††thanks<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
# The impact of inter-city mobility on urban welfare

Sayat Mimar  Department of Physics & Astronomy, University of Rochester,
Rochester, NY 14607, USA  David Soriano-Paños  GOTHAM Lab‚ Department of
Condensed Matter Physics and Institute for Biocomputation and Physics of
Complex Systems (BIFI), University of Zaragoza, E-50009 Zaragoza, Spain.  Alec
Kirkley  Department of Physics, University of Michigan, Ann Arbor, MI 48109,
USA  School of Data Science, City University of Hong Kong, Hong Kong  Hugo
Barbosa  Department of Computer Science, University of Exeter, Exeter, UK
Adam Sadilek  Google Inc., 1600 Amphitheatre Parkway, Mountain View, CA,
94043, USA  Alex Arenas  Department d’Enginyeria Informática i Matemàtiques,
Universitat Rovira i Virgili, 43007 Tarragona (Spain).  J. Gómez-Gardeñes
GOTHAM Lab‚ Department of Condensed Matter Physics and Institute for
Biocomputation and Physics of Complex Systems (BIFI), University of Zaragoza,
E-50009 Zaragoza, Spain.  Gourab Ghoshal  Department of Physics & Astronomy,
University of Rochester, Rochester, NY 14607, USA

###### Abstract

While much effort has been devoted to understand the role of intra-urban
characteristics on sustainability and growth, much remains to be understood
about the effect of inter-urban interactions and the role cities have in
determining each other’s urban welfare. Here we consider a global mobility
network of population flows between cities as a proxy for the communication
between these regions, and analyze how these flows impact socioeconomic
indicators that measure economic success. We use several measures of
centrality to rank cities according to their importance in the mobility
network, finding PageRank to be the most effective measure for reflecting
these prosperity indicators. Our analysis reveals that the characterization of
the welfare of cities based on mobility information hinges on their
corresponding development stage. Namely, while network-based predictions of
welfare correlate well with economic indicators in mature cities, for
developing urban areas additional information about the prosperity of their
mobility neighborhood is needed. For these developing cities, those that are
connected to sets of mature cities show markedly better socio-economic
indicators than those connected to less mature cities. We develop a simple
generative model for the allocation of population flows out of a city that
balances the costs and benefits of interaction with other cities that are
successful, finding that it provides a strong fit to the flows observed in the
global mobility network and highlights the differences in flow patterns
between developed and developing urban regions. Our results hint towards the
importance of leveraging inter-urban connections in service of urban
development and welfare.

##  I Introduction

Given the recent trend of rapid urbanization wherein the majority of the
global population now resides in urban centers DESAUN2018 ; urbanization ,
cities are at the center of innovation and technological advancement
berry2008urbanization ; Le2012 . The current relevance of cities has fueled
the birth of the so-called _science of cities_ aimed at uncovering the
physical and structural features driving their growth and function Batty2013 ;
Pan2013 ; Barthelemy2016 . Among the different topics addressed by this
discipline, one recurrent question has been that of quantifying how human
mobility shapes urban dynamics and provides reliable information on different
socioeconomic indicators. Along this line, recent studies have used intra-city
mobility flows to understand spatial city organization and its connection to
urban characteristics such as facilities, jobs and services that are crucial
for city livability and sustainability Bassolas2019 ; Lee2017 . Other works
have investigated accessibility in urban systems considering city topology and
infrastructure that mediate interactions and activities of inhabitants
Shelton2015 ; Xu2018 ; Barbosa_2021 . More generally, mobility flows have been
used to study the structure of urban areas and dynamics taking place within
them Louail2015 ; Roth2011 ; Barbosa2018 ; Kirkley2018 ; verbavatz2020growth .

The welfare of a city is just one example of the emergence of success within a
system composed of a large set of similar elements among which there are
relations of competition and cooperation. The study of the mechanisms driving
the emergence of cases of success has been recently addressed in a variety of
fields. For example, luck and randomness have been found to be crucial
features behind the prosperity of scientific Wang2013 ; Sinatra2016  or
creative careers Janosov2020 . Apart from the impact of intrinsically
stochastic events, the roots of individual success in different disciplines
has been addressed through the lens of network science. Specifically, in
Fraiberger2018 , the authors capture institutional prestige by studying a co-
exhibition network where nodes represent art galleries and the links are
determined by the movement of artists among institutions. Similarly, in a
recent work Bonaventura2020 , the network of professional relationships of
start-ups is built with the time-varying flows of employees among world-wide
companies. This network-based approach has shown to have predictive power of
the future success of companies. Finally, in Bonaventura2019 , the authors
construct a Workforce Mobility Network among metropolitan areas in the US and
predict diverse socioeconomic outputs of urban areas such as number of New
Patents, the number of R$\&$D establishments, and Total Wages.

“What makes a city successful?" is an intriguing and complicated question as
there are multiple factors shaping the success of urban systems. Here we aim
at addressing this question by leveraging the close relationship between human
mobility and the functioning of cities. To this end, we propose a network
science approach to shed light on the heterogeneous distribution of success,
as measured by various socioeconomic indicators, observed among cities
worldwide. We use anonymous and aggregated inter-city flows to construct a
global mobility network between cities, with the aim of capturing this success
using the observed flow patterns. We start by ranking cities with respect to
various network centrality measures—in particular connectivity, PageRank, and
eigenvector centrality—which capture not only the strength of connections but
also their quality, and compare these centralities with socioeconomic
indicators in $268$ global metropolitan areas. We demonstrate that the welfare
of an urban area is largely influenced by the welfare of its neighborhood,
that is, those cities with which it has the highest volume of inter-urban
population flows. This impact becomes even more visible when grouping cities
into various categories according to geographic region or development level,
to distinguish flow profiles of cities in different groups. We observe that
the international connections for a city in a less developed category (or
geographic region) contribute significantly to its prosperity. Finally, we
build a simple model for the outgoing flows from a city, finding that the
success of a city is a major element in attracting flows from others, and that
less developed cities place greater priority on directing flows to highly
successful cities.

##  II Results

###  II.1 Data Description

####  II.1.1 Mobility Data

The Google Aggregated Mobility Research Dataset contains anonymized mobility
flows aggregated over users who have turned on the Location History setting,
which is off by default. This is similar to the data used to show how busy
certain types of places are in Google Maps, which helps to identify when a
local business tends to be the most crowded. The dataset aggregates flows of
people from region to region weekly.

To produce this dataset, machine learning methods are applied to logs data to
automatically segment it into semantic trips
111https://www.nature.com/articles/s41467-019-12809-y. To provide strong
privacy guarantees, all trips are anonymized and aggregated using a
differentially private mechanism 222https://research.google/pubs/pub48778/ to
aggregate flows over time
333https://policies.google.com/technologies/anonymization. The analysis in
this paper is done on the resulting heavily aggregated and differentially
private data. No individual user data was ever manually inspected, only
heavily aggregated flows of large populations were handled.

All anonymized trips are processed in aggregate to extract their origin and
destination location and time. For example, if $n$ unique users traveled from
location $a$ to location $b$ within week $w$, the corresponding cell $(a,b,w)$
in the mobility tensor would have a value of $n\pm\eta$ where $\eta$ is noise
drawn from a Laplace distribution with mean $0$ and scale $1/0.66$. We then
remove all metrics for which the noisy number of users is lower than $100$,
following the process described in 444https://research.google/pubs/pub48778/.
This automated Laplace mechanism yields a ($\epsilon$, $\delta$)-differential
privacy guarantee of $\epsilon$ = 0.66 and $\delta$ =$2.1\times 10^{-29}$ per
metric. The parameter $\epsilon$ controls the noise intensity in terms of its
variance, while $\delta$ represents the deviation from pure
$\epsilon$-privacy. The closer each value is to zero, the stronger the privacy
guarantees. The resulting mobility flows are encoded in an origin-destination
matrix ${\bf T}$ whose elements $T_{ij}$ represent the flows from location $i$
to $j$ aggregated over the full year.

####  II.1.2 Socioeconomic indicators

To quantify the economic success of cities, we collected data of city-level
socioeconomic variables from the company Jones Lang LaSalle IP, Inc. (JLL) for
the year 2018 jll-data . The report published by JLL gives the GDP, Total Real
Estate Investment, and a Cross-Border Real Estate Investment and Commercial
Attraction Index (BHI) for $300$ cities worldwide. In particular, the BHI is a
composite measure that accounts for key real estate indicators (investment
volumes and commercial real estate stock), as well as socio-economic and
business indicators such as economic output, population, corporate presence,
and air connectivity.

Besides these continuous variables, there are categorical variables in the
dataset that divide cities into different subsets based on their level of
development and geographical region. The indicated levels of development (in
increasing order) are Early Growth, Developing, Transitional, and Mature,
which are assigned following information concerning the city’s real estate
liquidity, the depth of its corporate occupier base, and the quality and range
of its commercial stock. The geographical sub-regions indicated are North
America, LATAM $\&$ Caribbean, Asia, Australasia, Western Europe, CEE/CIS,
MENA, and Sub-Saharan Africa. For our analysis we consider 268 metropolitan
areas, excluding China for which we do not have mobility data. (See Figs. S1,
S2 and Table S1 for more details about the urban areas used in this study.)

###  II.2 Relating welfare and PageRank centrality

Figure 1: Connecting network position and welfare. Weights $W_{ij}$, of the
global mobility network for a the raw mobility flows ($\beta=0$ in Eq. (1))
and c when incorporating the effect of distance in the weights ($\beta=2.19$),
shown for North American cities. Node sizes are proportional to their PageRank
in both maps. By including distance in the flows, we give stronger weight to
long-range trips. b BHI vs PageRank for $\beta=0$, with a Spearman correlation
coefficient $\rho_{s}=0.53$. d BHI vs PageRank for $\beta=2.19$, yielding
$\rho_{s}=0.77$.

While we have metadata for 268 cities, for the purposes of our analysis, we
analyze the mobility network of $N=1,774$ nodes representing global
metropolitan areas with population $P>100,000$ and $E=39,868$ directed edges.
To incorporate, in a tunable way, the contribution of long-distance travel to
economic activity, we define the edge weights of the mobility network as:

$W_{ij}=T_{ij}\times d_{ij}^{\beta},$ (1)

where $\beta\geq 0$. In this expression $T_{ij}$ is the total flow from city
$i$ to $j$ and the $d_{ij}$ is the Euclidean distance between them, which
allows us to account for the effect of distance as a cost in population
movement Simini2012 ; Barthelemy2011 . Incorporating this allows for including
the fact that long-range connections indicate greater levels of implied
connectivity between the cities, given the associated cost of travel. The
optimal value of the exponent $\beta$ is determined later. The effect of
distance on the flow-weights is illustrated in Fig. 1 where panel a shows
mobility flows only ($\beta=0$), whereas in panel b we show an illustrative
example for $\beta>0$. As the figure shows, incorporating the effect of
distance makes long-range flows more prominent; colors indicate magnitude of
flow, increasing from yellow to red, and node sizes correspond to PageRank.
(Maps generated using Shapely 555https://pypi.org/project/Shapely/ and
GeoPandas 666https://geopandas.org/ packages in Python.)

Next we rank cities according to their importance based on their position in
the mobility network. An ideal metric for that is the PageRank centrality, a
network-based diffusion algorithm, used by Google and other web search engines
to estimate the importance or quality of web pages. The algorithm takes into
account both the number of connections as well as the importance of the
neighbors a node connects to Page1998 ; Ghoshal2011 . For a weighted directed
network, the PageRank of a node $i$ is computed as 10.5555/998669.998911 :

$PR_{i}=\frac{(1-\alpha)}{N}+\alpha\sum_{j}\frac{{W}_{ji}PR_{j}}{s^{\mathrm{out}}_{j}},$
(2)

where $s^{\mathrm{out}}_{j}$ is the total strength of outgoing links from node
$j$ and $\alpha$ is a “reset" parameter in the range $[0,1]$ (we use
$\alpha=0.85$). In addition to webpages, the metric has been successfully
employed to rank scientists based on their citation patterns Walker2007 ;
Chen2007 , disease-causing genes based on protein-protein interactions
Chen_2009 , roads or streets in terms of traffic Jiang_2008 , ecological
species based on their position in the food web Allesina2009 , highlight
cancer genes in proteomic data Gabor_2011  among other applications.

We start our analysis by ranking cities in the mobility network by their value
of PageRank and plot it against one of the composite socioeconomic indicators,
the BHI. The strength of association is measured by the Spearman’s correlation
coefficient $\rho_{s}$ that measures the strength and direction of association
between two ranked variables. In Fig. 1c we plot the case for $\beta=0$
finding a monotonic relationship with $\rho_{s}=0.53$. Including distance in
the edge-weights reveals an increase in the association between the two
variables for any non-zero $\beta$. In Fig. S3 panels a and b we plot
$\rho_{s}$ as well as the Pearson’s correlation coefficient $\rho_{p}$ between
$\log(\text{BHI})$ and $\log(PR)$ as a function of $\beta$, finding a peak
value for both correlation coefficients at $\beta=2.19$ ($\rho_{s}$) and
$\beta=1.88$ ($\rho_{p}$). In Fig. 1 panel d we plot the scatter plot for
PageRank and BHI for $\beta=2.19$, indicating a much stronger correlation
($\rho_{s}=0.77$). In Fig. S4 we show the connection of PageRank with other
socioeconomic indicators (total real estate investment, GDP, and cross-border
real estate investment) for a variety of choices of the edge-weights: the raw
flow $T_{ij}$, the euclidean distance $d_{ij}$ as well as Eq. (1). The results
indicate that (i) PageRank is correlated with all socioeconomic measures (the
strongest being BHI) and (ii) this association is enhanced when both the flow
and distance is incorporated in the edge-weights. We also show the results for
two other centrality measures (weighted degree and eigenvector centrality).
Taken together, it appears that both the position of a city in the global
mobility network, as well as the strength and number of connections is a
reasonable predictor of its welfare.

Figure 2: Rich-club organization and core-periphery structure of the mobility
network. The richness parameter $\rho(r)$ (Eq. (3)) a for the connectivity
$r=k^{out}$ and b weighted out-degree $r=s^{\text{out}}$ . The monotonically
increasing trend of $\rho(r)$ with $r$ indicates subsets of high-connectivity
cities that connect to each other more than would be expected as merely a
consequence of the distribution of links and edge-weights. c The number of
neighbors of each node, with a higher out-degree than the node itself, as a
function of the node’s out-degree. The scatter plot has a peak separating
cities into two clusters of core (in red) and peripheral (in blue) nodes. d
The distribution of BHI in the core and peripheral cities, showing that cities
in the core typically have much higher values of BHI than those in the
periphery.

Next, we dis-aggregate the cities based on their level of development and
geographical region to check whether the results are consistent across these
categories. In Fig. S5 we plot the PageRank against BHI for four levels of
development: Mature, Transitional, Developing and Early Growth. We find a
progressive decrease in the strength of association from Mature cities
($\rho_{s}=0.86$) through transitional and developing cities
($\rho_{s}=0.73,0.88$, respectively) and finally a weak correlation in early
growth cities ($\rho_{s}=0.38$). Indeed, the differences become rather stark
when looking at the other socioeconomic indicators as seen in Fig. S7. In
mature cities the PageRank is strongly correlated with all indicators, in
transitional cities the correlation with real-estate investment weakens
significantly, and moving to the other two categories the correlation is weak
with all indicators. Differences also exist when grouping cities based on
their geographical location as shown in Fig. S6. In prosperous regions (North
America, Australasia and Western Europe) the association between PageRank and
BHI is quite strong ($\rho_{s}=0.86,0.93,0.76$ respectively), in areas with
mixed levels of prosperity such as Latin America, Asia and the Middle East,
the correlations are slightly weaker ($\rho_{s}=0.72,0.86,0.73$) with the
weakest correlations being in Sub-Saharan Africa ($\rho_{s}=0.38$).

###  II.3 Effect of position in the mobility network

To check whether these observed differences based on maturity and location are
consequences of higher-order topological features of the mobility network, we
next employ the k-core decomposition, that identifies strongly connected core-
nodes and sparsely connected peripheral nodes, revealing meso-scale network
structure Zhang2015 ; Rombach2014 . Applying this analysis we find two core
clusters consisting of 49 Western European cities and 40 North American cities
as shown in Fig. S8. The rest of the clusters are much smaller (sizes between
5-20) and lie in the peripheral layers of the mobility network. The role of a
city’s position in the inner or outer-layers becomes clear when plotting
$\rho_{s}$ between PageRank and BHI as a function of the layer-number $k$
(numbered in increasing order from outer- to inner-layer) as shown in Fig. S9.
We see a monotonically decreasing trend of the strength of correlation as one
moves from the core to the peripheral layers indicating that PageRank become a
poor predictor of welfare for cities located more in the outer-layers of the
mobility network.

The existence of large core clusters concentrated in Western Europe and North
America (all Mature or Transitional cities) suggest that they form a densely
interconnected backbone of the mobility network and thus control most of human
capital and resource flow. This can be quantified by studying the weighted
rich-club organization Opsahl2008  which is defined as

$\rho(r)=\frac{\phi(r)}{\phi_{\text{null }}(r)},$ (3)

where $r$ is a _richness parameter_. Two flavors of the parameter that we
consider are the out-degree $k^{\text{out}}$ and the out-strength
$s^{\text{out}}$. Given this, $\phi(k)$ is a fraction that measures the number
of edges between nodes of degree $\geq k^{\text{out}}$ compared to the edges
they would share if all these nodes were connected to each other. The quantity
$\phi_{\text{null }}(k^{\text{out}})$ is the corresponding measure under out
degree-preserving reshuffling of links. For $s^{\text{out}}$,
$\phi_{\text{null}}(s^{\text{out}})$ is computed through both edge and weight
reshuffling. Values of $\rho(r)>1$ indicate a set of nodes that are connected
to more cities, with higher exchange of populations, than one would expect
merely as a consequence of the distribution of links and edge-weights. In Fig.
2a, we plot $\rho(k^{\text{out}})$ and in b $\rho(s^{\text{out}})$, finding
that nodes with high connectivity and strength of connections are between two
to three times more connected to each other than would be expected by random
chance. The results indicate that Mature and Transitional cities form a _rich-
club_ dominating mobility flows in the network.

Finally, while the $k$-core decomposition allows for assigning nodes to
layers, there is no clear way to define a boundary between the core and
periphery. To do so, we use a method introduced in Ma2015 : First, we sort
nodes with respect to $k^{\text{out}}$ in increasing order. Then, for each
node, we count the number of its neighbors that have out-degree higher than
itself. In Fig. 2c, we plot the sorted degrees of all nodes and their
corresponding number of connections to higher-degree nodes. There is a clear
peak in the scatter plot that allows us to distinguish 2 clusters of cities
being in the core (in red) and in the periphery (in blue) Guo2019 . The figure
indicates that the top $~{}30\%$ of nodes in terms of their out-degree are in
the core of the mobility network, consistent with the two dense cores
identified by the $k-$core method in Fig. S9. Having identified the core and
periphery we plot the distribution of BHI in the two regions of the network in
Fig. 2d finding that cities in the core are in general more prosperous than
peripheral cities.

###  II.4 Interplay between welfare and mobility neighborhood

The results thus far point towards strongly inter-connected sets of core
cities that dominate mobility flow (and therefore exchange of human resources)
that are successful in terms of socioeconomic indicators. This success can be
reasonably well predicted by their centrality, or just their topological
features in relation to the mobility network. For cities in the periphery, and
those that are not so tightly connected, their level of development cannot be
simply captured by topological aspects. In particular PageRank is a poor
measure to capture such cities’ welfare, given that it is noisy for nodes with
low connectivity and edge-weights Ghoshal2011 . Furthermore, it does not
capture an important feature, that is the maturity level of the cities from
which travelers come to a given city. For instance, it stands to reason that
for two developing cities with similar connectivity profiles, if one of them
has visitors from mature cities, and the other only from developing cities,
then the former may benefit more from the flow of intellectual and human
capital from more developed regions.

To account for this effect, we define the quantity

$\langle
S\rangle_{i}=\frac{\sum\limits_{j}W^{I}_{ji}S_{j}}{\sum\limits_{j}W^{I}_{ji}}\
,$ (4)

where $S_{j}$ denotes the success/welfare of a given city $j$ (measured
through the socioeconomic indicators) and ${\bf W^{I}}$ contains only
international flows from $j$ to $i$. We restrict ourselves to international
flows, given that in our dataset, cities within a country have by and large
the same maturity levels. The quantity $\langle S\rangle_{i}$ can then be
interpreted as the weighted average of the success of the international
visitors to city $i$. For instance if we take BHI as the measure of success,
then this number will be high if most visitors are from Mature cities, and low
if they are from Early Growth cities. Based on this we define an estimator for
a city $i$’s success

$\hat{S}_{i}=PR_{i}\times\langle S\rangle_{i}^{\gamma},$ (5)

where $PR_{i}$ is its PageRank and $\gamma$ is a tunable parameter that
measures the extent to which the level of development of a city’s neighbors
plays any role in predicting its own development. For $\gamma\simeq 0$, the
node’s network topological properties are the best indicator of its welfare,
whereas for $\gamma\geq 0$ incorporating information about the development
levels of its neighbors provides a better estimate.

Table 1: Role of mobility neighborhood for cities grouped by maturity Values of $\gamma^{opt}$ for cities based on their maturity, optimizing the Spearman’s correlation coefficient between the actual success $S_{i}$ (BHI) and the estimator $\hat{S}_{i}$ Eq. (5). Size denotes the number of cities in each category. $\rho_{s}(\gamma)$ and $\rho_{p}(\gamma)$ are the Spearman’s and Pearson’s correlation coefficients for a given $\gamma$. Note that Pearson’s correlation coefficients are computed by taking the logarithm of both $S$ and $\hat{S}$. Development-Stage | Size | $\gamma^{opt}$ | $\rho_{s}(\gamma^{opt})$ | $\rho_{p}(\gamma^{opt})$ | $\rho_{s}(0)$ | $\rho_{p}(0)$
---|---|---|---|---|---|---
[late after line= |  |  |  |  |  |

Table 1 contains the optimal value of the exponent $\gamma$ (see Figs. S10 and
S11), maximizing the Spearman correlation $\rho_{s}(\gamma)$, between the
estimator $\hat{S}_{i}$, and a city’s actual level of success $S_{i}$ as
quantified by the BHI which captures the city’s maturity level. A clear
pattern emerges, whereby we find a decreasing $\gamma$ with increasing levels
of development. The role of the average success of neighbors $\langle
S\rangle$ is the most prominent for Early growth cities, and practically non-
existent for Mature cities (see Fig. S12 for in-flow breakdown of Early Growth
and Developing cities originating from national and international cities).
Table 2 contains the same information for cities categorized by Geographical
Region. Mature cities in Western Europe and North America display the lowest
values for $\gamma$, whereas regions such as Asia, Latin America and the
Caribbean—that contain a mix of cities in terms of their development
levels—show intermediate values for $\gamma$. The highest values are seen for
the Middle East region and Central Europe that are home to fast rising cities.
Interestingly Sub-Saharan Africa shows the same trend as Asia, though it is an
outlier, as it seems neither the Pagerank nor the development levels of its
neighbors are enough to account for explaining its own levels of development
($\rho_{s}(\gamma)=0.448)$ unlike for the other regions ($\rho_{s}(\gamma)\geq
0.763$).

Table 2: Role of mobility neighborhood for cities grouped by geographic region. All quantities same as in Tab. 1.  Region | Size | $\gamma^{opt}$ | $\rho_{s}(\gamma^{opt})$ | $\rho_{p}(\gamma^{opt})$ | $\rho_{s}(0)$ | $\rho_{p}(0)$
---|---|---|---|---|---|---
[late after line= |  |  |  |  |  |

###  II.5 The influence of success on mobility flows

The enhanced ability to predict the true socioeconomic indicator of a city in
the less developed categories by incorporating the maturity level of its
neighborhood, suggests that perhaps success plays a role in how flows are
allocated. To check for this, we construct a simple model for the allocation
of the outgoing flows of a city among its destinations that accounts for both
the success of the target destinations, as well as the distance between these
targets and the source city. Suppose a city $i$ has a certain amount of total
outflow $c_{i}$, which it can allocate towards different cities $j$ within
some set of target cities $\partial_{i}$—its mobility network neighborhood. We
assume that $i$ allocates its outgoing flow $c_{i}$ to neighbors
$j\in\partial_{i}$ based on the success $S_{j}$ of city $j$, as well as the
distance $d_{ij}$, so that $\tilde{T}_{ij}(\mu,\nu)\propto
S_{j}^{\mu}d_{ij}^{\nu}$, where $\mu$ and $\nu$ are free parameters, and
$\tilde{T}_{ij}(\mu,\nu)$ is the predicted flow from $i$ to $j$ from the model
(whose dependence on $\mu$ and $\nu$ has been made explicit). More precisely,
we have

$\displaystyle\tilde{T}_{ij}(\mu,\nu)=\frac{S_{j}^{\mu}d_{ij}^{\nu}}{\sum_{j\in\partial_{i}}S_{j}^{\mu}d_{ij}^{\nu}}\times
c_{i}.$ (6)

If the success $S_{j}$ makes travel to city $j$ more attractive to city $i$,
then one would expect $\mu\geq 0$. On the other hand, if the cities are far
apart in distance $d_{ij}$, then it would make it more costly to travel from
$i$ to $j$, and in that case $\nu\leq 0$. The ratio $|\frac{\mu}{\nu}|$ then
determines the relative importance of these factors in determining
$\tilde{T}_{ij}$. Given that $\tilde{T}_{ij}(\mu,\nu)$ is invariant to the
scale of $S_{j}$ and $d_{ij}$, the relative importance of these measures can
be compared directly using this ratio regardless of units.

The outflow $c_{i}$ can be inferred from the raw (unweighted) origin-
destination matrix $\bm{T}$ using the expression
$c_{i}=\sum_{j\in\partial_{i}}T_{ij}$, where the neighborhood $\partial_{i}$
is restricted to the subset of cities $j$ that $i$ connects to that also have
BHI metadata $S_{j}$. We then fit Eq. 6 to the true observed flows $T_{ij}$ to
find the exponents $\hat{\mu}$ and $\hat{\nu}$ that optimize the Pearson
correlation between the true and predicted flows $T_{ij}$ and $\tilde{T}_{ij}$
for connected city pairs $i,j$. This optimization is nontrivial but can be
done approximately with a variety of methods, and here we choose a basin
hopping algorithm wales2003energy . Higher values of $|\hat{\mu}/\hat{\nu}|$
indicate that cities place high relative importance on success when allocating
their outflows, and low values of $|\hat{\mu}/\hat{\nu}|$ indicate that city
outflows are much more strongly influenced by distance than by success. To
determine whether the city subgroup (development stage or geographical region)
affects the influence of success on outgoing flows, we identify separate
exponents $\\{\hat{\mu},\hat{\nu}\\}$ for each subgroup by optimizing the
Pearson correlation between $\tilde{T}_{ij}$ and $T_{ij}$ for all flows
leaving cities $i$ in the subgroup.

We also determine whether or not the success covariate $S_{j}$ (BHI)
significantly improves our predictive model of outflows over a baseline model,
where only the distance between cities is considered. In this way, one can
check if cities tend to place substantial importance on success when choosing
how to allocate their flows. To do this, we identify the exponent $\nu_{0}$
that optimizes the correlation between the true and predicted flows while
ignoring the success parameter ($\mu=0$). Since the model with $\mu$ has one
more free parameter than the model with $\mu=0$, the fit between the true and
predicted flows will always be better according to the Pearson correlation
$\rho_{p}$. Thus for a fair comparison, the more complex model needs to be
penalized using some model selection criteria. Here we opt for both the Akaike
Information Criterion (AIC) and the Bayesian Information Criterion (BIC)
hurvich1990impact , which are derived from the log-likelihood of the model
fit, as well as the number of data points in the sample. Table S2 includes all
model fit results for each category of cities. Applying both penalties
indicate that the model performs significantly better with the inclusion of
$\mu$ for all subgroups. We find that $\hat{\mu}>0$ and $\hat{\nu}<0$ for all
city subgroups, indicating that cities allocate more flow to cities with
higher BHI, and allocate less flow to cities that are farther away, though the
extent to which this happens depends on the level of development, with the
effect most pronounced in Early Growth and Developing cities, and least so in
Mature and Transitional cities.

Figure 3: Interplay between benefit (success) and cost (distance)a
$|\hat{\mu}/\hat{\nu}|$ and coefficient of determination $r^{2}$ for all city
subgroups considered, using the model in Eq. 6 to fit the outflows of all
cities in the subgroup. Black circles indicate development subgroups, and red
triangles indicate geographical subgroups, while the size of the marker is
proportional to the number of cities in the subgroup. Cities with $r^{2}>0.8$
and $|\hat{\mu}/\hat{\nu}|>0.1$ are highlighted as having high relative flow
dependence on BHI due to high model correlations and contributions from
$S_{j}^{\mu}$. All fits demonstrated significant improvement through the
inclusion of the $S_{j}^{\mu}$ term, as evidenced by the corresponding AIC and
BIC values (Table. LABEL:sitab:modelsummary in SI). b Model schematic, showing
the variables involved in Eq. 6 around a central node $i$ for which outflows
$\tilde{T}_{ij}$ are being predicted. c Example predicted and true outflows
from cities in the Australasia subgroup, showing significant improvement from
including the BHI in the model fit. Only flows with $T_{ij}>100$ are
displayed, for clearer visualization, and the line of equality is shown for
reference.

In Fig. 3(a), the relative magnitude $|\hat{\mu}/\hat{\nu}|$ of the
coefficients $\hat{\mu}$ and $\hat{\nu}$ is plotted along with the coefficient
of determination $r^{2}$ for each city subcategory. As discussed, the ratio
$|\hat{\mu}/\hat{\nu}|$ quantifies the relative importance of success $S_{j}$
versus distance $d_{ij}$ in determining how a city $i$ allocates its flow
$c_{i}$. According to the model fits, in general developing cities place more
importance on success than more developed cities, as indicated by the shaded
red box. More specifically, cities in the Developing, Early Growth, and
Transitional subgroup place higher relative priority on the success of the
cities they allocate flows to as compared to cities in the Mature subgroup. We
also see a geographic dependence with North America and (Western and Eastern)
Europe placing relatively lower importance on BHI and having weaker model fits
in general. This suggests that other factors are more important in determining
the corresponding flows (perhaps consistent with these regions having many
developed cities). As insets, we illustrate the variables used in the flow
model (Fig. 3(b)), as well as show an example of the improved weight
prediction after the inclusion of BHI for the Australasia subgroup (Fig.
3(c)). In this third panel, we plot the flow as predicted by Eq. 6—with
$\hat{\mu}$ and $\hat{\nu}$ as the model parameters—along the x-axis, and the
true observed flow along the y-axis. The weights condense significantly around
the line of equality when including $S_{j}$ as a co-variate, as compared to
using only $d_{ij}$.

These results indicate that cities in earlier stages of development or in more
developing regions place greater importance on their neighbors’ success when
considering their mobility outflows. These results compliment our previous
analysis by suggesting that perhaps such cities strategically allocate their
outflows in order to maximize the benefits they receive by connecting with
more successful cities. On the other hand, for developed cities outflows are
more strongly influenced by travel distance.

##  III Discussion

We have presented here, a comprehensive analysis on the role of inter-city
global mobility flows apparent in driving the welfare and success of urban
areas. At the macroscopic level, we report the existence of a tightly
connected core of cities that form a rich-club dominating mobility flow in the
network. Cities in this core tend to have higher levels of development, and
are primarily located in Western Europe and North America. For these cities,
their development-level is well predicted by their network topological
properties, in particular centrality measures such as PagerRank. On the other
hand developing and early cities are located in the peripheral layers and are
scattered across multiple regions (Asia, Eastern Europe, Latin America, Middle
East and Sub-Saharan Africa). Their level of development is partially
connected to their network centrality in some cases, and very weakly so in
others, in particular in Africa.

For those regions, whose success is weakly connected to their centrality, we
find that adding information on their mobility neighborhood (that is the
average development level of the cities they are connected to) leads to a
marked increase in correlations with their socioeconomic indicators. This
effect is less pronounced for more developed cities, whose socioeconomic
indicators are connected primarily with their network centrality. Once again
Sub-Saharan Africa is an outlier, its development being weakly connected to
network properties and the socioeconomics of their mobility neighborhood. To
check whether the connection of cities to other cities is indeed influenced by
their development-level we propose a simple model that disentangles the
relevant aspects influencing the success/welfare of urban areas. The model
assumes that the outflow of cities is distributed among destinations in
proportion to their relative benefit (maturity) and cost (distance). The
predicted outflows are compared to the empirical outflows to determine the
contribution of each individual component of the model. We find that all
cities incorporate the development level of the cities they connect to, and
are less inclined to connect to cities at great distance. However, the former
is much more pronounced in developing regions as compared to developed
regions.

Taken together, our analysis indicates that the welfare of cities and the
inter-urban mobility network are strongly correlated, but that this
correlation depends on their level of maturity. The global urban ecosystem
appears to be a combination of a well-established set of core urban areas
whose interconnections describe their performance according to network science
principles, and a subset of developing cities whose welfare is influenced by
the extent of connections to the core. While we do not have data on individual
mobility-flows the results are akin to that seen in Fraiberger2018 , where
artists climb the ladder, as it were, by strategically positioning themselves
in the network of galleries exhibiting their work.

##  IV Limitations

These results should be interpreted in light of several important limitations.
First, the Google mobility data is limited to smartphone users who have opted
in to Google Location History feature, which is off by default. These data may
not be representative of the population as whole, and may vary by location.
Importantly, these limited data are only viewed through the lens of
differential privacy algorithms, specifically designed to protect user
anonymity and obscure fine detail. Moreover, comparisons across rather than
within locations are only descriptive since these regions can differ in
substantial ways.

##  V Data Availability

The Google Aggregated Mobility Research Data-set used for this study is
available with permission from Google LLC. Links to all other data sources
have been provided in the manuscript.

Acknowledgments

We thank A. Bassolas for useful discussions on the manuscript. G.G. and S.M.
acknowledge partial support from the U. S. Army Research Office under grant
number W911NF-18-1-0421 and National Science Foundation under Grant No
2029095. A.A. acknowledges financial support from Spanish Ministerio de
Ciencia e Innovación (grant PGC2018-094754-BC21), Generalitat de Catalunya
(grant No. 2017SGR-896 and 2020PANDE00098), and Universitat Rovira i Virgili
(grant No. 2019PFR-URVB2-41). A.A. also acknowledges support from Generalitat
de Catalunya ICREA Academia, and the James S. McDonnell Foundation (grant
220020325). D.S.P. and J.G.G. acknowledge financial support from Spanish
Ministerio de Ciencia e Innovación (projects FIS2017-87519-P and
PID2020-113582GB-I00), from the Departamento de Industria e Innovación del
Gobierno de Aragón y Fondo Social Europeo (FENOL group E-19), and from
Fundación Ibercaja and Universidad de Zaragoza (grant 224220). A.K. is
supported by the National Defense Science and Engineering Graduate Fellowship
through the Department of Defense.

## References

  * (1) Department of Economic and Social Affairs, United Nations.  The speed of urbanization around the world.  _Population Facts_ 1–2 (2018).
  * (2) Un (2018). World Urbanization Prospects: the 2018 revision – key findings – united nations (ser.a), population and vital statistics report.  ttps://esa.un.org/unpd/wup/Publications/Files/WUP2018-KeyFacts.pdf.  Accessed: 2019-01-30.
  * (3) Berry, B. J.  Urbanization.  In _Urban ecology_ , 25–48 (Springer, 2008).
  * (4) Le Néchet, F.  Urban spatial structure, daily mobility and energy consumption: a study of 34 european cities.  _Cybergeo: European Journal of Geography_ (2012).
  * (5) Batty, M.  _The new science of cities_ (MIT press, 2013).
  * (6) Pan, W., Ghoshal, G., Krumme, C., Cebrian, M. & Pentland, A.  Urban characteristics attributable to density-driven tie formation.  _Nature Communications_ 4, 1–7 (2013).  eprint 1210.6070.
  * (7) Barthelemy, M.  _The structure and dynamics of cities_ (Cambridge University Press, 2016).
  * (8) Bassolas, A. _et al._ Hierarchical organization of urban mobility and its connection with city livability.  _Nature Communications_ 10, 1–10 (2019).  URL http://dx.doi.org/10.1038/s41467-019-12809-y.
  * (9) Lee, M., Barbosa, H., Youn, H., Holme, P. & Ghoshal, G.  Morphology of travel routes and the organization of cities.  _Nature Communications_ 8 (2017).  URL http://dx.doi.org/10.1038/s41467-017-02374-7.
  * (10) Shelton, T., Poorthuis, A. & Zook, M.  Social media and the city: Rethinking urban socio-spatial inequality using user-generated geographic information.  _Landscape and Urban Planning_ 142, 198–211 (2015).
  * (11) Xu, Y., Belyi, A., Bojic, I. & Ratti, C.  Human mobility and socioeconomic status: Analysis of Singapore and Boston.  _Computers, Environment and Urban Systems_ 72, 51–67 (2018).
  * (12) Barbosa, H. _et al._ Uncovering the socioeconomic facets of human mobility.  _Scientific Reports_ 11, 8616 (2021).  URL https://doi.org/10.1038/s41598-021-87407-4.
  * (13) Louail, T. _et al._ Uncovering the spatial structure of mobility networks.  _Nature Communications_ 6 (2015).  eprint 1501.05269.
  * (14) Roth, C., Kang, S. M., Batty, M. & Barthélemy, M.  Structure of urban movements: Polycentric activity and entangled hierarchical flows.  _PLoS ONE_ 6, 2–9 (2011).  eprint 1001.4915.
  * (15) Barbosa, H. _et al._ Human mobility: Models and applications.  _Physics Reports_ 734, 1–74 (2018).  URL https://doi.org/10.1016/j.physrep.2018.01.001.  eprint 1710.00004.
  * (16) Kirkley, A., Barbosa, H., Barthelemy, M. & Ghoshal, G.  From the betweenness centrality in street networks to structural invariants in random planar graphs.  _Nature Communications_ 9 (2018).  URL http://dx.doi.org/10.1038/s41467-018-04978-z.
  * (17) Verbavatz, V. & Barthelemy, M.  The growth equation of cities.  _Nature_ 587, 397–401 (2020).
  * (18) Wang, D., Song, C. & Barabási, A. L.  Quantifying long-term scientific impact.  _Science_ 342, 127–132 (2013).
  * (19) Sinatra, R., Wang, D., Deville, P., Song, C. & Barabási, A. L.  Quantifying the evolution of individual scientific impact.  _Science_ 354, aaf5239 (2016).
  * (20) Janosov, M., Battiston, F. & Sinatra, R.  Success and luck in creative careers.  _EPJ Data Science_ 9 (2020).  URL http://dx.doi.org/10.1140/epjds/s13688-020-00227-w.
  * (21) Fraiberger, S. P., Sinatra, R., Resch, M., Riedl, C. & Barabási, A.-l.  Quantifying reputation and success in art.  _Science_ 362, 825–829 (2018).  URL http://www.sciencemag.org/lookup/doi/10.1126/science.aau7224.
  * (22) Bonaventura, M. _et al._ Predicting success in the worldwide start-up network.  _Scientific Reports_ 10, 1–6 (2020).
  * (23) Bonaventura, M., Aiello, L. M., Quercia, D. & Latora, V.  Predicting Urban Innovation from the Workforce Mobility Network in US 1–12 (2019).  URL http://arxiv.org/abs/1911.00436.  eprint 1911.00436.
  * (24) https://www.nature.com/articles/s41467-019-12809-y.
  * (25) https://research.google/pubs/pub48778/.
  * (26) https://policies.google.com/technologies/anonymization.
  * (27) https://research.google/pubs/pub48778/.
  * (28) Jones Lang LaSalle IP, Inc.  https://www.us.jll.com/en/trends-and-insights/research/global/crc/city-sunburst-chart.
  * (29) Simini, F., González, M. C., Maritan, A. & Barabási, A. L.  A universal model for mobility and migration patterns.  _Nature_ 484, 96–100 (2012).
  * (30) Barthélemy, M.  Spatial networks (2011).  eprint 1010.0302.
  * (31) https://pypi.org/project/Shapely/.
  * (32) https://geopandas.org/.
  * (33) Brin, S. & Page, L.  The anatomy of a large-scale hypertextual web search engine.  _Computer Networks_ 30, 107–117 (1998).  URL http://www-db.stanford.edu/~backrub/google.html.
  * (34) Ghoshal, G. & Barabási, A. L.  Ranking stability and super-stable nodes in complex networks.  _Nature communications_ 2, 394 (2011).  URL http://www.nature.com/articles/ncomms1396.
  * (35) Weighted pagerank algorithm.  In _Proceedings of the Second Annual Conference on Communication Networks and Services Research_ , CNSR ’04, 305–314 (IEEE Computer Society, USA, 2004).
  * (36) Walker, D., Xie, H., Yan, K. K. & Maslov, S.  Ranking scientific publications using a model of network traffic.  _Journal of Statistical Mechanics: Theory and Experiment_ 2007, P06010 (2007).  URL https://iopscience.iop.org/article/10.1088/1742-5468/2007/06/P06010https://iopscience.iop.org/article/10.1088/1742-5468/2007/06/P06010/meta.  eprint 0612122.
  * (37) Chen, P., Xie, H., Maslov, S. & Redner, S.  Finding scientific gems with Google’s PageRank algorithm.  _Journal of Informetrics_ 1, 8–15 (2007).
  * (38) Chen, J., Aronow, B. J. & Jegga, A. G.  Disease candidate gene identification and prioritization using protein interaction networks.  _BMC Bioinformatics_ 10, 73 (2009).  URL https://doi.org/10.1186/1471-2105-10-73.
  * (39) Jiang, B., Zhao, S. & Yin, J.  Self-organized natural roads for predicting traffic flow: a sensitivity study.  _Journal of Statistical Mechanics: Theory and Experiment_ 2008, P07008 (2008).  URL https://doi.org/10.1088/1742-5468/2008/07/p07008.
  * (40) Allesina, S. & Pascual, M.  Googling food webs: Can an eigenvector measure species’ importance for coextinctions?  _PLoS Computational Biology_ 5, e1000494 (2009).  URL https://dx.plos.org/10.1371/journal.pcbi.1000494.
  * (41) Gábor, I. & Grolmusz, V.  When the web meets the cell: using personalized pagerank for analyzing protein interaction networks.  _Bioinformatics_ 27, 405–407 (2011).
  * (42) Zhang, X., Martin, T. & Newman, M. E.  Identification of core-periphery structure in networks.  _Physical Review E - Statistical, Nonlinear, and Soft Matter Physics_ 91, 1–10 (2015).  eprint 1409.4813.
  * (43) Rombach, M. P., Porter, M. A., Fowler, J. H. & Mucha, P. J.  Core-periphery structure in networks.  _SIAM Journal on Applied Mathematics_ 74, 167–190 (2014).  URL https://epubs.siam.org/page/terms.
  * (44) Opsahl, T., Colizza, V., Panzarasa, P. & Ramasco, J. J.  Prominence and control: The weighted rich-club effect.  _Physical Review Letters_ 101, 1–4 (2008).  eprint 0804.0417.
  * (45) Ma, A. & Mondragón, R. J.  Rich-cores in networks.  _PLOS ONE_ 10, e0119678 (2015).  URL https://doi.org/10.1371/journal.pone.0119678.
  * (46) Guo, W. _et al._ Global air transport complex network: multi-scale analysis (2019).
  * (47) Wales, D. _et al._ _Energy landscapes: Applications to clusters, biomolecules and glasses_ (Cambridge University Press, 2003).
  * (48) Hurvich, C. M. & Tsai, C.  The impact of model selection on inference in linear regression.  _The American Statistician_ 44, 214–217 (1990).


#  Global Quenches after Localised Perturbations

Maurizio Fagotti  Université Paris-Saclay, CNRS, LPTMS, 91405, Orsay, France

###### Abstract

We investigate the effect of a single spin flip preceding a global quench
between translationally invariant local Hamiltonians in spin-$\frac{1}{2}$
chains. The effect of the localised perturbation does not fade away however
large the distance from the perturbation is. In particular, translational
invariance is not restored and the infinite time limit depends on whether the
spin was flipped or not. We argue that this phenomenon is more general than
the particular example considered and we conjecture that it is triggered by
topological properties, specifically, the existence of “semilocal charges”.

## Introduction.

In its simplest acceptation, “quantum quench” is a modern term indicating the
time evolution of the ground state of some physically meaningful Hamiltonian
of a quantum many-body system after a sudden change of a Hamiltonian
parameter. The quench was dubbed “global” (presumably around the appearance of
Refs [1, 2]) when the parameter is coupled to an extensive operator in such a
way that the initial state has a significant overlap with an exponentially
large number (with respect to the system size) of excited states. Probably the
first analytical study of such dynamics in a spin chain dates back to $1970$,
with Ref. [3]. Global quenches have attracted a lot of attention in the last
two decades for being a bridge between nonequilibrium and equilibrium physics
[4, 5, 6, 7, 8, 9]. A global quench results in complex dynamics, but at long
enough time most of the details become negligible and the local properties of
the state fit in the domain of statistical theories originally conceived to
describe systems at equilibrium[10, 11]. Arguably, most of our intuition about
global quenches comes from studies of translationally invariant systems.
Translational invariance significantly reduces complexity, and analytical
studies in particularly simple models become feasible. In this respect,
integrable models have been playing a key role [12, 13, 14, 15, 16, 17, 18],
both for providing a clue of which phenomena could be observed in a
nonequilibrium quantum many-body system and for the exceptional, exotic
physics they personify. Integrable systems do not thermalise: they approach
generalised Gibbs ensembles (GGE) [19, 20, 21, 22, 23, 24]. Subtleties aside,
this is now established, and it has strongly influenced the recent theoretical
attempts to relax the hypothesis of translational invariance. Localised
defects (see, e.g., [25, 26, 27, 28, 29, 30]) and more global inhomogeneities
(see, e.g., [31, 32, 33, 34, 35, 36, 37, 38]) in the Hamiltonian on one side
and inhomogeneities in the initial state (see, e.g., [39, 40, 41, 42, 43, 44])
on the other are often explained by establishing a connection with the
accepted phenomenon of local relaxation to a GGE. When this approach does not
work, it means that something is still not fully understood. For instance, we
remind the reader of the failure [45, 46, 47, 48] of the original formulation
[49, 50, 51] of GGE in the Heisenberg XXZ chain, which unveiled the importance
of so-called quasilocal conservation laws [23, 52, 53, 54]; or the failure
[55] of the GGE constructed with mode occupation numbers in noninteracting
chains [19], which unveiled the importance of conservation laws that do not
belong to a complete set of charges [55, 56, 57, 58].

In this Letter we report another piece of the puzzle that has been overlooked.
We consider a global quench from a very simple initial state that is
translationally invariant everywhere except for a finite region. Time
evolution is generated by a translationally invariant local spin-chain
Hamiltonian. Normally, we would expect the weak translational-symmetry
breaking in the initial state to be irrelevant at large times and distances.
We show that this is not the case.

## The example.

We consider a spin chain described by the Hamiltonian

$\boldsymbol{\bf
H}=\sum\nolimits_{j=1}^{L}\boldsymbol{\bf\sigma}_{j-1}^{x}\Bigl{(}J_{x}\boldsymbol{\bf
1}-J_{y}\boldsymbol{\bf\sigma}_{j}^{z}\Bigr{)}\boldsymbol{\bf\sigma}_{j+1}^{x}+J_{z}\boldsymbol{\bf\sigma}_{j}^{z}\,,$
(1)

where periodic boundary conditions are understood. This model is dual to the
Heisenberg XYZ one, so it is integrable. The duality transformation is
manifest in terms of the auxiliary Pauli matrices

$\boldsymbol{\bf\tau}_{j}^{z}=-\prod_{n=j}^{L-1}\boldsymbol{\bf\sigma}_{n}^{z}\boldsymbol{\bf\sigma}_{L}^{y}\
,\quad\boldsymbol{\bf\tau}_{j}^{x}=\begin{cases}-\boldsymbol{\bf\sigma}_{1}^{y}\prod_{n=2}^{L-1}\boldsymbol{\bf\sigma}_{n}^{z}\boldsymbol{\bf\sigma}_{L}^{y},&j=1\\\
\boldsymbol{\bf\sigma}_{j-1}^{x}\boldsymbol{\bf\sigma}_{j}^{x},&j>1\,.\end{cases}$
(2)

The Hamiltonian can indeed be written as follows

$\boldsymbol{\bf
H}=\sideset{}{{}^{\prime}}{\sum}_{j=1}^{L}(J_{x}\boldsymbol{\bf\tau}_{j}^{x}\boldsymbol{\bf\tau}_{j+1}^{x}+J_{y}\boldsymbol{\bf\tau}_{j}^{y}\boldsymbol{\bf\tau}_{j+1}^{y}+J_{z}\boldsymbol{\bf\tau}_{j}^{z}\boldsymbol{\bf\tau}_{j+1}^{z})$
(3)

where we introduced the notation

$\sideset{}{{}^{\prime}}{\sum}_{j=1}^{L}\boldsymbol{\bf
O}_{j}:=\tfrac{\boldsymbol{\bf
1}+\boldsymbol{\bf\Pi}_{\tau}^{x}}{2}\sideset{}{{}^{+}}{\sum}^{L}_{j=1}\\!\\!\boldsymbol{\bf
O}_{j}\tfrac{\boldsymbol{\bf
1}+\boldsymbol{\bf\Pi}_{\tau}^{x}}{2}+\tfrac{\boldsymbol{\bf
1}-\boldsymbol{\bf\Pi}_{\tau}^{x}}{2}\boldsymbol{\bf\tau}_{1}^{z}\sideset{}{{}^{-}}{\sum}_{j=1}^{L}\\!\\!\boldsymbol{\bf
O}_{j}\boldsymbol{\bf\tau}_{1}^{z}\tfrac{\boldsymbol{\bf
1}-\boldsymbol{\bf\Pi}_{\tau}^{x}}{2},$ (4)

with
$\boldsymbol{\bf\Pi}_{\tau}^{\alpha}=\prod_{j=1}^{L}\boldsymbol{\bf\tau}_{j}^{\alpha}$
and $\sum\nolimits^{\pm}$ standing for the sum in which
$\boldsymbol{\bf\tau}^{y,z}_{L+1}=\pm\boldsymbol{\bf\tau}^{y,z}_{1}$. Note
that the projectors into the distinct sectors can be easily written in terms
of the $\boldsymbol{\bf\sigma}$ operators, indeed we have
$\boldsymbol{\bf\Pi}_{\tau}^{x}=\boldsymbol{\bf\Pi}_{\sigma}^{z}$ and
$\boldsymbol{\bf\tau}_{1}^{z}\frac{\boldsymbol{\bf
1}-\boldsymbol{\bf\Pi}_{\tau}^{x}}{2}=i\boldsymbol{\bf\sigma}_{L}^{x}\frac{\boldsymbol{\bf
1}-\boldsymbol{\bf\Pi}^{z}_{\sigma}}{2}$. We warn the reader that duality
transformations affect the notion of locality and, in turn, the physical
properties of a system: (1) does not describe the Heisenberg model.

At the initial time we prepare the system in the state

$\ket{\Psi(0)}=\ket{\uparrow_{1}\cdots\uparrow_{L-1}\downarrow_{L}}_{\sigma}\equiv\ket{\Uparrow\downarrow_{L}}_{\sigma}\,,$
(5)

where the subscript $\sigma$ means that we are representing the spins with the
Pauli matrices $\boldsymbol{\bf\sigma}$, $\Uparrow$ ($\Downarrow$) stands for
a string of $\uparrow$’s ($\downarrow$’s) of whatever length. The last spin is
flipped to eventually simplify the boundary conditions in the dual
representation, but we stress that this is completely irrelevant in the
thermodynamic limit, as the Lieb-Robinson bounds [59] rule out any effect in
the bulk, provided that $t\ll L$. We then flip a spin in the bulk, let’s say,
at position $1\ll\ell\ll L$. This is achieved by applying
$\boldsymbol{\bf\sigma}_{\ell}^{x}$ to the state; we then have

$\ket{\Psi(0^{+})}=\ket{\Uparrow\downarrow_{\ell}\Uparrow\downarrow_{L}}_{\sigma}\,.$
(6)

This state is in the sector with $\boldsymbol{\bf\Pi}_{\sigma}^{z}=1$, and
hence $\boldsymbol{\bf\Pi}^{x}_{\tau}=1$, and in the $\tau$-representation has
the form of a cat state:

$\ket{\Psi(0^{+})}=\tfrac{1}{\sqrt{2}}(\ket{\Uparrow_{\ell}\Downarrow}_{\tau}+\ket{\Downarrow_{\ell}\Uparrow}_{\tau})\,.$
(7)

where the subscript $\ell$ indicates that the string ends at $\ell$. Since
$\boldsymbol{\bf\Pi}^{x}_{\tau}=1$, Hamiltonian (3) is equivalent to

$\tilde{\boldsymbol{\bf
H}}=\sum\nolimits_{j=1}^{L}\boldsymbol{\bf(}J_{x}\boldsymbol{\bf\tau}_{j}^{x}\boldsymbol{\bf\tau}_{j+1}^{x}+J_{y}\boldsymbol{\bf\tau}_{j}^{y}\boldsymbol{\bf\tau}_{j+1}^{y}+J_{z}\boldsymbol{\bf\tau}_{j}^{z}\boldsymbol{\bf\tau}_{j+1}^{z})\,,$
(8)

where periodic boundary conditions are understood. We are interested in the
expectation values of local (in the $\sigma$-representation) operators
$\boldsymbol{\bf O}$. Since the initial state is eigenstate of
$\boldsymbol{\bf\Pi}^{z}_{\sigma}$ and the latter commutes with the
Hamiltonian, operators anticommuting with $\boldsymbol{\bf\Pi}^{z}_{\sigma}$
have zero expectation value. We can therefore restrict to operators that
commute with
$\boldsymbol{\bf\Pi}^{z}_{\sigma}\equiv\boldsymbol{\bf\Pi}^{x}_{\tau}$. Such
operators are local also in the $\tau$-representation. Since
$\ket{\Uparrow_{\ell}\Downarrow}_{\tau}$ and
$\ket{\Downarrow_{\ell}\Uparrow}_{\tau}$ are macroscopically different (in the
$\tau$-representation), the off-diagonal matrix elements of the aforementioned
operators between $\ket{\Uparrow_{\ell}\Downarrow}_{\tau}$ and
$\ket{\Downarrow_{\ell}\Uparrow}_{\tau}$ vanish. In addition, since
$\ket{\Uparrow_{\ell}\Downarrow}_{\tau}=\boldsymbol{\bf\Pi}_{\tau}^{x}\ket{\Downarrow_{\ell}\Uparrow}_{\tau}$,
the diagonal matrix elements match, thus we have

$\braket{\Psi(t)}{\boldsymbol{\bf
O}}{\Psi(t)}=\braket{\Downarrow_{\ell}\Uparrow}{e^{i\tilde{\boldsymbol{\bf
H}}t}\boldsymbol{\bf O}e^{-i\tilde{\boldsymbol{\bf
H}}t}}{\Downarrow_{\ell}\Uparrow}_{\tau}\,.$ (9)

We now take the thermodynamic limit by fixing the position of the flipped spin
at a finite value $\ell$ and moving the boundaries to $\pm\infty$ (for
example, index $j$ in (8) now runs from $-\infty$ to $\infty$). Eq. (9) is the
expectation value of a local operator after a quench from a domain-wall state
in an integrable model, which is nowadays studied in the framework of
generalised hydrodynamics (GHD) [40, 41]. Given the GGEs describing the system
outside the lightcone, which in our case correspond to the homogeneous global
quenches with initial states $\ket{\Downarrow}_{\tau}$ and
$\ket{\Uparrow}_{\tau}$, GHD allows us to characterise the state inside the
lightcone. In interacting systems GHD relies on the assumption that at large
times the state becomes locally equivalent to a macro-state characterised by
the (quasi)local integrals of motion. In some specific instances the GHD
predictions have been confirmed by numerical simulations, but here we need to
be particularly prudent because the theory is somehow blind to localised
perturbations (in the domain-wall scenario they are hidden in the ambiguity of
how the states are joined), which we are instead questioning if potentially
relevant. Let us then focus on the case $J_{z}=0$. This corresponds to the
quantum XY model, which is a noninteracting system where generalised
hydrodynamics was shown to be exact [60, 61]. The initial state is a Slater
determinant (in terms of the usual Jordan-Wigner fermions $\boldsymbol{\bf
c}^{\dagger}_{j}=\boldsymbol{\bf\tau}_{j}^{+}\prod_{n<j}\boldsymbol{\bf\tau}^{z}_{n}$)
and the behaviour in the limit of large time is captured by first-order
generalised hydrodynamics. In the low-inhomogeneity limit the state becomes
locally equivalent to a stationary state that is completely characterised by
the occupation numbers $\braket{\boldsymbol{\bf n}(k)}=\braket{\boldsymbol{\bf
b}^{\dagger}_{k}\boldsymbol{\bf b}_{k}}$ of the Bogoliubov fermions
$\boldsymbol{\bf b}^{\dagger}_{k}$ diagonalising the Hamiltonian 111We warn
the reader that the XY model is non-abelian integrable [55], so this statement
is correct only if the additional charges do not play any role, like in the
present situation.. At the leading order one finds $\braket{\boldsymbol{\bf
n}(k)}_{x,t}\asymp\braket{\boldsymbol{\bf n}(k)}_{x-v(k)t,0}$, where $x$ is
the coarse-grained position of $\boldsymbol{\bf O}$ and $v(k)$ is the velocity
of the quasiparticle excitation with momentum $k$. In our specific case
$v(k)=-4J_{x}J_{y}\sin(2k)/[(J_{x}+J_{y})^{2}\cos^{2}k+(J_{x}-J_{y})^{2}\sin^{2}k]^{1/2}$.

The simplest observable that is local in both representations and has a
nonzero expectation value is $\boldsymbol{\bf\sigma}_{n}^{z}$, which is mapped
into $\boldsymbol{\bf\tau}_{n}^{z}\boldsymbol{\bf\tau}_{n+1}^{z}$. The latter
is a 4-fermion operator and its expectation value in a Gaussian state can be
computed using the Wick’s theorem. We finally find

$\braket{\Uparrow_{0}\Downarrow}{e^{i\tilde{\boldsymbol{\bf
H}}t}\boldsymbol{\bf\tau}_{n}^{z}\boldsymbol{\bf\tau}_{n+1}^{z}e^{-i\tilde{\boldsymbol{\bf
H}}t}}{\Uparrow_{0}\Downarrow}_{\tau}\xrightarrow{n\sim
t\rightarrow\infty}A^{2}(\tfrac{n}{t})-B^{2}(\tfrac{n}{t})$ (10)

with

$\displaystyle A(\zeta)=$
$\displaystyle\int\frac{\mathrm{d}k}{2\pi}\frac{\theta_{H}(\zeta^{2}-v^{2}(k))}{1+(\frac{J_{x}-J_{y}}{J_{x}+J_{y}})^{2}\tan^{2}k}$
(11) $\displaystyle B(\zeta)=$
$\displaystyle\int\frac{\mathrm{d}k}{2\pi}\frac{\mathrm{sgn}(v(k))\sin(2k)\theta_{H}(v^{2}(k)-\zeta^{2})}{2(\cos^{2}k+(\frac{J_{x}-J_{y}}{J_{x}+J_{y}})^{2}\sin^{2}k)^{1/2}}\,.$

Consequently, in the thermodynamic limit we have

$\braket{\Uparrow\downarrow_{\ell}\Uparrow}{e^{i\boldsymbol{\bf
H}t}\boldsymbol{\bf\sigma}_{n}^{z}e^{-i\boldsymbol{\bf
H}t}}{\Uparrow\downarrow_{\ell}\Uparrow}_{\sigma}\xrightarrow{|n-\ell|\sim
t\rightarrow\infty}\\\ A^{2}(\tfrac{n-\ell}{t})-B^{2}(\tfrac{n-\ell}{t})\,,$
(12)

which has the usual lightcone structure characterising the spreading of a
localised perturbation in integrable systems [59, 14] but, contrary to typical
cases, it does not decay in time – Fig. 1. Note that this is a macroscopic
effect, in the sense that also the total magnetisation in an arbitrarily large
subsystem is affected, provided that the time is at least comparable with the
subsystem’s length. In the limit of infinite time the expectation value
approaches a negative value

$\braket{\Uparrow\downarrow_{\ell}\Uparrow}{e^{i\boldsymbol{\bf
H}t}\boldsymbol{\bf\sigma}_{n}^{z}e^{-i\boldsymbol{\bf
H}t}}{\Uparrow\downarrow_{\ell}\Uparrow}_{\sigma}\rightarrow-(\tfrac{J_{x}+J_{y}}{\pi\max(J_{x},J_{y})})^{2}\,.$
(13)

This is different from the infinite-time limit without the initial spin flip,
which instead matches the value reached outside the lightcone

$\braket{\Uparrow}{e^{i\boldsymbol{\bf
H}t}\boldsymbol{\bf\sigma}_{n}^{z}e^{-i\boldsymbol{\bf
H}t}}{\Uparrow}_{\sigma}\rightarrow(\tfrac{J_{x}+J_{y}}{2\max(J_{x},J_{y})})^{2}\,:$
(14)

Even if we are just interested in the behaviour at late times, the spin flip
is still a relevant localised perturbation. Fig. 2 shows the validity of the
prediction.

Figure 1: Time evolution of $\braket{\boldsymbol{\bf\sigma}_{n}^{z}}$ in a
chain with $600$ spins with $J_{x}=1$, $J_{y}=0.5$, and $J_{z}=0$ from the
initial state
$\ket{\Uparrow\downarrow_{\ell}\Uparrow\downarrow_{L}}_{\sigma}$. Figure 2:
The discrepancy between the time evolution shown in Fig. 1 and the asymptotic
prediction (12). The lightcone is well visible because we truncated GHD at the
first order [60].

## Semilocal charges.

We have provided a counterexample to the belief that localised perturbations
in the initial state do not survive the limit of infinite time after a global
quench; we have not yet clarified why generally this is not the case. What is
so special in the model described by $\boldsymbol{\bf H}$? In this respect,
the duality transformation is again helpful. We have indeed an almost perfect
knowledge of $\tilde{\boldsymbol{\bf H}}$ with $J_{z}=0$. In particular we
know that the model is non-abelian integrable [55, 56] and the local
conservation laws are linear combinations of $\boldsymbol{\bf\tau}_{j}^{z}$
and operators of the form
$\boldsymbol{\bf\tau}_{j}^{\alpha}\prod_{m=j+1}^{j+n-1}\boldsymbol{\bf\tau}_{m}^{z}\boldsymbol{\bf\tau}_{j+n}^{\beta}$,
where $\alpha,\beta\in\\{x,y\\}$ [63, 64]. In order to have a local
$\sigma$-representation, they should commute with
$\boldsymbol{\bf\Pi}^{x}_{\tau}$, but this happens only if they consist
exclusively of operators for which $(\alpha=\beta\wedge n\text{\emph{ is
odd}})\vee(\alpha\neq\beta\wedge n\text{\emph{ is even}})$. This is realised
only in a subset of the local conservation laws. In fact, such charges are
completely irrelevant in the description of the long time limit: they have the
same expectation value on both sides of the junction and do not affect the
emerging locally quasi-stationary state. We are therefore left with charges
that are local only in the $\tau$-representation. In the
$\sigma$-representation they have instead the following form (in the
thermodynamic limit):

$\boldsymbol{\bf Q}=\sum\nolimits_{j}\boldsymbol{\bf
q}_{j}\boldsymbol{\bf\sigma}_{j+1}^{z}\boldsymbol{\bf\sigma}_{j+2}^{z}\cdots\,,$
(15)

where $\boldsymbol{\bf q}_{j}$ acts like the identity everywhere except for a
finite region ending at site $j$. By borrowing a terminology used in field
theory, we call them ‘semilocal’; note, in particular, that their density
$\boldsymbol{\bf q}_{j}\prod_{n=j+1}^{\infty}\boldsymbol{\bf\sigma}_{n}^{z}$
commutes with any local operator with support to the left of the support of
$\boldsymbol{\bf q}_{j}$ but anticommutes with any local operator to the right
if the latter anticommutes with $\boldsymbol{\bf\Pi}^{z}_{\sigma}$. We can
also define the corresponding semilocal currents, which, due to the locality
of the Hamiltonian, have the same semilocal form $\boldsymbol{\bf
J}_{j}[\boldsymbol{\bf q}]=\boldsymbol{\bf
j}_{j}\boldsymbol{\bf\sigma}_{j+1}^{z}\boldsymbol{\bf\sigma}_{j+2}^{z}\cdots$.
We believe that one could capture the long-time dynamics directly in the
$\sigma$-representation by including the semilocal charges in the generalised
hydrodynamic theory, but this question will be addressed in a separate
investigation.

## Interaction.

In the following we provide an argument that a similar phenomenology should be
expected also in the presence of interactions. To that aim we consider the
case $J_{x}=J_{y}\neq J_{z}$, which corresponds to the XXZ Heisenberg model.
We point out that this is not a genuine global quench because the initial
state is close to an eigenstate of the Hamiltonian. This case study is more
akin to investigations on geometric quenches [65, 66, 67, 68, 69, 70] and on
the effects of localised perturbations in ground states with symmetry breaking
[71, 72, 73] or in jammed states [74, 75]. It is however also the arguably
most studied domain-wall quench in recent literature, so it could give an
easily perceivable clue; we will discuss later how to transmute it into a
global quench. Taking the duality transformation into account, the potential
semilocal charges of $\boldsymbol{\bf H}$ should be associated with charges of
the XXZ model that anticommute with $\boldsymbol{\bf\Pi}_{\tau}^{x}$. For
$J_{z}>J_{x}=J_{y}$ only one charge of that kind is known: the component of
the total spin in the direction of the anisotropy. In that case, some short-
range correlators have been already studied [42]. In particular it was shown
that, in the ballistic scaling limit (also known as space-time scaling limit
or Euler limit) – cf. (10) – the sign of the magnetisation along $z$ in the
states that are joined together at the initial time affects only observables
odd under spin flip $z\rightarrow-z$. Since
$\boldsymbol{\bf\tau}_{j}^{z}\boldsymbol{\bf\tau}_{j+1}^{z}$ is an even
operator and our domain wall consists of two states that differ only in the
sign of the magnetisation, we argue that the phenomenon we are describing does
not present itself on the ballistic scale when $J_{z}>J_{x}=J_{y}$. We do not
exclude it to materialise in a different scaling limit, but we leave it as an
open question, which we can also extend to generic systems with a single
semilocal charge. There are instead other choices of the coupling constants
that are compatible with a ballistic spreading. Besides the $z$ component of
the total spin, indeed, an infinitely large family of charges anticommuting
with $\boldsymbol{\bf\Pi}_{\tau}^{x}$ has been identified at special values of
the anisotropy, known as roots of unity ($J_{z}=\cos(\pi q)J_{x}=\cos(\pi
q)J_{y}$, with $q$ a rational number) [76, 77]. In addition, just as the XY
model considered before exhibits a non-abelian structure, so does the XXZ
model at those values of the coupling constants [58, 78, 79]. Contrary to the
case $J_{z}>J_{x}=J_{y}$, for $J_{z}<J_{x}=J_{y}$ also the observables
commuting with $\boldsymbol{\bf\Pi}^{x}_{\tau}$ have a nontrivial profile in
the quench from the domain wall $\ket{\Downarrow_{\ell}\Uparrow}_{\tau}$
222Note that this problem could be solved exactly using the results of Refs
[84] and [85].; in the end one finds a behaviour analogous to the case
$J_{x}=J_{y}$ and $J_{z}=0$, in which the profile is qualitatively similar to
that shown in Fig. 1, except for the state not evolving outside the lightcone.

## Topological properties.

The typical properties of nonequilibrium states that are locally stationary
are generally fragile under global transformations of the initial state 333For
example, Refs [22, 86] point to the melting of order after global quenches, so
localised perturbations in ground states with symmetry breaking are not
expected to survive global quenches.. We now show that the interacting quench
considered above can instead be easily transmuted into a global one without
changing the phenomenology. To that aim, let us transform the initial
ferromagnetic state $\ket{\Uparrow}_{\sigma}$ with a unitary operator of the
form $e^{i\boldsymbol{\bf W}}$, where $\boldsymbol{\bf W}$ is a local,
translationally invariant operator commuting with
$\boldsymbol{\bf\Pi}^{z}_{\sigma}$ and acting nontrivally on
$\ket{\Uparrow}_{\sigma}$. Since the profile of local operators after the
quench from $\ket{\Downarrow_{\ell}\Uparrow}_{\tau}$ is expected to be
nontrivial, there must be a density $\boldsymbol{\bf q}_{\mathrm{sl}}$ of a
semilocal charge satisfying $\braket{\Uparrow}{\boldsymbol{\bf
q}_{\mathrm{sl}}}{\Uparrow}_{\tau}(=-\braket{\Downarrow}{\boldsymbol{\bf
q}_{\mathrm{sl}}}{\Downarrow}_{\tau})\neq 0$. The transformation
$e^{i\boldsymbol{\bf W}}$ can be interpreted as time evolution under
$-\boldsymbol{\bf W}$ for a unit of time. A necessary condition to kill the
effect would be $\braket{\Uparrow}{e^{-i\boldsymbol{\bf W}}\boldsymbol{\bf
q}_{\mathrm{sl}}e^{i\boldsymbol{\bf W}}}{\Uparrow}_{\tau}=0$, but the
expectation value of a quasilocal operator ($\boldsymbol{\bf q}_{\mathrm{sl}}$
is quasilocal in the $\tau$ representation) is a smooth function of the time,
which generally vanishes at a set of times with zero measure. Even if
$\braket{\Uparrow}{e^{-i\boldsymbol{\bf W}}\boldsymbol{\bf
q}_{\mathrm{sl}}e^{i\boldsymbol{\bf W}}}{\Uparrow}_{\tau}=0$, we can multiply
$\boldsymbol{\bf W}$ by a constant and obtain something different from zero.
This proves that the lightcone structure is not killed by the transformation
$e^{i\boldsymbol{\bf W}}$, which is nonetheless sufficient to make the quench
global. This possibility is exhibited in Fig. 3 for $J_{x}=J_{y}$ and
$J_{z}=0$, which points to the stability of the qualitative picture under a
change of the initial state. But there is much more.

Figure 3: Time evolution of $\braket{\boldsymbol{\bf\sigma}_{n}^{z}}$ in a
chain with $600$ spins with $J_{x}=J_{y}=1$, and $J_{z}=0$ after flipping the
$\ell$-th spin in $e^{i\boldsymbol{\bf W}}\ket{\Uparrow\downarrow_{L}}$, where
$\boldsymbol{\bf W}$ is a randomly generated translationally invariant local
Hermitian operator that, in the $\tau$-representation, is noninteracting and
commutes with $\boldsymbol{\bf\Pi}_{\tau}^{x}$.

Let us consider again the case $J_{z}=0$ (but arbitrary $J_{x}$ and $J_{y}$)
and imagine dressing the original perturbation (spin flip) with a unitary
operator $\boldsymbol{\bf U}_{\ell}$, acting nontrivially only around site
$\ell$ and commuting with $\boldsymbol{\bf\Pi}^{z}_{\sigma}$:
$\boldsymbol{\bf\sigma}_{\ell}^{x}\rightarrow\boldsymbol{\bf
U}_{\ell}\boldsymbol{\bf\sigma}_{\ell}^{x}$. This results in

$\braket{\Psi(t)}{\boldsymbol{\bf
O}}{\Psi(t)}=\braket{\Downarrow_{\ell}\Uparrow}{\boldsymbol{\bf
U}^{\dagger}_{\ell}e^{i\tilde{\boldsymbol{\bf H}}t}\boldsymbol{\bf
O}e^{-i\tilde{\boldsymbol{\bf H}}t}\boldsymbol{\bf
U}_{\ell}}{\Downarrow_{\ell}\Uparrow}_{\tau}\,.$ (16)

If $\boldsymbol{\bf U}_{\ell}$ is Gaussian in the $\tau$-representation, Ref.
[61] suggests that the local perturbation is not going to affect the long time
limit. In fact, it is reasonable to expect the same conclusion also for non-
Gaussian $\boldsymbol{\bf U}_{\ell}$ [82, 83]. Under this assumption, we
conclude that our prediction (12) is stable under a change of the
perturbation, provided that the latter connects the sectors with a different
value of $\boldsymbol{\bf\Pi}^{z}_{\sigma}$. We have checked it for a randomly
generated Gaussian unitary operator in the same system considered in Fig. 1.
We do not show the plot just because, very short times aside, it is
practically indistinguishable from Fig. 1. On the other hand, if we replace
the spin flip resulting in (6) by a unitary operator $\boldsymbol{\bf
U}_{\ell}$ with the same properties as above, we end up in the
$\tau$-representation with a global quench from the state with all spins in
the $z$ direction after a localised perturbation. Again, this perturbation is
not expected to affect the long time limit [82, 83], so we predict the
restoration of translational invariance. From this qualitative argument we
argue:

$\braket{\Uparrow}{\boldsymbol{\bf V}_{\ell}^{\dagger}e^{i\boldsymbol{\bf
H}t}\boldsymbol{\bf\sigma}_{n}^{z}e^{-i\boldsymbol{\bf H}t}\boldsymbol{\bf
V}_{\ell}}{\Uparrow}_{\sigma}\xrightarrow{|n-\ell|\sim t\rightarrow\infty}\\\
\tfrac{1+\delta_{\boldsymbol{\bf
V}}}{2}\left(\tfrac{J_{x}+J_{y}}{2\max(J_{x},J_{y})}\right)^{2}+\tfrac{1-\delta_{\boldsymbol{\bf
V}}}{2}[A^{2}(\tfrac{n-\ell}{t})-B^{2}(\tfrac{n-\ell}{t})]\,,$ (17)

where $\boldsymbol{\bf V}_{\ell}$ can be any unitary operator localised around
site $\ell$ (in the $\sigma$-representation) and $\delta_{\boldsymbol{\bf
V}}=\nicefrac{{(\braket{\Uparrow}{\boldsymbol{\bf\Pi}_{\sigma}^{z}\boldsymbol{\bf
V}_{\ell}^{\dagger}\boldsymbol{\bf\Pi}_{\sigma}^{z}\boldsymbol{\bf
V}_{\ell}}{\Uparrow}_{\sigma}+h.c.)}}{{2}}\,.$ Note that $\boldsymbol{\bf
V}_{\ell}$ could also represent multiple perturbations in different positions,
for example $\boldsymbol{\bf
V}_{\ell}=\boldsymbol{\bf\sigma}_{\ell-j}^{x}\boldsymbol{\bf\sigma}_{\ell+j}^{x}$.
In that case there can be multiple time scales in which lightcones emerge and
merge. In the end, however, only the parity of the number of perturbations
connecting the two sectors will be discriminating.

## Discussion.

We have shown that, even after global quenches, localised perturbations to the
initial state can have macroscopic effects. We have proved it in a model that
can be mapped to noninteracting fermions and provided evidence that analogous
behaviours can be seen in the presence of interactions preserving
integrability. We traced the phenomenon back to the existence of semilocal
charges, which should be incorporated in the generalised Gibbs ensemble, as
well as in the generalised hydrodynamic theory. How to do that remains an open
question. All the examples discussed in which we expect relevant localised
perturbations are described by Hamiltonians with a non-abelian set of
(quasi)local conservation laws. We did not use this hidden structure in the
case we solved analytically, but we still wonder whether a duality
transformation mapping non-abelian integrable systems into systems with
(infinitely many) semilocal charges should be generally expected.

## Acknowledgement.

I thank Bruno Bertini, Saverio Bocini and Lenart Zadnik for discussions and
Fabian Essler for insightful comments. I thank Kemal Bidzhiev for having
shared some data from DMRG simulations.

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

## References

  * Calabrese and Cardy [2005] P. Calabrese and J. Cardy, J. Stat. Mech. 2005, P04010 (2005).
  * Eisler and Peschel [2007] V. Eisler and I. Peschel, J. Stat. Mech. 2007, P06005 (2007).
  * Barouch  _et al._ [1970] E. Barouch, B. M. McCoy, and M. Dresden, Phys. Rev. A 2, 1075 (1970).
  * Polkovnikov  _et al._ [2011] A. Polkovnikov, K. Sengupta, A. Silva, and M. Vengalattore, Rev. Mod. Phys. 83, 863 (2011).
  * Gogolin and Eisert [2016] C. Gogolin and J. Eisert, Rep. Prog. Phys. 79, 056001 (2016).
  * Eisert  _et al._ [2015] J. Eisert, M. Friesdorf, and C. Gogolin, Nature Phys 11, 124 (2015).
  * Essler and Fagotti [2016] F. H. L. Essler and M. Fagotti, J. Stat. Mech. 2016, 064002 (2016).
  * Vidmar and Rigol [2016] L. Vidmar and M. Rigol, J. Stat. Mech. 2016, 064007 (2016).
  * Calabrese and Cardy [2016] P. Calabrese and J. Cardy, J. Stat. Mech. 2016, 064003 (2016).
  * Srednicki [1994] M. Srednicki, Phys. Rev. E 50, 888 (1994).
  * Rigol and Srednicki [2012] M. Rigol and M. Srednicki, Phys. Rev. Lett. 108, 110601 (2012).
  * Korepin  _et al._ [1993] V. E. Korepin, N. M. Bogoliubov, and A. G. Izergin,  _Quantum Inverse Scattering Method and Correlation Functions_, Cambridge Monographs on Mathematical Physics (Cambridge University Press, 1993).
  * Caux and Essler [2013] J.-S. Caux and F. H. L. Essler, Phys. Rev. Lett. 110, 257203 (2013).
  * Bonnes  _et al._ [2014] L. Bonnes, F. H. L. Essler, and A. M. Läuchli, Phys. Rev. Lett. 113, 187203 (2014).
  * Piroli  _et al._ [2017a] L. Piroli, B. Pozsgay, and E. Vernier, Nuclear Physics B 925, 362 (2017a).
  * Borsi  _et al._ [2020] M. Borsi, B. Pozsgay, and L. Pristyák, Phys. Rev. X 10, 011054 (2020).
  * Klobas  _et al._ [2021] K. Klobas, B. Bertini, and L. Piroli, Phys. Rev. Lett. 126, 160602 (2021).
  * Klobas and Bertini [2021] K. Klobas and B. Bertini, SciPost Phys. 11, 106 (2021).
  * Rigol  _et al._ [2007] M. Rigol, V. Dunjko, V. Yurovsky, and M. Olshanii, Phys. Rev. Lett. 98, 050405 (2007).
  * Cramer  _et al._ [2008] M. Cramer, C. M. Dawson, J. Eisert, and T. J. Osborne, Phys. Rev. Lett. 100, 030602 (2008).
  * Calabrese  _et al._ [2011] P. Calabrese, F. H. L. Essler, and M. Fagotti, Phys. Rev. Lett. 106, 227203 (2011).
  * Fagotti and Essler [2013a] M. Fagotti and F. H. L. Essler, Phys. Rev. B 87, 245107 (2013a).
  * Ilievski  _et al._ [2015a] E. Ilievski, J. De Nardis, B. Wouters, J.-S. Caux, F. H. L. Essler, and T. Prosen, Phys. Rev. Lett. 115, 157201 (2015a).
  * Essler  _et al._ [2015] F. H. L. Essler, G. Mussardo, and M. Panfil, Phys. Rev. A 91, 051602 (2015).
  * Bernard  _et al._ [2015] D. Bernard, B. Doyon, and J. Viti, J. Phys. A: Math. Theor. 48, 05FT01 (2015).
  * Fagotti [2016a] M. Fagotti, arXiv:1508.04401 (2016a).
  * Bertini and Fagotti [2016] B. Bertini and M. Fagotti, Phys. Rev. Lett. 117, 130402 (2016).
  * Ljubotina  _et al._ [2019] M. Ljubotina, S. Sotiriadis, and T. Prosen, SciPost Phys. 6, 4 (2019).
  * Bastianello and De Luca [2018] A. Bastianello and A. De Luca, Phys. Rev. Lett. 120, 060602 (2018).
  * Bastianello [2019] A. Bastianello, EPL 125, 20001 (2019).
  * Biella  _et al._ [2016] A. Biella, A. De Luca, J. Viti, D. Rossini, L. Mazza, and R. Fazio, Phys. Rev. B 93, 205121 (2016).
  * Doyon and Yoshimura [2017] B. Doyon and T. Yoshimura, SciPost Phys. 2, 014 (2017).
  * Dubail  _et al._ [2017] J. Dubail, J.-M. Stéphan, J. Viti, and P. Calabrese, SciPost Phys. 2, 002 (2017).
  * Cao  _et al._ [2018] X. Cao, V. B. Bulchandani, and J. E. Moore, Phys. Rev. Lett. 120, 164101 (2018).
  * Biella  _et al._ [2019] A. Biella, M. Collura, D. Rossini, A. D. Luca, and L. Mazza, Nat. Commun. 10, 4820 (2019).
  * Bastianello  _et al._ [2019] A. Bastianello, V. Alba, and J.-S. Caux, Phys. Rev. Lett. 123, 130602 (2019).
  * Ruggiero  _et al._ [2020] P. Ruggiero, P. Calabrese, B. Doyon, and J. Dubail, Phys. Rev. Lett. 124, 140603 (2020).
  * Malvania  _et al._ [2021] N. Malvania, Y. Zhang, Y. Le, J. Dubail, M. Rigol, and D. S. Weiss, Science 373, 1129 (2021).
  * Bernard and Doyon [2016] D. Bernard and B. Doyon, J. Stat. Mech. 2016, 064005 (2016).
  * Bertini  _et al._ [2016] B. Bertini, M. Collura, J. De Nardis, and M. Fagotti, Phys. Rev. Lett. 117, 207201 (2016).
  * Castro-Alvaredo  _et al._ [2016] O. A. Castro-Alvaredo, B. Doyon, and T. Yoshimura, Phys. Rev. X 6, 041065 (2016).
  * Piroli  _et al._ [2017b] L. Piroli, J. De Nardis, M. Collura, B. Bertini, and M. Fagotti, Phys. Rev. B 96, 115124 (2017b).
  * Bulchandani  _et al._ [2017] V. B. Bulchandani, R. Vasseur, C. Karrasch, and J. E. Moore, Phys. Rev. Lett. 119, 220604 (2017).
  * Bulchandani  _et al._ [2018] V. B. Bulchandani, R. Vasseur, C. Karrasch, and J. E. Moore, Phys. Rev. B 97, 045407 (2018).
  * Brockmann  _et al._ [2014] M. Brockmann, B. Wouters, D. Fioretto, J. D. Nardis, R. Vlijm, and J.-S. Caux, J. Stat. Mech. 2014, P12009 (2014).
  * Wouters  _et al._ [2014] B. Wouters, J. De Nardis, M. Brockmann, D. Fioretto, M. Rigol, and J.-S. Caux, Phys. Rev. Lett. 113, 117202 (2014).
  * Pozsgay  _et al._ [2014] B. Pozsgay, M. Mestyán, M. A. Werner, M. Kormos, G. Zaránd, and G. Takács, Phys. Rev. Lett. 113, 117203 (2014).
  * Mestyán  _et al._ [2015] M. Mestyán, B. Pozsgay, G. Takács, and M. A. Werner, J. Stat. Mech. 2015, P04001 (2015).
  * Fagotti and Essler [2013b] M. Fagotti and F. H. L. Essler, J. Stat. Mech. 2013, P07012 (2013b).
  * Pozsgay [2013] B. Pozsgay, J. Stat. Mech. 2013, P07003 (2013).
  * Fagotti  _et al._ [2014] M. Fagotti, M. Collura, F. H. L. Essler, and P. Calabrese, Phys. Rev. B 89, 125101 (2014).
  * Ilievski  _et al._ [2015b] E. Ilievski, M. Medenjak, and T. Prosen, Phys. Rev. Lett. 115, 120601 (2015b).
  * Ilievski  _et al._ [2016] E. Ilievski, M. Medenjak, T. Prosen, and L. Zadnik, J. Stat. Mech. 2016, 064008 (2016).
  * Piroli  _et al._ [2016] L. Piroli, E. Vernier, and P. Calabrese, Phys. Rev. B 94, 054313 (2016).
  * Fagotti [2014] M. Fagotti, J. Stat. Mech. 2014, P03016 (2014).
  * Fagotti [2016b] M. Fagotti, J. Phys. A: Math. Theor. 50, 034005 (2016b).
  * Doyon [2017] B. Doyon, Commun. Math. Phys. 351, 155 (2017).
  * Zadnik  _et al._ [2016] L. Zadnik, M. Medenjak, and T. Prosen, Nuclear Physics B 902, 339 (2016).
  * Lieb and Robinson [1972] E. H. Lieb and D. W. Robinson, Commun. Math. Phys. 28, 251 (1972).
  * Fagotti [2020] M. Fagotti, SciPost Phys. 8, 48 (2020).
  * Alba  _et al._ [2021] V. Alba, B. Bertini, M. Fagotti, L. Piroli, and P. Ruggiero, J. Stat. Mech. 2021, 114004 (2021).
  * Note [1] We warn the reader that the XY model is non-abelian integrable [55], so this statement is correct only if the additional charges do not play any role, like in the present situation.
  * Grady [1982] M. Grady, Phys. Rev. D 25, 1103 (1982).
  * Prosen [1998] T. Prosen, J. Phys. A 31, L397 (1998).
  * Mossel  _et al._ [2010] J. Mossel, G. Palacios, and J.-S. Caux, J. Stat. Mech. 2010, L09001 (2010).
  * Alba and Heidrich-Meisner [2014] V. Alba and F. Heidrich-Meisner, Phys. Rev. B 90, 075144 (2014).
  * Eisler and Maislinger [2018] V. Eisler and F. Maislinger, Phys. Rev. B 98, 161117 (2018).
  * Gruber and Eisler [2019] M. Gruber and V. Eisler, Phys. Rev. B 99, 174403 (2019).
  * Collura  _et al._ [2020] M. Collura, A. De Luca, P. Calabrese, and J. Dubail, Phys. Rev. B 102, 180409 (2020).
  * Scopa  _et al._ [2021] S. Scopa, P. Calabrese, and J. Dubail, arXiv:2109.05249 (2021).
  * Eisler and Maislinger [2020] V. Eisler and F. Maislinger, SciPost Phys. 8, 37 (2020).
  * Gruber and Eisler [2021] M. Gruber and V. Eisler, SciPost Phys. 10, 5 (2021).
  * Eisler [2021] V. Eisler, J. Phys. A: Math. Theor. 54, 424002 (2021).
  * Bidzhiev  _et al._ [2021] K. Bidzhiev, M. Fagotti, and L. Zadnik, arXiv:2108.03230 (2021).
  * [75] L. Zadnik, S. Bocini, K. Bidzhiev, and M. Fagotti, to appear.
  * Prosen [2014] T. Prosen, Nuclear Physics B 886, 1177 (2014).
  * Pereira  _et al._ [2014] R. G. Pereira, V. Pasquier, J. Sirker, and I. Affleck, J. Stat. Mech. 2014, P09037 (2014).
  * Medenjak  _et al._ [2020a] M. Medenjak, B. Buča, and D. Jaksch, Phys. Rev. B 102, 041117 (2020a).
  * Medenjak  _et al._ [2020b] M. Medenjak, T. Prosen, and L. Zadnik, SciPost Phys. 9, 3 (2020b).
  * Note [2] Note that this problem could be solved exactly using the results of Refs [84] and [85].
  * Note [3] For example, Refs [22, 86] point to the melting of order after global quenches, so localised perturbations in ground states with symmetry breaking are not expected to survive global quenches.
  * Gluza  _et al._ [2019] M. Gluza, J. Eisert, and T. Farrelly, SciPost Phys. 7, 38 (2019).
  * Murthy and Srednicki [2019] C. Murthy and M. Srednicki, Phys. Rev. E 100, 012146 (2019).
  * Pozsgay [2017] B. Pozsgay, J. Phys. A: Math. Theor. 50, 074006 (2017).
  * Collura  _et al._ [2018] M. Collura, A. De Luca, and J. Viti, Phys. Rev. B 97, 081111 (2018).
  * Collura and Essler [2020] M. Collura and F. H. L. Essler, Phys. Rev. B 101, 041110 (2020).