#  Mechanisms of scrambling and unscrambling of quantum information in the
ground state in spin chains: domain-walls, spin-flips and scattering phase
shifts

Samudra Sur1 and Diptiman Sen1,2 1Center for High Energy Physics, Indian
Institute of Science, Bengaluru 560012, India
2Department of Physics, Indian Institute of Science, Bengaluru 560012, India

###### Abstract

The spatiotemporal evolution of the out-of-time-order correlator (OTOC)
measures the propagation and scrambling of local quantum information. For the
transverse field Ising model with open boundaries, the local operator
$\sigma^{x}$ shows an interesting picture of the ground state OTOC where the
local information gets scrambled throughout the entire system and, more
strikingly, starts ‘unscrambling’ upon reflection at the other end. Earlier
discussions of OTOCs did not explain the physical processes responsible for
such scrambling and unscrambling of information. We explicitly show that in
the paramagnetic phase, the scrambling and unscrambling is due to the
scattering of a pair of low-energy spin-flip excitations, even in the presence
of small interactions. In the ferromagnetic phase the same phenomena are
explained by the motion of a domain-wall excitation. Thus, in different limits
of the system parameters, we have provided a simple and almost complete
understanding of the space-time pictures of the OTOCs, including the
unscrambling, in terms of the low-energy excitations like one and two spin-
flips or a single domain wall.

### Introduction.

The spreading of local quantum information in a quantum many-body system,
known as scrambling Roberts and Swingle (2016); Hosur  _et al._ (2016);
Swingle and Chowdhury (2017); Patel  _et al._ (2017); Campisi and Goold
(2017); Iyoda and Sagawa (2018); Pappalardi  _et al._ (2018); Klug  _et al._
(2018); Kirkby  _et al._ (2019); Alavirad and Lavasani (2019); Sahu  _et al._
(2019); Sur and Subrahmanyam (2022), is generally described by the growth of a
local operator under time evolution. The front of the operator (or the light-
cone) is known to spread ballistically Luitz and Bar Lev (2017); Bohrdt  _et
al._ (2017); Khemani  _et al._ (2018); Rakovszky  _et al._ (2018); Von
Keyserlingk  _et al._ (2018) in lattice systems with a bound on the speed of
propagation called the Lieb-Robinson velocity Lieb and Robinson (1972).
Quantitatively, scrambling is measured by the out-of-time-order correlator
(OTOC) Larkin and Ovchinnikov (1969); Shenker and Stanford (2014); Maldacena
_et al._ (2016); Dóra and Moessner (2017); Shen  _et al._ (2017); Heyl  _et
al._ (2018); Bao and Zhang (2020); McGinley  _et al._ (2019); Riddell and
Sørensen (2019); Dağ  _et al._ (2019); Zamani  _et al._ (2022); Shukla  _et
al._ (2021); Suchsland  _et al._ (2022),
$F(l,t)=\braket{W_{l}(t)V_{0}W_{l}(t)V_{0}}$ or more precisely the real part
of $F(l,t)$, where the local operators $V$ and $W$ are at the positions $0$
and $l$, and the expectation is taken in a suitable equilibrium ensemble.
Recently, it has been found that in non-interacting spin-$1/2$ chains, the
OTOCs for certain operators show scrambling of quantum information inside the
light-cone (marked by the value of $F(l,t)$ deviating considerably from $1$),
while for other operators they do not Lin and Motrunich (2018); Sur and Sen
(2022). This phenomenon was attributed to the locality or non-locality of
those operators in the Jordan-Wigner (JW) fermionic picture, but without a
quantitative explanation. So far, the approach to understand scrambling
through OTOCs has been attempted mainly by studying the Heisenberg time-
evolution of the local operator using the Baker-Campbell-Hausdorff commutator
expansion. In this context, several worksVon Keyserlingk  _et al._ (2018);
Khemani  _et al._ (2018); Rakovszky  _et al._ (2018); Gopalakrishnan  _et al._
(2018); Nahum  _et al._ (2018); Agrawal  _et al._ (2019); Lopez-Piqueres  _et
al._ (2021); Zhou  _et al._ (2023) in recent times have described the operator
spreading in one dimension in terms of a coarse-grained hydrodynamic picture
of quasiparticle diffusion. While these explain the ballistic spreading and
diffusive broadening of the operator front, an exact mechanism for scrambling
in terms of the excitations of a system has been elusive so far.

It has been reported Sur and Sen (2022) in a recent study that the initial
local operator starting from one end of the systems gets scrambled under time
evolution, and again becomes localized after reflection from the other end of
the system. We term this effect ‘unscrambling’, which is marked by the value
of the OTOC becoming close to 1 again. Interestingly, this feature disappears
gradually with increasing interactions. It is not well-understood yet if the
presence or absence of unscrambling is due of the non-integrability of the
system or only because of interactions between the excitations.

In an attempt to understand the mechanism of scrambling and unscrambling of
operators in terms of the nature of excitations of the system, we study the
ground state OTOC of JW non-local operators (local magnetization) in the
transverse field Ising model (TFIM) with and without interactions. The OTOC
for JW local operators in the TFIM, which shows only propagation of the front
without scrambling, can be evaluated exactly using Wick’s theorem. But the
calculation of the OTOC for JW non-local operators (see Fig. 1) in the
fermionic language is tricky and prevents a simple physical understanding of
the effects mentioned above. Therefore, we resort to a perturbative approach
directly in the spin language. We demonstrate that in the paramagnetic (PM)
phase of the TFIM, the operator creates spin-flip excitations above the ground
state. The scrambling and unscrambling of quantum information happens due to
the scattering phase shifts of two spin-flip excitations. Conversely, in the
ferromagnetic (FM) phase, the operator excites domain walls Suchsland  _et
al._ (2022) above the ground state which are responsible for the scrambling
and unscrambling. Additionally, if interactions are added, the unscrambling in
the ground state OTOC fades away. We find that this absence of unscrambling is
due to different reasons in the PM and the FM phases. In the PM phase, even a
small interaction alters the scattering phase shift significantly from the
non-interacting case and obstructs unscrambling. In the FM phase, a
comparatively larger interaction is needed to disrupt the unscrambling by
creating higher order domain wall excitations. In this letter, we will provide
a detailed quantitative analysis of this mechanism with a comparison of the
OTOCs calculated using numerical exact diagonalization and our effective
theory of low-energy excitations. The excellent match between the two confirms
that the operator spreading in the ground state is indeed governed by the low-
energy excitations.

Ground state OTOC. The Hamiltonian of the TFIM is

$H~{}=~{}-~{}J_{x}~{}\sum_{n=0}^{L-2}\sigma^{x}_{n}\sigma^{x}_{n+1}~{}-~{}h~{}\sum_{n=0}^{L-1}\sigma^{z}_{n},$
(1)

on a lattice with $L$ sites and open boundaries. As the transverse field $h$
is lowered from a large value, the model is known to exhibit a continuous
phase transition from paramagnetic to ferromagnetic at $h=J_{x}$. We consider
the OTOC of the local magnetization operator $\sigma^{x}$ starting from one
end,
$F^{xx}(l,t)=\langle\sigma^{x}_{l}(t)\sigma^{x}_{0}\sigma^{x}_{l}(t)\sigma^{x}_{0}\rangle$.
We will analyze the OTOCs in the PM and FM phases using a perturbative
treatment for large and small transverse fields respectively.

Figure 1: Spatiotemporal evolution of the ground state OTOC for (a) the JW
local operator $\sigma^{z}$ and (b) the JW non-local operator $\sigma^{x}$, at
the critical point of the TFIM given in Eq. (1) with $J_{x}=h=2$. The
$\sigma^{z}$-OTOC shows only light-cone lines of information propagation,
while the $\sigma^{x}$-OTOC shows alternating scrambling of quantum
information and unscrambling after reflections from the ends.

Paramagnetic phase: spin-flip excitations. In the paramagnetic phase, at large
values of the transverse field, $|h/J_{x}|\gg 1$, the ground state of the
model has all the spins aligned in the direction of the field,
$\ket{\uparrow\uparrow\uparrow\cdots\uparrow}_{z}$ in the $\sigma^{z}$-basis.
The lowest energy excitations are given by single spin-flips like
$\ket{\uparrow\downarrow\uparrow\cdots\uparrow}_{z}$. In the OTOC,
$\sigma^{x}_{0}$ acting on the ground state creates a spin-flip excitation at
the left-most (zero-th) site which can then hop through the chain due to the
$J_{x}\sigma^{x}_{n}\sigma^{x}_{n+1}$ term in the Hamiltonian. Subsequently,
$\sigma^{x}_{l}$ can either produce a second spin-flip excitation or de-excite
the system back to the ground state. Thus, the ground state OTOC is
approximately governed by a combination of two processes, one which involves
only single spin-flip excitations and the other involving two spin-flips (see
Fig. 2 (a)). This becomes clear if we write the OTOC using the intermediate
excited states by resolving the identity approximately as
$\mathbb{I}\approx\ket{\psi_{GS}}\bra{\psi_{GS}}+\sum_{q}\ket{q}\bra{q}+\sum_{q_{1},q_{2}}\ket{q_{1},q_{2}}\bra{q_{1},q_{2}}+$
higher order excitations.

$\displaystyle F^{xx}(l,t)\simeq$
$\displaystyle\sum_{q,q^{\prime}}\big{[}\bra{\psi_{GS}}e^{iHt}\sigma^{x}_{l}\ket{q^{\prime}}\bra{q^{\prime}}e^{-iHt}\sigma^{x}_{0}\ket{\psi_{GS}}\bra{\psi_{GS}}e^{iHt}\sigma^{x}_{l}\ket{q}$
$\displaystyle\bra{q}e^{-iHt}\sigma^{x}_{0}\ket{\psi_{GS}}\big{]}+\sum_{q_{1},q_{4},q_{2}<q_{3}}\big{[}\bra{\psi_{GS}}e^{iHt}\sigma^{x}_{l}\ket{q_{4}}$
$\displaystyle\bra{q_{4}}e^{-iHt}\sigma^{x}_{0}\ket{q_{2},q_{3}}\bra{q_{2},q_{3}}e^{iHt}\sigma^{x}_{l}\ket{q_{1}}\bra{q_{1}}e^{-iHt}\sigma^{x}_{0}\ket{\psi_{GS}}\big{]},$
(2)

where $\ket{q}$ and $\ket{q_{1},q_{2}}$ denote the eigenstates of the
Hamiltonian with single and two spin-flip excitations, respectively. The
quantum number $q$ takes the values $0,1,\dots,L-1$. It is worth noting that
it is enough to consider processes up to the second order if $J_{x}/h$ is
small. The matrix elements in Eq. (2) can be evaluated using the energies and
wave functions of the excited states $\ket{q}$ and $\ket{q_{1},q_{2}}$. The
single spin-flip excitations are described by an effective nearest-neighbour
tight-binding model on the $L$-site lattice with open boundaries, and
therefore have energies $\varepsilon_{q}=-2J_{x}~{}\cos(\frac{\pi q}{L+1})+2h$
above the ground state, and wave functions
$\psi_{q}(n)=\braket{n}{q}=\sqrt{2/(L+1)}~{}\sin[\pi(q+1)(n+1)/(L+1)]$, where
$\ket{n}=\sigma^{x}_{n}\ket{\psi_{GS}}$ denotes the state with a single
flipped spin at site $n$.

Figure 2: (a) Schematic of the mechanism of scrambling and unscrambling of
$F^{xx}(l,t)$ involving two spin-flip excitations in the paramagnetic phase of
the TFIM. (b) Spatiotemporal evolution of $F^{xx}(l,t)$ as obtained from a
numerical time evolution for $J_{x}=2$ and $h=4$. (c) $F^{xx}(l,t)$ for the
same parameter values obtained using the effective theory of two spin-flip
excitations as mentioned in the text. The exact numerics and the effective
theory calculations give remarkably similar results for the scrambling and
unscrambling of $\sigma^{x}$.

While the single-particle eigenstates are solved by considering the problem of
a particle in a box on a lattice, the case of two spin-flips is a bit subtle.
The two spin-flips should be on separate sites and follow commutation
relations, making it a system with two hard-core bosons. Due to the
indistinguishability, we can write the wave function for two spin-flip
excitations as
$\psi_{q_{1},q_{2}}(n_{1},n_{2})=\braket{n_{1},n_{2}}{q_{1},q_{2}}=\psi_{q_{1}}(n_{1})\psi_{q_{2}}(n_{2})-\psi_{q_{2}}(n_{1})\psi_{q_{1}}(n_{2})$,
where $\ket{n_{1},n_{2}}$ denotes the state with two spin-flips at sites
$n_{1}$ and $n_{2}$, and we choose $n_{1}<n_{2}$. The energy this state is the
sum of two single-particle energies,
$\varepsilon_{q_{1},q_{2}}=\varepsilon_{q_{1}}+\varepsilon_{q_{2}}$. Finally,
we arrive at SM

$\displaystyle
F^{xx}(l,t)\simeq\sum_{q,q^{\prime}}e^{-i(E_{q}+E_{q^{\prime}})t}\psi_{q}(l)\psi_{q}(0)\psi_{q^{\prime}}(l)\psi_{q^{\prime}}(0)$
$\displaystyle~{}+\sum_{q_{1},q_{4},q_{2}<q_{3}}e^{-i(E_{q_{1}}+E_{q_{4}}-E_{q_{2}}-E_{q_{3}})t}\psi_{q_{1}}(0)\psi_{q_{4}}(l)$
$\displaystyle\sum_{n_{1}}\psi_{q_{1}}(n_{1})\big{[}\theta(n_{1}-l)\psi_{q_{2},q_{3}}(l,n_{1})+\theta(l-n_{1})\psi_{q_{2},q_{3}}(n_{1},l)\big{]}$
$\displaystyle\sum_{n_{2}}\psi_{q_{4}}(n_{2})\psi_{q_{2},q_{3}}(0,n_{2}).$ (3)

We evaluate this expression exactly and contrast the spatiotemporal evolution
of $F^{xx}(l,t)$ calculated using exact numerical time-evolution with this
analysis in Figs. 2 (b) and (c). The striking agreement confirms that spin-
flip excitations indeed provide the mechanism responsible for the scrambling
of quantum information and the unscrambling after a reflection when $h$ is
large. Interestingly, this analysis agrees quite well with the exact result
even if $h$ is not much larger than $J_{x}$. In fact, the plot for OTOC looks
qualitatively the same for $h=J_{x}$ (Fig. 1 (b)) and $h>J_{x}$ (Fig. 2 (b)),
although an analysis in terms of the spin-flip excitations cannot be assumed
to be valid at the critical point.

Interactions in the presence of large field. We now look at the behavior of
the OTOC in the PM phase in the presence of an integrability-breaking
interaction term given by $J_{z}\sum_{n}\sigma^{z}_{n}\sigma^{z}_{n+1}$. The
full Hamiltonian is given by
$H=-J_{x}\sum_{n=0}^{L-2}\sigma^{x}_{n}\sigma^{x}_{n+1}-h\sum_{n=0}^{L-1}\sigma^{z}_{n}+J_{z}\sum_{n=0}^{L-2}\sigma^{z}_{n}\sigma^{z}_{n+1}$.
When $|h/J_{x}|>1$, a small value of $J_{z}$ can significantly change the
spatiotemporal plot for $F^{xx}(l,t)$, as shown in Fig. 3 (a). As usual, the
local quantum information starts scrambling after $\sigma^{x}_{0}$ acts at one
end. However, we do not see any unscrambling after reflection, unlike in the
model without interactions. As time progresses further, the local information
always stays scrambled and never becomes localized to a few sites. This can
also be explained using a simple picture of two-spin flips. We need to modify
Eq. (3) to account for two-spin flip states in the presence of interactions
and evaluate it numerically. The wave functions for the two-spin flip state
$\psi_{q_{1},q_{2}}(n_{1},n_{2})$ are replaced by two-body eigenstates
calculated numerically by considering an effective tight-binding model with
two hard-core bosons with a density-density interaction given by $J_{z}$. The
OTOC calculated using this analysis involving only up to two-spin flips shows
fairly good agreement with the plots obtained using exact numerics even with
interactions. We contrast the two plots in Figs. 3 $(i)$ and $(ii)$ for the
parameter values $J_{x}=2,~{}h=8$, and $J_{z}=1.8$.

We now introduce a simple space-time picture to understand the various
spatiotemporal plots of the OTOC. We rewrite the expression for the OTOC as
$F^{xx}(l,t)=\bra{\psi_{GS}}e^{iHt}\sigma^{x}_{l}e^{-iHt}\sigma^{x}_{0}e^{iHt}\sigma^{x}_{l}e^{-iHt}\sigma^{x}_{0}\ket{\psi_{GS}}=\braket{\psi_{2}}{\psi_{1}}$,
where $\ket{\psi_{1}}$ and $\ket{\psi_{2}}$ are both states with one particle
(spin-flip) and are defined respectively as
$\ket{\psi_{1}}=\sigma^{x}_{0}e^{iHt}\sigma^{x}_{l}e^{-iHt}\sigma^{x}_{0}\ket{\psi_{GS}}$
and $\ket{\psi_{2}}=e^{iHt}\sigma^{x}_{l}e^{-iHt}\ket{\psi_{GS}}$. The state
$\ket{\psi_{1}}$ can be interpreted as follows. Starting from the right, the
first $\sigma^{x}_{0}$ operator creates a spin-flip at the end site which then
propagates with some velocity for a time $t$ (due to $e^{-iHt}$). Next another
spin-flip is created at site $l$. The two flipped spins then propagate back in
time (due to $e^{iHt}$) to $t=0$ which involves zero, one or more scatterings
between them depending on the position $l$ and time $t$. Finally, at $t=0$ one
of the excitations is annihilated at site $l=0$ to produce a state with only
one spin-flip. In contrast, $\ket{\psi_{2}}$ has a simpler interpretation as
it involves a single spin-flip which is created at position $l$ and time $t$
on the time-evolved ground state, and which then propagates back in time to
$t=0$. The inner product $\braket{\psi_{2}}{\psi_{1}}$ then depends on the
scattering phase shift in the state $\ket{\psi_{1}}$ with respect to state
$\ket{\psi_{2}}$.

Some assumptions are now required to estimate the value of the OTOC using
scattering phase shifts. First, even though we are considering a system with
open boundaries, we will assume that the system is large enough such that away
from the edge, the momentum $q$ is a good quantum number. Second, we
approximate a spin-flip excitation to be a quasiparticle with a single momenta
$q_{0}$ with the largest velocity given by the Lieb-Robinson bound, although
actually it is a superposition of many momentum states around $q_{0}$,
propagating with the maximum possible group velocity. The entire region in the
spatiotemporal plot of the OTOC can now be divided into regions according to
the number of scatterings between the pair of quasiparticles with momenta
$q_{0}$ and $-q_{0}$ in the state $\ket{\psi_{1}}$. The value of the OTOC
(real part) in each region is approximately constant and is given by the real
part of the scattering phase shift obtained from the two-particle Bethe ansatz
SM .

Figure 3: Spatiotemporal plots of $F^{xx}(l,t)$ for the interacting TFIM with
$J_{x}=2,~{}h=8,~{}J_{z}=1.8$ obtained $(i)$ from exact numerics and $(ii)$
from the analysis using two spin-flip excitations. The value of $J_{z}$ is
such that a region which showed scrambling in the non-interacting case now
shows OTOC close to 1. $(iii)$ The point $(a)$ in the space-time plot
corresponds to a region where there is no phase difference due to scattering
between the states $\ket{\psi_{1}}$ and $\ket{\psi_{2}}$ (left panel), while
at point $(b)$ (right panel) there is a scattering event between the two
excitations in $\ket{\psi_{1}}$, and therefore the OTOC, defined as the inner
product $\braket{\psi_{2}}{\psi_{1}}$ picks out the phase difference between
the states.

The scrambling and unscrambling for the TFIM without interactions, given by
the successive dark (OTOC close to $1$) and bright (OTOC close to $-1$),
regions can be now understood as the regions having even and odd number of
scatterings respectively starting from the lowest dark region with zero
scattering. In the absence of interactions, every scattering event changes the
phase by $e^{i\theta}=-1$. For the interacting model, the absence of
unscrambling can be simply understood as the deviation of scattering phase
shifts from $-1$ due to $J_{z}$. However, it should be noted that for later
times the contributions from excitations not corresponding to the largest
group velocity become significant and they alter the value of the OTOC
obtained from the estimate given by scattering phase shift of two excitations.

To illustrate this, we have chosen two points $(a)$ and $(b)$ in the OTOC
plots in Fig. 3 $(i)$ where point $(a)$ is in a region of no scrambling and
$(b)$ is in a region of scrambling before any reflection. We now consider the
schematic shown in Fig. 3 $(iii)$. The red lines are guides to the eye for the
light-cone front. Then $\ket{\psi_{1}}$ at $(a)$ is given by a superposition
of one-particle states shown in $(a_{1})$ and $(a_{2})$ with an intermediate
process where a second excitation is first created at $(l,t)$ = $(0,0)$, then
propagated forward in time, propagated back in time, and finally annihilated
at the same point of creation. By contrast, $\ket{\psi_{2}}$ at $(a)$ is given
by superposition of one-particle states shown in $(a_{3})$ and $(a_{4})$
without any other particles created or annihilated. Since there is no
scattering event for point $(a)$, $F^{xx}=\braket{\psi_{2}}{\psi_{1}}=1$.
Similarly, for point $(b)$, $\ket{\psi_{1}}$ is made of a superposition of
states shown in $(b_{1})$ and $(b_{2})$ and $\ket{\psi_{2}}$ is made of a
superposition of states shown in $(b_{3})$ and $(b_{4})$. In contrast to point
$(a)$, we have one scattering phase shift between two excitations in the state
$\ket{\psi_{1}}$ with respect to the state $\ket{\psi_{2}}$. Hence the inner
product of the two states is given by the scattering phase shift which, for a
pair of non-interacting hard-core bosons, is equal to $-1$; this explains the
value of $F^{xx}(l,t)$ in the first scrambled region in the non-interacting
TFIM. For the interacting case also it has a value close to $-1$.
Subsequently, for larger times the scrambled and unscrambled regions have
values of $F^{xx}(l,t)$ given by the total number of scattering events
multiplied by the scattering phase shifts of one scattering event. In Figs. 3
$(i)$ and $(ii)$ we have considered a $J_{z}$ so that the region corresponding
to three scattering events has $F^{xx}(l,t)$ close to $1$, which, for the non-
interacting case, would have been close to $-1$.

The same effects are observed if we consider the integrable but interacting
$XXZ$ spin-1/2 model in a transverse field. Namely, for the Hamiltonian,
$H^{\prime}=-J\sum_{n=0}^{L-2}\sigma^{x}_{n}\sigma^{x}_{n+1}-J\sum_{n=0}^{L-2}\sigma^{y}_{n}\sigma^{y}_{n+1}-h\sum_{n=0}^{L-1}\sigma^{z}_{n}+J_{z}\sum_{n=0}^{L-2}\sigma^{z}_{n}\sigma^{z}_{n+1}$,
the spatiotemporal plot for the OTOC shows a similar behavior including the
absence of unscrambling as compared to the non-interacting $XX$ model without
the $J_{z}$-term (see Fig. 6 in Ref. SM ). Therefore, a similar analysis in
terms of the low-energy excitations holds true for the $XXZ$ spin-1/2 model
also, implying that the absence of unscrambling is not related to the the
integrability or non-integrability of the model.

Ferromagnetic phase: domain walls. We now discuss the scrambling in the FM
phase of the TFIM. For very small $h$, the ground states are the degenerate
states $\ket{I}=\ket{\uparrow\uparrow\cdots\uparrow}_{x}$ and
$\ket{II}=\ket{\downarrow\downarrow\cdots\downarrow}_{x}$ in the
$\sigma^{x}$-basis. The model has a $Z_{2}$ spin-flip symmetry, but the ground
states spontaneously break it and the true ground states are the
superpositions $\ket{+}=(\ket{I}+\ket{II})/\sqrt{2}$ and
$\ket{-}=(\ket{I}-\ket{II})/\sqrt{2}$ corresponding to the even and odd
fermion parity respectively. Setting $J_{x}=1$, and taking $h$ small, a
perturbative expansion up to first order in $h$ SM  yields the modified ground
states expressed in a product form
$\ket{I^{\prime}}=\ket{(\uparrow+\frac{h}{2}\downarrow)_{0}(\uparrow+\frac{h}{4}\downarrow)_{1}\cdots(\uparrow+\frac{h}{2}\downarrow)_{L-1}}_{x}$
and
$\ket{II^{\prime}}=\ket{(\downarrow+\frac{h}{2}\uparrow)_{0}(\downarrow+\frac{h}{4}\uparrow)_{1}\cdots(\downarrow+\frac{h}{2}\uparrow)_{L-1}}_{x}$
(up to a normalization constant), where the subscripts denote the site
indices. The lowest energy excitations are the two types of domain walls,
namely $\ket{m+1/2,I}$ and $\ket{m+1/2,II}$, where the site label
$m+\frac{1}{2}$ implies that the domain wall is situated between the $m$-th
and $(m+1)$-th sites, and $I$ ($II$) denotes that all the spins on the left of
the domain wall are down (up). Here $m$ can take values from $0$ to $L-2$. For
example, $\ket{0+1/2,I}=\ket{\downarrow\uparrow\uparrow\cdots\uparrow}_{x}$
denotes the state with a domain wall between the first and second spins. Now,
$\sigma^{x}_{0}$ acting on the state $\ket{I^{\prime}}$ creates the domain-
wall state $\ket{0+1/2,I}$ with an amplitude $h$ to leading order. Under time
evolution, this domain wall can move through the lattice with one less site
due to the $h\sum\sigma^{z}_{n}$ term in the Hamiltonian, allowing us to
construct domain-wall eigenstates with a quantum number label $\kappa$.
Accordingly, domain-wall states have an energy dispersion
$\epsilon_{\kappa,\alpha}=-2h$ sin$[\pi(\kappa+1)/L]+2J_{x}$ above the ground
state, and a wave function
$\phi_{\kappa,\alpha}(m)=\braket{m+1/2,\alpha\mid\kappa,\alpha}=\sqrt{2/L}~{}$sin$[\pi(\kappa+1)(m+1)/L]$,
where $\kappa=0,1,\cdots,L-2$, and $\alpha$ can be $I$ or $II$. Subsequently,
$\sigma^{x}_{l}$ for $l\neq 0$ does not create or destroy the domain-wall
excitation but only measures if the spin is up or down in the $x$-basis.
However, it changes the parity of the domain-wall excitation. The remaining
$\sigma^{x}_{0}$ and $\sigma^{x}_{l}$ operators first de-excite the domain
wall to a ground state with the opposite parity and then returns it to the
ground state we started with. In short, as the domain moves from one end to
the other end, it leaves flipped spins behind along its trajectory. This is
then measured by the operator $\sigma^{x}_{l}$ at different times. The entire
mechanism is described schematically in Fig. 4 (a). We have a different
situation when $l=0$, as the four $\sigma^{x}_{0}$ operators sequentially
excite and de-excite between the ground state and a domain-wall state,
resulting in a different value of the OTOC at the edge.

Figure 4: (a) Schematic of the mechanism of domain-wall assisted scrambling
and unscrambling in the ordered phase of the TFIM. Spatiotemporal plots of
$F^{xx}(l,t)$ for $J_{x}=2$ and $h=1$ obtained (b) using exact numerics in the
ordered phase, and (c) using the analysis of domain-wall excitations described
in the text. The remarkable agreement between these two plots suggests that
indeed scrambling and unscrambling in the FM phase is governed by domain-wall
excitations.

We now present a semi-analytical treatment of domain-wall dynamics to
quantitatively understand scrambling and unscrambling. To this end, we
construct the $\sigma^{x}$ operators at all sites which is perturbatively
correct up to order $h$ in the subspace consisting only of the ground state
and single domain-wall states. The operator $\sigma^{x}_{0}$ has non-trivial
matrix elements only in the subspace of states
$\\{\ket{I^{\prime}},\ket{0+1/2,I}\\}$ and
$\\{\ket{II^{\prime}},\ket{0+1/2,II}\\}$. On the other hand,
$\sigma^{x}_{L-1}$ has non-trivial elements in the subspaces spanned by the
states $\\{\ket{I^{\prime}},\ket{L-2+1/2,II}\\}$ and
$\\{\ket{II^{\prime}},\ket{L-2+1/2,I}\\}$. The $\sigma^{x}_{l}$ for all other
sites are diagonal and have trivial matrix elements ($\pm 1$) up to order $h$
as they only determine whether the spin is up or down. The Hamiltonian then
has a simple structure with only the ground state energy along the diagonal in
the ground state basis and a tight-binding structure in the subspace of domain
walls. Since the Hamiltonian only appears in the exponential in time
evolution, we do not need to consider the matrix elements between the ground
states and the domain walls to the lowest order.

The effective representations of $\sigma^{x}_{n}$ and the Hamiltonian $H$ in
the subspace spanned by the set of states
$\\{\ket{I^{\prime}},\\{\ket{m+1/2,I}\\}_{m=0}^{L-2},\ket{II^{\prime}},\\{\ket{m+1/2,II}\\}_{m=0}^{L-2}\\}$
are defined as $X_{n}$ and $\mathcal{H}$ (see Ref. SM  for the explicit matrix
forms). The OTOC in terms of domain-wall excitations can be written as

$\displaystyle F^{xx}(l,t)\approx$
$\displaystyle\frac{1}{2}\bra{I^{\prime}+II^{\prime}}e^{i\mathcal{H}t}X_{l}e^{-i\mathcal{H}t}X_{0}e^{i\mathcal{H}t}X_{l}e^{-i\mathcal{H}t}X_{0}\ket{I^{\prime}+II^{\prime}}.$
(4)

The plots for spatiotemporal evolution of $F^{xx}(l,t)$ using exact numerics
and our analysis agree very well as shown in Figs. 4 (b) and (c). Moreover, in
both the plots we observe the reflection of scrambled quantum information
happening slightly before it reaches the other end, which validates that it is
indeed governed by domain-wall excitations (which are defined midway between
sites). The OTOC expression given in Eq. (4) is correct up to order $h^{2}$.

Interactions in the presence of small field. In the FM phase, the introduction
of a small interaction $J_{z}$ (relative to $J_{x}$) in the Hamiltonian does
not alter the qualitative features of the $F^{xx}(l,t)$ much. We can see
scrambling and unscrambling much like the case without interactions in the
spatiotemporal plots in OTOC. A small $J_{z}$ also gives rise to dynamics of a
domain-wall by making it hop by two sites, in contrast to the transverse field
$h$ which leads to hopping by one site. Therefore, the presence of a small
interaction effectively only changes the dispersion of the domain walls to
$E_{k}=-2h\cos k+2J_{z}\cos(2k)$. The group velocity is then given by
$v_{k}=dE_{k}/dk=2h\sin k-4J_{z}\sin(2k)$. As the light-cone velocity is given
by the maximum group velocity of the quasiparticles, for small $J_{z}/J_{x}$,
the interacting model (Fig. 5 (a)) shows a different light-cone velocity than
its non-interacting counterpart.

Figure 5: $F^{xx}(l,t)$ showing a pronounced unscrambling effect for (a)
$J_{x}=2,~{}h=1,~{}J_{z}=0.5$, while for (b) $J_{x}=2,~{}h=1,~{}J_{z}=1$, it
shows much less unscrambling. As $J_{z}$ becomes comparable to $J_{x}$ the
perturbative expansion around the ground state becomes invalid. Also, higher
order excitations become important and the scrambling is no longer due to
single domain-wall motion.

If $J_{z}$ is increased further, we find that the unscrambling starts to go
away. Roughly, this happens around $J_{z}/J_{x}\gtrsim 0.4$, as observed
numerically. Fig. 5 (b) shows the absence of unscrambling for the parameter
values $J_{x}=2,~{}h=1,~{}J_{z}=1$. It can be reasonably expected that for
such values of $J_{z}$, the perturbative expansion starting from the states
$\ket{I}$ and $\ket{II}$ will no longer be accurate. Furthermore, the
assumption that only the lowest order excitations are responsible for
information scrambling also becomes a gross oversimplification. To explain the
absence of unscrambling when the interaction $J_{z}$ is comparable with
$J_{x}$, one needs to account for higher order excitations like dynamics of
three or more domain walls.

Conclusion. To summarize, we have studied the propagation of local quantum
information in a spin-1/2 chain with open boundary conditions using ground
state OTOCs of $\sigma^{z}$ and $\sigma^{x}$ operators which are local and
non-local in terms of JW fermions respectively. While both $\sigma^{z}$ and
$\sigma^{x}$ OTOCs show light-cone like propagation, only the $\sigma^{x}$
OTOC shows scrambling within the light-cone, in agreement with earlier
results. In addition, we discover remarkable unscramblings and scramblings of
the $\sigma^{x}$ OTOC after repeated reflections from the ends. We have
provided, for the first time, an analytical understanding of both scrambling
and unscrambling deep in the PM phase (when the transverse field $h$ is large)
and the FM phase (when the $XX$ coupling $J_{x}$ is large) phases in terms of
the low-energy excitations, namely, spin-flips when $h$ is large and domain
walls when $J_{x}$ is large. When an interaction between JW fermions ($ZZ$
coupling) is added, the unscrambling effect becomes weaker. In the PM phase,
even relatively weak interactions significantly change the scattering phase
shift for two spin-flips and thereby reduces the unscrambling. In the FM
phase, stronger interactions are required to destroy unscrambling and it
occurs due the the creation of multiple domain wall excitations.

To conclude, we point out a recent experimental measurement of the OTOC Green
_et al._ (2022) at finite temperature which, as the authors suggest, can also
be performed for the ground state. In this paper, finite temperature OTOCs of
the transverse field Ising model are studied for a trapped linear chain of
171Yb+ ions by creating a thermofield double state and then looking at its
time evolution. We believe that a similar route can be followed to
experimentally study the dynamics of low-energy excitations through the ground
state OTOCs of the spin chains discussed in our work.

Acknowledgements. S.S. thanks MHRD, India for financial support through the
PMRF. D.S. acknowledges funding from SERB, India (JBR/2020/000043).

## References

  * Roberts and Swingle (2016) D. A. Roberts and B. Swingle, Phys. Rev. Lett. 117, 091602 (2016).
  * Hosur  _et al._ (2016) P. Hosur, X.-L. Qi, D. A. Roberts, and B. Yoshida, Journal of High Energy Physics 2016, 1 (2016).
  * Swingle and Chowdhury (2017) B. Swingle and D. Chowdhury, Phys. Rev. B 95, 060201 (2017).
  * Patel  _et al._ (2017) A. A. Patel, D. Chowdhury, S. Sachdev, and B. Swingle, Phys. Rev. X 7, 031047 (2017).
  * Campisi and Goold (2017) M. Campisi and J. Goold, Phys. Rev. E 95, 062127 (2017).
  * Iyoda and Sagawa (2018) E. Iyoda and T. Sagawa, Phys. Rev. A 97, 042330 (2018).
  * Pappalardi  _et al._ (2018) S. Pappalardi, A. Russomanno, B. Žunkovič, F. Iemini, A. Silva, and R. Fazio, Phys. Rev. B 98, 134303 (2018).
  * Klug  _et al._ (2018) M. J. Klug, M. S. Scheurer, and J. Schmalian, Phys. Rev. B 98, 045102 (2018).
  * Kirkby  _et al._ (2019) W. Kirkby, J. Mumford, and D. H. J. O’Dell, Phys. Rev. Res. 1, 033135 (2019).
  * Alavirad and Lavasani (2019) Y. Alavirad and A. Lavasani, Phys. Rev. A 99, 043602 (2019).
  * Sahu  _et al._ (2019) S. Sahu, S. Xu, and B. Swingle, Phys. Rev. Lett. 123, 165902 (2019).
  * Sur and Subrahmanyam (2022) S. Sur and V. Subrahmanyam, Quantum Information Processing 21, 301 (2022).
  * Luitz and Bar Lev (2017) D. J. Luitz and Y. Bar Lev, Phys. Rev. B 96, 020406 (2017).
  * Bohrdt  _et al._ (2017) A. Bohrdt, C. B. Mendl, M. Endres, and M. Knap, New J. Phys. 19, 063001 (2017).
  * Khemani  _et al._ (2018) V. Khemani, A. Vishwanath, and D. A. Huse, Phys. Rev. X 8, 031057 (2018).
  * Rakovszky  _et al._ (2018) T. Rakovszky, F. Pollmann, and C. W. von Keyserlingk, Phys. Rev. X 8, 031058 (2018).
  * Von Keyserlingk  _et al._ (2018) C. Von Keyserlingk, T. Rakovszky, F. Pollmann, and S. L. Sondhi, Phys. Rev. X 8, 021013 (2018).
  * Lieb and Robinson (1972) E. H. Lieb and D. W. Robinson, Comm. Math. Phys. 28, 251 (1972).
  * Larkin and Ovchinnikov (1969) A. Larkin and Y. N. Ovchinnikov, JETP 28, 1200 (1969).
  * Shenker and Stanford (2014) S. H. Shenker and D. Stanford, Journal of High Energy Physics 2014, 1 (2014).
  * Maldacena  _et al._ (2016) J. Maldacena, S. H. Shenker, and D. Stanford, Journal of High Energy Physics 2016, 1 (2016).
  * Dóra and Moessner (2017) B. Dóra and R. Moessner, Phys. Rev. Lett. 119, 026802 (2017).
  * Shen  _et al._ (2017) H. Shen, P. Zhang, R. Fan, and H. Zhai, Phys. Rev. B 96, 054503 (2017).
  * Heyl  _et al._ (2018) M. Heyl, F. Pollmann, and B. Dóra, Phys. Rev. Lett. 121, 016801 (2018).
  * Bao and Zhang (2020) J.-H. Bao and C.-Y. Zhang, Communications in Theoretical Physics 72, 085103 (2020).
  * McGinley  _et al._ (2019) M. McGinley, A. Nunnenkamp, and J. Knolle, Phys. Rev. Lett. 122, 020603 (2019).
  * Riddell and Sørensen (2019) J. Riddell and E. S. Sørensen, Phys. Rev. B 99, 054205 (2019).
  * Dağ  _et al._ (2019) C. B. Dağ, K. Sun, and L.-M. Duan, Phys. Rev. Lett. 123, 140602 (2019).
  * Zamani  _et al._ (2022) S. Zamani, R. Jafari, and A. Langari, Phys. Rev. B 105, 094304 (2022).
  * Shukla  _et al._ (2021) R. K. Shukla, G. K. Naik, and S. K. Mishra, EPL 132, 47003 (2021).
  * Suchsland  _et al._ (2022) P. Suchsland, B. Douçot, K. Khemani, and R. Moessner, arXiv:2207.09502 (2022).
  * Lin and Motrunich (2018) C.-J. Lin and O. I. Motrunich, Phys. Rev. B 97, 144304 (2018).
  * Sur and Sen (2022) S. Sur and D. Sen, arXiv:2210.15302 (2022).
  * Gopalakrishnan  _et al._ (2018) S. Gopalakrishnan, D. A. Huse, V. Khemani, and R. Vasseur, Phys. Rev. B 98, 220303 (2018).
  * Nahum  _et al._ (2018) A. Nahum, S. Vijay, and J. Haah, Phys. Rev. X 8, 021014 (2018).
  * Agrawal  _et al._ (2019) U. Agrawal, S. Gopalakrishnan, and R. Vasseur, Phys. Rev. B 99, 174203 (2019).
  * Lopez-Piqueres  _et al._ (2021) J. Lopez-Piqueres, B. Ware, S. Gopalakrishnan, and R. Vasseur, Phys. Rev. B 104, 104307 (2021).
  * Zhou  _et al._ (2023) T. Zhou, A. Guo, S. Xu, X. Chen, and B. Swingle, Phys. Rev. B 107, 014201 (2023).
  * (39) Supplemental Material .
  * Green  _et al._ (2022) A. M. Green, A. Elben, C. H. Alderete, L. K. Joshi, N. H. Nguyen, T. V. Zache, Y. Zhu, B. Sundar, and N. M. Linke, Phys. Rev. Lett. 128, 140601 (2022).

##  I Supplemental Material

## A: Derivation of expression for OTOC in Eq. (3)

We will present here the derivation of the final expression of the OTOC in
terms of spin-flip excitations given in Eq. (3) for $|h/J_{x}|\gg 1$. The time
evolution of the spin-flip eigenstates $e^{-iHt}\ket{q}$ and
$e^{-iHt}\ket{q_{1},q_{2}}$ is trivial. The more difficult part is to find the
matrix elements $\bra{q_{1},q_{2}}\sigma^{x}_{l}\ket{q}$. We use the
resolution of identity,

$\displaystyle\bra{q_{2},q_{3}}\sigma^{x}_{l}\ket{q_{1}}=\big{(}\frac{2}{L+1}\big{)}^{3/2}\sum_{n_{1},n_{2}<n3}\braket{q_{2},q_{3}}{n_{2},n_{3}}\braket{n_{2},n_{3}}{\sigma^{x}_{l}}{n_{1}}\braket{n_{1}}{q_{1}},$
(S1)

and the wave functions
$\braket{q_{2},q_{3}}{n_{2},n_{3}}=\psi_{q_{1},q_{3}}(n_{2},n_{3})$ and
$\braket{n_{1}}{q_{1}}=\psi_{q}(n)$ as given in the main text. The matrix
element $\braket{n_{2},n_{3}}{\sigma^{x}_{l}}{n_{1}}$ depends on whether
$l<n_{1}$ or $l>n_{1}$ and we always choose $n_{2}<n_{3}$. Since
$\sigma^{x}_{l}$ creates a spin-flip, we should have $l\neq n_{1}$. Therefore,

$\displaystyle\braket{n_{2},n_{3}}{\sigma^{x}_{l}}{n_{1}}=\theta(n_{1}-l)\delta_{n_{2},l}\delta_{n_{1},n_{3}}+\theta(l-n_{1})\delta_{n_{3},l}\delta_{n_{1},n_{2}}.$
(S2)

Using these relations we obtain Eq. (3) in the main text.

## B: Perturbative ground state for $|h/J_{x}|<1$

k The degenerate ground states of the model in the limit of $h\to 0$ are
$\ket{I}=\ket{\uparrow\uparrow\cdots\uparrow}_{x}$ and
$\ket{II}=\ket{\downarrow\downarrow\cdots\downarrow}_{x}$. If a small $h$ is
turned on, we obtain, from degenerate perturbation theory,

$\displaystyle\ket{I^{\prime}}$
$\displaystyle=\ket{I}+h\sum_{n}\frac{\ket{n}\braket{n}{\sum_{l}\sigma^{z}_{l}}{I}}{2(\delta_{n,0}+\delta_{n,N-1})+4(1-\delta_{n,0}-\delta_{n,N-1})}~{}+~{}O(h^{2})$
$\displaystyle=\ket{I}+\frac{h}{2}\ket{\downarrow\uparrow\uparrow\cdots}_{x}+\frac{h}{4}\ket{\uparrow\downarrow\uparrow\cdots}_{x}+\frac{h}{4}\ket{\uparrow\uparrow\downarrow\uparrow\cdots}_{x}+\cdots+\frac{h}{2}\ket{\uparrow\uparrow\cdots\downarrow}_{x},$
(S3)

where $\ket{n}$ denotes the single spin-flip state in the $\sigma^{x}$-basis
at site $n$. Combining these together we can write down the following
perturbative ground state which is valid up to $O(h)$,

$\displaystyle\ket{I^{\prime}}=\ket{(\uparrow+\frac{h}{2}\downarrow)_{0}(\uparrow+\frac{h}{4}\downarrow)_{1}\cdots(\uparrow+\frac{h}{2}\downarrow)_{L-1}}_{x}.$
(S4)

In a similar way, we can derive the other perturbed ground state,

$\displaystyle\ket{II^{\prime}}=\ket{(\downarrow+\frac{h}{2}\uparrow)_{0}(\downarrow+\frac{h}{4}\uparrow)_{1}\cdots(\downarrow+\frac{h}{2}\uparrow)_{L-1}}_{x}.$
(S5)

## C: Matrix forms for the operators in the domain-wall picture

We will derive here the matrix forms of the $\sigma^{x}$ operators in the
ground state and the single domain-wall sector. We start with the
$\sigma^{x}_{0}$ operator acting on the perturbed ground state
$\ket{I^{\prime}}$, we will always make sure to consider processes which lie
in low-energy sector and to work only up to $O(h)$.

$\displaystyle\sigma^{x}_{0}\ket{I^{\prime}}=\ket{(\uparrow-\frac{h}{2}\downarrow)_{0}(\uparrow+\frac{h}{4}\downarrow)_{1}\cdots(\uparrow+\frac{h}{2}\downarrow)_{L-1}}_{x}=\ket{I^{\prime}}-h\ket{0+1/2,I}.$
(S6)

We now consider the perturbed domain-wall state $\ket{0+1/2,I}$ up to an
$O(h)$ correction. This can be written compactly as

$\displaystyle\ket{0+1/2,I^{\prime}}=\ket{(\downarrow-\frac{h}{2}\uparrow)_{0}(\uparrow)_{1}(\uparrow+\frac{h}{4}\downarrow)_{2}\cdots(\uparrow+\frac{h}{2}\downarrow)_{L-1}}_{x}.$
(S7)

Then the action of $\sigma^{x}_{0}$ on this state yields

$\displaystyle\sigma^{x}_{0}$
$\displaystyle\ket{0+1/2,I^{\prime}}=-\ket{(\downarrow)_{0}(\uparrow)_{1}(\uparrow+\frac{h}{4}\downarrow)_{2}\cdots(\uparrow+\frac{h}{2}\downarrow)_{L-1}}_{x}+h\ket{(\uparrow)_{0}(\uparrow+\frac{h}{4}\downarrow)_{1}\cdots(\uparrow+\frac{h}{2}\downarrow)_{L-1}}_{x}.$
(S8)

Up to order $h$ we can simply write

$\displaystyle\sigma^{x}_{0}\ket{0+1/2,I^{\prime}}=-\ket{0+1/2,I^{\prime}}+h\ket{I^{\prime}}.$
(S9)

Therefore, in this small basis $\\{\ket{I^{\prime}},\ket{0+1/2,I^{\prime}}\\}$
the matrix form of $\sigma^{x}_{0}$ is given by

$\displaystyle(X_{0})_{I}\approx\begin{pmatrix}\frac{1}{\sqrt{1+h^{2}}}&-\frac{h}{\sqrt{1+h^{2}}}\\\
-\frac{h}{\sqrt{1+h^{2}}}&-\frac{1}{\sqrt{1+h^{2}}}\\\ \end{pmatrix},$ (S9)

where we have normalized the matrix to have $X_{0}^{2}=\mathbb{I}$ as it is an
approximate representation of the unitary operator $\sigma^{x}_{0}$.
Similarly, it can be shown that the effective normalized matrix form for the
$\sigma^{x}_{0}$ operator in the basis
$\\{\ket{II^{\prime}},\ket{0+1/2,II^{\prime}}\\}$ is given by,

$\displaystyle(X_{0})_{II}\approx\begin{pmatrix}-\frac{1}{\sqrt{1+h^{2}}}&\frac{h}{\sqrt{1+h^{2}}}\\\
\frac{h}{\sqrt{1+h^{2}}}&\frac{1}{\sqrt{1+h^{2}}}\\\ \end{pmatrix}.$ (S10)

On the rest of the domain-wall states, $\sigma^{x}_{0}$ does not produce any
low-energy excitations up to $O(h)$. Therefore $\sigma^{x}_{0}$ only has
diagonal matrix elements which can be expressed as

$\displaystyle\braket{m+1/2,\alpha}{X_{0}}{m+1/2,\alpha}=(-1)^{\alpha},\text{for}~{}m>0.$
(S11)

Here, $\alpha$ is equal to $I,II$ when it appears as a label in the domain-
wall state and is equal to $1,2$ when it occurs in the power.

Combining all these, the $X_{0}$ matrix can be written as

$\displaystyle X_{0}$
$\displaystyle=\frac{1}{\sqrt{1+h^{2}}}\big{(}\outerproduct{I^{\prime}}{I^{\prime}}-h\outerproduct{I^{\prime}}{0+1/2,I}-h\outerproduct{0+1/2,I}{I^{\prime}}-\outerproduct{0+1/2,I}{0+1/2,I}\big{)}$
$\displaystyle~{}~{}~{}~{}-\frac{1}{\sqrt{1+h^{2}}}\big{(}\outerproduct{II^{\prime}}{II^{\prime}}-h\outerproduct{II^{\prime}}{0+1/2,II}-h\outerproduct{0+1/2,II}{II^{\prime}}-\outerproduct{0+1/2,II}{0+1/2,II}\big{)}$
$\displaystyle~{}~{}~{}~{}-\sum_{m=1}^{L-2}\outerproduct{m+1/2,I}{m+1/2,I}~{}+~{}\sum_{m=1}^{L-2}\outerproduct{m+1/2,II}{m+1/2,II}.$
(S12)

The matrix form of the $\sigma^{x}_{L-1}$ operator at the other end can be
derived in a similar fashion. The key thing is to note that $\sigma^{x}_{L-1}$
connects the pair of states $\ket{I^{\prime}}$ and $\ket{L-2+1/2,II}$ and also
the pair $\\{\ket{II^{\prime}},\ket{L-2+1/2,I}\\}$. For the other domain-wall
states, the matrix element of the operator is diagonal up to $O(h)$,

$\displaystyle\braket{m+1/2,\alpha}{X_{L-1}}{m+1/2,\alpha}=(-1)^{\alpha+1}~{}~{}~{}\text{for}~{}~{}~{}0<m<L-1.$
(S13)

Therefore the matrix representation of $\sigma^{x}_{L-1}$ in this low-energy
sector is given by

$\displaystyle X_{L-1}$
$\displaystyle=\frac{1}{\sqrt{1+h^{2}}}\big{(}\outerproduct{I^{\prime}}{I^{\prime}}-h\outerproduct{I^{\prime}}{L-2+1/2,II}-h\outerproduct{L-2+1/2,II}{I^{\prime}}-\outerproduct{L-2+1/2,II}{L-2+1/2,II}\big{)}$
$\displaystyle~{}~{}~{}~{}-\frac{1}{\sqrt{1+h^{2}}}\big{(}\outerproduct{II^{\prime}}{II^{\prime}}-h\outerproduct{II^{\prime}}{L-2+1/2,I}-h\outerproduct{L-2+1/2,I}{II^{\prime}}-\outerproduct{L-2+1/2,I}{L-2+1/2,I}\big{)}$
$\displaystyle~{}~{}~{}~{}+\sum_{m=1}^{L-3}\outerproduct{m+1/2,I}{m+1/2,I}~{}-~{}\sum_{m=1}^{L-3}\outerproduct{m+1/2,II}{m+1/2,II}.$
(S14)

The spin operators acting at sites other than the two ends do not produce or
destroy a single domain wall. They can connect the ground state to states with
two domain walls, but this is a higher order process and is therefore not
considered in our analysis. Therefore, considering only the lowest energy
excitations, the matrix elements of $\sigma^{x}_{l}$ for $l=1,\cdots,L-2$ are
all diagonal. Hence they can be compactly represented as

$\displaystyle X_{l}=$
$\displaystyle\outerproduct{I^{\prime}}{I^{\prime}}-\outerproduct{II^{\prime}}{II^{\prime}}~{}+~{}\sum_{m=0}^{L-2}(\theta(l-m)+\theta(n-l+1))\outerproduct{m+1/2,I}{m+1/2,I}$
$\displaystyle~{}~{}-~{}\sum_{m=0}^{L-2}(\theta(l-m)+\theta(n-l+1))\outerproduct{m+1/2,II}{m+1/2,II}.$
(S15)

## D: Two-body scattering phase shift from the Bethe ansatz

We consider the Hamiltonian

$H~{}=~{}\sum_{n=-\infty}^{\infty}~{}[-~{}J_{x}~{}\sigma^{x}_{n}\sigma^{x}_{n+1}~{}-~{}h~{}\sigma^{z}_{n}~{}+~{}J_{z}~{}\sigma^{z}_{n}\sigma^{z}_{n+1}],$
(S16)

where we assume for simplicity that the system is infinitely long. For $h\gg
J_{x},~{}J_{z}$, the ground state has all spins pointing up in the
$\sigma^{z}$ basis. Let us call this the vacuum state $|{\rm vac}\rangle$. The
lowest excited states are given by single spin-flip states where the spin at
site $n$ points down while all the other spins point up. We will denote this
state as $|n\rangle$, and it is related to the vacuum state as
$|n\rangle=c_{n}^{\dagger}|{\rm vac}\rangle$. Here $c_{n}$ and
$c_{n}^{\dagger}$ are annihilation and creation operators for hard-core
bosons, since $(c_{n}^{\dagger})^{2}|{\rm vac}\rangle=0$. We note that
$\sigma_{n}^{z}=1-2c_{n}^{\dagger}c_{n}$. In terms of these operators, the
Hamiltonian can be written as

$H_{c}~{}=~{}\sum_{n}~{}[2h~{}c_{n}^{\dagger}c_{n}~{}-~{}J_{x}~{}(c_{n}^{\dagger}c_{n+1}~{}+~{}c_{n+1}^{\dagger}c_{n})~{}+~{}4J_{z}~{}c_{n}^{\dagger}c_{n}c_{n+1}^{\dagger}c_{n+1}^{\dagger}],$
(S17)

where we have ignored a constant equal to the ground state energy. We see that
there is an interaction between spin-flip states on neighboring sites whose
strength is given by $4J_{z}$.

The first excited states consist of single spin-flips. Considering a momentum
eigenstate

$|k\rangle~{}=~{}\sum_{n}~{}e^{ikn}~{}|n\rangle,$ (S18)

we find from Eq. (S16) that the energy-momentum dispersion is
$E_{k}=2h-2J_{x}\cos k$, where $k$ takes values in the range $[-\pi,\pi]$.

We now consider states consisting of two spin-flips. Defining such states as
$|n_{1},n_{2}\rangle=c_{n_{1}}^{\dagger}c_{n_{2}}^{\dagger}|{\rm vac}\rangle$,
where $n_{1}<n_{2}$, we consider a state with a wave function given by the
Bethe ansatz

$|k_{1},k_{2}\rangle~{}=~{}\sum_{n_{1}<n_{2}}~{}e^{i(k_{1}n_{1}+k_{2}n_{2})}+e^{i(\theta+k_{1}n_{2}+k_{2}n_{1})}~{}|n_{1},n_{2}\rangle.$
(S19)

The momentum of this state is $k_{1}+k_{2}$ (since the wave function of the
state $|n_{1}+1,n_{2}+1\rangle$ is $e^{i(k_{1}+k_{2})}$ times the wave
function of $|n_{1},n_{2}\rangle$), and the energy is

$E_{k_{1},k_{2}}~{}=~{}4h~{}-~{}2J_{x}~{}(\cos k_{1}~{}+~{}\cos k_{2}).$ (S20)

The phase shift $\theta$ appearing in Eq. (S19) can be found in terms of
$k_{1},~{}k_{2}$ by demanding that
$H_{c}|k_{1},k_{2}\rangle=E_{k_{1},k_{2}}|k_{1},k_{2}\rangle$, and equating
the coefficients of each real space state $|n_{1},n_{2}\rangle$ on the two
sides of this equation. We discover that

$e^{i\theta}~{}=~{}-~{}\frac{\cos(\frac{k_{1}+k_{2}}{2})~{}+~{}\frac{2J_{z}}{J_{x}}~{}e^{i(k_{2}-k_{1})/2}}{\cos(\frac{k_{1}+k_{2}}{2})~{}+~{}\frac{2J_{z}}{J_{x}}~{}e^{i(k_{1}-k_{2})/2}}.$
(S21)

Note that in the absence of interactions, $J_{z}=0$, we get $e^{i\theta}=-1$
for all values of $k_{1},~{}k_{2}$.

In the main text, we are interested in the case where the two momenta
$k_{1},~{}k_{2}$ are equal and opposite to each other. We therefore set
$k_{1}=k_{0}$ and $k_{2}=-k_{0}$, where $k_{0}$ is determined by the condition
that the single-particle group velocity $v_{k}=dE_{k}/dk$ is maximum as a
function of $k$. The phase shift is then given by

$e^{i\theta}~{}=~{}-~{}\frac{1~{}+~{}\frac{2J_{z}}{J_{x}}~{}e^{-ik_{0}}}{1~{}+~{}\frac{2J_{z}}{J_{x}}~{}e^{ik_{0}}}.$
(S22)

## E: Absence of unscrambling in $XXZ$ model and comparison with $XX$ model,
both in a transverse field

Figure 6: $F^{xx}(l,t)$ showing (a) scrambling and unscrambling for the $XX$
spin chain with parameters $J_{x}=J_{y}=1,~{}h=8,~{}J_{z}=0$, and (b) absence
of unscrambling in the presence of interactions with the parameters
$J_{x}=J_{y}=1,~{}h=8,~{}J_{z}=0.25$.

We present numerical results for $F^{xx}(l,t)$ for the non-interacting $XX$
spin-1/2 chain and the interacting but integrable $XXZ$ chain with a
transverse field. Similar to the TFIM and the interacting non-integrable model
discussed in the main text, this pair of models also show scrambling and
unscrambling without interactions and absence of unscrambling with
interactions. The plots shown in Fig. 6 correspond to the parameters, $(a)$
$J_{x}=J_{y}=1,~{}h=8,~{}J_{z}=0$, and $(b)$
$J_{x}=J_{y}=1,~{}h=8,~{}J_{z}=0.25$. We note that even such a small value of
$J_{z}$ can completely destroy unscrambling as in the case discussed in the
main text. This confirms that the absence of unscrambling is not due to non-
integrability and can be simply explained by the scattering phase shift
argument given in the main text.


# Role of rapidity choice for the impact-parameter dependent Balitsky-
Kovchegov equation

Matej Vaculciak<EMAIL_ADDRESS> Jesus Guillermo Contreras  Jan
Cepila  Faculty of Nuclear Sciences and Physical Engineering, Czech Technical
University in Prague, Czech Republic

###### Abstract

Reaching higher energies of electron-ion collisions with facilities like EIC
is expected to provide a probe of a kinematic region where the parton
densities should start to exhibit signs of saturation. This phenomenon is
theoretically implemented by the Balitsky–Kovchegov (BK) equation, which,
within the colour dipole model, describes the evolution of the dipole
scattering amplitude with respect to rapidity. There are two possible
formulations of the BK equation based on which rapidity, projectile or target,
is considered. Besides this variable, there are four more degrees of freedom,
two of which have been so far incorporated into the numerical solutions. We
present a comparative solution of the two-dimensional BK equation formulated
in both projectile and target rapidity together with their impact on
quantities to be observed at EIC such as proton structure functions.
The contribution was presented at the Hot Quarks 2022 - Workshop for young
scientists on the physics of ultrarelativistic nucleus-nucleus collisions, Dao
House, Colorado, USA, October 11-17 2022.

###### keywords:

QCD phenomenology; parton saturation; Balitsky–Kovchegov equation; low-$x$ QCD

††journal: Journal of LaTeX Templates

##  1 Introduction

Theoretical studies of the hadron structure resulted in the need to restrict
the growth of parton densities at low Bjorken-$x$, so that the Froissart bound
is not violated. Such saturation mechanism can be implemented by the non-
linear Balitsky-Kovchegov (BK) evolution equation. See for example Ref. [1].

Experimentally, deeply inelastic electron-proton scattering is used to study
the hadron structure. The leading order of this interaction is mediated by the
exchange of a virtual photon, whose emission from the electron is a simple
process to describe within quantum electrodynamics. On the other hand, the
description of the effective vertex between the proton and the virtual photon
is non-trivial and is treated within the colour dipole model [1]. Here the
virtual photon is replaced with its simplest strongly interacting Fock
state—the colour dipole—and the interaction between the dipole and the proton
is then described by the so-called dipole scattering amplitude
$N(Y,\vec{r},\vec{b})$. This object depends on the rapidity $Y$ and two two-
dimensional vectors describing the dipole size $(\vec{r})$ and its transverse
position with respect to the proton $(\vec{b})$.

To calculate the dipole scattering amplitude, the BK equation

$\displaystyle\partial_{Y}N(\vec{r},\vec{b},Y)=\int\mathrm{d}\vec{r_{1}}K(r,r_{1},r_{2})\Big{[}$
$\displaystyle
N(\vec{r}_{1},\vec{b}_{1},Y)+N(\vec{r}_{2},\vec{b}_{2},Y)-N(\vec{r},\vec{b},Y)$
$\displaystyle-N(\vec{r}_{1},\vec{b}_{1},Y)N(\vec{r}_{2},\vec{b}_{2},Y)\Big{]},$
(1)

is used. It is an integro–differential equation and in the current models, it
is formulated in terms of the rapidity of the projectile (the virtual photon),
defined as

$\displaystyle Y=\ln{\frac{x_{0}}{x}},$ (2)

with $x$ being the Bjorken-$x$ and $x_{0}$ a parameter defining the evolution
starting point.

##  2 BK equation in projectile rapidity

Due to the complexity of the BK equation (1), only numerical solutions have
been obtained. To date a solution incorporating the dependence on all four
integration variables $\vec{r},\vec{b}$ (the 4D solution) is not yet known. So
far, the state of the art is to find 2D solutions depending on the norms of
the dipole size and the impact parameter $N(Y,r,b)$; results have been
presented in Ref. [2] exhibiting a reasonable agreement with experimental
data.

An example of the 2D solution is given in the left part of Figure 1, wehre the
BK equation was solved using the collinearly improved kernel

$\displaystyle
K(\vec{r},\vec{r}_{1},\vec{r}_{2})=\frac{\bar{\alpha}_{s}}{2\pi}\frac{r^{2}}{r_{1}^{2}r_{2}^{2}}\bigg{[}\frac{r^{2}}{\min\\{r_{1}^{2},r_{2}^{2}\\}}\bigg{]}^{\pm\alpha_{s}A_{1}}\frac{J_{1}\left(2\sqrt{\bar{\alpha}_{s}\ln{\frac{r_{1}^{2}}{r^{2}}}\ln{\frac{r_{2}^{2}}{r^{2}}}}\right)}{\sqrt{\bar{\alpha}_{s}\ln{\frac{r_{1}^{2}}{r^{2}}}\ln{\frac{r_{2}^{2}}{r^{2}}}}}.$
(3)

Here $\bar{\alpha}_{s}=\frac{N_{C}}{\pi}\alpha_{s}$ with
$\alpha_{s}=\alpha_{s}(\min\\{r,r_{1},r_{2}\\})$ being the running strong
coupling constant as described in [2] and $N_{C}=3$ being the number of QCD
colours. The constant $A_{1}=\frac{11}{12}$ and $J_{1}$ is the Bessel
function.

The initial condition for this solution was proposed in [2] and reads

$\displaystyle
N(Y=0,r,b)=1-\exp{\left[-\frac{Q_{s}^{2}}{4}r^{2}\exp{\left(-\frac{b^{2}}{2B}-\frac{r^{2}}{8B}\right)}\right]},$
(4)

with the process scale $Q_{s}^{2}=0.496\mathrm{~{}GeV}^{2}$ and parameter
$B=3.2258\mathrm{~{}GeV}^{-2}$.

Figure 1:  The dipole scattering amplitude evolved to rapidity $Y=\eta=5$
using the BK equation in projectile rapidity $Y$ with the collinearly improved
kernel in Eq. (3) (left) and the BK equation in target rapidity $\eta$ with
the collinearly improved kernel in Eq. (7) (right).

Figure 2:  One-dimensional cuts of the dipole scattering amplitude at fixed
impact parameter $b=1\mathrm{GeV}^{-1}$ (left) and fixed dipole size
$r=1\mathrm{GeV}^{-1}$ (right) at initial condition (black curves) and
subsequent steps of evolution. The evolution was driven by the collinearly
improved kernel in Eq. (7) for the target rapidity (full curves) and in Eq.
(3) for the projectile rapidity (dashed curves).

##  3 BK equation in target rapidity

Recently, it was pointed out (see Ref. [3]) that a slightly different
formulation of the BK equation should better describe the nature of the
interaction. Namely, there are two modifications that should be made.

First, the BK equation should be reformulated in terms of the rapidity of the
proton (target rapidity), defined as

$\displaystyle\eta=\ln{\frac{x_{0}}{x}},$ (5)

to take the form

$\displaystyle\partial_{\eta}N(\vec{r},\vec{b},\eta)=\int\mathrm{d}\vec{r}_{1}K(r,r_{1},r_{2})\Big{[}$
$\displaystyle
N(\vec{r}_{1},\vec{b}_{1},\eta_{1})+N(\vec{r}_{2},\vec{b}_{2},\eta_{2})-N(\vec{r},\vec{b},\eta)$
$\displaystyle-N(\vec{r}_{1},\vec{b}_{1},\eta_{1})N(\vec{r}_{2},\vec{b}_{2},\eta_{2})\Big{]},$
(6)

with the rapidities $\eta_{j}=\eta-\max\\{0,\ln{\frac{r^{2}}{r^{2}_{j}}}\\}$.
The collinearly improved kernel gets modified to

$\displaystyle
K(\vec{r},\vec{r}_{1},\vec{r}_{2})=\frac{\bar{\alpha}_{s}}{2\pi}\frac{r^{2}}{r_{1}^{2}r_{2}^{2}}\bigg{[}\frac{r^{2}}{\min\\{r_{1}^{2},r_{2}^{2}\\}}\bigg{]}^{\pm\alpha_{s}A_{1}}.$
(7)

Second, the projectile rapidity needs to be redefined as

$\displaystyle Y=\eta+\ln{\frac{Q^{2}}{Q_{s}^{2}}},$ (8)

where $Q^{2}$ is the virtuality and $Q_{s}^{2}$ is the scale of the process.

This formulation would then correspond better to the previous experience with
solving the BFKL equation and should exhibit better behaviour in terms of both
physics and numerics [3]. To compare both formulations, a solution of the
target rapidity BK equation, given by Eq. (6), is shown in the right plot of
Figure 1. For a better quantitative comparison, one-dimensional cuts of the
dipole scattering amplitude are shown in Figure 2 at a fixed impact parameter
$b$ (left) and a fixed dipole size $r$ (right).

##  4 Effect on observables

To explore the effect the rapidity change has on the final results of the
model, predictions of the electron-proton reduced cross section,

$\displaystyle\sigma_{r}(Q^{2},x,y)=F_{2}(Q^{2},x)-\frac{y^{2}}{1+(1-y)^{2}}F_{L}(Q^{2},x),$
(9)

are shown for both projectile and target rapidity formulations of the BK
equation in comparison with experimental data in Figure 3. The functions
$F_{2}(Q^{2},x)$ and $F_{L}(Q^{2},x)$ in Eq. (9) are the structure functions
and $y$ is the inelasticity.

Figure 3: Reduced cross section of the electron-proton scattering predicted by
models using both target (full) and projectile (dashed) rapidity BK equations
compared with data from HERA [4, 5]. Plots are shown at various fixed values
of virtuality $Q^{2}$.

As shown in Figure 3, the transition from projectile to target rapidity in the
formulation of the BK equation seems not to present any crucial numerical
difference at the level of observable quantities. The fact that the target
rapidity model exhibits overall a slightly worse data description comes
naturally as the same numerical parameters ($Q_{s}^{2},B$, etc, see [2]) are
used in both models, although they were fitted for the projectile rapidity
one.

At higher virtualities (e.g. $Q^{2}=60\mathrm{~{}GeV}^{2}$), the slope of the
full (target rapidity) curves even seems to follow the data better, giving a
good promise for future target rapidity BK studies.

##  5 Summary

The parton-saturation-implementing BK evolution equation was solved in two
different formulations: in terms of the target and the projectile rapidity.
The resulting evolution of the dipole scattering amplitude is shown for both
cases as two-dimensional graphs in Figure 1 with one-dimensional cuts in
Figure 2 for better quantitative comparison.

Both cases were used to calculate predictions of observables and an example of
the reduced electron-proton cross section is shown in Figure 3 to provide a
qualitative comparison. The transition from projectile to target rapidity does
not seem to present any crucial numerical difficulties and the data
description is satisfactory considering that for a clear comparison the same
model parameters were used in both cases, although they were fitted for the
projectile rapidity case.

## Acknowledgements

We thank Dagmar Bendova for useful discussion. The work has been partially
supported by the grant 22-27262S of the Czech Science Foundation (GACR). The
work was partially supported from European Regional Development Fund-Project
”Center of Advanced Applied Science” No. CZ.02.1.01/0.0/0.0/16-019/0000778.

## References

  * [1] Y. V. Kovchegov, E. Levin, Quantum Chromodynamics at High Energy, Vol. 33 of Cambridge Monographs on Particle Physics, Nuclear Physics and Cosmology (33), Cambridge University Press, 2022.  doi:10.1017/9781009291446.
  * [2] J. Cepila, J. G. Contreras, M. Matas, Collinearly improved impact-parameter dependent Balitsky-Kovchegov evolution, PoS DIS2019 (2019) 255.  doi:10.22323/1.352.0255.
  * [3] B. Ducloué, E. Iancu, A. H. Mueller, G. Soyez, D. N. Triantafyllopoulos, Non-linear evolution in QCD at high-energy beyond leading order, JHEP 04 (2019) 081.  arXiv:1902.06637, doi:10.1007/JHEP04(2019)081.
  * [4] H. Abramowicz, et al., Combination of measurements of inclusive deep inelastic ${e^{\pm}p}$ scattering cross sections and QCD analysis of HERA data, Eur. Phys. J. C 75 (12) (2015) 580.  arXiv:1506.06042, doi:10.1140/epjc/s10052-015-3710-4.
  * [5] S. Habib, Combined measurement and QCD analysis of the inclusive e+- p scattering cross sections at HERA, PoS DIS2010 (2010) 035.  doi:10.22323/1.106.0035.