Stauber (2017); Halimeh  _et al._ (2020a) In Fig.
2(a), we show the dynamics of the return rate and order parameter at
$h=0.044|J|<h_{\mathrm{cross}}(\alpha=1.6)\approx 0.051|J|$. The return rate
is smooth without any cusps for all evolution times we are able to access in
iMPS even though $h<h_{\mathrm{cross}}$. This can be explained by noting that
in the case of AF interactions, $h_{\mathrm{cross}}$ is considerably smaller
than its counterpart in the FM case for the same value of $\alpha$. This means
that anomalous cusps could potentially appear at extremely long evolution
times in the AF case, making them inaccessible in iMPS and also modern ion-
trap setups. Indeed, similar values of $h$ in the FM case have yielded no
anomalous cusps within accessible evolution times, even when larger values
$h<h_{\text{cross}}$ have due to faster dynamics.Halimeh and Zauner-Stauber
(2017) A spectral analysis of the return rate and the first rate-function
branchZauner-Stauber and Halimeh (2017) (see Appendix A.3) above it in Fig.
2(a) reveals that they oscillate at the same frequency within the precision of
our numerics, and therefore, we cannot ascertain whether they will intersect
at later times to yield nonanalyticities in the return rate. Instead, we see
in Fig. 2(a)—as we have also checked and found for various values of
$h\in\big{(}0,h_{\mathrm{cross}}\big{)}$—that no anomalous cusps arise even
after many cycles of the return rate. It is interesting to note, however, how
the dynamics of the order parameter is nevertheless quite constrained, showing
little decay over a significant temporal interval, while displaying persistent
oscillations at roughly the same frequency as the return rate. This is
qualitatively similar to the case of the FM-TFIC for small quenches within the
ordered phase.Halimeh  _et al._ (2017) This is expected from the point of view
of perturbation theory: The only eigenstates of the final Hamiltonian that the
initial state has significant overlap with, are those with very few single
spin flips. At low energies the spectrum of these excitations is roughly
linear, fixing the oscillation period, while the minimal magnetization is
limited by the number of spin flips in the state with the highest overlap.

The dynamical critical point
$h_{\mathrm{c}}^{\mathrm{d}}(h_{\mathrm{i}},\alpha)$ is the final quench value
of the transverse-field strength above which the order parameter crosses zero
during the time evolution.Halimeh  _et al._ (2020a) In models that host a
finite-temperature phase transition, this corresponds to
$h_{\mathrm{c}}^{\mathrm{d}}$ separating a long-time ordered steady state
($h<h_{\text{c}}^{\text{d}}$) from a paramagnetic one
($h>h_{\text{c}}^{\text{d}}$). In various models, this has been shown to
coincide with a transition in the return rate from exhibiting anomalous cusps
or no cusps at all for $h<h_{\mathrm{c}}^{\mathrm{d}}$ to regular cusps for
$h>h_{\mathrm{c}}^{\mathrm{d}}$.Heyl (2014); Halimeh  _et al._ (2020a);
Homrighausen  _et al._ (2017); Lang  _et al._ (2018a, b) This picture persists
in the AF-TFIC as well, where in Fig. 2(a,b) we see no cusps when
$h<h_{\mathrm{c}}^{\mathrm{d}}$ and the order parameter does not cross zero
within the evolution times accessible in iMPS. However, unlike in Fig. 2(a),
the dynamics in Fig. 2(b) is not constrained. The return rate seems close to
equilibration already during the accessible evolution times in iMPS, while the
order parameter shows a relatively fast decay.

The picture qualitatively changes again in Fig. 2(c) where
$h>h_{\mathrm{c}}^{\mathrm{d}}$, and a regular cusp appears during the
evolution times obtained in iMPS. As expected,Heyl (2014) this regular cusp
connects to a zero crossing of the order parameter. Within the accuracy of our
iMPS calculations, we find that for a given value of $\alpha$, the dynamical
and equilibrium critical points seem to be roughly the same,
$h_{\mathrm{c}}^{\mathrm{d}}(h_{\mathrm{i}},\alpha)\approx
h_{\mathrm{c}}^{\mathrm{e}}(\alpha)$, with the dynamical critical point
showing little dependence on $h_{\mathrm{i}}$. This is quite different from
the case of FM-TFIC, where
$h_{\mathrm{c}}^{\mathrm{d}}(h_{\mathrm{i}},\alpha)\leq
h_{\mathrm{c}}^{\mathrm{e}}(\alpha)$ for
$h_{\mathrm{i}}<h_{\mathrm{c}}^{\mathrm{e}}$, with their separation $\lvert
h_{\mathrm{c}}^{\mathrm{d}}(h_{\mathrm{i}},\alpha)-h_{\mathrm{c}}^{\mathrm{e}}(\alpha)\rvert$
growing larger the smaller $\alpha$ is.Halimeh and Zauner-Stauber (2017) In
both the FM-TFIC and AF-TFIC,
$h_{\mathrm{c}}^{\mathrm{d}}(h_{\mathrm{i}},\alpha)=h_{\mathrm{c}}^{\mathrm{e}}(\alpha)$
for $h_{\mathrm{i}}>h_{\mathrm{c}}^{\mathrm{e}}$, although in this work we do
not show results for this case since both models yield the same qualitative
picture already discussed for the FM-TFIC in Ref. Halimeh and Zauner-Stauber,
2017.

Figure 3: (Color online). Same as Fig. 2 but for $\alpha=2$. The conclusions
are identical with no cusps for $h<h_{\mathrm{c}}^{\mathrm{e}}$, and where the
return rate exhibits regular cusps corresponding to zero crossings of the
order parameter when $h>h_{\mathrm{c}}^{\mathrm{e}}$.

For completeness, we repeat these results at $\alpha=2$ in Fig. 3, and we
arrive at the same conclusions as those from Fig. 2 for $\alpha=1.6$. The
larger value of $\alpha=2$ allows us to push further in evolution times (see
Appendix A.1), but nevertheless no cusps can be seen for
$h<h_{\mathrm{c}}^{\mathrm{e}}$ as shown in Fig. 3(a,b), and the order
parameter is always positive for the evolution times we can access in iMPS.
Again worth noting is the slow decay in the order parameter for the small
quench $h<h_{\text{cross}}$ of Fig. 3(a), which exhibits persistent
oscillations at the same frequency of the return rate. Once again, we have
performed a spectral analysis of the return rate and first rate-function
branch above it for this case, and within our numerical precision, it seems
they oscillate at the same frequency within the accessible evolution times.
However, we cannot rule out that at later times they may incur a phase between
them that will lead to anomalous cusps at very long times. Once again, when
$h\in(h_{\text{cross}},h_{\mathrm{c}}^{\mathrm{e}})$, we find quick
equilibration of the return rate and a relatively fast decay of the order
parameter, indicating the absence of constrained dynamics; cf. Fig. 3(b). On
the other hand, regular cusps appear when $h>h_{\mathrm{c}}^{\mathrm{e}}$, as
shown in Fig. 3(c). Also in this case the regular cusps correspond to zero
crossings in the order parameter.

As a summary of the results presented in this section, the dynamics of the AF-
TFIC is fundamentally different from that of its FM counterpart in that small
quenches in the ordered phase to $h<h_{\text{cross}}$ do not yield anomalous
cusps during attainable evolution times in iMPS, and therefore, also times
currently accessible in modern trapped-ion experiments. This can be understood
by noting the much weaker domain-wall binding exhibited in this model compared
to the FM-TFIC at the same value of $\alpha$. Furthermore, whereas both dipole
and monopole excitations energetically favor smaller domains in the FM-TFIC
and can shrink at first-order processes in the transverse-field strength, in
the AF-TFIC monopole excitations with larger domains are energetically
favored. As such, in the single-domain picture of the AF-TFIC, dipole
excitations require at least second-order processes in the transverse-field
strength to shrink, and already $h_{\text{cross}}$ is very small. This weak
domain-wall binding behavior seems to at least significantly delay the
appearance of anomalous cusps to times currently inaccessible to both numerics
and trapped-ion experiments.

###  IV.2 Integrable case of $\alpha=0$

For $\alpha>0$, our initial state is a Néel state, i.e., a spin configuration
with alternating polarization. In a zero-dimensional system with infinite-
range interactions, such a state is not unique. A system of $N$ spin-$1/2$
degrees of freedom has $2^{N}$ eigenstates, which in the case of a system with
equal all-to-all interactions separate into only $\mathcal{O}(N^{2})$ distinct
eigenvalues. This implies that there are large equivalency classes of states
that behave indistinguishably during and after a quench. Consequently, it is
sufficient to initialize the system in a state that is equivalent to the Néel
state, which can be expressed in terms of total spin length $S$ and
magnetization $m$ along the $z$-axis. We detail the derivation in Appendix B,
which gives the normalized result

$\displaystyle\ket{\psi_{0}}=\sum_{S}\sqrt{\frac{D(S)}{\sum_{S^{\prime}}D(S^{\prime})}}\ket{S,m=1/2},$
(8)

where the degeneracy of the subspace with spin length $S$ is

$\displaystyle D_{N}(S)=\frac{2S+1}{N+1}{N+1\choose\frac{N}{2}-S}.$ (9)

Within each $\ket{S,m}$ subspace, there is only trivial dynamics, which thus
does not need to be resolved.

Following the initialization in a Néel-equivalent state, the system is evolved
in time with the Lipkin-Meshkov-Glick (LMG) Hamiltonian.Lipkin  _et al._
(1965) The latter is obtained, up to a trivial overall factor, in the limit of
infinite-range interactions ($\alpha=0$) of Eq. (3) and expressed in terms of
collective spin operators
$\hat{S}^{a}=\frac{1}{2}\sum_{i=1}^{N}\sigma_{i}^{a}$ as

$\displaystyle H=-\frac{J}{2N}\hat{S}^{z}\hat{S}^{z}-\Gamma\hat{S}^{x},$ (10)

with $\Gamma=h/4$, in keeping with the LMG notation.Lipkin  _et al._ (1965)

In the case of FM interactions ($J>0$), the LMG model has a second-order phase
transition at the quantum critical point
$\Gamma_{\text{c}}^{\text{e}}=J/2$,Botet  _et al._ (1982); Botet and Jullien
(1983) with a corresponding dynamical critical point
$\Gamma_{\text{c}}^{\text{d}}=(\Gamma_{\text{i}}+\Gamma_{\text{c}}^{\text{e}})/2$,
for quenches starting at $\Gamma_{\text{i}}\leq\Gamma_{\text{c}}^{\text{e}}$,
and $\Gamma_{\text{c}}^{\text{d}}=\Gamma_{\text{c}}^{\text{e}}$, for quenches
starting at $\Gamma_{\text{i}}\geq\Gamma_{\text{c}}^{\text{e}}$.Homrighausen
_et al._ (2017); Lang  _et al._ (2018a) In the case of AF interactions
($J<0$), the LMG model exhibits a first-order phase transition at the quantum
critical point $\Gamma_{\text{c}}^{\text{e}}=0$.Vidal  _et al._ (2004); Chen
_et al._ (2009)

We now focus on the case of AF interactions and, without loss of generality,
we set in the following $J=-1$. The conservation of total spin length $S$ for
$\alpha=0$ has a major impact on the return rate. In case of a thermal initial
state, it has been argued that the (interferometric) return rate employed here
is unsuitable for the description of dynamical phase transitions.Lang  _et
al._ (2018a); Sedlmayr  _et al._ (2018) Instead, the Loschmidt amplitude
should be replaced by the fidelity of the initial and final density matrix, or
similarly by the quantum return rate obtained from the sum of all Loschmidt
amplitudes within the subspaces of fixed spin.Lang  _et al._ (2018a) However,
for the present case of a pure initial state, the situation is quite
different. The fidelity return rate coincides with the definition (1), but the
geometric connection with the quantum return rate is lost as the corresponding
wave package on the Bloch sphere is extremely broad, thereby breaking the
semiclassical picture that connected the return rates in the first place.Lang
_et al._ (2018b) It is, however, possible to recover a geometric
interpretation as follows. Consider the Néel-equivalent state with all up-
spins sorted to the left,

$\displaystyle\ket{\psi_{0}}=\ket{\uparrow\dots\uparrow\downarrow\dots\downarrow}=\ket{\frac{N}{4},\frac{N}{4}}\ket{\frac{N}{4},-\frac{N}{4}}\,.$
(11)

The collective spin operators can be split into operators acting exclusively
on either the left or the right part of the system
($S^{\alpha}=S^{\alpha}_{l}+S^{\alpha}_{r}$) such that the Hamiltonian (10)
describes two coupled LMG models, each with half the original spin length

$\displaystyle
H=\sum_{s=l,r}\left(\frac{1}{2N}\hat{S}^{z}_{s}\hat{S}^{z}_{s}-\Gamma\hat{S}^{x}_{s}\right)+\frac{1}{N}\hat{S}^{z}_{l}\hat{S}^{z}_{r}\,.$
(12)

Parametrizing the classical spin vector
$\mathbf{S}=S\left(\sin{\theta}\cos{\phi},\sin{\theta}\sin{\phi},\cos{\theta}\right)^{\intercal}$,
we obtain the classical equations of motion in the form

$\displaystyle\frac{d\theta_{s}}{dt}$ $\displaystyle=-\Gamma\sin{\phi_{s}}$
(13a) $\displaystyle\frac{d\phi_{s}}{dt}$
$\displaystyle=-\Gamma\cos{\phi_{s}}\cot{\theta_{s}}+\frac{1}{4}\sum_{\eta=l,r}\cos{\theta_{\eta}}\,,$
(13b)

with $s\in\\{l,r\\}$, from which we can immediately read off the modulus of
the staggered magnetization in the thermodynamic limit as

$\displaystyle\mathcal{M}(t)=\lvert\cos(\Gamma t)\rvert\,.$ (14)

The advantage of the representation via two coupled systems lies in the well-
localized spin WKB wave function van Hemmen and Sütő (1986, 2003); Lang  _et
al._ (2018b) of the initial state (11), for which furthermore all return rates
coincide. It is now straightforward to apply the geometric interpretationLang
_et al._ (2018b) to obtain

$\displaystyle
r(t)\approx\min_{\theta,\phi}\frac{1}{2}\sum_{s=l,r}\frac{1}{4}\left[\sin^{2}{\vartheta_{s}(\theta_{l},\theta_{r},\phi_{l},\phi_{r}|t)}+\sin^{2}{\theta_{s}}\right]\,,$
(15)

where $\vartheta_{s}(\theta_{l},\theta_{r},\phi_{l},\phi_{r}|t)$ is the time-
evolved polar angle with initial coordinates
$\theta_{l},\theta_{r},\phi_{l},\phi_{r}$.

Figure 4: (Color online). Return rate in the LMG model given in Eq. (10) after
starting in the initial state (11) and quenching to $\Gamma=\lvert
J\rvert/15$. ED result (blue) is obtained for $N=1201$ spins. Geometric result
is shown in red. Regardless of the value of the latter, the return rate
exhibits regular cusps. The vertical lines indicate the snapshots in evolution
time $t\lvert J\rvert=10,\,30,\,95$ at which the corresponding Bloch sphere
representation is shown in Fig. 5 from left to right, respectively.

Figure 5: (Color online). Bloch sphere representation of the spin WKB wave
function (shown as a flock of initialization points) together with the
trajectories of the magnetization (blue) and the Loschmidt vector (red).
Dashed black circles indicate the equator. Depicted is a quench from the
initial state (11) to $\Gamma=\lvert J\rvert/15$ at times (a,d) $t=10/\lvert
J\rvert$, (b,e) $t=30/|J|$, and (c,f) $t=95/|J|$ for the (a,b,c) left and
(d,e,f) right subsystems of the WKB wave function. At each time the current
magnetization and Loschmidt vector are highlighted as blue and red vectors
respectively. The top row shows the part of the system initially aligned
parallel to the $z$-axis, while the bottom row displays the antiparallel half
with the initial orientation of both marked in yellow. The wave function
spreads quickly as fluctuations are enhanced compared to its ferromagnetic
counterpart.Lang  _et al._ (2018b) Interestingly, after an odd number of cusps
the Loschmidt echo is dominated by states with finite magnetization, despite
starting in an unpolarized initial state.

For the LMG model, every quench evolving the Néel state with $\Gamma\neq 0$
results in a regular return rate with two cusps in every period, as shown in
Fig. 4. This behavior is well captured by the spin WKB Ansatz, which shows
remarkable agreement with the corresponding ED result in the estimated
critical times. Note, however that the height of the first peak is
significantly underestimated since in the steep tails of the wave function
gradient corrections become increasingly important. We also show in Fig. 5 the
WKB wave function on the Bloch sphere of the left subsystem (top) and its
right counterpart (bottom), corresponding to evolution times marked by
vertical lines in Fig. 4. It is interesting to note that during the early
evolution shown in Fig. 5(a,d) the Loschmidt vector, i.e., the time-evolved
saddle-point coordinates of Eq. (15), evolves in the opposite direction of the
magnetization vector, exemplifying the enhanced spreading of the AF WKB wave
function. At the first cusp, the Loschmidt vector spontaneously breaks the
mirror symmetry $x\to-x$; see Fig. 5(b,e). As depicted in Fig. 5(c,f), only
shortly before the magnetization returns to its initial value does the
symmetric saddle point take over again, resulting in two cusps per period as
also observed in the ED result of Fig. 4.

Phenomenologically, it is not surprising that the return rate in the case of
the LMG model with AF interactions only regular cusps appear after quenching
from the Néel state, since at $T=0$ the AF LMG has no order at any
$\Gamma>\Gamma_{\text{c}}^{\text{e}}=0$ . As such, any quenches from a Néel
state will lead to regular cusps in the return rate, and correspondingly an
order parameter oscillating around zero, in accordance with Eq. (14).

##  V Experimental considerations

In this section, we discuss the observability of our results in ion-trap
setups. In this platform, spins-$1/2$ are represented by a suitable choice of
two internal electronic states, between which interactions are transmitted via
a phonon bus using lasers or microwave fields that are tuned close to the
vibrational modes of the ion crystal,Porras and Cirac (2004); Zippilli  _et
al._ (2014) e.g., using a Moelmer–Soerensen configuration.Sørensen and Mølmer
(2000)

An experimental sequence to study the cusps in the AF-TFIC may proceed as
follows: (i) First, the system is initialized in a Néel product state, which
can be done with near-perfect precision using optical pumping.Blatt and Roos
(2012); Schindler  _et al._ (2013); Monroe  _et al._ (2021) (ii) Then, the
electromagnetic fields are turned on that generate the effective spin–spin
Hamiltonian. (iii) After an adjustable evolution time, the Loschmidt amplitude
can be read out by projecting onto the initial state.Jurcevic  _et al._ (2017)
Since this is a product state, standard fluorescence imaging of the individual
electronic states of the ions is sufficient, which again can be performed with
high precision. Blatt and Roos (2012); Schindler  _et al._ (2013); Monroe  _et
al._ (2021) Although observing the overlap with a specific state is
exponentially inefficient, such measurements have been demonstrated for
systems up to $N=16$,Islam  _et al._ (2013) and in special cases even for more
than $N=100$ ions.Gärttner  _et al._ (2017) In the context of DQPTs, the
projectors have been measured for up to $N=10$ ions in a ferromagnetic
setting, from which cusps at $N\to\infty$ have been reliably
extrapolated.Jurcevic  _et al._ (2017)

The Hamiltonian governing step (ii) is worthy of some further discussion. The
natural spin–spin interactions in trapped ions are—due to the collective
character of the phonon modes—long ranged, with a decrease over distance that
can be adjusted by tuning the frequency of the Moelmer–Soerensen fields
relative to the phonon frequencies.Porras and Cirac (2004) In many
experiments, these interactions are well described by a spatial power-law
decay, Gärttner  _et al._ (2017); Islam  _et al._ (2013); Richerme  _et al._
(2014); Jurcevic  _et al._ (2014) an approximation that works rather well for
small systems.Nevado and Porras (2016); Trautmann and Hauke (2018) Increasing
the detuning to the phonon frequencies, larger power-law exponents can be
achieved, with a range of $\alpha$ accessible with current laser powers that
lies approximately between $0.6$ and $1.5$.Jurcevic  _et al._ (2014) In
addition, by tuning close to the lowest mode, which has the largest frequency
separation from the other modes, $\alpha=0$ can be achieved. In current
experiments, the power-law interactions are induced using the radial phonon
modes, which are closer spaced than the axial modes.Porras and Cirac (2004) In
such a configuration, the lowest-frequency mode is the zig-zag mode,Lechner
_et al._ (2016) which has opposite oscillation amplitude at neighboring ions.
This pattern is imprinted on the generated effective interactions, which as a
consequence are naturally of antiferromagnetic type. There have also been
various proposals to precisely engineer desired interaction patterns using
several laser fields Korenblit  _et al._ (2012); Hauke  _et al._ (2015);
Davoudi  _et al._ (2020) or segmented Paul traps,Zippilli  _et al._ (2014)
which could overcome any imprecision in the power-law couplings and extend the
range of accessible power-law exponents. As a final note, a transverse field
can easily be implemented in the Moelmer–Soerensen configuration using an
asymmetric detuning to the blue and red phonon sidebands.Jurcevic  _et al._
(2014)

In such experiments, a main potential restriction for the observability of the
cusps is the achievable evolution times, which are limited due to dissipative
processes such as spontaneous emission or fluctuations of magnetic fields or
laser intensities. Nevertheless, current experiments reach evolution times on
the order of several $\mathcal{K}_{\alpha}/J$,Richerme  _et al._ (2014);
Jurcevic  _et al._ (2014); Neyenhuis  _et al._ (2017); Gärttner  _et al._
(2017); Jurcevic  _et al._ (2017); Gärttner  _et al._ (2017) which are on the
order of the times studied here numerically. Hence, the physics discussed here
on the basis of numerical simulation should be observable in state-of-the-art
trapped-ion experiments.

##  VI Conclusions and outlook

We have investigated dynamical phase transitions in transverse-field Ising
chains with antiferromagnetic power-law-decaying interactions through exact
diagonalization and infinite matrix product states. Although the behavior is
similar to the ferromagnetic case when interactions are short-range, in the
long-range case the behavior is qualitatively different. Whereas anomalous
cusps are a prominent feature of quenches within the ordered phase in the case
of long-range ferromagnetic interactions, they are absent during all
accessible evolution times in the case of even the longest-range
antiferromagnetic interactions that we can numerically achieve. We attribute
this to the significantly weaker overall domain-wall binding in the
antiferromagnetic case, and additionally the fact that monopole excitations
are repulsive unlike in the ferromagnetic case where both dipoles and
monopoles are energetically favored to bind for small quenches.

This picture persists even in the integrable case of infinite-range
interactions. There, domain walls are not defined even in the ordered phase,
and local spin flips (equivalent to bound domain walls in $1$D) are the
lowest-lying quasiparticles. However, in this case the model hosts a first-
order phase transition at zero transverse-field strength in equilibrium at
zero temperature. Consequently, quenching from the ground-state manifold will
always lead to regular cusps and concomitantly an order parameter oscillating
around its long-time steady-state value of zero.

It is important to note that we cannot rule out anomalous cusps arising at
very long times for small quenches within the ordered phase in the regime with
domain-wall binding in the antiferromagnetic case. However, it seems certain
that these times are inaccessible in our numerics and, therefore, are likely
too long to be reached in modern quantum synthetic matter setups.

Our work further highlights the crucial role constrained dynamics plays in
dynamical phase transitions, and how long-range interactions are neither a
sufficient nor necessary condition for anomalous dynamical criticality.
Indeed, we find no anomalous cusps for all accessible evolution times in the
antiferromagnetic case even when long-range interactions are relevant
(exponent $\alpha\leq 2.25$). However, it is known that quantum Ising chains
with ferromagnetic exponentially decaying interactions, which are in the
short-range universality class in equilibrium, will still give rise to
anomalous cusps because one can achieve a profile there where domain-wall
binding is prominent.Halimeh  _et al._ (2020a)

The results presented in this paper can be observed in modern ion-trap setups
where long-range antiferromagnetic interactions are naturally induced. This is
due to the fact that the lowest-lying and best-separated mode of radial
phonons is the zig-zag one.Lechner  _et al._ (2016) This should make the
detection of dynamical phase transitions in long-range antiferromagnetic
quantum Ising chains a natural target following their ferromagnetic
counterparts, which have already been investigated.Jurcevic  _et al._ (2017)

###### Acknowledgements.

We acknowledge support by Provincia Autonoma di Trento, the ERC Starting Grant
StrEnQTh (project ID 804305), Q@TN — Quantum Science and Technology in Trento
— and the Collaborative Research Centre ISOQUANT (project ID 273811115),
Research Foundation Flanders (G0E1520N, G0E1820N), and ERC grants QUTE
(647905) and ERQUAF (715861).

##  Appendix A Details regarding iMPS calculations

In this Appendix, we explain how power-law interactions are realized in our
iMPS implementation, we discuss the backward time-evolution trick employed to
double the maximal time evolution reached in the return rate, and we define
rate-function branches from the eigenvalues of the MPS transfer matrix.

###  A.1 Implementation of long-range interactions

In a strict sense, power-law interactions are not possible to implement in
iMPS, because the latter is based on matrix product operator (MPO)
formulations of exponentials in Hamiltonian parts composed of commuting terms.
Naturally, this is ideal for systems where an MPO description is possible such
as when interactions are exponentially decaying. However, up to good accuracy,
a power-law decay can be approximated by a sum of exponentials, with the
number of the latter increasing the longer-range the power-law interactions
are. Consequently, a sum of the MPO representations of these exponentials will
accurately model the corresponding power-law interaction profile.Crosswhite
_et al._ (2008) The approximation of the power-law profile with exponentials
is given by

$\displaystyle\sum_{j<l}\frac{1}{\lvert
j-l\lvert^{\alpha}}\sigma_{j}^{z}\sigma_{l}^{z}\approx\sum_{j<l}\sum_{n=1}^{K}c_{n}u_{n}^{\lvert
j-l\lvert-1},$ (16)

where $K$ is the number of exponentials used. The coefficients $c_{n}$ and
$u_{n}$ are real numbers with $0\leq u_{n}<1$, and they are computed using a
nonlinear least-squares fit. Our fidelity threshold for this fit is an error
$\varepsilon\approx\mathcal{O}(10^{-10}-10^{-8})$, where $K$ is chosen over a
distance $d$ such that $\sum_{n=1}^{K}c_{n}u_{n}^{d-1}<\varepsilon$.
Consequently, $K$ increases the smaller $\alpha$ is, thereby limiting how
long-range the interactions we implement can be. For more details on the
implementation of long-range interactions in iMPS, we refer the reader to Ref.
Crosswhite  _et al._ , 2008.

###  A.2 Backward time-evolution trick

This trick entails noticing that the Loschmidt amplitude can be rewritten as
$G(t)=\bra{\psi_{0}}e^{-iHt}\ket{\psi_{0}}=\bra{\psi_{0}}e^{-iHt/2}e^{-iHt/2}\ket{\psi_{0}}$,
which means that to reach an evolution time $t$ in the Loschmidt return rate,
one need only time-evolve $\ket{\psi_{0}}$ up to $t/2$ to obtain
$\ket{\psi(t/2)}=\exp(-iHt/2)\ket{\psi_{0}}$, which is then complex conjugated
to obtain
$\ket{\psi(-t/2)}=\overline{\ket{\psi(t/2)}}=\exp(iHt/2)\ket{\psi_{0}}$. In
iMPS, we therefore evolve $\ket{\psi_{0}}$ forward in time to a maximal
evolution time $t_{\text{max}}$ (within convergence at a certain time-step and
bond dimension), allowing results for $r(t)$ up to $2t_{\text{max}}$. Since
this trick cannot be applied to the local order parameter $\mathcal{M}(t)$,
results for the latter are only available up to $t_{\text{max}}$.Zauner-
Stauber and Halimeh (2017)

###  A.3 MPS transfer matrix and rate-function branches

According to the definition of the return rate given in Eq. (2), the latter is
the negative of the natural logarithm of the dominant eigenvalue of the mixed
MPS transfer matrix $\mathcal{T}(t)$ between MPS tensors at evolution times
$0$ and $t$.Zauner-Stauber and Halimeh (2017) The spectral radius of
$\mathcal{T}(t)$ is bounded above by unity. Let us order the eigenvalues
$\epsilon_{n}(t)$ of $\mathcal{T}(t)$ in descending order by magnitude:
$\epsilon_{1}(t)\geq\epsilon_{2}(t)\geq\ldots\geq\epsilon_{\mathcal{D}}(t)$.
Then the rate-function branches are

$\displaystyle r_{n}(t)=-2\ln\lvert\epsilon_{n}(t)\rvert.$ (17)

Consequently, the return rate is $r(t)=r_{1}(t)\leq r_{2}(t)\leq
r_{3}(t)\leq\ldots\leq r_{\mathcal{D}}(t)$. At a critical time
$t_{\mathrm{c}}$, when a cusp appears in the return rate, the return rate and
the first rate-function branch are equal,
$r(t_{\text{c}})=r_{2}(t_{\text{c}})$. This is particularly useful since the
nonanalyticities of DPT-II are nothing but level crossings of the
eigenspectrum of $\mathcal{T}(t)$. Such level crossings are characteristic of
first-order phase transitions.

It is worth noting that $\mathcal{T}(t)$ is an approximation of the physical
quantum transfer matrix arising in a real-time path-integral formulation of
the Loschmidt amplitude $G(t)=\bra{\psi_{0}}e^{-iHt}\ket{\psi_{0}}$.

##  Appendix B Néel state with permutation invariance

A Néel state has minimal magnetization, thus for an odd number $N$, we can
choose (without loss of generality) $m=1/2$. The spin length, on the other
hand, is a bit more subtle, since the Néel state is no eigenstate of
$\hat{S}^{2}=\hat{S}_{x}^{2}+\hat{S}_{y}^{2}+\hat{S}_{z}^{2}$ with
$\hat{S}_{a}=\frac{1}{2}\sum_{i=1}^{N}\sigma_{i}^{a}$.

To explain the general procedure, let us first consider a system of only three
spins with the Néel state $\ket{\uparrow\downarrow\uparrow}$, which we want to
express in terms of the simultaneous eigenstates of $\hat{S}^{2}$ and
$\hat{S}^{z}$ denoted as

$\displaystyle\ket{\frac{3}{2},\frac{1}{2}}$
$\displaystyle=\frac{1}{\sqrt{3}}\big{(}\ket{\uparrow\uparrow\downarrow}+\ket{\uparrow\downarrow\uparrow}+\ket{\downarrow\uparrow\uparrow}\big{)}$
(18a) $\displaystyle\ket{\frac{1}{2},\frac{1}{2}}_{a}$
$\displaystyle=\sqrt{\frac{2}{3}}\Big{(}\ket{\uparrow\uparrow\downarrow}-\frac{1}{2}\ket{\uparrow\downarrow\uparrow}-\frac{1}{2}\ket{\downarrow\uparrow\uparrow}\Big{)}$
(18b) $\displaystyle\ket{\frac{1}{2},\frac{1}{2}}_{b}$
$\displaystyle=\frac{1}{\sqrt{2}}\big{(}\ket{\uparrow\downarrow\uparrow}-\ket{\downarrow\uparrow\uparrow}\big{)}.$
(18c)

This is achieved by

$\displaystyle\ket{\uparrow\downarrow\uparrow}=\frac{1}{\sqrt{3}}\ket{\frac{3}{2},\frac{1}{2}}-\frac{1}{\sqrt{6}}\ket{\frac{1}{2},\frac{1}{2}}_{a}+\frac{1}{\sqrt{2}}\ket{\frac{1}{2},\frac{1}{2}}_{b},$
(19)

which involves the states $\ket{\frac{1}{2},\frac{1}{2}}_{a}$ and
$\ket{\frac{1}{2},\frac{1}{2}}_{b}$, which are indistinguishable with respect
to the time-evolution. For the infinite-ranged system these behave
identically, such that we can write the equivalency statement

$\displaystyle\ket{\uparrow\downarrow\uparrow}=\frac{1}{\sqrt{3}}\ket{\frac{3}{2},\frac{1}{2}}+\sqrt{\frac{2}{3}}\ket{\frac{1}{2},\frac{1}{2}}.$
(20)

This construction can be extended to arbitrary system sizes with $N$ spins.
Since the Hamiltonian is integrable there exist $N$ conserved quantities,
i.e., the $N-1$ quantities $\vec{S}^{2}_{n}\equiv
S^{x2}_{n}+S^{y2}_{n}+S^{z2}_{n}$ ($n=2,\dots,N$) and the Hamiltonian itself
satisfy $[\vec{S}^{2}_{n},H_{\alpha=0}]=0$, where
$S^{\beta}_{n}=\sum_{i=1}^{n}s^{\beta}_{i}$ with $\beta=x,y,z$. Therefore, we
can label each eigenstate by
$\ket{S_{1},S_{2},\cdots,S_{n},\cdots,S_{N},S^{z}_{N}}$, where
$S_{n}\in\\{|S_{n-1}-\frac{1}{2}|,\cdots,S_{n-1}+\frac{1}{2}\\}$. For example,
the three states in Eqs. (18), $\ket{\frac{3}{2},\frac{1}{2}}$,
$\ket{\frac{1}{2},\frac{1}{2}}_{a}$ and $\ket{\frac{1}{2},\frac{1}{2}}_{b}$,
correspond to $\ket{1,\frac{3}{2},\frac{1}{2}}$,
$\ket{1,\frac{1}{2},\frac{1}{2}}$ and $\ket{0,\frac{1}{2},\frac{1}{2}}$
respectively. We can represent the LMG eigenstate
$\ket{S_{2},\cdots,S_{n},\cdots,S_{N},S^{z}_{N}}$ in the uncoupled tensor
basis $\ket{s^{z}_{1},s^{z}_{2},\cdots,s^{z}_{N-1},s^{z}_{N}}$ using the
following identity

$\displaystyle\ket{j_{1},j_{2},j,m}=$
$\displaystyle\sum_{m_{1},m_{2}}\ket{j_{1},m_{1}}\ket{j_{2},m_{2}}W^{j_{1}j_{2}}_{m_{1}m_{2},jm}\delta_{m,m_{1}+m_{2}}$
$\displaystyle=$
$\displaystyle\sum_{m_{2}}\ket{j_{1},m-m_{2}}\ket{j_{2},m_{2}}W^{j_{1}j_{2}}_{(m-m_{2})m_{2},jm}.$
(21)

Here, $W^{j_{1}j_{2}}_{m_{1}m_{2},jm}$ are the Clebsch-Gordan coefficients of
total angular momentum eigenstates $\ket{j_{1},j_{2},j,m}$ in the uncoupled
tensor product basis.

Therefore, we have

$\displaystyle\ket{S_{1},S_{2},\cdots,S_{N-1},S_{N},S^{z}_{N}}$
$\displaystyle=$
$\displaystyle\sum_{s^{z}_{N}}W^{S_{N-1}\frac{1}{2}}_{(S^{z}_{N}-s^{z}_{N})s^{z}_{N},S_{N}S^{z}_{N}}\ket{S_{N-1},S^{z}_{N}-s^{z}_{N}}\ket{\frac{1}{2},s^{z}_{N}}$
$\displaystyle=$
$\displaystyle\sum_{s^{z}_{N-1},s^{z}_{N}}W^{S_{N-2}\frac{1}{2}}_{(S^{z}_{N}-s^{z}_{N}-s^{z}_{N-1})s^{z}_{N-1},S_{N-1}(S^{z}_{N}-s^{z}_{N})}W^{S_{N-1}\frac{1}{2}}_{(S^{z}_{N}-s^{z}_{N})s^{z}_{N},S_{N}S^{z}_{N}}\ket{S_{N-2},S^{z}_{N}-s^{z}_{N}-s^{z}_{N-1}}\ket{\frac{1}{2},s^{z}_{N-1}}\ket{\frac{1}{2},s^{z}_{N}}$
$\displaystyle\vdots$ $\displaystyle=$
$\displaystyle\sum_{s^{z}_{2},\cdots,s^{z}_{N-1},s^{z}_{N}}\prod_{n=2}^{N}W^{S_{n-1}\frac{1}{2}}_{(S^{z}_{N}-\sum^{N}_{k=n}s^{z}_{k})s^{z}_{n},S_{n}(S^{z}_{N}-\sum^{N}_{k=n}s^{z}_{k}+s^{z}_{n})}\ket{s^{z}_{1},s^{z}_{2},\cdots,s^{z}_{N-1},s^{z}_{N}}$
$\displaystyle=$
$\displaystyle\sum_{s^{z}_{2},\cdots,s^{z}_{N-1},s^{z}_{N}}\prod_{n=2}^{N}W^{S_{n-1}\frac{1}{2}}_{\big{(}\sum^{n-1}_{k=1}s^{z}_{k}\big{)}s^{z}_{n},S_{n}\big{(}\sum^{n}_{k=1}s^{z}_{k}\big{)}}\ket{s^{z}_{1},s^{z}_{2},\cdots,s^{z}_{N-1},s^{z}_{N}}$
$\displaystyle=$
$\displaystyle\sum_{s^{z}_{2},\cdots,s^{z}_{N-1},s^{z}_{N}}\prod_{n=2}^{N}W^{S_{n-1}\frac{1}{2}}_{S^{z}_{n-1}s^{z}_{n},S_{n}S^{z}_{n}}\ket{s^{z}_{1},s^{z}_{2},\cdots,s^{z}_{N-1},s^{z}_{N}},$
(22)

where we have used the conservation of total angular momentum projection in
the $z$-direction, i.e., $S^{z}_{N}=\sum^{N}_{k=1}s^{z}_{k}$. Since the
Clebsch-Gordan coefficients can always be chosen as real numbers, we thus can
represent the uncoupled tensor state in the LMG basis

$\displaystyle\ket{s^{z}_{1},s^{z}_{2},\cdots,s^{z}_{N-1},s^{z}_{N}}$
$\displaystyle=\sum_{S_{1},\cdots,S_{N}}\prod_{n=2}^{N}W^{S_{n-1}\frac{1}{2}}_{S^{z}_{n-1}s^{z}_{n},S_{n}S^{z}_{n}}\ket{S_{1},S_{2},\cdots,S_{N-1},S_{N},S^{z}_{N}}.$
(23)

This expression is the general form for the three spins Eq. (19). Because the
flexibility of spin chain configurations $S_{1},\cdots,S_{N-1}$, the subspace
$\\{\ket{S_{N}=S,S^{z}_{N}=m}\\}$ is degenerate with its degeneracy $D_{N}(S)$
given by

$\displaystyle D_{N}(S)=\frac{2S+1}{N+1}{N+1\choose\frac{N}{2}-S}.$ (24)

Applying Eq. (B) to the Néel state with an odd number of spins and collecting
all the states in the degenerate subspace, we obtain a general expression for
arbitrary (odd) system sizes

$\displaystyle\ket{\text{N\'{e}el}}_{N}$
$\displaystyle\equiv\underbrace{\ket{\uparrow\downarrow\cdots\uparrow}}_{N}$
$\displaystyle=\sum_{S}\sqrt{\frac{D_{N}(S)}{\sum_{S^{\prime}}D_{N}(S^{\prime})}}\ket{S_{N}=S,\frac{1}{2}}.$
(25)

The above expression can be proved as follows. We first we assume Eq. (B) is
valid for Néel state of $N$ spins, which is true for $N=3$ from Eq. (20). By
representing the product state $\ket{\downarrow\uparrow}$ in the basis of
singlet and triplet states, we can obtain the Néel state of $N+2$ spins via
the following construction

$\displaystyle\ket{\text{N\'{e}el}}_{N+2}$
$\displaystyle=\ket{\text{N\'{e}el}}_{N}\otimes\ket{\downarrow\uparrow}$
$\displaystyle\propto\sum_{S}\sqrt{D_{N}(S)}\ket{S_{N}=S,\frac{1}{2}}\frac{1}{\sqrt{2}}\big{(}\ket{1,0}-\ket{0,0}\big{)}$
$\displaystyle=\sum_{S}\sqrt{\frac{D_{N}(S)}{2}}\Big{(}W^{S,1}_{\frac{1}{2},0;S-1,\frac{1}{2}}\ket{S-1,\frac{1}{2}}+W^{S,1}_{\frac{1}{2},0;S+1,\frac{1}{2}}\ket{S+1,\frac{1}{2}}+W^{S,1}_{\frac{1}{2},0;S,\frac{1}{2}}\ket{S,\frac{1}{2}}-W^{S,0}_{\frac{1}{2},0;S,\frac{1}{2}}\ket{S,\frac{1}{2}}\Big{)}$
$\displaystyle=\sum_{S}\sqrt{\frac{1}{2}\big{[}D_{N}(S+1)(W^{S+1,1}_{\frac{1}{2},0;S,\frac{1}{2}})^{2}+D_{N}(S-1)(W^{S-1,1}_{\frac{1}{2},0;S,\frac{1}{2}})^{2}+D_{N}(S)((W^{S,0}_{\frac{1}{2},0;S,\frac{1}{2}})^{2}+(W^{S,1}_{\frac{1}{2},0;S,\frac{1}{2}})^{2})\big{]}}\ket{S,\frac{1}{2}}$
$\displaystyle\propto\sum_{S}\sqrt{D_{N+2}(S)}\ket{S_{N+2}=S,\frac{1}{2}}.$
(26)

In the last two steps, we have regrouped the terms with the same total spin
$S_{N+2}=S$ and made use of the identities

$\displaystyle W^{S+1,1}_{\frac{1}{2},0;S,\frac{1}{2}}$
$\displaystyle=-\frac{1}{2}\sqrt{\frac{2S+1}{S+1}},$ (27a) $\displaystyle
W^{S,1}_{\frac{1}{2},0;S,\frac{1}{2}}$
$\displaystyle=\frac{1}{2}\sqrt{\frac{1}{S(S+1)}},$ (27b) $\displaystyle
W^{S-1,1}_{\frac{1}{2},0;S,\frac{1}{2}}$
$\displaystyle=\frac{1}{2}\sqrt{\frac{2S+1}{S}},$ (27c) $\displaystyle
W^{S,0}_{\frac{1}{2},0;S,\frac{1}{2}}$ $\displaystyle=1.$ (27d)

By normalising Eq. (B), the identity Eq. (B) is thus proved. For simplicity,
we drop the footnote of $D_{N}(S)$ and obtain Eq. (8) in the main text. The
present approach allows us to evaluate any expectation value of operators that
can be expressed in terms of $\hat{S}_{\alpha}$, which in particular includes
the Loschmidt echo and its quench dynamics. However, a staggered magnetization
is not covered by this class, which is easy to understand as staggering is not
uniquely defined in a zero-dimensional system.

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