# Kibble-Zurek scalings and coarsening laws in slowly quenched classical Ising
chains

Lakshita Jindal<EMAIL_ADDRESS>Kavita Jain<EMAIL_ADDRESS>Theoretical Sciences Unit,
Jawaharlal Nehru Centre for Advanced Scientific Research,
Bangalore 560064, India

(August 28, 2024)

###### Abstract

We consider a one-dimensional classical ferromagnetic Ising model when it is
quenched from a low temperature to zero temperature in finite time using
Glauber or Kawasaki dynamics. Most of the previous work on finite-time
quenches assume that the system is initially in equilibrium and focus on the
excess defect density at the end of the quench which decays algebraically in
quench time with Kibble-Zurek exponent. Here we are interested in
understanding the conditions under which the Kibble-Zurek scalings do not hold
and in elucidating the full dynamics of the defect density. We find that
depending on the initial conditions and quench time, the dynamics of the
defect density can be characterized by coarsening and/or the standard finite-
time quench dynamics involving adiabatic evolution and Kibble-Zurek dynamics;
the time scales for crossover between these dynamical phases are determined by
coarsening time and stationary state relaxation time. As a consequence, the
defect density at the end of the quench is either a constant or decays
following coarsening laws or Kibble-Zurek scaling. For the Glauber chain, we
formulate a low temperature scaling theory and find exact expressions for the
final defect density for various initial conditions. For the Kawasaki chain
where the dynamic exponents for coarsening and stationary state dynamics are
different, we verify the above findings numerically and also examine the
effect of unequal dynamic exponents.

##  I Introduction

The phenomenon of phase ordering in systems exhibiting thermal phase
transition between a disordered and an ordered phase has received considerable
attention in the past few decades [1]. Following a sudden quench from a
disordered phase to a low temperature symmetry-broken phase, the system does
not order instantaneously; instead, the domains of symmetry-broken phase grow
locally until the system reaches equilibrium at large times [1]. Even if the
system is quenched at a finite rate, it is not in equilibrium at the end of
the quench and there are more defects compared to the equilibrium state at the
quench temperature; one can then ask how the excess defect density decays with
the quench time and what the dynamics are while the system is being slowly
cooled.

Some of these questions can be addressed in the framework of a theory first
proposed by Kibble to describe the symmetry-breaking in the early universe [2,
3] and later extended by Zurek to condensed matter systems [4, 5]. For a
classical or quantum system that shows a continuous phase transition between a
disordered and an ordered phase, under a time-dependent change of a control
parameter such as temperature, the Kibble-Zurek (KZ) theory predicts that
there are more defects at the end of the slow quench than in equilibrium and
the residual density of defects, in general, decays algebraically in quench
time with an exponent that depends on equilibrium critical exponents and
quench protocol.

These results can be understood by noting that if the system is initially deep
in the high-symmetry phase, there is a competition between two timescales,
viz., the equilibrium relaxation time and the time remaining until the end of
the quench. Away from the critical region where the correlation length is
small, as the equilibrium relaxation time is much smaller than the time left
until the end of the quench, the system has sufficient time to relax in
response to the changing temperature and reaches the equilibrium state at the
instantaneous temperature (adiabatic phase). However, in the critical region,
as the equilibrium correlation length and hence the relaxation time diverges,
the system is unable to relax and falls out of equilibrium (KZ phase) so that
there are more defects than had the system been in equilibrium. Assuming that
the dynamics are ‘frozen’ in the nonequilibrium phase, the KZ argument
predicts how the excess defect density scales with the quench time [3, 5].

The slow annealing problem described above has been studied theoretically in
various condensed matter systems including classical [6, 7, 8, 9, 10, 11, 12,
13, 14, 15, 16, 17, 18, 19, 20, 21] and quantum [22, 23, 24, 25, 26, 27] Ising
models, and the KZ predictions have also been verified in experiments [28] on
a wide variety of systems such as non-Newtonian fluids [29], colloidal
monolayers [30] and ultra-cold atomic gases [31]. However, it has also been
shown in theoretical studies that if the system is quenched deep in the
ordered phase, the excess defect density follows coarsening (and not the KZ)
scaling laws [11, 14], and that the dynamics are not frozen in the KZ phase
which affects the amplitude (but not the KZ exponent) of the defect density at
the end of the quench [18].

The KZ argument described above assumes that the system is initially in
equilibrium state and far from the critical region. However, if the system
starts in a nonequilibrium state or it is initially equilibrated to a
temperature in the critical region and the quench time is not long enough for
the system to reach the adiabatic phase, the KZ scalings may not hold. For the
Ising chains studied here, we find that if the quench time is small compared
to the equilibrium relaxation time, the system initially equilibrated to a
temperature in the critical region can not relax to the perturbations arising
due to changing temperature and the defect density at the end of the quench
remains close to its initial value; on the other hand, if the system is
initially not in equilibrium and quench times are small relative to the
critical coarsening time scales, the defect density decreases following
critical coarsening laws. For larger quench time (relative to the appropriate
relaxation time), however, we find that the KZ scalings hold provided the
system size is large enough. Interestingly, for the infinitely long Glauber
chain, we show that the defect density at the end of quench can be captured
exactly by a single expression [see, (21), (27), (31) for different initial
conditions] for both small and large quench times.

In the following sections, we study these scenarios in detail; besides, the
results mentioned above for the defect density at the end of the quench, we
also obtain analytical expressions for its temporal evolution in the Glauber
Ising chain. We find that depending on the initial state and quench time, the
dynamical evolution can be characterized by coarsening and/or the standard
adiabatic evolution and KZ dynamics, and the crossover between these dynamical
phases occurs on time scales that depend on the nonequilibrium and equilibrium
relaxation times. Some of these results are also verified numerically for the
Kawasaki chain for which, unlike the Glauber model, the dynamic exponent
characterizing coarsening phenomenon is different from the stationary state
dynamic exponent.

##  II Glauber Ising chain

###  II.1 Model

We consider a one-dimensional ferromagnetic Ising model with nearest neighbor
interactions defined by the Hamiltonian

$H=-\sum_{i=1}^{L}\sigma_{i}\sigma_{i+1}$ (1)

where the spin variable, $\sigma_{i}=\pm 1$ at site $i$ and
$\sigma_{L+1}=\sigma_{1}$ as we assume periodic boundary condition for a
finite-sized system. In the equilibrium state, the correlation length
$\xi_{eq}(T)\sim e^{\frac{2}{T}}$ diverges at the critical temperature $T=0$.

To study the finite-time quench dynamics, in this section, we consider the
Glauber dynamics in which the system evolves via single spin-flip and the
total magnetization is not conserved [32]. The probability that the system is
in a configuration $\\{\sigma_{1},\sigma_{2},\ldots,\sigma_{L}\\}$ at time $t$
is described by

$\begin{split}\frac{d}{dt}p(\sigma_{1},\ldots,\sigma_{i},\ldots,\sigma_{L},t)=\sum_{i=1}^{L}\Big{[}w(-\sigma_{i}\to\sigma_{i},t)\;p(\sigma_{1},\ldots,-\sigma_{i},\ldots,\sigma_{L},t)-\\\
w(\sigma_{i}\to-\sigma_{i},t)\;p(\sigma_{1},\ldots,\sigma_{i},\ldots,\sigma_{L},t)\Big{]}\end{split}$
(2)

where $w(\sigma_{i}\to-\sigma_{i},t)$ is the transition rate at which the spin
$i$ flips at time $t$ and is given by [32, 18]

$w(\sigma_{i}\to-\sigma_{i},t)=1-\frac{\gamma(t)}{2}\sigma_{i}(\sigma_{i-1}+\sigma_{i+1})$
(3)

with

$\gamma(t)=\textrm{tanh}\left(\frac{2}{T(t)}\right)$ (4)

Previous work [12, 18] have shown that if the system is cooled faster than a
logarithmic decay in time (for example, if the temperature is decreased
algebraically in time), the dynamics are essentially the same as that for
infinitely rapid quench (up to logarithmic factors, see (23) and (24) of
[18]). We therefore consider the cooling protocol,
$T(t)\sim-4\left[\ln\\{(1-\gamma_{0})(1-\frac{t}{\tau})^{\alpha}\\}\right]^{-1}$,
or

$1-\gamma(t)=(1-\gamma_{0})\Big{(}1-\frac{t}{\tau}\Big{)}^{\alpha},\hskip
28.45274pt{\alpha>0}$ (5)

which states that the system is initially at a temperature $T_{0}$ where
$\gamma_{0}=\textrm{tanh}(2/T_{0})$ and then cooled to zero temperature in a
finite time $\tau$. Note that the above finite-time quench protocol reduces to
instantaneous quench problem when the parameter $\alpha\to\infty$.

###  II.2 Dynamics of spin-spin correlation function

In the following, we study the equal time spin-spin correlation function

$G_{k}(t)=\langle\sigma_{i}(t)\sigma_{i+k}(t)\rangle$ (6)

where the angular brackets denote the average with respect to the distribution
$p(\\{\sigma_{i}\\},t)$. One can write down the time-evolution equation for
$G_{k}(t)$ by multiplying both sides of (2) with $\sigma_{i}\sigma_{i+k}$ and
summing over all the possible configurations [32]. For time-dependent
$\gamma$, we then obtain [6, 9, 12, 18]

$\frac{dG_{k}}{dt}=-2G_{k}+\gamma(t)(G_{k-1}+G_{k+1}),\hskip
28.45274ptk=1,...,L-1$ (7)

with the boundary conditions, $G_{0}(t)=G_{L}(t)=1$ and the initial condition,
$G_{k}(0)$. The mean domain wall density is related to the correlation
function as

$D(t)=\frac{1-G_{1}(t)}{2}$ (8)

We first briefly summarize the known results that are pertinent to the
discussion in the following subsections. When an infinitely large system is
instantaneously cooled (or heated) to a time-independent temperature $T$, the
general solution for the two-point correlation function is given by (63) of
[32]:

$G_{k}(t)=G_{k,eq}+e^{-2t}\sum_{l=1}^{\infty}(G_{l}(0)-G_{l,eq})[I_{k-l}(2\gamma
t)-I_{k+l}(2\gamma t)]$ (9)

where $\gamma(T)$ is given by (4) for a constant temperature $T$ and
$I_{\nu}(z)$ is the modified Bessel function of the first kind. At
$t\to\infty$, the system reaches the equilibrium state at temperature $T$
where the correlation function is given by (54) and (56) of [32]:

$G_{k,eq}=\Bigg{(}\frac{1-\sqrt{1-\gamma^{2}}}{\gamma}\Bigg{)}^{k}$ (10)

from which we obtain the equilibrium defect density at a low temperature $T\ll
1$ to be

$D_{eq}(T)\approx\sqrt{\frac{1-\gamma}{2}}\leavevmode\nobreak\
,\leavevmode\nobreak\ \gamma\to 1$ (11)

As shown in Appendix A, if the system is instantaneously quenched from a high
temperature to a low temperature $T$, the defect density at large times decays
as [32]

$D(t)\approx\frac{1}{2\sqrt{\pi
t}}\left[1+2(1-\gamma)t\right]\leavevmode\nobreak\ ,\leavevmode\nobreak\ 1\ll
t\ll(1-\gamma)^{-1}$ (12)

Likewise, if the system is initially in the equilibrium state at zero
temperature and is instantaneously heated to $0<T\ll 1$, the defects in the
system increase as $\sqrt{t}$ (see Appendix A):

$D(t)\approx 2(1-\gamma)\sqrt{\frac{t}{\pi}}\leavevmode\nobreak\
,\leavevmode\nobreak\ 1\ll t\ll(1-\gamma)^{-1}$ (13)

On the other hand, if an infinitely large system is cooled to zero temperature
in a finite time $\tau$ using an arbitrary protocol $\gamma(t)$, an exact
expression for $G_{k}(t)$ can be written as [9, 18]

$\displaystyle G_{k,eq}(t)$ $\displaystyle-$ $\displaystyle
G_{k}(t)=\int_{0}^{t}dt^{\prime}\;e^{-2(t-t^{\prime})}\;\frac{d\gamma(t^{\prime})}{dt^{\prime}}\int_{0}^{\pi}\frac{dq}{\pi}\frac{\textrm{sin}(kq)\;\textrm{sin}(q)}{(1-\gamma(t^{\prime})\;\textrm{cos}(q))^{2}}\;e^{2\;\textrm{cos}(q)\int_{t^{\prime}}^{t}dy\;\gamma(y)}$
(14) $\displaystyle+$
$\displaystyle\frac{2}{\pi}\int_{0}^{\pi}dq\;\sin({kq})\;e^{-2\int_{0}^{t}dy(1-\gamma(y)\cos({q}))}\sum_{m=1}^{\infty}\sin({mq})\;(G_{m,eq}(0)-G_{m}(0))$

Using the above result, the dynamics of the spin-spin correlation function in
the KZ phase have been studied in detail for the cooling protocol (5) starting
from an infinite temperature ($\gamma_{0}=0$), and it is shown that for large
$\tau$, the mean defect density at the end of the quench is given exactly by
[18]

$D(\tau)=\frac{1}{2\sqrt{\pi}}\Big{(}\frac{2}{1+\alpha}\Big{)}^{\frac{1}{2(1+\alpha)}}\Gamma\Big{(}\frac{1+2\alpha}{2+2\alpha}\Big{)}\times\frac{1}{\tau^{\frac{\alpha}{2(1+\alpha)}}}$
(15)

which, for a given $\tau$, decreases monotonically with exponent $\alpha$. For
$\alpha\to 0$, the system stays close to the initial high temperature until
$\tau$ and therefore carries almost all the initial defects till the end of
the quench, while for $\alpha\to\infty$, the number of defects decrease via
coarsening and (15) coincides with (12) when $\gamma=1$.

The $\tau$-scaling in (15) can be understood using the Kibble-Zurek argument
[3, 5]: below a time scale $\hat{t}$, the system can relax to the equilibrium
state at the instantaneous temperature (adiabatic phase: $t\ll{\hat{t}}$) and
therefore, the relevant time scale in this regime is the equilibrium
relaxation time, $\xi^{z_{eq}}_{eq}(t)\sim
e^{\frac{2z_{eq}}{T(t)}}\sim(1-\gamma(t))^{-\frac{z_{eq}}{2}}\ll{\hat{t}}$.
But above $\hat{t}$ where the system can not relax due to diverging
correlation length (KZ phase: ${\hat{t}}\ll t<\tau$), the only time scale is
the time remaining until the quench ends, viz., $\tau-{\hat{t}}$. For the
cooling protocol (5), these two time scales are comparable when

$\tau-\hat{t}\sim(1-\gamma_{0})^{\frac{-z_{eq}}{2+\alpha
z_{eq}}}\;\tau^{\frac{\alpha z_{eq}}{2+\alpha z_{eq}}}$ (16)

Assuming that the dynamics during ${\hat{t}}<t<\tau$ can be neglected, the
mean defect density $D(\tau)\sim D({\hat{t}})\sim\xi^{-1}_{eq}({\hat{t}})$ and
yields

$D(\tau)\sim(1-\gamma_{0})^{\frac{1}{2+\alpha z_{eq}}}\;\tau^{-\beta}$ (17)

where the KZ exponent

$\beta=\frac{\alpha}{2+\alpha z_{eq}}$ (18)

which matches the $\tau$-dependence in (15) on using that the stationary state
dynamic exponent $z_{eq}=2$ [32, 33] for the Glauber chain.

The expression for the spin-spin correlation function given in (14) is valid
for infinitely large system and arbitrary initial temperature $T_{0}$, and has
been analyzed for large $\tau$ and high $T_{0}$ [18]. Here we are interested
in the scenario when $T_{0}\ll 1$. But as the double integrals appearing in
(14) are quite involved, in Appendix B, we develop a scaling theory for low
initial temperature or large initial correlation length. For infinitely long
chain, we find that for $\tau\to\infty,\gamma_{0}\to 1$ with finite
$\tau(1-\gamma_{0})$, the spin-spin correlation function is given by

$\displaystyle G(y,x)$ $\displaystyle=$
$\displaystyle\frac{2}{\pi}\int_{0}^{\infty}dq\;\sin(qy)\;q\;e^{-q^{2}x}\int_{0}^{x}dw\;e^{q^{2}w-\frac{2\lambda_{0}}{\alpha+1}\big{(}(1-w)^{\alpha+1}-(1-x)^{\alpha+1}\big{)}}$
(19) $\displaystyle+$
$\displaystyle\sqrt{\frac{2}{\pi}}\int_{0}^{\infty}dq\;\sin(qy)\;\tilde{G}(q,0)\;e^{-q^{2}x+\frac{2\lambda_{0}}{\alpha+1}\big{(}(1-x)^{\alpha+1}-1\big{)}}$

where

$y=\frac{k}{\sqrt{\tau}},x=\frac{t}{\tau},\lambda_{0}=\tau(1-\gamma_{0})$ (20)

and $\tilde{G}(q,0)$ is the sine transform of the initial condition $G(y,0)$.
Furthermore, the effect of finite system size is discussed using a scaling
argument. We also compare our analytical results with the numerical solution
of the exact equation (7).

####  II.2.1 Equilibrium initial condition

We first consider the situation when the system of size $L$ initially in the
equilibrium state at a low temperature $T_{0}$ is slowly quenched to zero
temperature using the cooling protocol (5). As the initial correlation length
$\xi_{0}\equiv\xi_{eq}(T_{0})$ is large, a perturbation in the equilibrium
state due to changing temperature will take time
$\sim\xi_{0}^{z_{eq}}=\xi_{0}^{2}$ to relax (see Supplemental Material Fig.
1(a)). Therefore if the quench time $\tau\ll\xi_{0}^{2}$ (regime I), as shown
in the inset of Fig. 1(a), the defect density stays close to its equilibrium
value at $T_{0}$ or the excess defect density increases with time (see Fig.
1(a) for $\tau=100$); thus the system can not enter the adiabatic phase and
the defect density is not expected to follow the KZ scaling (17) at late
times. For larger quench times (regime II), as shown in Fig. 1(a) for
$\tau>100$, the excess defect density initially increases as $D(t)\approx
D_{eq}(T_{0})$ for $t\ll\xi_{0}^{2}$, but for $\xi_{0}^{2}\ll t\ll{\hat{t}}$,
the system relaxes to the instantaneous temperature and the excess defect
density remains constant; this adiabatic phase is followed by the KZ phase
where the system can not keep up with the changing temperature due to
diverging correlation length and the excess defect density increases for
${\hat{t}}\ll t<\tau$. If the quench time is long enough that the finite-sized
system can relax (regime III), that is, $\tau-{\hat{t}}\gtrsim L^{z_{eq}}$,
the deviation of the defect density from its equilibrium value is essentially
zero, see Fig. 1(a) for $\tau=4\times 10^{6}$.

(a)

(b)

Figure 1: Glauber Ising chain when the system is initially equilibrated to a
low temperature $T_{0}$ and then slowly quenched to zero temperature: (a) The
inset and main figure, respectively, show the dynamics of defect density and
excess defect density for various quench times, and are obtained by
numerically solving the exact equation (7) (dots) which are compared for
representative values of $\tau$ with (19) from low temperature theory (black
solid lines). (b) The figure shows the defect density at the end of the quench
as a function of $\tau$ in three different regimes (represented by different
colors) for a fixed $T_{0}$ and $L$ where, the line depicts the analytical
solution (21). In these figures, the system size $L=2000$, $T_{0}=0.5$ and
corresponding $\xi_{0}\approx 27.3$, and $\alpha=3$ in the cooling protocol
(5).

Below we describe the dynamics of defect density quantitatively using the low
temperature theory discussed in Appendix B; Fig. 1(a) shows a comparison of
the defect density obtained by numerically solving the exact equation (7) and
the solution (19) from the low temperature theory for representative values of
$\tau$, and we find a good agreement. In view of the dynamical phases
discussed above, the mean defect density at the end of the quench falls in
three distinct regimes that are shown in Fig. 1(b). For an infinitely large
system, using (C.13) and (C.19), we find that in regime I and II, the exact
expression for $D(\tau)$ is given by

$\displaystyle D(\tau)$ $\displaystyle=$
$\displaystyle\frac{1}{2\sqrt{\pi\tau}}\left[\left(\frac{2\lambda_{0}}{\alpha+1}\right)^{\frac{1}{2\alpha+2}}\Gamma\left(\frac{2\alpha+1}{2\alpha+2}\right)+\frac{1}{2(\alpha+1)}E_{\frac{3+2\alpha}{2\alpha+2}}\left(\frac{2\lambda_{0}}{\alpha+1}\right)\right]$
(21) $\displaystyle-$
$\displaystyle\frac{e^{-\frac{2\lambda_{0}}{\alpha+1}}}{2\sqrt{\pi\tau}}\left[1-e^{2\lambda_{0}}\sqrt{2\pi\lambda_{0}}\;\text{erfc}\left(\sqrt{2\lambda_{0}}\right)\right]$

where $\lambda_{0}=\tau(1-\gamma_{0})\sim\frac{\tau}{\xi_{0}^{2}}$ and
$E_{n}(z)$ is the exponential integral function, and matches the numerical
results shown in Fig. 1(b). We now discuss these regimes in detail:

Regime I: for $\lambda_{0}\ll 1$ or $\tau\ll\xi_{0}^{2}$, as the system does
not get enough time to relax to the slowly changing temperature, it is always
in a nonequilibrium state. From (C.9a) and (C.18a), we find that at short
times

$D_{I}(t)\approx\sqrt{\frac{1-\gamma_{0}}{2}}\leavevmode\nobreak\
,\leavevmode\nobreak\ t\ll(1-\gamma_{0})^{-1}$ (22)

so that the defect density remains close to its initial value, viz., the
equilibrium defect density $D_{eq}(T_{0})$ given by (11). If the quench time
is not too small, the defect density evolves and decreases with time (see the
inset of Fig. 1(a)). At the end of the quench, from (21), we find that the
defect density is given by

$D_{I}(\tau)=\sqrt{\frac{1-\gamma_{0}}{2}}\left(1-\sqrt{\frac{2\lambda_{0}}{\pi}}\frac{4\alpha}{2\alpha+1}\right)\leavevmode\nobreak\
,\leavevmode\nobreak\ \lambda_{0}\ll 1$ (23)

The first factor on the RHS of the above equation is simply the equilibrium
defect density as the system stays close to the initial state due to diverging
correlation length and small quench time, and the second factor which depends
on the details of the cooling protocol captures the reduction in the defect
density from $D_{eq}(T_{0})$ due to changing temperature.

Regime II: For $\xi_{0}^{2}\ll\tau\ll L^{1/\beta}$, from (C.9b) and (C.18b),
we find that

$\displaystyle D_{II}(t)\approx$
$\displaystyle\sqrt{\frac{1-\gamma_{0}}{2}}\leavevmode\nobreak\ $ ,
$0<t\ll(1-\gamma_{0})^{-1}$ (24a) $\displaystyle D_{II}(t)\approx$
$\displaystyle\sqrt{\frac{(1-\gamma_{0})(1-\frac{t}{\tau})^{\alpha}}{2}}\leavevmode\nobreak\
$ , $(1-\gamma_{0})^{-1}\ll t\ll\frac{\tau}{2}$ (24b)

which, on comparing with (11), show that the defect density is close to its
initial value at very short times and then enters the adiabatic phase. At
later times ($t\gg{\hat{t}}$) where the system is in the KZ phase, the
dynamics are described by (34) of [18] for $\gamma_{0}=0$ and we do not
discuss them here. But at the end of the quench, from (21) [or, alternatively,
adapting the analyses of [18] to nonzero $\gamma_{0}$], we obtain

$D_{\textrm{II}}(\tau)=(1-\gamma_{0})^{\frac{1}{2(1+\alpha)}}D_{\textrm{II}}(\tau,\gamma_{0}=0)\leavevmode\nobreak\
,\leavevmode\nobreak\ \lambda_{0}\gg 1$ (25)

where $D_{\textrm{II}}(\tau,\gamma_{0}=0)$ is given by (15). As expected, the
defect density at the end of the quench is smaller when the system is
initially equilibrated to low temperatures than when one starts with high
temperatures.

Regime III: For $\tau\gg\xi_{0}^{2}$, in an infinitely large system, the
defect density at the end of the quench is inversely proportional to the
correlation length, $\xi({\hat{t}})$. But in a finite system, we expect that

$D_{\textrm{III}}(\tau)=\frac{1}{\xi({\hat{t}})}\;{\tilde{F}}\Big{(}\frac{\xi({\hat{t}})}{L}\Big{)}=\frac{1}{\tau^{\beta}}\;F\Big{(}\frac{\tau}{L^{1/\beta}}\Big{)}$
(26)

where the scaling function $F(w)$ is a constant for $\tau\ll L^{1/\beta}$
(regime II) and decays rapidly for $\tau\gg L^{1/\beta}$ towards the
equilibrium value (viz., zero) [see Supplemental Material Fig. 1(b)]. Naïvely,
one may expect that the quench time over which the finite system reaches the
equilibrium state scales as $L^{z_{eq}}=L^{2}$ but, as stated above, the
system relaxes to equilibrium if the quench time $\tau\sim L^{1/\beta}$. Thus
the quench time in which the system reaches the equilibrium state is non-
universal, and the scaling exponent $\frac{1}{\beta}=2+\frac{2}{\alpha}$ for
finite-time cooling is larger than that for instantaneous quench. Note,
however, that (26) assumes that $\xi_{0}\ll L$ but, if the initial correlation
length is as large as the system size, the system never reaches the adiabatic
phase (see Supplemental Material Fig. 2(a)) and the finite system relaxes to
equilibrium when $\tau\sim L^{2}$.

####  II.2.2 Nonequilibrium initial condition: I

We now consider a situation where the finite-sized system is initially not in
equilibrium at a low temperature $T_{0}$. Specifically, we assume that the
system is in the equilibrium state at a high temperature $T_{i}\gg 1$ and then
instantaneously cooled to a low temperature $T_{0}\ll 1$; starting from the
resulting nonequilibrium state, the system is slowly cooled from $T_{0}$ to
zero using the cooling protocol (5).

(a)

(b)

Figure 2: Glauber Ising chain when the system is first instantaneously cooled
from a high temperature $T_{i}$ to a low temperature $T_{0}$, and then slowly
quenched to zero temperature: (a) The inset and main figure, respectively,
show the dynamics of defect density and excess defect density for various
quench times, and are obtained by numerically solving the exact equation (7);
in the inset, the green points show the comparison with (19) from low
temperature theory. In these plots, the black line corresponds to the exact
solution (9) when the system is instantaneously cooled to $T_{0}$ and the red
dashed lines represent the excess defects obtained by numerically solving (7)
when the system is initially equilibrated to $T_{0}$. (b) Figure shows the
defect density at the end of the quench as a function of $\tau$ in three
different regimes (represented by different colors) for a fixed $T_{0}$ and
$L$ where, the line depicts the analytical solution (27). In these figures,
the parameters are $L=2000$, $T_{0}=0.5$ and corresponding $\xi_{0}\approx
27.3$, and $\alpha=3$ in the cooling protocol (5).

For $t\ll\tau$ where the variation in temperature can be neglected, the system
behaves as if it is instantaneously quenched from $T_{i}$ to $T_{0}$ and
undergoes coarsening dynamics; as Fig. 2(a) shows, the dynamics of defect
density under finite-time quench match those following a rapid quench from
$T_{i}$ to $T_{0}$ until a time $t_{0}\sim\xi_{0}^{z_{co}}=\xi_{0}^{2}$ as the
system reaches the equilibrium state at $T_{0}$ on this time scale. If the
quench time is small ($\tau\ll\xi_{0}^{z_{co}}$) so that the system can not
equilibrate to $T_{0}$, it stays in the coarsening phase until the end of the
quench (see the inset of Fig. 2(a)) and the KZ scaling (17) is not expected to
hold. But if the quench time is long enough that the system can equilibrate to
$T_{0}$, the dynamics are the same as discussed in the last subsection; in
fact, as shown in Fig. 2(a) for $\tau>100$, the finite-time quench curve for
nonequilibrium initial condition now matches the finite-time quench dynamics
when the system is initially equilibrated to $T_{0}$ where, as discussed in
Sec. II.2.1, the dynamics are in adiabatic phase ($t_{0}\ll t\ll{\hat{t}}$)
and KZ phase (${\hat{t}}\ll t<\tau$).

For an infinitely large system, from (C.13) and (C.20), we find that the mean
defect density at the end of quench is given exactly by

$\displaystyle D(\tau)$
$\displaystyle=\frac{1}{2\sqrt{\pi\tau}}\left[\left(\frac{2\lambda_{0}}{\alpha+1}\right)^{\frac{1}{2\alpha+2}}\Gamma\left(\frac{2\alpha+1}{2\alpha+2}\right)+\frac{1}{2(1+\alpha)}E_{\frac{3+2\alpha}{2\alpha+2}}\left(\frac{2\lambda_{0}}{\alpha+1}\right)\right]$
(27)

and matches the numerical results shown in Fig. 2(b). The dynamics of the
defect density are quantitatively described below:

Regime I: for $\tau\ll\xi_{0}^{2}$, the system is always in a nonequilibrium
state and from (C.9a) and (C.20), we find that at short times

$D_{I}(t)\approx\frac{1}{2\sqrt{\pi
t}}\left[1+2(1-\gamma_{0})t\right]\leavevmode\nobreak\ ,\leavevmode\nobreak\
t\ll(1-\gamma_{0})^{-1}$ (28)

which matches the result (12) for rapid quench to $T_{0}$. However, for
$t\lesssim\tau$ where the effect of changing temperature can not be neglected,
the defect density curve starts diverging from the instantaneous cooling curve
(see the inset of Fig. 2(a)). As a result, the defect density at the end of
the quench calculated from (27) is given by

$D(\tau)=\frac{1}{2\sqrt{\pi\tau}}\left(1+\frac{2\lambda_{0}}{2\alpha^{2}+3\alpha+1}\right)\leavevmode\nobreak\
,\leavevmode\nobreak\ \lambda_{0}\ll 1$ (29)

As explained for (15), the above expression shows that the defect density at
the end of the finite-time quench is larger than that for instantaneous quench
to zero temperature. But, as the inset of Fig. 2(a) and a comparison between
(12) and (29) show, it is smaller than when the system is instantaneously
quenched to $T_{0}$.

Regime II: for $\xi_{0}^{2}\ll\tau\ll L^{1/\beta}$, (C.9b) and (C.20) show
that

$\displaystyle D_{II}(t)\approx$ $\displaystyle\frac{1}{2\sqrt{\pi
t}}\left[1+2(1-\gamma_{0})t\right]\leavevmode\nobreak\ $ , $0\ll
t\ll(1-\gamma_{0})^{-1}$ (30a) $\displaystyle D_{II}(t)\approx$
$\displaystyle\sqrt{\frac{(1-\gamma_{0})(1-\frac{t}{\tau})^{\alpha}}{2}}\leavevmode\nobreak\
$ , $(1-\gamma_{0})^{-1}\ll t\ll\frac{\tau}{2}$ (30b)

and the defect density at the end of the quench is given by (25).

Regime III: For $\xi_{0}\ll L,\tau\gg L^{1/\beta}$, the defect density has the
same behavior as in regime III of Sec. II.2.1. But if $\xi_{0}\sim L$, the
defect density at the end of quench does not follow KZ scaling for any quench
time as the system can not enter the adiabatic phase and instead, it decays
according to the coarsening law until the system equilibrates (see
Supplemental Material Fig. 2(b)).

####  II.2.3 Nonequilibrium initial condition: II

(a)

(b)

Figure 3: Glauber Ising chain when the system in the equilibrium state at zero
temperature is instantaneously heated to a low temperature $T_{0}$ and then
slowly cooled to zero temperature: (a) The figure shows the dynamics of the
defect density for various quench times obtained by numerically solving the
exact equation (7), and the blue dots for $\tau=500$ show the comparison with
(19) from low temperature theory. As the quench time increases, the system is
able to reach the equilibrium state at $T_{0}$ as shown by the red dashed line
while the black line corresponds to the exact solution (9) when the system is
instantaneously heated to $T_{0}$. (b) The figure shows the defect density at
the end of the quench as a function of $\tau$ in three different regimes
(represented by different colors) for a fixed $T_{0}$ and $L$ where, the line
depicts the analytical solution (31). In these figures, the parameters are
$L=2000$, $T_{0}=0.5$ and corresponding $\xi_{0}\approx 27.3$, and $\alpha=3$
in the cooling protocol (5).

We now consider a situation where a finite-sized system in the critical state
(that is, zero temperature) is instantaneously heated to a low temperature
$T_{0}$ and then slowly cooled to zero temperature using the cooling protocol
(5). At short times $t\ll\tau$ where the effect of changing temperature can be
neglected, as in Sec. II.2.2, the system behaves as if it is instantaneously
heated from zero temperature to a finite temperature $T_{0}$; this is verified
in Fig. 3(a) where the dynamics of defect density under finite-time quench
match with those following a rapid heating from $T_{i}=0$ to $T_{0}$ till a
time $t_{0}\sim\xi_{0}^{2}$ as the system reaches the equilibrium state at
$T_{0}$ on this time scale. As in Sec. II.2.2, now depending on whether $\tau$
is smaller or larger than $\xi_{0}^{2}$, the defect density at the end of
quench shows different scalings.

From (C.13) and (C.25), we find that for this protocol, the mean defect
density at the end of quench is given exactly by

$\displaystyle D(\tau)$
$\displaystyle=\frac{1}{2\sqrt{\pi\tau}}\left[\left(\frac{2\lambda_{0}}{\alpha+1}\right)^{\frac{1}{2\alpha+2}}\Gamma\left(\frac{2\alpha+1}{2\alpha+2}\right)+\frac{1}{2(1+\alpha)}E_{\frac{3+2\alpha}{2\alpha+2}}\left(\frac{2\lambda_{0}}{\alpha+1}\right)-e^{-\frac{2\lambda_{0}}{\alpha+1}}\right]$
(31)

and matches the numerical results shown in Fig. 3(b).

Regime I: for $\tau\ll\xi_{0}^{2}$, from (C.9a) and (C.24a), we find that at
short times, the defect density increases as

$D_{I}(t)\approx\frac{2(1-\gamma_{0})\sqrt{t}}{\sqrt{\pi}}\leavevmode\nobreak\
,\leavevmode\nobreak\ t\ll\frac{\tau}{2}$ (32)

which matches the result (13) for instantaneous heating to temperature
$T_{0}$. For $t\lesssim\tau$, the defect density decreases so that a peak in
$D(t)$ occurs at a time that scales linearly with quench time. Furthermore,
from (31), we find that the defect density at the end of the quench is given
by

$D(\tau)=\frac{1}{\sqrt{\pi\tau}}\left(\frac{2\lambda_{0}}{2\alpha+1}\right)\leavevmode\nobreak\
,\leavevmode\nobreak\ \lambda_{0}\ll 1$ (33)

which, as expected, approaches zero as $\alpha\to\infty$.

Regime II: for $\tau\gg\xi_{0}^{2}$, the defect density initially increases
until it reaches $D_{eq}(T_{0})$ followed by the adiabatic phase so that from
(C.9b) and (C.24b), we have

$\displaystyle D_{II}(t)\approx$
$\displaystyle\frac{2(1-\gamma_{0})\sqrt{t}}{\sqrt{\pi}}\leavevmode\nobreak\ $
, $0\ll t\ll(1-\gamma_{0})^{-1}$ (34a) $\displaystyle D_{II}(t)\approx$
$\displaystyle\sqrt{\frac{(1-\gamma_{0})(1-\frac{t}{\tau})^{\alpha}}{2}}\leavevmode\nobreak\
$ , $(1-\gamma_{0})^{-1}\ll t\ll\frac{\tau}{2}$ (34b)

which is followed by the KZ phase where the defect density at the end of
quench is given by (25).

Regime III: For $\tau\gg L^{1/\beta}$, we obtain the same behavior as in
regime III described in Sec. II.2.1 provided $\xi_{0}\ll L$ otherwise the
regime II is absent and the defect density increases as $\sqrt{\tau}$ until
the finite system equilibrates (see Supplemental Material Fig. 2(c)).

###  II.3 Dynamics of auto-correlation function

In the last section, we discussed the equal time spin-spin correlation
function, and here we briefly consider the unequal time spin-spin correlation
function

$C_{n}(t,t_{w})=\langle\sigma_{i}(t_{w})\sigma_{i+n}(t)\rangle$ (35)

where $t_{w}\leq t$ is the waiting time. Using the conditional probability
$p(\sigma,t|\sigma^{\prime},t_{w})$ of finding an infinitely large system in
the state $\sigma$ at time $t$, given that it was in state $\sigma^{\prime}$
at time $t_{w}<t$, as for the equal time correlation function, we can write
the differential equation for the unequal time spin-spin correlation function
as

$\frac{\partial}{\partial
t}C_{n}(t,t_{w})=-C_{n}(t,t_{w})+\frac{\gamma(t)}{2}[C_{n-1}(t,t_{w})+C_{n+1}(t,t_{w})]$
(36)

where $-\infty<n<\infty$ with the boundary conditions $C_{-\infty}(t)=0$ and
$C_{\infty}(t)=0$ and the intial condition $C_{n}(t_{w},t_{w})=G_{n}(t_{w})$.

Figure 4: Glauber Ising chain when the system is quenched slowly from a high
temperature to zero temperature: The main figure show the auto-correlation
function between quench time $\tau$ and waiting time $t_{w}$, and the inset
figure shows the data collapse according to KZ scaling ansatz (37). The
parameters are $L=2000$ and $\alpha=3$ in the cooling protocol (5) with
$\gamma_{0}=0$.

When the system is instantaneously quenched from a high to a low temperature,
the auto-correlation function, $C_{0}(t,t_{w})$ decays as
$\sqrt{\frac{t_{w}}{t-t_{w}}},t\gg t_{w}$ [34, 35]. But if the system is
quenched at a finite rate, one expects that for $t_{w}<\hat{t}$, since the
system is far from the critical point, a spin at the end of the quench is
uncorrelated to its value at $t_{w}$ while for $t_{w}>\hat{t}$, the auto-
correlation function is expected to increase to one as the system is close to
the critical point where the correlation length is large. The auto-correlation
function, $C_{0}(\tau,t_{w})$ in Fig. 4 is indeed in agreement with these
expectations, and shows that unlike the defect density which is almost frozen
in the KZ phase (see, for example, inset of Fig. 3 in [18]), the auto-
correlation function undergoes a large change over the same range of time.
Furthermore, the inset of Fig. 4 shows that the data for different quench and
waiting times can be collapsed onto a single curve if we assume the following
scaling form

$C_{0}(\tau,t_{w})={\cal C}\left(Z=\frac{\tau-
t_{w}}{\tau-\hat{t}}\right)={\cal C}\left(Z=\frac{\tau-
t_{w}}{\tau^{\frac{\alpha z_{eq}}{2+\alpha z_{eq}}}}\right)$ (37)

which is in accordance with the KZ scaling (16).

##  III Kawasaki Ising chain

###  III.1 Model

In the last section, we have seen that both coarsening and stationary state
dynamics play an important role in the finite-time quench dynamics. However,
as the dynamic exponents for coarsening and stationary state dynamics are
identical for the Glauber chain, below we consider the Kawasaki chain, for
which these exponents are different, to understand how these affect the
finite-time quench dynamics.

Under Kawasaki dynamics [36], the neighboring anti-parallel spins exchange so
that the magnetization remains strictly conserved. For time-dependent
temperature, the master equation for the evolution of spin configurations can
be written as

$\displaystyle\frac{d}{dt}p(\sigma_{1},\ldots,\sigma_{i},\sigma_{i+1},\ldots,\sigma_{L},t)$
$\displaystyle=\sum_{i=1}^{L}\Big{[}w(\sigma_{i+1}\leftrightarrow\sigma_{i},t)\;p(\sigma_{1},\ldots,\sigma_{i+1},\sigma_{i},\ldots,\sigma_{L},t)$
(38)
$\displaystyle-w(\sigma_{i}\leftrightarrow\sigma_{i+1},t)\;p(\sigma_{1},\ldots,\sigma_{i},\sigma_{i+1},\ldots,\sigma_{L},t)\Big{]}$

where the transition probability for the $i$th and $(i+1)$th sites to exchange
their spins is given by [36]

$w(\sigma_{i}\leftrightarrow\sigma_{i+1},t)=\left(1-\frac{\gamma(t)}{2}(\sigma_{i-1}\sigma_{i}+\sigma_{i+1}\sigma_{i+2})\right)\times\frac{1}{2}(1-\sigma_{i}\sigma_{i+1})$
(39)

and, as in Sec. II.1, $\gamma(t)=\tanh{(2/T(t))}$ and its time-dependence is
described by (5). Thus in Kawasaki dynamics, the allowed moves are

$\displaystyle\uparrow\cdot\downarrow\cdot\uparrow\cdot\downarrow\hskip
14.22636pt$ $\displaystyle\stackrel{{\scriptstyle
1+\gamma}}{{\longrightarrow}}$ $\displaystyle\hskip
14.22636pt\uparrow\uparrow\cdot\downarrow\downarrow$ (40a)
$\displaystyle\uparrow\uparrow\cdot\downarrow\downarrow\hskip 14.22636pt$
$\displaystyle\stackrel{{\scriptstyle 1-\gamma}}{{\longrightarrow}}$
$\displaystyle\hskip
14.22636pt\uparrow\cdot\downarrow\cdot\uparrow\cdot\downarrow\hskip
28.45274pt\hskip 28.45274pt$ (40b)
$\displaystyle\uparrow\cdot\downarrow\cdot\uparrow\uparrow\hskip 14.22636pt$
$\displaystyle\stackrel{{\scriptstyle 1}}{{\longrightarrow}}$
$\displaystyle\hskip
14.22636pt\uparrow\uparrow\cdot\downarrow\cdot\downarrow\hskip
28.45274pt\hskip 28.45274pt$ (40c)

where the dot represents the domain wall. While the domain walls decrease and
increase, respectively, via the moves (40a) and (40b), the defect density
remains unchanged due to the diffusion move in (40c).

Before considering the finite-time quenches, we discuss the situation when the
system is instantaneously quenched to zero temperature; due to conserved
magnetization, at zero temperature, the system always gets stuck in a
metastable state which consists of domains of length two or more. Then, from
(40b), only the energy-raising transition is possible but that is not allowed
at zero temperature (as the rate $1-\gamma=0$). Hence, the Kawasaki chain
never reaches the equilibrium state of zero temperature. If now one quenches
the system to a temperature slightly above zero, energy-raising events are
allowed which can lead to domain growth (and hence equilibrium state) via
diffusion and annihilation moves, but as the domain wall creation rate
$1-\gamma\sim e^{-4/T}$ is very small at low temperatures, one can define a
new time scale $t^{\prime}=te^{-4/T}$ so that the move (40b) takes a finite
time but other processes occur instantaneously. Using these accelerated
dynamics [37, 38], it has been shown numerically and analytically that the
domain length grows as $t^{1/{z_{co}}}$ where the coarsening exponent
$z_{co}=3$. In contrast, in the stationary state, the relaxation time grows as
$\sim\xi_{eq}^{z_{eq}}$ where $z_{eq}=5$ [39]. Thus as a consequence of the
conservation, the Kawasaki dynamics are slower than the Glauber dynamics where
both these exponents are equal to two.

###  III.2 Dynamics of spin-spin correlation function

Using the master equation (38), we find that the evolution equation for the
$n$-point correlation function,
$\langle\sigma_{i_{1}}...\sigma_{i_{n}}\rangle$ is not closed as it depends on
the $(n+2)$-point correlation functions resulting in an infinite hierarchy of
equations [8] and it does not seem possible to obtain analytical expressions
for the defect density. Therefore, the results in the following subsections
are obtained by simulating long Kawasaki chains in continuous time. In our
simulations, an anti-parallel spin pair at site $i$ and $i+1$ exchange their
value at time $t$ with probability
$\frac{w(\sigma_{i}\leftrightarrow\sigma_{i+1},t)}{\sum_{j}w(\sigma_{j}\leftrightarrow\sigma_{j+1},t)}$,
and the time $t+\delta t$ at which the next update occurs is found using that
the increment time $\delta t$ is approximately exponentially-distributed with
rate $\sum_{j}w(\sigma_{j}\leftrightarrow\sigma_{j+1},t)$. When there are
large number of defects in the system, spin exchange occurs frequently, but
when very few defects are left, the time between successive updates becomes
large and therefore, close to zero temperature, the finite-time quench
dynamics grind to a halt and it seems difficult to obtain accurate numerical
results. However, for zero magnetization, we have measured the mean defect
density in simulations by averaging over $5000$ independent runs which are
discussed below.

####  III.2.1 Equilibrium initial condition

We first consider the situation when the system initially equilibrated to a
high temperature ($\gamma_{0}=0$) is slowly quenched to zero temperature
according to (5). Using Monte Carlo simulations described above, we measured
the defect density as a function of time for various quench times, and find
that at short times, the excess defect density remains close to zero and then
increases as the system falls out of equilibrium at time $\sim{\hat{t}}$ (data
not shown). We expect that in the KZ phase (${\hat{t}}\ll t<\tau$), the excess
defect density scales as [18]

$D(t)-D_{eq}(t)=\frac{1}{\tau^{\beta}}\;{\cal
K}\left(Z=\frac{\tau-t}{\tau-\hat{t}}\right)=\frac{1}{\tau^{\frac{\alpha}{2+5\alpha}}}\;{\cal
K}\left(Z=\frac{\tau-t}{\tau^{\frac{5\alpha}{2+5\alpha}}}\right)$ (41)

on using that the remaining time and the exponent $\beta$, respectively, are
given by (16) and (18), and the exponent $z_{eq}=5$ for these dynamics.

(a)

(b)

Figure 5: Kawasaki Ising chain when the system initially equilibrated to a
high temperature is slowly quenched to zero temperature: (a) The figure shows
the defect density at the end of the quench for two different $\alpha$ values
in the cooling protocol (5). Note that the exponents obtained are the best
fits from numerical simulations which do not match exactly but are in close
agreement with the KZ exponents quoted in the legend. (b) The figure shows the
collapse of scaled excess defect density with the scaling variable $Z$
according to (41) for $\alpha=3$ in the cooling protocol (5). The system size
$L=2000$ in both figures.

Fig. 5(a) shows that the exponent $\beta$ obtained from our simulations do not
match exactly with (18) but their values are in fair agreement with the KZ
predictions. Recently, the slowly quenched Kawasaki chain was studied
numerically in [20] where, for $\alpha=3$, the KZ exponent was found to be
$\approx 0.163$ which is closer to the exact exponent, $\beta=3/17\approx
0.176$ as compared to our best fit $0.15$ in Fig. 5(a), perhaps because much
larger values of quench time ($\tau\gtrsim 10^{7}$) were used in [20]. In Fig.
5(b), the scaling ansatz (41) is tested and we find that a fairly good data
collapse in the KZ phase is obtained so long as the temperature is not too
close to zero; however, we also note that close to zero temperature ($Z\approx
0$), the data collapse improves with increasing quench times. These results
therefore support the KZ scaling following slow quench in the Kawasaki chain.

We also simulated the case where the system is initially equilibrated to a low
temperature $T_{0}$ and then slowly quenched to zero temperature via the
cooling protocol (5). However, we were not able to check the scalings reliably
due to the inability of the system to evolve at low temperatures as it gets
stuck in the metastable states. But, as in Glauber chain (see Fig. 1(b)), we
expect that the defect density at the end of the quench has the following
scaling form

$\displaystyle
D_{eq}(\tau)=\frac{1}{\xi_{0}}f_{eq}\left(\frac{\tau}{\xi_{0}^{z_{eq}}}\right)\propto$
$\displaystyle\xi_{0}^{-1}\leavevmode\nobreak\ $ , $\tau\ll\xi_{0}^{z_{eq}}$
(42a) $\displaystyle
D_{eq}(\tau)=\frac{1}{\xi_{0}}f_{eq}\left(\frac{\tau}{\xi_{0}^{z_{eq}}}\right)\propto$
$\displaystyle\xi_{0}^{\beta z_{eq}-1}\tau^{-\beta}\leavevmode\nobreak\ $ ,
$\tau\gg\xi_{0}^{z_{eq}}$ (42b)

with $z_{eq}=5$. The behavior of the scaling function is deduced from the fact
that at small quench times, the defect density remains essentially close to
its initial value but for larger quench times, the system can enter the
adiabatic phase leading to KZ scaling at the end of the quench.

####  III.2.2 Nonequilibrium initial condition

Figure 6: Kawasaki Ising chain when the system is first instantaneously cooled
to a low temperature $T_{0}$, and then slowly cooled to zero temperature: The
figure shows the dynamics of excess defect density for two initial
temperatures $T_{0}$ at a fixed value of $\tau=5\times 10^{5}$. The black
dashed line corresponds to the dynamics when the system is instantaneously
cooled to $T_{0}$ and the red dashed lines represent the excess defects when
the system is initially equilibrated to $T_{0}$. The parameters are $L=2000$
and $\alpha=3$ in the cooling protocol (5).

We now consider the situation when the system initially at a high temperature
$T_{i}$ is instantaneously cooled to a low temperature $T_{0}$ and then slowly
quenched to zero temperature using cooling protocol (5). For initial
temperature $T_{0}\ll 1$, as discussed in Sec. II.2.2 for Glauber chain, we
expect that the defect density will decrease via coarsening and the system
will reach the equilibrium state at temperature $T_{0}$ at time
$t_{0}\sim\xi_{0}^{z_{co}}$. However, as $T_{0}$ is small and the correlation
length $\xi(t_{0})\sim\xi_{0}$ is large, it will take time
$\sim\xi_{0}^{z_{eq}}$ for a perturbation due to changing temperature to relax
and we therefore expect that $D(t)\sim\xi_{0}^{-1}$ for $\xi_{0}^{z_{co}}\ll
t\ll\xi_{0}^{z_{eq}}$. For $t\gg\xi_{0}^{z_{eq}}$, the system can enter the
adiabatic phase followed by the KZ phase. We further conjecture that for an
infinitely large system, at the end of the quench

$\displaystyle D_{neq}(\tau)\propto$
$\displaystyle\tau^{-1/z_{co}}\leavevmode\nobreak\ $ ,
$\tau\ll\xi_{0}^{z_{co}}$ (43a) $\displaystyle D_{neq}(\tau)\propto$
$\displaystyle\xi_{0}^{-1}\leavevmode\nobreak\ $ ,
$\xi_{0}^{z_{co}}\ll\tau\ll\xi_{0}^{z_{eq}}$ (43b) $\displaystyle
D_{neq}(\tau)\propto$ $\displaystyle\xi_{0}^{\beta
z_{eq}-1}\tau^{-\beta}\leavevmode\nobreak\ $ , $\tau\gg\xi_{0}^{z_{eq}}$ (43c)

For the Glauber chain, as $z_{co}=z_{eq}$, the regime (43b) with constant
defect density is not observed (see Fig. 2(b)).

To test the above expectations, using Monte Carlo simulations, we measured the
defect density for two initial temperatures as shown in Fig. 6. At short times
$t\ll\tau$, the system behaves as if it is instantaneously quenched from a
high temperature $T_{i}$ to $T_{0}$ and the finite-time quench curve matches
the instantaneous quench dynamics. As the temperature is varying with time,
the two curves start diverging at larger times, but the defects keep
decreasing via coarsening. Note that unlike for the Glauber chain, here at a
fixed $t$, on quenching the system from higher temperature results in lower
number of defects because at higher $T_{0}$, the system does not get stuck in
the metastable states and the spin updates occur more frequently (at least at
short times) resulting in fewer defects. The excess defect density reaches a
minimum when the system is in the adiabatic phase and then increases in the KZ
phase where the finite-time quench curve matches the corresponding curve if
the system started in equilibrium state at $T_{0}$.

As discussed above, due to different dynamic exponents, we expect that the
excess defect density will remain approximately constant (and close to zero)
for $\xi_{0}^{z_{co}}\ll t\ll\xi_{0}^{z_{eq}}$; however, we do not observe
this phase in Fig. 6 which, we believe, is because the initial correlation
length $\xi_{0}\sim 10$ is quite small and the scaling regimes have not set
in. Also, for the same reason, we have not been able to verify the scalings
for the defect density at the end of the quench stated in (43c). To observe
these scalings and dynamical phases, we need to consider initial temperatures
lower than those considered in Fig. 6. But even for $T_{0}=0.5,\xi_{0}\sim 25$
(as considered in Glauber chain), the time $\delta t$ between successive
updates is $\sim(1-\gamma_{0})^{-1}\sim 600$ which gets longer as the
temperature approaches zero and therefore, we need a better algorithm to
capture the low temperature dynamics of the slowly quenched Kawasaki chain.

##  IV Discussion

The Kibble-Zurek argument is a powerful and general theory that predicts the
density of defects when a classical or quantum system that exhibits second
order phase transition is quenched from the disordered phase to critical
region or ordered phase [3, 5]. It assumes that if a system starts in an
adiabatic phase, it will reach the KZ phase where the defect density decays as
a power-law with the quench time. In previous studies on finite-time quench
dynamics in the Ising model, the system is assumed to be initially
equilibrated to a high temperature and then cooled to the critical point [9,
12, 18, 21, 19] or deep in the ordered phase [11, 14] at a finite rate, and
one focuses on the defect density at the end of the quench (see, however,
[18]). In contrast, here we studied the effect of initial conditions specified
by the initial state and initial temperature on the full dynamics till the end
of the quench; we also elucidated how the system size affects the defect
density.

We find that depending on the initial condition, besides the well known
adiabatic and KZ phase, other dynamical phases such as coarsening are also
possible; these are observed when the system starts in a nonequilibrium
initial state which, to our knowledge, have not been considered in previous
work. We formulated a low temperature theory for the Glauber Ising chain using
which we obtained exact expressions (21), (27), (31) for the defect density at
the end of quench for different initial conditions that are shown in Figs.
1(b), 2(b), 3(b).

As an application and extension of the scaling ideas developed for the Glauber
chain, we also studied the Kawasaki Ising chain to understand the significance
of different stationary state dynamic exponent and coarsening exponent. Since
the equations do not close for these dynamics, we performed Monte Carlo
simulations but these simulations are also very hard as the system gets stuck
in the metastable states at low temperatures. Therefore it remains to be seen
if the scalings conjectured in (43c) can be tested at low temperatures in the
Kawasaki model or in some other model where the two dynamic exponents are
quite different.

Acknowledgements: LJ would like to thank CSIR for fellowship.

##  Appendix A Rapid heating and cooling of Glauber chain

When an infinitely long Glauber chain is rapidly cooled or heated to a low
temperature $T$ and then evolved at constant temperature $T$, the exact
equation (7) can be written as

$\frac{\partial G}{\partial x}=\frac{\partial^{2}G}{\partial y^{2}}-2G$ (A.1)

where, $x=t(1-\gamma),y=k\sqrt{1-\gamma}$ and the boundary conditions are
$G(0,x)=1,G(\infty,x)=0$. The defect density (8) is then given by

$D(x)=\frac{1-G(\sqrt{1-\gamma},x)}{2}\stackrel{{\scriptstyle\gamma\to
1}}{{\to}}-\frac{\sqrt{1-\gamma}}{2}\frac{\partial G}{\partial
y}\bigg{|}_{y=0}$ (A.2)

On taking the sine transform defined as
$\tilde{G}(q,x)=\sqrt{\frac{2}{\pi}}\int_{0}^{\infty}dy\;\sin{(qy)}\;G(y,x)$
of (A.1), we obtain

$\displaystyle{\tilde{G}}(q,x)=q\sqrt{\frac{2}{\pi}}\int_{0}^{x}dwe^{-(2+q^{2})(x-w)}+{\tilde{G}}(q,0)e^{-(2+q^{2})x}$
(A.3)

where ${\tilde{G}}(q,0)$ is the sine transform of the initial condition
$G(y,0)$. The inverse sine transform then yields

$\displaystyle{G}(y,x)$ $\displaystyle=$
$\displaystyle{\frac{2}{\pi}}\int_{0}^{\infty}dq\sin(qy)\int_{0}^{x}dwqe^{-(2+q^{2})(x-w)}+\sqrt{\frac{2}{\pi}}\int_{0}^{\infty}dq\sin(qy){\tilde{G}}(q,0)e^{-(2+q^{2})x}$
(A.4) $\displaystyle=$
$\displaystyle{\frac{1}{2\sqrt{\pi}}}\int_{0}^{x}dw\frac{ye^{\frac{-y^{2}}{4(x-w)}-2(x-w)}}{(x-w)^{3/2}}+\sqrt{\frac{2}{\pi}}\int_{0}^{\infty}dq\sin(qy){\tilde{G}}(q,0)e^{-(2+q^{2})x}$
(A.5)

where we have interchanged the order of integration in the first term. The
above integrals correspond to $\alpha=0,\lambda_{0}=1$ of (C.1) and (C.2)
which are analyzed in Appendix C.

Alternatively, if we work with $H(y,x)=G_{eq}(y)-G(y,x)$ with homogeneous
boundary conditions, $H(0,x)=H(\infty,x)=0$, we find that $H$ also obeys (A.1)
so that ${\tilde{H}}(q,x)={\tilde{H}}(q,0)e^{-(2+q^{2})x}$. The defect density
can be written as

$\displaystyle D(x)$ $\displaystyle=$
$\displaystyle\frac{\sqrt{1-\gamma}}{2}\left(\sqrt{2}+\sqrt{\frac{2}{\pi}}\int_{0}^{\infty}dqq{\tilde{H}}(q,x)\right)$
(A.6)

For quench from high temperature to a low temperature $T$, using the initial
condition ${\tilde{H}}(q,0)=\sqrt{\frac{2}{\pi}}\frac{q}{2+q^{2}}$, we obtain

$\displaystyle D(x)$ $\displaystyle=$
$\displaystyle\sqrt{\frac{1-\gamma}{2}}+\sqrt{\frac{1-\gamma}{\pi^{2}}}\left(\frac{\sqrt{\pi}e^{-2x}}{2\sqrt{x}}-\frac{\pi\text{erfc}\left(\sqrt{2x}\right)}{\sqrt{2}}\right)$
(A.7) $\displaystyle\stackrel{{\scriptstyle x\to 0}}{{\approx}}$
$\displaystyle\sqrt{\frac{1-\gamma}{\pi}}\left(\frac{1}{2\sqrt{x}}+\sqrt{x}\right)$
(A.8) $\displaystyle=$ $\displaystyle\frac{1}{2\sqrt{\pi
t}}+(1-\gamma)\sqrt{\frac{t}{\pi}}\leavevmode\nobreak\ ,\leavevmode\nobreak\
t\ll(1-\gamma)^{-1}$ (A.9)

Similarly, on heating the system from zero temperature to a low temperature
$T$, as
${\tilde{H}}(q,0)=\sqrt{\frac{2}{\pi}}\frac{q}{2+q^{2}}-\sqrt{\frac{2}{\pi}}\frac{1}{q}$,
we obtain

$\displaystyle D(x)$ $\displaystyle=$
$\displaystyle\sqrt{\frac{1-\gamma}{2}}\text{erf}\left(\sqrt{2x}\right)$
(A.10) $\displaystyle\stackrel{{\scriptstyle x\to 0}}{{\approx}}$
$\displaystyle 2(1-\gamma)\sqrt{\frac{t}{\pi}}\leavevmode\nobreak\
,\leavevmode\nobreak\ t\ll(1-\gamma)^{-1}$ (A.11)

##  Appendix B Low temperature scaling theory for Glauber chain

To describe the finite-time quench dynamics when an infinitely long Glauber
chain is quenched from a low temperature $T_{0}$ to zero, we first rewrite the
exact equation (7) as

$\displaystyle\frac{dG_{k}}{dt}$ $\displaystyle=$
$\displaystyle\gamma_{0}(G_{k-1}+G_{k+1}-2G_{k})-2G_{k}(1-\gamma(t))$ (B.1)
$\displaystyle+$ $\displaystyle(\gamma(t)-\gamma_{0})(G_{k-1}+G_{k+1}-2G_{k})$

The first term on the RHS of the above equation states that the dynamics are
the same as when the system evolves at a time-independent, low temperature for
which $\gamma_{0}\to 1$ (see (7) on replacing $\gamma(t)$ by $\gamma_{0}$)
which is expected to be true for $t\ll\tau$ as the temporal variation of the
temperature can be neglected. To take the effect of changing temperature into
account, we consider the above equation in continuous space by writing
$k^{\prime}=ka$ where $a$ is the lattice spacing, and define
$x=\frac{t}{\tau}$ to obtain

$\displaystyle\frac{\partial G(k^{\prime},x)}{\partial x}$ $\displaystyle=$
$\displaystyle a^{2}\tau\gamma_{0}\frac{\partial^{2}G(k^{\prime},x)}{\partial
k^{\prime 2}}-2\tau(1-\gamma_{0})(1-x)^{\alpha}G(k^{\prime},x)$ (B.2)
$\displaystyle+$ $\displaystyle
a^{2}(1-\gamma_{0})\tau\\{1-(1-x)^{\alpha}\\}\frac{\partial^{2}G(k^{\prime},x)}{\partial
k^{\prime 2}}$

For quenches from low temperatures, as $\gamma_{0}\to 1$, we choose the
lattice spacing $a=\sqrt{1-\gamma_{0}}\sim\xi_{0}^{-1}$. Then in the scaling
limit $\tau\to\infty,\gamma_{0}\to 1$ such that
$\lambda_{0}=\tau(1-\gamma_{0})$ and
$y=\frac{k^{\prime}}{a\sqrt{\tau}}=\frac{k}{\sqrt{\tau}}$ are finite, as the
last term on the RHS of the above equation is of order
$1-\gamma_{0}\sim\xi_{0}^{-2}$, it can be neglected and we finally arrive at

$\frac{\partial G(y,x)}{\partial x}=\frac{\partial^{2}G(y,x)}{\partial
y^{2}}-2\lambda_{0}(1-x)^{\alpha}G(y,x)$ (B.3)

The above equation is subject to boundary conditions $G(0,x)=1$ and
$G(\infty,x)=0$ (as the correlations are expected to vanish at large
distances) and initial condition, $G(y,0)$. The exact solution for $G(y,x)$
can be obtained by taking the sine transform of (B.3) with respect to $y$
which yields the following first order differential equation

$\frac{\partial\tilde{G}(q,x)}{\partial
x}+\left(2\lambda_{0}(1-x)^{\alpha}+q^{2}\right)\tilde{G}(q,x)=\sqrt{\frac{2}{\pi}}q$
(B.4)

where the sine transform is defined as
$\tilde{G}(q,x)=\sqrt{\frac{2}{\pi}}\int_{0}^{\infty}dy\;\sin{(qy)}\;G(y,x)$.
Solving the above equation, we obtain

$\displaystyle\tilde{G}(q,x)$ $\displaystyle=$
$\displaystyle\tilde{G}(q,0)\;e^{-q^{2}x+\frac{2\lambda_{0}}{\alpha+1}\left((1-x)^{\alpha+1}-1\right)}$
(B.5) $\displaystyle+$ $\displaystyle
e^{\frac{2\lambda_{0}}{\alpha+1}(1-x)^{\alpha+1}-q^{2}x}\int_{0}^{x}dw\;e^{q^{2}w-\frac{2\lambda_{0}}{\alpha+1}(1-w)^{\alpha+1}}\sqrt{\frac{2}{\pi}}q$

where ${\tilde{G}}(q,0)$ is the sine transform of the initial condition
$G(y,0)$. The inverse sine transform then yields

$\displaystyle G(y,x)$ $\displaystyle=$
$\displaystyle\frac{2}{\pi}\int_{0}^{\infty}dq\;\sin(qy)\;q\;e^{-q^{2}x}\int_{0}^{x}dw\;e^{q^{2}w-\frac{2\lambda_{0}}{\alpha+1}\big{(}(1-w)^{\alpha+1}-(1-x)^{\alpha+1}\big{)}}$
(B.6) $\displaystyle+$
$\displaystyle\sqrt{\frac{2}{\pi}}\int_{0}^{\infty}dq\;\sin(qy)\;\tilde{G}(q,0)\;e^{-q^{2}x+\frac{2\lambda_{0}}{\alpha+1}\big{(}(1-x)^{\alpha+1}-1\big{)}}$

and the defect density (8) is given by

$D(x)=\frac{G(0,x)-G(\tau^{-1/2},x)}{2}\stackrel{{\scriptstyle\tau\to\infty}}{{\to}}-\frac{1}{2\sqrt{\tau}}\frac{\partial
G(y,x)}{\partial y}\bigg{|}_{y=0}$ (B.7)

##  Appendix C Defect density dynamics for Glauber chain

As shown in Appendix B, the two-point correlation function can be written as
$G(y,x)=I_{1}+I_{2}$ where,
$y=\frac{k}{\sqrt{\tau}},x=\frac{t}{\tau},\lambda_{0}=\tau(1-\gamma_{0})$,

$\displaystyle I_{1}(y,x)$ $\displaystyle=$
$\displaystyle\frac{2}{\pi}\int_{0}^{\infty}dq\;q\sin(qy)\;e^{-q^{2}x}\int_{0}^{x}dw\;e^{q^{2}w-\frac{2\lambda_{0}}{\alpha+1}\big{(}(1-w)^{\alpha+1}-(1-x)^{\alpha+1}\big{)}}$
(C.1) $\displaystyle I_{2}(y,x)$ $\displaystyle=$
$\displaystyle\sqrt{\frac{2}{\pi}}\;e^{\frac{2\lambda_{0}}{\alpha+1}\big{(}(1-x)^{\alpha+1}-1\big{)}}\int_{0}^{\infty}dq\;\textrm{sin}(qy)\;\tilde{G}(q,0)\;e^{-q^{2}x}$
(C.2)

and $\tilde{G}(q,0)$ is the sine transform of the initial condition $G(y,0)$.

Integral $I_{1}$: We first analyze the double integral $I_{1}$ which is
independent of the initial condition. Interchanging the order of integration
and on carrying out the integral over $q$ exactly in (C.1), we obtain

$\displaystyle I_{1}(y,x)$ $\displaystyle=$
$\displaystyle\frac{y}{2\sqrt{\pi}}\int_{0}^{x}dw\;e^{-\frac{2\lambda_{0}}{\alpha+1}\big{(}(1-w)^{\alpha+1}-(1-x)^{\alpha+1}\big{)}}\frac{e^{\frac{-y^{2}}{4(x-w)}}}{(x-w)^{3/2}}$
(C.3) $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{\pi}}\int_{\frac{y^{2}}{4x}}^{\infty}du\;\frac{e^{-u}}{\sqrt{u}}\;e^{-\frac{2\lambda_{0}}{\alpha+1}\big{(}(1-x+\frac{y^{2}}{4u})^{\alpha+1}-(1-x)^{\alpha+1}\big{)}}$
(C.4)

Short time dynamics ($x<1/2$): The above integral is not exactly solvable but
for $x<1/2$, as $\frac{y^{2}}{4x}<\frac{y^{2}}{4(1-x)}<u$, on expanding the
integrand in the above expression in powers of $\frac{y^{2}}{4(1-x)u}$ and
retaining terms to leading order, we obtain

$\displaystyle I_{1}(y,x)$ $\displaystyle\stackrel{{\scriptstyle y\ll
1}}{{\approx}}$
$\displaystyle\frac{1}{\sqrt{\pi}}\int_{\frac{y^{2}}{4x}}^{\infty}du\;\frac{e^{-u}}{\sqrt{u}}\;e^{-\lambda_{0}(1-x)^{\alpha}\frac{y^{2}}{2u}}$
(C.5) $\displaystyle=$ $\displaystyle\frac{1}{2}\sum_{\epsilon=\pm
1}e^{\epsilon
y\sqrt{2(1-x)^{\alpha}\lambda_{0}}}\textrm{erfc}\Big{(}\frac{y+2\epsilon
x\sqrt{2(1-x)^{\alpha}\lambda_{0}}}{2\sqrt{x}}\Big{)}$ (C.6)

Hence

$\displaystyle\frac{\partial I_{1}(y,x)}{\partial y}\bigg{|}_{y=0}$
$\displaystyle=$
$\displaystyle-\frac{e^{-2\lambda_{0}x(1-x)^{\alpha}}}{\sqrt{\pi
x}}+\frac{1}{2}\sqrt{2\lambda_{0}(1-x)^{\alpha}}\sum_{\epsilon=\pm
1}\epsilon\;\textrm{erfc}\Big{(}\epsilon\sqrt{2x(1-x)^{\alpha}\lambda_{0}}\Big{)}$
(C.7) $\displaystyle\approx$
$\displaystyle-\frac{e^{-2\lambda_{0}x}}{\sqrt{\pi
x}}+\frac{1}{2}\sqrt{2\lambda_{0}(1-x)^{\alpha}}\sum_{\epsilon=\pm
1}\epsilon\;\textrm{erfc}\Big{(}\epsilon\sqrt{2x\lambda_{0}}\Big{)}$ (C.8)

where we have written $x(1-x)^{\alpha}\approx x$ for $x<1/2$ which gives

$\displaystyle\frac{\partial I_{1}(y,x)}{\partial y}\bigg{|}_{y=0}\approx$
$\displaystyle-\frac{1}{\sqrt{\pi
x}}-\frac{2\lambda_{0}\sqrt{x}}{\sqrt{\pi}}\leavevmode\nobreak\
,\leavevmode\nobreak\ \lambda_{0}x\ll 1$ (C.9a) $\displaystyle\frac{\partial
I_{1}(y,x)}{\partial y}\bigg{|}_{y=0}\approx$
$\displaystyle-\sqrt{2\lambda_{0}(1-x)^{\alpha}}\leavevmode\nobreak\
,\leavevmode\nobreak\ \lambda_{0}x\gg 1$ (C.9b)

At the end of quench ($x=1$): For $1/2<x<1$, we have not been able to find a
suitable approximation but we can obtain an expression for $I_{1}$ when $x=1$
(that is, at the end of the quench). For arbitrary $\lambda_{0}$ and
$u_{0}=(\frac{2\lambda_{0}}{\alpha+1})^{\frac{1}{\alpha+1}}(\frac{y^{2}}{4})$,
we rewrite (C.4) as

$\displaystyle I_{1}(y,1)$ $\displaystyle=$ $\displaystyle
1+\sqrt{\frac{u_{0}}{\pi}}\int_{0}^{\infty}du\;\frac{e^{-uu_{0}}}{\sqrt{u}}\;(e^{-(\frac{1}{u})^{\alpha+1}}-1)-\frac{1}{\sqrt{\pi}}\int_{0}^{\frac{y^{2}}{4}}du\;\frac{e^{-u}}{\sqrt{u}}\;e^{-(\frac{u_{0}}{u})^{\alpha+1}}$
(C.10) $\displaystyle\stackrel{{\scriptstyle y\to 0}}{{\approx}}$
$\displaystyle
1+\sqrt{\frac{u_{0}}{\pi}}\int_{0}^{\infty}du\;\frac{e^{-(\frac{1}{u})^{\alpha+1}}-1}{\sqrt{u}}-\frac{1}{\sqrt{\pi}}\int_{0}^{\frac{y^{2}}{4}}du\;\frac{1}{\sqrt{u}}\;e^{-(\frac{u_{0}}{u})^{\alpha+1}}$
(C.11) $\displaystyle=$ $\displaystyle
1-\frac{y}{\sqrt{\pi}}\left(\frac{2\lambda_{0}}{\alpha+1}\right)^{\frac{1}{2\alpha+2}}\Gamma\left(\frac{2\alpha+1}{2\alpha+2}\right)-\frac{y}{2(\alpha+1)\sqrt{\pi}}E_{\frac{3+2\alpha}{2\alpha+2}}\left(\frac{2\lambda_{0}}{\alpha+1}\right)$
(C.12)

We therefore have

$\frac{\partial I_{1}(y,1)}{\partial
y}\bigg{|}_{y=0}=-\frac{1}{\sqrt{\pi}}\left[\left(\frac{2\lambda_{0}}{\alpha+1}\right)^{\frac{1}{2\alpha+2}}\Gamma(\frac{2\alpha+1}{2\alpha+2})+\frac{1}{2(\alpha+1)}E_{\frac{3+2\alpha}{2\alpha+2}}\left(\frac{2\lambda_{0}}{\alpha+1}\right)\right]$
(C.13)

where, $E_{n}(z)=\int_{1}^{\infty}dww^{-n}e^{-zw}$ is the exponential integral
function. We then obtain

$\displaystyle\frac{\partial I_{1}(y,1)}{\partial y}\bigg{|}_{y=0}=$
$\displaystyle-\frac{1}{\sqrt{\pi}}\left(1+\frac{2\lambda_{0}}{2\alpha^{2}+3\alpha+1}\right),$
$\lambda_{0}\ll 1$ (C.14a) $\displaystyle\frac{\partial I_{1}(y,1)}{\partial
y}\bigg{|}_{y=0}=$
$\displaystyle-\frac{1}{\sqrt{\pi}}\left[\left(\frac{2\lambda_{0}}{\alpha+1}\right)^{\frac{1}{2\alpha+2}}\Gamma\left(\frac{2\alpha+1}{2\alpha+2}\right)+\frac{e^{-\frac{2\lambda_{0}}{\alpha+1}}}{4\lambda_{0}}\right],$
$\lambda_{0}\gg 1$ (C.14b)

Integral $I_{2}$: We now analyze the integral $I_{2}$ for different initial
conditions:

Equilibrium state at $T_{0}$: For $G(k,0)=G_{k,eq}\approx
e^{-k\sqrt{2(1-\gamma_{0})}}=e^{-y\sqrt{2\lambda_{0}}}$, we obtain

$\displaystyle
I_{2}(y,x)=\frac{2}{\pi}\;e^{\frac{2\lambda_{0}}{\alpha+1}\big{(}(1-x)^{\alpha+1}-1\big{)}}\int_{0}^{\infty}dq\;\textrm{sin}(qy)\;\frac{q}{q^{2}+2\lambda_{0}}\;e^{-q^{2}x}$
(C.15)

which gives

$\displaystyle\frac{\partial I_{2}(y,x)}{\partial y}\bigg{|}_{y=0}$
$\displaystyle=$ $\displaystyle
e^{-\frac{2\lambda_{0}}{\alpha+1}\big{(}1-(1-x)^{\alpha+1}\big{)}}\left[\frac{1-e^{2\lambda_{0}x}\sqrt{2\pi\lambda_{0}x}\;\text{erfc}\left(\sqrt{2\lambda_{0}x}\right)}{\sqrt{\pi
x}}\right]$ (C.16) $\displaystyle\stackrel{{\scriptstyle x<1/2}}{{\approx}}$
$\displaystyle
e^{-2\lambda_{0}x}\left[\frac{1-e^{2\lambda_{0}x}\sqrt{2\pi\lambda_{0}x}\;\text{erfc}\left(\sqrt{2\lambda_{0}x}\right)}{\sqrt{\pi
x}}\right]$ (C.17)

and therefore

$\displaystyle\frac{\partial I_{2}(y,x)}{\partial y}\bigg{|}_{y=0}\approx$
$\displaystyle-\sqrt{2\lambda_{0}}+\frac{1}{\sqrt{\pi
x}}+\frac{2\lambda_{0}\sqrt{x}}{\sqrt{\pi}}\leavevmode\nobreak\ $ ,
$\lambda_{0}x\ll 1$ (C.18a) $\displaystyle\frac{\partial I_{2}(y,x)}{\partial
y}\bigg{|}_{y=0}\approx$ $\displaystyle{\cal
O}(e^{-2\lambda_{0}x})\leavevmode\nobreak\ $ , $\lambda_{0}x\gg 1$ (C.18b)

However, at the end of the quench ($x=1$)

$\frac{\partial I_{2}(y,1)}{\partial
y}\bigg{|}_{y=0}=e^{-\frac{2\lambda_{0}}{\alpha+1}}\left[\frac{1-e^{2\lambda_{0}}\sqrt{2\pi\lambda_{0}}\;\text{erfc}\left(\sqrt{2\lambda_{0}}\right)}{\sqrt{\pi}}\right]\\\
$ (C.19)

Paramagnetic state at $T_{0}$: If the system is in a paramagnetic state,
$G(k,0)=\delta_{k,0}$ or $G(y,0)\sim\delta(y)$ but the sine transform
${\tilde{G}}(q,0)=0$ so that

$I_{2}(y,x)=0$ (C.20)

at all times.

Critical state at $T_{0}$: If the system is initially in the critical state,
the correlation function $G_{k}(0)=1$ for all $k$ and its sine transform
$\tilde{G}(q,0)=\sqrt{\frac{2}{\pi}}\frac{1}{q}$. Using this in (C.2), we
obtain

$\displaystyle I_{2}(y,x)$ $\displaystyle=$ $\displaystyle
e^{-\frac{2\lambda_{0}}{\alpha+1}\big{(}1-(1-x)^{\alpha+1}\big{)}}\textrm{erf}\left(\frac{y}{2\sqrt{x}}\right)$
(C.21)

which yields

$\displaystyle\frac{\partial I_{2}(y,x)}{\partial y}\bigg{|}_{y=0}$
$\displaystyle=$ $\displaystyle\frac{1}{\sqrt{\pi
x}}e^{-\frac{2\lambda_{0}}{\alpha+1}\big{(}1-(1-x)^{\alpha+1}\big{)}}$ (C.22)
$\displaystyle\stackrel{{\scriptstyle x<1/2}}{{\approx}}$
$\displaystyle\frac{1}{\sqrt{\pi x}}e^{-2\lambda_{0}x}$ (C.23)

and therefore

$\displaystyle\frac{\partial I_{2}(y,x)}{\partial y}\bigg{|}_{y=0}\approx$
$\displaystyle\frac{1}{\sqrt{\pi
x}}-\frac{2\lambda_{0}\sqrt{x}}{\sqrt{\pi}}\leavevmode\nobreak\ $ ,
$\lambda_{0}x\ll 1$ (C.24a) $\displaystyle\frac{\partial I_{2}(y,x)}{\partial
y}\bigg{|}_{y=0}\approx$ $\displaystyle{\cal
O}(e^{-2\lambda_{0}x})\leavevmode\nobreak\ $ , $\lambda_{0}x\gg 1$ (C.24b)

However, at the end of the quench ($x=1$)

$\frac{\partial I_{2}(y,1)}{\partial
y}\bigg{|}_{y=0}=\frac{1}{\sqrt{\pi}}e^{-\frac{2\lambda_{0}}{\alpha+1}}\\\ $
(C.25)

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##  Supplemental Material D Glauber chain for equilibrium initial condition

(a) Effect of initial temperature

(b) Effect of system size

Figure 1: Glauber Ising chain when the system initially equilibrated to a low
temperature $T_{0}$ is slowly quenched to zero temperature: (a) The main
figure shows the density of defects at the end of the quench for various
quench times for two different $T_{0}$ values. The black dashed lines
correspond to the equilibrium value (11) at the respective initial
temperatures. The inset figure shows the scaling collapse with
$\tau\sim\xi_{0}^{2}$ in the regime I. (b) The main figure shows the effect of
finite system size on the defect density at the end of the quench for various
quench rates. The inset figure shows the scaling collapse in accordance with
(26). The parameters are $L=2000$, $T_{0}=0.5$ with $\xi_{0}\approx 27.3$, and
$\alpha=3$ in the cooling protocol (5).

##  Supplemental Material E Glauber chain for various initial conditions

(a) Equilibrium

(b) Nonequilibrium (cooling)

(c) Nonequilibrium (heating)

Figure 2: Glauber Ising chain when quenched to zero temperature for various
initial conditions and small system size: The figure (a) shows that the KZ
phase is absent in the defect density at the end of the quench because the
system is initially in equilibrium at a very low temperature where the
correlation length $\xi_{0}$ is comparable to the system size. In (b) and (c)
the system is in nonequilibrium state at $T_{0}$, the initial correlation
length $\xi_{i}\sim{\cal O}(1)$ but $\xi_{0}\sim L$ due to which system
behaves as if it is instantaneously cooled or heated to zero temperature. The
numerical data is obtained by solving the exact differential equation (7) and
the analytical data is obtained from (21), (27) and (31), respectively, for
figures (a)-(c). The parameters are $L=2000$, $T_{0}=0.2$ and corresponding
$\xi_{0}\approx 11000$, and $\alpha=3$ in the cooling protocol (5).