Measurement catastrophe and ballistic spread of charge density with vanishing
current

L. Zadnik1, S. Bocini1, K. Bidzhiev1, and M. Fagotti1

1 Université Paris-Saclay, CNRS, LPTMS, 91405, Orsay, France

*<EMAIL_ADDRESS>
## Abstract

One of the features of many-body quantum systems with Hilbert-space
fragmentation are stationary states manifesting quantum jamming. It was
recently shown that these are “states with memory”, in which, e.g., measuring
a localised observable has everlasting macroscopic effects. We study such a
measurement catastrophe with an example that stands out for its clarity. We
show in particular that at late times the expectation value of a charge
density becomes a nontrivial function of the ratio between distance and time
notwithstanding the corresponding current approaching zero.

###### Contents

  1. 1 Introduction
    1. 1.1 Overview of the model
  2. 2 Initial state and spin-flip protocol
  3. 3 Exact solution to the dynamics
    1. 3.1 Macrosite fluctuations of particles
    2. 3.2 Local magnetisation
    3. 3.3 Emergence of a locally quasi-jammed state
    4. 3.4 Local conservation laws and currents
    5. 3.5 Entanglement in a weakly interacting setting
  4. 4 From the dual folded XXZ to the XXZ model
  5. 5 Discussion
  6. A Proofs
    1. A.1 Time evolution of the state and $\braket{\sigma_{\ell}^{z}}_{t}$
    2. A.2 Equal and non-equal time correlations
    3. A.3 Matrix elements of $\sigma_{\ell}^{z}$
      1. A.3.1 Increasing $n$: the right-moving impurity
      2. A.3.2 Decreasing $n$: the left-moving impurity
    4. A.4 Asymptotic profiles of magnetisation
  7. B Correlation functions in the weakly-interacting case
    1. B.1 The idea behind the computation
    2. B.2 List of the two-point correlation functions
  8. C Details on numerical simulations

##  1 Introduction

Complexity is behind the most intriguing phenomena observed in many-body
systems: phases of matter, spontaneous symmetry breaking, relaxation, etc. On
the other hand, our personal understanding of a phenomenon is almost
inevitably reductive, being based on simplifications and analogies. Undressing
a problem of the complicated details that do not affect the behaviour under
investigation is one of the most important steps towards the development of a
decent physical picture. Just as the framework of renormalisation group allows
one to identify what is relevant at a phase transition, so the conception of a
simple model helps identify what is essential.

Exactly solvable models are arguably the main source of procedural knowledge
in physics: by solving a problem exactly, we can grasp the significance of
what could be seen in nature. These models, which are generally integrable,
are clearly special. As such, they exhibit exceptional features that could be
absent in real systems. A phenomenon identified in an exactly solvable model
is nevertheless always potentially observable, if not exactly like predicted
in the model, as a somehow related phenomenon. Pre-thermalisation is a
remarkable example of this kind: integrable systems are known to relax
differently than generic ones [1, 2, 3, 4, 5], and yet such special, atypical
dynamics leave a mark also on the relaxation of generic systems [6], which can
exhibit pre-thermalisation plateaux [7].

In fact, even working in the framework of integrability can present formidable
complications that undermine our qualitative understanding. This is one of the
motivations behind the recent interest in identifying the “simplest”
interacting integrable systems. In this respect, a spin-$1/2$ chain model with
a three-site interaction termed the “dual folded XXZ model” was recently
highlighted [8, 9, 10]. It corresponds to a special point of a rather
complicated integrable system called Bariev model [11, 12], and is equivalent
to the celebrated Heisenberg XXZ model in the limit of large anisotropy. The
dual folded XXZ model was recently solved by two of us from scratch by means
of a coordinate Bethe Ansatz that has been adapted to the special symmetries
emerging in the large-anisotropy limit of the XXZ model [8]. In particular,
the Hilbert space splits into an exponentially large number of sectors with a
given configuration of particles, each particle being associated with a spin
up in either an even or an odd position. Similar structures arise in what was
recently dubbed “Hilbert-space fragmentation” [13, 14, 15]. In each sector the
model is almost free, with a diagonal scattering matrix in which only one
diagonal element is different from $-1$ and equal to $-e^{iq}$, where $q$ is
the difference between the momenta of the scattering particles, in a
remarkable analogy with the scattering phase modification factor in hard-rod
deformed models (see, e.g., Refs [16, 17]). A striking feature found in
fragmented models, such as the folded XXZ one, is the existence of an
exponentially large sector consisting of jammed states: particles are stuck
and can not move [18, 19, 20, 21, 22, 23]. In Ref. [24] it was shown that such
states have memory: the effect of a localised perturbation does not fade away
but rather becomes visible at macroscopic scales for arbitrarily long times.
This is similar to the memory effects observed when perturbing ground states
with spontaneously broken symmetries [25, 26] (see also Refs [26, 27, 28, 29]
for a perturbation local in the fermionic basis), where however the dynamics
are effectively constrained in a finite-dimensional space. In the setting of
Ref. [24], instead, an exponentially large number of states are involved.
Considering how dramatic the change in the state is, it seems appropriate to
refer to it as a “measurement catastrophe”. In particular, this effect was
reported in a class of states that allow for an exact analytic analysis but
are, however, not simple enough to make us gain a clear understanding of the
phenomenon. This paper is first of all a solution to that problem.

We consider a family of jammed states for which it is possible to write down
the time evolving state after a spin flip in a simple, elegant way. We then
focus on observables whose expectation values can be predicted with minimal
information about the initial state. Among them there will be the densities of
two conserved charges: the magnetisation and the staggered magnetisation. They
will be our key monitor of the measurement catastrophe: indeed, they turn out
to be affected by the original spin flip in a macroscopic way. This sets the
example herein apart from the one considered in Ref. [24], where instead, in
the long time limit, charge densities and currents become independent of their
positions. The central part of the paper will be devoted to devising a
physical picture that could reconcile the nontrivial profile of the
(staggered) magnetisation at the Euler/ballistic scale with the fact that the
corresponding currents vanish in the limit of infinite time. First we will
present a set of rules that allow us to predict the asymptotic behaviour of
any spin solely from the knowledge of the initial configuration of few
neighbouring spins. Using the rules we will then show that the asymptotic
state of the system is locally jammed and that the currents of the
magnetisation and the staggered magnetisation, as well as all other
conventional charges of the folded XXZ model, vanish in it. By computing the
statistics of the particles’ positions, we will show that the macroscopic
reorganisation of the spin configuration coincides with small correlated
microscopic fluctuations of particles (spins up) that do not exceed two
lattice sites. Finally, we will focus on a special class of weakly interacting
initial jammed states, which allow for the preparation of correlated spin
pairs in an antiferromagnetic background. We will study, in particular, the
entanglement between spins.

###  1.1 Overview of the model

The dual folded XXZ model describes the time evolution of states in the large-
anisotropy limit of the Heisenberg model. It is obtained in the “folded
picture”, which was defined in Ref. [8] as a representation of time evolution
with Hamiltonians that have a large coupling constant. Based on the strong
coupling expansion of Ref. [30], the “folded picture” refers to a
decomposition of time evolution into a periodic high-frequency evolution of
operators and a “gentle” evolution of the state with a time-independent
Hamiltonian. The dual folded XXZ Hamiltonian reads

$\displaystyle
H=\mathcal{J}\sum_{\ell}\frac{1-\sigma_{\ell+1}^{z}}{2}(\sigma_{\ell}^{x}\sigma_{\ell+2}^{x}+\sigma_{\ell}^{y}\sigma_{\ell+2}^{y})\,.$
(1)

It is integrable and describes constrained hopping on a lattice, summarised by
the following local dynamical rule:
$\ket{\uparrow\downarrow\downarrow}\,\leftrightarrow\,\ket{\downarrow\downarrow\uparrow}$.
Such a rule preserves the sequence of parities of the positions of spins up.
This in turn implies conservation of the configuration of two species of
particles defined by that parity. The presence of multiple species of
particles typically signals that an integrable model is solvable via a
complicated nested Bethe Ansatz. And indeed, the dual folded XXZ model
corresponds to a strong-repulsion regime of the two-component Bariev model
[11, 12], which is of that type. Nevertheless, at the special point described
by Eq. (1) the nested structure of the Bariev model is reduced to such a
degree that Bethe equations can be solved explicitly. Perhaps the most
striking remnant of the nested structure is the conservation of the
configuration, which results in the breakdown of one-site shift invariance
into two-site shift invariance.

The constrained hopping leads to a very rich Hilbert-space structure combined
with an exceptional simplicity of the Bethe equations. It enables exact
analytical and numerical analysis of various phenomena that have recently been
in the spotlight: Hilbert-space fragmentation [19], pre-relaxation [31],
generalised hydrodynamics [9], spin transport phenomena [32], the effect of
non-abelian symmetries, time-translation symmetry breaking [10], and the
macroscopic effects of local measurements [24]. Most of these phenomena are
accessible in sectors that allow for an even simpler description; we mention,
for example, the non-interacting sectors, where only one species of particles
is present, or the exponentially large jammed sector, where nearest-neighbour
pairs of spins down are prohibited and the basis states can be chosen to be
eigenstates of all $\sigma^{z}_{\ell}$ simultaneously.

The local conserved charges of the dual folded XXZ model have been classified
in Ref. [8]. Their place in the Yang-Baxter integrability structure of the
model [33, 34, 35] was described in Refs. [36, 37]. We mention that the
fragmentation of the Hilbert space in the folded XXZ model could be related to
the existence of additional non-abelian conservation laws, potentially
associated with the emergence of the integrable analogue (cf. Refs [38, 39,
40]) of quantum many-body scars [15, 41, 42, 43, 44, 45].

##  2 Initial state and spin-flip protocol

We consider a protocol that consists of the following steps:

  1. 1.

Prepare the system in a _jammed_ product state with spins aligned along the
$z$-axis (in either direction).

In a jammed state fragments of the form $\cdots\downarrow\downarrow\cdots$ are
prohibited. We use the notations of Ref. [8]: spins up correspond to particles
of two species determined by the parity of their positions. Specifically, a
spin up at site $\ell$ corresponds to a particle of species
$b=2\lceil\ell/2\rceil-\ell\in\\{0,1\\}$ at macrosite
$\ell^{\prime}=\lceil\ell/2\rceil$. A jammed state is completely characterised
by the configuration $\underline{b}=(\ldots,b_{n},b_{n+1},\ldots)$ of spins up
as follows:

$\displaystyle\ket{\underline{b}}=\prod_{j}\sigma_{2\ell_{j}^{\prime}-b_{j}}^{+}\ket{\downarrow\cdots\downarrow}\,.$
(2)

Here, the macrosites satisfy the recurrence relation

$\displaystyle\ell^{\prime}_{j+1}=\ell^{\prime}_{j}+1-b_{j}(1-b_{j+1})\,,$ (3)

according to which two subsequent particles of species $1$ and $0$,
respectively, occupy the same macrosite. This prevents state fragments of the
form
$\cdots\underline{\uparrow\downarrow}\,\underline{\downarrow\uparrow}\cdots$
(underlined are the macrosites), which violate the jamming condition.

  2. 2.

Flip a spin up that was close to a spin down (otherwise the state remains
jammed). This step can be interpreted as a result of a double projective
measurement, e.g., measuring $\sigma^{z}$ after $\sigma^{x}$.

The spin flip destroys a particle. We choose its index to be $0$, so that the
state now reads

$\Bigl{(}\prod_{j\leq-1}\sigma^{+}_{2\ell^{\prime}_{j}-b_{j}}\Bigr{)}\Bigl{(}\prod_{j\geq
1}\sigma^{+}_{2\ell^{\prime}_{j}-b_{j}}\Bigr{)}\ket{\downarrow\cdots\downarrow}\,.$
(4)

Recurrence (3) can now be solved with respect to the position of the spin
flipped. First we note that either $b_{-1}$ or $b_{1}$ should be equal to
$b_{0}$: we flipped a spin close to a spin down (the site of the next spin up
has the same parity as the site of the spin flipped). We choose the macrosites
in such a way that at $\ell^{\prime}_{0}=0$ there are two spins down. As a
result we either have $b_{-1}=b_{0}=0$ or $b_{1}=b_{0}=1$. In fact, since we
removed a particle (a spin up), it is convenient to redefine the index of the
particles, $j$, so as to fill the hole at $j=0$. We choose the convention
$j_{\rm new}=j_{\rm old}+\theta(j_{\rm old}<0)$ and then find

$\ell^{\prime}_{j}=j-\theta(j\leq
0)+\begin{cases}-\sum^{j-1}_{m=1}b^{(0)}_{m}(1-b^{(0)}_{m+1})\,,&\text{if
}j>0\\\ \sum^{|j|}_{m=1}(1-b_{1-m}^{(0)})b_{-m}^{(0)}\,,&\text{if }j\leq
0\,.\end{cases}$ (5)

Here, $\underline{b}^{(0)}$ is the configuration obtained by removing $b_{0}$
from $\underline{b}$: $b^{(0)}_{j}=b_{j-\theta(j\leq 0)}$. We indicate this
state by

$\ket{\Psi(t=0)}=\ket{0;\underline{b}^{(0)}}\,,$ (6)

where $\ket{n;\underline{c}}$ denotes a state which is jammed everywhere
except between the particles $c_{n}$ and $c_{n+1}$, where either two or three
consecutive spins down are present. We call this excess of spins down
“impurity”.

  3. 3.

Time evolve with $H$.

While the folded XXZ model is interacting, time evolution is particularly
simple in the sector of our initial state. As shown in Appendix A.1, this
sector is spanned by states $\ket{n;\underline{b}^{(0)}}$, in which the
particle macrosites read

$\displaystyle\ell^{\prime(n)}_{j}=j-\theta(j\leq
n)+\begin{cases}-\sum^{j-1}_{m=1}b^{(0)}_{m}(1-b^{(0)}_{m+1})\,,&\text{if
}j>0\\\ \sum^{-j}_{m=1}(1-b_{1-m}^{(0)})b_{-m}^{(0)}\,,&\text{if }j\leq
0\,.\end{cases}$ (7)

They are obtained from Eq. (5) by noting that, when the impurity jumps across
a particle, the macrosite of the latter shifts by $1$ in the opposite
direction.

##  3 Exact solution to the dynamics

The configuration of particles in the state is preserved under the action of
the Hamiltonian (1). As a result, the impurity can be interpreted as a fermion
freely hopping in the background $\underline{b}^{(0)}$ of particles. As shown
in Appendix A.1, the state at time $t$ reads

$\ket{\Psi(t)}=\sum_{n}(-i)^{n}J_{n}(4\mathcal{J}t)\ket{n;\underline{b}^{(0)}}\,,$
(8)

where $J_{n}(x)$ are Bessel functions.

Let us denote the expectation value $\bra{\Psi(t)}A\ket{\Psi(t)}$ by
$\braket{A}_{t}$ for any operator $A$ at arbitrary time $t$. For the
expectation values of operators $D$ that are diagonal in the basis of states
$\ket{n;\underline{b}^{(0)}}$, for example $D=\sigma_{\ell}^{z}$, we now have

$\braket{D}_{t}=\sum_{n}|J_{n}(4\mathcal{J}t)|^{2}\braket{n;\underline{b}^{(0)}}{D}{n;\underline{b}^{(0)}}\,.$
(9)

If $D^{2}=1$, like for a string of Pauli matrices $\sigma_{\ell}^{z}$, the
expectation values $\braket{n;\underline{b}^{(0)}}{D}{n;\underline{b}^{(0)}}$
can be either $1$ or $-1$.

The subtle difference between these dynamics and the ones in the XX model, in
which the initial state has a single spin up, is hidden in the mapping (7)
between the positions in the chain and those in the configuration. As will be
shown in the following, this mapping has striking effects on the dynamics of
local observables, such as the $z$-component of the spin.

###  3.1 Macrosite fluctuations of particles

We embrace here the Lagrangian perspective of following the motion of
particles. A particle is characterised by its species and its macro-position.
The species is fixed whereas the macro-position is a dynamical variable. In
the following we compute the statistics of the particles’ macro-positions. To
that aim, let us consider the macrosite (7) corresponding to the particle
labeled by $j$ (i.e., the $j$-th spin up). This can be thought of as the
eigenvalue of an operator that is diagonal in the basis of states
$\ket{n;\underline{b}^{(0)}}$, in which it assumes values $\theta(j>n)+c(j)$,
where $c(j)$ denotes is its state-independent part. Using Eq. (9) we can now
compute the characteristic function $\braket{e^{ik\ell^{\prime}_{j}}}_{t}$. We
find

$\braket{e^{ik\ell^{\prime}_{j}}}_{t}=e^{ikc(j)}[(e^{ik}-1)f(j,t)+1]\,,$ (10)

where we introduced the auxiliary quantity

$\displaystyle
f(x,t):=\sum_{n}\theta(x>n)|J_{n}(4\mathcal{J}t)|^{2}\sim\frac{1}{2}+\frac{1}{\pi}\arcsin\Big{(}\frac{x}{4\mathcal{J}t}\Big{)}\,.$
(11)

The characteristic function (10), in particular, implies

$\displaystyle\mathrm{Prob}(\ell^{\prime}_{j})=\begin{cases}1-f(j,t)\,,&\text{if
}\ell^{\prime}_{j}=c(j)\\\ f(j,t)\,,&\text{if
}\ell^{\prime}_{j}=c(j)+1\,,\end{cases}$ (12)

as well as

$\displaystyle\braket{\ell^{\prime}_{j}}_{t}=f(j,t)+c(j)\,,\qquad\braket{\ell_{j}^{\prime
2}}_{t}^{\rm c}=f(j,t)-f(j,t)^{2}\,,$ (13)

where $\langle AB\rangle^{\rm c}_{t}:=\langle AB\rangle_{t}-\langle
A\rangle_{t}\langle B\rangle_{t}$ denotes the connected correlation function.

The particles therefore experience fluctuations of order $O(1)$ around their
average positions. The fluctuations of different particles are moreover
coherent, as can be inferred from the following non-negative equal-time
connected correlation function

$\displaystyle\braket{\ell^{\prime}_{m}\ell^{\prime}_{n}}_{t}^{\rm
c}=\theta(m\geq n)f(n,t)+\theta(n>m)f(m,t)-f(m,t)f(n,t)\,,$ (14)

plotted along with $\braket{\ell_{j}^{\prime 2}}_{t}^{\rm c}$ in Fig. 1.

Figure 1: Fluctuations of the macrosite corresponding to the $j$-th particle,
and the equal-time connected correlation function between the macrosites of
particle $0$ and particle $j$. Both are plotted as functions of the ballistic
ray $j/(\mathcal{J}t)$.

###  3.2 Local magnetisation

We now adopt an Eulerian point of view and study the properties of the state
at a given position or ray. The Hamiltonian commutes with
$\prod_{j}\sigma_{j}^{z}$ and the initial state is an eigenstate of that
operator, therefore
$\braket{\sigma_{\ell}^{x}}_{t}=\braket{\sigma_{\ell}^{y}}_{t}=0$. The local
magnetisation along the $z$-axis instead exhibits a nontrivial ballistic-scale
profile. Indeed, as shown in Appendices A.3 and A.4, the asymptotic behaviour
of the local magnetisation along the $z$-axis is determined by the following
rules:

$\displaystyle\braket{\sigma_{\ell}^{z}}_{t}\sim\begin{cases}1\,,&\text{for
}\ket{\cdots\bullet\cdots\uparrow_{\ell}\updownarrow\uparrow\cdots}\vee\ket{\cdots\uparrow\updownarrow\uparrow_{\ell}\cdots\bullet\cdots}\\\
\frac{2}{\pi}\arcsin\big{(}\tfrac{x(\ell)}{4\mathcal{J}t}\big{)}\,,&\text{for
}\ket{\cdots\bullet\cdots\uparrow_{\ell}\uparrow\downarrow\cdots}\vee\ket{\cdots\uparrow\uparrow\downarrow_{\ell}\cdots\bullet\cdots}\\\
-\frac{2}{\pi}\arcsin\big{(}\tfrac{x(\ell)}{4\mathcal{J}t}\big{)}\,,&\text{for
}\ket{\cdots\bullet\cdots\downarrow_{\ell}\uparrow\uparrow\cdots}\vee\ket{\cdots\downarrow\uparrow\uparrow_{\ell}\cdots\bullet\cdots}\\\
-1\,,&\text{for
}\ket{\cdots\bullet\cdots\downarrow_{\ell}\uparrow\downarrow\cdots}\vee\ket{\cdots\downarrow\uparrow\downarrow_{\ell}\cdots\bullet\cdots}\,.\end{cases}$
(15)

Here, $\ket{\cdots\bullet\cdots\updownarrow_{\ell}\cdots}$ and
$\ket{\cdots\updownarrow_{\ell}\cdots\bullet\cdots}$ stand for fragments of
the initial state in which $\ell$ is positive or negative, respectively. For
large-enough $|\ell|$, $x(\ell)$ can be computed from

$\displaystyle\frac{1}{\ell}\int_{0}^{x(\ell)}\mathrm{d}n\,\xi(n)=\frac{1}{2}\,,$
(16)

where $\xi(n)$ is the coarse-grained macro-distance between particles, namely
the average of $1-b^{(0)}_{j}(1-b^{(0)}_{j+1})$ over a large-enough interval
of $j$ including $n$. For example, $\xi(n)=3/4$ for the fragment
$\ldots,1,1,0,0,1,1,0,0,1,1,0,0,\ldots$ The remarkable feature of Eq. (15) is
that the asymptotic value of the local magnetisation at site $\ell$ is
determined solely by a configuration of three consecutive spins around that
site. The only sites in which the spins are asymptotically not aligned along
the $z$-axis are those that start in one of the following spin configurations:
$\bullet\cdots\underline{\uparrow}\\!\uparrow\downarrow$,
$\bullet\cdots\underline{\downarrow}\\!\uparrow\uparrow$,
$\uparrow\uparrow\\!\underline{\downarrow}\cdots\bullet$, and
$\downarrow\uparrow\\!\underline{\uparrow}\cdots\bullet$. Their common feature
is the existence of both species of particles and therefore the presence of
interaction [8]. Indeed, as shown in Fig. 2, initial states with a single
species of particles do not exhibit nontrivial ballistic profiles.

Figure 2: Panel (a) shows $\braket{\sigma^{z}_{\ell}}_{t}$ as a function of
the ballistic ray coordinate $\ell/(\mathcal{J}t)$. The initial state
(immediately after the spin flip) is
$\ket{\cdots\uparrow\uparrow\downarrow\uparrow\uparrow\downarrow\uparrow\downarrow\downarrow\uparrow\uparrow\downarrow\uparrow\uparrow\downarrow\cdots}$.
Blue points correspond both to numerical implementation of Eq. (9) with
$D=\sigma_{\ell}^{z}$, as well as to the DMRG data (they are in perfect
agreement). The orange line corresponds to the asymptotic behaviour (15), in
which $x(\ell)=2\ell/3$, as follows from Eq. (16). Panel (b) shows
$\braket{\sigma^{z}_{\ell}}_{t}$ for $\mathcal{J}t=10$ for a state with one
species of particles, i.e., a noninteracting state. For the initial
configuration we chose a Néel state with one of the spins up flipped down
$\ket{\cdots\uparrow\downarrow\uparrow\downarrow\downarrow\downarrow\uparrow\downarrow\uparrow\downarrow\cdots}$.
Panel (c) shows $\braket{\sigma^{z}_{\ell}}_{t}$ for $\mathcal{J}t=10$ in the
state given in Panel (b) with complementary flipped up spin at $\ell=10$. Two
blue points describe the spins not aligned with the $z$-axis.

###  3.3 Emergence of a locally quasi-jammed state

Since the spin flip at time $t=0$ creates an excess of spins down, for $t>0$
the state is not jammed anymore. However, the weaker condition

$\displaystyle
P_{\downarrow\downarrow}(\ell)=\bigl{\langle}\frac{1-\sigma_{\ell}^{z}}{2}\frac{1-\sigma_{\ell+1}^{z}}{2}\bigr{\rangle}_{t}\xrightarrow{\mathcal{J}t\rightarrow\infty}0\,,\quad\text{for
all $\ell$}\,,$ (17)

used in Ref. [24] to define locally quasi-jammed states (LQJS), still holds.
Expressed in the basis of states $\ket{n;\underline{b}^{(0)}}$, the
expectation value used in the LQJS condition reads (see Appendix A.2)

$\displaystyle
P_{\downarrow\downarrow}(\ell)=\sum_{n}|J_{n}(4\mathcal{J}t)|^{2}\braket{n;\underline{b}^{(0)}}{\frac{1-\sigma_{\ell}^{z}}{2}}{n;\underline{b}^{(0)}}\braket{n;\underline{b}^{(0)}}{\frac{1-\sigma_{\ell+1}^{z}}{2}}{n;\underline{b}^{(0)}}\,.$
(18)

From Eq. (18) it follows that the LQJS condition is satisfied up to
corrections of order $O(1/t)$, a feature corroborated by exact numerics – see
Fig. 3 (an exact analytical calculation in a weakly interacting scenario is
presented in Section 3.5). Indeed, only few terms of the sum in Eq. (18) are
different from zero. Let us call $j(\ell)$ the index of the particle
represented by a spin up at site $\ell$, or at one of the neighbouring sites
if the spin at $\ell$ is down. For $|n-j(\ell)|>n_{0}$, with $n_{0}$ a finite
positive integer, the matrix elements of $\sigma^{z}_{\ell}$ and
$\sigma_{\ell+1}^{z}$ have simple asymptotic forms derived in Appendix A.3.
Using those, we can compute the eigenvalues of
$\tfrac{1}{2}(1-\sigma^{z}_{\ell})$ and $\tfrac{1}{2}(1-\sigma^{z}_{\ell+1})$
for all possible configurations of spins in the vicinity of $\ell$: they are
shown in Table 1 for a positive $\ell$. Observing that their product is always
zero, we conclude that $P_{\downarrow\downarrow}(\ell)$ is a finite sum of
squared Bessel functions, which decay to $0$ as $1/t$.

  1. (a)

$n-j(\ell)>n_{0}$:
$\uparrow_{\ell}\uparrow\downarrow\uparrow$
$\uparrow_{\ell}\uparrow\uparrow\downarrow$
$\uparrow_{\ell}\uparrow\uparrow\uparrow$
$\uparrow_{\ell}\downarrow\uparrow\uparrow$
$\uparrow_{\ell}\downarrow\uparrow\downarrow$
$\downarrow_{\ell}\uparrow\uparrow\uparrow$
$\downarrow_{\ell}\uparrow\uparrow\downarrow$
$\downarrow_{\ell}\uparrow\downarrow\uparrow$
$\braket{n;\underline{b}^{(0)}}{\tfrac{1}{2}(1\\!-\\!\sigma_{\ell}^{z})}{n;\underline{b}^{(0)}}$
1 0 0 0 0 0 0 1
$\braket{n;\underline{b}^{(0)}}{\tfrac{1}{2}(1\\!-\\!\sigma_{\ell+1}^{z})}{n;\underline{b}^{(0)}}$
0 1 0 0 1 0 1 0

  2. (b)

$n-j(\ell)<-n_{0}$:
$\uparrow_{\ell}\uparrow\downarrow\uparrow$
$\uparrow_{\ell}\uparrow\uparrow\downarrow$
$\uparrow_{\ell}\uparrow\uparrow\uparrow$
$\uparrow_{\ell}\downarrow\uparrow\uparrow$
$\uparrow_{\ell}\downarrow\uparrow\downarrow$
$\downarrow_{\ell}\uparrow\uparrow\uparrow$
$\downarrow_{\ell}\uparrow\uparrow\downarrow$
$\downarrow_{\ell}\uparrow\downarrow\uparrow$
$\braket{n;\underline{b}^{(0)}}{\tfrac{1}{2}(1\\!-\\!\sigma_{\ell}^{z})}{n;\underline{b}^{(0)}}$
0 0 0 0 0 1 1 1
$\braket{n;\underline{b}^{(0)}}{\tfrac{1}{2}(1\\!-\\!\sigma_{\ell+1}^{z})}{n;\underline{b}^{(0)}}$
0 0 0 1 1 0 0 0

Table 1: The expectation values of the projectors
$\tfrac{1}{2}(1-\sigma_{\ell}^{z})$ and $\tfrac{1}{2}(1-\sigma_{\ell+1}^{z})$
in the states $\ket{n;\underline{b}^{(0)}}$ depend only on the initial
configuration of the four spins on sites $\ell$, $\ell+1$, $\ell+2$, and
$\ell+3$. For $|n-j(\ell)|>n_{0}$, with $n_{0}$ a finite positive integer, the
products of these expectation values are always zero. Figure 3: Rescaled
jamming condition profile
$\mathcal{J}t\mathcal{P}_{\downarrow\downarrow}(\ell,t)$ for $\mathcal{J}t=25$
(blue) and $\mathcal{J}t=50$ (orange) in the initial state
$\ket{\cdots\uparrow\uparrow\downarrow\uparrow\uparrow\downarrow\uparrow\downarrow\downarrow\uparrow\uparrow\downarrow\uparrow\uparrow\downarrow\cdots}$.
The DMRG data confirm the $O(1/t)$ approach towards an LQJS:
$\mathcal{J}tP_{\downarrow\downarrow}(\ell,t)$ forms a profile with a scale
invariant envelope, conjectured to be $a/\sqrt{1-[\zeta/v_{\pm}]^{2}}$, where
$a\approx 0.16$, $v_{\pm}=\pm
8\mathcal{J}/\left(1+\tfrac{2}{L}\braket{S^{z}}_{0}\right)=\pm 6$
($\braket{S^{z}}_{0}$ is the total magnetisation in the initial state) – see
Ref. [24].

###  3.4 Local conservation laws and currents

The spin along the $z$-axis is related to the macrosite densities of two local
conservation laws

$\displaystyle
S^{z}=\sum_{\ell^{\prime}}\tfrac{1}{2}(\sigma_{2\ell^{\prime}-1}^{z}+\sigma_{2\ell^{\prime}}^{z})\,,\qquad
S_{-}^{z}=\sum_{\ell^{\prime}}\tfrac{1}{2}(\sigma_{2\ell^{\prime}}^{z}-\sigma_{2\ell^{\prime}-1}^{z})\,,$
(19)

i.e., the magnetisation and the staggered magnetisation, respectively. Using
the asymptotic rules (15) we obtain

$\Bigl{\langle}\frac{\sigma_{2\ell^{\prime}}^{z}+\sigma_{2\ell^{\prime}-1}^{z}}{2}\Bigr{\rangle}_{t}\sim\begin{cases}1\,,&\text{for
}\ket{\cdots\bullet\cdots(\uparrow\uparrow)_{\ell^{\prime}}\uparrow\uparrow\cdots}\vee\ket{\cdots\uparrow\uparrow(\uparrow\uparrow)_{\ell^{\prime}}\cdots\bullet\cdots}\\\
\frac{1}{\pi}\arccos\big{(}-\tfrac{\tilde{x}(\ell^{\prime})}{4\mathcal{J}t}\big{)}\,,&\text{for
}\begin{cases}\ket{\cdots\bullet\cdots(\uparrow\uparrow)_{\ell^{\prime}}\uparrow\downarrow\cdots}\vee\ket{\cdots\uparrow\uparrow(\uparrow\downarrow)_{\ell^{\prime}}\cdots\bullet\cdots}\\\
\ket{\cdots\bullet\cdots(\uparrow\uparrow)_{\ell^{\prime}}\downarrow\uparrow\cdots}\vee\ket{\cdots\uparrow\uparrow(\downarrow\uparrow)_{\ell^{\prime}}\cdots\bullet\cdots}\end{cases}\\\
\frac{1}{\pi}\arccos\big{(}\tfrac{\tilde{x}(\ell^{\prime})}{4\mathcal{J}t}\big{)}\,,&\text{for
}\begin{cases}\ket{\cdots\bullet\cdots(\uparrow\downarrow)_{\ell^{\prime}}\uparrow\uparrow\cdots}\vee\ket{\cdots\uparrow\downarrow(\uparrow\uparrow)_{\ell^{\prime}}\cdots\bullet\cdots}\\\
\ket{\cdots\bullet\cdots(\downarrow\uparrow)_{\ell^{\prime}}\uparrow\uparrow\cdots}\vee\ket{\cdots\downarrow\uparrow(\uparrow\uparrow)_{\ell^{\prime}}\cdots\bullet\cdots}\end{cases}\\\
0\,,&\text{for
}\begin{cases}\ket{\cdots\bullet\cdots(\uparrow\downarrow)_{\ell^{\prime}}\uparrow\downarrow\cdots}\vee\ket{\cdots\uparrow\downarrow(\uparrow\downarrow)_{\ell^{\prime}}\cdots\bullet\cdots}\\\
\ket{\cdots\bullet\cdots(\downarrow\uparrow)_{\ell^{\prime}}\uparrow\downarrow\cdots}\vee\ket{\cdots\downarrow\uparrow(\uparrow\downarrow)_{\ell^{\prime}}\cdots\bullet\cdots}\\\
\ket{\cdots\bullet\cdots(\downarrow\uparrow)_{\ell^{\prime}}\downarrow\uparrow\cdots}\vee\ket{\cdots\downarrow\uparrow(\downarrow\uparrow)_{\ell^{\prime}}\cdots\bullet\cdots}\end{cases}\end{cases}$
(20)

for the magnetisation, and

$\Bigl{\langle}\frac{\sigma_{2\ell^{\prime}}^{z}-\sigma_{2\ell^{\prime}-1}^{z}}{2}\Bigr{\rangle}_{t}\sim\begin{cases}-\frac{1}{\pi}\arccos\big{(}\tfrac{\tilde{x}(\ell^{\prime})}{4\mathcal{J}t}\big{)}\,,&\text{for
}\ket{\cdots\bullet\cdots(\uparrow\uparrow)_{\ell^{\prime}}\uparrow\downarrow\cdots}\vee\ket{\cdots\uparrow\uparrow(\uparrow\downarrow)_{\ell^{\prime}}\cdots\bullet\cdots}\\\
-\frac{1}{\pi}\arccos(-\tfrac{\tilde{x}(\ell^{\prime})}{4\mathcal{J}t})\,,&\text{for
}\ket{\cdots\bullet\cdots(\uparrow\downarrow)_{\ell^{\prime}}\uparrow\uparrow\cdots}\vee\ket{\cdots\uparrow\downarrow(\uparrow\uparrow)_{\ell^{\prime}}\cdots\bullet\cdots}\\\
\frac{1}{\pi}\arccos(\tfrac{\tilde{x}(\ell^{\prime})}{4\mathcal{J}t})\,,&\text{for
}\ket{\cdots\bullet\cdots(\uparrow\uparrow)_{\ell^{\prime}}\downarrow\uparrow\cdots}\vee\ket{\cdots\uparrow\uparrow(\downarrow\uparrow)_{\ell^{\prime}}\cdots\bullet\cdots}\\\
\frac{1}{\pi}\arccos(-\tfrac{\tilde{x}(\ell^{\prime})}{4\mathcal{J}t})\,,&\text{for
}\ket{\cdots\bullet\cdots(\downarrow\uparrow)_{\ell^{\prime}}\uparrow\uparrow\cdots}\vee\ket{\cdots\downarrow\uparrow(\uparrow\uparrow)_{\ell^{\prime}}\cdots\bullet\cdots}\\\
\frac{2}{\pi}\arcsin(\tfrac{\tilde{x}(\ell^{\prime})}{\mathcal{J}t})\,,&\text{for
}\ket{\cdots\bullet\cdots(\downarrow\uparrow)_{\ell^{\prime}}\uparrow\downarrow\cdots}\vee\ket{\cdots\downarrow\uparrow(\uparrow\downarrow)_{\ell^{\prime}}\cdots\bullet\cdots}\\\
-1\,,&\text{for
}\ket{\cdots\bullet\cdots(\uparrow\downarrow)_{\ell^{\prime}}\uparrow\downarrow\cdots}\vee\ket{\cdots\uparrow\downarrow(\uparrow\downarrow)_{\ell^{\prime}}\cdots\bullet\cdots}\\\
0\,,&\text{for
}\ket{\cdots\bullet\cdots(\uparrow\uparrow)_{\ell^{\prime}}\uparrow\uparrow\cdots}\vee\ket{\cdots\uparrow\uparrow(\uparrow\uparrow)_{\ell^{\prime}}\cdots\bullet\cdots}\\\
1\,,&\text{for
}\ket{\cdots\bullet\cdots(\downarrow\uparrow)_{\ell^{\prime}}\downarrow\uparrow\cdots}\vee\ket{\cdots\downarrow\uparrow(\downarrow\uparrow)_{\ell^{\prime}}\cdots\bullet\cdots}\end{cases}$
(21)

for the staggered magnetisation: in both cases $\tilde{x}(\ell^{\prime})\sim
x(2\ell^{\prime})\sim x(2\ell^{\prime}-1)$ is the coarse-grained position in
the chain.

The currents of the magnetisation and of the staggered magnetisation are
obtained from the time derivative of $\sigma_{\ell}^{z}$ in the Heisenberg
picture

$\displaystyle\frac{{\rm d}}{{\rm
d}t}\sigma^{z}_{\ell}(t)\Bigr{|}_{t=0}=-\mathcal{J}D_{\ell,\ell+2}\frac{1-\sigma_{\ell+1}^{z}}{2}+\mathcal{J}D_{\ell-2,\ell}\frac{1-\sigma_{\ell-1}^{z}}{2}\,,$
(22)

where
$D_{\ell,m}=\sigma_{\ell}^{x}\sigma_{m}^{y}-\sigma_{\ell}^{y}\sigma_{m}^{x}$.
In a jammed state, each term on the right-hand side vanishes. We can therefore
conclude that the two currents approach zero in any locally quasi-jammed state
in which the jamming condition is satisfied up to $O(1/t)$ corrections.

The reader could be surprised at this result as we showed earlier that the
corresponding charges have a nontrivial profile in the ballistic scaling limit
$x\sim\mathcal{J}t$: in these situations, using the continuity equation, one
could infer that also the differences of currents should be functions of $x/t$

$\partial_{t}q(\tfrac{x}{t})+\partial_{x}j(x,t)=0\quad\Rightarrow\quad
j(x,t)-j(0,t)=\int_{0}^{x/t}{\rm d}\zeta q^{\prime}(\zeta)\zeta\,.$ (23)

Here, $q$ is the expectation value of the charge density and $j$ that of the
corresponding current. This argument can however be applied only if the charge
density is a (piece-wise) continuous function of $x/t$. In our case we have an
additional microscopic dependence on the lattice position $q=q(\ell,t)$: the
charge expectation value depends on the pattern of spins around the site
$\ell$ – cf. (15). As we move from site to site, $q$ jumps between four smooth
asymptotic curves that can be recognised only on large (ballistic) scales. The
continuity equation on the chain then reads

$\partial_{t}q(\ell,t)=j(\ell+1,t)-j(\ell,t)$ (24)

and, due to the microscopic dependence on $\ell$, it now becomes possible to
have a nontrivial ballistic profile for the charge notwithstanding the
corresponding current approaching zero. The macroscopic reconfiguration of the
spin profiles after a localised perturbation is thus not a typical transport
phenomenon: as shown in Section 3.1, the positions of the spins up only
experience “collective” fluctuations of magnitude $O(1)$. The phenomenon is
somewhat reminiscent of the travelling surface waves in which the particles of
water undergo only a short-scale oscillatory motion. The distinguishing and
perhaps most surprising feature of the phenomenon shown here is that it is a
result of an instantaneous perturbation.

Figure 4: Spin current $q_{1}^{-}(\ell,t)$ and higher charge
$q_{2}^{+}(\ell,t)$ for $\mathcal{J}t=25$ (blue points) and $\mathcal{J}t=50$
(orange points), obtained from the DMRG data for the initial state
$\ket{\cdots\uparrow\uparrow\downarrow\uparrow\uparrow\downarrow\uparrow\downarrow\downarrow\uparrow\uparrow\downarrow\uparrow\uparrow\downarrow\cdots}$.
In the large-time limit both profiles vanish as $1/t$, whereas the charges
$q_{1}^{+}$ and $q_{2}^{-}$ are strictly zero during the time evolution.

Using the thermodynamic Bethe Ansatz description developed in Ref. [9], it can
also be shown that the expectation values of the remaining (known) local
conservation laws (reported in Refs [8, 9]) asymptotically vanish. In the
thermodynamic limit their expectation values per macrosite read [9]

$\displaystyle q_{n}^{\pm}(x,t)=\int{\rm
d}p\,\rho_{x,t}(p)\mathcal{Q}_{n}^{\pm}(p)\,,$ (25)

where the root density $\rho_{x,t}(p)$ encodes the distribution of the
occupied momenta at $x$ and $t$, while
$\mathcal{Q}_{n}^{+}(p)=4\mathcal{J}\cos(np)$ and
$\mathcal{Q}_{n}^{-}(p)=4\mathcal{J}\sin(np)$ are the single-particle charge
expectation values. The root density $\rho$ and the total density $\rho_{\rm
t}$ of vacancies (i.e., particles and holes) in the momentum space satisfy
$\int{\rm d}p\,\rho(p)=\xi^{-1}$ and $2\pi\rho_{\rm t}=1+\mu/\xi$,
respectively, where $\xi^{-1}$ is the particle density per macrosite, while
$\mu$ is the thermodynamic limit of the ratio $\sum_{j}b_{j}(1-b_{j+1})/N$,
$N$ being the number of spins up (see Ref. [9]). In a jammed state of an
infinite system one has $\xi+\mu=1$, whence $\int{\rm d}p\,\rho(p)=1/(1-\mu)$
and $\rho_{\rm t}=1/(2\pi[1-\mu])$ [8]. The occupancy ratio
$n=\rho(p)/\rho_{\rm t}$ then satisfies $\int{\rm d}p\,n(p)=2\pi$ and, since
$0\leq n(p)\leq 1$, one has $n(p)=1$. This implies that $\rho(p)$ is constant,
whence all local charge expectation values, expect for the magnetisation (and
the staggered one, albeit for a different reason111The staggered magnetisation
is not related to the root density $\rho$, so the argument above does not
pertain to it.), vanish:

$\displaystyle q_{n}^{\pm}(x,t)=0,\quad\text{for all $n>0$}\,.$ (26)

A similar conclusion can be drawn for the currents, whose expectation values
read

$\displaystyle j_{n}^{\pm}(x,t)=\int{\rm
d}p\,\rho_{x,t}(p)v_{x,t}(p)\mathcal{Q}_{n}^{\pm}(p)+\text{const.}\,,$ (27)

where $v(p)=-4\mathcal{J}\sin
p-4\mathcal{J}\mu\int\frac{\mathrm{d}k}{2\pi}n(k)(\sin k-\sin p)$ is the
dressed velocity [9]. By restricting to jammed states, we immediately see that
all currents become independent of the state and hence approach trivially
constant profiles in locally quasi-jammed states like those emerging in our
setting.

###  3.5 Entanglement in a weakly interacting setting

In this section we identify both classical and pure quantum contributions to
the two-point spin correlation functions. We focus on a weakly interacting
scenario in which the background consists of jammed particles that are of the
same species almost everywhere except for those in a special region. In the
latter, both species of particles are present in the maximal number. In other
words, the initial jammed state is obtained from the Néel state by flipping up
a sequence of $M$ spins down. We denote the corresponding macrosites by
$m^{\prime},m^{\prime}+1,\ldots,m^{\prime}+M-1$, so we have

$\ket{\Psi(0^{-})}=\ket{\cdots\downarrow\uparrow\downarrow\uparrow\underset{m^{\prime}}{\underline{\uparrow\uparrow}}\uparrow\uparrow\cdots\uparrow\uparrow\underset{m^{\prime}+M-1}{\underline{\uparrow\uparrow}}\downarrow\uparrow\downarrow\uparrow\cdots}\,,$
(28)

where we are going to assume $m^{\prime}>0$. Note that for this specific state
$M=\sum_{j}b_{j}(1-b_{j+1})$. In the light of the Bethe Ansatz solution of
Ref. [8], a nonzero sum $\sum_{j}b_{j}(1-b_{j+1})$ is always associated with
interaction. Since, however, the sum is finite and we are in the thermodynamic
limit, the state (28) can be considered weakly interacting. By flipping the
spin in position $0$ according to the protocol described in Section 2, we end
up with the state

$\ket{\Psi(0)}=\ket{\cdots\downarrow\uparrow\underset{0}{\underline{\downarrow\downarrow}}\downarrow\uparrow\cdots\downarrow\uparrow\downarrow\uparrow\underset{m^{\prime}}{\underline{\uparrow\uparrow}}\uparrow\uparrow\cdots\uparrow\uparrow\underset{m^{\prime}+M-1}{\underline{\uparrow\uparrow}}\downarrow\uparrow\downarrow\uparrow\cdots}\,,$
(29)

which is no longer jammed, but evolves in time towards a locally quasi-jammed
state. One can indeed readily show (see Appendix B)

$P_{\downarrow\downarrow}(\ell)=\left\\{\begin{aligned}
&J^{2}_{\lceil\ell/2\rceil}(4\mathcal{J}t)\,,&&\text{if
}\lfloor\ell/2\rfloor<m^{\prime}-1\\\
&J^{2}_{\ell-m^{\prime}+2}(4\mathcal{J}t)\,,&&\text{if
}m^{\prime}-1\leq\lfloor\ell/2\rfloor<m^{\prime}+M-2\\\
&J^{2}_{\lceil\ell/2\rceil+M}(4\mathcal{J}t)\,,&&\text{if
}\lfloor\ell/2\rfloor\geq m^{\prime}+M-2\,,\\\ \end{aligned}\right.$ (30)

hence the jamming condition is asymptotically satisfied up to $O(1/t)$
corrections – cf. (17).

For the local magnetisation we obtain

$\displaystyle\begin{aligned}
\braket{\sigma^{z}_{2\ell^{\prime}-1}}_{t}&=\left\\{\begin{aligned}
&2f(m^{\prime}+2M-2,t)-1\,,&&\text{if }\ell^{\prime}=m^{\prime}+M-1\\\
&1-2f(m^{\prime},t)\,,&&\text{if }\ell^{\prime}=m^{\prime}-1\\\
&1-2J^{2}_{m^{\prime}-2\ell^{\prime}-1}(4\mathcal{J}t)-2J^{2}_{m^{\prime}-2\ell^{\prime}}(4\mathcal{J}t)\,,\phantom{ii}&&\text{if
}m^{\prime}\leq\ell^{\prime}\leq m^{\prime}+M-2\\\
&-1\,,&&\text{otherwise}\,,\end{aligned}\right.\\\
\braket{\sigma^{z}_{2\ell^{\prime}}}_{t}&=\left\\{\begin{aligned}
&1-2J^{2}_{\ell^{\prime}}(4\mathcal{J}t)\,,&&\text{if
}\ell^{\prime}<m^{\prime}-1\\\
&1-2J^{2}_{\ell^{\prime}+M}(4\mathcal{J}t)\,,&&\text{if
}\ell^{\prime}>m^{\prime}+\Delta-2\\\
&1-2J^{2}_{m^{\prime}-2\ell^{\prime}-2}(4\mathcal{J}t)-2J^{2}_{m^{\prime}-2\ell^{\prime}-1}(4\mathcal{J}t)\,,&&\text{if
}m^{\prime}-1\leq\ell^{\prime}\leq
m^{\prime}+M-2\,,\end{aligned}\right.\end{aligned}$ (31)

where $f(x,t)$ is defined in Eq. (11), while
$\braket{\sigma^{x}_{\ell}}_{t}=0$ and $\braket{\sigma^{y}_{\ell}}_{t}=0$ (see
Appendix B.1 for details). Note that, except for the sites $2m^{\prime}-3$ and
$2m^{\prime}+2M-3$, the long-time limit of the expectation value of
$\sigma^{z}$ is either $1$ or $-1$. Using the asymptotic expression for the
function $f$, reported in Eq. (11), we instead obtain

$\displaystyle\braket{\sigma^{z}_{2m^{\prime}-3}}_{t}$
$\displaystyle\sim-\frac{2}{\pi}\arcsin\Bigl{(}\frac{m^{\prime}}{4\mathcal{J}t}\Bigr{)},\qquad\braket{\sigma^{z}_{2m^{\prime}+2M-3}}_{t}$
$\displaystyle\sim\frac{2}{\pi}\arcsin\Bigl{(}\frac{m^{\prime}}{4\mathcal{J}t}\Big{[}1+2\frac{M-1}{m^{\prime}}\Big{]}\Bigr{)}\,,$
(32)

for the $z$-components of the spins at the sites $2m^{\prime}-3$ and
$2m^{\prime}+2M-3$ (see Fig. 5). This implies that, in the infinite-time
limit, the reduced density matrix of any spin with site index
$j\nin\\{2m^{\prime}-3,2m^{\prime}+2M-3\\}$ is pure. The reduced density
matrix of each of the two spins at sites $2m^{\prime}-3$ and
$2m^{\prime}+2M-3$ is instead maximally mixed (proportional to the identity).
Hence, the spins $2m^{\prime}-3$ and $2m^{\prime}+2M-3$ are maximally
entangled with the rest of the system (the entanglement entropy of their
reduced density matrices is maximal).

On the other hand, the large-time limit of the correlation
$\braket{\sigma^{z}_{2m^{\prime}-3}\sigma^{z}_{2m^{\prime}+2M-3}}_{t}$ is
$-1$, i.e., the two spins are asymptotically maximally correlated. One might
be tempted to conclude that these two spins are entangled with one another
even in the infinite-time limit, i.e., that they are in a cat-state. Instead,
it can be shown that their entanglement goes to zero as
$\log(\mathcal{J}t)/(\mathcal{J}t)^{2}$, as does the entanglement between any
other two spins. To see this, we compute the entanglement of formation [46,
47] $E$ between two given spins:

$E(\hat{\rho})=h\Bigl{(}\frac{1+\sqrt{1-C^{2}(\hat{\rho})}}{2}\Bigr{)}\,.$
(33)

Here, $\hat{\rho}$ is the reduced density matrix of the two spins,

$h(x)=-x\log_{2}x-(1-x)\log_{2}(1-x)\,,$ (34)

while $C(\hat{\rho})$ is the so-called concurrence, defined as

$C(\hat{\rho})=\max\\{0,\lambda_{1}-\lambda_{2}-\lambda_{3}-\lambda_{4}\\},$
(35)

where $\lambda_{j}$ are the square roots of the eigenvalues of
$\hat{\rho}(\sigma^{y}\otimes\sigma^{y})\hat{\rho}^{\ast}(\sigma^{y}\otimes\sigma^{y})$
in descending order, and $\hat{\rho}^{\ast}$ is the complex conjugate of
$\hat{\rho}$. Even though
$\hat{\rho}(\sigma^{y}\otimes\sigma^{y})\hat{\rho}^{\ast}(\sigma^{y}\otimes\sigma^{y})$
is not necessarily self-adjoint, its eigenvalues are real and non-negative (it
is the product of two positive semi-definite matrices).

Figure 5: Magnetisation profile $\braket{\sigma^{z}_{\ell}}_{t}$ for the
initial state described in Eq. (29) with $m^{\prime}=9$ and $M=10$. Colors of
the points correspond to different fixed times $\mathcal{J}t=10$ (green) and
$\mathcal{J}t=20$ (blue). The spins at sites $2m^{\prime}-3$ and
$2m^{\prime}+2M-3$ are the only ones for which the $z$-components of the
magnetisation are not asymptotically equal to $\pm 1$. Instead, as time $t$ is
increased, they move towards the zero ray (time increases while the lattice
positions are fixed) along the trajectories (orange dashed lines) given in Eq.
(32).

Note that the concurrence (and hence the entanglement) is automatically zero
whenever the density matrix is diagonal because the matrix
$\hat{\rho}(\sigma^{y}\otimes\sigma^{y})\hat{\rho}^{\ast}(\sigma^{y}\otimes\sigma^{y})$
has two pairs of degenerate positive eigenvalues. One can show that the off-
diagonal elements of the two-spin reduced density matrix $\hat{\rho}$ go to
zero as $1/t$ for large times, thus lifting the degeneracy of the matrix
$\hat{\rho}(\sigma^{y}\otimes\sigma^{y})\hat{\rho}^{\ast}(\sigma^{y}\otimes\sigma^{y})$
by a term of order $O(1/t)$. This makes the entanglement of formation approach
zero as $\log(\mathcal{J}t)/(\mathcal{J}t)^{2}$. Nonetheless, the two spins at
sites $2m^{\prime}-3$ and $2m^{\prime}+2M-3$ are special, since they are
maximally correlated: in the infinite-time limit their reduced density matrix
reads

$\hat{\rho}\xrightarrow{t\rightarrow\infty}\frac{\ket{\uparrow\downarrow}\bra{\uparrow\downarrow}+\ket{\downarrow\uparrow}\bra{\downarrow\uparrow}}{2}\,,$
(36)

which is a classical mixture. Note that the macro-distance $M$ between the
correlated spins can be arbitrarily large.

In general, the spread of the rescaled spin-spin entanglement with time is
shown in Fig. 6. At $t=0$ there are no entangled spins (the initial state is a
product state). For $t>0$ the region containing entangled spins grows linearly
in time, consistently with the light-cone dynamics. At first the entanglement
spreads from the initial position of the impurity in all directions – cf. Fig.
6(a) – with the maximal velocity $8J$. When the edge of the light-cone reaches
macro-site $m$, a “leakage” of the entanglement between the wedges occurs
through a corridor, in which only the next-nearest neighbour spins become
entangled – cf. Fig. 6(b,c). We observe that the couples of spins $(i,j)$ with
$i,j<2m^{\prime}-1$ are not affected by the presence of the domain wall in the
initial state: their behaviour is the same as if the initial state were a Néel
state. Instead, the entanglement properties of the pairs of spins $(i,j)$ with
$i,j>2m^{\prime}+M-5$ are shifted: their entanglement corresponds to that in
the Néel state in which the impurity is initially located at macro-site $-M$.
Finally, in our case, all the spins outside those two regions are decoupled
unless they are next-nearest neighbours, which is not the case when the
dynamics start from the Néel state.

(a) $\mathcal{J}t=1.5$(b) $\mathcal{J}t=5.1$(c)
$\mathcal{J}t=9.9$$0.6$$0.0$$0.3$ Figure 6: Spin-spin entanglement for an
initial state with $m^{\prime}=12$, $M=9$ at different times, rescaled by
$\log(\mathcal{J}t)/(\mathcal{J}t)^{2}$. The numbers on both axes refer to
spin indices: the element $(i,j)$ of the plot characterises the entanglement
between the spins at the sites $i$ and $j$. The plot is symmetric w.r.t. the
bisector, i.e., the line containing the points $(i,i)$, since the entanglement
between the spins $i$ and $j$ equals that between the spins $j$ and $i$. The
blue (orange) lines refer to the spin $2m^{\prime}-1$ (resp.
$2m^{\prime}+2M-5$), i.e., the one close to the start (resp. end) of the
domain wall of spins up. They delimit the nonzero entanglement regions. Panels
(a), (b), and (c) show the spread of the entanglement before the domain of
spins up is reached by the light cone, during the spread of the entanglement
through the domain, and after the edge of the light cone has passed through
the domain, respectively.

##  4 From the dual folded XXZ to the XXZ model

The dual folded XXZ model was originally envisioned as an effective model
describing the large-anisotropy limit of the Heisenberg XXZ spin-$1/2$ chain.
In this respect it is natural to wonder how well it acts as such.
Specifically, is there a sign of the measurement catastrophe studied herein
and in Ref. [24] in the XXZ model with large anisotropy? To answer this
question, one needs to consider the inverse of the duality transformation and
invert an additional unitary transformation that acts on the initial state and
characterises the so-called “folded picture” (see Ref. [8]). The latter
unitary transformation only contributes $O(1/\Delta)$ corrections and will be
neglected in the following discussion. The inverse duality transformation maps
$\sigma_{\ell}^{z}\mapsto\sigma_{\ell}^{z}\sigma_{\ell+1}^{z}$, so it
essentially acts as follows:

$\displaystyle\ket{\cdots\uparrow\uparrow\downarrow\uparrow\uparrow\downarrow\uparrow\downarrow\downarrow\uparrow\uparrow\downarrow\uparrow\uparrow\downarrow\cdots}\xrightarrow[\text{transformation}]{\text{inverse
duality}}\ket{\cdots\uparrow\uparrow\uparrow\downarrow\downarrow\downarrow\uparrow\uparrow\downarrow\uparrow\uparrow\uparrow\downarrow\downarrow\downarrow\uparrow\cdots}\,.$
(37)

Time-evolving this initial state with the Heisenberg Hamiltonian $H_{\rm
XXZ}(\Delta)$ is expected to yield a scaling profile of
$\braket{\sigma_{\ell}^{z}\sigma_{\ell+1}^{z}}_{t}$ converging, as $\Delta$ is
increased, to the one of $\braket{\sigma_{\ell}^{z}}_{t}$ in the dual folded
XXZ model. This is indeed observed in numerical simulations based on tensor
network techniques – cf. Fig. 7, and supports the validity of the strong-
coupling expansion described in Ref. [8]. Besides that, transforming back to
XXZ helps us see why the local measurement could have such a relevant effect:
in the XXZ setting, it creates a domain-wall structure. If the two halves of
the chain had different integrals of motion, the theory of generalised
hydrodynamics [48, 49, 50] would predict nontrivial ballistic profiles. This
is not the case. One could then wonder whether we are seeing the effects of
quasi-conserved operators that are exactly conserved only in the large-
anisotropy limit. So far, our investigations have been inconclusive, so the
question is still open.

Figure 7: Panel (a) shows correlation function
$\braket{\sigma^{z}_{\ell}\sigma^{z}_{\ell+1}}_{t}$ in the Heisenberg XXZ
model (blue and orange points) compared to the magnetization of the folded
model Eq. (15) (black line). Data points quickly converge to the prediction of
the folded model for a relatively large values of $\Delta$, whereas for small
anisotropy they exhibit a disagreement with the prediction during the whole
time evolution. Panel (b) shows the difference
$\braket{\sigma^{z}_{\ell}}_{\rm
folded}-\braket{\sigma^{z}_{\ell}\sigma^{z}_{\ell+1}}_{\rm XXZ}$ outside of
the light cone as a function of time. For short times the predictions in both
models linearly deviate from each other. In the inset we plot a slope $\alpha$
of this difference as a function of the anisotropy $\Delta$. Data points are
fitted with $a\Delta^{-2}+b\Delta^{-3}$ , where $a\approx 0.241$, $b\approx
1.31$.

##  5 Discussion

The simplicity of the calculations carried out in this paper is rather
surprising in view of the fact that we are dealing with a genuinely
interacting model. It goes even beyond the simplicity of the Bethe equations
of the dual folded XXZ model [8]. It seems rather related to the setting
considered – that of locally perturbed jammed states, in which particles are
closely packed. Such a scenario can be mapped into the problem of a free
particle moving in a jammed background. We have exploited this mapping to
solve the measurement catastrophe discussed in Ref. [24], i.e., the
macroscopic change of spin profiles after a localised perturbation. We have
established that no genuine spin transport occurs: the particles’ positions
exhibit small fluctuations around their expectation values, and the current of
the local magnetisation is asymptotically zero. This is a result of the
emergent scale separation in which the smooth asymptotic scaling of the
sublattice local magnetisation coexists with an underlying discontinuous
dependence of $\braket{\sigma^{z}_{\ell}}$ on the position $\ell$.
Interactions play a key role in the setting we studied: the phenomenon is not
observed if particles do not interact. On the other hand, a finite number of
interacting particles is enough to trigger the effect. In particular,
perturbing a weakly interacting initial state enables isolation of classically
correlated pairs of spins.

Jammed states typically arise in kinetically constrained models, and it
remains unclear whether the measurement catastrophe described in this work is
a universal feature of such models [51].

During the finalisation of this work and in the attempt to unveil the physical
mechanism behind the measurement catastrophe, one of us has realised that an
analogous sensitivity to localised perturbations can be present also after
global quenches [52]. In that case the existence of infinitely many “semilocal
charges” was identified as crucial. That mechanism, however, does not seem to
account for the phenomenon discussed in this paper, which appears to be more
related to the exceptionality of the jammed sector, consisting of states with
memory. Indeed, to the best of our knowledge the dual folded Hamiltonian only
has a single semilocal charge, first identified in Ref. [8], which corresponds
to the total spin along the $z$-axis when mapping back to the XXZ Heisenberg
model. That charge has a nontrivial profile in some of the examples that we
considered, but it is irrelevant in others, including in the example
investigated in Ref. [24].

## Acknowledgements

##### Funding information.

This work was supported by the European Research Council under the Starting
Grant No. 805252 LoCoMacro.

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##  Appendix A Proofs

These sections contain the proofs of the equations stated in the main part of
the text.

###  A.1 Time evolution of the state and $\braket{\sigma_{\ell}^{z}}_{t}$

Recall that the state $\ket{n;\underline{b}^{(0)}}$ is defined in such a way
that the impurity lies on the immediate right-hand side of the particle (spin
up) with index $n$. The state is jammed everywhere, except at the position of
the impurity, which consists of either two or three consecutive spins down.
The width of the impurity, i.e., the number of spins down that it contains, is
not preserved under the action of the Hamiltonian. Nevertheless, the types of
particles represented by spins up are well defined and their sequence is
preserved, as shown in Ref. [8] (e.g., the particle corresponding to a spin up
at the site $\ell$ is of the type $b=2\lceil\ell/2\rceil-\ell$).

Let us define the impurity creation operator $c_{n}^{\dagger}$ and the
reference state $\ket{\emptyset;\underline{b}^{(0)}}$, so that

$\displaystyle
c_{n}^{\dagger}\ket{\emptyset;\underline{b}^{(0)}}:=\ket{n;\underline{b}^{(0)}}\,.$
(A.1)

Since the background $\underline{b}^{0}$ is preserved under the action of the
Hamiltonian (1), the impurity effectively behaves as a freely hopping fermion.
States $\ket{n;\underline{b}^{(0)}}$ span an invariant subspace, on which the
dynamics can be readily solved. Employing a diagonalisation procedure for a
free hopping Hamiltonian in the thermodynamic limit, we obtain

$\displaystyle\begin{gathered}e^{-iHt}c^{\dagger}_{\ell}\ket{\emptyset;\underline{b}^{(0)}}=\sum_{n}\int\frac{{\rm
d}k}{2\pi}e^{-ik(n-\ell)-i\varepsilon(k)t}c_{n}^{\dagger}\ket{\emptyset;\underline{b}^{(0)}}=\sum_{n}(-i)^{n-\ell}J_{n-\ell}(4\mathcal{J}t)\ket{n;\underline{b}^{(0)}}\end{gathered}$
(A.3)

for the evolution of the state $\ket{\ell;\underline{b}^{(0)}}\equiv
c_{\ell}^{\dagger}\ket{\emptyset;\underline{b}^{(0)}}$. Here, the energy
dispersion relation reads $\varepsilon(k)=4\mathcal{J}\cos k$, and $J_{n}(x)$
are Bessel functions. In particular, the evolution of the initial state
$\ket{\Psi(0)}=\ket{0;\underline{b}^{(0)}}\equiv
c_{0}^{\dagger}\ket{\emptyset;\underline{b}^{(0)}}$ in the protocol described
in the main text reads

$\displaystyle\ket{\Psi(t)}=e^{-iHt}c^{\dagger}_{0}\ket{\emptyset;\underline{b}^{(0)}}=\sum_{n}(-i)^{n}J_{n}(4\mathcal{J}t)\ket{n;\underline{b}^{(0)}}\,.$
(A.4)

Consider now the expectation value
$\braket{\sigma_{\ell}^{z}}_{t}=\bra{\Psi(t)}\sigma_{\ell}^{z}\ket{\Psi(t)}$,
where $\ell$ is the site on the lattice. We have

$\displaystyle\braket{\sigma_{\ell}^{z}}_{t}=\sum_{n,m}\int\frac{{\rm
d}k}{2\pi}\int\frac{{\rm d}p}{2\pi}e^{ikn-
ipm}e^{i[\varepsilon(k)-\varepsilon(p)]t}\bra{n;\underline{b}^{(0)}}\sigma_{\ell}^{z}\ket{m;\underline{b}^{(0)}}.$
(A.5)

The operator $\sigma_{\ell}^{z}$ is diagonal in the basis
$\ket{n;\underline{b}^{(0)}}$, therefore the only non-vanishing matrix
elements are
$\braket{n;\underline{b}^{(0)}}{\sigma_{\ell}^{z}}{n;\underline{b}^{(0)}}$,
whence

$\displaystyle\braket{\sigma_{\ell}^{z}}_{t}=\sum_{n}|J_{n}(4\mathcal{J}t)|^{2}\bra{n;\underline{b}^{(0)}}\sigma_{\ell}^{z}\ket{n;\underline{b}^{(0)}}\,.$
(A.6)

In Eq. (A.6) $\sigma_{\ell}^{z}$ can be substituted by any operator $D$,
diagonal in the basis of states $\ket{n;\underline{b}^{(0)}}$. For observables
$A$ that are not diagonal in this basis, we instead have

$\displaystyle\braket{A}_{t}=\sum_{n_{1},n_{2}}e^{i\frac{\pi}{2}(n_{1}-n_{2})}J_{n_{1}}(4\mathcal{J}t)\overline{J_{n_{2}}(4\mathcal{J}t)}\bra{n_{1};\underline{b}^{(0)}}A\ket{n_{2};\underline{b}^{(0)}}\,.$
(A.7)

###  A.2 Equal and non-equal time correlations

Let $\langle\bullet\rangle\equiv\langle\bullet\rangle_{t=0}$ denote the
average in the initial state $\ket{\Psi(0)}=\ket{0;\underline{b}^{(0)}}$. In
this section we report different-time correlation functions of operators that
are diagonal in the basis of states $\ket{n;\underline{b}^{(0)}}$. In
particular, the general two-point correlation function of two diagonal
operators $D_{1}$ and $D_{2}$ reads

$\displaystyle\begin{gathered}\braket{D_{1}(t_{1})D_{2}(t_{2})}=\sum_{m,n}J_{m}(4\mathcal{J}t_{1})J_{m-n}(4\mathcal{J}[t_{1}-t_{2}])J_{n}(4\mathcal{J}t_{2})\times\\\
\times\braket{m;\underline{b}^{(0)}}{D_{1}}{m;\underline{b}^{(0)}}\braket{n;\underline{b}^{(0)}}{D_{2}}{n;\underline{b}^{(0)}}\,.\end{gathered}$
(A.10)

At equal times the correlation function becomes

$\displaystyle\begin{gathered}\braket{D_{1}(t)D_{2}(t)}=\sum_{n}|J_{n}(4\mathcal{J}t)|^{2}\braket{n;\underline{b}^{(0)}}{D_{1}}{n;\underline{b}^{(0)}}\braket{n;\underline{b}^{(0)}}{D_{2}}{n;\underline{b}^{(0)}}\,,\end{gathered}$
(A.12)

while at $t_{2}=0$ it factorises

$\displaystyle\braket{D_{1}(t)D_{2}}=\big{(}\sum_{m}|J_{m}(4\mathcal{J}t)|^{2}\bra{m;\underline{b}^{(0)}}D_{1}\ket{m;\underline{b}^{(0)}}\Big{)}\bra{0;\underline{b}^{(0)}}D_{2}\ket{0;\underline{b}^{(0)}}=\braket{D_{1}}_{t}\braket{D_{2}}\,.$
(A.13)

###  A.3 Matrix elements of $\sigma_{\ell}^{z}$

In this section we compute the diagonal matrix elements
$\braket{n;\underline{b}^{(0)}}{\sigma_{\ell}^{z}}{n;\underline{b}^{(0)}}$.

####  A.3.1 Increasing $n$: the right-moving impurity

First we consider the case of increasing $n$, so that the impurity passes from
the left- to the right-hand side of the site $\ell$. The local Hamiltonian
density

$\displaystyle
h_{\ell,\ell+1,\ell+2}=\mathcal{J}\frac{1-\sigma^{z}_{\ell+1}}{2}(\sigma_{\ell}^{x}\sigma_{\ell+2}^{x}+\sigma_{\ell}^{y}\sigma_{+2}^{y})$
(A.14)

acts on three subsequent sites. The impurity is identified with two or three
consecutive spins down: once it reaches the left-hand side of the site $\ell$,
two spins down occupy the sites $\ell-2$ and $\ell-1$. Three subsequent
actions of the local Hamiltonian density are then required for it to pass to
the right-hand side of the site $\ell$. Specifically, the actions are those of
$h_{\ell-2,\ell-1,\ell}$, $h_{\ell-1,\ell,\ell+1}$, and
$h_{\ell,\ell+1,\ell+2}$ (only two steps are needed, for example, when a spin
down already occupies the $\ell$-th site). To determine the matrix element of
$\sigma^{z}_{\ell}$ we thus need to consider eight processes, determined by
the spins on sites $\ell$, $\ell+1$, and $\ell+2$ on the right-hand side of
the right-moving impurity. Three of these processes do not occur, since the
state is jammed everywhere but at the position of the impurity itself: we can
not have $\uparrow\downarrow\downarrow$, $\downarrow\downarrow\uparrow$, or
$\downarrow\downarrow\downarrow$ on sites $\ell$, $\ell+1$, and $\ell+2$,
while the impurity is on the left-hand side of the site $\ell$. The remaining
five processes that determine the matrix elements
$\braket{n;\underline{b}^{(0)}}{\sigma_{\ell}^{z}}{n;\underline{b}^{(0)}}$ are
presented in Fig. A.1, which serves as a proof of the following rules
determining the matrix element:

$\displaystyle\braket{n;\underline{b}^{(0)}}{\sigma_{\ell>0}^{z}}{n;\underline{b}^{(0)}}=\begin{cases}1-2\delta_{n,j_{R}}-2\delta_{n,j_{R}+1}\,,&\text{if
}\ket{0;\underline{b}^{(0)}}=\ket{\cdots\uparrow^{(0)}\bullet\uparrow^{(1)}\cdots\uparrow^{(j_{R})}_{\ell}\uparrow\uparrow\cdots}\\\
1-2\delta_{n,j_{R}}\,,&\text{if
}\ket{0;\underline{b}^{(0)}}=\ket{\cdots\uparrow^{(0)}\bullet\uparrow^{(1)}\cdots\uparrow^{(j_{R})}_{\ell}\downarrow\uparrow\cdots}\\\
1-2\theta(n\geq j_{R})\,,&\text{if
}\ket{0;\underline{b}^{(0)}}=\ket{\cdots\uparrow^{(0)}\bullet\uparrow^{(1)}\cdots\uparrow_{\ell}^{(j_{R})}\uparrow\downarrow\cdots}\\\
1-2\theta(n\leq j_{R})\,,&\text{if
}\ket{0;\underline{b}^{(0)}}=\ket{\cdots\uparrow^{(0)}\bullet\uparrow^{(1)}\cdots\downarrow_{\ell}\uparrow^{(j_{R})}\uparrow\cdots}\\\
-1\,,&\text{if
}\ket{0;\underline{b}^{(0)}}=\ket{\cdots\uparrow^{(0)}\bullet\uparrow^{(1)}\cdots\downarrow_{\ell}\uparrow^{(j_{R})}\downarrow\cdots}\,.\end{cases}$
(A.15)

Here, $j_{R}$ is the index of the particle represented by a spin up at the
site $\ell$, or, if the latter is occupied by a spin down, at the site
$\ell+1$. The impurity is represented by $\bullet$.

$\ell$$n=j_{R}-1$$n=j_{R}$$n=j_{R}+1$$n=j_{R}+2$$\bra{n;\underline{b}^{(0)}}\sigma_{\ell}^{z}\ket{n;\underline{b}^{(0)}}=1-2\delta_{n,j_{R}}-2\delta_{n,j_{R}+1}$

$\ell$$n=j_{R}-1$$n=j_{R}$$n=j_{R}+1$$\bra{n;\underline{b}^{(0)}}\sigma_{\ell}^{z}\ket{n;\underline{b}^{(0)}}=1-2\delta_{n,j_{R}}$

$\ell$$n=j_{R}-1$$n=j_{R}$$n=j_{R}+1$$\bra{n;\underline{b}^{(0)}}\sigma_{\ell}^{z}\ket{n;\underline{b}^{(0)}}=1-2\theta(n\geq
j_{R})$

$\ell$$n=j_{R}-1$$n=j_{R}$$n=j_{R}+1$$\bra{n;\underline{b}^{(0)}}\sigma_{\ell}^{z}\ket{n;\underline{b}^{(0)}}=1-2\theta(n\leq
j_{R})$

$\ell$$n=j_{R}-1$$n=j_{R}$$\bra{n;\underline{b}^{(0)}}\sigma_{\ell}^{z}\ket{n;\underline{b}^{(0)}}=-1$

Figure A.1: The five processes that determine the rules (A.15) for the right-
moving impurity. Dark blue  and orange  colors represent spins up and down,
respectively: two or three consecutive spins down (orange boxes) constitute an
impurity, which starts on the left-hand side of the site $\ell$. The white dot
represents the spin up corresponding to the particle with index $j_{R}$. The
consequent steps (actions of the local Hamiltonian densities) follow in the
downwards direction, and the site $\ell$ is framed.

####  A.3.2 Decreasing $n$: the left-moving impurity

Let us now consider the case of decreasing $n$, in which the impurity passes
from the right- to the left-hand side of the site $\ell$. The rules proven in
Fig. A.2 now read

$\displaystyle\braket{n;\underline{b}^{(0)}}{\sigma_{\ell<-1}^{z}}{n;\underline{b}^{(0)}}=\begin{cases}1-2\delta_{n,j_{L}-1}-2\delta_{n,j_{L}-2}\,,&\text{if
}\ket{0;\underline{b}^{(0)}}=\ket{\cdots\uparrow\uparrow\uparrow_{\ell}^{(j_{L})}\cdots\uparrow^{(0)}\bullet\uparrow^{(1)}\cdots}\\\
1-2\delta_{n,j_{L}-1}\,,&\text{if
}\ket{0;\underline{b}^{(0)}}=\ket{\cdots\uparrow\downarrow\uparrow_{\ell}^{(j_{L})}\cdots\uparrow^{(0)}\bullet\uparrow^{(1)}\cdots}\\\
1-2\theta(n\geq j_{L}-1)\,,&\text{if
}\ket{0;\underline{b}^{(0)}}=\ket{\cdots\uparrow\uparrow^{(j_{L})}\downarrow_{\ell}\cdots\uparrow^{(0)}\bullet\uparrow^{(1)}\cdots}\\\
1-2\theta(n<j_{L})\,,&\text{if
}\ket{0;\underline{b}^{(0)}}=\ket{\cdots\downarrow\uparrow\uparrow^{(j_{L})}_{\ell}\cdots\uparrow^{(0)}\bullet\uparrow^{(1)}\cdots}\\\
-1\,,&\text{if
}\ket{0;\underline{b}^{(0)}}=\ket{\cdots\downarrow\uparrow^{(j_{L})}\downarrow_{\ell}\cdots\uparrow^{(0)}\bullet\uparrow^{(1)}\cdots}\,.\end{cases}$
(A.16)

Here, $j_{L}$ is the index of the particle represented by a spin up at the
site $\ell$, or, if the latter is occupied by a spin down, at the site
$\ell-1$.

$\ell$$n=j_{L}$$n=j_{L}-1$$n=j_{L}-2$$n=j_{L}-3$$\bra{n;\underline{b}^{(0)}}\sigma_{\ell}^{z}\ket{n;\underline{b}^{(0)}}=1-2\delta_{n,j_{L}-1}-2\delta_{n,j_{L}-2}$

$\ell$$n=j_{L}$$n=j_{L}-1$$n=j_{L}-2$$\bra{n;\underline{b}^{(0)}}\sigma_{\ell}^{z}\ket{n;\underline{b}^{(0)}}=1-2\delta_{n,j_{L}-1}$

$\ell$$n=j_{L}$$n=j_{L}-1$$n=j_{L}-2$$\bra{n;\underline{b}^{(0)}}\sigma_{\ell}^{z}\ket{n;\underline{b}^{(0)}}=1-2\theta(n\geq
j_{L}-1)$

$\ell$$n=j_{L}$$n=j_{L}-1$$n=j_{L}-2$$\bra{n;\underline{b}^{(0)}}\sigma_{\ell}^{z}\ket{n;\underline{b}^{(0)}}=1-2\theta(n<j_{L})$

$\ell$$n=j_{L}$$n=j_{L}-1$$\bra{n;\underline{b}^{(0)}}\sigma_{\ell}^{z}\ket{n;\underline{b}^{(0)}}=-1$

Figure A.2: The five processes that determine the rules (A.16) for the left-
moving impurity. Dark blue  and orange  colors represent spins up and down,
respectively: two or three consecutive spins down (orange boxes) constitute an
impurity, which now starts on the right-hand side of the site $\ell$. The
white dot represents the spin up corresponding to the particle with index
$j_{L}$. The consequent steps (actions of the local Hamiltonian densities)
follow in the downwards direction, and the site $\ell$ is framed.

###  A.4 Asymptotic profiles of magnetisation

In the previous section we have established the rules for the computation of
the matrix elements of $\sigma_{\ell}^{z}$. With this let us return to its
expectation value (A.6). For the asymptotic profiles only large values of $n$
are relevant, whence the rules (A.15) and (A.16) reduce to

$\displaystyle\braket{n;\underline{b}^{(0)}}{\sigma_{\ell}^{z}}{n;\underline{b}^{(0)}}\sim\begin{cases}1\,,&\text{for
}\ket{\cdots\bullet\cdots\uparrow_{\ell}\updownarrow\uparrow\cdots}\vee\ket{\cdots\uparrow\updownarrow\uparrow_{\ell}\cdots\bullet\cdots}\\\
1-2\theta(n\geq x(\ell))\,,&\text{for
}\ket{\cdots\bullet\cdots\uparrow_{\ell}\uparrow\downarrow\cdots}\vee\ket{\cdots\uparrow\uparrow\downarrow_{\ell}\cdots\bullet\cdots}\\\
1-2\theta(n\leq x(\ell))\,,&\text{for
}\ket{\cdots\bullet\cdots\downarrow_{\ell}\uparrow\uparrow\cdots}\vee\ket{\cdots\downarrow\uparrow\uparrow_{\ell}\cdots\bullet\cdots}\\\
-1\,,&\text{for
}\ket{\cdots\bullet\cdots\downarrow_{\ell}\uparrow\downarrow\cdots}\vee\ket{\cdots\downarrow\uparrow\downarrow_{\ell}\cdots\bullet\cdots}\,.\end{cases}$
(A.17)

Here, $\updownarrow$ stands for either a spin up or a spin down, while
$x(\ell)$ corresponds to the index of the particle represented by the spin up
at the site $\ell$ (or the neighbouring site). Asymptotically we compute it by
observing $\ell^{\prime}_{x(\ell)}\sim\ell/2$ and using Eq. (7). For example,
for large $x(\ell)>0$ one has

$\displaystyle\ell^{\prime}_{x(\ell)}=x(\ell)-\theta(x(\ell)\leq
n)-\sum^{x(\ell)-1}_{m=1}b^{(0)}_{m}(1-b^{(0)}_{m+1})\sim\frac{\ell}{2}\,,$
(A.18)

where any terms of magnitude $O(1)$ can be neglected. For large $|\ell|$ we
can thus compute $x(\ell)$ from the integral equation

$\displaystyle\frac{1}{\ell}\int_{0}^{x(\ell)}{\rm d}n\,\xi(n)=\frac{1}{2}\,,$
(A.19)

where $\xi(n)$ is the average $1-b^{(0)}_{j}(1-b^{(0)}_{j+1})$ around the
particle with index $n$.

Using the asymptotic rules (A.17) we now readily obtain Eq. (15). Consider,
for example, the second rule in Eq. (A.17):

$\displaystyle\begin{gathered}\sum_{n}\bra{n;\underline{b}^{(0)}}\sigma_{\ell}^{z}\ket{n;\underline{b}^{(0)}}|J_{n}(4\mathcal{J}t)|^{2}=1-2\sum_{n}\theta(n\geq
x(\ell))|J_{n}(4\mathcal{J}t)|^{2}=\\\ =1-2\\!\int\\!\frac{{\rm
d}k}{2\pi}\\!\int\\!\frac{{\rm d}p}{2\pi}\\!\sum_{n\geq
x(\ell)}\\!e^{-i(k-p)n-i4\mathcal{J}t(\cos k-\cos p)}=1-2\\!\int\\!\frac{{\rm
d}k}{2\pi}\\!\int\\!\frac{{\rm
d}p}{2\pi}\frac{e^{-i(k-p)x(\ell)-i4\mathcal{J}t(\cos k-\cos
p)}}{1-e^{-i(k-p)}}\sim\\\ \sim 1-\frac{1}{\pi}\int{\rm d}p\,\theta\Big{(}\sin
p-\frac{x(\ell)}{4\mathcal{J}t}\Big{)}=\frac{2}{\pi}\arcsin\Big{(}\frac{x(\ell)}{4\mathcal{J}t}\Big{)}\,.\end{gathered}$
(A.23)

##  Appendix B Correlation functions in the weakly-interacting case

###  B.1 The idea behind the computation

In this section we show the idea behind the exact computation of any
correlation function for the weakly-interacting scenario discussed in the main
text, in which the unperturbed state at time $t=0^{-}$ is

$\ket{\Psi(0^{-})}=\ket{\cdots\downarrow\uparrow\downarrow\uparrow\underset{m^{\prime}}{\underline{\uparrow\uparrow}}\uparrow\uparrow\cdots\uparrow\uparrow\underset{m^{\prime}+M-1}{\underline{\uparrow\uparrow}}\downarrow\uparrow\downarrow\uparrow\cdots}\,.$
(B.1)

Essentially, the sums over $n_{1}$ and $n_{2}$, e.g., in Eq. (A.7), are split
in such a way that the indices referring to eigenstates with the same
characteristics are grouped together. In particular, we can distinguish four
different classes of eigenstates $\ket{n;\underline{b}^{(0)}}$:

  * •

$n<m^{\prime}$: the macrosite $n$ is in the configuration
$\downarrow\downarrow$, the macrosites
$m^{\prime},m^{\prime}+1,...,m^{\prime}+M-1$ are each in the configuration
$\uparrow\uparrow$, all the other macrosites are in the configuration
$\downarrow\uparrow$;

  * •

$n=m^{\prime}+2a,a\in\\{0,1,...,M-1\\}$: the macrosite $m^{\prime}+a-1$ is in
the configuration $\uparrow\downarrow$, the macrosites
$m^{\prime}-1,m^{\prime},...,m^{\prime}+a-2,m^{\prime}+a+1,...,m^{\prime}+M-1$
are each in the configuration $\uparrow\uparrow$ (note that the latter set is
empty for $M=1$), all the other macrosites are in the configuration
$\downarrow\uparrow$;

  * •

$n=m^{\prime}+2b+1,b\in\\{0,1,...,M-2\\}$: (note that this set is empty for
$M=1$) the macrosite $m^{\prime}+b$ is in the configuration
$\downarrow\downarrow$, the macrosites
$m^{\prime}-1,m^{\prime},...,m^{\prime}+b-1,m^{\prime}+b+1,...,m^{\prime}+M-1$
are each in the configuration $\uparrow\uparrow$, all the other macrosites are
in the configuration $\downarrow\uparrow$;

  * •

$n\geq m^{\prime}+2M-1$: the macrosite $n-M$ is in the configuration
$\downarrow\downarrow$, the macrosites
$m^{\prime}-1,m^{\prime},...,m^{\prime}+M-2$ are each in the configuration
$\uparrow\uparrow$, all the other macrosites are in the configuration
$\downarrow\uparrow$.

Using Eq. (A.6), we obtain

$\displaystyle\begin{aligned}
\braket{\sigma^{z}_{2\ell^{\prime}-1}}_{t}=&\sum_{a=0}^{M-2}|J_{m^{\prime}+2a}(4\mathcal{J}t)|^{2}\left(2\chi_{[m^{\prime}-1,m^{\prime}+a-1]}(\ell^{\prime})+2\chi_{[m^{\prime}+a+1,m^{\prime}+M-1]}(\ell^{\prime})-1\right)\\\
&+\sum_{b=0}^{M-2}|J_{m^{\prime}+2b+1}(4\mathcal{J}t)|^{2}\left(2\chi_{[m^{\prime}-1,m^{\prime}+b-1]}(\ell^{\prime})+2\chi_{[m^{\prime}+b+1,m^{\prime}+M-1]}(\ell^{\prime})-1\right)\\\
&+\sum_{n=-\infty}^{m^{\prime}-1}|J_{n}(4\mathcal{J}t)|^{2}\left(2\chi_{[m^{\prime},m^{\prime}+M-1]}(\ell^{\prime})-1\right)\\\
&+\sum_{n=m^{\prime}+2M-2}^{\infty}|J_{n}(4\mathcal{J}t)|^{2}\left(2\chi_{[m^{\prime}-1,m^{\prime}+M-2]}(\ell^{\prime})-1\right)\,,\end{aligned}$
(B.2)

for the local magnetisation at the odd sites. Here, $\chi_{[a,b]}(x)$ is the
characteristic function on the interval $[a,b]$, i.e., it is equal to one for
$x\in[a,b]$ and zero otherwise. A straightforward calculation now yields

$\braket{\sigma^{z}_{2\ell^{\prime}-1}}_{t}=\left\\{\begin{aligned}
&2f(m^{\prime}+2M-2,t)-1\,,&&\text{if }\ell^{\prime}=m^{\prime}+M-1\\\
&1-2f(m^{\prime},t),&&\text{if }\ell^{\prime}=m^{\prime}-1\\\
&1-2|J_{m^{\prime}-2\ell^{\prime}-1}(4\mathcal{J}t)|^{2}-2|J_{m^{\prime}-2\ell^{\prime}}(4\mathcal{J}t)|^{2}\,,&&\text{if
}\ell^{\prime}\in\\{m^{\prime},m^{\prime}+1,...,m^{\prime}+M-2\\}\\\
&-1\,,&&\text{otherwise}\,,\end{aligned}\right.\,$ (B.3)

where $f(x,t)$ is defined in Eq. (11).

Let us now consider an example of a non-diagonal operator, and compute it
using Eq. (A.7). The operators $\sigma^{x}$ and $\sigma^{y}$ have zero
expectation value, since the states $\bra{n_{1};\underline{b}^{(0)}}$ and
$\ket{n_{2};\underline{b}^{(0)}}$ in Eq. (A.7) contain the same number of
spins up. To give an example of a non-trivial non-diagonal operator, we need
to consider correlation functions involving an even number of $\sigma^{x}$ and
$\sigma^{y}$, such as
$\braket{\sigma^{x}_{2\ell^{\prime}-2d}\sigma^{x}_{2\ell^{\prime}}}$, for
$d\in\mathbb{Z}^{+}$. In order to get a non-zero contribution from a generic
term of the sum, the overall action of the spin-flips on
$\ket{n_{2};\underline{b}^{(0)}}$ should not change the background. By
dividing the sum over $n_{2}$ in the sets that we introduced above, the only
options are

$\sigma^{x}_{2\ell^{\prime}-2d}\sigma^{x}_{2\ell^{\prime}}\ket{n_{2};\underline{b}^{(0)}}\rightarrow\\\
\left\\{\begin{aligned}
&\delta_{n_{2},\ell^{\prime}-d}\ket{\ell^{\prime};\underline{b}^{(0)}}+\delta_{n_{2},\ell^{\prime}}\chi_{(-\infty,m^{\prime})}(\ell^{\prime})\ket{\ell^{\prime}-d;\underline{b}^{(0)}}\,,&&\text{if
}n_{2}<m^{\prime}\\\
&\delta_{n_{2}\\!-\\!M,\ell^{\prime}\\!-\\!d}\ket{\ell^{\prime}\\!+\\!M\\!;\\!\underline{b}^{(0)}}\\!+\\!\delta_{n_{2}\\!-\\!M,\ell^{\prime}}\chi_{[m^{\prime}\\!+\\!d\\!+\\!M\\!-\\!2,\infty)}(\ell^{\prime})\ket{\ell^{\prime}\\!+\\!M\\!-\\!d;\underline{b}^{(0)}},&&\text{if
}n_{2}\geq m^{\prime}+2M-2\\\
&\delta_{d,1}\delta_{\ell^{\prime},m^{\prime}+a}\ket{m^{\prime}+2a+1;\underline{b}^{(0)}}\,,&&\text{if
}n_{2}=m^{\prime}+2a\\\
&\delta_{d,1}\delta_{\ell^{\prime},m^{\prime}+a}\ket{m^{\prime}+2a;\underline{b}^{(0)}},&&\text{if
}n_{2}=m^{\prime}+2a+1\,,\end{aligned}\right.\,$ (B.4)

where $a\in\\{1,2,...,M-2\\}$ and we reported only the cases in which the
background is unchanged (whence we used “$\rightarrow$” instead of “$=$”). By
inserting back such result into the initial expression, we finally have

$\braket{\sigma^{x}_{2(\ell^{\prime}-d)}\sigma^{x}_{2\ell^{\prime}}}_{t}=2\cos\bigl{(}\frac{\pi}{2}d\bigr{)}\Bigl{(}\theta(\ell^{\prime}<m^{\prime})J_{\ell^{\prime}}(4\mathcal{J}t)J_{\ell^{\prime}-d}(4\mathcal{J}t)+\\\
+\theta(\ell^{\prime}\geq
m^{\prime}+M+d-2)J_{\ell^{\prime}+M}(4\mathcal{J}t)J_{\ell^{\prime}+M-d}(4\mathcal{J}t)\Bigr{)}\,.$
(B.5)

All the other one- and two-point functions are obtained analogously.

###  B.2 List of the two-point correlation functions

$\displaystyle\braket{\sigma^{\alpha}_{2\ell^{\prime}-j}\sigma^{\beta}_{2m^{\prime}-1+j}}=$
$\displaystyle 0\,,$ (B.6)
$\displaystyle\braket{\sigma^{\alpha}_{2(\ell^{\prime}-d)-1}\sigma^{\alpha}_{2\ell^{\prime}-1}}=$
$\displaystyle 0\,,$
$\displaystyle\braket{\sigma^{\alpha}_{2(\ell^{\prime}-d)}\sigma^{\beta}_{2\ell^{\prime}}}=$
$\displaystyle
2\cos\Bigl{(}\frac{\pi}{2}(\beta-\alpha+d)\Bigr{)}[\theta(\ell^{\prime}<m^{\prime})J_{\ell^{\prime}}(4Jt)J_{\ell^{\prime}-d}(4\mathcal{J}t)$
$\displaystyle\qquad+\theta(\ell^{\prime}\geq
m^{\prime}+M+d-2)J_{\ell^{\prime}+M}(4Jt)J_{\ell^{\prime}+M-d}(4\mathcal{J}t)$
$\displaystyle\qquad+\delta_{d,1}\theta(m^{\prime}\leq\ell^{\prime}\leq
m^{\prime}+M-2)J_{2\ell^{\prime}-m^{\prime}+1}(4\mathcal{J}t)J_{2\ell^{\prime}-m^{\prime}}(4\mathcal{J}t)]\,,$
$\displaystyle\braket{\sigma^{3-\alpha}_{2(\ell^{\prime}-d)-1}\sigma^{\alpha}_{2\ell^{\prime}-1}}=$
$\displaystyle
2(-1)^{\alpha}\delta_{d,1}\theta(m^{\prime}\leq\ell^{\prime}\leq
m^{\prime}+M-1)J_{m^{\prime}-2\ell^{\prime}+1}(4\mathcal{J}t)J_{m^{\prime}-2\ell^{\prime}}(4\mathcal{J}t)\,,$
$\displaystyle\braket{\sigma^{z}_{2(\ell^{\prime}-d)-1}\sigma^{z}_{2\ell^{\prime}-1}}=$
$\displaystyle 1-2\delta_{m^{\prime}+M-1,\ell^{\prime}-d}f(m^{\prime}+2M-2,t)$
$\displaystyle+2\delta_{m^{\prime}-1,\ell^{\prime}-d}(-1+f(m^{\prime},t)+\theta(d<M)(2f(m^{\prime},t)-1))$
$\displaystyle-2\delta_{\ell^{\prime},m^{\prime}+M-1}(f(m^{\prime}+2M-2,t)+2\theta(d<M)f(m^{\prime}+2M-2,t))$
$\displaystyle+2\delta_{\ell^{\prime},m^{\prime}-1}(-1+f(m^{\prime},t))+4\delta_{d,M}\delta_{m^{\prime}-1,\ell^{\prime}-d}(f(m^{\prime}+2M-2,t)-f(m^{\prime},t))$
$\displaystyle-2\theta(d\geq
M)(\theta(m^{\prime}\leq\ell^{\prime}<m^{\prime}+M-1)+\theta(m^{\prime}+d\leq\ell^{\prime}<m^{\prime}+d+M-1))$
$\displaystyle-2\theta(d<M)(\theta(m^{\prime}\leq\ell^{\prime}<m^{\prime}+d-1)+\theta(m^{\prime}+M-1\leq\ell^{\prime}<m^{\prime}+d+M-1))$
$\displaystyle+2(\theta(m^{\prime}\leq\ell^{\prime}-d\leq
m^{\prime}+M-2)-2\theta(m^{\prime}+d\leq\ell^{\prime}\leq m^{\prime}+M-1))$
$\displaystyle\qquad\times(J^{2}_{m^{\prime}-2\ell^{\prime}+2d}(4\mathcal{J}t)+J^{2}_{m^{\prime}-2\ell^{\prime}+2d-1}(4Jt))$
$\displaystyle+2(\theta(m^{\prime}\leq\ell^{\prime}\leq
m^{\prime}+M-2)-2\theta(m^{\prime}+d-1\leq\ell^{\prime}\leq m^{\prime}+M-2))$
$\displaystyle\qquad\times(J^{2}_{m^{\prime}-2\ell^{\prime}}(4\mathcal{J}t)+J^{2}_{m^{\prime}-2\ell^{\prime}-1}(4\mathcal{J}t))\,,$
$\displaystyle\braket{\sigma^{z}_{2(\ell^{\prime}-d)}\sigma^{z}_{2\ell^{\prime}}}=$
$\displaystyle
1-2\theta(\ell^{\prime}-d<m^{\prime})J^{2}_{\ell^{\prime}-d}(4\mathcal{J}t)-2\theta(\ell^{\prime}<m^{\prime})J^{2}_{\ell^{\prime}}(4\mathcal{J}t)$
$\displaystyle-2\theta(\ell^{\prime}-d\geq
m^{\prime}+M-2)J^{2}_{\ell^{\prime}+M-d}(4\mathcal{J}t)-2\theta(\ell^{\prime}\geq
m^{\prime}+M-2)J^{2}_{\ell^{\prime}+M}(4\mathcal{J}t)$
$\displaystyle-2\theta(m^{\prime}-1\leq\ell^{\prime}-d<m^{\prime}+M-2)J^{2}_{m^{\prime}-2\ell^{\prime}+2d-2}(4\mathcal{J}t)$
$\displaystyle-2\theta(m^{\prime}-1\leq\ell^{\prime}<m^{\prime}+M-2)J^{2}_{m^{\prime}-2\ell^{\prime}-2}(4\mathcal{J}t)$
$\displaystyle-2\theta(m^{\prime}\leq\ell^{\prime}-d<m^{\prime}+M-1)J^{2}_{m^{\prime}-2\ell^{\prime}+2d-1}(4\mathcal{J}t)$
$\displaystyle-2\theta(m^{\prime}\leq\ell^{\prime}<m^{\prime}+M-1)J^{2}_{m^{\prime}-2\ell^{\prime}-1}(4\mathcal{J}t)\,,$
$\displaystyle\braket{\sigma^{z}_{2a^{\prime}}\sigma^{z}_{2\ell^{\prime}-1}}=$
$\displaystyle-1+2\theta(a^{\prime}<m^{\prime})J^{2}_{a^{\prime}}(4\mathcal{J}t)+2\theta(a^{\prime}\geq
m^{\prime}+M-2)J^{2}_{a^{\prime}+M}(4\mathcal{J}t)$
$\displaystyle+2\theta(m^{\prime}-1\leq a^{\prime}\leq
m^{\prime}+M-3)J^{2}_{m^{\prime}-2a^{\prime}-2}(4\mathcal{J}t)$
$\displaystyle+2\theta(m^{\prime}\leq a^{\prime}\leq
m^{\prime}+M-2)J^{2}_{m^{\prime}-2a^{\prime}-1}(4\mathcal{J}t)$
$\displaystyle+2\delta_{\ell^{\prime},m^{\prime}-1}[1-f(m^{\prime},t)-2\theta(a^{\prime}\geq
m^{\prime}+M-2)J^{2}_{a^{\prime}+M}(4\mathcal{J}t)]$
$\displaystyle+2\delta_{\ell^{\prime},m^{\prime}+M-1}[f(m^{\prime}+2M-2,t)-2\theta(a^{\prime}<m^{\prime})J^{2}_{a^{\prime}}(4\mathcal{J}t)]$
$\displaystyle+2\theta(m^{\prime}\leq\ell^{\prime}\leq
m^{\prime}+M-2)[1-J^{2}_{m^{\prime}-2\ell^{\prime}}(4\mathcal{J}t)-J^{2}_{m^{\prime}-2\ell^{\prime}-1}(4\mathcal{J}t)$
$\displaystyle\qquad-2\theta(a^{\prime}<m^{\prime})J^{2}_{a^{\prime}}(4\mathcal{J}t)-2\theta(a^{\prime}\geq
m^{\prime}+M-2)J^{2}_{a^{\prime}+M}(4\mathcal{J}t)$
$\displaystyle\qquad+2\delta_{\ell^{\prime},a^{\prime}+1}J^{2}_{m^{\prime}-2\ell^{\prime}}(4\mathcal{J}t)+2\delta_{a^{\prime},\ell^{\prime}}J^{2}_{m^{\prime}-2a^{\prime}-1}(4\mathcal{J}t)]$
$\displaystyle-4\theta(m^{\prime}-1\leq\ell^{\prime}\leq
m^{\prime}+M-1)[\theta(m^{\prime}-1\leq a^{\prime}\leq
m^{\prime}+M-3)J^{2}_{m^{\prime}-2a^{\prime}-2}(4\mathcal{J}t)$
$\displaystyle\qquad+\theta(m^{\prime}\leq a^{\prime}\leq
m^{\prime}+M-2)J^{2}_{m^{\prime}-2a^{\prime}-1}(4\mathcal{J}t)]\,,$

where $\alpha,\beta\in\\{1,2\\}$, $\sigma^{1}=\sigma^{x}$,
$\sigma^{2}=\sigma^{y}$, $j\in\\{0,1\\}$, and $d\geq 1$.

##  Appendix C Details on numerical simulations

Figure C.3: Bipartite entanglement entropy $S^{vN}_{\ell}(t)$ for $t=25$ (blue
points) and $t=50$ (orange points). The inhomogeneity in the initial state
$\ket{\cdots\uparrow\uparrow\downarrow\uparrow\uparrow\downarrow\uparrow\downarrow\downarrow\uparrow\uparrow\downarrow\uparrow\uparrow\downarrow\cdots}$
spreads with time and forms a nontrivial ballistic profile. The inset shows
the entanglement entropy $S^{vN}_{0}(t)$ in the center of the system.
Oscillations vanish with time, while the entanglement reaches its limiting
value.

Numerical calculations are performed using C++ Itensor library [53]. We use
the time evolving block decimation (TEBD) as a time-evolution scheme, where
$e^{-i\delta tH}$ for the folded model is approximated by Trotter-Suzuki gates
[54, 55, 56, 9]. In order to obtain higher approximation, we represent the
Hamiltonian (1) as a sum

$\displaystyle
H=\sum_{\ell}h_{3\ell,3\ell+1,3\ell+2}+\sum_{\ell}h_{3\ell+1,3\ell+2,3\ell+3}+\sum_{\ell}h_{3\ell+2,3\ell+3,3\ell+4}=H_{1}+H_{2}+H_{3}\,,$
(C.1)

with commuting terms within each $H_{n}$. The time evolution operator reads

$e^{-i\delta t(H_{1}+H_{2}+H_{3})}=e^{-i\frac{\delta
t}{2}H_{1}}e^{-i\frac{\delta t}{2}H_{2}}e^{-i\delta tH_{3}}e^{-i\frac{\delta
t}{2}H_{2}}e^{-i\frac{\delta t}{2}H_{1}}+\mathcal{O}(\delta t^{3})\,,$ (C.2)

where the time step is chosen as $\delta t=0.01\mathcal{J}^{-1}$. Numerical
simulations for the XXZ model are performed using the matrix-product-based
approximation to $e^{-i\delta tH}$ of the third order, denoted by $W^{\rm II}$
in Ref [57, 58]. As a time step in this case we chose $\delta
t=0.001\mathcal{J}^{-1}$ to reduce the potential errors from the large-
anisotropy term.

The level of matrix-product state truncation is determined by some maximal
value of the discarded weight, which we chose to be $\delta\lambda=10^{-12}$.
It turns out that for the most of the calculations in the folded XXZ model the
maximal bond dimension is $\chi_{\rm max}=3$ during the whole time evolution,
meanwhile the bipartite entanglement entropy $S_{vN}$ forms a nontrivial
scaling profile in the ballisitc limit – see Fig. C.3.