# Universality in the tripartite information after global quenches

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

###### Abstract

We consider macroscopically large 3-partitions $(A,B,C)$ of connected
subsystems $A\cup B\cup C$ in infinite quantum spin chains and study the
Rényi-$\alpha$ tripartite information $I_{3}^{(\alpha)}(A,B,C)$. At
equilibrium in clean 1D systems with local Hamiltonians it generally vanishes.
A notable exception is the ground state of conformal critical systems, in
which $I_{3}^{(\alpha)}(A,B,C)$ is known to be a universal function of the
cross ratio $x=|A||C|/[(|A|+|B|)(|C|+|B|)]$, where $|A|$ denotes $A$’s length.
We identify different classes of states that, under time evolution with
translationally invariant Hamiltonians, locally relax to states with a nonzero
(Rényi) tripartite information, which furthermore exhibits a universal
dependency on $x$. We report a numerical study of $I_{3}^{(\alpha)}$ in
systems that are dual to free fermions, propose a field-theory description,
and work out their asymptotic behaviour for $\alpha=2$ in general and for
generic $\alpha$ in a subclass of systems. This allows us to infer the value
of $I_{3}^{(\alpha)}$ in the scaling limit $x\rightarrow 1^{-}$, which we call
“residual tripartite information”. If nonzero, our analysis points to a
universal residual value $-\log 2$ independently of the Rényi index $\alpha$,
and hence applies also to the genuine (von Neumann) tripartite information.

The concept of entanglement was introduced to distinguish quantum systems from
classical ones [1, 2]. With the development of quantum information theory [3,
4], such a peculiarity of the quantum world was recognised as a resource, and
the quantification of the entanglement became a key question [5]. Various
measures of entanglement have then been put forward. Besides their original
purpose, the tools studied in quantum information attracted the attention of
the community working on quantum many-body systems. It was indeed realised
that the entanglement measures unveil universal properties. A famous example
in 1D is the von Neumann entropy of an interval in the ground state of a
(conformal) critical system, which has a typical logarithmic growth with the
length, proportional to the central charge of the underlying conformal field
theory (CFT) [6, 7, 8]. Another quantity that attracted some attention
(unfortunately under different names) is the _tripartite information_ , which
contains more information about the underlying CFT[9, 10, 11]; but not only
that. In 2D the tripartite information was shown to be sensitive to
topological order and renamed for that reason “topological entanglement
entropy” [12]. Concerning higher dimensions, we mention that Ref. [13]
investigated the tripartite information in generic quantum field theories
(QFT) (see also Ref. [14]), among which those with holographic duals hold a
special place [15]. More recently, a type of tripartite information was
proposed as a diagnostic of scrambling [16, 17, 18].

The tripartite information is defined as [4]

$I_{3}(A,B,C)=I_{2}(A,B)+I_{2}(A,C)-I_{2}(A,B\cup C)$ (1)

where $I_{2}(A,B)=S(A)+S(B)-S(A\cup B)$ denotes the mutual information and
$S(A)\equiv S_{1}[\rho_{A}]=-\mathrm{tr}[\rho_{A}\log\rho_{A}]$ is the von
Neumann entropy of subsystem $A$ with density matrix $\rho_{A}$. It is defined
so as to cancel the extensive and the boundary contributions to the entropies.
Moreover, just as the mutual information quantifies the extensiveness of the
von Neumann entropy, so $I_{3}(A,B,C)$ quantifies the (bi)extensiveness of the
mutual information after fixing one of the subsystems. We focus on the case in
which $A$, $B$, and $C$ are adjacent intervals in an infinite spin chain and
assume that their lengths are asymptotically large:

$A$$B$$C$

In noncritical 1D systems at equilibrium (and with clustering properties),
$I_{3}(A,B,C)$ approaches $0$ as the lengths approach infinity (we are not
aware of exceptions). A more interesting behavior is observed in the ground
state of a critical system with a low-energy CFT description, where conformal
symmetry forces $I_{3}(A,B,C)$ to be a function of the cross ratio
$x=\frac{|A||C|}{(|A|+|B|)(|B|+|C|)}$ [11]. That is to say, the limit

$I_{3}(A,B,C)\xrightarrow[|A||C|/[(|A|+|B|)(|B|+|C|)]=x]{|A|,|B|,|C|\rightarrow\infty}G(x)$
(2)

exists, is universal, and can in principle be computed within the underlying
CFT. As a matter of fact, some difficulties of the calculation have not yet
been overcome. The most important intermediate results concern the Rényi
generalisation of the tripartite information, which we will refer to as
Rényi-$\alpha$ tripartite information and indicate by
$I_{3}^{(\alpha)}(A,B,C)$. This has the same definition of $I_{3}(A,B,C)$ with
the von Neumann entropy replaced by the Rényi-$\alpha$ entropy
$S_{\alpha}[\rho]=\tfrac{1}{1-\alpha}\log\mathrm{tr}[\rho^{\alpha}]$. Ideally,
the tripartite information is recovered in the limit $\alpha\rightarrow
1^{+}$. Provided that the intervals $A,B,C$ are adjacent111If the intervals
are disjoint, the CFT prediction depends instead on three cross ratios — see,
e.g., Refs [19, 20], in a CFT also the Rényi-$\alpha$ tripartite information
depends only on the cross ratio $x$, i.e.,
$I_{3}^{(\alpha)}(A,B,C)=G_{\alpha}(x)$, and, in some theories, it has been
computed exactly for generic integer $\alpha>1$ [10, 21, 22, 23]. Just as
$G(x)$ does for $I_{3}$, so $G_{\alpha}(x)$ describes the scaling limit of
$I_{3}^{(\alpha)}$ in any spin chain with the same underlying CFT [24, 25, 26,
27, 28].

In this Letter we investigate the limit of _infinite time_ of
$I_{3}^{(\alpha)}$ after global quenches with _local_ Hamiltonians. The
entropies are known to become extensive [29, 30, 31], and often the system
exhibits typical features of thermal states [32, 33, 34]. We discuss when
$I_{3}^{(\alpha)}$ should be expected not to vanish (in contrast to thermal
states) and argue that it captures universal properties. We point out, in
particular, that a residual tripartite information with a discrete value
$-\log 2$ can emerge.

## The model.

We focus on two classes of noninteracting spin chains. The first is the
generalised XY model [35], which is mapped into free fermions by a Jordan-
Wigner transformation
$\bm{\mathbf{a}}_{2\ell-1}=\prod_{j<\ell}\bm{\mathbf{\sigma}}_{j}^{z}\
\bm{\mathbf{\sigma}}^{x}_{\ell}$,
$\bm{\mathbf{a}}_{2\ell}=\prod_{j<\ell}\bm{\mathbf{\sigma}}_{j}^{z}\
\bm{\mathbf{\sigma}}^{y}_{\ell}$, where $\bm{\mathbf{a}}_{\ell}$ are Majorana
fermions satisfying
$\\{\bm{\mathbf{a}}_{\ell},\bm{\mathbf{a}}_{n}\\}=2\delta_{\ell
n}\bm{\mathbf{I}}$ and $\bm{\mathbf{\sigma}}^{\alpha}_{\ell}$ are Pauli
operators. The most studied models of this class are described by the quantum
XY Hamiltonian [36] in a transverse field

$\bm{\mathbf{H}}=\sum\nolimits_{\ell}J_{x}\bm{\mathbf{\sigma}}_{\ell}^{x}\bm{\mathbf{\sigma}}_{\ell+1}^{x}+J_{y}\bm{\mathbf{\sigma}}_{\ell}^{y}\bm{\mathbf{\sigma}}_{\ell+1}^{y}+h\bm{\mathbf{\sigma}}_{\ell}^{z}\,.$
(3)

This includes, in particular, the XX model ($J_{x}=J_{y}$) and the transverse-
field Ising model ($J_{y}=0$).

The second class of systems is mapped into free fermions by the Kramers-
Wannier transformation
$\bm{\mathbf{\tau}}_{\ell}^{x}=\prod_{j\leq\ell}\bm{\mathbf{\sigma}}_{j}^{x}$,
$\bm{\mathbf{\tau}}_{\ell}^{y}=(\prod_{j<\ell}\bm{\mathbf{\sigma}}_{j}^{x})\bm{\mathbf{\sigma}}_{\ell}^{y}\bm{\mathbf{\sigma}}_{\ell+1}^{z}$
(for the sake of clarity, we have used a different notation,
$\bm{\mathbf{\tau}}_{\ell}^{\alpha}$, for the Pauli operators), followed by
the aforementioned Jordan-Wigner transformation. An example is the dual XY
model [37, 38]

$\bm{\mathbf{H}}=\sum\nolimits_{\ell}\bm{\mathbf{\tau}}_{\ell-1}^{x}(J_{x}\bm{\mathbf{I}}-J_{y}\bm{\mathbf{\tau}}_{\ell}^{z})\bm{\mathbf{\tau}}_{\ell+1}^{x}\,.$
(4)

Hamiltonians like (4) posses semilocal conserved operators [38], which enable
symmetry-protected topological order after global quenches [39]. Since the
tripartite information was recognised as an indicator of topological order in
2D [12], how $I_{3}$ behaves after a quench in this second class of models is
a compelling question.

## Homogeneous quench from a critical ground state.

Figure 1: Rényi-2 tripartite information for the quench in the XY model
$(J_{x},J_{y},h):(1,1,1)\rightarrow(1,0.5,0.5)$ for
$|A|=|C|\in\\{45,90,180,360\\}$ and variable $|B|$. The solid curve is
prediction (9). Inset: the relative error $\eta_{3}^{(2)}$ as a function of
$l=\min(|A|,|B|,|C|)$ for all configurations with lengths multiple of $5$ in
the range $[5,120]$; the color and the size of the points varies linearly in
$x$; the dashed line is a guide for the eye $\sim 1/l$.

The first example we consider is the paradigm of global quench: the ground
state of a translationally invariant Hamiltonian is let to evolve under a
different translationally invariant Hamiltonian [40]. We focus on generalised
XY models. In generic situations $I_{3}^{(\alpha)}$ vanish (both at the
initial time and) at infinite time after the quench (cf. Refs [24, 41]). We
are about to uncover exceptions when the initial state is the ground state of
a conformal critical system.

In noninteracting models the corresponding central charge, $c$, is a multiple
of $\frac{1}{2}$. Our numerical analysis shows that generically
$I_{3}^{(\alpha)}$ are nonzero also at late times, independently of whether
the post-quench Hamiltonian is critical or not, just provided that $c\geq 1$
in the initial state. This condition seems to be related to how slow the
slowest spatial connected correlations decay. Specifically, after quenches
from critical ground states there are 2-point correlation functions that decay
with the distance as a power-law. In all the cases investigated with the
initial state in the Ising universality class ($c=\frac{1}{2}$) we find that
the spin connected correlations in the stationary state do not decay more
slowly than $1/r^{4}$, where $r$ is the distance; this is not enough to
generate nonzero tripartite information. With $c\geq 1$ in the initial state
we find instead that the slowest spin correlations generally decay as
$1/r^{2}$, and $I_{3}^{(\alpha)}$ become nonzero (negative). Fig. 1 shows
$I_{3}^{(2)}$ at late times after a quench in the XY model. Remarkably,
$I_{3}^{(2)}$ remains a function of the cross ratio. This is observed also for
larger values of $\alpha$ and other choices of the Hamiltonian parameters; in
addition, the data seem to approach curves that depend on few system’s details
[42].

## Bipartitioning protocol.

Another type of global quench that has attracted a lot of attention is the
time evolution after joining two globally different states [43]. We consider
here the basic case in which the initial state consists of a domain of spins
aligned along $z$ joined with a domain of spins aligned in the opposite
direction. If we take the Hamiltonian of the XY model with $J_{x}\neq J_{y}$
_the quench is global_ , indeed the initial state is locally different from
any excited state of the Hamiltonian. We stress that the initial state is
_not_ critical and we are not aware of any CFT description of the infinite
time limit. Fig. 2 shows the Rényi-$3$ and Rényi-$4$ tripartite information in
the nonequilibrium steady state emerging at infinite time. In agreement with
the previous discussion, the slowest connected correlations decay as $1/r^{2}$
and we obtain nonzero $I_{3}^{(\alpha)}$. Again, the latter become functions
of the cross ratio and seem to remain so even in more sophisticated
bipartitioning protocols [42]. In the XY model considered here the asymptotic
curves do not even seem to depend on $J_{x}$ and $J_{y}$.

Figure 2: Rényi-$\alpha$ tripartite information at infinite time after the
quench from $\ket{\ldots\uparrow\uparrow\downarrow\downarrow\ldots}$ under the
XY Hamiltonian (3) with $(J_{x},J_{y},h)=(1,0.5,0)$ for
$|A|=|C|\in\\{50,100,150,200\\}$ and variable $|B|$. The solid curves are
predictions (6).

## Quench with symmetry-protected topological order.

We consider time evolution under (4) of two initial states: (a), the product
state with all spins aligned in the $z$ direction (cf. Ref. [39]), (b), the
same as (a) with one spin flipped (cf. Ref. [38]). In case (a),
$I_{3}^{(\alpha)}$ vanish. In case (b), $I_{3}^{(\alpha)}$ become nonzero and
seem to approach the same curves as in the domain-wall quench above. Since the
latter is dual to case (b) [38], our analysis suggests that $I^{(\alpha)}_{3}$
are not affected by the Kramers-Wannier duality. A more detailed analysis of
global quenches in this kind of systems will be reported in Ref. [44].

## Towards universality.

All systems investigated with nonzero $I_{3}^{(\alpha)}$ exhibit extensive
entropies with logarithmic corrections, which can in turn be traced back to
the presence of discontinuities in the “filling function” (or Fermi weight)
$\vartheta(k)$ [45, 46]. We remind the reader that, in an integrable model
with a thermodynamic Bethe Ansatz description [47, 48], filling functions
characterise excited states in the thermodynamic limit and represent the
fraction of particle excitations per given momentum [49]; in noninteracting
models
$\vartheta(k)\sim\braket{\bm{\mathbf{b}}^{\dagger}_{k}\bm{\mathbf{b}}^{\phantom{{\dagger}}}_{k}}$
is the coarse grained fermion occupation number. For example, in the Fermi sea
equivalent to the ground state of the XX model in zero field we have
$\vartheta(k)=\frac{1+\mathrm{sgn}(\cos(k))}{2}$. We are not aware of theorems
connecting discontinuities in $\vartheta(k)$ with conformal invariance when
$|2\vartheta(k)-1|\neq 1$, but we stress that discontinuities produce
nevertheless algebraically decaying correlations.

For the sake of simplicity, we restrict ourselves to filling functions with
symmetric discontinuities $\lim_{k\rightarrow
k_{F}^{\pm}}(2\vartheta(k)-1)=\tanh(\eta_{k_{F}}^{\pm})$, with
$\eta_{k_{\textsc{f}}}^{-}=-\eta_{k_{\textsc{f}}}^{+}$. We find that the
large-distance properties of the state can be described by a quantum field
theory of massless Dirac fermions, whose Euclidian action reads [42]

$S=\int\mathrm{d}x\int\mathrm{d}\tau\sum_{k_{\textsc{f}}}\sum_{s=\pm}\psi_{s,k_{\textsc{f}}}^{\dagger}(\partial_{\tau}+isv(k_{\textsc{f}})\partial_{x})\psi_{s,k_{\textsc{f}}}\,,$
(5)

where $k_{\textsc{f}}$ distinguishes theories emerging in the expansion around
different discontinuities, $v(k_{\textsc{f}})$ is the velocity, and the fields
satisfy standard anticommutation relations
$\\{\psi_{s,k_{\textsc{f}}}(x),\psi^{\dagger}_{s^{\prime},k_{\textsc{f}}^{\prime}}(y)\\}=\delta_{ss^{\prime}}\delta_{k_{\textsc{f}},k_{\textsc{f}}^{\prime}}\delta(x-y)\mathrm{I}$,
$\\{\psi_{s,k_{\textsc{f}}}(x),\psi_{s^{\prime},k_{\textsc{f}}^{\prime}}(y)\\}=0$.
The standard procedure to compute the Rényi entropies of subsystems starts
with reinterpreting the moments of the reduced density matrix,
$\mathrm{tr}[\rho_{X}^{\alpha}]$, as partition functions (cf. Fig. 1 of Ref.
[11]) in models formed by $\alpha$ copies of the original one under the
condition that each field is identified with the successive copy of itself
when crossing the space-time lines corresponding to $X$ at fixed (imaginary)
time $\tau=0$. The moments are finally conveniently identified with the
correlation functions of _local_ branch twist fields (associated with the
global symmetry of exchange of the copies) [50], which are localised at the
boundaries of $X$ and implicitly defined through the partition functions.

In spin chains, if the subsystem consists of two disjoint blocks, there is an
additional complication related to the non-locality of the Jordan-Wigner
transformation, indeed there are spin operators in $X$ whose fermionic
representation includes fermions lying in between the blocks. This problem can
be overcome by writing the reduced density matrix as a linear combination of
four density matrices [24]. One should then generalise the field theory
description so as to capture such a richer structure — see, e.g., (Sec. 4 of)
Ref. [28]. For example, one encounters also terms of the form
$\mathrm{tr}[\cdots P_{A}\rho_{A\cup C}^{n_{j}}P_{A}\rho_{A\cup
C}^{n_{j+1}}\cdots]$, where $P_{A}$ counts the parity of the fermions in the
first block $A$. In the QFT language, $P_{A}$ corresponds to the
transformation that changes the sign of the field when crossing $A$. In
general there are also other contributions with the same root, but they are
multiplied by expectation values of strings of Pauli matrices, which in our
nonequilibrium setting decay exponentially with the separation. Accordingly,
these terms can be dropped (see [42] for more details). This is why, despite
the similarity with Fermi seas, condition
$|\eta_{k_{\textsc{f}}}^{\pm}|\rightarrow\infty$ after global quenches
generally leads to an unusual tripartite information: the value of the filling
function at the discontinuities does not characterise the long-distance
properties of all the relevant degrees of freedom, as a nontrivial behavior of
the filling function between the discontinuities has the strong effect to turn
some algebraically decaying correlations into exponentially decaying ones.

In our situation, in turn, a part of the structure behind the QFT procedure
sketched above is lost. Specifically, the action of the $\alpha$-copy model is
still simply the sum of $\alpha$ copies of $S$ (5), but only a “physical part”
of the fields, namely,
$\Psi^{phys}_{k_{\textsc{f}}}(x)=\sum_{s}[1+\exp(2s\eta_{k_{F}}^{+})]^{-1/2}\psi_{s,k_{\textsc{f}}}(x)$
satisfies the standard conditions relating different copies on $X$ at zero
imaginary time (whereas the “unphysical parts” of the copies are independent).
And the same subtlety applies to $P_{A}$, which changes only the sign of the
physical fields. A direct consequence of having physical and unphysical fields
is that the energy density of the $\alpha$-copy model does change when
crossing $X$ at $\tau=0$, undermining, e.g., the interpretation of
$\mathrm{tr}[\rho_{X}^{\alpha}]$ as the correlation function of _local_ branch
twist fields [50]. As detailed in Ref. [42], however, it is anyway possible to
work out the asymptotic behaviour of the Rényi entropies of connected and
disconnected spin blocks using the resolvent method, similarly to what was
done in Refs [51, 52, 53, 54] for fermions.

## Results.

We announce here the simplest results, as they are already sufficient to
unveil the most striking feature of the tripartite information. To start with,
we focus on the limit $|\eta_{k_{\textsc{f}}}^{\pm}|\rightarrow\infty$. Since
the same condition is satisfied in a Fermi sea, we can use the correspondence
with the ground state of a CFT and take advantage of the results of Ref. [28],
which computed the terms $\mathrm{tr}[\cdots P_{A}\rho_{A\cup
C}^{n_{j}}P_{A}\rho_{A\cup C}^{n_{j+1}}\cdots]$ contributing to the tripartite
information in the CFT ground state one by one (together with terms that do
not contribute in our case).

The resulting prediction reads

$G_{\alpha}(x)\xrightarrow{|\eta_{k_{\textsc{f}}}^{\pm}|\rightarrow\infty}\frac{\log\Bigl{[}\sum_{\delta_{j}\in\\{0,\frac{1}{2}\\}\atop
j=1,\dots,\alpha-1}\bigl{(}\tfrac{\Theta(\vec{\delta}|\hat{\tau}_{x})}{\Theta(\vec{0}|\hat{\tau}_{x})}\bigr{)}^{\nu}\Bigr{]}}{\alpha-1}-\log
2$ (6)

where $\hat{\tau}_{x}$ is the $(\alpha-1)\times(\alpha-1)$ period matrix of
the Riemann surface $\mathcal{R}_{\alpha}$ with elements

$[\hat{\tau}_{x}]_{\ell
n}=\frac{2i}{\alpha}\sum_{k=1}^{\alpha-1}\sin(\tfrac{\pi
k}{\alpha})\cos(\tfrac{2\pi
k(\ell-n)}{\alpha})\tfrac{P_{(k/\alpha)-1}(2x-1)}{P_{(k/\alpha)-1}(1-2x)}\,.$
(7)

Here $P_{\mu}(z)$ denotes the Legendre polynomials,
$\Theta(\vec{z},M)=\sum_{\vec{m}\in\mathbb{Z}^{\alpha-1}}e^{i\pi\vec{m}^{t}M\vec{m}+2\pi
i\vec{m}\cdot\vec{\delta}}$ is the Siegel theta function, and $\nu$ is the
number of discontinuities of the filling function (assuming
$|\eta_{k_{\textsc{f}}}^{\pm}|\rightarrow\infty$ for each one of them). The
quench in Fig 2 can be used to check this prediction, indeed
$|\eta_{k_{\textsc{f}}}^{\pm}|\rightarrow\infty$. The agreement between
numerical data and prediction is excellent. We stress that, contrary to the
CFT analogue, our system does _not_ exhibit crossing symmetry, indeed
$G_{\alpha}(x)\neq G_{\alpha}(1-x)$.

Remarkably, $\lim_{x\rightarrow 1^{-}}G_{\alpha}(x)=-\log 2$ for every
$\alpha$, therefore (by the replica trick) we conclude that also the genuine
tripartite information approaches the same value

$\boxed{\mathfrak{I}_{3}=\lim_{x\rightarrow 1^{-}}G(x)=-\log 2}\,.$ (8)

This limit corresponds to small separation between the blocks compared to
their size, i.e. to the limit $1\ll|B|\ll|A|,|C|$. We call it _residual
tripartite information_ , because $x$ is exactly equal to $1$ only when
$|B|=0$, for which the tripartite information is zero by definition. Such
unusual nonzero residual tripartite information should be contrasted to the
ordinary zero value, which is found both in other nonequilibrium settings,
such as after translationally invariant quenches from ground states of gapped
Hamiltonians and critical Hamiltonians with $c=\frac{1}{2}$, and in
equilibrium at any temperature, independently of criticality.

Equation (8) is the main result of these notes. In all investigated systems
with a nonzero tripartite information our analysis points to a universal
residual tripartite information equal to $-\log 2$, irrespectively of the
quench protocol. In support of it, we also announce $G_{2}(x)$ with generic
$\eta_{k_{\textsc{f}}}^{-}$ and $\eta_{k_{\textsc{f}}}^{+}$:

$G_{2}(x)=\log\big{(}\tfrac{1+(1-x)^{\gamma}}{2}\big{)}\quad\Rightarrow
G_{2}(1^{-})=-\log 2\,,$ (9)

where
$\gamma=\sum_{k_{\textsc{f}}}[\tfrac{1}{\pi}\arg(\sin(\frac{\pi}{4}+i\eta_{k_{\textsc{f}}}^{+})/\sin(\frac{\pi}{4}+i\eta_{k_{\textsc{f}}}^{-}))]^{2}$.
Fig. 1 shows the excellent agreement of prediction(9) with numerical data.

## Numerical method.

The Rényi entropies have been numerically evaluated using their expressions in
terms of the fermionic correlation matrix $\Gamma_{\ell n}=\delta_{\ell
n}-\braket{\bm{\mathbf{a}}_{\ell}\bm{\mathbf{a}}_{n}}$. The latter has been
computed directly in the generalised Gibbs ensemble emerging in the limit of
infinite time [55, 56]. For the entropy of connected subsystems we used
$S_{\alpha}(X)=\mathrm{tr}\bigl{[}\log\bigl{(}(\frac{\mathrm{I}_{X}+\Gamma_{X}}{2})^{\alpha}+(\frac{\mathrm{I}_{X}-\Gamma_{X}}{2})^{\alpha}\bigr{)}\bigr{]}/[2(1-\alpha)]$,
where $\Gamma_{X}$ is the correlation matrix in $X$ [57, 58]. For the entropy
of disjoint blocks in the generalised XY model we used the algorithm proposed
in Ref. [24], which allows one to express $S_{\alpha}(A\cup C)$ in terms of
four matrices: $\Gamma_{1}\equiv\Gamma_{A\cup C}$, $\Gamma_{2}\equiv
P_{A}\Gamma_{1}P_{A}$, $\Gamma_{3}\equiv\Gamma_{1}-\Gamma_{A\cup
C,B}\Gamma_{B}^{-1}\Gamma_{B,A\cup C}$, $\Gamma_{4}=P_{A}\Gamma_{3}P_{A}$,
where $\Gamma_{A,A^{\prime}}$ is the correlation matrix in which the row and
column indices run in $A$ and $A^{\prime}$, respectively (we refer the reader
to Sec. 3 of Ref. [24] for the formula with $\alpha=2,3,4$ 222We warn the
reader of a typo in Ref. [24]: $\det\Gamma_{B_{1}}$ in Eqs (57)-(59) should be
replaced by $|\det\Gamma_{B_{1}}|$). The terms claimed before to survive the
infinite time limit are those constructed with $\Gamma_{1}$ and $\Gamma_{2}$
only. For disjoint blocks in the dual XY model we have generalised the
previous algorithm following Ref. [39], and it will be detailed elsewhere
[44].

## Discussion.

We have shown that the Rényi-$\alpha$ tripartite information captures
universal properties in the limit of infinite time after a global quench
whenever the correlations in the stationary state decay sufficiently slow with
the distance. We have provided evidence that $I_{3}^{(\alpha)}$ can be
obtained within a quantum field theory and, in some cases, we have been able
to predict their asymptotic behaviour. We think that the proposed framework
could be generalised at least to interacting integrable systems that can be
described by thermodynamic Bethe Ansatz, which have similar relaxation
properties. If nonzero, we found always negative $I_{3}^{(\alpha)}$; we wonder
how general in our setting this property is. Finally, we defined a residual
tripartite information $\mathfrak{I}_{3}$, which we found either equal to $0$
or to $-\log 2$. We leave the question open of which values $\mathfrak{I}_{3}$
could attain with interactions.

###### Acknowledgements.

We are grateful to Leonardo Mazza for collaboration at an early stage of this
project. We thank Raoul Santachiara for valuable discussions. We also thank
Pasquale Calabrese and Erik Tonni for useful correspondence. This work was
supported by the European Research Council under the Starting Grant No. 805252
LoCoMacro.

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