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The State University of New Jersey, Piscataway, New Jersey, USA B. Chiarito,
J.P. Chou0000-0001-6315-905X, A. Gandrakota0000-0003-4860-3233, Y.
Gershtein0000-0002-4871-5449, E. Halkiadakis0000-0002-3584-7856, A. Hart, M.
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Stone, S.A. Thayil0000-0002-1469-0335, S. Thomas, H. Wang0000-0002-3027-0752
University of Tennessee, Knoxville, Tennessee, USA H. Acharya, A.G.
Delannoy0000-0003-1252-6213, S. Fiorendi0000-0003-3273-9419, S.
Spanier0000-0002-8438-3197 Texas A&M University, College Station, Texas, USA
O. Bouhali940000-0001-7139-7322, M. Dalchenko0000-0002-0137-136X, A.
Delgado0000-0003-3453-7204, R. Eusebi, J. Gilmore, T. Huang, T. Kamon95, H.
Kim0000-0003-4986-1728, S. Luo0000-0003-3122-4245, S. Malhotra, R. Mueller, D.
Overton, D. Rathjens0000-0002-8420-1488, A. Safonov0000-0001-9497-5471 Texas
Tech University, Lubbock, Texas, USA N. Akchurin, J. Damgov, V. Hegde, S.
Kunori, K. Lamichhane, S.W. Lee0000-0002-3388-8339, T. Mengke, S.
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Wang, A. Whitbeck Vanderbilt University, Nashville, Tennessee, USA E.
Appelt0000-0003-3389-4584, S. Greene, A. Gurrola0000-0002-2793-4052, W. Johns,
A. Melo, H. Ni, K. Padeken0000-0001-7251-9125, F. Romeo0000-0002-1297-6065, P.
Sheldon0000-0003-1550-5223, S. Tuo, J. Velkovska0000-0003-1423-5241 University
of Virginia, Charlottesville, Virginia, USA M.W. Arenton0000-0002-6188-1011,
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Neu0000-0003-3644-8627, B. Tannenwald0000-0002-5570-8095, S.
White0000-0002-6181-4935, E. Wolfe0000-0001-6553-4933 Wayne State University,
Detroit, Michigan, USA N. Poudyal0000-0003-4278-3464 University of Wisconsin -
Madison, Madison, WI, Wisconsin, USA K. Black0000-0001-7320-5080, T.
Bose0000-0001-8026-5380, J. Buchanan0000-0001-8207-5556, C. Caillol, S.
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Smith0000-0003-3195-0909, D. Teague, S. Trembath-Reichert, W.
Vetens0000-0003-1058-1163 †: Deceased
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# Tunable superdiffusion in integrable spin chains using correlated initial
states

Hansveer Singh1, Michael H. Kolodrubetz3, Sarang Gopalakrishnan2 and Romain
Vasseur1 1Department of Physics, University of Massachusetts, Amherst,
Massachusetts 01003, USA
2Department of Electrical and Computer Engineering, Princeton University,
Princeton NJ 08544, USA
3Department of Physics, The University of Texas at Dallas, Richardson, Texas
75080, USA

###### Abstract

Although integrable spin chains only host ballistically propagating particles
they can still feature diffusive spin transport. This diffusive spin transport
originates from quasiparticle charge fluctuations inherited from the initial
state’s magnetization Gaussian fluctuations. We show that ensembles of initial
states with quasi-long range correlations lead to superdiffusive spin
transport with a tunable dynamical exponent. We substantiate our prediction
with numerical simulations and explain how deviations arise from finite time
and finite size effects.

_Introduction_ —In lattice quantum systems, typical initial states far from
equilibrium relax according to diffusive hydrodynamics. After a local
equilibration step, one expects that the only data from the initial state that
survive in local observables are those which determine conjugate thermodynamic
quantities, e.g. local temperature, chemical potential. The remaining
evolution from local to global equilibrium is governed by the classical theory
of hydrodynamics, which generically predicts diffusive transport for lattice
systems. However this intuition breaks down in kinetically constrained
dynamics Garrahan (2018); Fredrickson and Anderson (1984); Garrahan  _et al._
(2010); Ritort and Sollich (2003); Everest  _et al._ (2016); Iaconis  _et al._
(2019); Feldmeier  _et al._ (2020); Gromov  _et al._ (2020); Morningstar  _et
al._ (2020); Guardado-Sanchez  _et al._ (2020); Sala  _et al._ (2020); Khemani
_et al._ (2020); Ljubotina  _et al._ (2023) which can exhibit both sub- and
superdiffusion, or avoid thermalization altogether Moudgalya  _et al._ (2022),
and in integrable systems where initial state fluctuations in densities of
infinitely-many conserved quantities play a prominent role in governing
hydrodynamics.

In integrable systems one has an extensive number of extensive conserved
quantities or, equivalently, stable ballistically propagating quasiparticles
at finite energy density Castro-Alvaredo  _et al._ (2016); Bertini  _et al._
(2016); Bulchandani  _et al._ (2017); Bastianello  _et al._ (2022). Although
such systems are in principle fine tuned, they are of considerable interest
since current experiments can engineer systems that are approximately
integrable Kinoshita  _et al._ (2006); Scheie  _et al._ (2021); Zundel  _et
al._ (2019); Hild  _et al._ (2014); Wei  _et al._ (2022); Gring  _et al._
(2012); Jepsen  _et al._ (2020a); Tang  _et al._ (2018); Erne  _et al._
(2018). Naively, stable ballistically-propagating quasiparticles should always
lead to ballistic transport. However this does not always hold, as highlighted
by the $|\Delta|>1$ regime of the spin-1/2 XXZ chain where one finds diffusive
spin transport Damle and Sachdev (1998); Sachdev and Damle (1997); De Nardis
_et al._ (2018); Gopalakrishnan  _et al._ (2018); Gopalakrishnan and Vasseur
(2019); Nardis  _et al._ (2019); Bulchandani  _et al._ (2021); Gopalakrishnan
and Vasseur (2023).

One can understand the origin of this diffusion as coming from two
ingredients: ballistic motion of the quasiparticles (magnons and boundstates
thereof) and fluctuations in the quasiparticle’s charge which is “screened” by
the magnetization fluctuations of the initial state. That is to say if in a
region of size $\ell$, the magnetization density fluctuations scale as
$\ell^{-w}$, where $w$ is known as the wandering exponent. Then, if the
quasiparticle travels a distance $\ell$ the fluctuations of the
quasiparticle’s charge over that distance also scales as $\ell^{-w}$.

For initial states drawn from a thermal ensemble, these fluctuations obey the
central limit theorem ($w=1/2$) and using this fact along with the ballistic
motion of the quasiparticles one can show that this leads to diffusive spin
transport Gopalakrishnan and Vasseur (2019). This suggests that tuning the
wandering exponent of the initial state should modify the nature of spin
transport.

In this work we demonstrate that this is indeed the case by constructing an
ensemble of initial states with tunable wandering exponent which have
fluctuations that grow faster than a thermal ensemble’s. Based on the charge
screening argument outlined above, we show that these states should display
superdiffusive transport and indeed find that this is the case by numerically
computing the variance of the charge transfer. The variance of the charge
transfer grows in a power law fashion and provide an argument for the expected
exponent of this power law. We find relatively good agreement with numerical
results and explain how deviations arise from finite size effects.

Figure 1: Folded XXZ Automaton. (a) Circuit geometry for the XXZ automaton
follows a staircase pattern. (b) The rules for the updates in the automaton.
One can see that they conserve the number of domain walls. (c) A highlighted
magnon trajectory traversing ballistically through domains.

_Screening Argument_.— To illustrate our argument, we study transport in the
folded XXZ automaton Pozsgay (2021) which qualitatively captures transport
features of the easy axis regime and can be thought of as the
$\Delta\rightarrow\infty$ limit of the quantum spin-1/2 XXZ chain Zadnik and
Fagotti (2021); Zadnik  _et al._ (2021). A desirable feature of this model
compared to the spin-1/2 XXZ chain is that product states in the occupation
basis are mapped to product states and thus we can perform simulations to long
times with large system sizes. Our argument will rely solely on a
quasiparticle picture, and we expect it to generalize to the entire diffusive
regime $|\Delta|>1$ of the XXZ spin chain.

The system is comprised of $L$ qubits whose individual basis states are
$|\bullet\rangle$ (particle) and $|\circ\rangle$ (hole). The unitary governing
the dynamics is given by $\mathcal{U}=V_{3}V_{2}V_{1}$ where
$V_{j}=\prod_{i\equiv j\\!\\!\\!\mod 3}U_{i,i+1,i+2,i+3}$ and

$\begin{split}U_{i,i+1,i+2,i+3}&=P^{\bullet}_{i}\mathrm{SWAP}_{i+1,i+2}P^{\bullet}_{i+3}\\\
&+P^{\circ}_{i}\mathrm{SWAP}_{i+1,i+2}P^{\circ}_{i+3}\\\
&+P^{\circ}_{i}P^{\bullet}_{i+3}+P^{\bullet}_{i}P^{\circ}_{i+3},\end{split}$
(1)

where $P^{\circ}=|\circ\rangle\langle\circ|$ and
$P^{\bullet}=|\bullet\rangle\langle\bullet|$, conserving both particle and
domain wall number. A pictorial representation of the circuit along with the
update rule performed by the gates are shown in Fig. 1. This model is
integrable and has three types of quasiparticles: left- and right-moving
magnons (single isolated particles or holes), and frozen domains which arise
from domain wall conservation.

To understand transport in integrable systems we essentially have to keep
track of how much charge has been spread by the magnons. To track the amount
of charge a magnon spreads we compute $\delta
x_{\mathrm{charge}}(t)^{2}=\langle(q_{\mathrm{mag}}(t)x_{\mathrm{mag}}(t))^{2}\rangle$
where $q_{\mathrm{mag}}$ denotes the charge of the magnon, $x_{\mathrm{mag}}$
denotes the distance the magnon has traveled, and $\langle\cdot\rangle$
denotes averaging over the ensemble of initial statess. In integrable systems
magnons traverse ballistically with a well-defined velocity, $v$ (we imagine
tracking a single species), so

$\delta x_{\mathrm{charge}}(t)^{2}=v^{2}t^{2}\langle
q_{\mathrm{mag}}(t)^{2}\rangle.$ (2)

On average the magnon’s charge will be zero at infinite temperature but the
variance will non-zero. As shown in Fig. 1, magnons change their charge by
passing through frozen domains. For example, if a magnon is a particle prior
to collision, it will change to a hole while traversing the next domain (see
Fig. 1). This means a magnon’s charge fluctuations are entirely specified by
the pattern of frozen domains in the initial state. Fluctuations in the
pattern of frozen domains are determined by the initial state magnetization
density, $m$, defined such that a particle/hole has charge $\pm 1$. Thus,
fluctuations are characterized by the wandering exponent, $w$, i.e. in a
region of size $\ell$, $m\sim\ell^{-w}$.

Thus we have that the charge of the magnon should scale in the same way as the
initial state magnetization density fluctuations. So if a magnon moves a
distance $\ell$ in a time $t$ then $\langle
q^{2}_{\mathrm{mag}}\rangle\sim\ell^{-w}\sim t^{-w}$. Therefore

$\delta x_{\mathrm{charge}}(t)\sim t^{1-w}.$ (3)

This argument was first used in Ref. Gopalakrishnan and Vasseur (2019) and was
applied to thermal states where $w=1/2$ where one correctly predicts diffusive
spin transport. In the next section we will provide a procedure to generate
states with tunable wandering exponent with $w<1$ which implies superdiffusive
spin transport.

Figure 2: Charge Variance vs Stability Parameter. (a) Variance of the charge
transfer as a function of time shown for stability parameter, $\alpha=1.4$,
compared to initial states drawn from an infinite temperature, i.e.
uncorrelated, ensemble. Note that the fluctuations for $\alpha=1.4$ grow
faster than those in the thermal ensemble indicating superdiffusive behavior.
(b) Dynamical exponents, $z_{\mathrm{var}}$, versus time obtained via log
derivatives for two different values of $\alpha$. (c) $z_{\rm var}$ obtained
via log-derivatives at a late time, $t=500$. The orange point $\alpha=2$ is
obtained by sampling states from an infinite temperature ensemble. One sees
that the prediction, $z_{\mathrm{var}}=\alpha^{2}/2$ is roughly consistent
with the numerics. The data is averaged over $10^{7}$ initial states.

_Correlated States_.— Based on the screening argument, generating states which
do not obey the central limit theorem should lead to anomalous transport. One
way to do so is to create states that are spatially correlated. This can be
achieved by forming product states which are comprised of contiguous fully
polarized domains (e.g.
$|\bullet\bullet\bullet\bullet\circ\circ\circ\circ\bullet\bullet\bullet\circ\circ\rangle$)
that have unbounded lengths. To do so we draw the lengths of these domains
from Lévy alpha stable distributions (although any distribution with power law
tails should work) Bouchaud and Georges (1990). These are distributions with
power law tails that can be tuned via the so-called stability parameter that
characterizes the distribution. To be more precise, let
$S(y;\alpha,c,\beta,\mu)$ denote the stable distribution with stability
parameter, $\alpha$; scale parameter, $c$; skewness, $\beta$; mean, $\mu$.

In this work $\mu=0$ and $\beta=0$ and $1<\alpha\leq 2$. For large $y$,
$S(y)\sim y^{-(\alpha+1)}$. Given a random variable $y$ distributed according
to $S$ then the domain length of the $i$th domain is given by
$x_{i}=\mathrm{floor}(|y|)$. For $1<\alpha\leq 2$ the average of $y$ is well
defined and we note that the average of $|y|$ for $\beta=\mu=0$ is given by
$\frac{2}{\pi}c^{1/\alpha}\Gamma(1-1/\alpha)$ where $\Gamma(z)$ is the usual
gamma function and so the scale parameter determines the average domain size.
We randomly choose a domain to be fully filled (empty) with equal probability.

To determine the wandering exponent of these states we compute the
magnetization ((un)occupied sites have charge $\pm 1$) in a region of size
$\ell$, i.e. $M=\sum_{i=1}^{\ell}\sigma_{i}$ and $\sigma_{i}=\pm 1$. Let
$\tau_{i}$ denote the polarization of the $i$th domain (this is $\pm 1$ for
occupied and unoccupied respectively). We can rewrite the sum as
$M=\sum_{i=1}^{k}x_{i}\tau_{i}+(\ell-\sum_{i=1}^{k}x_{i})\tau_{k+1}$ where the
last term accounts for the fact not all $k$ domains may fit inside the region.
Let the magnetization of a domain be $m_{i}=x_{i}\tau_{i}$. Then the
distribution of $m_{i}$ is symmetric since $\tau_{i}=\pm 1$ with equal
probability and has power law tails inherited from the domain length
distribution. Thus we can invoke the generalized central limit theorem to show
that $\langle M^{2}\rangle\sim k^{2/\alpha}$. For the stability parameters we
consider, the mean is well-defined and so the typical domain lengths will be
controlled by $c$. For $c\sim O(1)$, we expect that the number of domains
$k\sim\ell$. Thus $\langle M^{2}\rangle\sim\ell^{2/\alpha}$. We conclude that
the wandering exponent is given by $w=1-1/\alpha$.

For $1<\alpha\leq 2$, this means that magnetization fluctuations decay slower
than thermal states consistent with the fact that these states have larger
correlations. Based on the screening argument this would imply that charge
spreads as $t^{1/\alpha}$ which would indicate superdiffusive transport. In
the next section we demonstrate that this is indeed the case by computing
transport observables.

_Transport and numerics_.— To characterize transport in this special class of
initial states, we cannot use the standard linear-reponse Kubo formalism
(which relies on special properties of thermal states). Instead, we diagnose
charge transport via the variance of the so-called charge transfer $Q(t)$,

$Q(t)=\sum_{x\leq 0}(n_{x}(t)-n_{x}(0))-\sum_{x>0}(n_{x}(t)-n_{x}(0)),$ (4)

with $n_{x}=0,1$ the particle density on site $x$. The charge transfer keeps
track of how much charge has been transferred across the origin at time $t$.
The full distribution of the charge transfer is known as the full counting
statistics and it not only captures linear response, but also captures details
about fluctuations on top of the average hydrodynamic behavior Levitov and
Lesovik (1993); Levitov  _et al._ (1996); Ivanov  _et al._ (1997); Belzig and
Nazarov (2001); Börlin  _et al._ (2002); Levitov and Reznikov (2004); Beaud
_et al._ (2013); Krajnik  _et al._ (2022a, b, c, 2023); Gopalakrishnan  _et
al._ (2022); McCulloch  _et al._ (2023). Experimental setups can access the
full distribution via snapshots corresponding to projective measurements on
the whole system Mazurenko  _et al._ (2017); Gross and Bloch (2017); Bakr  _et
al._ (2009); Hofferberth  _et al._ (2008); Weitenberg  _et al._ (2011); Bohrdt
_et al._ (2021); Parsons  _et al._ (2016); Hilker  _et al._ (2017); Mitra  _et
al._ (2018); Haller  _et al._ (2015); Sherson  _et al._ (2010); Bloch  _et
al._ (2012); Bernien  _et al._ (2017); Islam  _et al._ (2011); Zhang  _et al._
(2017); Gärttner  _et al._ (2017); Song  _et al._ (2017); Arute  _et al._
(2019); Blais  _et al._ (2021); Wendin (2017); Wei  _et al._ (2022). In this
work, we will only be focusing on the behavior of the variance, $\langle
Q^{2}\rangle-\langle Q\rangle^{2}$, for states at half filling, i.e. $\mu=0$.
When $\mu=0$, $\langle Q\rangle=0$ but $\langle Q^{2}\rangle\neq 0$ and is
generally expected to grow in a power law fashion, $\langle Q^{2}\rangle\sim
t^{1/z_{\mathrm{var}}}$ and $z_{\mathrm{var}}=2$ for diffusive systems. In
this section we numerically compute $\langle Q^{2}\rangle$ and extract
$z_{\mathrm{var}}$ via a log-derivative, i.e.
$z_{\mathrm{var}}(t)=2\bigg{(}\frac{d\log\langle Q^{2}\rangle}{d\log
t}\bigg{)}^{-1}$.

Our results are shown in Fig. 2. One can see that when initial states are
drawn from the correlated ensemble $\langle Q^{2}\rangle$ grows faster than
initial states drawn from a thermal ensemble at infinite temperature, which is
consistent with the screening argument. We extracted $z_{\mathrm{var}}$ for
various $\alpha$ and one can clearly see that tuning $\alpha$ does indeed
change the behavior of $\langle Q^{2}\rangle$ which is consistent with the
fact that tuning fluctuations of the initial state should correspond to tuning
transport.

_Theory of charge transfer_.— To understand the behavior of $Q(t)$ we first
consider initial states where the magnetization density $m_{x}=(2n_{x}-1)/2$
for $x\leq 0$ ($x>0$) is $m/2$ ($-m/2$) and the imbalance, $m$, is large. In
this limit one has a low density of magnons in each half of the chain so there
is a well-defined domain wall that separates the two halves of the chain. For
example an initial state might look like

,

where the squiggly line indicates the location of the domain wall which
separates the halves of the chain. Due to domain wall conservation the only
time charge can be transferred is if a particle or hole moves across the
origin. At some later time the magnon will cross the origin and our state will
look like

.

Notice that the domain wall that separated the halves of the chain moves two
sites in the opposite direction in which the magnon moved. The key observation
that was made in Ref. Gopalakrishnan  _et al._ (2022) is that charge transfer
is linked to the motion of this domain wall that separates the two halves. If
at time $t$, the domain wall is at $x_{\mathrm{dw}}(t)$ and if
$m_{[0,x_{\mathrm{dw}}]}$ is the magnetization density in the region between
the origin and the location of the domain wall then

$Q(t)=-2m_{[0,x_{\mathrm{dw}}(t)]}x_{\mathrm{dw}}(t).$ (5)

When $m\rightarrow 0$ there is not a well defined notion of a domain wall that
separates the two halves of the chain. However one can imagine a fictitious
domain wall which is located at the origin and moves in the same fashion as
one would expect when $m\simeq 1$. For instance, suppose we have the following
initial state,

,

where the squiggly line indicates the location of our fictitious domain wall
initially at the origin. At the next time step the hole next to the origin
crosses over to the left half of the system and our state will look like

.

The domain wall moves two steps in the opposite direction in which the magnon
propagated. Demanding that the fictitious domain wall move in this manner
implies that the charge transfer satisfies Eq. 5.

From Eq. 5, we can now relate the dynamical exponent of the charge transfer
variance to the motion of the domain wall. On average the domain wall will be
at the origin at half-filling, i.e. $\langle x_{\mathrm{dw}}(t)\rangle=0$,
since the number of left and right movers is equal, but $\langle
x_{\mathrm{dw}}(t)^{2}\rangle\neq 0$ and is expected to grow as a power law,
$\langle x_{\mathrm{dw}}(t)^{2}\rangle\sim t^{2/z_{\mathrm{dw}}}$. Using the
fact that the magnetization density over a distance $|x_{\mathrm{dw}}|$ scales
$m\sim|x_{\mathrm{dw}}|^{-1+1/\alpha}$, then the variance of the charge
transfer should obey the following scaling relation,

$\langle Q(t)^{2}\rangle\sim t^{2/\alpha z_{\mathrm{dw}}}.$ (6)

Figure 3: Dynamical Exponent Finite Size Effects. (a) One sees that at short
times the dynamical exponent of the variance of the charge differs for
different scaling parameters but at longer times converges to the same value.
Here data was averaged $10^{7}$ initial states. (b) The domain wall dynamical
exponent as function of time features much stronger finite sizes as the
exponent does not appear to converge until $\sim 10^{3}$ which is an order of
magnitude larger than the variance of the charge transfer. Here results were
averaged over $10^{6}$ realizations.

To obtain what $z_{\mathrm{dw}}$ is we note that the domain wall only moves
when a magnon “hits” it. And whenever the magnon “hits” the domain wall its
charge changes thus the variance in time of $x_{\mathrm{dw}}$ should follow
the same time dependence as the variance in the amount of charge a magnon has
carried, i.e. $\delta x_{\mathrm{mag}}(t)$. Since $\delta
x_{\mathrm{mag}}^{2}\sim t^{2-2w}=t^{2/\alpha}$ then $\langle
x_{\mathrm{dw}}(t)^{2}\rangle\sim t^{2/\alpha}$. Using this result along with
Eq. 6 implies $z_{\mathrm{var}}=\alpha^{2}/2$. From Fig. 2, we see relatively
good agreement with our prediction.

Numerical deviations from this prediction can be understood from two finite
size/time effects: the motion of the domain wall and the dependence of the
dynamical exponents on $c$. The first comes from the fact that in a time $t$
the domain wall will have only moved a distance $t^{1/\alpha}$ and thus the
domain wall cannot efficiently probe the tails of the distribution. The second
comes from the fact that the mean domain size of is tuned by the scaling
parameter, $c$, and at long times and distances one expects the dependence on
$c$ to drop out and only the tails of the distributions should matter. As
shown in Fig. 3, we see that $z_{\mathrm{var}}$ and $z_{\mathrm{dw}}$ display
a dependence on the scale parameter with the more severe dependence in the
latter case presumably coming from the fact that the domain wall does not
probe the tails of the distribution well enough and thus is much more prone to
experience stronger effects from the scaling parameter.

_Discussion_.— In this work we demonstrated that anisotropic integrable spin
chains provide a route towards tunable transport by tuning initial state
magnetization fluctuations. We gave a procedure to generate states which have
quasi-long range correlations using Lévy alpha stable distributions. And we
demonstrated that the fluctuations in this ensemble of correlated states is
tuned by the stability parameter that changes the tails of these stable
distributions.

Using the variance of the charge transfer we found superdiffusive transport in
these initial states which is tuned by the stability parameter consistent with
the screening argument set forth in Ref. Gopalakrishnan and Vasseur (2019).
Furthermore we are able to provide a theoretical prediction for the dynamical
exponent controlling the growth of the charge variance via the motion of a
“domain wall” and the magnetization density in the region between the position
of the domain wall and the position at which the charge transfer is being
measured. We find good agreement with our prediction and numerical simulations
with some finite size effects.

We showed that these finite size effects come about from the domain wall’s
inability to efficiently probe the tails of these stable distributions at
short times. As such the domain wall motion is severely affected by the
average domain size that it probes which is controlled by the scaling
parameter of the Lévy alpha stable distributions. Surprisingly the time scale
for which the dependence on $c$ goes away is different for the charge variance
and the domain wall motion. This suggests that finite size effects in the
magnetization density cancel out the finite size effects in the motion of the
“domain wall.” It would be interesting to find different ensembles of
correlated states that are able to probe the tails more effectively.

While we only presented results for superdiffusive transport in principle one
can also achieve tunable subdiffusive transport. For this to happen one would
need to generate states which have fluctuations that drop off faster than
those satisfying the central limit theorem. Such states, also known as
hyperuniform states, would need to have domains that are anti-correlated.

A second interesting route would be to go away from the
$\Delta\rightarrow\infty$ limit and probe transport in the $\Delta>1$ regime
of the spin-1/2 XXZ chain using these correlated initial states. In this
regime fully polarized domains are no longer frozen but are only frozen for a
time $t\sim\Delta^{s-1}$ where $s$ is the domain length. The dynamics of the
spin-1/2 XXZ chain is accessible in experiments Jepsen  _et al._ (2020b) and
so it would also be interesting to explore this tunable transport
experimentally since these correlated initial states are product states.
Approaching the isotropic $\Delta=1$ limit should be particularly interesting,
as it already shows superdiffusive transport with $z=3/2$ for thermal states
Žnidarič (2011); Ljubotina  _et al._ (2019); Bulchandani  _et al._ (2021);
Gopalakrishnan and Vasseur (2023); Wei  _et al._ (2022).

_Acknowledgements_.— We thank Vedika Khemani for stimulating discussions. This
work was supported by the National Science Foundation under NSF Grants No.
DMR-2103938 (S.G.), DMR-2104141 (R.V.), DMR-1945529 (M.K.), the Welch
Foundation through award number AT-2036-20200401 (M.K.), and the Alfred P.
Sloan Foundation through a Sloan Research Fellowship (R.V.) .

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