# Signatures of quantum phase transitions after quenches in
quantum chaotic one-dimensional systems

Asmi Haldar  Indian Association for the Cultivation of Science (School of
Physical Sciences), 2A & 2B Raja S. C. Mullick Road, Kolkata 700032, India
Krishnanand Mallayya  Department of Physics, The Pennsylvania State
University, University Park, Pennsylvania 16802, USA  Markus Heyl  Max Planck
Institute for the Physics of Complex Systems, Nöthnitzer Straße 38, D-01187,
Dresden, Germany  Frank Pollmann  Department of Physics, Technische
Universität München, James-Franck-Str. 1/I 85748 Garching b. München, Germany
Marcos Rigol  Department of Physics, The Pennsylvania State University,
University Park, Pennsylvania 16802, USA  Arnab Das  Indian Association for
the Cultivation of Science (School of Physical Sciences), 2A & 2B Raja S. C.
Mullick Road, Kolkata 700032, India

###### Abstract

Quantum phase transitions are central for the understanding of the equilibrium
low-temperature properties of quantum matter. Locating them can be challenging
both by means of theoretical techniques as well as for experiments. Here, we
show that the antithetic strategy of forcing a system strongly out of
equilibrium can provide a route to identify signatures of quantum phase
transitions. By quenching a quantum chaotic (nonintegrable) spin chain, we
find that local observables can exhibit distinct features in their
intermediate-time dynamics, when the quench parameter is close to its critical
value, where the ground state undergoes a quantum phase transition. We find
that the effective temperature in the expected thermal-like states after
equilibration exhibits a minimum in the vicinity of the quantum critical value
of the quench parameter, correlating with the features in the real-time
dynamics of observables. We also explore dynamical nonequilibrium signatures
of a quantum critical point in a model with a topological transition, and
discuss how to access our results experimentally in systems of Rydberg atoms.

##  I Introduction

Quantum phase transitions constitute a key concept for the perception and
description of the low-temperature properties of quantum many-body systems
ranging across various branches in physics from quantum magnets, high-
temperature superconductivity, neutron stars, to the quark-gluon plasma
Sachdev (2011); Carr (2010). The microscopic description of the underlying
complex many-body models can, however, be challenging theoretically. Quantum
simulators promise to overcome such challenges by experimental means through
providing pristine and controllable realizations of the respective microscopic
models Bloch  _et al._ (2008); Esslinger (2010). Importantly, many of these
quantum simulator platforms provide access to real-time dynamics, which has
been very recently used to identify unique real-time aspects of quantum phase
transitions in a system of Rydberg atoms by probing the Kibble-Zurek mechanism
of universal defect production for slow parameter sweeps Keesling  _et al._
(2019). Recent theoretical works have provided evidence that nonequilibrium
quantum evolution can be utilized to probe quantum phase transitions in
integrable systems Bhattacharyya  _et al._ (2015); Roy  _et al._ (2017), in
prethermal states for models close to integrability Titum  _et al._ (2019), or
through out-of-time-order correlators Heyl  _et al._ (2018). However,
identifying real-time signatures of quantum phase transitions in generic
quantum chaotic many-body systems accessible within current experimental
technology has remained a challenge.

In this work we show that ergodic quantum matter can exhibit dynamical
signatures of quantum phase transitions by the antithetic strategy of forcing
these systems far out of equilibrium and therefore beyond the ground state
manifold. We find that the intermediate-time dynamics of local observables and
the entanglement entropy exhibit distinct features after quantum quenches in
the anisotropic next-nearest neighbor Ising (ANNNI) chain upon tuning the
quench parameter across an underlying quantum phase transition. We observe
that the derivatives of these quantities with respect to the quench parameter
develop strong dips/peaks in the vicinity of the quantum phase transition. We
determine the quantum real-time evolution by means of the infinite-Time
Evolved Block Decimation (iTEBD), which provides numerically exact results for
the transient to intermediate-time dynamics in the thermodynamic limit Vidal
(2003); *Vidal_2; *Vidal_3; Schollwock (2011); Hauschild and Pollmann (2018).

In order to access the asymptotic long-time properties of the considered
quantum chaotic system after the expected thermalization, we employ a
numerical linked cluster expansion (NLCE) for thermal equilibrium states Rigol
_et al._ (2006); *rigol2007numerical. Most importantly, we again find distinct
signatures of the quantum phase transition in derivatives of the correlation
functions. Also, the effective temperature exhibits a marked minimum as
function of the quench parameter in close vicinity to the quantum phase
transition. Since the considered one-dimensional system does not support
singular behavior after equilibration, upon assuming that eigenstate
thermalization occurs Deutsch (1991); Srednicki (1994); Rigol  _et al._
(2008); D’Alessio  _et al._ (2016), these prominent features are not
associated with nonanalytic properties (in contrast to the integrable systems
studied in Refs. Bhattacharyya  _et al._ (2015); Roy  _et al._ (2017)), but
nevertheless represent distinct signatures of quantum phase transitions.
Finally, we discuss similar phenomena for a quantum phase transition in a
model with a topological transition. We also discuss how our findings can be
accessed experimentally with current technology in systems of Rydberg atoms.

The presentation is organized as follows. In Sec. II, we introduce the
Hamiltonian of the ANNNI chain, and introduce the protocol used to probe the
ferromagnetic to paramagnetic quantum phase transition via dynamics following
quantum quenches. The results obtained for dynamics after the quenches are
presented in Sec. III, while the results after thermalization are presented in
Sec. IV. Combining results from the dynamics and thermalization, in Sec. V we
report the estimated phase diagram for the ferromagnetic to paramagnetic
quantum phase transition for a wide range of parameters of the ANNNI chain. In
Sec. VI we summarize our results, discuss their implications, and report
evidence of qualitatively similar behavior in the short-time dynamics after
quenches across a topological transition.

##  II ANNNI Hamiltonian and
Quench Protocol

The ANNNI chain is a very well studied spin model (see, e.g., Ref. Suzuki  _et
al._ (2013)). Its Hamiltonian in a chain with $L$ sites can be written as

$\hat{H}\doteq-\sum_{i}^{L}\sigma_{i}^{x}\sigma_{i+1}^{x}+\kappa\sum_{i}^{L}\sigma_{i}^{x}\sigma_{i+2}^{x}-\Gamma\sum_{i}^{L}\sigma_{i}^{z}.$
(1)

When mapped onto a fermionic Hamiltonian using Jordan-Wigner transformation
Cazalilla  _et al._ (2011), the next-nearest-neighbor term (with strength
$\kappa$) maps onto a four-fermion interaction. At $T=0$, this model has a
rich (and still partly controversial) phase diagram in the $\kappa$-$\Gamma$
plane. The quantum phase transition line from the ferromagnetic to the
paramagnetic phase, which occurs as the antiferromagnetic next-nearest
neighbor coupling $\kappa>0$ crosses a critical value for a fixed
$|\Gamma|<1$, is a second order phase transition (see, e.g., Ref. Beccaria
_et al._ (2006)). In the quadrant ($\kappa>0$, $\Gamma>0$), this line is well
described using second order perturbation theory, with the critical parameters
satisfying (see Fig. 11) Beccaria  _et al._ (2006):

$1-2\kappa_{c}=\Gamma_{c}-\Gamma_{c}^{2}\frac{\kappa_{c}}{2(1-\kappa_{c})}.$
(2)

To probe this ferromagnetic to paramagnetic quantum phase transition at a
fixed value of $\Gamma$, we generate a family of Hamiltonians
$\hat{H}(\kappa)$. We then generate a family of nonequilibrium states via
quenches with $\hat{H}(\kappa)$. The protocol (straightforward to generalize
to other models) consists of following steps (see Fig. 1):

  1. (i)

The initial state is fixed to be the ground state $|\psi(\kappa_{I})\rangle$
of $\hat{H}(\kappa_{I})$, where $\kappa_{I}$ is deep in the ferromagnetic
phase.

  2. (ii)

We suddenly change (quench) $\kappa_{I}\rightarrow\kappa$ at $t=0$, and study
the unitary time evolution of the system under the time-independent
Hamiltonian $\hat{H}(\kappa)$, so that
$|\psi(t,\kappa)\rangle=\exp[-i\hat{H}(\kappa)t]|\psi(\kappa_{I})\rangle$,
where we have set $\hbar=1$.

  3. (iii)

We compute expectation values of observables ${\cal
O}(t,\kappa)=\langle\psi(t,\kappa)|\hat{\cal O}|\psi(t,\kappa)\rangle$.

  4. (iv)

For a fixed value of $t$, we study how ${\cal O}(t,\kappa)$ changes with
$\kappa$, focusing on the behavior in the vicinity of $\kappa_{c}$, where
$\kappa_{c}$ is the critical value of $\kappa$ for the transition given the
selected value of $\Gamma$.

Figure 1: Schematic representation of our quench protocol, superimposed on a
schematic ground-state phase diagram of the ANNNI chain.

##  III Quantum Dynamics

Here we study the time evolution of observables after quantum quenches in the
infinite ANNNI chain, which we compute using iTEBD Vidal (2003); *Vidal_2;
*Vidal_3; Schollwock (2011); Hauschild and Pollmann (2018). Following the
protocol introduced in Sec. II (see Fig. 1), we fix $\Gamma$ (we take
$\Gamma=0.2$) and then fix $\kappa_{I}$ so that the initial state is a ground
state of the ANNNI chain deep in the ferromagnetic phase (we take
$\kappa_{I}=0$). In our iTEBD calculations, we introduce a very small ($\sim
10^{-6}$) longitudinal field to pin one of the two degenerate maximally
polarized ground states. The critical value of $\kappa$ for the ferromagnetic
to paramagnetic quantum phase transition for $\Gamma=0.2$ is
$\kappa_{c}\approx 0.41$.

We first focus on the dynamics of two local observables, the nearest and next
nearest neighbor longitudinal correlators

$C^{x}_{1(2)}=\frac{1}{L}\sum_{i=1}^{L}\langle\sigma^{x}_{i}\sigma_{i+1(2)}^{x}\rangle.$
(3)

Figure 2: Time evolution of (a) $C^{x}_{1}$ and (b) $C^{x}_{2}$ for six values
of $\kappa$ after quenches starting from the ground state of the ANNNI
Hamiltonian with $\Gamma=0.2$ and $\kappa_{I}=0$. Figure 3: Ferromagnetic to
paramagnetic quantum phase transition in the ANNNI chain as revealed via real
time dynamics of local observables. (a) $C^{x}_{1}$ and (b) $C^{x}_{2}$ at
different times plotted as functions of $\kappa$ after the quench (the legend
is the same for both observables). (Insets) Derivative with respect to
$\kappa$ of the results shown in the main panels. The initial state is the
ground state of the ANNNI Hamiltonian with $\Gamma=0.2$ and $\kappa_{I}=0$.
The vertical dashed lines mark the critical $\kappa_{c}\approx 0.41$.

In Fig. 2, we show results for the time evolution of $C^{x}_{1}$ [Fig. 2(a)]
and $C^{x}_{2}$ [Fig. 2(b)] for six values of $\kappa$ after the quench. The
dynamics of both longitudinal correlators is qualitatively similar for the
values of $\kappa$ shown. Their decrease with time speeds up as $\kappa$
increases about $\kappa_{c}$. How the closeness to $\kappa_{c}$ affects the
dynamics is better seen by plotting the correlations for fixed times $t$ after
the quench as functions of $\kappa$ [step (iv) in the protocol introduced in
Sec. II]. This is done in Fig. 3, where we show results for $C^{x}_{1}$ [Fig.
3(a)] and $C^{x}_{2}$ [Fig. 3(b)]. At all times reported, $C^{x}_{1}$ and
$C^{x}_{2}$ decrease rapidly with increasing the value of $\kappa$ for
$\kappa\gtrsim\kappa_{c}$. In addition, with increasing time, the decrease in
the correlators becomes more prominent when $\kappa\approx\kappa_{c}$. This is
better seen in the insets, where we show the derivative of the correlators.
They develop sharper dips close to $\kappa_{c}$ as the evolution time
increases.

Figure 4: Ferromagnetic to paramagnetic quantum phase transition in the ANNNI
chain as revealed via real time dynamics of the half-chain entanglement
entropy $S_{1/2}$. (a) Time evolution of $S_{1/2}$ for six values of $\kappa$.
(b) $S_{1/2}$ at different times plotted as a function of $\kappa$. (Inset)
Derivative with respect to $\kappa$ of the results shown in the main panel.
The initial state for the dynamics is the ground state of the ANNNI
Hamiltonian with $\Gamma=0.2$ and $\kappa_{I}=0$. The vertical dashed lines
mark the critical $\kappa_{c}\approx 0.41$.

In Refs. Bhattacharyya  _et al._ (2015); Roy  _et al._ (2017) it was proved
that following the same protocol discussed here but for noninteracting models
(or models mappable to them) results in nonanalytic behavior of local
observables at the quantum phase transition in the limit $t\rightarrow\infty$
(after having taken the thermodynamic limit first). While this is not the case
in the quenches in generic models studied here (see Sec. IV), the prominent
features seen in Fig. 3 at finite times are promising for an experimental
determination of $\kappa_{c}$.

We also studied the dynamics of the half chain entanglement entropy
$S_{1/2}=-\text{Tr}{[\rho_{1/2}\ln{\rho_{1/2}}]}$, where $\rho_{1/2}$ is the
density matrix of the half chain (obtained by tracing out the other half).
This is a nonlocal observable that is expected to increase linearly with time
in quantum chaotic systems Kim and Huse (2013). In Fig. 4(a), we plot the time
evolution of $S_{1/2}$ for six values of $\kappa$. As for the local operators
in Fig. 2(a), the change of $S_{1/2}$ with time speeds up as $\kappa$
increases about $\kappa_{c}$. Figure 4(b) shows $S_{1/2}$ at fixed times $t$
after the quench plotted vs $\kappa$, and the inset in Fig. 4(b) shows the
derivative with respect to $\kappa$ of the results in the main panel. Like the
local operators in Fig. 3, the behavior of the half chain entanglement entropy
carries a marker of the quantum phase transition.

###  III.1 Changing $\kappa_{I}$

Figure 5: Results for $dC^{x}_{1}/d\kappa$, as those in the inset in Fig.
3(a), obtained at a fixed time $t=15$ after the quench for different values of
$\kappa_{I}$ (=-0.2, 0, 0.2) in the initial ground state. The vertical dashed
line marks the critical $\kappa_{c}\approx 0.41$.

In Fig. 5, we show the derivative of $C^{x}_{1}$ with respect to $\kappa$ at a
fixed time after the quench, plotted as a function of $\kappa$, for different
values of $\kappa_{I}$ in the initial ground state. We recall that as
$\kappa_{I}$ departs from $\kappa_{c}$, for $\kappa_{I}<\kappa_{c}$, the
ground state of the system is deeper in the ferromagnetic phase. The results
in Fig. 5 show that starting deeper in the ferromagnetic phase results in a
slightly shallower dip in $dC^{x}_{1}/d\kappa$, while its position remains
unchanged. This might be expected as the departure of $\kappa_{I}$ from
$\kappa_{c}$ increases the magnitude of the quench, and hence increases the
final energy density, thereby blunting the signature of the quantum phase
transition. This is consistent with the results in Sec. IV.3, where we discuss
the effect that the increase in the magnitude of the quench has in observables
after thermalization.

##  IV Long-time thermal results

Because of the linear growth of the entanglement entropy seen in Fig. 4, the
iTEBD technique only allows one to study dynamics at short and intermediate
times. To explore the fate of observables after thermalization, we use a
numerical linked cluster expansion (NLCE) Rigol  _et al._ (2006);
*rigol2007numerical. We broaden the class of initial states to explore how
initial nonzero temperatures modify the behavior of observables after
thermalization.

Here we consider more general quenches within the ANNNI Hamiltonian involving
initial states $\hat{\rho}_{I}$ that are thermal equilibrium states of the
initial Hamiltonian $\hat{H}(\kappa_{I})$. For an initial temperature $T_{I}$,
$\hat{\rho}_{I}$ has the form

$\displaystyle\hat{\rho}_{I}=\dfrac{e^{-\hat{H}(\kappa_{I})/T_{I}}}{\text{Tr}[e^{-\hat{H}(\kappa_{I})/T_{I}}]}.$
(4)

When $T_{I}=0$, $\hat{\rho}_{I}$ is the ground state of $\hat{H}(\kappa_{I})$.
As in the previous section, we quench $\kappa_{I}\rightarrow\kappa$, while
$\Gamma$ is kept unchanged ($\Gamma=0.2$). In Secs. IV.1 and IV.2 we fix
$\kappa_{I}=0$. In Sec. IV.3, we explore what changes when $\kappa_{I}$ is
varied within the ferromagnetic phase ($\kappa_{I}<\kappa_{c}\approx 0.41$).

Since the energy after the quench is the only conserved quantity, at
sufficiently long times in the thermodynamic limit, observables are expected
to be described by a Gibbs ensemble D’Alessio  _et al._ (2016)

$\displaystyle\hat{\rho}_{\text{GE}}(\kappa)=\dfrac{e^{-\hat{H}(\kappa)/T(\kappa)}}{\text{Tr}[e^{-\hat{H}(\kappa)/T(\kappa)}]},$
(5)

with a temperature $T(\kappa)>0$ (which is nonzero even when $T_{I}=0$)
determined by the energy $E(\kappa)$ set by the initial state
$\hat{\rho}_{I}$, as dictated by:

$\displaystyle\text{Tr}[\hat{\rho}_{\text{GE}}(\kappa)\hat{H}(\kappa)]=\text{Tr}[\hat{\rho}_{I}\hat{H}(\kappa)].$
(6)

We use the numerical linked cluster expansion (NLCE) technique introduced in
Refs. Rigol  _et al._ (2006); *rigol2007numerical to study the thermal
expectation values of observables in the thermodynamic limit. All the NLCE
results reported in what follows are obtained using 15 orders of the maximally
connected cluster expansion introduced in Refs. Rigol (2014); *rigol_16. To
gauge how well the series has converged, we estimate the convergence error for
an observable by computing the relative difference between the last two orders
(14 and 15) of the NLCE Rigol (2014); *rigol_16. We only report results whose
convergence error for the energy is less than $10^{-5}$. $T(\kappa)$ is
obtained by numerically matching the energies in the left and right side of
Eq. (6). Both energies are evaluated using NLCE to 15 orders, and $T(\kappa)$
is computed by enforcing that their relative difference is less than
$10^{-11}$ (see Ref. Rigol (2014); *rigol_16). For observables other than the
energy, we only report results whose convergence errors are less than $5\times
10^{-5}$ (except for the entropy, for which we set the cut off to be $7\times
10^{-5}$). The convergence errors are small enough to be unimportant for the
discussions that follow.

###  IV.1 Observables

As mentioned before, in the thermodynamic limit at sufficiently long times
after the quench, thermalization is expected to occur in the nonintegrable
systems considered here D’Alessio  _et al._ (2016). Next we study the expected
thermal equilibrium results that observables $\hat{O}$ reach after
equilibration following the quench.

In the space of all possible thermal equilibrium ensembles parameterized by
the coordinates $(T,\kappa)$, the initial state $\hat{\rho}_{I}$ sets a
trajectory $T(\kappa)$ determined by Eq. (6). One can then write

$\displaystyle\dfrac{dO}{d\kappa}$ $\displaystyle=$
$\displaystyle\dfrac{dT}{d\kappa}\left(\dfrac{\partial O}{\partial
T}\right)_{\kappa}+\left(\dfrac{\partial O}{\partial\kappa}\right)_{T}.$ (7)

Since $O(T,\kappa)$ is an analytic function whenever $T>0$, and since
$dT/d\kappa$ is expected to be a smooth function of $\kappa$ (we discuss this
in Sec. IV.2), then $dO/d\kappa$ must be a smooth function of $\kappa$ after
equilibration following the quench. Still, for observables that are indicators
of the quantum phase transition in nonintegrable systems (e.g., order
parameters and related observables), $(\partial O/\partial T)_{\kappa}$ and
$(\partial O/\partial\kappa)_{T}$ can be large if $T$ is low when $\kappa$ is
close to $\kappa_{c}$ (we show the latter to be the case for our quenches in
Sec. IV.2). This means that, even in thermal equilibrium, it is possible to
have prominent (but smooth) features in $dO/d\kappa$ as observed at
intermediate times in the previous section. In integrable systems, in which
all possible states after equilibration are described by generalized Gibbs
ensembles that are parameterized by extensive numbers of quantities Rigol  _et
al._ (2007b); Vidmar and Rigol (2016), nonanalytic behavior is possible and
has in fact been observed in Refs. Bhattacharyya  _et al._ (2015); Roy  _et
al._ (2017).

Figure 6: The nearest (next nearest) neighbor longitudinal spin-spin
correlation per site $C_{1}^{x}$ ($C_{2}^{x}$), see Eq. (3), evaluated in
thermal equilibrium using NLCE following quenches
$\kappa_{I}=0\rightarrow\kappa$, with $T_{I}=0,\,0.1,\,0.5,$ and 1.0. We also
show $C_{1}^{x}$ and $C_{2}^{x}$ in the ground state of $\hat{H}(\kappa)$
(dotted lines) computed using iTEBD. The main panels in (a) and (b) show
$dC_{1}/d\kappa$ and $dC_{2}/d\kappa$, respectively, while the corresponding
insets show $C_{1}^{x}$ and $C_{2}^{x}$. The vertical dashed lines mark the
critical $\kappa_{c}\approx 0.41$.

In Fig. 6, we show the thermal equilibrium results obtained for the nearest
$C_{1}^{x}$ and next nearest $C_{2}^{x}$ neighbor longitudinal spin
correlations per site [see Eq. (3)] as functions of $\kappa$ after the quench,
as well as their expectation values in the ground state of $\hat{H}(\kappa)$
computed with iTEBD. The main panels show $dC_{1(2)}/d\kappa$, while the
insets show $C_{1(2)}(\kappa)$, for various initial temperatures $T_{I}$ and
in the ground state (dotted lines, computed with iTEBD). In the ground state,
$C_{1}^{x}$ and $C_{2}^{x}$ are nearly one in the ferromagnetic phase and
exhibit a rapid decrease when crossing the quantum phase transition (prominent
minima can be seen in $dC_{1(2)}/d\kappa$ at $\kappa_{c}$), i.e., they serve
as indicators of the quantum phase transition. (They also serve as indicators
of the ferromagnetic to paramagnetic quantum phase transition in the
integrable transverse field Ising model, see Appendix A.) This zero-
temperature behavior is the precursor of the behavior of $C_{1}^{x}$ and
$C_{2}^{x}$ observed in the insets for low initial $T_{I}$, which, in turn,
produces the prominent minima in $dC_{1(2)}/d\kappa$ near $\kappa_{c}$
observed in the main panels. Figure 6 shows that the position of the minima
drift away from $\kappa_{c}$, and they become shallower, with increasing
$T_{I}$. Note that the results for $T_{I}=0$ and $T_{I}=0.1$ are
indistinguishable in the plots.

Figure 7: The transverse spin magnetization (von-Neumann entropy) per site
$m^{z}$ ($s$), see text, evaluated using NLCE in thermal equilibrium following
quenches $\kappa_{I}=0\rightarrow\kappa$ with $T_{I}=0,\,0.1,\,0.5,$ and 1.0.
The main panels in (a) and (b) show $dm^{z}/d\kappa$ and $ds/d\kappa$,
respectively, while the corresponding insets show $m^{z}$ and $s$. In (a), we
also show results for $m^{z}$ in the ground state of $\hat{H}(\kappa)$ (dotted
lines) computed using iTEBD. The vertical dashed line marks the critical
$\kappa_{c}\approx 0.41$.

Qualitatively similar results were obtained for other local observables, such
as the transverse magnetization per site $m^{z}=\sum_{i}\sigma_{i}^{z}/L$,
shown in Fig. 7(a), and for the (von-Neumann) entropy per site $s$ of the
thermal state $\hat{\rho}_{\text{GE}}(\kappa)$, shown in Fig. 7(b). In the
ground state, $m^{z}$ increases rapidly when transitioning from the
ferromagnetic to the paramagnetic phase, as shown in Fig. 7(a) (dotted lines,
computed with iTEBD). Hence, $m^{z}$ serves as an indicator of the quantum
phase transition, and its behavior at zero temperature is the reason there are
prominent maxima in $dm^{z}/d\kappa$ near $\kappa_{c}$ for quenches at low
$T_{I}$. (See Appendix A for ground state results of $m^{z}$ across the
ferromagnetic to paramagnetic quantum phase transition in the integrable
transverse field Ising model.) The entropy, on the other hand, is strictly
zero at zero temperature, i.e., it does not change at the phase transition
[the entanglement entropy does change, as shown in Fig. 4(b)]. However, as we
show in Sec. IV.2, when $T_{I}$ is low, the temperature after quench increases
rapidly when $\kappa$ crosses $\kappa_{c}$ and this produces the rapid
increase of $s$ seen in Fig. 7(b).

###  IV.2 Temperature

Let us now show that $dT/d\kappa$ is a smooth function of $\kappa$. The ANNNI
Hamiltonian can be written as $\hat{H}(\kappa)=\hat{H}_{0}+\kappa\hat{V}$, so
that keeping the initial state $\hat{\rho}_{I}$ fixed and changing $\kappa$
after the quench results in $E(\kappa)$ being a linear function of $\kappa$

$\displaystyle
E(\kappa)=\left(\text{Tr}[\hat{\rho}_{I}\hat{H}_{0}]\right)+\kappa\left(\text{Tr}[\hat{\rho}_{I}\hat{V}]\right),$
(8)

with a slope $A\equiv dE(\kappa)/d\kappa=\text{Tr}[\hat{\rho}_{I}\hat{V}]$.

As in the previous section for $dO/d\kappa$, for the energy one can write

$\displaystyle\dfrac{dE}{d\kappa}$ $\displaystyle=$
$\displaystyle\dfrac{dT}{d\kappa}\left(\dfrac{\partial E}{\partial
T}\right)_{\kappa}+\left(\dfrac{\partial E}{\partial\kappa}\right)_{T},$ (9)

where $\left(\partial E/\partial T\right)_{\kappa}=C_{\kappa}(T)$ is the
specific heat. Combining Eqs. (8) and (9), we have that

$\displaystyle\dfrac{dT(\kappa)}{d\kappa}=\dfrac{A-\left(\dfrac{\partial
E}{\partial\kappa}\right)_{T(\kappa)}}{C_{\kappa}[T(\kappa)]}.$ (10)

All functions in the r.h.s. of Eq. (10) are smooth, and
$C_{\kappa}[T(\kappa)]>0$, because $T(\kappa)>0$ after the quench. This shows
that $T(\kappa)$ is also a smooth function. Next, we use numerical
calculations to explore whether quenches $\kappa_{I}\rightarrow\kappa$
spanning across $\kappa_{c}$ produce temperatures $T(\kappa)$ with signatures
of the quantum phase transition, as shown to be the case in Sec. IV.1 for
local observables.

Figure 8 shows $T(\kappa)$ for quenches with $\kappa_{I}=0\rightarrow\kappa$
for various initial temperatures $T_{I}$, including the ground state of
$\hat{H}(\kappa_{I})$. For very low initial temperatures $T_{I}\lesssim 0.1$,
the temperatures $T(\kappa)$ after the quench are essentially
indistinguishable from those for $T_{I}=0$. This explains why all the results
reported in Sec. IV.1 are indistinguishable for $T_{I}=0$ and $T_{I}=0.1$. For
those very low $T_{I}$, the temperatures $T(\kappa)$ exhibit a low-temperature
minimum in the vicinity of $\kappa_{c}$ [notice that $T(\kappa)$ lies between
0.05 and 0.1]. At $T_{I}=0.5$, a temperature at which $T(\kappa)$ after the
quench departs from the $T_{I}=0$ result, a minimum in $T(\kappa)$ still
remains visible close to $\kappa_{c}$. The locus of minima in $T(\kappa)$,
shown as a dotted line for a large number of $T_{I}$, makes apparent that the
minima remain close to $\kappa_{c}$ as long as $T_{I}$ remains low
($T_{I}\lesssim 1.0$). At higher initial temperatures, the minima depart from
$\kappa_{c}$ indicating that the information about $\kappa_{c}$ is washed out.

Figure 8: NLCE results for the temperature of the Gibbs ensemble describing
observables after equilibration, following quantum quenches
$\kappa_{I}=0\rightarrow\kappa$ within the ANNNI Hamiltonian, for initial
thermal states at different temperatures $T_{I}$. The locus of minima
[$\kappa_{m},T(\kappa_{m})$] for a large number of initial temperatures
$T_{I}$ is also shown. The vertical dashed lines mark the critical
$\kappa_{c}\approx 0.41$.

Overall it is remarkable that, due to the presence of the phase transition
(and the corresponding closing of the gap above the ground state), when
quenching to the same (ordered) side of the critical point, the effective
temperature decreases as the size of the quench increases and $\kappa$
approaches the critical point. This trend sharply reverses as $\kappa$ crosses
the critical point.

Examining Eq. (10) in the context of our numerical results allows us to
understand why a minimum develops near $\kappa_{c}$ at very low
($T_{I}\lesssim 0.1$) and low ($T_{I}\lesssim 1.0$) initial temperatures. At
the minimum, we have that

$\displaystyle a=\left(\dfrac{\partial e}{\partial\kappa}\right)_{T},$ (11)

where we defined the intensive counterparts of the extensive quantities in Eq.
(10) as $a=A/L$ and $e=E/L$.

The main panel in Fig. 9 shows $(\partial e/\partial\kappa)_{T}$ at different
temperatures. For $T=0$, we also show iTEBD results (the NLCE results do not
converge close to $\kappa=\kappa_{c}$). Notice that, in the region in which
the NLCE results converge to the precision mentioned in the introduction of
this section, they are indistinguishable from the iTEBD ones. The iTEBD
results for $(\partial e/\partial\kappa)_{T=0}$ exhibit a rapid decrease about
$\kappa_{c}$ (resulting in a singularity in
$(\partial^{2}e/\partial\kappa^{2})_{T=0}$ at $\kappa_{c}$, Fig. 9, right
inset), reflecting the nonanalytic behavior of the energy at the (second
order) quantum phase transition. That rapid decrease leaves its signature in
the low temperature behavior of $(\partial e/\partial\kappa)_{T>0}$, and this
is what makes possible for Eq. (11) to be satisfied close to $\kappa_{c}$ for
low initial temperatures.

Figure 9: Thermal equilibrium energy per site $e$ and its derivatives at
different $T$. (Main panel) $(\partial e/\partial\kappa)_{T}$, (left inset)
$e$ versus $\kappa$ at constant temperature, and (right inset)
$(\partial^{2}e/\partial\kappa^{2})_{T}$. Results are shown for different
values of $T$ (the main panel and the insets share the legend). The solid
(black) curve ($T=0$) shows the iTEBD results for the ground quantities, while
the other curves show NLCE results. Also depicted in the main panel is
$a=\langle\sigma_{i}^{x}\sigma_{i+2}^{x}\rangle_{\hat{\rho}_{I}}/L\approx
0.99$ for $T_{I}\lesssim 0.1$. The vertical dashed lines mark the critical
$\kappa_{c}\approx 0.41$.

In Fig. 9, $a=\langle\sigma^{x}_{i}\sigma^{x}_{i+2}\rangle_{\hat{\rho}_{I}}/L$
is shown as a horizontal line for $T_{I}\lesssim 0.1$. For those very low
initial temperatures, $a$ is very close to 1 ($a\approx 0.99$) since
$\kappa_{I}=0$ is deep in the ferromagnetic phase, and Fig. 9 shows that the
condition $(\partial e/\partial\kappa)_{T}=a$ is satisfied for $\kappa$ very
close to $\kappa_{c}$ for temperatures between $T=0.05$ and $T=0.1$ (effective
temperatures after the quench for $\kappa$ close to $\kappa_{c}$ when
$T_{I}\lesssim 0.1$, see Fig. 8). This explains why the minimum in $T(\kappa)$
vs $\kappa$ occurs very close to $\kappa_{c}$ for $T_{I}\lesssim 0.1$.
Increasing the initial temperature beyond $T_{I}=0.1$ increases $T$ but also
reduces the value of $a$. This results in the minimum remaining close (and
actually slightly approaching) $\kappa_{c}$ in Fig. 8 when $T_{I}$ departs
from 0.1 but still remains low ($T_{I}\lesssim 1.0$). Since the slope of
$(\partial e/\partial\kappa)_{T}$ at the crossing point near $\kappa_{c}$ is
negative, it follows from Eq. (10) that the extremum in $T(\kappa)$ near
$\kappa_{c}$ is a minimum.

###  IV.3 Changing $\kappa_{I}$

Motivated by the results discussed in Sec. III.1, we explore next what happens
to the thermal equilibrium results after equilibration when one changes
$\kappa_{I}$ within the ferromagnetic regime, keeping $T_{I}=0$ fixed. In Fig.
10(a), we show $T(\kappa)$ vs $\kappa$ for $\kappa_{I}=-0.2$, 0, and $0.2$. As
expected from the fact that the initial state remains a nearly perfect
ferromagnet, the minima in $T(\kappa)$ close $\kappa_{c}$ are robust to the
choice of initial $\kappa_{I}$. However the minimum value of $T(\kappa)$
attained decreases as $\kappa_{I}$ approaches $\kappa_{c}$. As a result, the
signature of the presence of a quantum critical point in observables after
thermalization becomes sharper as $\kappa_{I}\rightarrow\kappa_{c}$. This is
apparent in Fig. 10(b) in which we plot $dC_{1}^{x}/d\kappa$.

Note that in Fig. 10(a) there is a singularity in $T(\kappa)$ at $\kappa=0.2$
for $\kappa_{I}=0.2$, as well as at $\kappa=0$ for $\kappa_{I}=0$. These are
trivial consequences of performing no quench, which means that the system
remains in the ground state. The fact that $|dT/d\kappa|\rightarrow\infty$ at
those points follows from Eq. (10) due to specific heat
$C_{\kappa}(T\rightarrow 0)\rightarrow 0$ in the denominator. These
singularities have no consequence in the expectation values of observables.

Figure 10: (a) Equilibrium temperature $T(\kappa)$ and (b)
$dC_{1}^{x}/d\kappa$ in thermal equilibrium, after quenches
$\kappa_{I}\rightarrow\kappa$ from initial ground states of
$\hat{H}(\kappa_{I})$ for three different values of $\kappa_{I}$. The vertical
dashed lines mark the critical $\kappa_{c}\approx 0.41$.

##  V Phase Diagram

Here we combine results obtained for $C_{1}^{x}$ at intermediate times after
the quench (from iTEBD calculations), and after equilibration (from NLCE
calculations), to identify the phase boundary in the $(\kappa,\Gamma)$ plane
separating the ferromagnetic and paramagnetic phases in the ground state. We
estimate $\kappa_{c}$ by carrying out quenches $\kappa_{I}=0\rightarrow\kappa$
for different values of $\Gamma$ ($\Gamma$ is not changed during the quench).
Qualitatively similar results were obtained for other local observables such
as $C_{2}^{x}$ and $m^{z}$ and are not reported here.

Figure 11: Phase boundary for the ground state quantum phase transition in the
$(\kappa,\Gamma)$ plane separating the ferromagnetic and paramagnetic phases.
Unbiased results for the boundary were obtained using ground-state iTEBD
[iTEBD${}_{\text{GS}}$ in the legend, obtained locating the singularity in
$(\partial^{2}e/\partial\kappa^{2})_{T=0}$] and are closely followed by the
predictions of second order perturbation theory (continuous line). The phase
boundary is well described by $\kappa_{c}$ estimated from the extrema of
$dC_{1}^{x}/d\kappa$ obtained in finite-time iTEBD calculations after the
quench (for $\tau=25$) and in the (expected) long-time thermal results
obtained using NLCE. In all quenches $\kappa_{I}=0\rightarrow\kappa$, $\Gamma$
is not changed during the quench, and we show results for $T_{I}=0$ (iTEBD and
NLCE), and for $T_{I}=0.3$ (NLCE).

In the main panel of Fig. 11, we show $\kappa_{c}$ extracted from the extrema
of $dC_{1}^{x}/d\kappa$ obtained using iTEBD results at $t=25$ after quenches
starting from the ground state, and NLCE thermal equilibrium results after
quenches starting from the ground state ($T_{I}=0$) and from an initial
temperature $T_{I}=0.3$. As $\Gamma$ increases, the NLCE convergence errors
are higher for quenches starting from the ground state. This occurs because
the critical point gets closer to $\kappa_{I}=0$ and the effective temperature
after the quench becomes too small (see Fig. 10 and related discussion). This
is the reason no NLCE points are reported for quenches with $\Gamma\geq 0.4$
and $T_{I}=0$. On the other side of the phase diagram, when $\Gamma$ is small,
the quenches in $\kappa$ result in fewer excitations ($\Gamma\rightarrow 0$
becomes the classical Ising chain) thereby bringing the thermal equilibrium
ensemble about $\kappa_{c}$ close to the ground state critical point. This
also affects the NLCE convergence, resulting in no NLCE data points for
$\Gamma\lesssim 0.2$. The results in Fig. 11 show that both the intermediate-
time and (expected) long-time extrema follow very closely the phase boundary
calculated using iTEBD for the ground state [locating the singularity in
$(\partial^{2}e/\partial\kappa^{2})_{T=0}$], which is well described by the
second order perturbation theory results.

##  VI Summary and discussion

In summary, we have shown that local observables can be used to locate the
ferromagnetic to paramagnetic quantum phase transition in the ANNNI chain (a
nonintegrable model) both at intermediate times after a quench and at long
times after thermalization. The initial states for our quenches were chosen to
be ground states of the ANNNI chain deep in the ferromagnetic phase. We
explored the effect that changing the magnitude of the quench and starting
from initial finite temperature states has in many of our conclusions, and
showed that our conclusions are robust against those changes.

The fact that intermediate-time dynamics, following quenches whose initial
states are ground states far from a quantum phase transition, provide a way to
locate the quantum phase transition is promising for experiments with
ultracold quantum gases Bloch  _et al._ (2008); Esslinger (2010) and ions
Leibfried  _et al._ (2003); Islam  _et al._ (2011). In those experiments, it
is usually straightforward to prepare ground states far away from quantum
phase transitions but it is much more challenging to prepare them close to the
transitions. The latter is needed to locate the quantum critical point via
traditional measurements of the system in equilibrium. Also, not needing to
wait long times to observe signatures of the quantum phase transition in the
dynamics after the quench is important because, due to heating and other
undesirable effects, keeping the dynamics coherent in the experiments becomes
increasingly challenging as the evolution time increases.

We stress that the dynamics studied in this work is within reach in current
experiments. In particular, the considered quantum quenches in the ANNNI model
appear feasible using Rydberg dressing for ultracold atoms in optical lattices
Zeiher  _et al._ (2016, 2017). Rydberg-dressed atoms exhibit a soft-core
interaction potential, which is approximately constant below a threshold
distance between two atoms and decays quickly beyond the threshold in a
$d^{-6}$ fashion as a function of distance $d$ Zeiher  _et al._ (2016). Thus,
it is possible to tune the relative strength of nearest and next nearest
neighbor interactions in such a way that further distant interactions can be
approximately neglected due to the fast decay of the interaction potential at
larger separations. Further, it is straightforward to also realize the
necessary transverse field, to initialize the system in fully-polarized
product states Zeiher  _et al._ (2016), and to measure the correlation
functions defined in Eq. (3) Zeiher  _et al._ (2017).

An important question that we leave open to further exploration is how
generally one can use our protocol to locate quantum phase transitions in
other one-dimensional models. Given our results and insights gained within the
ANNNI chain, we believe our protocol should be widely applicable to other one-
dimensional models with traditional quantum phase transitions. A different
question is whether it can be used to locate topological quantum phase
transitions, as shown for noninteracting models in Ref. Roy  _et al._ (2017).
In what follows, we report results from a preliminary exploration of dynamics
after quantum quenches about a topological transition in a quantum chaotic
model.

Figure 12: Topological transition. Signature of the Néel to Haldane quantum
phase transition in the anisotropic XXZ chain. Derivatives with respect to
$\Delta$ of (a) $C_{1}^{x}$ and (b) $C_{1}^{z}$ at different fixed times after
the quench, and of the results in the ground state (black solid line). The
gray vertical line shows the critical $\Delta_{c}\approx 1.183$

We explored the quantum phase transition from the Néel to the symmetry
protected topological “Haldane” phase in the spin-1 anisotropic (XXZ)
Heisenberg chain model. The Hamiltonian for this model reads

$\hat{H}_{\text{XXZ}}=\sum_{i}^{L}\left(\hat{S}_{i}^{x}\hat{S}_{i+1}^{x}+\hat{S}_{i}^{y}\hat{S}_{i+1}^{y}+\Delta\hat{S}_{i}^{z}\hat{S}_{i+1}^{z}\right),$
(12)

where $\hat{S}^{x,y,z}_{i}$ denote the $x,\,y$ and $z$ components of the
spin-1 operator at site $i$. Four different phases occur in this model when
one changes the anisotropy parameter $\Delta$ (see, e.g., Refs. Chen  _et al._
(2003); Heng Su  _et al._ (2012) and references therein). Here we focus on the
transition that occurs upon decreasing $\Delta$ from $\Delta>1$, a limit in
which $\hat{H}_{\text{XXZ}}$ reduces to the spin-1 Ising antiferromagnet. With
decreasing $\Delta$, the ground state of $\hat{H}_{\text{XXZ}}$ undergoes a
quantum phase transition from the antiferromagnet to the Haldane phase at
$\Delta_{c}\approx 1.183$. The Haldane phase is a topological phase, protected
by any one of the following three global symmetries: $D_{2}$ spin rotation,
time-reversal, and bond centered inversion Pollmann  _et al._ (2012). This
transition is of second order, and belongs to the 2D Ising universality class
Sakai and Takahashi (1990); Nomura (1989).

In our protocol, we start from the ground state at large $\Delta_{I}=2$ and
quench $\Delta$ across the neighborhood of $\Delta_{c}$. Due to the high
computational cost of the iTEBD calculations for the spin-1 anisotropic
Heisenberg chain, we were only able to study dynamics at short times ($t\leq
7$) after the quench. Still, for these short times, Fig. 12(a) shows that a
peak appears to develop with time in the derivative of the correlator
$C^{x}_{1}$ [see Eq. (3)], and Fig. 12(b) shows that a dip appears to develop
in the derivative of the correlator

$C^{z}_{1}=\frac{1}{L}\sum_{i=1}^{L}\langle\hat{S}_{i}^{z}\hat{S}_{i+1}^{z}\rangle,$
(13)

with respect to $\Delta$ at fixed times. In Figs. 12(a) and 12(b), we also
plot the derivatives of $C^{x}_{1}$ and $C^{x}_{2}$, respectively, with
respect to $\Delta$ in the ground state. The peak [Fig. 12(a)] and dip [Fig.
12(b)] seen there are expected to be the precursors of the features observed
in the time evolution of those observables.

Thus the above results suggest that our protocol can also be used to locate
topological quantum phase transitions (here transition from a Nëel ordered
phase to a topological phase) through the study of the quench dynamics of
local operators. Further studies are needed to further explore this
possibility.

###### Acknowledgements.

We thank F. Essler and R. Moessner for useful discussions. A.D. and A.H.
acknowledge support from DST-MPI partner group program “Spin liquids:
correlations, dynamics and disorder” between MPI-PKS (Dresden) and IACS
(Kolkata), and the visitor’s program of MPI-PKS. K.M. and M.R. were supported
by the National Science Foundation under Grant No. PHY-1707482. This research
was supported in part by the International Centre for Theoretical Sciences
(ICTS) during a visit for the program - Thermalization, Many body localization
and Hydrodynamics (Code: ICTS/hydrodynamics2019/11).FP is funded by the
European Research Council (ERC) under the European Unions Horizon 2020
research and innovation program (grant agreement No. 771537). FP acknowledges
the support of the DFG Research Unit FOR 1807 through grants no.PO 1370/2-1,
TRR80, and the Deutsche Forschungsgemeinschaft (DFG, German Research
Foundation) under Germany’s Excellence Strategy EXC-2111- 390814868. This
project has also received funding from the European Research Council (ERC)
under the European Union’s Horizon 2020 research and innovation programme
(grant agreement No. 853443), and M. H. further acknowledges support by the
Deutsche Forschungsgemeinschaft via the Gottfried Wilhelm Leibniz Prize
program.

##  Appendix A Integrable Transverse Field Ising Chain

Figure 13: Ground state results for (a) $C_{1(2)}^{x}$, and $m_{z}$, and (b)
their derivatives, as functions of the strength of the transverse magnetic
field.

The integrable transverse field Ising chain (TFIM) is probably the most
studied exactly solvable model in the context of quantum phase transitions
Sachdev (2011); Suzuki  _et al._ (2013). Its Hamiltonian reads

$\hat{H}\doteq-\sum_{i}^{L}\sigma_{i}^{x}\sigma_{i+1}^{x}-\Gamma\sum_{i}^{L}\sigma_{i}^{z}.$
(14)

It is the noninteracting limit ($\kappa=0$) of our ANNNI Hamiltonian [Eq.
(1)].

In Fig. 13, we report ground state results for $C_{1(2)}$ and $m_{z}$ [Fig.
13(a)], and their derivatives [Fig. 13(b)], across the ferromagnetic to
paramagnetic phase transition, which occurs in this model at $\Gamma=1$.

## References

  * Sachdev (2011) S. Sachdev,  _Quantum Phase Transitions_ (Cambridge University Press, 2011).
  * Carr (2010) L. D. E. Carr,  _Understanding Quantum Phase Transitions (1st Ed.)_ (Taylor and Francis, 2010).
  * Bloch  _et al._ (2008) I. Bloch, J. Dalibard, and W. Zwerger, Rev. Mod. Phys. 80, 8855964 (2008).
  * Esslinger (2010) T. Esslinger, Annu. Rev. Condens. Matter Phys. 1, 129152 (2010).
  * 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, Nature (London) 568, 207211 (2019).
  * Bhattacharyya  _et al._ (2015) S. Bhattacharyya, S. Dasgupta, and A. Das, Scientific Reports 5, 16490 (2015).
  * Roy  _et al._ (2017) S. Roy, R. Moessner, and A. Das, Phys. Rev. B 95, 041105(R) (2017).
  * Titum  _et al._ (2019) P. Titum, J. T. Iosue, J. R. Garrison, A. V. Gorshkov, and Z.-X. Gong, Phys. Rev. Lett. 123, 115701 (2019).
  * Heyl  _et al._ (2018) M. Heyl, F. Pollmann, and B. Dora, Phys. Rev. Lett. 121, 016801 (2018).
  * Vidal (2003) G. Vidal, Phys. Rev. Lett. 91, 147902 (2003).
  * Vidal (2004) G. Vidal, Phys. Rev. Lett. 93, 040502 (2004).
  * Vidal (2007) G. Vidal, Phys. Rev. Lett. 98, 070201 (2007).
  * Schollwock (2011) U. Schollwock, Annals of Physics 326, 96 (2011).
  * Hauschild and Pollmann (2018) J. Hauschild and F. Pollmann, arXiv:1805.00055 (2018).
  * Rigol  _et al._ (2006) M. Rigol, T. Bryant, and R. R. P. Singh, Phys. Rev. Lett. 97, 187202 (2006).
  * Rigol  _et al._ (2007a) M. Rigol, T. Bryant, and R. R. P. Singh, Phys. Rev. E 75, 061118 (2007a).
  * Deutsch (1991) J. M. Deutsch, Phys. Rev. A 43, 204662049 (1991).
  * Srednicki (1994) M. Srednicki, Phys. Rev. E 50, 8888901 (1994).
  * Rigol  _et al._ (2008) M. Rigol, V. Dunjko, and M. Olshanii, Nature 452, 854 (2008).
  * D’Alessio  _et al._ (2016) L. D’Alessio, Y. Kafri, A. Polkovnikov, and M. Rigol, Adv. Phys. 65, 233362 (2016).
  * Suzuki  _et al._ (2013) S. Suzuki, J.-i. Inoue, and B. K. Chakrabarti,  _Quantum Ising Phases and Transitions in Transverse Ising Models_ (Springer, Heidelberg, 2013).
  * Cazalilla  _et al._ (2011) M. A. Cazalilla, R. Citro, T. Giamarchi, E. Orignac, and M. Rigol, Rev. Mod. Phys. 83, 14051466 (2011).
  * Beccaria  _et al._ (2006) M. Beccaria, M. Campostrini, and A. Feo, Phys. Rev. B 73, 052402 (2006).
  * Kim and Huse (2013) H. Kim and D. A. Huse, Phys. Rev. Lett. 111, 127205 (2013).
  * Rigol (2014) M. Rigol, Phys. Rev. Lett. 112, 170601 (2014).
  * Rigol (2016) M. Rigol, Phys. Rev. Lett. 116, 100601 (2016).
  * Rigol  _et al._ (2007b) M. Rigol, V. Dunjko, V. Yurovsky, and M. Olshanii, Phys. Rev. Lett. 98, 050405 (2007b).
  * Vidmar and Rigol (2016) L. Vidmar and M. Rigol, J. Stat. Mech. 2016, 064007 (2016).
  * Leibfried  _et al._ (2003) D. Leibfried, R. Blatt, C. Monroe, and D. Wineland, Rev. Mod. Phys. 75, 281324 (2003).
  * Islam  _et al._ (2011) R. Islam, E. E. Edwards, K. Kim, S. Korenblit, C. Noh, H. Carmichael, G.-D. Lin, L.-M. Duan, C.-C. J. Wang, J. K. Freericks, and C. Monroe, Nat. Comm. 2, 377 (2011).
  * Zeiher  _et al._ (2016) J. Zeiher, R. van Bijnen, P. Schauß, S. Hild, J.-Y. Choi, T. Pohl, I. Bloch, and C. Gross, Nature Physics 12, 10951099 (2016).
  * Zeiher  _et al._ (2017) J. Zeiher, J.-y. Choi, A. Rubio-Abadal, T. Pohl, R. van Bijnen, I. Bloch, and C. Gross, Physical Review X 7, 041063 (2017).
  * Chen  _et al._ (2003) W. Chen, K. Hida, and B. C. Sanctuary, Phys. Rev. B 67, 104401 (2003).
  * Heng Su  _et al._ (2012) Y. Heng Su, S. Young Cho, B. Li, H.-L. Wang, and H.-Q. Zhou, Journal of the Physical Society of Japan 81, 074003 (2012).
  * Pollmann  _et al._ (2012) F. Pollmann, E. Berg, A. M. Turner, and M. Oshikawa, Phys. Rev. B 85, 075125 (2012).
  * Sakai and Takahashi (1990) T. Sakai and M. Takahashi, Journal of the Physical Society of Japan 59, 26882693 (1990).
  * Nomura (1989) K. Nomura, Phys. Rev. B 40, 91429146 (1989).