#  Driven one-dimensional noisy Kitaev chain

Manvendra Singh, Santanu Dhara, and Suhas Gangadharaiah  Department of
Physics, Indian Institute of Science Education and Research, Bhopal, India

###### Abstract

We study one dimensional Kitaev chain driven by the anisotropy parameter,
$J_{-}=t/\tau_{Q}$, in the presence of weak Gaussian noise in the parameter
$J_{+}$. The system is prepared in the ground state of the Hamiltonian at the
initial time $t\rightarrow-\infty$. The defect density and the residual energy
at the end of the drive protocol reveals anti-Kibble Zurek (AKZ) behavior
which deviates from the earlier reported linear in quench time dependence for
the defect density. The entropy density at the end of the protocol is found to
exhibit the signature of the AKZ behavior. In the context of the Kitaev chain,
the two-point spin correlators are short ranged. The hidden topological order
across the quantum phase transition point is probed via the non-local string
order parameters. The two-point Majorana correlator and the hidden string
correlator calculated in the final decohered state at the end of the noisy
drive protocol is consistent with the AKZ picture. We have analyzed the kink
statistics in the dual spin space, the first three cumulants at the end of the
noisy drive also exhibit the AKZ scaling behavior for slower sweeps similar to
the defect density. The kink distribution function is well approximated with
the normal distribution with non-universal noise dependent mean and variance.

###### pacs:

##  I Introduction

In the context of the drive through a second order phase transition, the
Kibble-Zurek mechanism (KZM) predicts the production of topological defects
across the phase transition point where the defect density, $n$, scales with
the quench time $\tau_{Q}$ as $n\sim\tau^{-\beta}_{Q}$, with the KZM exponent
$\beta=d\nu/(1+\nu z)$ expressed in terms of the universal critical exponents
$\nu$ and $z$ which are the correlation length and the dynamical exponent,
respectively, while $d$ is the dimension of the system [1, 2, 3, 4, 5, 6]. The
applicability of the KZM has been extended to describe non-equilibrium quantum
dynamics in the context of quantum phase transition (QPT) in various scenarios
[7, 8, 9, 10, 11, 12, 13, 14, 15]. Even for infinitesimally slow drive the
creation of the defects across the quantum critical point occurs due to the
vanishing energy gap and diverging relaxation time at the quantum critical
point (QCP) i.e., the system is unable to follow the instantaneous ground
state in the drive through the QCP resulting in the creation of excitations.

Recent studies [16, 17, 18, 19, 21, 20, 22, 23, 24, 25, 26] have established
that the drive protocols through the QCP that have noisy component to it are
more susceptible towards defect generation and increased residual energy
specifically in the slow sweep regime. In particular, it was shown numerically
for the transverse field quantum Ising model (TFQIM) [18] and the XY-spin
chain model [19], that for certain protocols the defect density can be
approximated as, $n\approx c\tau_{Q}^{-\beta}+d\eta^{2}_{0}\tau_{Q}$ for the
weak Gaussian noise case, with $c$ and $d$ being system dependent parameters
and $\eta_{0}$ the noise strength. The result was further corraborated by the
analytical studies on XY-spin chain with transverse drive protocol [25]. The
linear in $\tau_{Q}$ term owes its origin to the presence of noise and is the
Anti-Kibble-Zurek (AKZ) term which is responsible for the increased defect
generation in the slow-sweep regime. In the present work, we study the driven
one dimensional Kitaev chain across a QCP. The protocol considered here is
qualitatively different from those considered in [18, 25] and results in
significantly different behavior for the AKZ term. For the noisy Kitaev chain
driven by the Hamiltonian parameter $J_{-}$, we find that the noise dependent
defect density has linear (in quench time) term multiplied by a log-term in
the weak noise limit a result different from all the earlier studies [18, 19,
25]. As a consequence, the optimal quench time has a completely different
dependence on the noise strength from the previously reported $\eta^{-4/3}$
power. However, the residual energy exhibits conventional behavior with the
noise dependent term being linear in quench time.

The consequences of the (noiseless) drive through the QCP and the KZM picture
of defect formation is also reflected in the von Neumann entropy density [27].
In the KZM scenario, slower sweep results in lower entropy density. In both
the fast and the slow sweep regimes of the noiseless drive, the entropy
density is strongly suppressed while the entropy density is maximized at an
intermediate sweep speed. However, for the noisy drive case, there is a
characteristic quench time scale related to the optimal quench time beyond
which the entropy density increases. Interestingly, the entropy density also
signifies the crossover behavior of the correlation functions [27, 28, 29].

For the Kitaev chain, the two point spin-spin correlators are short ranged
(i.e., the spin correlators vanishes for more than one lattice constant
separation which signifies that both phases across the QCP are disordered)
[31, 30, 32]. Unlike the prediction from Landau’s theory of the second order
continuous phase transition, a QPT can also occur between the two disordered
phases without any symmetry breaking related to the change of topological
order. Many low dimensional quantum systems exhibit such QPT of topological
nature without any symmetry breaking associated with the local order parameter
(LOP). Examples include topological and Mott insulators, spin liquids,
frustrated magnetic systems, etc. [30, 31, 32, 33, 34, 35, 36, 37, 38, 39].
Therefore, the conventional Landau theory based on the LOP is not directly
applicable to characterize such QPT. In the quantum phase transitions of the
topological type, the hidden topological order across the QCP can in principle
be probed via various non-local string order parameters (SOP) by extending the
Landau theory to accommodate these non-local string operators and their
correlation functions as suggested in various works [35, 36, 37]. These non-
local SOP show either long range order or short range order across the QCP
thus successfully describing hidden order or disorder in different phases.
Motivated from these studies, we calculate the two-point Majorana correlator
and a hidden string operator to probe hidden order or disorder for the driven
one dimensional noisy Kitaev chain. In the noiseless drive case, both the two-
point Majorana correlator and a hidden string correlator show long range order
for slower drives which is consistent with KZM predictions [40, 29, 30].
Furthermore, the two-point Majorana correlator and the non-local hidden SOP
exhibit the signature of the AKZ behavior, i.e., correlation length-scale
related to the ordered domain quantified by these non-local correlators
decreases due to the presence of noise.

Next, we considered the kink/defect distribution in the dual spin space at the
end of the noisy drive protocol. In particular, we calculate the first three
cumulants (mean, variance and third central moment), all of which exhibit the
universal KZ scaling behavior ($\propto 1/\sqrt{\tau_{Q}}$) with the quench
time in the slower drive regime for the noiseless case as discussed in Ref.
[41]. For the noisy drive cases we find that all the three cumulants exhibit
the KZ as well as the noise dependent AKZ scaling behavior similar to what we
find for the defect density. Similar to the noiseless case, the kink
distribution for the noisy drives in the slow sweep regime approaches the
normal distribution $\mathcal{N}(\kappa_{1},\kappa_{2})$ having non-universal
noise dependent mean ($\kappa_{1}=\langle n\rangle$) and variance
($\kappa_{2}=\langle n^{2}\rangle-\langle n\rangle^{2}$), however unlike the
noiseless drive case the variance and other higher cumulants no longer remain
proportional to the kink density.

The outline of this article is as follows. In Sec. II, we introduce the model
Hamiltonian for the one dimensional Kitaev model and study the dynamics of the
system for the case of the noisy drive through the QCP in the thermodynamic
and in the long time limit. We solve the dynamics of the system by mapping the
time dependent Hamiltonian in momentum space to the set of independent noisy
Landau-Zener (LZ) problems. In Sec. III, we calculate the defect density and
the residual energy which show AKZ behavior for the noisy drive protocol. In
Sec. IV we calculate the entropy density for the final decohered state due to
noisy drive through the QCP. In Sec. V, we study the non-local string operator
correlators, in particular we calculate the two-point Majorana correlator and
a hidden SOP (at finite separation) to characterize the hidden (topologically)
ordered or disordered phase of the Kitaev chain at the end of the noisy drive.
These non-local correlators also exhibit the signature of AKZ picture of
defect generation in the noisy drive protocol. In Sec. VI we show that the
lower order cumulants corresponding to the kink distribution function exhibit
AKZ behavior similar to that in the defect density. We conclude our results in
the Sec. VII.

##  II Model Hamiltonian

The one-dimensional Kitaev model Hamiltonian is given by the Hamiltonian [29,
42, 43, 44],

$H=\sum^{N/2}_{j=1}\left(J_{1}\,\sigma^{x}_{2j}\sigma^{x}_{2j+1}+J_{2}\,\sigma^{y}_{2j-1}\sigma^{y}_{2j}\right),$
(1)

where $\sigma^{x,y}$ are the Pauli matrices, and the parameters $J_{1}$ and
$J_{2}$ are the bond dependent spin couplings. Performing the Jordan-Wigner
transformation with,

$a_{j}=(\Pi^{2j-1}_{i=-\infty}\sigma^{z}_{i})\sigma^{y}_{2j}\,\,,\,\,b_{j}=(\Pi^{2j}_{i=-\infty}\sigma^{z}_{i})\sigma^{x}_{2j+1},$
(2)

where $a_{j}$ and $b_{j}$ are the two independent Majorana fermions ( i.e.,
$a^{\dagger}_{j}=a_{j}$, $b^{\dagger}_{j}=b_{j}$) at site $j$ satisfying
$\\{a_{i},a_{j}\\}=2\delta_{ij}$ , $\\{b_{i},b_{j}\\}=2\delta_{ij}$ and
$\\{a_{i},b_{j}\\}=0$. The Hamiltonian given by Eq. 1 expressed in terms of
the Fourier transformed variables acquires the following form [38],

$H=\sum^{\pi}_{k=0}\psi^{\dagger}_{k}\mathcal{H}_{k}\psi_{k}$ (3)

with,

$\mathcal{H}_{k}=2i\begin{bmatrix}0&-J_{1}-J_{2}e^{-ik}\\\
J_{1}+J_{2}e^{ik}&0\end{bmatrix},\\\ $ (4)

where $\psi_{k}=(a_{k},b_{k})^{T}$, note $a_{k}$ and $b_{k}$ are Fourier
components which satisfy the standard anti-commutation relations. We consider
the quenching scheme wherein, $J_{-}=J_{1}-J_{2}$, is varied linearly with
time as: $J_{-}(t)=t/\tau_{Q}$, with $t$ running from $-\infty$ to $\infty$
and $J_{+}=J_{1}+J_{2}$. We next perform unitary transformation so as to
recast the Hamiltonian $\mathcal{H}_{k}$ in Eq. [4] into the Landau-Zener type
Hamiltonian of the following form

$\tilde{\mathcal{H}}_{k}=2\,\begin{bmatrix}J_{-}(t)\sin(k/2)&J_{+}\cos(k/2)\\\
J_{+}\cos(k/2)&-J_{-}(t)\sin(k/2)\end{bmatrix}\\\ ,$ (5)

where the $k$ runs from $0$ to $\pi$. The energy gap,
$4\sqrt{J^{2}_{-}\sin^{2}{\frac{k}{2}}+J^{2}_{+}\cos^{2}{\frac{k}{2}}}$,
vanishes at the critical point $J_{-}=0$ for the critical mode $k=\pi$, and
corresponds to the quantum phase transition of topological nature with the
critical exponents, $\nu=z=1$ [30, 44]. Note that although the Kitaev spin
chain can be mapped to TFQIM in the dual spin space, however the quenching
scheme considered here which involves $J_{-}$ leads to the fundamentally
different fermionized Hamiltonian as compared to that of the Hamiltonian of
the TFQIM [18] or the TFQIM limit of the XY spin chain case considered for the
transverse protocol [25].

Apart from the linear drive term $J_{-}(t)=vt=t/\tau_{Q}$, we consider
Gaussian noise dependence in $J_{+}$ i.e., $J_{+}(t)=J_{+}+\eta(t)$, the noisy
LZ type Hamiltonian can thus be written as,

$\mathcal{H}^{\eta}_{k}(t)=\tilde{\mathcal{H}}_{k}+[2\eta(t)\cos{\frac{k}{2}}]\tau_{1},$
(6)

where $\tau_{1}$ is the $x$-Pauli matrix. The noise term $\eta(t)$ is Gaussian
correlated and characterized by zero mean and
$\overline{\eta(t)\eta(t_{1})}=\eta_{0}^{2}e^{-\Gamma|t-t_{1}|}$, where
$\eta_{0}$ is the noise strength and $\Gamma$ is the inverse time scale
corresponding to the noise. We would like to point out that the noise term
does not change the original translational invariance of the problem.
Therefore, the k-modes remain decoupled even in the presence of noise.
Moreover, since the initial state that we consider has a tensor product
structure, the k-modes remain decoupled (since there are neither interactions
nor impurities present), a statement which holds even under noise averaging. A
second point we will briefly discuss here is the difference between the drive
protocol considered for the noisy Kitaev chain and the quench protocol
considered in $XY$-spin chain [25] while taking the $\gamma=1$ limit and the
noisy TFQIM considered in [18]. As discussed in Sec. V, the duality
transformation $\sigma^{x}_{j}=\tau^{x}_{j-1}\tau^{x}_{j}$ and
$\sigma^{y}_{j}=\Pi^{N}_{l=j}\tau^{y}_{l}$, maps the Kitaev Hamiltonian to the
TFQIM, i.e.,
$H_{d}=\sum^{N/2}_{j=1}(J_{1}\tau^{x}_{2j-1}\tau^{x}_{2j+1}+J_{2}\tau^{y}_{2j-1})$.
Since we consider the drive by $J_{-}$ while the noise is present in the
$J_{+}$ term, therefore both the spin terms include the driving and the noise
term. While in our model the drive and the noise term in the $J_{1}$ term come
with the same relative sign, in the $J_{2}$ term they come with the opposite
signs. On the other hand, in the TFQIM considered in [18] both the $J_{1}$ and
$J_{2}$ have the noise and the drive terms with the same relative sign, in
other words the driving term has noise in it. For the TFQIM obtained by the
$\gamma=1$ limit of the $XY$-spin chain [25], as opposed to the Kitaev model,
the noise term is present only in the spin-spin term while the drive parameter
is on the transverse term.

A general time evolved state $\ket{\psi^{\eta}_{k}(t)}$ can be written as,
$\ket{\psi^{\eta}_{k}(t)}=c^{\eta}_{1k}(t)\ket{\bar{e}_{1k}}+c^{\eta}_{2k}(t)\ket{\bar{e}_{2k}}$,
for a single realisation of the Gaussian noise, where $c^{\eta}_{ik}(t)$ are
time dependent probability amplitudes for the (diabatic) states
$\ket{\bar{e}_{ik}}$ (which are the eigenstates of Hamiltonian, Eq. [4] in the
limit $J_{-}\rightarrow\infty$). The time evolution of the state is given by
the stochastic Schrodinger equation,

$i\frac{d}{dt}\ket{\psi^{\eta}_{k}(t)}=\mathcal{H}^{\eta}_{k}(t)\ket{\psi^{\eta}_{k}(t)},$
(7)

where the corresponding density matrix obtained for the single realization of
noise,
$\hat{\rho}^{\eta}_{k}(t)=\ket{\Psi^{\eta}_{k}(t)}\bra{\Psi^{\eta}_{k}(t)}$,
is averaged over the noise to yield $\overline{\hat{\rho}^{\eta}_{k}(t)}$.
With the initial conditions, $c^{\eta}_{1k}(-\infty)=1$ and
$c^{\eta}_{2k}(-\infty)=0$, we set up the Bloch equation for the population
inversion ($\rho^{\eta}_{k}$) after considering the transformation,
$\tau=2\Delta_{k}t$ and
$v_{{}_{\mathrm{LZ}}}=v\sin{\frac{k}{2}}/\Delta^{2}_{k}$ (where,
$\Delta_{k}=2J_{+}\cos{\frac{k}{2}}$),

$\displaystyle\frac{d}{d\tau}\rho^{\eta}_{k}(\tau)=-\frac{1}{2}\int^{\tau}_{-\infty}d\,\tau_{1}e^{i\,\int^{\tau}_{\tau_{1}}d\tau_{2}\,\frac{v_{{}_{\mathrm{LZ}}}\tau_{2}}{2}}\,\rho^{\eta}_{k}(\tau_{1}){}$
$\displaystyle-\frac{1}{2\,J^{2}_{+}}\int^{\tau}_{-\infty}d\,\tau_{1}e^{i\,\int^{\tau}_{\tau_{1}}d\tau_{2}\,\frac{v_{{}_{\mathrm{LZ}}}\tau_{2}}{2}}\eta(\tau)\eta(\tau_{1})\,\rho^{\eta}_{k}(\tau_{1})+h.c..$
(8)

Following the approaches similar to that given in [45, 46], the reduced master
equation is solved by averaging the Bloch equation over the fast Gaussian
noise to yield in the long time limit,

$\overline{\rho_{k}}(\infty)=e^{-2\pi\,\eta_{0}^{2}/(2\,v_{{}_{\mathrm{LZ}}}\,J^{2}_{+})}\Big{(}2\,e^{-\pi/(2\,v_{{}_{\mathrm{LZ}}})}-1\Big{)},$

and the noise averaged excitation probability is given by,

$p_{k}=\frac{1}{2}\left[1+e^{-\chi\frac{\cos^{2}{k/2}}{\sin{k/2}}}(2e^{-2\pi\tau_{Q}\frac{\cos^{2}{k/2}}{\sin{k/2}}}-1)\right]$
(9)

where $\chi=4\pi\eta_{0}^{2}\tau_{Q}$ and we have set $J_{+}=1$. Notice that
the non-adiabatic region responsible for increased excitations opens up around
$k=0$ (as can be seen from the Fig. [1] of the excitation probability Eq. [9])
instead of $k=\pi/2$ region which is the case for the TFQIM or XY spin-chian
case with transverse drive protocol [18, 19, 25]. For $\eta_{0}=0$, one
retrieves the excitation probability as given by the LZ formula for the
noiseless drive case [42, 43],

$p^{0}_{k}=e^{-2\pi\tau_{Q}\frac{\cos^{2}{k/2}}{\sin{k/2}}}.$ (10)

Figure 1: In the inset we have plotted the excitation probability $p^{0}_{k}$
as a function of $k$ for the noiseless drive scenario. For the slower drive,
i.e., large $\tau_{Q}$ case, the excitation probability is non-zero around the
critical point $k=\pi$ only. This suggests that the non-adiabatic contribution
to the defect density comes around the critical point only in the slower sweep
regime. In the main figure we have plotted the excitation probability $p_{k}$
vs. $k$, for $\tau_{Q}=100$ (which is the slow drive regime), in the presence
of noisy drive. The increase of the noise strength, $\eta_{0}$ opens up a new
non-adiabatic region around the $k=0$ point.

In the thermodynamic limit ($N\rightarrow\infty$) and in the long time limit
($t\rightarrow\infty$), the noise averaged decohered density matrix can be
written as, $\overline{\rho}_{D}=\Pi_{k>0}\overline{\rho}^{k}_{D}$, with
$\overline{\rho}^{k}_{D}$ given by,

$\overline{\rho}^{k}_{D}=p_{k}\ket{\bar{e}_{1k}}\bra{\bar{e}_{1k}}+(1-p_{k})\ket{\bar{e}_{2k}}\bra{\bar{e}_{2k}},$
(11)

which characterizes the mixed state i.e., the noise averaged coherences of the
density matrix, $\overline{\rho^{k}_{12}}$ and $\overline{\rho^{k}_{21}}$
involving terms like $\ket{\bar{e}_{1k}}\bra{\bar{e}_{2k}}$ and
$\ket{\bar{e}_{2k}}\bra{\bar{e}_{1k}}$ are highly fluctuating and vanishes in
the long time limit.

##  III Defect Density and Residual Energy

For the noiseless drive through the QCP, the defect density is obtained by
integrating the excitation probability i.e.,
$n_{0}=\frac{1}{\pi}\int^{\pi}_{0}dk\,p^{0}_{k}$. For the slow sweep regime,
the dominant contribution for the defect generation comes from the non-
adiabatic region around the critical point at $k=\pi$ (see the inset of Fig.
[1]). Therefore, the exponent of $p_{k}^{0}$ acquires a quadratic form and the
integral yields $n_{0}\approx 1/\pi\sqrt{2\tau_{Q}}$, implying that the defect
density follows the KZ scaling behavior, $n_{0}\sim\tau^{-\beta}_{Q},$ with
$\beta=d\nu/(1+z\nu)=1/2$ (where dimension $d=1$ and the critical exponents
are $\nu=z=1$) [43].

The defect density for the noisy drive,

$n=\frac{1}{\pi}\int^{\pi}_{0}p_{k}dk,$ (12)

with $p_{k}$ given by Eq. 9 is evaluated numerically. Unlike the noiseless
case ($\eta_{0}=0$), the defect density for noisy drive exhibits AKZ behavior
i.e., slower drive results in increased defect creation with the increase of
noise strength as seen from Fig.[2]. The behavior of the defect density can be
split in to two asymptotic regimes depending on the noise strength $\eta_{0}$.
In either case we will focus on the slow sweep scenarios ($\tau_{Q}\gg 1$).
For $\chi\ll 1$, the defect density can be approximated as

$n\approx\frac{1}{\pi\sqrt{2\tau_{Q}}}-\frac{\chi}{\pi}\log(\chi),$ (13)

where the first term is the usual KZ behavior. The second term exhibits AKZ
behavior which is present due to the contribution from the $k\approx 0$ region
of the excitation probability (see Fig. [1]), in particular from the term,
$\int dk[1-\exp(-\chi\cos^{2}k/2/\sin(k/2))]$. It is multiplied by $\log\chi$
and hence is more dominant than the AKZ terms obtained in earlier studies [18,
19, 25]. The quench time which minimizes the defect density, Eq. 13 , also
referred to as the optimal quench time can be estimated to be,

$\tau^{n}_{O}\approx\big{[}8\sqrt{2}\pi\eta_{0}^{2}\big{(}\log\frac{1}{4\pi\eta_{0}^{2}\tau_{0}}-1\big{)}\big{]}^{-2/3},$
(14)

where
$\tau_{0}=\big{[}8\sqrt{2}\pi\eta_{0}^{2}(\log\frac{1}{4\pi\eta_{0}^{2}}-1)\big{]}^{-2/3}.$
In the second limit, $\chi\gg 1$, the defect density can be approximated as

$n\approx\frac{1}{2}\Big{(}1-\frac{1}{\sqrt{\pi\chi}}\Big{)}+\frac{1}{\pi\sqrt{2\pi\tau_{Q}+\chi}},$
(15)

thus for very slow sweep rates the noise tends to completely scramble the
initial state i.e, $n\rightarrow\frac{1}{2}$.

Figure 2: (a) Defect density $n$ as a function of quench time shows AKZ
behavior for non-zero noise amplitude. (b) The log of optimal quench time
extracted from the defect density as a function of $\log\eta_{0}^{2}$ exhibits
strong agreement between the one obtained via the numerical approach and by
using the approximate analytical result for $\tau^{n}_{O}$. The best fit line
exhibits deviation from the usual $\eta_{0}^{-4/3}$ optimal quench time
scaling behavior. (c) The residual Energy per site, $E_{r}$, as a function of
quench time also exhibits AKZ behavior for non-zero noise amplitude. (d) The
quench time, $\tau_{O}^{e}$, to minimize the residual energy exhibits,
$\eta_{0}^{-4/3}$, power-law scaling behaviour with the noise strength.

The effect of the quench through the critical points is also imprinted on the
residual energy. The residual energy is the excess energy relative to the
ground state of the Hamiltonian at the end of the drive protocol. The
expression for the residual energy per site in the units of $J_{-}$ is given
by [47],

$E_{r}=\text{lim}_{t,N\rightarrow\infty}\frac{\text{tr}\,[\bar{\rho}^{D}\,H]-E_{0}}{N|J_{-}(t)|}=\frac{4}{\pi}\int^{\pi}_{0}dk~{}p_{k}\sin{\frac{k}{2}},$
(16)

where $\text{tr}\,[\bar{\rho}^{D}\,H]$ is the expectation value of the
Hamiltonian with respect to the final decohered state,
$\bar{\rho}^{D}=\Pi_{k>0}\overline{\rho}^{k}_{D}$ and $E_{0}$ is the total
energy of the ground state. The numerically evaluated residual energy shows
the signature of AKZ behavior for slower drives, as seen from Fig.[2].
Analogous to the derivation of the limiting behavior of the defect density, in
the limit $\chi\gg 1$, $E_{r}\approx 4n$. In the opposite limit, $\chi\ll 1$,
$E_{r}\approx 2\sqrt{2}/\pi\sqrt{\tau_{Q}}+\chi$, i.e., the KZ term has the
same form as that for $n$, except now it is multiplied by a factor of $4$,
while the AKZ term has the conventional linear in $\chi$ dependence.
Consequently, the quench time which minimizes the residual energy has a
conventional scaling behavior as $\eta_{0}^{-4/3}$ [18], which is also
confirmed by the Fig. 2.

Figure 3:  (a) Entropy density as a function of the quench time exhibits AKZ
behavior at slower quench times (the entropy density exhibits an upturn) due
to the noise induced increased entropy production. (b)The log-log plot of the
optimal quench time, $\tau^{n}_{O}$ and $\tau^{s}_{O}$ (which is extracted
from the minimized noisy entropy density) has been plotted with respect to the
noise strength ($\log\eta_{0}^{2}$) which clearly shows that $\tau^{n}_{O}$
and $\tau^{s}_{O}$ are proportional. (c) The behavior of $|\log(\pi|\langle
O_{0}\rangle|/2)|$ is qualitatively similar to that of the entropy density.

##  IV Entropy Density

A finite non-zero entropy density is a signature of mixed state which can
result from either the internal or external decoherence during the drive
through the critical points. The von Neumann entropy density in the final
decohered state $\bar{\rho}_{D}$ is given by,

$s=-\frac{1}{\pi}\int^{\pi}_{0}dk\,[p_{k}\,\text{ln}\,p_{k}+(1-p_{k})\
\text{ln}\,(1-p_{k})],$ (17)

where the numerically integrated entropy density as a function of the quench
time has been plotted in Fig. [3]. In the noiseless drive scenario consistent
with the KZ picture, the entropy density is small in both the fast
($\tau_{Q}\ll 1$) and the slow sweep regime ($\tau_{Q}\gg 1$). Furthermore,
there is an intermediate $\tau_{Q}$-regime between the fast and the slow sweep
regimes where the entropy density maximizes. For the noisy drive case, we
observe (see Fig. [3]) that for slow sweep regime the entropy density shows an
upturn for quench times of the order of the noise strength dependent optimal
quench time. In the $\chi\gg 1$ limit, the entropy density can be approximated
by, $s\approx\text{ln}2-\frac{1}{4\eta_{0}\sqrt{2\tau_{Q}}}$, which approaches
the maximum entropy state ($s\rightarrow\text{ln}\,2$) for sufficiently slow
sweeps, i.e., the noise randomizes the system completely. The noisy drive is
characterized by two quench time scales, first, the intermediate quench time
scale at which the entropy density locally maximizes and the second quench
time scale/regime characterized by the optimal quench time ($\tau^{e}_{O}$)
beyond which results in increased entropy production due to the presence of
the noise, i.e., AKZ behavior [25]. As can be seen from the Fig. [3], the
optimal quench time scale ($\tau^{e}_{O}$) related to entropy density exhibits
similar power law dependence with the noise strength as shown by the optimal
quench time ($\tau^{n}_{O}$) associated with the defect density.

Interestingly, this behavior has qualitative similarities with the behavior of
$|\log(\pi|\langle O_{0}\rangle|/2)|$ (where $\langle O_{0}\rangle$ is the
two-point Majorana correlator at zero separation as defined in Sec. V) as can
be seen Fig. [3] and can also be associated with the crossover behavior of
$\langle|O_{0}|\rangle$ [29]. For $\tau_{Q}\ll 1$, the excitation probability
$p_{k}$ for the noiseless drive tends to $1$ for all the $k$-modes, therefore
the entropy density will be small and one notices that $|\log(\pi|\langle
O_{0}\rangle|/2)|$ also vanishes. On the other hand, for the $\tau_{Q}\gg 1$
and the noiseless case, the excitation probability $p_{k}$ will be close to
zero for the all the modes except near the non-adiabatic region around $k=\pi$
point, where the entropy density and at the same time $|\log(\pi|\langle
O_{0}\rangle|/2)|$ obtains vanishingly small contribution. Moreover, in the
noisy regime similar to the entropy, $|\log(\pi|\langle O_{0}\rangle|/2)|$,
exhibits upturn for the slow sweep regimes.

##  V String Correlations

The transition at $J_{1}=J_{2}$ across the two gapped fermionic state
corresponds to the change of topological order rather than the change of
symmtery [30, 44]. It turns out that the two-point spin-spin correlator
vanishes for larger than one separation in either regimes [31], while in the
“ordered phase” the highly non-local string operator correlations acquire non-
vanishing values [30, 32, 35, 36, 37], moreover, it was shown by performing
the spin-duality transformation that one can map the $1$-D Kitaev chain to
$1$-D Ising chain. The local operators underlying order to disorder transition
in the dual space are highly non-local in terms of the original spin
operators. These are referred to as the hidden string order parameters (SOP)
[30].

We begin by calculating the two-point Majorana operators of the type
$O_{r}=ib_{n}a_{n+r}$, which in the spin language corresponds to the string
operators. Our aim is to understand the long range order for slower sweeps in
the final decohered/mixed state both for the noiseless and the noisy
scenarios. We will next calculate the spin-spin correlations in the dual
space, these are expressed in terms of the product of two-point Majorana
correlators.

###  V.1 Two-point Majorana correlator

The two-point Majorana correlator denoted by $O_{r}=ib_{n}a_{n+r}$ has the
following expression in terms of the fermion operators $a_{k}$ and $b_{k}$ ,

$O_{r}=\frac{4i}{N}\sum_{k=0}^{\pi}(b_{k}^{\dagger}a_{k}e^{ikr}+b_{k}a_{k}^{\dagger}e^{-ikr}+b_{k}a_{k}e^{-ikr}+b_{k}^{\dagger}a_{k}^{\dagger}e^{ikr}).$
(18)

In the limit $J_{-}\rightarrow\infty$, keeping $J_{+}$ fixed, the eigenvectors
of the Hamiltonian Eq. [4] are given by,

$\displaystyle\ket{\bar{e}_{1}}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}(a^{\dagger}_{k}e^{\frac{-ik}{2}}+b^{\dagger}_{k})\ket{0}{}$
$\displaystyle\ket{\bar{e}_{2}}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}(-a^{\dagger}_{k}e^{\frac{-ik}{2}}+b^{\dagger}_{k})\ket{0}$
(19)

where the eigenvalues are, $\bar{E}_{1k,2k}=\pm 2J_{-}\sin{\frac{k}{2}}$ for
the the excited state $\ket{\bar{e}_{1k}}$ and the ground state
$\ket{\bar{e}_{2k}}$, respectively. The final decohered state has the
following form $\overline{\rho}_{D}=\Pi_{k>0}\overline{\rho}^{k}_{D}$, where
$\overline{\rho}^{k}_{D}$ is the density matrix corresponding to the
$k^{\text{th}}$-mode given by,
$\overline{\rho}^{k}_{D}=p_{k}\ket{\bar{e}_{1k}}\bra{\bar{e}_{1k}}+(1-p_{k})\ket{\bar{e}_{2k}}\bra{\bar{e}_{2k}}$,
which is a mixed state.

The expectation value of $O_{r}$, with respect to the final decohered state
i.e., $\langle O_{r}\rangle=\text{tr}(\overline{\rho}_{D}O_{r})$ is given by,

$\langle
O_{r}\rangle=\sum_{k=0}^{\pi}p_{k}\bra{\bar{e}_{1k}}O_{r}\ket{\bar{e}_{1k}}+(1-p_{k})\bra{\bar{e}_{2k}}O_{r}\ket{\bar{e}_{2k}}.$
(20)

Using the result,
$\bra{\bar{e}_{1k}}b_{k}a_{k}^{\dagger}\ket{\bar{e}_{1k}}=-\frac{1}{2}e^{\frac{ik}{2}}$,
and
$\bra{\bar{e}_{2k}}b_{k}a_{k}^{\dagger}\ket{\bar{e}_{2k}}=\frac{1}{2}e^{\frac{ik}{2}}$,
the expectation values of $\langle O_{r}\rangle$ yields the following
expression in the thermodynamic limit ($N\rightarrow\infty$)

$\langle
O_{r}\rangle=\frac{2}{\pi}\int^{\pi/2}_{0}\,dk(1-2\,p_{k})\sin{k(2r-1)}.$ (21)

Following similar approach we find,

$\langle ia_{n}a_{n+r}\rangle=i\int_{0}^{\pi}dk\cos{kr}=i\delta_{r,0},$ (22)

and $\langle ib_{n}b_{n+r}\rangle=-i\delta_{r,0}$.

Figure 4:  (a) $\langle\bar{O}_{r}\rangle$ as a function of $r$ exhibits
damped oscillatory behavior in the absence of noise. For the slower sweep
speeds the spatial extension of the two-point Majorana correlator is long
ranged. (b) As observed from the rescaled figure, the correlation length
associated with the two point Majorana correlator $\langle\bar{O}_{r}\rangle$
exhibits the following scaling behaviour,
$e^{-(2r-1)^{2}/8\pi\tau_{Q}}/\sqrt{\tau_{Q}}$, with the correlation length
proportional to $\sqrt{\tau_{Q}}$.

Figure 5: (a). In the regime $\chi=4\pi\eta_{0}^{2}\tau_{Q}\ll 1$ with
$\tau_{Q}=1000$, or the effectively weak noise regime,
$\langle\bar{O}_{r}\rangle$ vs. $r$-separation continues to exhibit damped
oscillatory behavior, however there is a downward shift as predicted by Eq.
23. (b) In the $\chi\gg 1$ regime (here $\eta_{0}=0.009$) $\langle
O_{r}\rangle$ exhibits exponential scaling with respect to
$r^{2}/\eta_{0}^{2}\tau_{Q}$ in confirmation with Eq. 24.

In the noiseless regime, $\langle O_{0}\rangle$ for $\tau_{Q}\ll 1$ and
$\tau_{Q}\gg 1$ differs simply by a sign. This is because $p_{k}\rightarrow 1$
for $\tau_{Q}\ll 1$ while it vanishes for the slow sweep regime. Similar
observation was made in Ref. [29] although over there the quenching scheme
involved changing $J_{1}$, moreover the expression for $\langle O_{r}\rangle$
is different from that given in Eq. 21. We note that the first term
corresponds to the defect free part of the expectation value of $\langle
O_{r}\rangle$ and yields a power law decay with distance,
$\frac{1}{\pi(r-1/2)}$. The second term on the other hand accounts for the
measure of spatial extension of the defect correlations in the final decohered
state and for the slow sweep and the noiseless scenario is given by,
$(-1)^{r}\sqrt{2/\pi^{2}\tau_{Q}}~{}e^{-(2r-1)^{2}/8\pi\tau_{Q}}.$ The plot of
$\langle\bar{O}_{r}\rangle=\langle O_{r}\rangle-(\text{defect free part})$,
shown in Fig. [4] for the noiseless case, shows the spatial extension of the
defect correlation which clearly has oscillatory damped behavior (long ranged
order) in the slow sweep regime. For the noisy drive scenario, there is an
increment in the defect generation for the slow sweep regime therefore we
expect shift of $\langle\bar{O}_{r}\rangle$ towards negative values as can
also be observed from the Fig. 5. In particular, for $\chi\ll 1$ and $r$ such
that $\chi(2r-1)\leq 1$, $\langle\bar{O}_{r}\rangle$ is given by

$\langle\bar{O}_{r}\rangle\approx-\chi[1-\frac{\chi(4r-2)}{\pi}]+\sqrt{\frac{2}{\pi^{2}\tau_{Q}}}e^{-\frac{(2r-1)^{2}}{8\pi\tau_{Q}}+i\pi
r}.$ (23)

For the second scenario, $\chi\gg 1$ and therefore $p_{k}\approx 1/2$, the
power-law decay of $\langle O_{r}\rangle$ present in the weak noise case is
completely absent and instead replaced by exponential decay terms as can also
be verified from rescaled plot in Fig. [5] and has the following form

$\langle
O_{r}\rangle\approx-(-1)^{r}\frac{e^{-(2r-1)^{2}/4\chi}}{2\pi\sqrt{\chi}}.$
(24)

We note that the correlation length attributed to the exponential decay is
reduced by a factor of $\sqrt{\chi/2\pi\tau_{Q}}$ as compared to the noiseless
case.

###  V.2 Hidden String Correlator

Figure 6:  (a) $\langle\tau^{x}_{0}\tau^{x}_{2j}\rangle$ vs. $j$ for the
noisless scenario exhibits monotonic decay behavior at fast sweep speeds, at
slower sweeps it exhibits oscillatory decay. (b)For fixed $\tau_{Q}=500$, the
increased noise strength results in stronger decay. (c) For fixed noise
strength $\eta_{0}=0.02$, in agreement with the AKZ behavior slower sweep
speeds results in faster decay of the correlation.

Performing the spin duality transformation [39, 30],
$\sigma^{x}_{j}=\tau^{x}_{j-1}\tau^{x}_{j}$ and
$\sigma^{y}_{j}=\Pi^{N}_{l=j}\tau^{y}_{l}$, the Hamiltonian as given in Eq.
[1] acquires the following form in the dual lattice space,

$H_{d}=\sum^{N/2}_{j=1}(J_{1}\tau^{x}_{2j-1}\tau^{x}_{2j+1}+J_{2}\tau^{y}_{2j-1}),$
(25)

which is the usual one dimensional TFQIM.

For $J_{1}>J_{2}$ (i.e., $J_{-}>0$), the order parameter
$\langle\tau^{x}_{j}\rangle$ acquires a finite value with respect to the
ground state. Thus at the same time the expectation value of the operator
$\tau^{x}_{1}\tau^{x}_{2j+1}$ with respect to the final decohered state
acquires long range order implying the presence of hidden string order
parameter in the original spin space. Our objective is to calculate the
expectation value of this string operator

$\tau^{x}_{1}\tau^{x}_{2j+1}=\Pi^{2j}_{l=1}\,\sigma^{x}_{l}=(-i)^{j}(a_{1}b_{1}a_{2}b_{2}.....a_{j}b_{j}),$
(26)

where $a_{j}$ and $b_{j}$ are the Majorana fermions. As discussed above, the
two point correlators which survive at the end of the protocol are of the form
$\langle a_{n}b_{n+r}\rangle$. Therefore the expectation value of the operator
$\langle\tau^{x}_{1}\tau^{x}_{2j+1}\rangle$ can be recast in to the Toeplitz
matrix determinant form [48, 49],

$\langle\tau^{x}_{1}\tau^{x}_{2j+1}\rangle=(-1)^{j}\begin{vmatrix}f_{0}&f_{-1}&....&f_{-(j-2)}&f_{-(j-1)}\\\
f_{1}&f_{0}&....&f_{-(j-3)}&f_{-(j-2)}\\\ .&f_{1}&....&.&.\\\ .&.&....&.&.\\\
.&.&....&.&f_{-1}\\\ f_{j-1}&f_{j-2}&....&f_{1}&f_{0}\end{vmatrix},$ (27)

where the elements of the Toeplitz matrix are given by,

$f_{r}=\langle ia_{n}b_{n+r}\rangle\,\,\text{and}\,\,f_{-r}=\langle
ia_{n+r}b_{n}\rangle.$ (28)

The numerically evaluated expectation values with respect to the finite
separation have been plotted in the Fig. [6]. One notices that the expectation
value of $\langle\tau^{x}_{0}\tau^{x}_{2j}\rangle$ shows monotonic decay at
the fast sweep speeds and exhibits damped oscillatory behavior in the slower
sweep regime. Meanwhile, in the noisy drive scenario, we find that the
correlator is suppressed in the slow sweep regime with the increased noise
strengths, as shown in the Fig. [6], which is in qualitative agreement with
the AKZ picture.

##  VI Kinks statistics

The kink number distribution $P(n)$ describes the probability of observing $n$
kinks. As discussed in the previous section, one obtains TFQIM after the spin
duality transformation of the Kitaev model as given by Eq. [25]. In the limit
of large time $J_{1}$ remains greater than $J_{2}$, therefore the system will
be in the ferromagnetic state where the defects or kinks are the domain walls
in the dual spin space and the operator measuring the number of kinks can be
written as,

$\mathcal{N}=\frac{1}{2}\sum^{N/2}_{j=1}(1-\tau^{x}_{2j-1}\tau^{x}_{2j+1}).$
(29)

By considering the periodic boundary condition, $\tau^{x}_{N+1}=\tau^{x}_{1}$
(with assumption, i.e., $N/4$ is even) and applying the standard procedure of
the JW transformation along with the Fourier transformation and the Bogoliubov
transformation, the TFQIM can be diagonalized yielding,
$H=\sum_{k}E_{k}(\gamma_{k}^{\dagger}\gamma_{k}+\gamma_{-k}^{\dagger}\gamma_{-k}-1)$,
where $\gamma_{k}$ are the Bogoliubov modes satisfying the standard fermionic
anti-commutation relation [38]. The energy for the $k$-mode is given by,
$E_{k}=2\sqrt{J^{2}_{-}\sin^{2}{\frac{k}{2}}+J^{2}_{+}\cos^{2}{\frac{k}{2}}}$.
Furthermore, the kink number operator in terms of the Bogoliubov modes has a
simple representation as a sum of number of quasiparticles,
$\mathcal{N}=\sum_{k>0}\gamma^{\dagger}_{k}\gamma_{k}$, where the average
$\langle\gamma^{\dagger}_{k}\gamma_{k}\rangle$ is the excitation probability
of the quasiparticles,i.e.,
$p_{k}=\langle\gamma^{\dagger}_{k}\gamma_{k}\rangle$ with respect to the final
state and will remain the same as given by Eq. [9] solved for the Hamiltonian
Eq. [6] driven linearly with respect to time by varying the parameter $J_{-}$
and noise in the $J_{+}$ parameter.

Following Ref. [41], the kink number distribution can be written as the
expectation value of the operator $\delta(\mathcal{N}-n)$ with respect to the
the final state $\hat{\rho}_{D}$ at the end of the quench protocol,
$P(n)=\text{Tr}\,[\hat{\rho}_{D}\,\delta(\mathcal{N}-n)],$ where the
characteristic function of $P(n)$ is defined by taking the Fourier transform
of $P(n)$,

$\tilde{P}(\theta)=\int^{\pi}_{-\pi}d\theta P(n)e^{in\theta}.$ (30)

Furthermore, the cumulants generating function are obtained by taking the
logarithm of the characteristic function, i.e.,
$\text{log}\,\tilde{P}(\theta)$, where
$\tilde{P}(\theta)=\Pi_{k>0}\left[1+(e^{i\theta}-1)p_{k}\right].$ The
cumulants $\kappa_{q}$ of the distribution $P(n)$ can be obtained via the
expansion of the cumulant generating function,
$\text{log}\,\tilde{P}(\theta)=\sum^{\infty}_{q=1}\,\frac{(i\theta)^{q}}{q!}\,\kappa_{q}.$
The cumulants of the Poisson binomial distribution function are given by,

$k_{q}=(-i)^{q}\frac{d^{q}}{d\theta^{q}}\text{log}\,\tilde{P}(\theta)|_{\theta=0}.$
(31)

The first cumulant $\kappa_{1}=\langle n\rangle=\sum_{k>0}p_{k}$ is the mean,
the second cumulant is the variance

$\kappa_{2}=\langle n^{2}\rangle-\langle
n\rangle^{2}=\sum_{k>0}p_{k}(1-p_{k}),$

(a) For $\eta_{0}=0$

(b) For $\eta_{0}=0.001$

(c) For $\eta_{0}=0.003$

(d) For $\eta_{0}=0.005$

Figure 7:  Dependence of first three cumulants (density) of the kink number
distribution on the quench time, $\tau_{Q}$, have been plotted for different
noise strengths. Here, the connected black line shows the slope proportional
to $1/\sqrt{\tau_{Q}}$ while the dashed black line is proportional to
$-\chi\log(\chi)$. As can be observed from (a) all three kink cumulants follow
the universal KZ scaling behavior in the slow sweep regime as expected for the
noiseless drive. The noisy drive scenarios as shown in (b), (c), and (d), the
kink cumulants exhibit nonmonotonic behavior with the quench time consistent
with the AKZ picture.

and,

$\kappa_{3}=\langle(n-\langle
n\rangle)^{3}\rangle=\sum_{k>0}p_{k}(1-p_{k})(1-2p_{k}),$

is the third central moment where $p_{k}$ is given by Eq. [9]. For the
Gaussian distribution the first two cumulants are non-zero while the higher
cumulants vanish, i.e., $\kappa_{q}=0$ for $q>2$. Note that any non-zero
higher cumulants tells us about the non-normal features of the distribution.
In the thermodynamic limit, by numerically integrating over $k$, we plot in
Fig [7] the cumulants as a function of the quench time $\tau_{Q}$ for
different noise strengths. For the noiseless limit the scaling behavior of the
cumulants is consistent with the KZM picture, i.e., as shown in the Fig.
[7](a) the first three cumulants exhibits $1/\sqrt{t_{Q}}$ universal KZ
scaling with the quench time in the slow sweep regime as has also been
observed in Refs. [26, 41, 50]. However, the deviations from the KZ scaling
behavior are apparent in the fast sweep regime when $\tau_{Q}<1$ (see Fig.
[7](a)). Another reason for the deviations from the KZM for the noiseless case
can be attributed to a scenario wherein the system size becomes comparable to
the effective separation between two nearest defects [41]. While the above
argument is not relevant for our work, it acquires significance for
experimental scenarios. In the noisy drive, the cumulants exhibit noise
dependent AKZ scaling behavior proportional to the $-\chi\text{log}(\chi)$
term for slower sweeps (see Fig [7]). We find analytically that the source of
the anamolous term is the same as that for the defect density, i.e., from
$\int dkp_{k}$. On the other hand, the AKZ terms arising from $\int
dkp_{k}^{2}$ and $\int dkp_{k}^{3}$ are of the form, $\chi$, which are of the
conventional type. For relatively fast sweep sweep speeds (quench time smaller
than the optimal quench time, but still $\tau_{Q}>1$), the kink cumulants are
unaffected by the noise and follows the KZ scaling behavior (similar to the
case of coherent dynamics in an the isolated quantum system [26, 50]).

To better understand the dependence of the kink cumulants on the quench time,
we plot the ratios of the cumulants for different noise strengths in Fig. [8].
As can be observed in Fig. [8](a) for the noiseless case when $\eta_{0}=0$,
the cumulants $\kappa_{2}\propto\kappa_{1}$ and $\kappa_{3}\propto\kappa_{1}$,
specifically $\kappa_{2}/\kappa_{1}=0.29$ and $\kappa_{3}/\kappa_{1}=0.033$,
which turn out to be independent of the quench time in the slow sweep regime
($\tau_{Q}\gg 1$) as expected from the universal KZM scenario, i.e.,
$\kappa_{q}\propto t^{-1/2}_{Q}$ for all cumulants [41, 50, 51, 26]. For
$\eta_{0}=0.001$ case as seen from Fig. [8](b), the ratios of the cumulants
develop non-universal dependence on the quench time for slower sweeps and
moreover, $\kappa_{3}/\kappa_{1}$ is affected first at relatively small quench
time due to the noise which also shows that the higher cumulants are more
susceptible towards noise, however the amplitude of the higher cumulants
remain very small compared to the first two cumulants. Consider next
$\eta_{0}=0.01$ (see Fig. [8](c)) for which the ratio $\kappa_{2}/\kappa_{1}$
at small quench times are similar to that for the noisless case but increases
at intermediate quench times finally saturating at $0.5$ for very long quench
times. At the same time the ratio $\kappa_{3}/\kappa_{1}$ acquires a maxima
(with quite small value as compared to the ratios $\kappa_{2}/\kappa_{1}$) at
an intermediate quench time beyond which it monotonically decreases with the
ratio approaching zero. In the limit of large quench times and in the presence
of noise the state of the system get maximally scrambled or $p_{k}\rightarrow
1/2$ leading to the above asymptotic values for the ratios of the cumulants.

Figure 8: The ratios of the cumulants $\kappa_{2}/\kappa_{1}$ and
$\kappa_{3}/\kappa_{1}$ have been plotted with the quench time for different
noise strengths, i.e., (a) for $\eta_{0}=0$, (b) for $\eta_{0}=0.001$, and (c)
for $\eta_{0}=0.01$ cases. Figure 9: Plotted are the kink number distribution
for different quench times (taking the system size- $N=800$). The solid lines
represents the Gaussian approximation, i.e., the normal distribution with mean
$\kappa_{1}$ and variance $\kappa_{2}$. Figure 10: Kink distribution has been
plotted for different noise strengths for quench time $t_{Q}=200$ (slow sweep
regime). The solid lines represents normal distribution approximation with
non-universal mean density $\kappa_{1}$ and variance $\kappa_{2}$, which
depend on the quench time as well as the noise strength as described by the
AKZ picture (see Fig. [7]).

Besides the cumulants, we numerically calculate the kink distribution function
by evaluating the inverse Fourier transform of the characteristic function
(Eq. [30]). They are plotted in Figs. [9] and [10] for the noiseless drive
scenario and for the noisy drive case, respectively. The kink distribution
function is well approximated by the normal distribution
$\mathcal{N}(\kappa_{1},\kappa_{2})$ with $\kappa_{2}\propto\kappa_{1}$ (in
spite of having nonzero higher cumulants $\kappa_{q}>0$ for $q>2$) for the
noiseless case (see Fig. [9]) and away from the onset of the adiabaticity,
i.e., $\langle n\rangle>1$ and in slow sweep regime $\tau_{Q}>1$. This was
first reported by the authors in Ref. [41] and later verified in various
experimental studies in Refs. [50, 51, 26]. For the noisy case as can be seen
from Fig. [7], the noise affects the cumulants most in the slow sweep regime
($\tau_{Q}\gg 1$). We plot the kink distribution for different noise strengths
at fixed quench time $\tau_{Q}=200$ (see Fig. [10]), the kink distribution of
the noiseless limit shifts towards higher mean density values and broadens
with increased variance values due to increased noise strengths consistent
with the AKZ picture. Interestingly, the kink distribution is still well
approximated by the normal distribution (depicted by the solid lines) but with
non-universal noise dependent mean $\kappa_{1}$ and variance $\kappa_{2}$
inspite of the noise dependent nonzero higher cumulants (since they still
remain relatively quite small in comparison to first two cumulants even in the
presence of noise for slow sweeps). However, note that $\kappa_{2}$ and
$\kappa_{3}$ are no longer proportional to $\kappa_{1}$ unlike for the
noiseless case.

The recent experimental studies simulating one dimensional TFQIM using
superconducting flux quanta [26] and the quantum annealing protocol for the
two dimensional TFQIM carried out with D-Wave device [21] have also found
deviations from the universal KZM theory. The signature of the AKZ behavior
are clearly visible in the kink density ($\kappa_{1}$) in the slow sweep
regime (large annealing times) which can be attributed to nonzero temperature
effects or due to the coupling of the system to noise source or environment.
These studies indicate that the higher cumulants will also be susceptible to
environmental noise or temperature specifically in the slow sweep regime,
which we have shown explicitly in Fig [7] considering the noisy drive through
the QCP. In the experimental work of Bando et al. [51] the quantum annealing
of the one dimensional TFQIM using the D-Wave device was studied and it was
reported that the cumulants of the kink distribution follow universal KZ
scaling behavior (with $\kappa_{q}\propto t^{-\alpha}_{a}$ , however the KZM
exponent $\alpha$ turns out to be sensitive to the noise-strength or
environmental coupling). Interestingly, the absence of the nonmonotonicity in
the kink density in their measurements was attributed to the fact that their
annealing time range was too short for the noise or temperature related
effects to become significant at the end the drive protocol. All of the above
experimental studies suggest that the effect of noise on the driven system
strongly depends on the problem type as well as on the quench time range which
is also verified from our present study.

##  VII Discussion

In this article, we have studied driven one dimensional Kitaev chain in the
presence of noise. In particular, the Hamiltonian is driven linearly with time
by the parameter, $J_{-}(t)$, while the time dependent Gaussian noise is
present in the parameter, $J_{+}(t)$. In the quenching scheme involving the
parameter $J_{-}$, the quantum phase transition of topological type occurs at
the QCP, $J_{-}=0$, which is related to the change of topological order across
the QCP [30, 44]. The vanishing energy gap at the QCP leads to the diverging
relaxation time which therefore results in the creation of defects/excitations
across the QCP at the end of the quench protocol. We have analytically
calculated the momentum-dependent LZ excitation probability for the quenching
scheme by $J_{-}$ at the final time ($t\rightarrow\infty$) in the limit of
fast noise which clearly differs from the transverse drive protocol of the
TFQIM or the XY chain, i.e., here the new non-adiabatic region opens up around
$k=0$ instead of the $k=\pi/2$ region. This has been utilized to evaluate the
defect density numerically and also analytically which shows the AKZ behavior,
i.e., the slower noisy drives creating more defects however the AKZ term,
$-\chi\text{log}\chi$, in weak noise limit differs from the usual AKZ scaling
behavior reported in all the previous studies. As a result, the optimal quench
time required for the minimization of the defect density exhibits a rather
complicated power law scaling behavior with the noise strength given by Eq.
[14] (see also Fig. [2](b)). The residual energy calculated in the final
decohered state also exhibits the AKZ behavior with AKZ term proportional to
$\chi$ resulting in the conventional power law scaling with the noise strength
of the optimal quench time scale associated with the minimized energy density,
i.e., $\tau^{e}_{O}\propto\eta^{-4/3}_{0}$ (see Fig. [2](d)). In addition, the
numerically evaluated von Neumann entropy density which gives the measure of
decoherence due to the drive through the QCP exhibits a local maxima at an
intermediate quench time scale, either side of which the entropy density
decreases. In the large quench time regime, the von Neumann entropy exhibits
an upturn consistent with the AKZ behavior arising due to the noise. This
behavior of the entropy density can be linked with the crossover behavior of
two-point Majorana correlator at zero separation.

It turns out that certain non-local string correlators can capture the hidden
topological order across the QCP. The two-point Majorana correlator and the
hidden string correlator exhibit signature of KZ scaling behavior in the
noiseless case i.e., slower drive leads to larger correlation lengths. In
particular, we also analytically approximate the two-point Majorana correlator
in the limiting cases for $\chi\ll 1$ and $\chi\gg 1$ as given by Eq. [23] and
Eq. [24], respectively. We next consider the expectation value of the hidden
string correlator with respect to the final decohered state which can be
written as product of two-point Majorana correlators. The product in turn can
be expressed in the Toeplitz matrix determinant form. Evaluating the Toeplitz
determinant numerically, we find that the hidden string correlator exhibits
signature of the AKZ picture of defect generation i.e. the correlator becomes
short ranged in the presence of noise for the slow drive case. The analysis of
kink statistics is performed in the dual space, where cumulants till the third
order were considered. These also exhibit AKZ scaling behavior in the noisy
drive scenario similar to the defect density in the weak noise limit, i.e.,
dominant contribution comes from the AKZ term proportional to,
$-\chi\text{log}\chi$ in $\chi\ll 1$ limit for slower sweeps as shown in Fig
[7]. Furthermore, the kink distribution is well approximated by the normal
distribution with non-universal noise dependent mean density and variance for
slower sweeps as shown in Fig. [10], but unlike the KZM scenario of noiseless
drive case, for the noisy case the higher cumulants are not proportional to
the first cumulant as can also be seen from Fig. [8].

##  VIII  Acknowledgement

S. G. and M. S. are grateful to the SERB, Government of India, for the support
via the Core Research Grant Number CRG/2020/002731. We would like to thank
Prof. S. Chaturvedi and Prof. D. Sen for the useful discussion.

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