text:
\section{Introduction}
Non-interacting disordered systems in one-dimension, isolated from external environment, are known to exhibit Anderson localization. \cite{AndersonPhysRev.109.1492}. One expects that if a generic initial state is allowed to evolve in such a system, at long time scales, the wavefunction will not show significant change. Among the several quantifiers of this phenomenon\cite{MirlinRevModPhys.80.1355}, one of the most commonly used is imbalance, a quantity easy to measure in experiments. Imbalance $I$ is the staggered magnetization and captures the difference in orientation of spins on adjacent sites,
\begin{equation}
I=(1/L)\sum_{j=1}^L(-1)^j \langle\sigma_j^z\rangle,
\end{equation}
where $L$ is the length of the lattice, $\sigma^z$ is the Pauli spin z operator, and the expectation is taken with the state we want to measure the quantity. It should be evident that for computational states, the state with the largest $I$, which equals $1$, is the N\'eel state.

One can also surmise that for an Anderson localized system, if one starts from this state, one should see $I\sim 1$ at large timescales. However, if the localized system is no longer isolated, then the external degrees of freedom typically serve to break Anderson localization. The system eventually forgets the memory of its initial state. Accordingly, $I$ would also evolve with time. In the previous works, the focus has mainly been to find the non equilibrium steady state (NESS) in such systems\cite{Znidaric_2010,ScarPhysRevB.103.184202,PolettiPhysRevA.95.052107,PhysRevB.98.020202Flach,Wu_2019,Monthus_2017,PhysRevLett.118.070402Iva,Vershinina_2017}, or perturbations around NESS\cite{SchiroPhysRevB.104.144301}, while recently there has been some work studying how different observables relax to NESS in such systems\cite{MariyaMarkoPhysRevB.93.094205,GopalKnapPhysRevLett.119.046601,PolettiPhysRevLett.114.170401,PhysRevLett.116.237203Garrahan,10.21468/SciPostPhys.12.5.174BarLev,PhysRevB.95.024310Garrahan,PhysRevB.100.165144Poletti} or in Stark localized systems\cite{EckardtPhysRevLett.123.030602}. Of particular focus was the decay of imbalance. A slow stretched exponential ($A \exp(-t^{\alpha})$) decay of $I(t)$ has been reported\cite{AltmanPhysRevLett.116.160401,CaiBarthelPhysRevLett.111.150403,CaiPhysRevLett.124.130602,SchneiderPhysRevB.106.134211,PhysRevLett.116.237203Garrahan}, though the value of $\alpha$ obtained has some dispute. For example, a short theoretical analysis in Ref.~\onlinecite{CaiPhysRevLett.124.130602} finds $\alpha\sim 0.33$ whereas a different analysis in Ref.~\onlinecite{AltmanPhysRevLett.116.160401} puts $\alpha\sim 0.5$. Numerical fits in Refs.~\onlinecite{AltmanPhysRevLett.116.160401} puts $\alpha \sim 0.38$ and Refs.~\onlinecite{SchneiderPhysRevB.106.134211,PhysRevLett.116.237203Garrahan} put $\alpha \sim 0.42$.  Additionally, the analytical expressions suggested usually require at least one fitting parameter to be matched with the numerical results.

In our work, we seek to understand and resolve this inconsistency. Upon carefully analyzing the relaxation, we see that several time scales emerge. Broadly speaking one has a regime where off-diagonal matrix elements of $\rho(t)$ are large, then a regime in which $\rho(t)$ becomes increasingly diagonal due to dephasing and the scaling variable is~\cite{MariyaMarkoPhysRevB.93.094205} $\tau=8\gamma t/W^2$, and finally a regime where the system starts to feel its finite size $L$. In fact, we argue that the `stretched exponential' is not a true (asymptotic) description for the relaxation. It holds only in a finite window, and is universal and independent of disorder type just in a linear-expansion regime, after which a non-generic disorder-dependent behaviour follows. Additionally, beyond a timescale, $\tau=8 \gamma t/W^2\sim1$, we show that the relaxation smoothly changes to $1/(L\sqrt{t})$ from the non-generic behaviour in open boundary conditions, which is also the standard result for clean systems with dephasing. Finally, for finite size systems, an exponential decay occurs at the longest timescale, governed by the finite Liouvillian gap.

Using a toy model involving three lattice sites, we find an expression of $I(\tau)$ analytically, using second order perturbation theory, that matches with numerics very accurately, till $\tau\sim 1$, without any fitting parameters. From the model, we show that the universal behaviour $I(\tau)=(1-\beta \sqrt{\tau})$ occurs for very short times $\tau\ll 1$, beyond which the non universal behaviour follows till $\tau\sim 1$.

We first provide a brief description of our main results in Sec.~\ref{summary} before going into the details later.

\section{Summary of Results}
\label{summary}
We begin by summarizing our findings. Starting from the N\'eel initial state, in the XX model with on-site disorder and dephasing, the behaviour of $-\ln I(t)$ can be schematically represented by Fig.~\ref{fig1}(a), where $\ln$ denotes the natural logarithm, $\gamma$ is the strength of dephasing, $W$ is the strength of disorder and $J$ is the exchange interaction strength of XX model. We use open boundary conditions (obc) unless specified.

\begin{figure}
\centering
\includegraphics[width=0.85 \columnwidth]{plots/schematic.pdf}
 \includegraphics[width=0.9 \columnwidth]{plots/fig1b.pdf}
 \caption{(a) A schematic figure denoting the five timescales of imbalance decay for an initial N\'eel state. (a) shows schematic diagram with timescales $t_0\sim {\rm Min}(1/\gamma,1/W)$, $t_1\sim 0.1 W^2/(8 \gamma J^2 )$, $t_2 \sim  W^2/(8 \gamma J^2 )$ and $t_3 \sim L^2$, whereas in (b) disorder averaged exact numerical data for different system sizes $L$ is plotted with $\tau=8 \gamma t/W^2$ and $J=1$, while $\beta$ is obtained from Taylor expansion of Eq.~\ref{box} with $\gamma=1$ and $W=8$. The green vertical lines denote the timestamps of the schematic diagram for $L=50$ (open boundary conditions are used).
 }
\label{fig1}
\end{figure}
As evident from the figure, five timescales emerge during the evolution of $I(t)$.  They are,
\begin{enumerate}[I.]
\item $t < t_0 \sim {\rm Min} (1/\gamma,1/W)$: The shortest timescale in the problem. Here, $ I(t) \sim 1-t^2$ and hence $-\ln I(t) \sim t^2$. During this time period, off diagonal correlations first develop in the system and then start decaying after reaching a maxima.\footnote{Recall that since we start from the N\'eel state, a computational state, and hence at $t=0$ the system is completely uncorrelated.} When the disorder strength $W$ is small compared to the dephasing $\gamma$, $t_0$ is given by $\sim 1/\gamma$, akin to clean systems. However when $W \gg \gamma$, then this behaviour exists till $t \sim 1/W$.
\item $t_0 < t < t_1 \sim 0.1 W^2/(8 \gamma J^2 )$: This is where local relaxation of $\sigma^z$ starts, and most (but not all) off-diagonal correlations become negligible. In this timescale one sees $-{\rm \ln} I(t)\sim -{\rm \ln}(1-\beta \sqrt{t}) \sim \beta \sqrt{t}$ behaviour irrespective of the nature of disorder chosen ($\beta$ can depend on nature of disorder). This is the short time, linear regime of the stretched exponential behaviour noted in previous works\cite{AltmanPhysRevLett.116.160401,CaiBarthelPhysRevLett.111.150403,CaiPhysRevLett.124.130602,SchneiderPhysRevB.106.134211,PhysRevLett.116.237203Garrahan}. Unlike previous results, we find that this regime is not asymptotic and is only valid till $\tau\sim\tau_1=(8 \gamma J^2 t_1)/W^2 = 0.1$. Therefore, the ``stretched exponential'' is not really a true stretched exponential decay as it does not hold asymptotically, i.e., at arbitrarily small values of $I$.
\item  $ t_1<t<t_2\sim W^2/( 8 \gamma J^2)$: In this regime, which holds till $\tau\sim\tau_2= (8 \gamma J^2 t_2)/W^2  = 1$ a non-generic decay dependent on the choice of disorder distribution is seen. $t_2$ also marks the end of local relaxation in the system.
\item $t_2<t< t_3$: In this regime,  the decay smoothly changes to a power law due to the bunching of low magnitude eigenvalues of Liouvillian.\cite{CaiBarthelPhysRevLett.111.150403,MedvKehrinPhysRevB.90.205410} This is the asymptotic behaviour in the thermodynamic limit for finite $W$. Here, $I(t)\propto 1/(L\sqrt{t})$. The exponent -- $1/2$ in our case -- can depend on boundary conditions. The change of the nature of decay from Region III to IV occurs slowly at a timescale $\propto L$, not shown in the schematic diagram. For finite systems, due to the discrete nature of the Liouvillian spectrum, this behaviour is not asymptotic and there is another timescale.
\item $t> t_3 \sim L^2$: The final timescale in finite sized systems is governed by the Liouvillian gap, i.e. the largest real non zero eigenvalue of the Liouvillian (since it has an eigenvalue of zero). Here the decay is given by $I(t)\sim \exp(-|\lambda_1| t)$ where $\lambda_1$ is the Liouvilian gap. This timescale begins around $t_3 \sim 1/|\lambda_1|$. As will be discussed in Sec.~\ref{region5}, $\lambda_1\sim 1/L^2$, hence this regime starts around $t_3 \sim L^2$.
\end{enumerate}

It is worth noting that till $t \sim t_2$, i.e. till Region III, there is no system size dependence of the results, neither the time windows nor the behaviour depends on $L$. However, both the behaviour of $I(t)$ and the time span of Region IV is dependent on $L$. This feature is clearly visible in Fig.~\ref{fig1}(b) where we plot numerical results obtained for evolution of $-{\rm \ln} I(t)$ for a disorder strength of $W=8$ for different system sizes. We perform disorder averaging over $10^2$ realizations for $L=50,500$ and $10$ for $L=3000$.  We also mark the various timescales in this plot, now  with respect to $\tau$. Clearly, for $\tau>1$, different system sizes start showing different  behaviour.

Since till $\tau\sim 1$, the rescaled data for the different system sizes clearly overlap with one another and is almost linear in $\ln-\ln$ scale, this prompted the stretched exponential fits in previous works. In Fig.~\ref{fig1}(b), we have added a black dashed line showing the stretched exponential expression obtained from a theoretical computation in Sec.~\ref{region23}. As will be clear from the analysis in that section, this behaviour is expected to hold best till small $\tau \sim 0.1$, which is what is demonstrated in the plot. \footnote{Hence numerical fits of the form $\exp(-t^{\alpha})$ in Refs. \onlinecite{AltmanPhysRevLett.116.160401,CaiBarthelPhysRevLett.111.150403,CaiPhysRevLett.124.130602,SchneiderPhysRevB.106.134211} found different $\alpha$s depending on fit windows as the behaviour is not a true stretched exponential. } Finally for $L=50$ one sees a fast growth of $-\ln I(t)$ (see a slight bend upwards) at large times, which is due to the exponential decay of $I(t)$ from the finite size Liouvillian gap.

In what follows, we shall discuss the above findings in details and provide a theoretical understanding for the same. In the next section, we describe the model in more details and elaborate on the technique used to compute the numerical results. Then in Sec.~\ref{timescales} we elaborate on the behaviour of $I(t)$ in different timescales. Finally, in Sec.~\ref{discussion} we provide some final remarks on our results and possible extensions of this work.
\section{Model}
\label{model}
We take the one dimensional disordered XX chain with spin 1/2 particles,
\begin{equation}
H=-J\sum_{j=1}^{L-1}(\sigma^x_j\sigma^x_{j+1}+\sigma^y_j\sigma^y_{j+1})+\sum_{j=1}^L h_j\sigma^z_j
\label{lind1}
\end{equation}
where $h_j$ are the on-site disorders and $\sigma^{x,y,z}$ denotes the Pauli matrices. In this work, unless otherwise mentioned, $h_j$s are chosen from a uniform distribution in $(-W,W)$, $W$ being the strength of disorder. We set $J=1$ for the rest of our work. To simulate an open system, each spin is exposed to a dephasing. The evolution of the system's density matrix is given by,
\begin{equation}
\frac{{\rm d}\rho}{{\rm d} t}=i [\rho,H]+\mathcal{L}^{{\rm deph}}(\rho)=\mathcal{L}(\rho)
\end{equation}
The non-unitary part can be written in terms of Lindblad operators as, $\mathcal{L}^{{\rm deph}}(\rho)=\sum_k([L_k \rho,L_k^{\dagger}]+[L_k, \rho L_k^{\dagger}])$. We choose $L_k=\sqrt{\frac{\gamma_k}{2}}\sigma_k^z$ to represent the on site dephasing term. For the rest of this work we shall put $\gamma_k=\gamma$, i.e. have a spatially uniform dephasing. This choice of dephasing causes an exponential decay of the off diagonal elements of the density matrix with a strength $\gamma$, in the diagonal basis of $\sigma^z$ in absence of disorder.

If we want to solve Eq.~\ref{lind1} directly, numerically or otherwise, we need to solve a set of $4^L-1$ coupled differential equations. Fortunately due to the choice of dephasing and the quadratic nature of the Hamiltonian, the exponentially many equations can be decoupled into blocks of polynomial complexity. \cite{Znidaric2013,Znidaric_2010,PolettiPhysRevA.98.052126,Eisler_2011,Temme_2012,CiracPhysRevA.87.012108,arxiv.2210.10856,SchiroPhysRevB.104.144301} In other words, observables follow a hierarchy based on the number of Fermionic operators they contain. For example, the block of two point correlators decouples from the rest, and one can write a closed set of equations for these observables. Then this solution serves as a source term for the three point correlations and so on.  Since we are interested in imbalance, which can be extracted from a two-point correlator in the Fermionic language $(\sigma^z \sim c^{\dagger} c)$, we will just consider the subspace of two point correlation functions. Following the consideration of. Ref.~\onlinecite{Znidaric_2010,Znidaric2013}, we define the operators,
\begin{eqnarray}
A(t)&=&\sum_{r=1}^L \sum_{j=1}^{L+1-r}a_j^{(r)}(t)A_j^{(r)}\nonumber \\
B(t)&=&\sum_{r=2}^L \sum_{j=1}^{L+1-r}b_j^{(r)}(t)B_j^{(r)},
\label{hierarchy}
\end{eqnarray}
where $A_j^{(r+1)}=\sigma_x^j Z_{j+1}^{(r-1)} \sigma_x^{j+r}+\sigma_y^j Z_{j+1}^{(r-1)} \sigma_y^{j+r}$ and $B_j^{(r+1)}=\sigma_x^j Z_{j+1}^{(r-1)} \sigma_y^{j+r}-\sigma_y^j Z_{j+1}^{(r-1)} \sigma_x^{j+r}$ for $r\ge 2$. $Z_j^{(r)}=\sigma_z^j\hdots \sigma_z^{j+r-1}$ are strings of $\sigma_z$ operators, and $A_j^{(1)}=-\sigma_z^j$. Then the equation governing the time evolution  of the set of two point correlation functions can be written compactly as,
\begin{equation}
\frac{{\rm d}\mathbf{C}(t)}{{\rm d} t}+2 {\rm i} (\mathbf{P}\mathbf{C}(t)-\mathbf{C}(t)\mathbf{P}^T)+2(\mathbf{\Gamma}\tilde{\mathbf{C}}(t)+\tilde{\mathbf{C}}(t)\mathbf{\Gamma})=0,
\label{diffeq}
\end{equation}
where $\mathbf{C},\tilde{\mathbf{C}},\mathbf{P},\mathbf{\Gamma}$ are $L \times L$ matrices. Their elements are defined as, $C_{jk}(t)=a_j^{(k-j+1)}(t)+{\rm i} b_j^{(k-j+1)}(t)$ for $k>j$, $C_{jj}(t)=a_j^{(1)}(t)$ and $C_{jk}=C_{kj}^*$. $\tilde{\mathbf{C}}=\mathbf{C}-{\rm diag} (\mathbf{C})$. $\mathbf{P}=\mathbf{W}-\mathbf{T}$ where, $W_{jk}=h_k \delta_{jk}$, the on-site disorders and for our model. Also for our model, $T_{jk}=J(\delta_{j,k-1}+\delta_{j,k+1})$, as we consider only nearest neighbour couplings \footnote{For Fermionic Hamiltonians, if longer range couplings are present then in the most general case,  $T_{jk}=J_{jk}$ where $J_{jk}$ denotes the hopping strength between two different sites at positions $j$ and $k$.} and  $\Gamma_j^k=\gamma \delta_{jk}$. Clearly, $I(t)=(1/L)\sum_j(-1)^{j+1}C_{jj}(t)$. Hence for the N\'eel initial state $C_{jj}(0)=(-1)^{j+1} $ and $I(0)=1$.

Eq.~\ref{diffeq} can be recast into a linear differential equation with $L^2$ variables, in the form,
\begin{equation}
\frac{d\mathbf{f}}{dt}=\mathcal{Q} \mathbf{f}
\label{linearised}
\end{equation}
where $\mathbf{f}=(C_{11},C_{12},\hdots,C_{1L},C_{21},\hdots,C_{LL})$ and $\mathcal{Q}$ is the $L^2 \times L^2$ matrix governing the evolution. In fact, due to the hierarchical structure of observables, the eigenvalues of $\mathcal{Q}$ are exactly the eigenvalues of the one particle sector of the Liouvillian, $\mathcal{L}$, while the eigenvectors of both are connected via an appropriate rotation. This linearised equation will be useful in understanding the behaviour of $I(t)$ in different regimes.

Unlike most of the previous results for this model which are generated by either an effective Hamiltonian or DMRG based techniques \cite{AltmanPhysRevLett.116.160401,CaiPhysRevLett.124.130602}, we use Eq.~\ref{diffeq} or Eq.~\ref{linearised} for the analysis that follows. This not only allows us to obtain an exact description of the correlators but also provides access to sizes and times an order of magnitude larger than usually accessed by DMRG techniques. Thus, we can make better statements about system size dependence of the results.

\section{Different timescales}
\label{timescales}

Let us now discuss each timescale in more details.
\subsection{Region I}
\label{reg1}
\begin{figure}
\includegraphics[width=0.75 \columnwidth]{plots/ultrashorttime.pdf}
\caption{Figure showing the initial $t^2$ behaviour of $-\ln I(t)$ in region I for different disorder strengths $W$.}
\label{fig2}
\end{figure}
Doing a simple power series expansion of $I(t)$ in $t$, the first non-zero term turns out to be quadratic. Hence, we expect $I(t)\sim 1-\eta t^2$ behaviour at the smallest timescale. As evident from Fig.~\ref{fig2}, this is the case. This behaviour continues till time $t_0$ which is dependent on $\gamma$ and $W$. To be more precise, let us first consider the simpler case, $W\rightarrow 0$. At $t=0$, since we start from a pure state, $\rho(t=0)$ has only one non zero element, which is in the diagonal, and $C(t=0)$ is diagonal as well. Correlations then spread rapidly throughout the system, reach a maxima and start decaying around $t \sim 1/\sqrt{|\gamma^2-4|} \sim 1/\gamma$ for $\gamma \gg 1$. As we see from Fig.~\ref{fig2}, indeed below $t_0 \sim 0.5$ for $\gamma=1$ and $W=0$, one can approximate $I(t)$ as $1-\eta t^2$, and hence $-\ln I(t)\sim \eta t^2$. From the numerical fit in Fig.~\ref{fig2} shown by the black dashed line, we find $\eta \sim 16$. This behaviour can also be qualitatively extracted from a simple two-site model (details in Appendix~\ref{appA1}), and we get, for small $t$, $ I(t) \sim 1-8 t^2 +O(t^3)$. One can then numerically check that on addition of a few more sites to the system  $\eta$ approaches $16$ rapidly.

However, on addition of disorder, $t_0$ reduces with increasing $W$, as can be seen Fig.~\ref{fig2}. This can be qualitatively understood from Eq.~\ref{diffeq}, where we see that correlations can develop in the system either due to $\mathbf{P}$, i.e. the terms involving disorder or due to $\mathbf{\Gamma}$ which involves dephasing. The dominant term would then determine the smallest timescale. Also, it can be shown using the two-state model again that, for $W \gg \gamma$, the $t_0$ now becomes $1/\delta$, $\delta$ being the difference of on-site disorders in the two-site problem.  Hence several timescales emerge due to different $\delta$s at different lattice sites, and we will get $t_0 \ll 1/\gamma$. If we approximate the width of the distribution of $\delta$s as $W$, we can say that $t_0$ is around $ {\rm Min}(1/\gamma,1/W)$. This is Region I.

\begin{figure*}
\includegraphics[width=0.9 \columnwidth]{plots/parametertheorycompare2.pdf}
\includegraphics[width=0.9 \columnwidth]{plots/distributiontheorycompare.pdf}
\caption{ Comparison of analytical predictions with exact numerics in Regions II and III. (a) Plot showing how the rescaled time causes complete collapse of data with different $L,\gamma, W$, and agreement of the said numerical results for the box distributed random on site disorder, with the theoretical predictions. $f(\tau)$ is defined in Eq.~\ref{firstapprox} and $g(\tau)$ is defined in Eq.~\ref{box}. (b) Plot showing agreement of our theoretical prediction with numerical results derived from for two other disorder  distributions, viz.  gaussian random and alternating disorder, see Eq.~\ref{gaussian} and Eq.~\ref{alternating} respectively.  $f_G(\tau)$ is obtained from Eq.~\ref{firstapprox} where the random numbers are picked from a Gaussian distribution.}
\label{fig3}
\end{figure*}
\subsection{Regions II and III}
\label{region23}
Next, we shall understand the decay of imbalance in Regions II and III. Local evolution is dominant in these regimes and we can use the same analysis to describe both. Hence we discuss them together. A timescale separating the two regimes will emerge naturally from the computation that follows.

We consider $W \gg \gamma$, a regime where localization effects would be prominent in the system. Then, to understand the evolution of $\langle\sigma_j^z \rangle$ for site $j$, we need to take into account the influence of the neighbouring sites, $j-1$ and $j+1$. Matrix $\mathbf{C}$ in Eq.~\ref{hierarchy} for this $3$-site system, becomes a $3 \times 3$ matrix. We further ignore $C_{ij}$s where $|j-i|>1$, as they are negligible compared to the rest in this timescale. We can then use Eq.~\ref{linearised} with
\begin{equation}
 \mathbf{f}=\begin{pmatrix}
 C_{11}, & C_{12}, & C_{21}, & C_{22}, & C_{23}, & C_{32}, & C_{33},
 \end{pmatrix}
 \label{threesitef}
 \end{equation}
 where we have taken $j=2$ without loss of generality. The form of $\mathcal{Q}$ for this system in Eq.~\ref{linearised} is given in Appendix~\ref{appA2}. We use second order perturbation theory to find the eigenvalues as outlined in Appendix~\ref{appA2}, since exact analytical diagonalization is not tractable. We also just need to diagonalize the subspace where we have the smaller modulus eigenvalues, as the larger modulus eigenvalues dictate the behaviour in Regime I. Doing so, we obtain the relevant eigenvalues as given in Eq.~\ref{eigenvalues} in Appendix.~\ref{appA2}. These in the regime $ W \gg \gamma$ can be approximated as $\Delta^{(0)}=0$ and,
\begin{equation}
\Delta^{(\pm)}\sim\frac{8 \gamma  (\delta_1^2+\delta_2^2\pm\sqrt{\delta_1^4+\delta_2^4-\delta_1^2\delta_2^2})}{\delta_1^2 \delta_2^2},
\label{approxeigen}
\end{equation}
where $\delta_1=|h_{1}-h_2|$ and $\delta_2=|h_2-h_{3}|$.
Hence the long time behaviour of $C_{22}(t)=-\langle\sigma_2^z(t)\rangle$, can be generically written as,
\begin{equation}
C_{22}(t)=\sum_{m=\pm,0}d^{(m)}_2 \exp(-\Delta^{(m)} t)
\label{C22three}
\end{equation}
where $d^{(m)}_2$ is the corresponding element of the $m^{th}$ eigenvector, weighed by the factors arising from  the initial conditions. We shall drop the subscript $2$ in what follows.

Further simplification of Eq.~\ref{C22three} can be made by observing that typically (see Appendix~\ref{appA2}) $d^{(+)}$ is the largest coefficient, hence the principal mode of relaxation is $\Delta^{(+)}$. The contribution from $\Delta^{(0)}$ and $\Delta^{(-)}$ modes weighted by $d^{(0)}$ and $d^{(-)}$ respectively effectively act as perturbations around the principal mode\footnote{This is difficult to rigorously prove analytically, but can be seen numerically} (see Appendix~\ref{appA2}). In the random disorder case since $\Delta^{(+)}$s are different for different sites, we reintroduce the subscript $j$, and write for the N\'eel initial state, $ I(t)=(1/L)\sum_{j=1}^L|C_j(t)| \sim(1/L)\sum_{j=1}^L |d_j^{(+)}| e^{-\Delta^{(+)}_j t}$, where the perturbative corrections due to the other modes can be neglected due to averaging.
 Additionally, since $d_j^{(+)}$s are $\mathcal{O}(1)$(see Appendix~\ref{appA2}), it is not essential to keep track of them. Hence, given a disorder distribution one can calculate the evolution of imbalance for the N\'eel initial state in the thermodynamic limit as,
 \begin{equation}
 I(t)= f(t,W,\gamma)=\lim_{L \to \infty}(1/L)\sum_{j=1}^L e^{-\Delta^{(+)}_j t},
 \label{firstapprox}
\end{equation}
where different $\Delta^{(+)}_j$s are obtained from the distribution of $h_j$s, using Eq.~\ref{approxeigen}. From Eq.~\ref{firstapprox} we can see that in this timescale, $I(t)$ is the average over the decay of magnetization at individual sites with effective decay rates at each site controlled by the disorders in its nearest neighbours. From Eq.~\ref{approxeigen}, we also see an emergent energy scale $8 \gamma/W^2$ (recall $J=1$), which gives us the rescaled time $\tau=8 \gamma t/W^2$.

  In Fig.~\ref{fig3}(a) we show comparison between exact numerical results with Eq.~\ref{firstapprox} and other approximations described later in the section. The plots for different system sizes $L$, disorder strengths $W$ and dephasing $\gamma$ obtained via exact numerics collapse on each other when we use rescaled $\tau=8 \gamma t/W^2$, confirming the presence of the universal timescale\cite{MariyaMarkoPhysRevB.93.094205}. As shown in Fig.~\ref{fig3}(a) with the black dashed line, computing Eq.~\ref{firstapprox} by sampling a large number of random numbers from the box disorder distribution, we get a result that is a very accurate match with exact numerics till $\tau \sim 1$. Note how a numerical result obtained via solving many coupled differential equations can be replicated by correctly sampling the underlying disorder distribution, due to locality of the evolution.

However, it is difficult to perform the summation analytically to obtain a closed form expression for different disorder distributions. To proceed further we need to go to the continuum limit and write Eq.~\ref{firstapprox} as $I(t)=\int d \Delta^{(+)} p(\Delta^{(+)}) e^{-\Delta^{(+)}t}$, $ p(\Delta^{(+)})$ being the probability distribution of $\Delta^{(+)}$. Even so, computing $p(\Delta^{(+)})$ analytically is a very difficult task. Hence, we need to make further simplifying assumptions. If we approximate $ \delta_1^4+\delta_2^2-\delta_1^2\delta_2^2\sim(\delta_1^2+\delta_2^2)^2$ and hence $\Delta^{(+)} \sim 16\gamma ( \frac{1}{\delta_1^2}+\frac{1}{\delta_2^2})$, the integral becomes tractable. This approximation is valid in the limit of $|\delta_1^2- \delta_2^2| \gg 0$, where we have $\delta_1^2\delta_2^2 \ll \delta_1^4+\delta_2^4$ and hence works in the large $\Delta^{(+)}$ tail of $p(\Delta^{(+)})$ very well, see Fig.~\ref{fig6a} in Appendix.~\ref{appA2}. \footnote{This choice of approximation is used to obtain a tractable form for $p(\Delta^{(+)})$. Other choices are also possible in the said limit, but they do not simplify the calculation. We show a comparison of the probability distribution of the approximated and exact $\Delta^{(+)}$ in Appendix~\ref{appA2}}. Then, we consider that $\delta_1$ and $\delta_2$ are independently drawn from different sets of random numbers to avoid calculating the complicated convolution term. This allows us to write,
\begin{equation}
I(t)=\left(\int d\delta p(\delta) e^{-16 \gamma t/\delta^2}\right)^2
\label{master1}
\end{equation}
where $p(\delta)$ is the probability distribution of $\delta$s. Eq.~\ref{master1} allows us to compute the evolution of $I(t)$ analytically for any given disorder distribution. For example, for a disorder distributed uniformly in $(-W,W)$, i.e. the box distribution, Eq.~\ref{master1} gives the result in terms of known mathematical functions as,
\begin{eqnarray}
I(\tau=8 \gamma t/W^2)=g(\tau)&=&[-\sqrt{2 \pi } \sqrt{\tau } \text{erfc}\left(\sqrt{\frac{\tau }{2}}\right)\nonumber \\
&&+e^{-\frac{\tau }{2}}+\frac{1}{2} \tau  \Gamma \left(0,\frac{\tau }{2}\right)]^2
\label{box}
\end{eqnarray}
where $\text{erfc(z)}=1-\text{erf}(z)$, ${\rm erf}$ denotes the Error function and $\Gamma$ is the incomplete gamma function defined by $\Gamma(0,z)=\int_z^{\infty}e^{- t}/t dt$. The green dashed line in Fig.~\ref{fig3}(a), obtained from Eq.~\ref{box} shows good agreement till $\tau \sim 0.1$, which is expected since our approximation works best in the large $\Delta^{(+)}$ tail.

\paragraph*{\textbf{$1-\beta \sqrt{t}$ scaling regime:}} Actually, in the regime of $\tau \ll 1$ or $t \ll W^2$, Taylor expanding Eq.~\ref{box} in $\tau$, we get $I(\tau)=1-\beta\sqrt{\tau} +O(\tau)$ with $\beta=\sqrt{8\pi}$, which is exactly the linear term in the stretched exponential\cite{CaiBarthelPhysRevLett.111.150403}  $e^{-\beta \sqrt{\tau}}$. In Fig.~\ref{fig3}(a), we plot $-\ln I(\tau)\sim \beta \sqrt{\tau}$ as the brown dashed line, and it also agrees well with exact numerics till $\tau \sim 0.1$.

Generally, if $p(\delta)$ is an analytic function of $\delta$,  then it can be expanded as $p(\delta)=\frac{1}{\mathcal{N}}(c+O(\delta))$, where $\mathcal{N}=\int d\delta (c+O(\delta)) $. Since $\int d\delta e^{-16 t/\delta^2}\sim t^{1/2}+O(t)$, the lowest order term obtained from $p(\delta)$ in this form would be $\sqrt{t}$ (for $c=0$ the result would be different).  For many different disorder distributions this is a good approximation of results till higher order terms become important with larger $t$.

To highlight this special scaling, we label the regime where the linear approximation of $\beta \sqrt{\tau}$ is valid as Region II which is approximately till $\tau=0.1$, and denote the non-generic regime during $0.1\lesssim\tau \lesssim 1$ as Region III.

\paragraph*{\textbf{Other disorder distributions:}} To further consolidate our claims, we repeat the above calculations for other disorder distributions, viz. the Gaussian random distribution and staggered on-site or alternating potential, shown in Fig.~\ref{fig3}(b). We have also checked the quasiperiodic Aubre Andre distribution, data not shown to avoid cluttering. In Fig.~\ref{fig3}(b), results from Eq.~\ref{firstapprox} is represented by the green dashed line for the Gaussian disorder case, and the black dashed line for the alternating potential case. In both cases (and the qausiperiodic one not shown), we see that Eq.~\ref{firstapprox}, obtained from the second order perturbation result of the $3$-site model, captures the nuances in the evolution very well till $\tau=1$ (longer for alternating potential see Appendix~\ref{appA2}). In general, our analysis is valid for any disorder distribution where typical $|h_j-h_{j+1}| \gg0$, to make the second order perturbation theory hold.\footnote{One can still numerically solve the small size Hamiltonians to obtain the eigenenergies non-perturbatively and continue with the analysis.}

For the Gaussian distribution it is also possible to compute Eq.~\ref{master1} analytically and the result is,
\begin{eqnarray}
I(\tau)=&e^{- \sqrt{8\tau}},
\label{gaussian}
\end{eqnarray}
a different result to Eq.~\ref{box} and exactly the stretched exponential\footnote{Ref.~\onlinecite{AltmanPhysRevLett.116.160401} arrived at this result using a different approach effectively leading to the same integral. However, under their approximations one arrives at this result irrespective of the disorder chosen and then needs to fix a free parameter via numerical fitting to fit to different disorders}. This is shown as the purple dot-dashed line in Fig.~\ref{fig3}(b) which we again see to be valid till $\tau\sim 0.1$, thus showing a separation of timescales similar to box distribution. Other standard disorder distributions such as the exponential and student's T distribution also show similar timescale separation.

Finally, for the staggered potential, i.e when $h_j=(-1)^j W$, Eq.~\ref{firstapprox} has a simple form. Since we have $\delta_1=-\delta_2=\delta/2= W$, $\Delta^{(+)}=6 \gamma/W^2$ for all sites. In this case the weight of the $\Delta^{(-)}$ mode, $d^{(-)}=0$, hence $I(t)\sim d^{(0)}+d^{(+)} e^{-\Delta^{(+)} t}$. For large $W$, this can be approximated as $f_{{\rm alt}}=e^{-\Delta_{{\rm eff}} t}$, where  $\Delta_{{\rm eff}}=\frac{8 \gamma}{W^2}$ and hence
\begin{equation}
I(\tau) \sim e^{-\tau}.
\label{alternating}
\end{equation} See Appendix~\ref{appA2} for more details. In Fig.~\ref{fig3}(b), we see exactly the expected behaviour. The almost perfect agreement between our theoretical predictions with the numerical data provides a good benchmark about the generality of our theory.

\paragraph*{\textbf{Effective Hamiltonian approach:}} Before we end this section we make a digression to briefly discuss an alternative approach.  In Ref. \onlinecite{MariyaMarkoPhysRevB.93.094205} it was shown that, for the interacting XX chain with dephasing and strong disorder, beyond a timescale given by $1/\gamma$, off-diagonal terms of the density matrix are negligible and the evolution of the diagonal terms of the density matrix is given by the differential equation, $\frac{d \bm{\rho}_D}{dt}=-\mathcal{H}_{\rm eff}\bm{\rho}_D$, where $\bm{\rho}_D$ denotes the diagonal part of $\bm{\rho}$ and
\begin{equation}
\mathcal{H}_{\rm eff}=\sum_{j=1}^L 2 \frac{(\gamma_j+\gamma_{j+1})}{(h_j-h_{j+1})^2+(\gamma_j+\gamma_{j+1})^2}(1-\bm{\sigma}_j.\bm{\sigma}_{j+1}),
\label{effham}
\end{equation}
 where $\bm{\sigma}=(\sigma^x,\sigma^y,\sigma^z)$. An emergent rescaled time, $ 4\gamma t/(W^2+4 \gamma^2)$ is seen the overall prefactor of $\mathcal{H}_{\rm eff}$, if we consider $\gamma_{j}=\gamma$ and $\delta_j=h_j-h_{j+1}\sim W$, reminiscent of the one we extracted from the three site model, since this is also a second order perturbative description. In our non-interacting XX model, this effective description still holds as there is no term involving any interaction in Eq.~\ref{effham}. Beyond a cutoff timescale, which for $L=24$ and $W=30$ is given by $\tau \sim 0.01$ (check Appendix~\ref{appB}) the agreement of exact numerics and evolution of $I(t)$ with $\mathcal{H}_{\rm eff}$ is excellent. This means the effective Hamiltonian describes the system approximately for $\tau \gg \tau_0$ in our model ($\tau_0 \sim 0.001$ for $W=30$), while before that timescale the off diagonal elements of the density matrix still has significant contribution to the evolution. While, a direct use of $\mathcal{H}_{\rm eff}$ to repeat the analysis of this section to understand Region II and III are a bit involved, we shall use this effective Hamiltonian to explain the behaviour in Region IV and V in the following sections.

Finally, it is worth noting that For $W=0$, Regions II and III no longer exist and there is an oscillatory behaviour in $-{\rm \ln} I(t)$ as can be seen in Fig.~\ref{fig2}, with an envelope growing as $4 \gamma t$. For large $\gamma$ one can again apply the effective Heisenberg model to obtain these results for this system as in Ref.~\onlinecite{CaiBarthelPhysRevLett.111.150403}. Upon analysis via our three site model, we observe that the magnitude of the real part of eigenvalues involved in the evolution $\sim 4\gamma$ as the corresponding eigenvectors carry almost all the weight, a distinct shift from what happens in the disordered case, where lower magnitude eigenvalues carry most of the weight. Near the end of this time-scale the system begins to realize its non-local nature, and the evolution slowly changes to what is seen in Region IV, discussed below, irrespective of the presence of disorder.

\subsection{Region IV}
\label{region4}
\begin{table*}
\centering
\begin{tabular}{|c|c|c|c|c|c|}
\hline
Case & disorder & $\alpha$ & $\beta$& $I(t)$ &$-\ln I(t)$\\
\hline
pbc, even $L$ &$0$& $2$ &no overlap & $e^{-4 \gamma t}$& $4 \gamma$\\
\hline
pbc, odd $L$ &$0$& $2$ & 0 & $\frac{1}{L\sqrt{8 \pi t}}$ &$\frac{1}{2}\ln ( t L^2)+\frac{1}{2}\ln (8 \pi)$\\
\hline
obc, any $L$ &$0$& $2$ & 0 & $\frac{1}{L\sqrt{8 \pi t}}$ &$\frac{1}{2}\ln ( t L^2)+\frac{1}{2}\ln (8 \pi)$\\
\hline
obc, any $L$ &$W$& $2$ & 0 & $\frac{\sqrt{\pi}}{2 L\sqrt{a t}}$&$\frac{1}{2}\ln (tL^2)+\frac{1}{2}\ln \frac{4a}{\pi}$\\
\hline
pbc, any $L$ &$W$& $2$ & 2 & $\frac{\sqrt{\pi}}{2 (at)^{3/2}}$ &$\frac{3}{2}\ln t+\frac{1}{2}\ln \frac{4 a^3}{\pi}$\\
\hline
\end{tabular}
\caption{Table showing the decay of $I(t)$ in Regime IV for different cases, $\alpha$ and $\beta$ are exponents defined in Eq.~\ref{alphabeta}. For obc, the scaling of $I(t)$ with $t$ do not qualitatively change on addition of disorder, whereas for pbc they are remarkably different. See text for details}
\label{table1}
\end{table*}
In this section, we shall discuss the behaviour of $I(t)$ in the fourth time-window, where we observe an asymptotic power law decay irrespective of the nature of disorder distribution. This is the first regime where we see $L$ dependent behaviour, as evident from Fig.~\ref{fig1}(b), where the evolution of $-\ln I(t)$ is plotted for different $L$s with open boundary conditions. Additionally, unlike the previous regimes which did not depend on the boundary conditions, behaviour of $I(t)$ in Region IV is strongly dependent on such factors. As shown in Fig.~\ref{fig4}(c), for open boundary conditions, we see a $I(t)\propto 1/(L\sqrt{t})$ or $-\ln I(t)\sim \frac{1}{2}\ln (tL^2)$ behaviour(dependence on $\gamma$ is more complicated, discussed later in the section). However for periodic boundary conditions (pbc), as plotted in Fig.~\ref{fig4}(d), $-\ln I(t)\sim \frac{3}{2}\ln t$, show a completely different behaviour with a different exponent and no system size dependence!

 To explain this behaviour, we first realize that in this regime charge is transported on longer scales and the system can no longer be described by three site models of the previous section. Hence, we need to study the eigenspectrum of the Liouvillian. Additionally, we need to focus on the low magnitude eigenvalues as this regime is asymptotic. However, since we have the hierarchical structure of observables, we do not need to study the full $4^L \times 4^L$ Liouvillian, but the eigenspectrum of $\mathcal{Q}$ in Eq.~\ref{linearised}, which, as mentioned earlier, is related to the one-particle Liouvillian of the problem. We can write the solution of Eq.~\ref{linearised} generically as,
\begin{equation}
\mathbf{f}(t)=\mathbf{f}_{{\rm NESS}}+\sum_j \mathbf{\delta f}(j) e^{(\lambda_j+i \Omega_j) t},
\label{rhoness}
\end{equation}
where $\mathbf{f}$ is defined under Eq.~\ref{linearised}, $\mathbf{f}_{{\rm NESS}}$ is the steady state value of the observables, $\lambda_j$ and $\Omega_j$ are the real and imaginary parts of the eigenvalues of $\mathcal{Q}$ and $\mathbf{\delta f}(j)$ contains the corresponding eigenvectors weighted by the initial conditions. Typically, the weights are similar to the corresponding elements of the eigenvector, i.e. $\mathbf{\delta f}(j) \sim m_j^2$ where $m_j$ is the $j^{{\rm th}}$ eigenvector (See Appendix~\ref{appC} for more details).

From Eq.~\ref{rhoness}, it might naively seem that any observable should show exponential decay with different rates at different timescales, depending on $\lambda_j$s. However, in reality, in the thermodynamic limit the eigenspectrum becomes continuous, and this leads to a power law approach of observables towards NESS. To demonstrate this, assume without loss of generality that
\begin{equation}
 \lambda_j\sim -j^\alpha, \hspace{0.2in} \mathbf{\delta f}(j) \sim j^{\beta}
 \label{alphabeta}
\end{equation} and\footnote{$\Omega_j$ gives an oscillatory term which is usually $\sim 0$ for the relevant low magnitude eigenvalues of the Liouvillian spectrum. If they are included the power law reduces by 1\cite{MedvKehrinPhysRevB.90.205410}} $\Omega_j \ll \lambda_j$. Then Eq.~\ref{rhoness} can be written in the continuum limit as\cite{CaiPhysRevLett.124.130602,CaiBarthelPhysRevLett.111.150403,MedvKehrinPhysRevB.90.205410},
\begin{equation}
\mathbf{f}(t)-\mathbf{f}_{{\rm NESS}}\sim\int dj j^{\beta} e^{-j^\alpha t}\sim t^{-\frac{\beta+1}{\alpha}}.
\label{scaling}
\end{equation}
 In what follows, we shall understand the behaviour of $I(t)$ for different systems using Eq.~\ref{rhoness}.

In Table~\ref{table1} we first summarize the results of various cases which shall be discussed in this section.

\begin{figure*}
\includegraphics[width=0.9 \columnwidth]{plots/pbcobceigenvalues.pdf}
\includegraphics[width=0.9 \columnwidth]{plots/pbcobceigenvectoroverlap.pdf}
\includegraphics[width=0.9 \columnwidth]{plots/inversesqrtt.pdf}
\includegraphics[width=0.9 \columnwidth]{plots/periodicpowerlaw.pdf}
\caption{Power law decay of $I(t)$ in Region IV. (a) Plot showing $j^2$ scaling of low magnitude Liouvillian eigenvalues averaged over $100$ disorder realizations compared with approximate theoretical prediction in Eq.~\ref{eigenvaluesapp}. (b) Plot showing scaling of $\mathcal{I}(j)$ defined in Eq.~\ref{support} averaged over $100$ disorder realizations. obc and pbc denote open and periodic boundary conditions respectively. See text for details. (c) Plots showing the scaling of $-\ln I(t)$ with $t$ for obc at different system sizes $L$. (d) Same as (c) but for periodic boundary conditions.}
\label{fig4}
\end{figure*}
Since $\mathbf{f}$ contains all the two particle Fermionic operators in the system, while $I(t)=\frac{1}{L}\sum_{k=1}^L(-1)^{k+1} C_{kk}(t)$, we can rewrite Eq.~\ref{rhoness} in the continuum for $I(t)$ as,
\begin{equation}
I(t)=I_{{\rm NESS}}+\int dj \mathcal{I}(j) e^{\lambda_j t}
\end{equation}
where
\begin{equation}
\mathcal{I}(j)=\frac{1}{L}\sum_{k=1}^L (-1)^{k+1} [\delta \mathbf{f}(j)]_{(k-1)L+k},
\label{support}
\end{equation} sums over the relevant operator subspace for the finite $L$ system. Eq.~\ref{scaling} gets appropriately modified.

Let us first look at the simpler case of systems without disorder. It is known that the low lying eigenenergies in the one-particle Liouvillian, in open boundary conditions are approximately \cite{MarkoPhysRevE.92.042143},
$\lambda_j=2 \left(\sqrt{4 \cos \left(\frac{ \pi  j}{L}\right)-3}-1\right), j=0\hdots L-1$.
For periodic boundary conditions the expression becomes, $2 \left(\sqrt{4 \cos \left(\frac{ 2 \pi  j}{L}\right)-3}-1\right), j=0\hdots L-1$, i.e. there is a degeneracy of $2$. Since we are interested in the regime $j/L \ll 1$, we can approximate  obc eigenvalues as $\frac{2 \pi^2 j^2}{L^2 \gamma}$. For pbc this gets multiplied by a factor of $4$, and hence in both cases $\alpha=2$.

$\mathcal{I}(j)$ can be computed from the relevant elements of the eigenvectors of the one-particle Liouvillian. It turns out that the relevant elements of the $j^{{\rm th}}$ eigenvector can be approximated to be exactly the free fermion wavefunction with the corresponding boundary condition, i.e. for pbc, they are  $\frac{1}{\sqrt{L}}\sum_{k=1}^{L}\exp(2 i \pi j k/L)$. Hence $\mathcal{I}(j)\sim\frac{1}{L}[\sum_{k=1}^{L}\frac{1}{\sqrt{L}} \cos(\pi k)\exp(2 i \pi j k/L)]^2$ is identically $0$ for even $L$ due to the symmetry of the wavefunction.\footnote{The square comes from the weightage of the initial conditions, whose weights are of the same order as the element of the eigenfunction. Hence we use $\sim$ instead of $=$.} Since $I$ has no support on the low magnitude eigenspectrum of the Liouvillian, it does not show a power law decay. Instead it decays exponentially with the the largest $\lambda_j=4 \gamma$.

The situation for odd $L$ in pbc is different as the one site is unpaired. In this case $\mathcal{I}(j)\sim\frac{1}{L}[\sum_{k=1}^{L} \frac{1}{\sqrt{L}} \cos(\pi k)\exp(2 i \pi j k/L)]^2 \sim \frac{1}{L^2}$ for $j \ll L$. This means $\beta=0$, and hence $I(t)\sim 2\int dj \frac{1}{L^2} e^{-8 \pi^2 j^2 /L^2}= \frac{1}{L\sqrt{8 \pi t}}$, where the factor $2$ in front comes from the degeneracy of the eigenvalues.

For obc the eigenvectors are given by, $\sqrt{\frac{2}{L}}\sum_{k=1}^{L}\cos( \pi j k/L)$. Hence $\mathcal{I}(j)=\frac{1}{L}[\sum_{k=1}^{L}\sqrt{\frac{2}{L}} \cos(\pi k)\cos( \pi j k/L)]^2$. This equals  $0$ when both $L$ and $j$ are even or odd, and $\frac{2}{L^2}$ otherwise. This gives $I(t)\sim \int dj \frac{2}{ L^2} e^{-2 \pi^2 (2j)^2 /L^2}= \frac{1}{L\sqrt{8 \pi t}}$. These results for the clean systems have been matched with exact numerics, data not shown.

Now we focus our attention to disordered systems. Intuitively, we can predict that since the behaviour in this regime involves the low energy eigenspectrum of the Liouvillian, it should not drastically change on addition of disorder. If we look into the obc case, we indeed see $ \propto 1/\sqrt{t}$ behaviour for both clean and disordered systems, but not so for the pbc case where we see a $t^{-3/2}$ behaviour. Let us investigate why.

 Unfortunately it is very difficult to find an analytical expression for the eigenspectrum when we add the disorder in the system, so we have to resort to exact diagonalization. However, we can make some crude approximations about the low lying eigenvalues and their scaling with $L,W$ and $\gamma$ since the evolution is described by the effective Heisenberg Hamiltonian, Eq.~\ref{effham} in this time regime. Firstly, since the effective hamiltonian is a Heisenberg model, we conclude that the low energy distribution would be $\propto 1/L^2$. We make a further educated guess by borrowing the result $\lambda_j=\frac{2 \pi^2 j^2}{L^2 \gamma}$ for clean systems from Ref.~\onlinecite{MarkoPhysRevE.92.042143} and then introducing the relevant prefactor in $\mathcal{H}_{{\rm eff}}$ for disordered systems. Thus, we have disorder averaged
 \begin{equation}
 \lambda_j \sim -a(\gamma,W) j^2/L^2,
 \label{eigenvaluesapp}
\end{equation}
for $j \ll L$, where
 \begin{equation}
 a(\gamma, W)=\frac{8 \pi^2 \gamma }{ (\frac{2W^2}{3}+4 \gamma^2)}
 \label{a}
\end{equation}
and we have used $\langle (h_{j}-h_{j+1})^2 \rangle\sim 2 W^2/3$ for the box distribution $(-W,W)$.  Henceforth we shall write $a(\gamma, W)$ as $a$.
\begin{figure}
\includegraphics[width=0.9 \columnwidth]{plots/finalgapimbalance.pdf}
\caption{Plot showing the linear growth of $-\ln I(t)$ in Region V for finite size systems of different lengths $L$ and at disorder strengths $W$. The black dashed lines denote the expected growth of $-\ln I(t)$ with the rate given by corresponding $\lambda_1$ dependent on $L,W$ and $\gamma$.}
\label{fig5a}
\end{figure}
 \begin{figure*}
\includegraphics[width=0.9 \columnwidth]{plots/lowestgapvsL.pdf}
\includegraphics[width=0.9 \columnwidth]{plots/gapvsWandgamma.pdf}

\caption{ Scaling of $\lambda_1$, the smallest Liouvillian gap, with different parameters. (a) Plot showing $1/L^2$ scaling of $\lambda_1$, for different disorder strengths, $W$. $L_c$ from Eq.~\ref{criticallength} is marked by appropriate coloured gridlines in the plot. Averaging is done over $100$ realizations.  (b) Plot showing dependence of $\ln |\lambda_1 L^2|$ with $W$ and $\gamma$ for $L=3000$ averaged over 100 disorder realizations. The red dashed lines indicate three constant contours at $0,2.5$ and $3.5$ for $\ln a$ obtained from Eq.~\ref{a}.}
\label{fig5}
\end{figure*}
In Fig.~\ref{fig4}(a) we see a very good agreement between our approximate prediction in Eq.~\ref{eigenvaluesapp} and numerical results for the low magnitude eigenvalues of the Liouvillian. Now we turn our attention to the eigenvectors. As shown before, imbalance has a constant support which was significant for odd/even $j$ in clean obc systems. Small perturbations around this value due to disorder, does not affect the overall scheme of things upon averaging over many disorder realizations, as can be verified from Fig.~\ref{fig4}(b). As seen in Fig.~\ref{fig4}(b), for odd $j$, $\mathcal{I}_j$ is almost the same for clean and disordered obc systems with even $L$. For even $j$ on the other had this quantity is no longer identically $0$ as the symmetry of the system is broken due to disorder, and shows a $(\sqrt 2j)^2/L^2$ scaling. However, this does not seriously affect the evolution as their magnitude is much smaller than for odd $j$. Consequently, for disordered obc systems,
\begin{eqnarray}
I(t)\sim&& \frac{1}{L} \int dj \frac{2}{L} e^{-a (2j)^2 /L^2}\nonumber \\
&&= \frac{\sqrt{\pi}}{2L\sqrt{a t}}.
\end{eqnarray}
This is the behaviour seen in Fig.~\ref{fig4}(c), where disorder, $W=4$ , is chosen to be smaller to highlight Region IV (data for $W=10$ is similar at larger times, not shown).

  However, for the periodic case, we have a different scenario. Previously $\mathcal{I}(j)$ had no support for any $j$ in this regime for clean systems, but now, due to breaking of symmetry, $\mathcal{I}(j)\sim (\sqrt{2}j)^2/L^2$ as evident from Fig.~\ref{fig4}(b). The eigenvalues as seen in Fig.~\ref{fig4}(a) follow the same scaling law as obc (the degeneracy in the smallest magnitude eigenvalue is resolved as we increase $j$). Hence, we get,
  \begin{eqnarray}
  I(t)\sim&& \frac{1}{L} \int dj \frac{2j^2}{L^2} e^{-a (j)^2 /L^2}\nonumber \\
  =&& \frac{\sqrt{\pi}}{2 (a t)^{3/2}}.
   \end{eqnarray} Indeed this is what we see in Fig.~\ref{fig4}(d), including the lack of $L$ dependence in the result.

\subsection{Region V}
\label{region5}
As mentioned in Sec.~\ref{summary}, Region IV becomes asymptotic in the thermodynamic limit, when we have a continuum in the eigenspectrum. For finite size systems though, the spectrum eventually gets resolved and hence for larger $t$ we observe a different behaviour, which we classify as region V. In this region, $I(t)$ shows an exponential decay, with the rate governed by the Liouvillian gap $\lambda_1$.\footnote{the exception being clean pbc systems with even $L$ where imbalance has no support on the eigenfunction as shown before} $\lambda_1$ is thelargest real part of non-zero Liouvillian eigenvalue for even $L$ systems. For odd $L$ systems, it is the next largest as the imbalance operator does not have support for the largest non-zero eigenvalue in this case. In Fig.~\ref{fig5a}, we see that $-\ln I(t)$ grows linearly with time at large times, with the slope $|\lambda_1(L,\gamma,W)|$ verifying our expectation. As before, analytical results for clean systems are well known\cite{MarkoPhysRevE.92.042143}, but upon addition of disorder one can only resort to numerics for exact results. Nevertheless, the description given by Eq.~\ref{effham} is still valid in this time scale, and explains the scaling behaviour of $\lambda_1$ with $L,W$ and $\gamma$ as discussed below.

In Fig.~\ref{fig5}(a) we show the change of $\lambda_1$ with $L$ for constant $W$ and $\gamma$. It is known that the $1/L^2$ scaling of $\lambda_1$ works for sufficiently large $L$ for clean systems. In fact, it has already been shown in Ref.~\onlinecite{MarkoPhysRevE.92.042143} that the critical length required to observe this scaling is $L_c \sim \frac{\pi \sqrt{2}}{\gamma}$. For $\gamma=0.1$ this is at $L\sim 45$ and is shown by the blue gridline in the Fig.~\ref{fig5}(a). This behaviour also continues when we introduce disorder in the system. It is evident from the figure that upon increasing $W$, $L_c$ decreases. This can be explained as in the previous section using Eq~\ref{effham} and Eq.~\ref{eigenvaluesapp}, which tells us that on addition of disorder of strength $W$, $\gamma$ can be effectively replaced by $(4\gamma^2+\frac{2 W^2}{3})/(4\gamma)$. This  gives
\begin{equation}
L_c(W)\sim \frac{ 4 \sqrt{2} \pi \gamma}{4\gamma^2+\frac{2 W^2}{3}},
\label{criticallength}
\end{equation}  shown as different coloured gridlines in the plot, and capture the transition point between the two regimes quite well.

Finally in Fig.~\ref{fig5}(b) we show the change of $\lambda_1$ on changing $W$ and $\gamma$ for constant $L=3000$ via a contour plot. We expect $|\lambda_1 L^2| \sim a$ and hence the equation of constant contours can be obtained from Eq.~\ref{a} as
\begin{equation}
\ln a={\rm constant}.
\label{fitgap}
\end{equation}
Three such contours are plotted as red dashed lines in Fig.~\ref{fig5}(b) and they match very well with the contours obtained from exact numerics.

\section{Discussion}
\label{discussion}
In this work we have studied the evolution of imbalance, $I(t)$ for the disordered XX chain with on-site dephasing, starting from the N\'eel initial state. Using the hierarchical nature of equations for the observables, we have computed $I(t)$ for large system sizes ($L\sim 10^3$) and long times ($t \sim 10^3$). Our analysis showed the emergence of five timescales in disordered systems.

The shortest time scale, $t < t_0 \sim {\rm Min(1/\gamma,1/W)}$ where $ I(t) \sim (1-\eta t^2)$ constitutes the  linear response regime of the system. Then, due to the localization via disorder, two timescales denoted by Region II and III emerge, which are absent in clean systems. Region II is the linear regime of what has been called a stretched exponential decay in previous works\cite{AltmanPhysRevLett.116.160401,CaiBarthelPhysRevLett.111.150403,CaiPhysRevLett.124.130602,SchneiderPhysRevB.106.134211,PhysRevLett.116.237203Garrahan}, where $I(t) \sim (1-\beta \sqrt{t})$, and is universal irrespective of the nature of disorder chosen, and continues till $t_1 \sim 0.1 W^2/8\gamma$, before smoothly transitioning to a disorder dependent behaviour in region III which continues till$t_2 \sim W^2/(8\gamma J^2)$. We thus conclude that the stretched exponential fits are neither universal nor asymptotic for the system under study. Simple local three-site models describe the behaviour in these two regimes, and a rescaled time emerges during the analysis $\tau= 8 \gamma t/W^2$. An effective Heisenberg model given by Eq.~\ref{effham} also provides the correct description of the evolution from Region II.

For $t > t_2$, where Region IV begins, dephasing breaks localization and the system shows size and boundary conditions dependent power law decay of $I(t)$. In fact for obc it shows a $1/(L\sqrt{t})$ decay and for pbc a $1/\sqrt{t^3}$ decay. These were explained from the continuum limit of the low magnitude eigenspectrum of the Liouvillian. Finally, we discussed that for finite sized systems due to resolution of the eigenspectrum, the decay of $I(t)$ in final Region V at $t>t_3$ is exponential with the rate governed by the relevant Liouvillian gap $\lambda_1$. We also provided analysis of how $\lambda_1$ typically scales with $W,L,\gamma$ using the effective Heisenberg Hamiltonian approach.

Our analysis gives insight to the mechanism behind the evolution of imbalance in such a system. It demystifies the behaviour of the system in Region II and III clearly showing its local origin and gives us the ability to make predictions about the evolution for many different disorder distributions. Additionally, the proper analysis of Liouvillian eigenspectrum provides us with the correct asymptotic description of evolution for different models and boundary conditions.

We would also like to briefly mention the nuances of crossover from one regime to another. Since the transition is smooth, there is a always a finite time taken by the system to go from one regime to another. For example, from Region III to IV, where the finite size effect first appears, the system typically takes $t\sim L$ to transition to a power law for open boundary conditions. Hence, if the thermodynamic limit is taken first, we would see that even if region III ends at $L$ independent $\tau_2=8 \gamma t_2/W^2 \sim 1$, the power law in Region IV is only reached asymptotically. However, this does not seem to be the case for periodic boundary conditions. This is also consistent with the system size dependence of the results in Region IV.

 Additionally, it is worthy to remark that the effective Hamiltonian description\cite{MariyaMarkoPhysRevB.93.094205} of Eq.~\ref{effham} is also valid for weakly interacting systems where the interaction strength is much smaller than disorder strength. This means weak interaction does not play much role in evolution in such timescales, so we expect most of our results to hold for weakly interacting systems as well.

Furthermore, while we have discussed the N\'eel initial state in this work, decay of $I(t)$ from other generic computational initial states also share some similar features. The behaviour in regions I, IV and V seen for the N\'eel state is still observed for other typical states. The differences occur in $t_2$ and the behaviour in Regions II and III. The first difference in other initial states is that $I(t=0)< 1$ i.e. $-\ln I(t=0)>0$. Since $t_2$ is defined by a finite non-zero value of $I$ universal for a given disorder distribution, relaxation from different $I(t=0)$ typically reaches this value at different times. Secondly, all the sites of the initial state are not locally equivalent unlike the N\'eel state. Hence, the scaling in Regions II and III show a difference due to variation in local behaviours. A detailed analysis of this aspect is beyond the scope of the present work.

There are still a few open questions left in the context of this work. One natural extension would be to study the evolution of other observables such as current in such a model. The question of whether the scaling laws depend on the type of dephasing is also worth studying. Indeed, the hierarchical structure of observables or the effective Heisenberg model might break down if we choose a different model of dephasing. Additionally, while Region I-III has been studied under different disorder settings, we have provided a general explanation for the power law decay in Region IV, discussing the two cases where it shows different exponents. Specific potentials may give rise to unique behaviour in this time scale, and an analysis of that is left for a future work.

\section*{acknowledgement}
We would like to acknowledge support by Grants No.~J1-1698, J1-4385 and
No.~P1-0402 from the Slovenian Research Agency. RG would also like to acknowledge support from UKRI grant EP/R029075/1.

redpajama_set_name: RedPajamaArXiv


Darya Danesh is the co-founder of StudioBinder, a leading film production software company that helps production companies manage their crew, schedules and call sheets. With over 10 years of experience as a Producer, LP & UPM, Darya has worked on large projects including PBS, Direct TV, Syfy, Animal Planet, Fox, YouTube, Sony, Atlantic, Island Def Jam, Gaiam & BMW.
So your script is done, and you are about to begin the script breakdown process for production. In order for the script to import correctly into scheduling software, you must review the script for formatting errors. In this article, we'll review some of the most common culprits.
A script breakdown is when a producer or 1st AD imports a script into scheduling software. During this process, the elements of a scene — characters, locations, key props, stunt, vehicles, sound effects, etc. — are identified in what is called a script breakdown sheet.
Since few screenwriters are producers, and even fewer have experience as a 1st AD, they don't understand the work that goes into a script breakdown. As a result, scene elements are rarely tagged (so they must be inputted manually), scene numbers may be missing, and formatting errors abound. Inconsistencies in character names, locations, and time of day can cause some of the biggest issues when importing. It's best to avoid the issues altogether by making sure that the script is first, and foremost, formatted correctly.
To start, you'll need to have a script in its source format with whatever screenwriting software you use. If all you have is a PDF, you can use WriterDuet for free to convert it to your screenwriting software of choice.
Actually read the script. Start to finish. Once you begin the script breakdown you won't be able to experience the script from an audience member's perspective again. Leave your pencil and pens aside so you can focus on the characters and story.
Since budgets usually have their way, you'll inevitably need to identify which scenes should be trimmed, consolidated, or removed. Understanding the narrative themes, character arcs, and controlling ideas will help you prioritize the shooting schedule without compromising the story.
This is most common with family relationships, boyfriends, girlfriends, roommates, etc.
A new scene occurs whenever a location in the script changes. As mentioned earlier in this post, if the scene goes from outside to inside, or one room to another, you must treat these as different scenes.
This blog was originally posted on studiobinder.com. StudioBinder a leading film production software company that helps production companies manage their crew, schedules and call sheets. Learn more about StudioBinder by following them on with @studiobinder on Twitter and Facebook.


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A very rare opportunity to acquire a two bedroom detached bungalow located on a superb corner plot on the Bewdley side of Kidderminster. Entrance Porch. Reception Hall. Lounge. Two Bedrooms. Fitted Kitchen. Pantry/Utility. Shower Room. Detached Garage and gated driveway to the rear. Potential to extend subject to normal planning permissions. No Upward Chain. 'Energy Rating E'.
Two ceiling light points. Central heating radiator. Telephone point. Smoke alarm. Picture rail. Doors radiate to dining room, lounge, bedrooms, shower room and kitchen.
Front facing UPVC double glazed bay window. Three wall light points. Central heating radiator. Picture rail.
UPVC double glazed bay window. Ceiling light point. Fireplace with ornate chimney breast. Central heating radiator. Picture rail.
Rear facing UPVC double glazed window. Ceiling light point. Central heating radiator. Picture rail. A double bedroom.
Side facing window. Ceiling light point. Central heating radiator.
Side facing UPVC double glazed window. Central heating radiator. Majority tiled with brick effect tiling and mosaic trim. White suite comprising w.c., wash hand basin and shower cubicle with 'Mira Sport' shower over.
Side facing window. Ceiling light point. Space for fridge freezer. Space for tumble dryer. Shelving to walls. Fuse board and meters.
Up and over door. Side facing window. Door opens to the rear gardens and access to the road.
Fore Garden - The property sits well back from the kerbside behind a gated fore garden offering block paved driveway for parking. Established borders with mature shrubs. Gravelled areas. Gated access leading to the side and rear gardens.
Enclosed Rear Gardens - Paved patio area. Adjacent lawned area. Established borders with mature shrubs. Access leading to the detached garage and additional driveway.


text:
Nintendo Stock Down -5.27%, Nikkei Believes Lack Of New Switch Information And Sluggish Sales Are To Blame
Dr. Serkan Toto has tweeted that Nintendo stock is dramatically down 5.27% which is slightly concerning. Japanese publication The Nikkei believes this is down to a lack of new Nintendo Switch information on upcoming plans and games. The news publication also says sluggish sales are likely to blame. As we know from the charts, Japanese software and hardware sales are nowhere near what they used to be as more and more consumers play games on smartphones and tablets. The Switch has been performing well in the west but it is struggling against Sony's PlayStation 4 which has become the dominant console this generation.
-5.27%. Nintendo stock again got slaughtered at the Tokyo Stock Exchange today.
The Nikkei thinks it's because of a lack of new information on the game pipeline and sluggish sales of the Switch, causing foreign shareholders to get rid of the stock.
— Dr. Serkan Toto / Kantan Games Inc. (@serkantoto) July 4, 2018
Tags:nikkeiNintendonintendo switchstock
27 thoughts on "Nintendo Stock Down -5.27%, Nikkei Believes Lack Of New Switch Information And Sluggish Sales Are To Blame"
The Man of Brisk July 4, 2018 at 10:03 am
While I strongly believe Nintendo's decision to focus too much on Smash Ultimate is a deciding factor, I hardly think this is anything to be concerned about. As soon as Smash launches or more new games are announced, this will shoot back up again.
The Collector (@Collect_Or_Die) July 4, 2018 at 10:20 am
Why do people make such a big deal every time stocks go down? Stocks always go up, down, up, down, all the time.
skeletons420 @ www.getcontinuum.com 🐙 July 4, 2018 at 1:06 pm
read about it and you'll know.
The Collector (@Collect_Or_Die) July 4, 2018 at 6:37 pm
It's too boring to read about stocks.
frogopus July 4, 2018 at 3:00 pm
Because video game sites aren't exactly full of investors.
jockmahon July 4, 2018 at 10:59 am
im not sure where ur getting sluggish sales from, last wk 49K switches where sold, more than double any other console in japan, aces was #1 game with 123k and Nintendo had 10 in the top 20
mylifewithhiv July 4, 2018 at 2:16 pm
Exactly, the Switch has been dominating Japan sales the whole year. So the only thing that could be meant that even that Switch is dominating the sales are still too low for investors. Worldwide Switch has been behind PS4, but still a strong 2nd place, with Xbox One trailing behind on 3rd with cosiderable distance
Kintoki July 4, 2018 at 11:01 am
The Switch is selling between 30.000 and 40.000 units per week in Japan. I wouldn't call this exactly "sluggish".
And for the rest of the world: I would wait, until we get official numbers.
luminoth64 July 5, 2018 at 10:33 am
I know right? I remember when the Wii U would get 30-40K sales WORLDWIDE on a good week. The sales aren't the break neck success they were at the start, but they're by no means bad, or even average.
P Edwards July 4, 2018 at 12:30 pm
This is nonsense
IMO July 4, 2018 at 5:25 pm
Curious that SSB for wii u couldn't save the console and now the announcement of a new entry is hurting the company.
SSB is good but this new one should not have been the focus of the E3, sadly almost everybody wants a BOTW type of game every E3, i am not hating, i sm going to buy the new smash as soon as it arrives.
The Wii U couldn't have been saved by Smash because that idiot Sakurai decided to release it on the much more popular 3DS as well. Nearly 2/3rds of the games entire sales went to the 3DS version. Nintendo honestly should have slapped him when he suggested putting the game on both systems and said "Do you SEE the situation with the Wii U right now?!" It would've come out earlier as well, which would've been better, but oh well, can't change the past.
But I would agree that this smash is not getting the same amount of buzz as previous games. Smash was pretty much at a breaking point in terms of making things different and new (which is why Sm4sh sucked compared to the previous titles) and now another one that also looks nearly exactly like the previous two entries with seemingly not much to separate it is just not going to make the same amount of buzz.
Rosa Parks July 4, 2018 at 5:46 pm
Technically speaking, the switch is the only 9th gen console out. So they win there. I can see why sales are a little lower, but I see it heavily picking up come late August or early September.
negatorous July 4, 2018 at 5:48 pm
Nintendo should be able to deliver on that, instead they are thinking of a replacement for the 3DS and they will have to assign crews to make games for it. Which of course means as usual one rmain triple A Mario game and one triple A Zelda game per console cycle. Which is ridiculous IMO.
Cambez July 4, 2018 at 5:56 pm
i do not know what the hell the were thinking.
Sure, focusing an entire E3 to the biggest game worked for 2 straight years.
this one was a smash bros focused event and people got up their hype because they teased metroid prime the past year, everyone knew smash bros was the focus.
there is something new in this game?.
Maybe, instead, they focused their time in the classic free for all instead of showing single player modes and stuff.
What a waste of time their direct was.
Exactly. And due to them not showing more single player modes, they didn't make me want to buy it. It just looked like the same old thing to me. And I hated the Wii U and 3DS versions. Worst Smash games ever. They even had the nerve to ruin my favorite side game, Break The Targets. How dare them.
Anonymous July 4, 2018 at 8:06 pm
Lemme guess, Smash 4 is not like Melee meaning if any Smash game is not like Melee then its trash, sorry Mr.Entitlement but not every Smash has to be like Melee. This is my unpopular opinion, I liked Smash 4 and yes it has its flaws and I don't care if its not like Melee because then it won't have its own identity.
Infinite Ridley July 5, 2018 at 12:03 am
You are wrong to assume he's referring to Melee. He's a big fan of Brawl actually because it had the story mode Subspace Emissary.
ChaseThePlayer July 4, 2018 at 6:21 pm
2018 seems to be a bit of a slow year for Nintendo… they have some decent games coming out, but the only heavy hitter they have is Smash Ultimate, which probably isn't making as big an impact as Mario and Zelda last year because it's more or less an expected addition. Things will probably pick up in 2019 with Prime 4 and Pokemon, though.
Nintendo Messiah July 4, 2018 at 7:10 pm
Umm Let's Go Pikachu/ Eevee are a thing as well. I think people underestimate how well these two games are going to sell.
Nintendo First Order 𝓡𝓮𝓪𝓿𝓮𝓻 July 4, 2018 at 7:31 pm
Why do I have the feel that Nintendo will show something cool at September?
Provided September's reveals aren't letdowns, this stock will just go right back up after or during TGS.
CurlyJ July 5, 2018 at 3:14 pm
Sluggish sales!! LOL based on what?!
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Rev Esp Cardiol. 2010 Feb;63(2):141-8.
Percutaneous implantation of the CoreValve self-expanding aortic valve prosthesis in patients with severe aortic stenosis: early experience in Spain.
Avanzas P1, Muñoz-García AJ, Segura J, Pan M, Alonso-Briales JH, Lozano I, Morís C, Suárez de Lezo J, Hernández-García JM.
Area del Corazon, Hospital Universitario Central de Asturias, Oviedo, Spain.
The aim of the study was to describe early experience and medium-term follow-up with the CoreValve self-expanding aortic prosthesis at three Spanish hospitals.
The study included patients with severe symptomatic aortic stenosis. Other inclusion criteria were: aortic valve area <1 cm(2) (<0.6 cm(2)/m(2)); aortic valve annulus diameter in the range 20-27 mm; diameter of the ascending aorta at the level of the sinotubular junction < or = 40 mm (small prosthesis) or < or = 43 mm (large prosthesis), and femoral artery diameter >6 mm.
The study included 108 patients with a mean age of 78.6 + or - 6.7 years, a mean aortic valve area of 0.63 + or - 0.2 cm(2) and a mean logistic EuroSCORE of 16% + or - 13.9% (range, 2.27%-86.4%). After valve implantation, the maximum echocardiographic transaortic valve gradient decreased from 83.8 + or - 23 to 12.6 + or - 6 mmHg. No patient presented with greater than grade-2 residual aortic regurgitation on angiography. The procedural success rate was 98.1%. No patient died during the procedure. Definitive pacemaker implantation was carried out for atrioventricular block in 38 patients (35.2%). At 30 days, all-cause mortality and the rate of the combined endpoint of death, stroke, myocardial infarction or referral for surgery were 7.4% and 8.3%, respectively. The estimated 1-year survival rate calculated using the Kaplan-Meier method was 82.3% (for a median follow-up period of 7.6 months).
Our early experience indicates that percutaneous aortic valve replacement is a safe and practical therapeutic option for patients with severe aortic stenosis who are at a high surgical risk.