# The square negative correlation property for generalized Orlicz balls

Jakub Onufry Wojtaszczyk
Department of Mathematics, Computer Science and Mechanics
University of Warsaw
ul. Banacha 2, 02-097 Warsaw, Poland
email<EMAIL_ADDRESS>
(September 11, 2005)

###### Abstract

Recently Antilla, Ball and Perissinaki proved that the squares of coordinate
functions in $l_{p}^{n}$ are negatively correlated. This paper extends their
results to balls in generalized Orlicz norms on ${\mathbb{R}}^{n}$. From this,
the concentration of the Euclidean norm and a form of the Central Limit
Theorem for the generalized Orlicz balls is deduced. Also, a counterexample
for the square negative correlation hypothesis for 1-symmetric bodies is
given.

##  1 Introduction

Given a convex, central-symmetric body $K\subset{\mathbb{R}}^{n}$ of volume 1,
consider the random variable $X=(X_{1},X_{2},\ldots,X_{n})$, uniformly
distributed on $K$. We are interested in determining whether the vector has
the square negative correlation, i.e. if

${\rm
cov}(X_{i}^{2},X_{j}^{2}):={\mathbb{E}}(X_{i}^{2}X_{j}^{2})-{\mathbb{E}}X_{i}^{2}{\mathbb{E}}X_{j}^{2}\leq
0.$

We assume that $K$ is in isotropic position, i.e. that

${\mathbb{E}}X_{i}=0\hbox{\ \ \ and \ \ \ }{\mathbb{E}}X_{i}\cdot
X_{j}=L_{K}^{2}\delta_{ij},$

where $\delta_{ij}$ is the Kronecker delta and $L_{K}$ is a positive constant.
Since any convex body not supported on an affine subspace has an affine image
which is in isotropic position, this is not a restrictive assumption.

The motivation in studying this problem comes from the so-called central limit
problem for convex bodies, which is to show that most of the one-dimensional
projections of the uniform measure on a convex body are approximately normal.
It turns out that the bounds on the square correlation can be crucial to
estimating the distance between the one-dimensional projections and the normal
distribution (see for instance [ABP03], [MM05]). A related problem is to
provide bounds for the quantity $\sigma_{K}$, defined by

$\sigma_{K}^{2}=\frac{{\rm Var}(|X|^{2})}{nL_{K}^{4}}=\frac{n{\rm
Var}(|X|^{2})}{({\mathbb{E}}|X|^{2})^{2}},$

where $X$ is uniformly distributed on $K$. It is conjectured (see for instance
[BK03]) that $\sigma_{K}$ is bounded by a universal constant for any convex
symmetric isotropic body. Recently Antilla, Ball and Perissinaki (see [ABP03])
observed that for $K=l_{p}^{n}$ the covariances of $X_{i}^{2}$ and $X_{j}^{2}$
are negative for $i\neq j$, and from this deduced a bound on $\sigma_{K}$ in
this class.

In this paper we shall study the covariances of $X_{i}^{2}$ and $X_{j}^{2}$
(or, more generally, of any functions depending on a single variable) on a
convex, symmetric and isotropic body. We will show a general formula to
calculate the covariance for given functions and $K$, and from this formula
deduce the covariance of any increasing functions of different variables, in
particular of the functions $X_{i}^{2}$ and $X_{j}^{2}$, has to be negative on
generalized Orlicz balls. Then we follow [ABP03] to arrive at a concentration
property and [MM05] to get a Central Limit Theorem variant for generalized
Orlicz balls.

The layout of this paper is as follows. First we define notations which will
be used throughout the paper. In Section 2 we transform the formula for the
square correlation into a form which will be used further on. In Section 3 we
use the formula and the Brunn-Minkowski inequality to arrive at the square
negative correlation property for generalized Orlicz balls. In Section 4 we
show the corollaries, in particular a central-limit theorem for generalized
Orlicz balls. Section 5 contains another application of the formula from
Section 2, a simple counterexample for the square negative correlation
hypothesis for 1-symmetric bodies.

### Notation

Throughout the paper $K\subset{\mathbb{R}}^{n}$ will be a convex central-
symmetric body of volume 1 in isotropic position. Recall that by isotropic
position we mean that for any vector $\theta\in S^{n-1}$ we have
$\int_{K}\left\langle\theta,x\right\rangle^{2}dx=L_{K}^{2}$ for some constant
$L_{K}$. For $A\subset{\mathbb{R}}^{n}$ by $|A|$ we will denote the Lebesgue
volume of $A$. For $x\in{\mathbb{R}}^{n}$, $|x|$ will mean the Euclidean norm
of $x$. We assume that ${\mathbb{R}}^{n}$ is equipped with the standard
Euclidean structure and with the canonic orthonormal base
$(e_{1},\ldots,e_{n})$. For $x\in{\mathbb{R}}^{n}$ by $x_{i}$ we shall denote
the $i$th coordinate of $x$, i.e. $\left\langle e_{i},x\right\rangle$. We will
consider $K$ as a probability space with the Lebesgue measure restricted to
$K$ as the probability measure. If there is any danger of confusion, then
$\mathbb{P}_{K}$ will denote the probability with respect to this measure,
${\mathbb{E}}_{K}$ will denote the expected value with respect to
$\mathbb{P}_{K}$, and so on. By $X$ we will usually denote the $n$-dimensional
random vector equidistributed on $K$, while $X_{i}$ will denote its $i$th
coordinate. By the covariance ${\rm cov}(Y,Z)$ for real random variables $Y$,
$Z$ we mean ${\mathbb{E}}(YZ)-{\mathbb{E}}Y{\mathbb{E}}Z$. By an 1-symmetric
body $K$ we mean one that is invariant under reflections in the coordinate
hyperplanes, or equivalently, such a body that $(x_{1},x_{2},\ldots,x_{n})\in
X{\Longleftrightarrow}({\varepsilon}_{1}x_{1},{\varepsilon}_{2}x_{2},\ldots,{\varepsilon}_{n}x_{n}\in
X)$ for any choice of ${\varepsilon}_{i}\in\\{-1,1\\}$. The parameter
$\sigma_{K}$, as in [BK03], will be defined by

$\sigma_{K}^{2}=\frac{{\rm Var}(|X|^{2})}{nL_{K}^{4}}=\frac{n{\rm
Var}(|X|^{2})}{({\mathbb{E}}|X|^{2})^{2}}.$

For any $n\geq 1$ and convex increasing functions
$f_{i}:[0,\infty){\rightarrow}[0,\infty)$, $i=1,\ldots,n$ satisfying
$f_{i}(0)=0$ (called the Young functions) we define the generalized Orlicz
ball $K\subset{\mathbb{R}}^{n}$ to be the set of points
$x=(x_{1},\ldots,x_{n})$ satisfying

$\sum_{i=1}^{n}f_{i}(|x_{i}|)\leq 1.$

This is easily proven to be convex, symmetric and bounded, thus

$\|x\|=\inf\\{\lambda:x\in\lambda K\\}$

defines a norm on ${\mathbb{R}}^{n}$. In the case of equal functions $f_{i}$
the norm is called an Orlicz norm, in the general case a generalized Orlicz
norm. Examples of Orlicz norms include the $l_{p}$ norms for any $p\geq 1$
with $f(t)=|t|^{p}$ being the Young functions. The generalized Orlicz spaces
are also referred to as modular sequence spaces (I thank the referee for
pointing this out to me).

##  2 The general formula

We wish to calculate ${\rm cov}(f(X_{i}),g(X_{j}))$, where $f$ and $g$ are
univariate functions, $i\neq j$ and $X_{i},X_{j}$ are the coordinates of the
random vector $X$, equidistributed on a convex, symmetric and isotropic body
$K$. For simplicity we will assume $i=1$, $j=2$ and denote $X_{1}$ by $Y$ and
$X_{2}$ by $Z$. For any $(y,z)\in{\mathbb{R}}^{2}$ let $m(y,z)$ be equal to
the $n-2$-dimensional Lebesgue measure of the set
$(\\{(y,z)\\}\times{\mathbb{R}}^{n-2})\cap K$. We set out to prove:

######  Theorem 2.1.

For any symmetric, convex body $K$ in isotropic position and any functions
$f$, $g$ we have

${\rm
cov}(f(Y),g(Z))=\int_{{\mathbb{R}}^{4},|y|>|{\bar{y}}|,|z|>|{\bar{z}}|}\big{(}m(y,z)m({\bar{y}},{\bar{z}})-m(y,{\bar{z}})m({\bar{y}},z)\big{)}\big{(}f(y)-f({\bar{y}})\big{)}\big{(}g(z)-g({\bar{z}})\big{)}.$

Furthermore, for 1-symmetric bodies and symmetric functions we will have the
following corrolary:

######  Corollary 2.2.

For any symmetric, convex, uncondtitional body $K$ in isotropic position and
symmetric functions $f$, $g$ we have

${\rm cov}(f(Y),g(Z))=16\
\int_{{\mathbb{R}}^{4},y>{\bar{y}}>0,z>{\bar{z}}>0}\big{(}m(y,z)m({\bar{y}},{\bar{z}})-m(y,{\bar{z}})m({\bar{y}},z)\big{)}\big{(}f(y)-f({\bar{y}})\big{)}\big{(}g(z)-g({\bar{z}})\big{)}.$

The corollary is a simple consequence of the fact that for symmetric functions
$f$ and $g$ and an 1-symmetric body $K$ the integrand is invariant under the
change of the sign of any of the variables, so we may assume all of them are
positive.

As concerns the sign of ${\rm cov}(f,g)$, which is what we set out to
determine, we have the following simple corollary:

######  Corollary 2.3.

For any central-symmetric, convex, 1-symmetric body $K$ in isotropic position
and symmetric functions $f$, $g$ that are non-decreasing on $[0,\infty)$ if
for all $y>{\bar{y}}>0$, $z>{\bar{z}}>0$ we have

$m(y,{\bar{z}})m({\bar{y}},z)\geq m(y,z)m({\bar{y}},{\bar{z}}),$ (1)

then

${\rm cov}(f,g)\leq 0.$

Similarly, if the opposite inequality is satisfied for all $y>{\bar{y}}>0$ and
$z>{\bar{z}}>0$, then the covariance is non-negative.

###### Proof.

The second and third bracket of the integrand in Corollary 2.2 is positive
under the assumptions of Corollary 2.3. Thus if we assume the first bracket is
negative, then the whole integrand is negative, which implies the integral is
negative, and vice-versa.∎

###### Proof of Theorem 2.1.

We have

${\rm cov}(f(Y),g(Z))={\mathbb{E}}f(Y)g(Z)-{\mathbb{E}}f(Y){\mathbb{E}}g(Z).$

From the Fubini theorem we have

${\mathbb{E}}f(Y)g(Z)=\int_{R^{2}}m(y,z)f(y)g(z),$

and similar equations for ${\mathbb{E}}f(Y)$ and ${\mathbb{E}}g(Z)$.

For any function $h$ of two variables $a,b\in A$ we can write
$\int_{A^{2}}h(a,b)=\int_{A^{2}}h(b,a)=\frac{1}{2}\int_{A^{2}}h(a,b)+h(b,a)$.
We shall repeatedly use this trick to transform the formula for the covariance
of $f$ and $g$ into the required form:

$\displaystyle{\mathbb{E}}f(Y){\mathbb{E}}g(Z)$ $\displaystyle=$
$\displaystyle\int_{{\mathbb{R}}^{2}}m(y,z)f(y)\int_{{\mathbb{R}}^{2}}m({\bar{y}},{\bar{z}})g({\bar{z}})$
$\displaystyle=$
$\displaystyle\int_{{\mathbb{R}}^{4}}m(y,z)m({\bar{y}},{\bar{z}})f(y)g({\bar{z}})=\int_{{\mathbb{R}}^{4}}m({\bar{y}},{\bar{z}})m(y,z)f({\bar{y}})g(z)=$
$\displaystyle=$
$\displaystyle\frac{1}{2}\int_{{\mathbb{R}}^{4}}m({\bar{y}},{\bar{z}})m(y,z)\big{(}f({\bar{y}})g(z)+f(y)g({\bar{z}})\big{)}.$

We repeat this trick, exchanging $z$ and ${\bar{z}}$ (and leaving $y$ and
${\bar{y}}$ unchanged):

$\displaystyle{\mathbb{E}}f(Y){\mathbb{E}}g(Z)$ $\displaystyle=$
$\displaystyle\frac{1}{4}\int_{{\mathbb{R}}^{4}}m({\bar{y}},{\bar{z}})m(y,z)\big{(}f(y)g({\bar{z}})+f({\bar{y}})g(z)\big{)}+m({\bar{y}},z)m(y,{\bar{z}})\big{(}f(y)g(z)+f({\bar{y}})g({\bar{z}})\big{)}.$

We perform the same operations on the second part of the covariance. To get a
integral over ${\mathbb{R}}^{4}$ we multiply by an ${\mathbb{E}}1$ factor
(this in effect will free us from the assumption that the body’s volume is 1):

$\displaystyle{\mathbb{E}}f(Y)g(Z){\mathbb{E}}1$ $\displaystyle=$
$\displaystyle\int_{{\mathbb{R}}^{4}}m(y,z)m({\bar{y}},{\bar{z}})f(y)g(z)$
$\displaystyle=$
$\displaystyle\frac{1}{4}\int_{{\mathbb{R}}^{4}}m(y,z)m({\bar{y}},{\bar{z}})\big{(}f(y)g(z)+f({\bar{y}})g({\bar{z}})\big{)}+m(y,{\bar{z}})m({\bar{y}},z)\big{(}f(y)g({\bar{z}})+f({\bar{y}})g(z)\big{)}.$

Thus:

$\displaystyle{\rm
cov}(f(Y),g(Z))={\mathbb{E}}(f(Y)g(Z)){\mathbb{E}}1-{\mathbb{E}}f(Y){\mathbb{E}}g(Z)=$
$\displaystyle=$
$\displaystyle\frac{1}{4}\bigg{(}\int_{{\mathbb{R}}^{4}}m(y,z)m({\bar{y}},{\bar{z}})\big{(}f(y)g(z)+f({\bar{y}})g({\bar{z}})\big{)}+m(y,{\bar{z}})m({\bar{y}},z)\big{(}f(y)g({\bar{z}})+f({\bar{y}})g(z)\big{)}-$
$\displaystyle-m({\bar{y}},{\bar{z}})m(y,z)\big{(}f(y)g({\bar{z}})+f({\bar{y}})g(z)\big{)}-m({\bar{y}},z)m(y,{\bar{z}})\big{(}f(y)g(z)+f({\bar{y}})g({\bar{z}})\big{)}\bigg{)}=$
$\displaystyle=$
$\displaystyle\frac{1}{4}\int_{{\mathbb{R}}^{4}}\bigg{(}\big{(}m(y,{\bar{z}})m({\bar{y}},z)-m(y,z)m({\bar{y}},{\bar{z}})\big{)}\big{(}f(y)g({\bar{z}})+f({\bar{y}})g(z)\big{)}+$
$\displaystyle+\big{(}m(y,z)m({\bar{y}},{\bar{z}})-m({\bar{y}},z)m(y,{\bar{z}})\big{)}\big{(}f(y)g(z)+f({\bar{y}})g({\bar{z}})\big{)}\bigg{)}=$
$\displaystyle=$
$\displaystyle\frac{1}{4}\int_{{\mathbb{R}}^{4}}\big{(}m(y,{\bar{z}})m({\bar{y}},z)-m(y,z)m({\bar{y}},{\bar{z}})\big{)}\big{(}f(y)g({\bar{z}})+f({\bar{y}})g(z)-f(y)g(z)-f({\bar{y}})g({\bar{z}})\big{)}=$
$\displaystyle=$
$\displaystyle\frac{1}{4}\int_{{\mathbb{R}}^{4}}\big{(}m(y,{\bar{z}})m({\bar{y}},z)-m(y,z)m({\bar{y}},{\bar{z}})\big{)}\big{(}f(y)-f({\bar{y}})\big{)}\big{(}g({\bar{z}})-g(z)\big{)}$

Finally, notice that if we exchange $y$ and ${\bar{y}}$ in the above formula,
then the formula’s value will not change — the first and second bracket will
change signs, and the third will remain unchanged. The same applies to
exchanging $z$ and ${\bar{z}}$. Thus

${\rm
cov}(f,g)=\int_{{\mathbb{R}}^{4},|y|>|{\bar{y}}|,|z|>|{\bar{z}}|}\big{(}m(y,z)m({\bar{y}},{\bar{z}})-m(y,{\bar{z}})m({\bar{y}},z)\big{)}\big{(}f(y)-f({\bar{y}})\big{)}\big{(}g(z)-g({\bar{z}})\big{)}.$

∎

##  3 Generalized Orlicz spaces

Now we will concentrate on the case of symmetric, non-decreasing functions on
generalized Orlicz spaces. We will prove the inequality (1):

######  Theorem 3.1.

If $K$ is a ball in an generalized Orlicz norm on ${\mathbb{R}}^{n}$, then for
any $y>{\bar{y}}>0$ and $z>{\bar{z}}>0$ we have

$m(y,{\bar{z}})m({\bar{y}},z)\geq m(y,z)m({\bar{y}},{\bar{z}}).$ (2)

From this Theorem and Corollary 2.3 we get

######  Corollary 3.2.

If $K$ is a ball in an generalized Orlicz norm on ${\mathbb{R}}^{n}$ and $f,g$
are symmetric functions that are non-decreasing on $[0,\infty)$, then ${\rm
cov}_{K}(f,g)\leq 0$.

It now remains to prove the inequality (2).

###### Proof of Theorem 3.1.

Let $f_{i}$ denote the Young functions of $K$. Let us consider the ball
$K^{\prime}\subset{\mathbb{R}}^{n-1}$, being an generalized Orlicz ball
defined by the Young functions $\Phi_{1},\Phi_{2},\ldots,\Phi_{n-1}$, where
$\Phi_{i}(t)=f_{i+1}(t)$ for $i>1$ and $\Phi_{1}(t)=t$ — that is, we replace
the first two Young functions of $K$ by a single identity function.

For any $x\in{\mathbb{R}}$ let $P_{x}$ be the set
$(\\{x\\}\times{\mathbb{R}}^{n-2})\cap K^{\prime}$, and $|P_{x}|$ be its
$n-2$-dimensional Lebesgue measure. $K^{\prime}$ is a convex set, thus, by the
Brunn-Minkowski inequality (see for instance [G02]) the function
$x\mapsto|P_{x}|$ is a logarithmically concave function. This means that
$x\mapsto\log|P_{x}|$ is a concave function, or equivalently that

$|P_{tx+(1-t)y}|\geq|P_{x}|^{t}\cdot|P_{y}|^{1-t}.$

In particular, for given real positive numbers $a$, $b$, $c$ we have

$|P_{a+c}|\geq|P_{a}|^{b/(b+c)}|P_{a+b+c}|^{c/(b+c)},$
$|P_{a+b}|\geq|P_{a}|^{c/(b+c)}|P_{a+b+c}|^{b/(b+c)},$

and as a consequence when we multiply the two inequalities,

$|P_{a+b}|\cdot|P_{a+c}|\geq|P_{a}|\cdot|P_{a+b+c}|.$ (3)

Now let us consider the ball $K$. Let us take any $y>{\bar{y}}>0$ and
$z>{\bar{z}}>0$. Let $a=f_{1}({\bar{y}})+f_{2}({\bar{z}})$,
$b=f_{1}(y)-f_{1}({\bar{y}})$, and $c=f_{2}(z)-f_{2}({\bar{z}})$. The numbers
$a$, $b$ and $c$ are positive from the assumptions on $y$, $z$, ${\bar{y}}$
and ${\bar{z}}$ and because the Young functions are increasing. Then
$m({\bar{y}},{\bar{z}})$ is equal to the measure of the set

$\\{x_{3},x_{4},\ldots,x_{n}:f_{1}({\bar{y}})+f_{2}({\bar{z}})+\sum_{i=3}^{n}f_{i}(x_{i})\leq
1\\}=\\{x_{3},x_{4},\ldots,x_{n}:a+\sum_{i=2}^{n}\Phi_{i}(x_{i})\leq
1\\}=P_{a}.$

Similarly $m(y,{\bar{z}})=|P_{a+b}|$, $m({\bar{y}},z)=|P_{a+c}|$ i
$m(y,z)=|P_{a+b+c}|$.

Substituting those values into the inequality (3) we get the thesis:

$m(y,{\bar{z}})m({\bar{y}},z)\geq m(y,z)m({\bar{y}},{\bar{z}}).$

∎

##  4 The consequences

For the consequences we will take $f(t)=g(t)=t^{2}$. The first simple
consequence is the concentration property for generalized Orlicz balls. Here,
we follow the argument of [ABP03] for $l_{p}$ balls.

######  Theorem 4.1.

For every generalized Orlicz ball $K\subset{\mathbb{R}}^{n}$ we have

$\sigma_{K}\leq\sqrt{5}.$

###### Proof.

From the Cauchy-Schwartz inequality we have

$n^{2}L_{K}^{4}=\bigg{(}\sum_{i=1}^{n}{\mathbb{E}}_{K}X_{i}^{2}\bigg{)}^{2}=\bigg{(}{\mathbb{E}}_{K}|X|^{2}\bigg{)}^{2}\leq{\mathbb{E}}_{K}|X|^{4}.$

On the other hand from Corollary 3.2 we have

$\displaystyle{\mathbb{E}}_{K}|X|^{4}$ $\displaystyle=$
$\displaystyle{\mathbb{E}}_{K}\bigg{(}\sum_{i=1}^{n}X_{i}^{2}\bigg{)}^{2}=\sum_{i=1}^{n}{\mathbb{E}}_{K}X_{i}^{4}+\sum_{i\neq
j}{\mathbb{E}}_{K}X_{i}^{2}X_{j}^{2}$ $\displaystyle\leq$
$\displaystyle\sum_{i=1}^{n}{\mathbb{E}}_{K}X_{i}^{4}+\sum_{i\neq
j}{\mathbb{E}}_{K}X_{i}^{2}{\mathbb{E}}_{K}X_{j}^{2}$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{n}{\mathbb{E}}_{K}X_{i}^{4}+n(n-1)L_{K}^{4}.$

As for 1-symmetric bodies the density of $X_{i}$ is symmetric and log-concave,
we know (see e.g. [KLO96], Section 2, Remark 5)

${\mathbb{E}}_{K}X_{i}^{4}\leq
6\bigg{(}{\mathbb{E}}_{K}X_{i}^{2}\bigg{)}^{2}=6L_{K}^{4},$

thence

$n^{2}L_{K}^{4}\leq{\mathbb{E}}_{K}|X|^{4}\leq(n^{2}+5n)L_{K}^{4}.$

This gives us

${\rm Var}(|X|^{2})={\mathbb{E}}_{K}|X|^{4}-n^{2}L_{K}^{4}\leq 5nL_{K}^{4},$

and thus

$\sigma_{K}^{2}=\frac{{\rm Var}|X|^{2}}{nL_{K}^{4}}\leq 5.$

∎

######  Corollary 4.2.

For every generalized Orlicz ball $K\subset{\mathbb{R}}^{n}$ and for every
$t>0$ we have

$\mathbb{P}_{K}\bigg{(}\bigg{|}\frac{|X|^{2}}{n}-L_{K}^{2}\bigg{|}\geq
t\bigg{)}\leq\frac{5L_{K}^{4}}{nt^{2}}$

and

$\mathbb{P}_{K}\bigg{(}\bigg{|}\frac{|X|}{\sqrt{n}}-L_{K}\bigg{|}\geq
t\bigg{)}\leq\frac{5L_{K}^{2}}{nt^{2}}$

###### Proof.

From the estimate on the variance of $|X|^{2}$ and Chebyshev’s inequality we
get

$t^{2}\mathbb{P}_{K}\bigg{(}\bigg{|}\frac{|X|^{2}}{n}-L_{K}^{2}\bigg{|}\geq
t\bigg{)}\leq{\mathbb{E}}_{K}\bigg{(}\frac{|X|^{2}}{n}-L_{K}^{2}\bigg{)}^{2}\leq\frac{1}{n^{2}}{\rm
Var}(|X|^{2})\leq\frac{5}{n}L_{K}^{4}.$

For the second part let $t>0$. We have

$\displaystyle\mathbb{P}_{K}(|X|-\sqrt{n}L_{K}|\geq t\sqrt{n})$
$\displaystyle\leq$ $\displaystyle\mathbb{P}_{K}(|X|^{2}-nL_{K}^{2}|\geq
tnL_{K})$ $\displaystyle\leq$
$\displaystyle\frac{5L_{K}^{4}}{t^{2}nL_{K}^{2}}=\frac{5L_{K}^{2}}{t^{2}n}.$

∎

This result confirms the so-called concentration hypothesis for generalized
Orlicz balls. The hypothesis, see e.g. [BK03], states that the Euclidean norm
concentrates near the value $\sqrt{n}L_{K}$ as a function on $K$. More
precisely, for a given ${\varepsilon}>0$ we say that $K$ satisfies the
${\varepsilon}$-concentration hypothesis if

${\mathbb{P}}_{K}\bigg{(}\bigg{|}\frac{|X|}{\sqrt{n}}-L_{K}\bigg{|}\geq{\varepsilon}L_{K}\bigg{)}\leq{\varepsilon}.$

From Corollary 4.2 we get that the class of generalized Orlicz balls satisfies
the ${\varepsilon}$-concentration hypothesis with
${\varepsilon}=\sqrt{5}n^{-1/3}$.

A more complex consequence is the Central Limit Property for generalized
Orlicz balls. For $\theta\in S^{n-1}$ let $g_{\theta}(t)$ be the density of
the random variable $\left\langle X,\theta\right\rangle$. Let $g$ be the
density of ${\mathcal{N}}(0,L_{K}^{2})$. Then for most $\theta$ the density
$g_{\theta}$ is very close to $g$. More precisely, by part 2 of Corollary 4 in
[MM05] we get

######  Corollary 4.3.

There exists an absolute constant $c$ such that

$\sup_{t\in{\mathbb{R}}}\bigg{|}\int_{-\infty}^{t}\big{(}g_{\theta}(s)-g(s)\big{)}ds\bigg{|}\leq
c\|\theta\|_{3}^{3/2}.$

##  5 The counterexample for 1-symmetric bodies

It is generally known that the negative square correlation hypothesis does not
hold in general in the class of 1-symmetric bodies. However, the formula from
section 2 allows us to give a counterexample without any tedious calculations.
Let $K\subset{\mathbb{R}}^{3}$ be the ball of the norm defined by

$\|(x,y,z)\|=|x|+\max\\{|y|,|z|\\}.$

The quantity $m(y,z)$ considered in Corollary 2.3, defined as the volume of
the cross-section $({\mathbb{R}}\times\\{y,z\\})\cap K$ is equal to
$2(1-\max\\{|y|,|z|\\})$ for $|y|,|z|\leq 1$ and $0$ for greater $|y|$ or
$|z|$. To check the inequality (1) for $y>{\bar{y}}>0$ and $z>{\bar{z}}>0$ we
may assume without loss of generality that $y\geq z$ (as $K$ is invariant
under the exchange of $y$ and $z$). We have

$\displaystyle m(y,{\bar{z}})m({\bar{y}},z)$ $\displaystyle-$ $\displaystyle
m(y,z)m({\bar{y}},{\bar{z}})=$ $\displaystyle=$ $\displaystyle
4(1-\max\\{y,{\bar{z}}\\})(1-\max\\{{\bar{y}},z\\})-4(1-\max\\{y,z\\})(1-\max\\{{\bar{y}},{\bar{z}}\\})$
$\displaystyle=$ $\displaystyle
4(1-y)(1-\max\\{{\bar{y}},z\\})-4(1-y)(1-\max\\{{\bar{y}},{\bar{z}}\\})$
$\displaystyle=$ $\displaystyle
4(1-y)(\max\\{{\bar{y}},{\bar{z}}\\}-\max\\{{\bar{y}},z\\}).$

As $y\leq 1$ all we have to consider is the sign of the third bracket.
However, as $z>{\bar{z}}$, the third bracket is never positive, and is
negative when $z>{\bar{y}}$. Thus from Corollary 2.3 the covariance ${\rm
cov}(f,g)$ is positive for any increasing symmetric functions $f(Y)$ and
$g(Z)$, in particular for $f(Y)=Y^{2}$ and $g(Z)=Z^{2}$.

## References

  * [ABP03] M. Anttila, K. Ball and I. Perissinaki, The central limit problem for convex bodies. Trans. Amer. Math. Soc., 355 (2003), pp. 4723–-4735.
  * [BK03] S. G. Bobkov and A. Koldobsky, On the Central Limit Property of Convex Bodies. GAFA Seminar, Lecture Notes in Math. 1807 (2003), pp. 44–52.
  * [G02] R. J. Gardner, The Brunn-Minkowski Inequality, Bull. Amer. Math. Soc. 39 (2002), pp. 355-405
  * [KLO96] S. Kwapień, R. Latała and K. Oleszkiewicz, Comparison of Moments of Sums of Independent Random Variables and Differential Inequalities. Journal of Functional Analysis, 136 (1996), pp. 258–268.
  * [MM05] E. Meckes and M. Meckes, The Central Limit Problem for Random Vectors with Symmetries. Preprint. Available at http://arxiv.org/abs/math.PR/0505618.


# Quasiballistic transport in long-range anisotropic Heisenberg model

M. Mierzejewski  J. Wronowicz  J. Pawłowski  J. Herbrych  Department of
Theoretical Physics, Faculty of Fundamental Problems of Technology, Wrocław
University of Science and Technology, 50-370 Wrocław, Poland

(August 27, 2024)

###### Abstract

Purely ballistic transport is a rare feature even for integrable models. By
numerically studying the Heisenberg chain with the power-law exchange,
$J\propto 1/r^{\alpha}$, where $r$ is a distance, we show that for spin
anisotropy $\Delta\simeq\exp(-\alpha+2)$ the system exhibits a quasiballistic
spin transport and the presence of fermionic excitation which do not decay up
to extremely long times $\sim 10^{3}/J$. This conclusion is reached on the
base of the dynamics of spin domains, the dynamical spin conductivity,
inspecting the matrix elements of the spin-current operator, and by the
analysis of most conserved operators. Our results smoothly connects two models
where fully ballistic transport is present: free particles with nearest-
neighbor hopping and the isotropic Haldane-Shastry model.

##  I Introduction

Range and type of interaction play the most crucial role in determining the
properties of quantum many-body systems. Phenomena like ballistic propagation
in integrable one-dimensional (1D) Heisenberg model Zotos1997 ; Bertini2021 ,
exotic magnetism due to frustration Nisoli2017 , unusual phase transitions
Sandvik2010 ; Yang2021 , highly entangled spin liquids Balents2010 , to name a
few, essentially depend on the type of exchange present in the system.
Nowadays, the experimental progress in quantum simulators, i.e., cold-atoms in
optical lattice Bloch2006 ; Hild2014 ; Gross2017 ; Zhang2017 ; Schafer2020 ;
Jepsen2020 ; Wei2022 , also with Rydberg states Weimer2010 ; Anderson2011 ;
Labuhn2016 ; Bernien2017 ; Sanchez2018 ; Browaeys2020 ; Borish2020 ; Geier2021
; Hollerith2022 ; Scholl2022 ; Steinert2022 , allow for the study of such
systems with unprecedented precision. Furthermore, it allows also for the
discovery of new phenomena. Especially the systems with direct long-range
exchange (much beyond the nearest-neighbor spacing) are of great interest
since they appear only as an effective low-energy description in the solid-
state setups. For example, the transverse Ising model with power-law decaying
long-range interaction was successfully created Britton2012 ; Joshi2022  and
studied Richerme2014 ; Jurcevic2014  in the context of the Lieb-Robinson bound
on the information propagation in the system.

Although in recent years, there was a lot of work devoted to the nonlocality
in the long-range models Hauke2013 ; Feug2015 ; Maghrebi2016 ; Lepori2017 ;
Frerot2017 ; Vanderstraeten2018 ; Cevolani2018 ; Kloss2019 ; Lerose2019 ;
Ren2020 ; Schneider2021 ; Birnkammer2022 ; Bulchandani2022 , the dynamical
properties of such systems are mostly unknown. Since the latter are often
directly probed in the experimental setups (e.g., in the density expansion
experiments), in this work we discuss the spin transport in the anisotropic
Heisenberg chain with long-range exchange, $J(r)=J/r^{\alpha}$. Studying the
dynamics of spin domains (Sec. II), the dynamical spin conductivity (Sec.
III), inspecting the matrix elements of the spin-current operator (Sec. IV),
and analyzing the most conserved current operators (Sec. V), we show that the
model exhibits a transient ballistic transport persisting up to very long
times $t/J\sim 10^{2}-10^{3}$. Such quasiballistic behaviour is shown to
exists along a sharp line in the space of model parameters that smoothly
interpolates between two purely ballistic models: free particles with nearest-
neighbor hopping for $\alpha\to\infty$ and the isotropic Haldane-Shastry model
for $\alpha=2$. The former is trivially exactly solvable Lieb1961  via the
Fourier transform to the momentum space. The latter was originally introduced
independently by Haldane Haldane1988  and Shastry Shastry1988  as a solution
of the 1D resonating-valence-bond state Anderson1973 . Interestingly, not only
these models are integrable, but also the spin current, $j$, is a constant of
motion of these Hamiltonians, $[H,j]=0$ in the thermodynamic limit Essler1995
; Greiter2005 ; Sechin2018 . One should contrast this behaviour with the
nearest-neighbor anisotropic Heisenberg model (nn-AHM) which is integrable but
$[H,j]\neq 0$.

In the following, we study the 1D spin-$1/2$ anisotropic Heisenberg model with
long-range exchange (long-range AHM),

$H=\sum_{\ell,r}J(r)\left[\frac{1}{2}\left(S^{+}_{\ell}S^{-}_{\ell+r}+S^{-}_{\ell}S^{+}_{\ell+r}\right)+\Delta
S^{z}_{\ell}S^{z}_{\ell+r}\right]\,,$ (1)

with $\Delta$ as an anisotropy in $z$-direction and $J(r)=J/r^{\alpha}$ as a
long-range power-law spin exchange, where $J=1$ sets the unit of energy (as
well as $\hbar=k_{B}=1$) and $\alpha>1$ controls the decay of exchange. The
$\alpha\to\infty$ limit recovers the nn-AHM, which is integrable and can be
mapped via the Jordan-Wigner transformationzvyagin  to spinless fermions
interacting with the strength $\Delta$. Note that for any finite $\alpha$,
this transformation maps the spin chain on interacting fermions also for
$\Delta=0$.

Figure 1: (a) Time-dependence of the spin density, $\langle
S^{z}_{\ell}\rangle$, for the expansion from the nearly down-polarized state
with $5$ up-spins at the boundary of a $L=45$ sites chain. (b,c) Time-
dependence of the first density moment, i.e., of the center of the mass,
$r_{\mathrm{cm}}(t)$ for (b) $\Delta=0.2$ and (c) $\Delta=0.5$ anisotropy, and
various values of exchange decay parameter $\alpha\in\\{2.0,2.5,\dots,5.0\\}$.
Insets of (b,c) depict comparison between $r_{\mathrm{cm}}(t)$ and extracted
$\overline{\nu}$ for the optimal values of parameter $\alpha$. (d,e)
Corresponding dependence of the velocity $\nu(t)$ obtained as a time-
derivative of $r_{\mathrm{cm}}(t)$ from panels (c,d) . Shaded region depict
the time window from which average velocity $\bar{\nu}$ is obtained, i.e.,
$t\in[20,30]$. (f) $\alpha$-dependence of $\bar{\nu}$ for various anisotropy
$\Delta$. In the same panel we present also the average velocity for the
optimal anisotropy $\Delta_{\mathrm{O}}$. Error bars depict the standard
deviation of $\overline{\nu}$. Inset: $\alpha$\- and $\Delta$-dependence the
averaged velocity $\bar{\nu}$. Points represent the anisotropy
$\Delta_{\nu_{\mathrm{max}}}$ at which we find the largest velocity, while
solid line depicts optimal anisotropy $\Delta_{\mathrm{O}}=\exp(-\alpha+2)$.

##  II Spin density expansion

Motivated by the setups studied in the cold-atoms experiments Ronzheimer2013 ;
Vidmar2013 ; Neyenhuis2017 ; Joshi2022 , let us first investigate the
expansion of the spin density starting from the nearly down-polarized product
state (spin domain)

$|\psi(t=0)\rangle=|\underbrace{\uparrow\uparrow\uparrow\uparrow\uparrow}_{x}\underbrace{\downarrow\cdots\downarrow}_{L-x}\rangle\,,$
(2)

with $x=5$ up-spins in a chain with $L=45$ sites (here we assume open boundary
conditions). In order to access longer times of spin expansion, the up-sins
are initially located at the left boundary of the studied system. Utilizing
the Lanczos time evolution method PrelBonLan  we monitor the magnetization
$\langle S^{z}_{\ell}(t)\rangle$ at each site, $\ell$, as exemplified by
results presented in Fig. 1(a). Since the Hamiltonian conserves that total
spin projection $S^{z}_{\rm tot}$, the calculations have been carried out in
the subspace with $S^{z}_{\rm tot}=-35/2$ spanned by $\sim 10^{6}$ basis
states. In order to quantify the dynamics of the system we analyze the
velocity $\nu$ obtained from the time derivative of the first density moment,
i.e., from the “center of the mass”,

$\nu(t)=\frac{\mathrm{d}\,r_{\mathrm{cm}}(t)}{\mathrm{d}t}\,,\quad
r_{\mathrm{cm}}(t)=\frac{\sum_{\ell}\,\ell\left(\langle
S^{z}_{\ell}(t)\rangle+1/2\right)}{\sum_{\ell}\,\left(\langle
S^{z}_{\ell}(t)\rangle+1/2\right)}\,,$ (3)

where the denominator equals $S^{z}_{\rm tot}/2+L/2$. The detailed
$r_{\mathrm{cm}}(t)$ results for exemplary $\Delta=0.2$ and $\Delta=0.5$ and
various values of exchange decay parameter $\alpha\in\\{2.0,2.5,\dots,5.0\\}$
are presented in Fig. 1(b,c), while the corresponding velocity $\nu(t)$ in
Fig. 1(d,e). Presented results reveal that for given $\Delta$ there exist
$\alpha$ for which the $r_{\mathrm{cm}}(t)$ expands with the largest velocity.

In order to quantify the behavior of the system for various parameters
$(\Delta,\alpha)$, in Fig. 1(f) we present $\alpha$-dependence of the velocity
$\overline{\nu}$ averaged over the time interval $t\in[20,30]$ [shaded region
in Fig. 1(d,e)] for various $\Delta\leq 1$ [see also inset of Fig. 1(f)]. As
evident, $\overline{\nu}$ for fixed $\Delta$ has a nonmonotonic dependence on
the decay parameter $\alpha$ with maximum at $\alpha_{\nu_{\mathrm{max}}}$
(equivalently, for fixed $\alpha$ there exist a maximum at some
$\Delta=\Delta_{\nu_{\mathrm{max}}}$). E.g., for a modest value of the
anisotropy $\Delta=0.2$, we find the maximum of $\overline{\nu}$ at
$\alpha_{\nu_{\mathrm{max}}}\simeq 3.5$, while $\Delta\to 1$ has maximum at
$\alpha_{\nu_{\mathrm{max}}}\to 2$. Furthermore, in the inset of Fig. 1(f) we
present also the position of $\Delta_{\nu_{\mathrm{max}}}$ as a function of
the decay parameter $\alpha$. As one of the main results of this work, we show
that the maximum of the velocity at finite times can be found at
$\Delta_{\nu_{\mathrm{max}}}\simeq\Delta_{\mathrm{O}}=\exp(-\alpha+2)$
(presented also in Fig. 6). Interestingly, the velocity is approximately
constant $\overline{\nu}\simeq J/\sqrt{2}$ along the $\Delta_{\mathrm{O}}$
line for $\alpha\geq 2$, see black line in Fig 1(f). Our results indicate that
the largest velocity, $\nu_{\mathrm{max}}\simeq J/\sqrt{2}$, characteristic
for the nn free-fermion model Langer2012 ; Vidmar2013 , is preserved in the
transient dynamics of an interacting chain with the optimal anisotropy
$\Delta_{\mathrm{O}}(\alpha)$.

Figure 2: (a) Integrated spin conductivity $I(\omega)$ of long-range AHM,
calculated for $L=20$ and $\Delta=0.5$. Thick line represents results for the
optimal decay parameter $\alpha_{\sigma}\simeq 2.5$. (b) Size-dependence of
$I(\omega)$ for $\Delta=0.5$ below ($\alpha=2.3$), at ($\alpha=2.5$), and
above ($\alpha=4.0$) the optimal value $\alpha_{\sigma}$. For clarity,
$\alpha=2.5$ and $\alpha=2.0$ results have $0.4$ and $1.2$ offset,
respectively.

##  III Spin conductivity

In order to gain understanding on the spin dynamics in the long-range AHM, we
study the spin conductivity at infinite temperature, see Refs. Bertini2021, ;
jaklic,  for review. This quantity probes the whole eigenspectrum containing
$Z$ states,

$\sigma(\omega)=\frac{\pi}{LZ}\sum_{n,m}\langle
n|j|m\rangle^{2}\,\delta(\omega-\epsilon_{m}+\epsilon_{n})\,,$ (4)

where the current operator is obtained from continuity relation,
$j=i\sum_{\ell}\ell[H,S^{z}_{\ell}]=i/2\sum_{\ell,r}r\,J(r)\left(S^{+}_{\ell}S^{-}_{\ell+r}-\mathrm{H.c.}\right)$
and $H|n\rangle=\epsilon_{n}|n\rangle$. The results for $\sigma(\omega)$
presented in Fig. 2 and Fig. 3 were obtained with help of exact
diagonalization on the system of $L=20$ sites with periodic boundary
conditions PBC  in the largest magnetization sector, i.e., $S^{z}_{\rm
tot}=0$. Furthermore, in order to avoid an arbitrary binning of the finite-
size spectra we present the integrated conductivity normalized to the sum rule

$\displaystyle I(\omega)$ $\displaystyle=$ $\displaystyle\frac{1}{{\cal
S}_{\rm
tot}}\int_{-\omega}^{\omega}\mathrm{d}\omega^{\prime}\sigma(\omega^{\prime})=\frac{\pi}{LZ{\cal
S}_{\rm tot}}\sum_{n,m}\theta_{mn}\langle n|j|m\rangle^{2}\,,$
$\displaystyle{\cal S}_{\rm tot}$ $\displaystyle=$
$\displaystyle\int_{-\infty}^{\infty}\mathrm{d}\omega^{\prime}\sigma(\omega^{\prime})=\frac{\pi}{LZ}\sum_{n,m}\langle
n|j|m\rangle^{2}\,,$ (5)

so that $I(\omega\to\infty)=1$. Here,
$\theta_{mn}=\theta(\omega-|\epsilon_{m}-\epsilon_{n}|)$, where $\theta(x)$ is
the step function.

Figure 3: (a-h) $I(\omega)$ in the long-range AHM, for $L=20$ vs. the exchange
decay parameter $\alpha$ for various $\Delta$. Note that $\omega$ is shown
using the log-scale. Solid lines represent the frequency, $\omega^{*}$, for
which $I(\omega^{*})=0.9$. (i-l) $I(\omega)$ in the generalized HSM for
$L=20$. The dip shown by the line in panel (k) is consistent with the
ballistic transport in the integrable HSM for $\Delta=1$ and $\alpha=2$.

In Fig. 2(a) we show $\alpha$-dependence of $I(\omega)$ for exemplary
$\Delta=0.5$. In the $\alpha\to\infty$ limit we recover the standard behavior
of the integrable nn-AHM, i.e., the dissipationless Drude $\delta$-peak at
$\omega=0$ and a pronounced incoherent (regular) spectrum at $\omega/J\sim
0.1-1$ Prelovsek2021 . Upon increasing the range of spin exchange ($\alpha$
decreases down to $\alpha\simeq 10$) we observe broadening of the Drude
$\delta$-peak, with no effect on the incoherent part of the spectrum. Such
behavior is consistent with the integrability-breaking of the nn-AHM. However,
for $\alpha<10$, the incoherent part of $\sigma(\omega)$ shifts towards lower
frequencies. E.g., for $\alpha=2.5$ the most of the spectral weight is
contained below $\omega/J<0.01$, i.e., $I(\omega/J>0.01)\simeq 1$. It means
that up to extremely long times, $t\sim 100/J$, the spin dynamics in the
studied systems is the same as the ballistic particle dynamics in a system of
noninteracting particles where $\sigma(\omega)\propto\delta(\omega)$ so that
$I(\omega>0)=1$. Surprisingly, further reduction of $\alpha$ reverses this
behavior and the frequency range of the incoherent part increases.

In order to quantify this behavior, in Fig. 3(a-h) we present the frequency-
dependence of $I(\omega)$ for various $\alpha$ and $\Delta$. The solid lines
mark the frequencies, $\omega^{*}$, for which $I(\omega^{*})$ covers $90\%$ of
the total sum rule. For appropriately tuned parameters $\alpha$ and $\Delta$,
$\omega^{*}$ becomes unexpectedly small, $\sim 10^{-3}-10^{-2}$, hence the
studied systems exhibit a transient ballistic transport up to very long times
$t^{*}\sim J/\omega^{*}$ (consistent with the results discussed in Sec. II).
We define the optimal $\Delta_{\sigma}$ as a value of anisotropy when
$\omega^{*}$ is the smallest for fixed $\alpha$ or, equivalently, the optimal
$\alpha_{\sigma}$ for fixed $\Delta$. The latter quantity can be identified
via the minima of the black curves presented in Fig. 3. Figure 4 shows that
$\Delta_{\sigma}$ obtained from low-frequency dynamics of a system with
periodic boundary conditions is in perfect agreement with
$\Delta_{\mathrm{O}}(\alpha)$ established previously from the short-time
expansion of the spin domains in a chain with open boundary conditions. The
latter clearly shows that the presence of the optimal anisotropy does not
emerge as a finite-size effect that might have been expected in a finite-size
chains with long-range interactions (the issue to which we will come back in
Sec. IV). Furthermore, our finite-size analysis, presented in Fig. 2(b)
indicates that the size-dependence is rather inessential for $\alpha\sim 2$
whereas for larger $\alpha$, the incoherent part of $\sigma(\omega)$ is
shifted with increasing $L$ to even smaller frequencies.

Interestingly, the exponential dependence,
$\Delta_{\mathrm{O}}=\exp(-\alpha+2)$, extends also to the regime with
$\alpha<2$ when $\Delta_{\sigma}>1$, see Fig. 3(g) and 3(h), whereas the
ballistic transport in the integrable nn-AHM is limited to $\Delta<1$
Znidaric2011 ; Herbrych2011 ; Bertini2021 . Note however that our available
system sizes (i.e., $L=20$ for ED method) don’t allow for accurate
investigation of $\alpha<2$ region.

Figure 4: Exchange decay $\alpha$ dependence of the optimal anisotropy
$\Delta$ as extracted from analysis of the spin domain expansion (discussed in
Sec. 1), i.e., $\Delta_{\nu_{\mathrm{max}}}$, and from behaviour of optical
conductivity, i.e., $\Delta_{\sigma}$. Black solid line depicts
$\Delta_{\mathrm{O}}(\alpha)=\exp(-\alpha+2)$ prediction. Error bars depict
accuracy of the data grid on base of which given curve was obtained.

### Haldane-Shastry-like model

Related to our investigations is the integrable Haldane-Shastry model (HSM)
Haldane1988 ; Shastry1988  where $\Delta=1$ and
$J(r)=\sin^{2}(\pi/L)/\sin^{2}(\pi r/L)$. Within this model, the spin current
operator $j$ is fully conserved in $L\to\infty$ limit Sirker2011 ; Defenu2021
, and one obtains a purely ballistic transport as it is the case for
noninteracting particles [$\alpha\to\infty$, $\Delta=0$ of (1)]. Note also
that in the thermodynamic limit, the spin exchange in HSM is identical to our
investigations, since $\sin^{-2}(r/L)\propto r^{-2}$ for $r\ll L$. However,
for any finite $L$ (especially, for numerically accessible $L\simeq 20$),
$J(r)\propto 1/r^{2}$ breaks the integrability of underling model.

Here, we generalize the HSM model allowing for a finite spin anisotropy
$\Delta\neq 1$ and $\alpha$-dependent exchange interaction, i.e., we use
$J(r)=J_{\mathrm{HS}}(r)$ in Eq. (1) with

$J_{\mathrm{HS}}(r)=J\frac{\sin^{\alpha}(\pi/L)}{\sin^{\alpha}(\pi\,r/L)}\,.$
(6)

Figure 3(i-l) shows the integrated optical conductivity for the generalized
HSM, which is very similar to the conductivity in the long-range AHM with
$J(r)\propto 1/r^{\alpha}$. In particular, we observe that the HSM reproduces
all main features of $I(\omega)$ which were obtained for AHM and interpreted
in terms of the quasiballistic transport in the latter model. The only
exception concerns the point $\Delta=1$ and $\alpha=2$ for which we recover
the integrability of HSM with the Drude-peak at $\omega=0$. The latter peak
shows up as a dip in the solid line in Fig. 3(k). For the studied system with
$L=20$, the Drude contribution at the integrable point contains $94\%$ of the
total sum-rule, whereas it should contain the entire spectral weight in the
$L\to\infty$ limit. Note that in the generalized HSM the specific form of the
anisotropy which corresponds to the optimal quasiballistic transport,
$\Delta_{\mathrm{O}}(\alpha)=\exp(-\alpha+2)$, smoothly interpolates between
two models where the transport is purely ballistic: the free-particle model
for $\Delta=0$, $\alpha\to\infty$ and the integrable HSM for $\Delta=1$ and
$\alpha=2$. Since the spin conductivity is almost identical between the
$1/\sin(\pi r/L)^{\alpha}$ and $1/r^{\alpha}$ models, they share the same set
of parameters which are optimal for the ballistic transport.

Figure 5: $I(\omega)$ in (a-h) the $\alpha$-dependent $J_{1}$-$J_{2}$ AHM
model, Eq. (7), and in (i-p) the long-range $t-V$ model of spinless fermions,
Eq. (8), for $L=18$ vs. the exchange decay parameter $\alpha$. Note that
$\omega$ is shown using the log-scale. Solid lines represent the frequency,
$\omega^{*}$, for which $I(\omega^{*})=0.9$.

### Spin conductivity of related models

In Fig. 3 we have shown the detailed frequency, $\omega$, dependence of the
spin conductivity $\sigma(\omega)$ for the anisotropic Heisenberg model with
long-range exchange $J(r)=1/r^{\alpha}$ and for the generalized Haldane-
Shastry model (HSM) with $J(r)\propto 1/\sin^{\alpha}(\pi\,r/L)$. To verify
that the phenomenon presented is unique to the long-range AHM, we have
investigated the behavior of $\sigma(\omega)$ in two other models relevant for
our analysis. First, we have considered the $J_{1}-J_{2}$ AHM, i.e., the model
(1) limited to the nearest- ($r=1$) and next-to-nearest ($r=2$) neighbor spin
exchange,

$\displaystyle H_{J_{1}-J_{2}}=$
$\displaystyle\sum_{\ell}J\left[\frac{1}{2}\left(S^{+}_{\ell}S^{-}_{\ell+1}+S^{-}_{\ell}S^{+}_{\ell+1}\right)+\Delta
S^{z}_{\ell}S^{z}_{\ell+1}\right]$ $\displaystyle+$
$\displaystyle\sum_{\ell}\frac{J}{2^{\alpha}}\left[\frac{1}{2}\left(S^{+}_{\ell}S^{-}_{\ell+2}+S^{-}_{\ell}S^{+}_{\ell+2}\right)+\Delta
S^{z}_{\ell}S^{z}_{\ell+2}\right]\,.$ (7)

Results presented in Fig. 5(a-b) for $\Delta=0.0-0.2$ are akin to the full
long-range AHM. Such behavior is expected since for small value of anisotropy
$\Delta$, the optimal decay parameter is $\alpha\sim 4$ and, consequently,
$r\gtrsim 3$ terms in (1) have small overall contribution. Further increasing
$\Delta$ yields rather featureless conductivity $\sigma(\omega)$ with most of
the spectral weight at $\omega/J\sim 0.1$. As a consequence we conclude that
the long-range exchange $J(r)$ is necessary for the quasiballistic transport
at extremely small $\omega$-scales. Second, we have examined the fermionic
version of the model, i.e., the long-range $t-V$ model of interacting spinless
fermions,

$\displaystyle H_{t-V}=\sum_{\ell,r}\frac{1}{r^{\alpha}}$
$\displaystyle\bigg{[}\frac{t}{2}\left(c^{\dagger}_{\ell}c^{\phantom{\dagger}}_{\ell+r}+c^{\phantom{\dagger}}_{\ell}c^{\dagger}_{\ell+r}\right)$
$\displaystyle+V(n_{\ell}-1/2)(n_{\ell+r}-1/2)\bigg{]}\,,$ (8)

with $t=1$ and $V$ as the strength of interaction. Here,
$c^{\left(\dagger\right)}_{\ell}$ obey the fermionic commutation relations.
Note that in the $\alpha\to\infty$ limit the above model and long-range AHM
(1) are identical, i.e. they are related to each via the Jordan-Wigner
transformation. Our results reveal that the quasiballistic transport is absent
in this model. As a test, we confirm in Fig. 5(i) that the free particles
($V=0$) are always ballistic [with $\sigma(\omega)=D\delta(\omega)$] for all
values of the decay parameter $\alpha$. On the other hand, for finite
interaction $0<V<1$ our results show that in the fermionic model, the optimal
behavior occurs in the $\alpha\to\infty$ limit, i.e., for the nearest-neighbor
$t-V$ model (for nn-AHM) where the ballistic transport of particles originates
from the integrability. We recall that for $V>1$, the transport in the
integrable nn-AHM chain ($\alpha\to\infty$) is diffusive Steinigeweg2009 ;
Karrasch2014 ; Steinigeweg2015 ; Prelovsek2022 . Figs. 5(o) and 5(p) suggest
that for $V>1$, the conductivity in the long-range $t$-$V$ model weakly
depends on $\alpha$. Therefore, we expect that for $V>1$ the transport is
diffusive for all studied $\alpha>1$.

Figure 6: (a) Leading contribution to the spectral sum of spin conductivity,
${\cal S}_{1}$, accounting for $Z$ (out of total $Z^{2}$) matrix elements.
${\cal S}_{1}$ is obtained from $L=20$ and normalized to the total spectral
sum ${\cal S}_{\rm tot}$. (b) Color background shows the average frequency
$\delta_{1}$ of dominating contribution ${\cal S}_{1}$ to $\sigma(\omega)$.
Here, the values of $\delta_{1}$ are normalized to the average level spacing.
In both panels the optimal anisotropy $\Delta_{\mathrm{O}}=\exp(-\alpha+2)$ is
also presented. (c) Partial contributions to the spectral sum, ${\cal S}_{1}$,
vs. the corresponding frequencies $\delta_{i}$. Points connected by each line
correspond to ${\cal S}_{1}$, ${\cal S}_{2}$,…,${\cal S}_{10}$ ordered from
left to right. Here, ($\Delta=0.5\,,\alpha=2.7$) corresponds to the optimal
case.

##  IV Similarity to noninteracting fermions

Spin conductivity in the integrable nn-AHM with $\Delta\lesssim 1$ contains
significant regular part that extends over a broad range of frequencies
Prelovsek2021 . On the other hand, the regular part is absent in the nn-AHM
for $\Delta=0$ (noninteracting fermions) as well as in the thermodynamic limit
of the HSM Sirker2011 . In the case of the long-range AHM with optimal
$\Delta_{\sigma}$, the regular part of $\sigma(\omega)$ is accumulated at very
small albeit nonzero frequencies, see Fig. 3. Consequently a decay of the spin
current in long-range spin model corresponds to a time-scale that is much
larger than the times-scale for the (partial) decay of the current in the nn-
AHM. Therefore the spin transport in the long-range AHM with
$\Delta_{\mathrm{O}}(\alpha)$ resembles the ballistic transport of
noninteracting particles rather then a more generic case of nn-AHM model. In
oder to demonstrate this similarity in more detail, in this Section we show
that for each eigenstate of the Hamiltonian $|n\rangle$ there exist only a few
matrix elements $\langle m|j|n\rangle$ which significantly contribute to the
optical spin conductivity and to the corresponding sum rule, ${\cal S}_{\rm
tot}$, in Eq. (5).

In the long-range AHM with $\Delta=\Delta_{\mathrm{O}}$ the current operator
$j$ has large matrix elements only for states close in energy. This behavior
resembles the case of purely ballistic systems where $[H,j]=0$ and the latter
states have equal energies. For each $|n\rangle$ we sort the eigenstates
$\\{|m\rangle\\}$ in descending order of $\langle n|j|m\rangle^{2}$ and denote
the sorted states as $|n_{i}\rangle,\;i=1,\dots,Z$. The spectral sum can be
represented as the sum of partial contributions ${\cal S}_{\rm tot}={\cal
S}_{1}+{\cal S}_{2}+\dots+{\cal S}_{Z}$ where ${\cal
S}_{i}=\pi(LZ)^{-1}\sum_{n}\langle n|j|n_{i}\rangle^{2}$ and ${\cal
S}_{1}\geq{\cal S}_{2}\dots\geq{\cal S}_{Z}$. We determine also the (weighted)
average frequencies of these contributions, cf. Eq. (4),

$\delta_{i}=\frac{\sum_{n}\langle
n|j|n_{i}\rangle^{2}\;|\epsilon_{n}-\epsilon_{n_{i}}|}{\sum_{n}\langle
n|j|n_{i}\rangle^{2}}\,.$ (9)

Figure 6(a) depicts the ratio ${\cal S}_{1}/{\cal S}_{\rm tot}$ whereas the
color background in Fig. 6(b) shows the frequency $\delta_{1}$ normalized to
the average level spacing. For the optimal anisotropy $\Delta_{\mathrm{O}}$
one observes that ${\cal S}_{1}$ covers a majority of the spectral sum, see
bright colors in fig. 6(a). Results in fig. 6(b) demonstrate that ${\cal
S}_{1}$ originates from the matrix elements $\langle n|j|m\rangle$ between
eigenstates $|n\rangle$ and $|m\rangle$ for which the energy difference
$|\epsilon_{n}-\epsilon_{m}|$ is of the order of several level spacings (for
accessible $L$). It is not the case for other $\Delta$, when ${\cal S}_{1}$ is
significantly smaller and the corresponding frequency $\delta_{1}$ is
significantly larger than for the optimal $\Delta_{\rm O}$. In order to
visualize the finite-size effects, in Fig. 6(c) we show the ratio ${\cal
S}_{i}/{\cal S}_{\rm tot}$ vs. $\delta_{i}$ for various $L$. Points on each
line correspond to ${\cal S}_{1}$, ${\cal S}_{2}$,…,${\cal S}_{10}$ which are
ordered from left to right. Upon increasing $L$, the contributions to
$\sigma(\omega)$ which come from the matrix elements entering ${\cal S}_{i>1}$
becomes more significant also for the optimal anisotropy. However, we observe
that $\delta_{i}$ remains small whenever ${\cal S}_{i}$ is large. Therefore
for the optimal anisotropy, the regular part of the optical conductivity is
confined to the low frequency regime for all accessible $L$. The same message
arises also from results in Fig. 2(b).

##  V Slowly-decaying fermionic currents

In previous Sections we have shown that at the optimal anisotropy
$\Delta_{\mathrm{O}}=\exp(-\alpha+2)$ the spin current
$j\propto\sum_{\ell,r}(S^{+}_{\ell}S^{-}_{\ell+r}-\mathrm{H.c.})$ is decaying
slowly. Here we will demonstrate that on the $\Delta_{\mathrm{O}}(\alpha)$
line one can find a broad class of physically relevant observables which
display similar dynamics. In order to identify such operators, we apply a
numerical algorithm that was originally applied for identifying an orthogonal
set of local (and quasilocal) integrals of motion Mierzejewski2015-1 ;
Mierzejewski2015-2 . Except for the known integrable points, $\alpha\to\infty$
and $\Delta=1$ with $\alpha=2$ for HSM, we do not expect the presence of local
integrals of motion in the thermodynamic limit. Nevertheless, one can expect
that found operators will exhibit finite stiffness, i.e., nondecaying
component in finite-size system, which should indicate slow dynamics in
macroscopic chains.

For the sake of completeness, we recall the main steps of the numerical
algorithm. First, we construct all translationally-invariant current-like
operators supported on up to $M$ consecutive lattice sites

$A_{\gamma}=i\frac{1}{\sqrt{L}}\sum_{\ell}\left(o_{\ell+1}o_{\ell+2}...o_{\ell+M}-{\rm
H.c.}\right)\,,$ (10)

where
$o_{\ell}\in\\{1_{\ell},\sqrt{2}S^{-}_{\ell},\sqrt{2}S^{+}_{\ell},2S^{z}_{\ell}\\}$
and the index $\gamma$ enumerates the resulting operators. We then expressed
all $A_{\gamma}$ in the basis of the eigenstates of the Hamiltonian,
$H|n\rangle=E_{n}|n\rangle$. The conserved part of $A_{\gamma}$ is determined
by the matrix elements between the degenerate eigenstates,

$\bar{A}_{\gamma}=\sum_{m,n:\;E_{m}=E_{n}}\langle
m|A_{\gamma}|n\rangle|m\rangle\langle n|,\quad\quad[\bar{A}_{\gamma},H]=0\,.$
(11)

The final step is the numerical diagonalization of a matrix that contains
averaged products of the latter operators,
$M_{\gamma\gamma^{\prime}}=\langle\bar{A}_{\gamma}\bar{A}_{\gamma^{\prime}}\rangle=(1/Z){\rm
Tr}(\bar{A}_{\gamma}\bar{A}_{\gamma^{\prime}})$, where the trace is carried
out over all many-body states in all $S^{z}_{\rm tot}$ sectors. Namely, we
numerically solve the eigenproblem

$\sum_{\gamma,\gamma^{\prime}}(O^{T})_{a\gamma}M_{\gamma\gamma^{\prime}}O_{\gamma^{\prime}a^{\prime}}=\lambda_{a}\delta_{a,a^{\prime}}\,,$
(12)

where $O$ is an orthogonal matrix.

Figure 7: (a-c) The eigenvalues (stiffnesses) $\lambda_{a}$ of three local
operators $I_{a}$ with the largest conserved part. The remaining panels (d-l)
depict projections of $I_{a}$ on the local operators defined in Eq. (13),
$\langle A_{\gamma}I_{a}\rangle^{2}=|O_{\gamma a}|^{2}$ for $a=1,2,3$ and
$\gamma=1,2,3$. See text for details.

The eigenvectors of $M$ define a sequence of local operators,
$I_{a}=\sum_{\gamma}O_{\gamma a}A_{\gamma}$, which are mutually orthogonal,
$\langle I_{a}I_{a^{\prime}}\rangle=\delta_{a,a^{\prime}}$, and the squared
norms of their conserved parts (called stiffnesses from now on) are equal the
corresponding eigenvalues,
$||\bar{I}_{a}||^{2}=\langle\bar{I}_{a}\bar{I}_{a}\rangle=\lambda_{a}$. All
stiffnesses are bounded $1\geq\lambda_{1}\geq\lambda_{2}\geq\dots\geq 0$ while
local integrals of motion correspond to $\lambda_{a}=1$. Again, we want to
point out that one does not expect the latter for nonintegrable
$J(r)=1/r^{\alpha}$ system studied here (except in $\alpha\to\infty$ limit).

Figure 7 depicts the first three largest eigenvalues $\lambda_{a}$ and
corresponding operators $I_{a}$, obtained for $M=4$ sites support. In Fig.
7(a-c) we present the value of stiffnesses itself obtained for various
$\Delta$ and $\alpha$. It is evident from the presented results that second
and third eigenvalue, $\lambda_{2}$ and $\lambda_{3}$, respectively, have the
largest value at $\Delta_{\mathrm{O}}$. On the other hand, $\lambda_{1}$ does
not have clear maximum at $\Delta_{\mathrm{O}}$, but rather constant value for
$\Delta(\alpha)>\Delta_{\mathrm{O}}$ and sharp drop for
$\Delta(\alpha)<\Delta_{\mathrm{O}}$.

Figure 8 shows the finite-size dependence of the stiffness $\lambda_{2}$ and
$\lambda_{3}$ for a set of parameters corresponding to the optimal anisotropy,
$\Delta=3.5,\alpha=0.2$ as well as for parameters which are shifted away from
it. One may observe that not only the stiffnesses are maximal for
$\Delta=\Delta_{\mathrm{O}}$, but also the finite-size effect are least
visible. We are not aware of any mechanism that might explain the presence of
nonzero stiffnesses in the studied chains in the thermodynamic limit. For this
reason we expect that the stiffnesses vanish also for the optimal anisotropy,
however their decay becomes visible for much larger systems than for other
model parameters.

Finally, our analysis of the most conserved operators $I_{a}$ [corresponding
to the eigenvalues in Fig. 7(a-c)] indicated that the largest contributions
come from the local current-like operators of the form:

$\displaystyle A_{1}$ $\displaystyle=$ $\displaystyle
i\frac{2}{\sqrt{L}}\sum_{\ell}\left(S^{+}_{\ell}S^{-}_{\ell+1}-{\rm
H.c.}\right)\,,$ $\displaystyle A_{2}$ $\displaystyle=$ $\displaystyle
i\frac{4}{\sqrt{L}}\sum_{\ell}\left(S^{+}_{\ell}S^{z}_{\ell+1}S^{-}_{\ell+2}-{\rm
H.c.}\right)\,,$ $\displaystyle A_{3}$ $\displaystyle=$ $\displaystyle
i\frac{8}{\sqrt{L}}\sum_{\ell}\left(S^{+}_{\ell}S^{z}_{\ell+1}S^{z}_{\ell+2}S^{-}_{\ell+3}-{\rm
H.c.}\right)\,.$ (13)

The latter contributions are formally quantified via projections $\langle
A_{\gamma}I_{a}\rangle^{2}=|O_{\gamma a}|^{2}$ and are shown in Fig. 7(d-l).
Since $\sum_{\gamma}|O_{\gamma a}|^{2}=1$, the most conserved operator,
$I_{1}$, has the largest projection on the current $A_{2}$ which is even under
the spin-flip transformation, whereas $I_{2}$ and $I_{3}$ have the largest
projections on $A_{1}$ and $A_{3}$, respectively. The latter two operators are
odd under the spin-flip transformation, which maps $S^{z}_{i}$ to
$-S^{z}_{i}$, $S^{+}_{i}$ to $S^{-}_{i}$ and $S^{-}_{i}$ to $S^{+}_{i}$. The
Wigner-Jordan transformation maps the spin-current operators Eq. (13) to
fermionic currents which describe hoppings between the first, the second and
the third nearest-neighbors. Despite the studied spin-system can not be mapped
on a fermionic model with two-body interactions, numerical results discussed
in the present Section consistently support the presence of slowly decaying
odd fermionic currents which are most stable at $\Delta_{\mathrm{O}}(\alpha)$.
We also note that one of the studied local currents, $A_{1}$, has a large
projection on the actual spin current that was studied in Sec. III.

Figure 8: Finite-size dependence of (a) second $\lambda_{2}$ and (b) third
$\lambda_{3}$ eigenvalue (stiffness) at optimal anisotropy
($\Delta=0.2\,,\alpha=3.5$) and away from this point ($\Delta=0.0$ and
$\Delta=0.8$).

##  VI Conclusion

We have studied the spin transport in a Heisenberg chain with long-range
exchange $J(r)=J/r^{\alpha}$ and spin anisotropy $\Delta$. We have found that
the system may be tuned to a quasiballistic regime, where the spectral sum of
the spin conductivity is accumulated at frequencies smaller than
$\omega^{*}/J\sim 10^{-3}-10^{-2}$. Such systems exhibits a ballistic spin
transport that is transient but persists up to unexpectedly long times $t\sim
1/\omega^{*}$, consistent with our domain wall expansion analysis. The
quasiballistic transport occurs not just in the neighborhood of known
ballistic systems but rather along a sharp line in the space of model
parameters, $\Delta\simeq\exp(-\alpha+2)$. The latter line smoothly
interpolates between two models in which the transport is ballistic, i.e., the
regular part of conductivity is absent. Namely, the line interpolates between
free particles for $\alpha=\infty$, $\Delta=0$ and the Haldane-Shastry-like
model for $\alpha=2$, $\Delta=1$. Furthermore, the presence of a long-living
quasiballistic transport resembles the phenomenon of prethermalization
Berges2004 ; Moeckel2008 ; Kollar2011  that is present in nearly-integrable
models. Indeed, our results may suggest that the studied Heisenberg chain
could be viewed as nearly-integrable for all parameters satisfying the
relation $\Delta\simeq\exp(-\alpha+2)$, and not in just small parameter space
close to the known integrable points. Our numerical studies indicate that the
quasiballistic dynamics can be observed not only from the spin current but
also from a broader class of fermionic-like operators, despite the studied
spin chain can not be mapped on a fermionic model with two-body interaction.

In the case of large $\alpha\gg 1$ we have found that the optimal anisotropy
$\Delta\simeq\exp(-\alpha+2)\ll 1$ can be identified also in the spin chain
that includes first nearest neighbor ($J$) and second nearest neighbor
($J_{2}=J/2^{\alpha}$) terms. However, the long-lasting ballistic transport
discussed above is specific to the spin chains and does not occur in a
fermionic system with long-range hoppings and interactions. The origin of the
optimal anisotropy and its exponential dependence on $\alpha$ remain open
problems. A possible (but speculative) scenario is that the studied spin chain
with $\Delta\simeq\exp(-\alpha+2)$ is a good approximation to a more complex
spin model that is integrable and conserves the spin current. Then, the values
$\omega^{*}$ would be determined by the quality of this approximation.

###### Acknowledgements.

The authors thank R. Steinigeweg and P. Prelovšek for fruitful discussions. M.
Mierzejewski, J. Wronowicz and J. Pawłowski acknowledge support by the
National Science Centre (NCN), Poland via project 2020/37/B/ST3/00020. J.
Herbrych acknowledges support by the Polish National Agency for Academic
Exchange (NAWA) under contract PPN/PPO/2018/1/00035 and by the National
Science Centre (NCN), Poland via project 2019/35/B/ST3/01207. The calculations
were carried out using resources provided by the Wrocław Centre for Networking
and Supercomputing.

## References

  * (1) X. Zotos, F. Naef, and P. Prelovsek, Transport and conservation laws., Phys. Rev. B 55, 11029 (1997).
  * (2) B. Bertini, F. Heidrich-Meisner, C. Karrasch, T. Prosen, R. Steinigeweg, and M. Žnidarič, Finite-temperature transport in one-dimensional quantum lattice models, Rev. Mod. Phys. 93, 025003 (2021).
  * (3) C. Nisoli, V. Kapaklis, and P. Schiffer, Deliberate exotic magnetism via frustration and topology., Nat. Phys. 13, 200 (2017).
  * (4) A. W. Sandvik, Ground States of a Frustrated Quantum Spin Chain with Long-Range Interactions., Phys. Rev. Lett. 104, 137204 (2010).
  * (5) L. Yang, A. E. Feiguin, From deconfined spinons to coherent magnons in an antiferromagnetic Heisenberg chain with long range interactions., SciPost Phys. 10, 110 (2021).
  * (6) L. Balents, Spin liquids in frustrated magnets., Nature 464, 119 (2010).
  * (7) I. Bloch and P.Zoller, Focus on Cold Atoms in Optical Lattices., New J. Phys. 8, E02 (2006).
  * (8) S. Hild, T. Fukuhara, P. Schauß, J. Zeiher, M. Knap, E. Demler, I. Bloch, and C. Gross, Far-from-Equilibrium Spin Transport in Heisenberg Quantum Magnets, Phys. Rev. Lett. 113, 147205 (2014).
  * (9) C. Gross and I. Bloch, Quantum simulations with ultracold atoms in optical lattices., Science 357, 995 (2017).
  * (10) J. Zhang, G. Pagano, P. W. Hess, A. Kyprianidis, P. Becker, H. Kaplan, A. V. Gorshkov, Z.-X. Gong, and C. Monroe , Observation of a many-body dynamical phase transition with a 53-qubit quantum simulator, Nature 551, 601 (2017).
  * (11) F. Schäfer, T. Fukuhara, S. Sugawa, Y. Takasu, and Y. Takahashi, Tools for quantum simulation with ultracold atoms in optical lattices., Nat. Rev. Phys. 2, 411 (2020).
  * (12) P. N. Jepsen, J. Amato-Grill, I. Dimitrova, W. W. Ho, E. Demler, and W. Ketterle, Spin transport in a tunable Heisenberg model realized with ultracold atoms, Nature 588, 403 (2020).
  * (13) D. Wei, A. Rubio-Abadal, B. Ye, F. Machado, J. Kemp, K. Srakaew, S. Hollerith, J. Rui, S. Gopalakrishnan, N. Y. Yao, I. Bloch, J. Zeiher, Quantum gas microscopy of Kardar-Parisi-Zhang superdiffusion, Science 376, 716 (2022).
  * (14) H. Weimer, M. Müller, I. Lesanovsky, P. Zoller, H. P. Buüchler, A Rydberg quantum simulator., Nat. Phys. 6, 382 (2010).
  * (15) S. E. Anderson, K. C. Younge, and G. Raithel, Trapping Rydberg Atoms in an Optical Lattice., Phys. Rev. Lett. 107, 263001 (2011).
  * (16) H. Labuhn, D. Barredo, S. Ravets, S. de Léséleuc, T. Macrí, T. Lahaye, and A. Browaeys, Tunable two-dimensional arrays of single Rydberg atoms for realizing quantum Ising models, Nature 534, 667 (2016).
  * (17) H. Bernien, S. Schwartz, A. Keesling, H. Levine, A. Omran, H. Pichler, S. Choi, A. S. Zibrov, M. Endres, M. Greiner, V. Vuletić, and M. D. Lukin , Probing many-body dynamics on a 51-atom quantum simulator, Nature 551, 579 (2017).
  * (18) E. Guardado-Sanchez, P. T. Brown, D. Mitra, T. Devakul, D. A. Huse, P. Schauß, and W. S. Bakr, Probing the Quench Dynamics of Antiferromagnetic Correlations in a 2D Quantum Ising Spin System, Phys. Rev. X 8, 021069 (2018).
  * (19) A. Browaeys and T. Lahaye, Many-body physics with individually controlled Rydberg atoms, Nat. Phys. 16, 132 (2020).
  * (20) V. Borish, O. Marković, J. A. Hines, S. V.. Rajagopal, and M. Schleier-Smith, Transverse-Field Ising Dynamics in a Rydberg-Dressed Atomic Gas, Phys. Rev. Lett. 124, 063601 (2020).
  * (21) S. Geier, N. Thaicharoen, C. Hainaut, T. Franz, A. Salzinger, A. Tebben, D. Grimshandl, G. Zürn, M. Weidemüller, Floquet Hamiltonian engineering of an isolated many-body spin system, Science 374, 1149 (2021).
  * (22) S. Hollerith, K. Srakaew, D. Wei, A. Rubio-Abadal, D. Adler, P. Weckesser, A. Kruckenhauser, V. Walther, R. van Bijnen, J. Rui, C. Gross, I. Bloch, and J. Zeiher, Realizing Distance-Selective Interactions in a Rydberg-Dressed Atom Array, Phys. Rev. Lett. 128, 113602 (2022).
  * (23) P. Scholl, H. J. Williams, G. Bornet, F. Wallner, D. Barredo, T. Lahaye, A. Browaeys, L. Henriet, A. Signoles, C. Hainaut, T. Franz, S. Geier, A. Tebben, A. Salzinger, G. Zürn, M. Weidemüller, Microwave-engineering of programmable XXZ Hamiltonians in arrays of Rydberg atoms, PRX Quantum 3, 020303 (2022).
  * (24) L.-M. Steinert, P. Osterholz, R. Eberhard, L. Festa, N. Lorenz, Z. Chen, A. Trautmann, C. Gross, Spatially programmable spin interactions in neutral atom arrays, arXiv:2206.12385 (2022).
  * (25) J. W. Britton, B. C. Sawyer, A. C. Keith, C.-C. J. Wang, J. K. Freericks, H. Uys, M. J. Biercuk, and J. J. Bollinger, Engineered two-dimensional Ising interactions in a trapped-ion quantum simulator with hundreds of spins., Nature 484, 489 (2012).
  * (26) M. K. Joshi, F. Kranzl, A. Schuckert, I. Lovas, C. Maier, R. Blatt, M. Knap, C. F. Roos, Observing emergent hydrodynamics in a long-rangequantum magnet, Science 376, 720 (2022).
  * (27) P. Richerme, Z.-X. Gong, A. Lee, C. Senko, J. Smith, M. Foss-Feig, S. Michalakis, A. V. Gorshkov, and C. Monroe, Non-local propagation of correlations in quantum systems with long-range interactions., Nature 511, 198 (2014).
  * (28) P. Jurcevic, B. P. Lanyon, P. Hauke, C. Hempel, P. Zoller, R. Blatt, and C. F. Roos. Quasiparticle engineering and entanglement propagation in a quantum many-body system., Nature 511, 202 (2014).
  * (29) P. Hauke and L. Tagliacozzo, Spread of Correlations in Long-Range Interacting Quantum Systems., Phys. Rev. Lett. 111, 207202 (2013).
  * (30) M. Foss-Feig, Z.-X. Gong, C. W. Clark, and A. V. Gorshkov, Nearly Linear Light Cones in Long-Range Interacting Quantum Systems., Phys. Rev. Lett. 114, 157201 (2015).
  * (31) M. F. Maghrebi, Z.-X. Gong, M. Foss-Feig, and A. V. Gorshkov, Causality and quantum criticality in long-range lattice models., Phys. Rev. B 93, 125128 (2016).
  * (32) L. Lepori, A. Trombettoni, and D. Vodola, Singular dynamics and emergence of nonlocality in long-range quantum models., J. Stat. Mech. 2017, 033102 (2017).
  * (33) I. Frérot, P. Naldesi, and T. Roscilde, Entanglement and fluctuations in the XXZ model with power-law interactions., Phys. Rev. B 95, 245111 (2017).
  * (34) L. Vanderstraeten, M. Van Damme, H. P. Büchler, and F. Verstraete, Quasiparticles in Quantum Spin Chains with Long-Range Interactions., Phys. Rev. Lett. 121, 090603 (2018).
  * (35) L. Cevolani, J. Despres, G. Carleo, L. Tagliacozzo, and L. Sanchez-Palencia, Universal scaling laws for correlation spreading in quantum systems with short- and long-range interactions, Phys. Rev. B 98, 024302 (2018).
  * (36) B. Kloss and Y. B. Lev, Spin transport in a long-range-interacting spin chain., Phys. Rev. A 99, 032114 (2019).
  * (37) A. Lerose, B. Žunkovič, A. Silva, and A. Gambassi, Quasilocalized excitations induced by long-range interactions in translationally invariant quantum spin chains., Phys. Rev. B 99, 121112(R) (2019).
  * (38) J, Ren, W.-L. You, and X. Wang, Entanglement and correlations in a one-dimensional quantum spin-1/2 chain with anisotropic power-law long-range interactions, Phys. Rev. B 101, 094410 (2020).
  * (39) J. T. Schneider, J. Despres, S. J. Thomson, L. Tagliacozzo, and L. Sanchez-Palencia, Spreading of correlations and entanglement in the long-range transverse Ising chain, Phys. Rev. Research 3, L012022 (2021).
  * (40) S. Birnkammer, A. Bohrdt, F. Grusdt, and M. Knap, Characterizing topological excitations of a long-range Heisenberg model with trapped ions, Phys. Rev. B 105, L241103 (2022).
  * (41) V. B. Bulchandani and D. A. Huse, Hot band sound, arXiv:2208.13767 (2022).
  * (42) E. Lieb, T. Schultz, and D. Mattis, Two soluble models of an antiferromagnetic chain., Ann. Phys. 16, 407 (1961).
  * (43) F. D. M. Haldane, Exact Jastrow-Gutzwiller resonating-valence-bond ground state of the spin-$1/2$ antiferromagnetic Heisenberg chain with $1/r^{2}$ exchange., Phys. Rev. Lett. 60, 635 (1988).
  * (44) B. S. Shastry, Exact solution of an $S=1/2$ Heisenberg antiferromagnetic chain with long-ranged interactions., Phys. Rev. Lett. 60, 639 (1988).
  * (45) P. W. Anderson, Resonating valence bonds: A new kind of insulator?, Mater. Res. Bull. 8, 153 (1973).
  * (46) F. H. L. Eßler, Dressed $S$ matrices in models with long-range interactions, Phys. Rev. B 51, 13357 (1995).
  * (47) M. Greiter and D. Schuricht, No attraction between spinons in the Haldane-Shastry model, Phys. Rev. B 71, 224424 (2005).
  * (48) I. Sechin and A. Zotov, $R$-matrix-valued Lax pairs and long-range spin chains, Phys. Lett. B, 781, 1 (2018)
  * (49) A.A. Zvyagin, Quantum Theory of One-dimensional Spin Systems, Cambridge Scientific Publishers, (2010).
  * (50) J. P. Ronzheimer, M. Schreiber, S. Braun, S. S. Hodgman, S. Langer, I. P. McCulloch, F. Heidrich-Meisner, I. Bloch, and U. Schneider, Expansion Dynamics of Interacting Bosons in Homogeneous Lattices in One and Two Dimensions., Phys. Rev. Lett. 110, 205301 (2013).
  * (51) L. Vidmar, S. Langer, I. P. McCulloch, U. Schneider, U. Schollwöck, and F. Heidrich-Meisner, Sudden expansion of Mott insulators in one dimension., Phys. Rev. B 88, 235117 (2013).
  * (52) B. Neyenhuis, J. Zhang, P. W. Hess, J. Smith, A. C. Lee, P. Richerme, Z.-X, Gong, A. V. Gorshkov, and C. Monroe, Observation of prethermalization in long-range interacting spin chains., Sci. Adv. 3, e1700672 (2017).
  * (53) P. Prelovšek and J. Bonča, Ground state and finite temperature Lanczos methods, in Strongly Correlated Systems–Numerical Methods, edited by A. Avella and F. Mancini (Springer, Berlin,2013).
  * (54) S. Langer, M. J. A. Schuetz, I. P. McCulloch, U. Schollwöck, and F. Heidrich-Meisner, Expansion velocity of a one-dimensional, two-component Fermi gas during the sudden expansion in the ballistic regime, Phys. Rev. A 85, 043618 (2012).
  * (55) In order to avoid the double counting of $J(r)$ on the system with periodic boundary condtions we have restricted the sum over $r$ in (1) to $L/2-1$.
  * (56) J. Jaklič, P. Prelovšek, Finite-temperature properties of doped antiferromagnets., Advances in Physics 49, 1 (2000).
  * (57) P. Prelovšek, M. Mierzejewski, and J. Herbrych, Coexistence of diffusive and ballistic transport in integrable quantum lattice models., Phys. Rev. B 104, 115163 (2021).
  * (58) Marko Žnidarič, Spin Transport in a One-Dimensional Anisotropic Heisenberg Model., Phys. Rev. Lett. 106, 220601 (2011)
  * (59) J. Herbrych, P. Prelovšek, and X. ZotosFinite-temperature Drude weight within the anisotropic Heisenberg chain., Phys. Rev. B 84, 155125 (2011).
  * (60) J. Sirker, R.G. Pereira, and I. Affleck, Conservation laws, integrability, and transport in one-dimensional quantum systems., Phys. Rev. B 83, 035115 (2011).
  * (61) N. Defenu, Metastability and discrete spectrum of long-range systems, Proc. Natl. Acad. Sci. U.S.A., 118(30)e2101785118 (2021).
  * (62) R. Steinigeweg and J. Gemmer, Density dynamics in translationally invariant spin-$\frac{1}{2}$ chains at high temperatures: A current-autocorrelation approach to finite time and length scales, Phys. Rev. B 80, 184402 (2009).
  * (63) C. Karrasch, J. E. Moore, and F. Heidrich-Meisner, Real-time and real-space spin and energy dynamics in one-dimensional spin-$\frac{1}{2}$ systems induced by local quantum quenches at finite temperatures, Phys. Rev. B 89, 075139 (2014).
  * (64) R. Steinigeweg, J. Gemmer, and W. Brenig, Spin and energy currents in integrable and nonintegrable spin-$\frac{1}{2}$ chains: A typicality approach to real-time autocorrelations, Phys. Rev. B 91, 104404 (2015).
  * (65) P. Prelovšek, S. Nandy, Z. Lenarčič, M. Mierzejewski, and J. Herbrych, From dissipationless to normal diffusion in the easy-axis Heisenberg spin chain, Phys. Rev. B 106, 245104 (2022).
  * (66) M. Mierzejewski, T. Prosen, and P. Prelovšek, Identifying Local and Quasilocal Conserved Quantities in Integrable Systems, Phys. Rev. Lett. 114, 140601 (2015).
  * (67) M. Mierzejewski, T. Prosen, and P. Prelovšek, Approximate conservation laws in perturbed integrable lattice models, Phys. Rev. B 92, 195121 (2015).
  * (68) J. Berges, Sz. Borsányi, and C. Wetterich, Prethermalization., Phys. Rev. Lett. 93, 142002 (2004).
  * (69) M. Moeckel and S. Kehrein, Interaction Quench in the Hubbard Model., Phys. Rev. Lett. 100, 175702 (2008).
  * (70) M. Kollar, F. A. Wolf, and M. Eckstein, Generalized Gibbs ensemble prediction of prethermalization plateaus and their relation to nonthermal steady states in integrable systems, Phys. Rev. B 84, 054304 (2011).