aainstitutetext: Université Paris-Saclay, CNRS, LPTMS, 91405, Orsay, France

# Universality in the tripartite information after global quenches:
(generalised) quantum XY models

Vanja Marić a Maurizio Fagotti<EMAIL_ADDRESS>
###### Abstract

We consider the Rényi-$\alpha$ tripartite information $I_{3}^{(\alpha)}$ of
three adjacent subsystems in the stationary state emerging after global
quenches in noninteracting spin chains from both homogeneous and bipartite
states. We identify settings in which $I_{3}^{(\alpha)}$ remains nonzero also
in the limit of infinite lengths and develop an effective quantum field theory
description of free fermionic fields on a ladder. We map the calculation into
a Riemann-Hilbert problem with a piecewise constant matrix for a doubly
connected domain. We find an explicit solution for $\alpha=2$ and an implicit
one for $\alpha>2$. In the latter case, we develop a rapidly convergent
perturbation theory that we use to derive analytic formulae approximating
$I_{3}^{(\alpha)}$ with outstanding accuracy.

##  1 Introduction

Nonequilibrium time evolution in isolated quantum many-body systems has been
intensively investigated for its capability to turn a ground state into an
effective finite-temperature one Polkovnikov2011Colloquium ; Eisert2015Quantum
; Gogolin2016Equilibration . Such a remarkable feature, closely related to the
eigenstate thermalisation hypothesis, is expected quite generally if the
interactions decay sufficiently fast to zero with the distance Rigol2008 ;
deutsch91 ; Srednicki1994Chaos ; Deutsch_2018 . The main exception is found in
integrable systems, in which the stationary properties of observables could
differ even significantly from those of thermal states Kinoshita2006 ;
Rigol2007Relaxation ; Vidmar2016Generalized . For example, in one dimensional
systems with local interactions the correlation lengths are generally finite
at nonzero temperature (which can be connected with the absence of phase
transitions) and the entropy of subsystems is extensive (which can be traced
back to the thermodynamic laws); on the other hand, at zero temperature the
correlation lengths can diverge (in critical systems) and the entropy is sub-
extensive Eisert2010Area_laws . Thus, at equilibrium in 1D systems the
divergence of a correlation length and the extensivity of the entropy seem to
be incompatible features. In the stationary state emerging after a quench in
an integrable system, such two properties can instead coexist
Maric2022Universality , suggesting the potential genesis of exotic states of
matter.

In the last three decades the measures of entanglement have acquired an
increasingly important role in quantifying the universal properties of a
quantum many-body system, especially for their mild dependence on the system’s
details Amico2008 ; Eisert2010Area_laws ; Horodecki2009Quantum . We also
mention that, while probing entanglement experimentally is still a difficult
problem, progress has been made also in that respect Kaufman2016 . The best
example of entanglement measure is arguably the von Neumann entropy of
subsystems. Restricting to 1D systems, which are the ones we consider in this
paper, universality is usually associated with the zero temperature properties
at quantum phase transitions; in this regard the entanglement entropy turns
out to be sensitive to criticality at the leading order of an asymptotic
expansion in the subsystem’s length Calabrese2004Entanglement ;
Calabrese2009Entanglement ; Holzhey1994Geometric ; Korepin2004PRL . It could
however also happen that the universal properties of interest are subleading
with respect to non-universal ones. This is the case for the topological
contribution to the entanglement entropy in 2D, which is concealed behind a
term proportional to the area of the subsystem Kitaev2006Topological . It
becomes then desirable to undress the quantities of the unwanted non-universal
details. In the aforementioned example, this is done by considering a special
combination of entanglement entropies, known as “tripartite
information”Cerf1998Information , that in the context of topological order is
usually referred to as “topological entanglement entropy”Kitaev2006Topological
.

The measures of entanglement have been thoroughly studied also after global
quenches, establishing for example that the entropy saturates to an extensive
value, that is to say, proportional to the size of the subsystem
Calabrese2005Evolution ; Sotiriadis2008 ; Bastianello2018Spreading ;
Bertini2022Growth ; Alba2017Entanglement ; Alba2018Entanglement ;
Casini2016spread ; Liu2014 ; Skinner2019 . One could argue that such extensive
parts are mainly due to classical correlations, therefore, it becomes of
interest to study combinations of entropies that simplify such contributions,
in complete analogy with the topological entanglement entropy. For this
reason, quantities such as the mutual information have attracted special
attention Alba2019QuantumInformationScrambling ;
Bertini2022EntanglementNegativity ; Eisler2014 ; Parez2022 .

At late times after a global quench, correlations often exhibit an exponential
decay in the distance Calabrese2011Quantum ; Calabrese2012 ; Sotiriadis2008 ,
but this is not a strict rule (see, e.g., Maric2022Universality ) and there
are many examples in which they decay algebraically, i.e., there are divergent
correlation lengths. In this paper we are interested exactly in such exotic
states that emerge after global quenches and that display both extensive
entropy and infinite correlation lengths. Similarly to the topological case
mentioned above, in those states the leading asymptotic behaviour of the
entanglement entropy is not universal and is not qualitatively different from
what one would find in any other stationary state with finite correlation
lengths: the universal properties that we envisage to exist are subleading.
Inspired by the definition of the topological entanglement entropy, we turn
our attention to the tripartite information. This is a special combination of
entanglement entropies that simplifies both extensive and boundary
contributions (of the subsystem), leaving a potentially universal quantity. We
show that this is indeed the case. Specifically, we derive and generalise the
results announced in Ref. Maric2022Universality . We study the Rényi-$\alpha$
tripartite information of three adjacent intervals in the stationary states
emerging after global quenches in noninteracting spin chains. We identify and
discuss a wide class of global quantum quench protocols which at infinite time
reach stationary states with a non-zero tripartite information. We derive the
asymptotic behaviour of the Rényi-$\alpha$ tripartite information by taking a
continuum limit and mapping the calculation to a Riemann-Hilbert problem with
a piecewise constant matrix for a doubly-connected domain. We show that the
Rényi-$\alpha$ tripartite information depends on few system details and
discuss a limit, which Ref. Maric2022Universality  dubbed “residual tripatrite
information”, that seems to be even independent of them. Technically speaking,
the methods that we employ build on Refs Casini2009reduced_density ;
Casini2009Entanglement ; Fries2019 ; Blanco2022 , which we enrich to take into
account the important differences Igloi2010 ; Fagotti2010disjoint  between the
entanglement of spins and that of fermions.

The paper is organized as follows. In section 2 we introduce the tripartite
information and discuss its behavior in different equilibrium settings.
Section 3 introduces the non-interacting models and quantum quench protocols
we study, discusses the differences between the entanglement of spins and
fermions and introduces the essentials of the free fermionic techniques that
we will use. Section 4 is a summary of the results obtained in this work, and
it has been postponed up to that point only because some quantities are
defined in the preceding sections; a reader familiar with the concepts
introduced in Sections 2 and 3 is encouraged to read Section 4 before the
previous ones. Section 4 shows also examples where the asymptotic formulae,
which will be derived later on, are applied to concrete quench protocols. The
derivations of the results are presented afterwards. Section 5 introduces a
continuum description, shows that stationary states of the studied quench
protocols can be described by an effective quantum field theory of free
fermionic fields on a ladder, presents a brief overview of the formalism for
computing the entanglement entropies in quantum field theory and applies it to
derive expressions that enable mapping the calculation to a Riemann-Hilbert
problem. The mapping is the subject of section 6. Section 7 deals with the
solution of the Riemann-Hilbert problem, which will be exact for $\alpha=2$
and perturbative for $\alpha>2$. Conclusions are drawn in section 8.

##  2 Tripartite information and its residual value

The von Neumann entropy $S_{1}(A)$ of a subsystem $A$ described by a reduced
density matrix $\rho_{A}$ is the Shannon entropy of $\rho_{A}$’s eigenvalues:

$S_{1}(A)=-\mathrm{tr}[\rho_{A}\log\rho_{A}]\,.$ (2.1)

If the state is pure, $S_{1}(A)$ measures the entanglement between $A$ and the
rest. Additional information about the entanglement is provided by the Rényi
entropies

$S_{\alpha}(A)=\frac{1}{1-\alpha}\log\mathrm{tr}[\rho_{A}^{\alpha}]\,,$ (2.2)

with $\alpha=2,3,4,\ldots$, which completely characterise the distribution of
eigenvalues of $\rho_{A}$ (also known as entanglement spectrum). If the state
is mixed, the von Neumann entropy and the Rényi entropies, which we will also
generically refer to as “entanglement entropies”, are also affected by
classical correlations, that is to say, they are not good measures of
entanglement.

$A$$B$$C$ Figure 1: By $A$, $B$ or $C$ we denote a connected block of spins,
i.e. an interval, in an infinite spin chain. In this work we study the
tripartite information of three adjacent blocks.

In this work we will indicate the subsystems with words of letters, $A$, $B$,
and $C$, that refer to connected blocks of spins, and we will always assume
that $A$, $B$, and $C$ are adjacent, as shown in Figure 1. In such a geometry,
at finite temperature the entropies become extensive with the size of the
subsystem and carry a leading classical contribution. Such extensive terms can
be cancelled in combinations of entropies, such as in the mutual information.
The mutual information of two subsystems $A$ and $C$ is defined as
Cerf1998Information

$I_{2}(A,C)=S_{1}(A)+S_{1}(C)-S_{1}(A\cup C)$ (2.3)

and its Rényi analogue reads

$I_{2}^{(\alpha)}(A,C)=S_{\alpha}(A)+S_{\alpha}(C)-S_{\alpha}(A\cup C)\,.$
(2.4)

The simplification of the extensive contributions is manifest in thermal
states when $C$ is the complement of $A$ (i.e., $B=\emptyset$ and $A$ and $C$
extend over the entire chain in Fig. 1), where it was shown that the mutual
information satisfies an area law Wolf2008 ; Kuwahara2021 ;
Eisert2010Area_laws ; Bernigau2015 ; Lemm2022 ; Alhambra2022 .

The mutual information is a natural quantity to study also in quantum field
theory, and it has received substantial attention both in 1+1 Casini2005 ;
Calabrese2009Entanglement1 ; Casini2009reduced_density ; Casini2007mutual ;
Furukawa2009Mutual ; Headrick2010 ; Chen2013JHEP ; Chen2014Holographic ;
Headrick2015 ; Casini2015Mutual ; Blanco2011 ; Fries2019 ; Arias2018 ;
Blanco2019 ; Blanco2022 ; Ares2022 ; Asplund2014 ; Molina-Vilaplana2011 ;
Lepori2022  and higher dimensions Casini2009Remarks ; Shiba2012 ; Cardy2013 ;
Allais2012 ; Agon2016 ; Agon2016Large ; Chen2017PRD ; Casini2015area ;
Tonni2011 : while the entanglement entropies are ultraviolet divergent, such
divergences cancel out in the mutual information, provided that the subsystems
are not adjacent.

A related quantity is the tripartite information. For three subsystems $A,B,C$
the tripartite information Cerf1998Information  is defined as

$I_{3}(A,B,C)=I_{2}(A,B)+I_{2}(A,C)-I_{2}(A,B\cup C)=\\\
S_{1}(A)+S_{1}(B)+S_{1}(C)-S_{1}(A\cup B)-S_{1}(B\cup C)-S_{1}(A\cup
C)+S_{1}(A\cup B\cup C)$ (2.5)

and its Rényi analogue reads

$I_{3}^{(\alpha)}(A,B,C)=I_{2}^{(\alpha)}(A,B)+I_{2}^{(\alpha)}(A,C)-I_{2}^{(\alpha)}(A,B\cup
C)\,,$ (2.6)

thus it can be viewed as a measure of the extensivity of the mutual
information. While the mutual information can be interpreted Groisman2005  as
a measure of the total amount of correlations, classical and quantum, between
the two subsystems, to the best of our knowledge there is no such a simple
interpretation for the tripartite information. There have been numeorus
investigations into the tripartite information in disparate settings. Besides
the remarkable role in quantifying topological properties
Kitaev2006Topological , which we have already mentioned, it has received a
substantial attention in quantum field theories in 1+1 dimensions
Casini2009Remarks ; Caraglio2008Entanglement ; Furukawa2009Mutual ;
Rajabpour2012 ; Calabrese2009Entanglement1 ; Calabrese2011Entanglement ;
Coser2014OnRenyi ; Ruggiero2018 ; Alba2010Entanglement ; Blanco2011 ;
Alba2011Entanglement ; Fagotti2010disjoint ; Fagotti2012New ; Fries2019 ;
Balasubramanian2011 ; Grava2021 ; Ares2021Crossing , mainly conformal, as well
as in higher dimensions Casini2009Remarks ; Agon2016 ; Agon2022 ;
AliAkbari2021 . There is also a general result Hayden2013Holographic  that in
holographic theories the tripartite information is never positive, implying
that the mutual information is always extensive or superextensive, and
suggesting that quantum entanglement dominates over classical correlations. En
passant, we note that the tripartite information was recently studied in
continuously monitored chains Carollo2022 , on Hamming graphs
Parez2022Multipartite , and also as a diagnostic for quantum scrambling
Hosur2016Chaos ; Schnaack2019Tripartite ; Sunderhauf2019Quantum ; Kuno2022 .

The key reason why we focus on the tripartite information is that it remains
bounded in the limit $a\ll|B|\ll|A|,|C|$, allowing us to unveil a contribution
that is subleading in the mutual information and that exhibits universal
properties. Here and in the following $|A|$ stands for the size of $A$. To be
more quantitative, let us consider a connected block $A$. In many states of
interest in spin chains (e.g., at equilibrium), the entanglement entropies of
such a block are captured by the Ansatz

$S_{\alpha}(A)\sim\kappa_{\alpha}\frac{|A|}{a}+\omega_{\alpha}\log\frac{|A|}{a}+d_{\alpha}\,,$
(2.7)

where $a$ is the lattice spacing and $\sim$ denotes the limit in which the
subsystem size is much larger than $a$. In the rest of the paper we will use
$\sim$ to denote the limit in which all subsystem’s sizes are much larger than
the lattice spacing. In the ground state of a local Hamiltonian
$\kappa_{\alpha}$ vanishes, while $\omega_{\alpha}$ is nonzero only if there
are divergent correlation lengths Hastings2007 . In equilibrium at finite
temperature $\kappa_{\alpha}$ becomes nonzero and $\omega_{\alpha}$ is
expected to vanish (this was explicitly shown in conformal field theories
Calabrese2004Entanglement , and, using the approach of Ref. Jin2004 , could be
rather easily proved in the absence of interactions; it can also be argued by
the absence of phase transitions at finite temperature, but we are not aware
of a general proof). A non-zero value of both $\kappa_{\alpha}$ and
$\omega_{\alpha}$ was instead observed in non-equilibrium steady states
Eisler2014 ; Fraenkel2021 ; FagottiMaricZadnik2022  and in excited states
Alba2009Entanglement ; Ares2014 , as well as after global quenches from ground
states of critical systems Maric2022Universality .

Assuming translational invariance, from (2.7) it follows that, if $A$ and $B$
are adjacent intervals, the mutual information behaves as

$I_{2}^{(\alpha)}(A,B)\sim\omega_{\alpha}\log\frac{|A||B|}{a(|A|+|B|)}+d_{\alpha}\,,$
(2.8)

which explicitly shows the dependency on the UV cut-off $a$ as well as on the
boundary contributions included in $d_{\alpha}$. Note that the mutual
information diverges in the limit of infinite size of the blocks $A$ and $B$.
For three adjacent intervals $A,B,C$ (see Figure 1), the behaviour of the
tripartite information follows from that of the mutual information (cf. (2.8)
and (2.6))

$I_{3}^{(\alpha)}(A,B,C)\sim\omega_{\alpha}\log(1-x)+I_{2}^{(\alpha)}(A,C),$
(2.9)

where $x=|A||C|/((|A|+|B|)(|B|+|C|))$ is the cross ratio. In equilibrium at
nonzero temperature, $I_{2}(A,C)$ vanishes exponentially fast with $|B|$
irrespective of $A,C$ Bluhm2022exponentialdecayof ; Alhambra2022  hence, since
$\omega_{\alpha}$ is zero, $I_{3}^{(\alpha)}(A,B,C)$ approaches zero in the
limit of large lengths. This applies also to the ground state of noncritical
systems, which satisfy the area law Hastings2007  and exhibit exponential
decay of correlations Hastings2006 . On the other hand, in the ground state of
a conformal critical system, $\omega_{\alpha}$ is proportional to the central
charge and we need more refined arguments to infer the values of the
tripartite information. In a 1+1 dimensional conformal field theory for the
configuration of Figure 1 one has Calabrese2009Entanglement1

$\mathrm{tr}\bm{\mathbf{\rho}}_{A\cup C}^{\alpha}\sim
c_{\alpha}^{2}\left(\frac{a^{2}(|A|+|B|)(|B|+|C|)}{|A||B||C|(|A|+|B|+|C|)}\right)^{\frac{c}{6}(\alpha-\frac{1}{\alpha})}\mathcal{F}_{\alpha}(x),$
(2.10)

where $c$ is the central charge, $c_{\alpha}$ is a non-universal normalization
factor and $\mathcal{F}_{\alpha}(x)$ is a universal function of the cross
ratio $x=|A||C|/((|A|+|B|)(|A|+|C|))$ which depends on the full operator
content of the theory. Consequently, the tripartite information is a universal
function of $x$, directly related to $\mathcal{F}$ through

$I_{3,\,CFT}^{(\alpha)}(A,B,C)\sim\mathcal{G}_{\alpha}(x)\,,\quad\mathcal{G}_{\alpha}(x)=\frac{1}{\alpha-1}\log[\mathcal{F}_{\alpha}(x)].$
(2.11)

We mention that $\mathcal{G}_{\alpha}(x)$ can be positive, zero, or negative
Casini2009Remarks , the latter case occurring, in particular, when the central
charge is large enough Fagotti2012New .

Generally, however, $\mathcal{G}_{\alpha}(x)$ exhibits two main properties:
$\mathcal{G}_{\alpha}(0^{+})=0$, which manifests cluster decomposition, and
$\mathcal{G}_{\alpha}(x)=\mathcal{G}_{\alpha}(1-x)$, which is a consequence of
crossing symmetry (it follows from the fact that, in a pure state, the entropy
of a subsystem matches the entropy of its complement). These two properties
together imply $\mathcal{G}_{\alpha}(1^{-})=0$. That is to say, in all the
situations covered in this brief summary the tripartite information is zero in
the limit $a\ll|B|\ll|A|,|C|$, which corresponds to the limit $x\to 1^{-}$.
Note also that the value of the tripartite information for $|B|=0$ is zero by
definition111This follows from the convention that the entanglement entropy of
an empty set is zero. Defining it differently would lead to a violation of
some desirable properties, such as the strong subadditivity of the Von Neumann
entanglement entropy. Moreover, we want to keep the property that in pure
states the entropy of a subsystem is equal to the entropy of its complement.
Since the entropy of the whole system is zero, we should also set it to zero
for its complement, the empty set.. A nonzero value of
$I_{3}^{(\alpha)}(A,B,C)$ in the limit $a\ll|B|\ll|A|,|C|$, which was called
“residual tripartite information” Maric2022Universality , is therefore an
indicator of properties that are not generally found in systems at
equilibrium. Ref. Maric2022Universality  has identified some nonequilibrium
settings which are expected to exhibit a residual tripartite information.
Focusing on those quench protocols, which we introduce in the next section, we
will analytically compute the Rényi-$\alpha$ tripartite information of
adjacent subsystems in the limit in which the subsystem’s lengths approach
infinity.

Figure 2: Schematic representation of a global quench from the ground state of
a local Hamiltonian $\bm{\mathbf{H}}_{0}$. We study the tripartite information
of $A$, $B$, and $C$ in the limit of infinite time.

##  3 The model

We consider a class of Hamiltonians, known as generalised XY model
Suzuki1971The ; Lieb1961 , which are mapped into quadratic forms of fermions
by the Jordan-Wigner transformation

$\bm{\mathbf{a}}_{2\ell-1}=\prod_{j<\ell}\bm{\mathbf{\sigma}}_{j}^{z}\bm{\mathbf{\sigma}}_{\ell}^{x}\qquad\bm{\mathbf{a}}_{2\ell}=\prod_{j<\ell}\bm{\mathbf{\sigma}}_{j}^{z}\bm{\mathbf{\sigma}}_{\ell}^{y}\,.$
(3.1)

The fermions defined by Eq. (3.1) are self-adjoint (Majorana) and satisfy the
algebra

$\\{\bm{\mathbf{a}}_{j},\bm{\mathbf{a}}_{\ell}\\}=2\delta_{j\ell}\;.$ (3.2)

The most studied example belonging to this class is the quantum XY model

$\bm{\mathbf{H}}=\sum\nolimits_{\ell}\left(J_{x}\bm{\mathbf{\sigma}}_{\ell}^{x}\bm{\mathbf{\sigma}}_{\ell+1}^{x}+J_{y}\bm{\mathbf{\sigma}}_{\ell}^{y}\bm{\mathbf{\sigma}}_{\ell+1}^{y}+h\bm{\mathbf{\sigma}}_{\ell}^{z}\,\right),$
(3.3)

of which the XX model ($J_{x}=J_{y}$) and the transverse-field Ising model
($J_{y}=0$) are special cases. The generalisation, which is described in
Appendix A, can be dealt with analogously to the XY model. It allows for
interactions of any range, though with a particular structure. For the sake of
simplicity we shall assume that the Hamiltonian is local, i.e., that the
interactions have a finite range222All the results we are going to present are
expected to hold true even with quasilocal Hamiltonians in which the coupling
constants decay exponentially fast with the range..

By translational invariance, $\bm{\mathbf{H}}$ can be expressed as
follows333More precisely, with periodic boundary conditions the mapping would
result in two distinct noninteracting sectors, which however turn out to be
equivalent for our purposes.

$\bm{\mathbf{H}}=\frac{1}{4}\sum_{j,\ell}\vec{\bm{\mathbf{a}}}_{j}^{\dagger}\int_{-\pi}^{\pi}\frac{dk}{2\pi}\mathcal{H}(k)e^{ik(j-l)}\
\vec{\bm{\mathbf{a}}}_{\ell},\quad\vec{\bm{\mathbf{a}}}_{\ell}\equiv\begin{pmatrix}\bm{\mathbf{a}}_{2\ell-1}\\\
\bm{\mathbf{a}}_{2\ell}\end{pmatrix},$ (3.4)

where the $2\times 2$ matrix $\mathcal{H}(k)$ generates the coupling constants
through its Fourier coefficients and, from now on, will be referred to as the
symbol of the Hamiltonian Fagotti2016Charges . To be explicit, Hamiltonian
(3.3) has the following symbol

$\mathcal{H}(k)=2(J_{x}-J_{y})\sin(k)\sigma^{x}+[2h-2(J_{x}+J_{y})\cos(k)]\sigma^{y}\;.$
(3.5)

The properties of the symbol in view of Hamiltonian’s locality are given in
Table 1. In particular, local Hamiltonians are characterized by a smooth
symbol.

We consider two kinds of quenches:

Quench from a critical point:

The initial state is the ground state of a critical Hamiltonian
$\bm{\mathbf{H}}_{0}$ with local interactions, such as the XX model. In the
symbol of $\bm{\mathbf{H}}_{0}$ criticality is manifested in the existence of
(at least) a momentum $\bar{k}$ for which the kernel of
$\mathcal{H}_{0}(\bar{k})$ is non-empty.

The state is let to evolve with a different local Hamiltonian
$\bm{\mathbf{H}}$ for an arbitrarily long time (see Fig. 2 for a sketch).

If we relax the constraint of criticality in the initial state, this is the
standard example of global quench and, as such, has been thoroughly
investigated (we refer the reader to Refs Calabrese2016Quantum ;
Essler2016Quench ; Vidmar2016Generalized ; Caux2016  for some reviews on the
topic that are relevant to our work). In integrable systems, like the
generalised XY model, local observables approach stationary values described
by generalised Gibbs ensembles (GGEs), which retain memory of infinitely many
integrals of motion.

In this paper we will assume that the time is so large that the observables in
the subsystems can be described by a GGE.

Bipartitionig protocol:

The initial state is separated in two parts with macroscopically different
properties, for example, a domain-wall of spins or two thermal states of a
local Hamiltonian $\bm{\mathbf{H}}_{0}$ at different temperatures.

The state is let to evolve with a different local Hamiltonian
$\bm{\mathbf{H}}$ for an arbitrarily long time (see Fig. 3 for a sketch).

Similar bipartitioning protocols in integrable systems have recently received
a lot of attention, we mention Refs Sabetta2013 ; Eisler2014Entanglement ;
Mazza2018 ; Bertini2018Entanglement ; Alba2019Entanglement ;
Gruber2019Magnetization ; Collura2020Domain  as an incomplete list of works
focussed on integrable systems including also investigations into the
entanglement properties; the interested reader is encouraged to consult the
Collection Bastianello2022Introduction  for an overview (Ref.
Alba2021Generalized  in particular reviews the case considered here in which
the quench is global).

In integrable systems, the abrupt change of properties of the initial state
experiences a ballistic spreading in time in which the expectation values of
local observables approach values that depend on the ratio between the
distance from the inhomogeneity and the time. This is the paradigm of quench
protocol captured by the theory of generalised hydrodynamics
Bertini2016Determination ; Bertini2016Transport ; Castro-Alvaredo2016Emergent
. The limit of infinite time, in particular, is described by a nonequilibrium
stationary state, usually called NESS, which can be represented by a
generalised Gibbs ensemble notwithstanding the entire state not being
translationally invariant.

In this paper we will assume that the time is so large that the observables in
the subsystems can be described by a NESS.

Figure 3: Schematic representation of a bipartitioning protocol in which the
quench is global. We study the limit of infinite time in the tripartite
information of $A$, $B$, and $C$ in which the lengths $|A|,|B|,|C|$ will be
assumed large but not enough to be affected by the inhomogeneity of the state.

###  3.1 Correlation matrices and filling functions

$2$-by-$2$ symbol of $M$ ($M_{2\ell+i,2n+j}=M_{2(\ell-n)+i,j}$, $\ell,n\in\mathbb{Z}$): | $M_{ij}(k)=\sum\limits_{n\in\mathbb{Z}}e^{ikn}M_{2(\ell-1)+i,2(\ell+n-1)+j}$
---|---
Jordan-Wigner transformation: | $\vec{\bm{\mathbf{a}}}_{\ell}\equiv\begin{pmatrix}\bm{\mathbf{a}}_{2\ell-1}\\\ \bm{\mathbf{a}}_{2\ell}\end{pmatrix}=\begin{pmatrix}\prod_{j<\ell}\bm{\mathbf{\sigma}}_{j}^{z}\bm{\mathbf{\sigma}}_{\ell}^{x}\\\ \prod_{j<\ell}\bm{\mathbf{\sigma}}_{j}^{z}\bm{\mathbf{\sigma}}_{\ell}^{y}\end{pmatrix}$
Hamiltonian and its symbol: | $\bm{\mathbf{H}}=\frac{1}{4}\sum_{j,\ell}\vec{\bm{\mathbf{a}}}_{j}^{\dagger}\int_{-\pi}^{\pi}\frac{dk}{2\pi}\mathcal{H}(k)e^{ik(j-l)}\ \vec{\bm{\mathbf{a}}}_{\ell}$
quasilocality: |  $\mathcal{H}(k)$ is smooth
locality: |  $\mathcal{H}(k)$ has a finite number of nonzero
| Fourier coefficients
dispersion relation (reflection symmetric $\bm{\mathbf{H}}$): | $\varepsilon(k)=\frac{1}{2}\sqrt{\mathrm{tr}[\mathcal{H}^{2}(k)]}$
correlation matrix: | $\Gamma_{2(\ell-1)+i,2(n-1)+j}=\delta_{\ell n}\delta_{ij}-\braket{\vec{\bm{\mathbf{a}}}_{\ell}\vec{\bm{\mathbf{a}}}_{n}^{\dagger}}_{ij}$
ground state of $\bm{\mathbf{H}}$ (w/o symmetry breaking): | $\Gamma(k)=-\mathrm{sgn}\left(\mathcal{H}(k)\right)$
thermal state of $\bm{\mathbf{H}}$: | $\Gamma(k)=-\tanh\left(\frac{\beta\mathcal{H}(k)}{2}\right)$
GGE after a quench $\Gamma_{0}\rightarrow\bm{\mathbf{H}}$ (refl. sym. $\bm{\mathbf{H}}$): | $\Gamma(k)=\frac{\\{\Gamma_{0}(k),\mathcal{H}(k)\\}}{2\varepsilon^{2}(k)}\mathcal{H}(k)$
NESS after a quench $\Gamma_{-}\otimes\Gamma_{+}\rightarrow\bm{\mathbf{H}}$ (refl. sym. $\bm{\mathbf{H}}$): | $\Gamma(k)=\sum_{s=\pm 1}\bigg{(}\frac{\Gamma_{s}(k)}{2}$
|
$-\mathrm{sgn}[\varepsilon^{\prime}(k)]s\frac{\mathrm{tr}[\Gamma_{s}(k)\mathcal{H}(k)]+\mathrm{tr}[\Gamma_{s}(k)]\mathcal{H}(k)}{4\varepsilon(k)}\bigg{)}$
Table 1: Free-fermion dictionary. A list of the main ingredients of the free-
fermion techniques used in this paper. The symbol of a reflection-symmetric
Hamiltonian of the generalised XY model can be any traceless 2-by-2 matrix
function $\mathcal{H}(k)$ satisfying the usual property of symbols, i.e.,
$\mathcal{H}^{\dagger}(k)=\mathcal{H}(k)=-\mathcal{H}^{t}(-k)$.

Since we restrict ourselves to noninteracting quenches, the states are
Gaussian at any time, i.e., their density matrix is the exponential of a
quadratic form of fermions. In such states the expectation values of operators
are completely determined by the fermionic two-point correlation matrix
$\Gamma_{2\ell+i,2n+j}=\delta_{\ell
n}\delta_{ij}-\braket{\vec{\bm{\mathbf{a}}}_{\ell}\vec{\bm{\mathbf{a}}}_{n}^{\dagger}}_{ij}$.
In a one-site shift invariant state the correlation matrix is block Toeplitz
and can be expressed in terms of a $2$-by-$2$ matrix function $\Gamma(k)$, its
symbol, by Fourier transform

$\Gamma_{2\ell+i,2n+j}=\int_{-\pi}^{\pi}\frac{dk}{2\pi}e^{ik(\ell-n)}\Gamma_{ij}(k)\;.$
(3.6)

We remind the reader that the algebra of the Majorana fermions implies the
property $\Gamma^{\dagger}(k)=\Gamma(k)=-\Gamma^{t}(-k)$, where
${\bullet}^{t}$ denotes transposition.

The correlation matrices for the quenches of interest are shown in Table 1. In
all quenches considered, the late time dynamics can be captured by a
generalised Gibbs ensemble; in this respect, we remind the reader that a GGE
is locally equivalent to a family of excited states, which, in this context,
are called “representative states” Caux2013Time ; Essler2016Quench . Thus, we
briefly introduce also the standard language used to describe excited states
in noninteracting models, as well as in a class of interacting integrable
systems. Given an excited state, the filling function $\vartheta(k)$ is a
regularised characteristic function of the set of occupied momenta, i.e., it
represents the average occupation of momenta around $k$444 We warn the reader
that the ambiguity in the definition of what is a particle and what is a hole
allows for alternative definitions of filling functions. In particular, one is
in principle free to choose any excited state as a reference state
characterised by $\vartheta(k)=0$. In order to attach some physical meaning to
$\vartheta(k)$, it is however convenient to choose a reference state with low
entanglement. If the ground state is non-critical, the most natural choice is
to promote the ground state to the reference state (the other possibility
being the state with maximal energy). If the ground state is critical,
instead, choosing the ground state of a conservation law with a non-critical
ground state would do the job. For example, the ground state of the XX model
in $0$ field is critical. Since, however, the total magnetization in the $z$
direction commutes with the Hamiltonian, we can choose as reference state the
ground state of the model with a large enough magnetic field, which is the
state in which all spins are aligned in the $z$ direction. This is a customary
choice in models with a $U(1)$ symmetry. In this paper we will use such
standard conventions.. Filling functions, together with the closely related
root densities555In noninteracting systems, in particular, the filling
function $\vartheta(k)$ and the root density $\rho(k)$ satisfy
$\vartheta(k)=2\pi\rho(k)$., are the main ingredients of the theory of
generalised hydrodynamics. Since they satisfy simple dynamical equations (see
Ref. Fagotti2020  for a thorough discussion in noninteracting spin chains),
they are convenient to work with especially in the presence of
inhomogeneities, where other formalisms struggle with. Assuming that there is
a smooth unitary matrix function $U(k)$ of $k$ diagonalising the symbol of the
correlation matrix of a stationary state, the filling function can be
identified with $\Gamma(k)$’s eigenvalues as follows:

$U^{\dagger}(k)\Gamma(k)U(k)=\mathrm{diag}\left[2\vartheta(k)-1,-2\vartheta(-k)+1\right].$
(3.7)

We refer the reader to Section 2 of Ref. Alba2021Generalized  for a
pedagogical introduction to filling functions (or root densities) in
noninteracting systems; for the purpose of this paper, it is sufficient to
regard (3.7) as a definition of $\vartheta(k)$.

###  3.2 Rényi entropies

The tripartite information of adjacent subsystems is a linear combination of
entropies of spin blocks with the entropy of two disjoint blocks (2.5). In our
system the entropy of a spin block is equal to the entropy of the
corresponding block of Jordan-Wigner fermions Jin2004 ; Vidal2003  and reads

$S_{\alpha}(A)=\tfrac{1}{1-\alpha}\tfrac{1}{2}\mathrm{tr}\Bigl{[}\log\Bigl{(}\bigl{(}\tfrac{\mathrm{I}+\Gamma_{A}}{2}\bigr{)}^{\alpha}+\bigl{(}\tfrac{\mathrm{I}-\Gamma_{A}}{2}\bigr{)}^{\alpha}\Bigr{)}\Bigr{]}\,,$
(3.8)

where $\Gamma_{A}$ is the correlation matrix in $A$. In the case of disjoint
blocks, instead, the non-locality of the Jordan-Wigner transformation
complicates the correspondence. A procedure for computing the Rényi entropies
of two disjoint blocks on the lattice that takes into account these non-
locality effects has been proposed in Fagotti2010disjoint  and is briefly
reviewed below.

The entropy $S_{\alpha}(A\cup C)$ is expressed in terms of four correlation
matrices: $\Gamma_{1}\equiv\Gamma_{A\cup C,A\cup C}$,
$\Gamma_{2}\equiv\mathrm{P}\Gamma_{1}\mathrm{P}$,
$\Gamma_{3}\equiv\Gamma_{1}-\Gamma_{A\cup C,B}\Gamma_{B,B}^{-1}\Gamma_{B,A\cup
C}$ and $\Gamma_{4}\equiv P\Gamma_{3}P$, where
$\Gamma_{A^{\prime},A^{\prime\prime}}$ for some subsystems
$A^{\prime},A^{\prime\prime}$ is the correlation matrix with the indices
running in $A^{\prime},A^{\prime\prime}$ respectively,

$\begin{split}&\left(\Gamma_{A^{\prime},A^{\prime\prime}}\right)_{2(\ell-1)+i,2(n-1)+j}=\delta_{A^{\prime}(\ell),A^{\prime\prime}(n)}\delta_{ij}-\braket{\vec{\bm{\mathbf{a}}}_{A^{\prime}(\ell)}\vec{\bm{\mathbf{a}}}^{\dagger}_{A^{\prime\prime}(n)}}_{ij}\;,\\\
\quad&\ell\in\\{1,2,\ldots,|A^{\prime}|\\},\
n\in\\{1,2,\ldots,|A^{\prime\prime}|\\},\ i,j\in\\{1,2\\},\\\
&A^{\prime}(1)<A^{\prime}(2)<\ldots<A^{\prime}(|A^{\prime}|)\textrm{ are the
elements of $A^{\prime}$ in ascending order (analogously for
$A^{\prime\prime}$)},\end{split}$ (3.9)

and

$\mathrm{P}_{\ell n}\equiv\delta_{\ell n}\begin{cases}-1&1\leq\ell\leq 2|A|\\\
1&\text{otherwise}\end{cases}$ (3.10)

introduces a minus sign for each fermion in block $A$. Specifically, we have

$S_{\alpha}(A\cup
C)=\frac{1}{1-\alpha}\log\left[\frac{1}{2^{\alpha}}\sum_{j_{1},j_{2},\ldots
j_{\alpha}=1,2,3,4\atop(-1)^{N_{3}+N_{4}}=1}(-1)^{N_{4}}|\det\Gamma_{B,B}|^{\frac{N_{3}+N_{4}}{2}}\\{\Gamma_{j_{1}},\Gamma_{j_{2}},\ldots,\Gamma_{j_{\alpha}}\\}\right].$
(3.11)

Here $N_{3}$ ($N_{4}$) is the total number of $j_{1},j_{2},\ldots j_{\alpha}$
equal to $3$ ($4$). Only the terms in which the total number of indices equal
to $3$ or $4$ is even appear. The symbol $\\{\ldots\\}$ inside the logarithm
is defined as the trace of a product of normalized Gaussians,

$\\{\Gamma_{j_{1}},\Gamma_{j_{2}},\ldots,\Gamma_{j_{\alpha}}\\}\equiv\mathrm{tr}\left[\rho(\Gamma_{j_{1}})\rho(\Gamma_{j_{2}})\ldots\rho(\Gamma_{j_{\alpha}})\right],$
(3.12)

where $\rho(\Gamma)\propto\exp{\left(\sum_{j,\ell\in
A,C}\vec{\bm{\mathbf{a}}}_{j}^{\dagger}W_{j,l}\vec{\bm{\mathbf{a}}}_{\ell}/4\right)}$
has correlation matrix $\Gamma$ ($\Gamma$ and $W$ are related to each other
through $\Gamma=\tanh(W/2)$).

The entropy of two disjoint blocks of fermions consists solely of the first
term,

$S_{\alpha}^{\mathrm{(fermions)}}(A\cup
C)=\frac{1}{1-\alpha}\log\\{\Gamma_{1},\Gamma_{1},\ldots,\Gamma_{1}\\}.$
(3.13)

The other terms appearing in (3.11) account for the non-locality of the
Jordan-Wigner fermions with respect to the spin degrees of freedom. For
instance, the operator $\sigma^{x}_{j\in A}\sigma^{x}_{\ell\in C}$, when
expressed in terms of fermions, contains an undesired string in between the
two blocks. In accordance, terms with $N_{3}+N_{4}\neq 0$ in (3.11) are
accompanied by a factor equal to a power of the expectation value of the
string of $\bm{\mathbf{\sigma}}_{\ell}^{z}$ operators between the two blocks,

$\left|\det\Gamma_{B,B}\right|^{\frac{N_{3}+N_{4}}{2}}=\braket{\prod_{j\in
B}\bm{\mathbf{\sigma}}_{j}^{z}}^{N_{3}+N_{4}}\,.$ (3.14)

After global quenches in systems without semilocal charges, string order
doesn’t survive. That is to say, the expectation value of strings of Pauli
matrices decays exponentially with the length of the string (see Ref.
FagottiMaricZadnik2022  for a more detailed discussion). Thus, it is
reasonable to expect the string expectation value to kill all the
contributions in (3.11) from the terms with non-zero $N_{3}+N_{4}$. And indeed
our numerical checks confirm the validity of this statement. In the limit in
which all lengths approach infinity, the entropy of disjoint blocks after a
global quench is therefore captured by the following simplified expression

$S_{\alpha}(A\cup
C)\sim\frac{1}{1-\alpha}\log\left[\frac{1}{2}\sum_{j_{1},j_{2},\ldots,j_{\alpha}=1}^{2}\\{\Gamma_{j_{1}},\Gamma_{j_{2}},\ldots,\Gamma_{j_{\alpha}}\\}\right]+\log
2.$ (3.15)

Each term of the sum can be evaluated using the recursive formula derived in
Ref. Fagotti2010disjoint , which expresses a product of two normalised
Gaussians as a Gaussian

$\rho(\Gamma)\rho(\Gamma^{\prime})=\\{\Gamma,\Gamma^{\prime}\\}\rho\left(\Gamma\times\Gamma^{\prime}\right),\quad\Gamma\times\Gamma^{\prime}\equiv\mathrm{I}-(\mathrm{I}-\Gamma^{\prime})(\mathrm{I}+\Gamma\Gamma^{\prime})^{-1}(\mathrm{I}-\Gamma)\,,$
(3.16)

together with a formula for the trace of a product of normalised Gaussians

$\\{\Gamma,\Gamma^{\prime}\\}^{2}=\mathrm{tr}[\rho(\Gamma)\rho(\Gamma^{\prime})]^{2}=\det\bigg{(}\frac{\mathrm{I}+\Gamma\Gamma^{\prime}}{2}\bigg{)}\;.$
(3.17)

Alternatively, (3.16) could be replaced by the formula for the trace of a
product of an arbitrary number of Gaussians reported in Ref. Klich2014 . Note
that (3.17) fixes the value of $\\{\Gamma,\Gamma^{\prime}\\}$ only up to a
sign; in general such a sign ambiguity can be lifted by computing the product
of the eigenvalues with halved degeneracy Fagotti2010disjoint .

Finally, similarly to what happens in the ground state of free fermions
Casini2009Entanglement , we anticipate that also in our nonequilibrium setting
the fermionic tripartite information asymptotically vanishes. Consequently,
the asymptotic behaviour of the spin tripartite information after the quench
is captured by the following formula

$I_{3}^{(\alpha)}(A,B,C)\sim\frac{1}{\alpha-1}\log\left[\frac{1}{2}\sum_{j_{1},j_{2},\ldots,j_{\alpha}=1}^{2}\frac{\\{\Gamma_{j_{1}},\Gamma_{j_{2}},\ldots,\Gamma_{j_{\alpha}}\\}}{\\{\Gamma_{1},\Gamma_{1},\ldots,\Gamma_{1}\\}}\right]-\log
2,$ (3.18)

which we will conveniently use in the analytical calculations but refrain from
assuming in the numerical checks.

##  4 Results

###  4.1 Tripartite information and Rényi entropies of disjoint blocks

Universality.

We find that, when nonzero, the tripartite information
$I_{3}^{(\alpha)}(A,B,C)$ of adjacent subsystems at late times after global
quenches and in the limit of large subsystems ($|A|,|B|,|C|\to\infty$) depends
on the quench through few system details and on the subsystems through the
cross ratio

$x=\frac{|A||C|}{(|A|+|B|)(|C|+|B|)}\,.$ (4.1)

As far as we can see, there is no underlying conformal symmetry that could
explain the dependency on just $x$. We prove it analytically for $\alpha=2$
and for some special contributions to $I_{3}^{(\alpha)}(A,B,C)$ with
$\alpha>2$. More generally, we show it to be consistent with our analytical
and numerical analysis.

The most universal behaviour that we observe is in the limit $x\rightarrow
1^{-}$ (i.e., $|A|,|C|\gg|B|\gg a$, where $a$ is the lattice spacing) in which
the tripartite information (see Fig 4 for a sketch) is either equal to $0$ (in
the trivial cases) or equal to $-\log 2$, independently of $\alpha$. Since the
result is independent of the Rényi index, it also applies to the (von Neumann)
tripartite information $I_{3}(A,B,C)$ ($\alpha\to 1^{+}$). Following the
terminology of Ref. Maric2022Universality , in all the interesting cases we
have then found a universal “residual tripartite information” equal to $-\log
2$. This should be contrasted with the zero value found in the ground states
of conformal systems and in thermal states.

Figure 4: Residual tripartite information: $I_{3}(A,B,C)$ with
$a\ll|B|\ll|A|,|C|$, where $a$ is the lattice spacing.

Asymptotic predictions.

Besides asymptotically depending on the cross ratio, the tripartite
information depends only on the discontinuities of the filling function
characterising the stationary state after the quench. Denoting the filling
function by $\vartheta(k)$, as in (3.7), we parametrize the discontinuties
about momentum $k=k_{F}$ as follows

$\gamma_{\pm}=\lim_{k\to k_{F}^{\pm}}[2\vartheta(k)-1]\,.$ (4.2)

Incidentally, we will also call $k_{F}$ Fermi momentum, for the role it plays
in the ground state of critical systems. We obtain the analytic predictions

$I_{3}^{(2)}(A,B,C)=-\log
2+\log\left[1+(1-x)^{\sum_{k_{F}}\frac{[\arctan(\gamma_{+})-\arctan(\gamma_{-})]^{2}}{\pi^{2}}}\right]$
(4.3)

for $\alpha=2$ and

$I_{3}^{(\alpha)}(A,B,C)=-\log
2+\frac{1}{\alpha-1}\log\left[\frac{1}{2}\sum_{\Sigma}(1-x)^{\sum_{k_{F}}\mathrm{tr}[(R_{0}\Sigma)^{2}-R_{0}^{2}]}Q(x;\mathrm{R}_{0},\Sigma)\right]$
(4.4)

for higher integer $\alpha$. Here the sum is over all $\alpha\times\alpha$
diagonal matrices $\Sigma$ with the elements on the diagonal equal either to
$1$ or $-1$ (in total $2^{\alpha}$ terms), $\mathrm{R}_{0}$ is the
$\alpha\times\alpha$ circulant matrix defined in (7.4) and
$Q(x;\mathrm{R}_{0},\Sigma)$ is a “non-abelian correction”, always very close
to $1$. Specifically, each ratio
$\frac{\\{\Gamma_{j_{1}},\Gamma_{j_{2}},\ldots,\Gamma_{j_{\alpha}}\\}}{\\{\Gamma_{1},\Gamma_{1},\ldots,\Gamma_{1}\\}}$
appearing in (3.18) is described by the term in (4.4) with $\Sigma_{\ell
n}=-\delta_{\ell n}e^{i\pi j_{\ell}}$. Formula (4.3) was first conjectured in
Ref. Maric2022Universality  and we prove here its validity. Formula (4.4)
without $Q(x;\mathrm{R}_{0},\Sigma)$ can be obtained as the lowest order of a
rapidly convergent perturbation theory based on an implicit solution to a
Riemann-Hilbert problem (Section 7.3). We call it "Abelian approximation"
since it neglects all the effects of non-commutativity of $\mathrm{R}_{0}$ and
$\Sigma\mathrm{R}_{0}\Sigma$. Although a priori crude, this approximation
gives surprisingly accurate results. We lack an explicit expression for the
correction to the formula, parametrised here by $Q(x;\mathrm{R}_{0},\Sigma)$,
but we have computed it at the leading order in the perturbation theory. The
result reads

$Q(x;\mathrm{R}_{0},\Sigma)\approx
k_{3,1}(x)^{-\sum_{k_{F}}\frac{1}{4}\mathrm{tr}[(i[\Sigma\mathrm{R}_{0}\Sigma,\mathrm{R}_{0}])^{2}]}\,,$
(4.5)

where $k_{3,1}(x)$ is worked out in Section 7 and is shown in Fig. 5. More
generally, each term of the expansion of $\log Q(x;\mathrm{R}_{0},\Sigma)$ is
factorised in a term that depends only on $\alpha$ and $\gamma_{\pm}$ (e.g.,
the exponent of (4.5)) and a term that depends only on the cross ratio $x$
(e.g., $\log k_{3,1}(x)$). In practice, going beyond the third order (4.5)
seems to be a mathematical problem rather than a physical one, as it is almost
impossible to see a disagreement from the third-order approximation.

Predictions apart, it should be noticed that our results imply a non-zero
value of the tripartite information after quenches, similarly to the ground
states of conformal systems and unlike any thermal (Gibbs) state. A property
that makes the behavior of the tripartite information after quenches under
consideration different from the former is the absence of the crossing
symmetry $x\to 1-x$, which is manifest in the possibility to have a nonzero
"residual tripartite information" in a state with clustering properties.

Figure 5: The function $k_{3,1}(x)$ characterizing the dependency of the
leading non-Abelian correction (4.5) on the subsystem (through the cross ratio
$x$).

Determinant representation.

As a side product of this work, we produce simple formulas that can be used as
an alternative to those worked out in Ref. Fagotti2010disjoint  (discussed in
section 3.2) for computing the terms with
$j_{1},j_{2},\ldots,j_{\alpha}\in\\{1,2\\}$ in (3.11), i.e., all terms
appearing in (3.18). Specifically, we argue (see Appendix B.1) that any such
term can be evaluated with the formula

$\\{\Gamma_{j_{1}},\ldots,\Gamma_{j_{\alpha}}\\}=\sqrt{\det\left|2^{\frac{1-\alpha}{\alpha}}\mathrm{I}_{\alpha}\otimes\mathrm{I}_{2(\ell_{A}+\ell_{C})}-i2^{\frac{1-\alpha}{\alpha}}\left[\frac{\mathrm{M}+\Sigma\mathrm{M}\Sigma}{2}\otimes\Gamma_{1}-\frac{\mathrm{M}-\Sigma\mathrm{M}\Sigma}{2}\otimes(P\Gamma_{1})\right]\right|},$
(4.6)

where $\mathrm{M}$ is the $\alpha\times\alpha$ Hermitian circulant matrix
defined in (5.42). This representation is more useful than the one proposed in
Ref. Fagotti2010disjoint : on the one hand, the matrix in the determinant is
linear in $\Gamma_{1}$ and, on the other hand, it allows one to evaluate the
Rényi entropy even for arbitrarily large values of $\alpha$ without the need
of working out the recursive procedure of Ref. Fagotti2010disjoint . We expect
similar representations for any term appearing in formula (3.11) for the Rényi
entropy of disjoint blocks, but we have not developed further in that
direction.

###  4.2 Examples

(a) Thermal state $\bm{\mathbf{\rho}}\propto\exp(-\beta\bm{\mathbf{H}})$ with
$\beta=1$ in the XX model with $(J_{x},J_{y},h)=(1,1,0.7)$.

(b) (Critical) Ground state of the XX model with $(J_{x},J_{y},h)=(1,1,0.7)$.

(c) Quench from the ground state of a gapped Ising Hamiltonian with
$(J_{x},J_{y},h)=(1,0,5)$.

(d) Quench from the ground state of the critical Ising model
$(J_{x},J_{y},h)=(1,0,{1})$.

(e) Quench from the ground state of a critical XX model with
$(J_{x},J_{y},h)=(1,1,0.3)$.

(f) Bipartitioning protocol from $\bm{\mathbf{\rho}}\propto
e^{-\beta_{-}\sum_{\ell<0}{\bm{\mathbf{\sigma}}_{\ell}^{z}}}\otimes
e^{\beta_{+}\sum_{\ell\geq 0}{\bm{\mathbf{\sigma}}_{\ell}^{z}}}$ with
$\beta_{-}=2$ and $\beta_{+}=1$.

Figure 6: Rényi-$2$ tripartite information vs the cross ratio
$x=|A||C|/((|A|+|B|)(|B|+|C|))$ for $|A|=|C|\in\\{10,20,50,100\\}$ and
variable $|B|$ in different situations. Time evolution in all nonequilibrium
settings is under the XY Hamiltonian (3.3) with $(J_{x},J_{y},h)=(1,0.5,0.7)$.
The solid curves are exact analytical predictions for the limit of large
subsystems $A,B,C$. In particular, (a) shows the standard vanishing of the
tripartite information in equilibrium at nonzero temperature and (b) is the
CFT prediction Calabrese2009Entanglement1 ; Fagotti2012New ;
Furukawa2009Mutual  $I_{3}(A,B,C)=-\log 2+\log(1+\sqrt{1-x}+\sqrt{x})$. The
remaining predictions have been conjectured in Ref. Maric2022Universality  and
are proved in this paper.

We report explicit examples in which we have numerically evaluated the
tripartite information. The numerical data are obtained using (3.8) and (3.11)
with the correlation matrices computed using the formulas collected in Table
1.

####  4.2.1 Equilibrium

According to our previous discussion, the tripartite information is expected
to vanish in equilibrium at nonzero temperature as the lengths approach
infinity. In the noninteracting systems we are considering, this can be argued
by the fact that the symbol of the correlation matrix is smooth. We show it
explicitly in Fig. 6, where we consider a thermal state of the XX model. In
contrast, Fig. 6 displays the Rényi-$2$ tripartite information at zero
temperature, which is nonzero only because we chose a model with a conformal
critical ground state.

####  4.2.2 Global quenches with translational invariance

Figure 7: Rényi-$\alpha$ tripartite information at infinite time after the
quench from the ground state of the XX model with $(J_{x},J_{y},h)=(1,1,0.3)$
under the XY Hamiltonian (3.3) with $(J_{x},J_{y},h)=(1,0.5,0.7)$ for
$|A|=|C|\in\\{25,50,100,200\\}$ and variable $|B|$. The solid curves are the
Abelian approximation, i.e., (4.4) with $Q(x;\mathrm{R}_{0},\Sigma)=1$. See
Fig. 8 for a comparison with more accurate approximations.  Figure 8:
Rényi-$\alpha$ tripartite information at infinite time after the same quench
as in Figure 7 vs |B|. The value of the cross ratio $x$ is fixed, while the
subsystem sizes are varied as follows:
$(|A|,|B|,|C|)=n(10,1,10),n(10,1,20),n(20,1,20)$ with integer $n$. The blue
dashed lines are the Abelian approximation, while the black solid lines
include the leading non-Abelian correction.

We consider here three global quenches from the ground state of local
Hamiltonians. In the first case, shown in Fig. 6, the pre-quench Hamiltonian
is gapped, so the correlations lengths are finite. Since the post-quench
Hamiltonian is local, this property is preserved even at infinite time after
the quench. We do not expect a nonzero tripartite information in this case and
indeed the numerical data show quite clearly that $I_{3}^{(2)}(A,B,C)$
approaches zero. In the second case, shown in Fig. 6, the pre-quench
Hamiltonian is gapless and there are correlations that decay algebraically
even at infinite time after the quench. Nevertheless, the tripartite
information approaches zero as the lengths are increased. As pointed out in
Ref. Maric2022Universality , this happens because correlations do not decay to
zero slowly enough.

In the third example we replaced the initial state with the ground state of a
critical Hamiltonian with central charge equal to $1$. According to our work,
the Rényi-$\alpha$ tripartite information should be nonzero. In Fig. 6 we show
the Rényi-$2$ tripartite information, while in Figs. 7 and 8 we discuss higher
values of $\alpha$. Predictions (4.3) and (4.4) are in excellent agreement
with the numerical data. One can clearly see that the finite size effects
related to the lengths of the subsystems disappear as their size is increased.

####  4.2.3 Bipartitioning protocols

Figure 9: Rényi-$\alpha$ tripartite information at infinite time after the
quench from $\bm{\mathbf{\rho}}\propto
e^{-\beta_{-}\sum_{\ell<0}{\bm{\mathbf{\sigma}}_{\ell}^{z}}}\otimes
e^{\beta_{+}\sum_{\ell\geq 0}{\bm{\mathbf{\sigma}}_{\ell}^{z}}}$, with
$\beta_{-}=2$ and $\beta_{+}=1$, under the XY Hamiltonian (3.3) with
$(J_{x},J_{y},h)=(1,0.5,0.7)$ for $|A|=|C|\in\\{25,50,100,200\\}$ and variable
$|B|$. The solid curves are the Abelian approximation, i.e., (4.4) with
$Q(x;\mathrm{R}_{0},\Sigma)=1$. See Fig. 10 for a comparison with more
accurate approximations. Figure 10:  The same as in Figure 8 for the quench of
Figure 9.

As an example of bipartitioning protocol we consider time evolution under the
XY Hamiltonian (3.3) of the junction of two thermal states,
$\bm{\mathbf{\rho}}_{-}\propto\exp{(-\beta_{-}\bm{\mathbf{H}}_{0,-}})$ and
$\bm{\mathbf{\rho}}_{+}\propto\exp{(-\beta_{+}\bm{\mathbf{H}}_{0,+}})$
described by the simplified Hamiltonian
$\bm{\mathbf{H}}_{0,-}=\sum_{\ell<0}\bm{\mathbf{\sigma}}_{\ell}^{z}$ and
$\bm{\mathbf{H}}_{0,+}=-\sum_{\ell\geq 0}\bm{\mathbf{\sigma}}_{\ell}^{z}$. In
similar settings correlations are expected to decay algebraically in the NESS
as a manifestation of the abrupt change of properties in the initial state
(independently of how correlations decay there). Figures 6, 9 and 10 show an
excellent agreement the between predictions (4.3) and (4.4) and the numerical
data.

##  5 Field theory description

###  5.1 Fermionic correlations at late times

The low-energy properties of a critical spin-chain Hamiltonian can be
described by an effective quantum field theory. That can be obtained by
reintroducing the lattice spacing $a$ in such a way that only the low-energy
part of the spectrum survives the continuum limit $a\rightarrow 0$. In
particular, in noninteracting spin chains it is sufficient to expand around
the momenta with the lowest excitation energies; the fermionic operators are
then written in terms of fast oscillating factors and slow fields, which have
a well defined continuum limit (see, e.g., Affleck1988LesHouches ). The fast
oscillating factors cancel out in the effective description of the
Hamiltonian, which can indeed be expressed in terms of slow fields only.

After a global quench, an extensive number of particles is excited, and the
excitations with the lowest energy are not sufficient to describe the state.
In the next section we sketch a trick to describe anyway the problem within a
low-energy field theory. Here however we adopt another approach, which allows
us to identify the relevant properties of the chain system more quickly in
view of our immediate access to the late-time correlation matrix.
Specifically, we group spin sites into blocks and define lattice fermions
whose correlations are slow and for which a continuum limit exists. In this
respect, the procedure we present has some similarities to the renormalization
group Kadanoff1966 ; Wilson1974 .

For the sake of simplicity, we start assuming that the limit of infinite time
can be described by a stationary state of the XX model, whose correlation
matrix has the symbol

$\Gamma(k)=[\vartheta(k)-\vartheta(-k)]\mathrm{I}+[\vartheta(k)+\vartheta(-k)-1]\sigma^{y}\,.$
(5.1)

The generalisation to other correlation matrices is easy and will be discussed
at the end of the section. First, we note that $\vartheta(k)$ can be
identified with the filling function. Second, from (5.1) it follows that the
Jordan-Wigner fermions
$\bm{\mathbf{c}}_{\ell}=(\bm{\mathbf{a}}_{2\ell-1}+i\bm{\mathbf{a}}_{2\ell})/2$
exhibit the correlations

$\braket{\bm{\mathbf{c}}_{j}\bm{\mathbf{c}}_{\ell}^{\dagger}}=\int_{-\pi}^{\pi}\frac{dk}{2\pi}\vartheta(k)e^{ik(\ell-j)},\quad\braket{\bm{\mathbf{c}}_{j}\bm{\mathbf{c}}_{\ell}}=0.$
(5.2)

If the filling function is smooth, correlations decay exponentially. A more
interesting case arises when the filling function $\vartheta(k)$ is
discontinuous. Let us assume that such discontinuities $k_{r}$,
$r=1,2,\ldots,\Upsilon$ are rational multiples of $\pi$; hence, there is a
positive integer $N$ and integers $m_{r}$, for $r=1,2,\ldots,\Upsilon$, such
that $k_{r}=2\pi m_{r}/N$. Note that, even if the condition of rationality is
not satisfied, any Fermi momentum $k_{r}$ could be approximated to arbitrary
precision by taking sufficiently large $N$. From now on, in analogy with the
ground-state physics, we will call Fermi momenta the momenta at the
discontinuities. The next step is to divide the chain in blocks of length $N$
and define $N$ species of fermions $\bm{\mathbf{d}}_{\ell}^{\nu}$ as the
discrete Fourier transform of the original $\bm{\mathbf{c}}_{\ell}$ in the
blocks

$\bm{\mathbf{d}}_{\ell}^{\nu}=\frac{1}{\sqrt{N}}\sum_{m=0}^{N-1}\bm{\mathbf{c}}_{N\ell+m}e^{i\frac{2\pi\nu
m}{N}}\,,\quad\nu=0,1,\ldots,N-1.$ (5.3)

We then define $N$ fields $\bm{\mathbf{\psi}}_{\nu}(x)$ as the continuum limit
of $\bm{\mathbf{d}}_{\ell}^{\nu}$ at position $x=Na\ell$, by letting the
lattice spacing $a$ to zero and promoting $x$ to a continuum variable,

$\bm{\mathbf{\psi}}_{\nu}(x)\sim\frac{1}{\sqrt{Ma}}\bm{\mathbf{d}}^{\nu}_{\frac{x}{Ma}},\quad
a\to 0.$ (5.4)

The reader will forgive us for the abuse of notations, in that from now on $x$
will indicate a position rather than the cross ratio as in the introduction.
Note that $\bm{\mathbf{\psi}}_{\nu}(x)$ satisfy the standard fermionic
anticommutation relations

$\\{\bm{\mathbf{\psi}}_{\mu}(x),\bm{\mathbf{\psi}}_{\nu}^{\dagger}(y)\\}=\delta_{\mu,\nu}\delta(x-y),\qquad\\{\bm{\mathbf{\psi}}_{\mu}(x),\bm{\mathbf{\psi}}_{\nu}(y)\\}=0.$
(5.5)

The 2-point correlation functions of the fields can be readily obtained by
taking the continuum limit of the fermionic correlations. Specifically we have

$\braket{\bm{\mathbf{\psi}}_{\mu}(x)\bm{\mathbf{\psi}}_{\nu}^{\dagger}(y)}\sim\frac{1}{Na}\braket{\bm{\mathbf{d}}^{\mu}_{\frac{x}{Na}}\bm{\mathbf{d}}^{\nu{\dagger}}_{\frac{y}{Na}}}=\frac{1}{N^{2}a}\sum_{m,n=0}^{N-1}e^{i\frac{2\pi\mu
m}{N}}e^{-i\frac{2\pi\nu
n}{N}}\sum_{r=1}^{\Upsilon}\int_{k_{r}}^{k_{r+1}}\frac{dk}{2\pi}\vartheta(k)e^{ik(\frac{y-x}{a}+n-m)}\\\
=\delta_{\mu,\nu}\frac{\vartheta(\frac{2\pi\mu}{N}^{+})-\vartheta(\frac{2\pi\mu}{N}^{-})}{2}\frac{\chi_{K}(\frac{2\pi\mu}{N})}{i\pi(x-y)}+O(a)$
(5.6)

where $\chi_{K}$ is the indicator function of the set
$K\equiv\\{k_{1},k_{2},\ldots k_{R}\\}$ of Fermi points, i.e., the correlation
is nonzero only if $2\pi\mu/N$ is equal (up to $2\pi$) to some of the Fermi
momenta $k_{r}$. The short-distance behaviour of the correlation
(corresponding to distances approaching $0$ as $a\rightarrow 0$) can be
obtained with a symmetric integration over one position about the other (so
that the contribution above vanishes)

$\mathrm{p.v.}\int\mathrm{d}y\braket{\bm{\mathbf{\psi}}_{\mu}(x)\bm{\mathbf{\psi}}_{\nu}^{\dagger}(y)}\sim\lim_{\epsilon\rightarrow
0^{+}}\sum_{d=-\infty}^{\infty}\braket{\bm{\mathbf{d}}^{\mu}_{\frac{x}{Na}}\bm{\mathbf{d}}^{\nu{\dagger}}_{\frac{x}{Na}+d}}e^{-\epsilon|d|}=\\\
\frac{1}{N^{2}}\sum_{m,n=0}^{N-1}e^{i\frac{2\pi\mu m}{N}}e^{-i\frac{2\pi\nu
n}{N}}\sum_{r=1}^{\Upsilon}\int_{k_{r}}^{k_{r+1}}dk\vartheta(k)\sum_{q=-\infty}^{\infty}\delta(k-\tfrac{2\pi
q}{N})e^{\frac{2\pi
iq(n-m)}{N}}=\delta_{\mu,\nu}\tfrac{\vartheta(\frac{2\pi\mu}{N}^{+})+\vartheta(\frac{2\pi\mu}{N}^{-})}{2}$
(5.7)

where $\vartheta(\frac{2\pi\mu}{N}^{+})\neq\vartheta(\frac{2\pi\mu}{N}^{-})$
only if $\frac{2\pi\mu}{N}\in K$. Putting all together we get

$\braket{\bm{\mathbf{\psi}}_{\mu}(x)\bm{\mathbf{\psi}}_{\nu}^{\dagger}(y)}=\delta_{\mu,\nu}\Bigl{[}\frac{\vartheta(\frac{2\pi\mu}{N}^{+})+\vartheta(\frac{2\pi\mu}{N}^{-})}{2}\delta(x-y)+\frac{\vartheta(\frac{2\pi\mu}{N}^{+})-\vartheta(\frac{2\pi\mu}{N}^{-})}{2}\frac{\chi_{K}(\frac{2\pi\mu}{N})}{i\pi(x-y)}\Bigr{]}\,.$
(5.8)

From this expression we see that each discontinuity is associated with a field
with non-trivial correlations but still decoupled from the rest; all the
remaining fields with $\frac{2\pi\mu}{N}\notin K$ have trivial correlations,
proportional to the identity. Such a factorisation allows us to focus on the
contribution from a particular Fermi point with $k_{F}\in K$, taking care only
eventually of the presence of more fields. For the sake of compactness, we
introduce the parametrization $\gamma_{\pm}=\lim_{k\to
k_{F}^{\pm}}[2\vartheta(k)-1]$, in terms of which the 2-point function of the
field $\bm{\mathbf{\psi}}$ corresponding to a particular discontinuity $k_{F}$
(omitting the index of the field) reads

$\braket{\bm{\mathbf{\psi}}(x)\bm{\mathbf{\psi}}^{\dagger}(y)}=\frac{2+\gamma_{+}+\gamma_{-}}{4}\delta(x-y)+\frac{\gamma_{+}-\gamma_{-}}{2}\frac{1}{2\pi
i(x-y)}\,.$ (5.9)

The final step is to explain why these expressions hold true even when the
symbol of the correlation matrix does not take the simple form (5.1). In
general $\Gamma(k)$ can not be diagonalised by a momentum-independent change
of basis; it takes instead the form

$\Gamma(k)=e^{iW(k)}\Bigl{\\{}[\vartheta(k)-\vartheta(-k)]\mathrm{I}+[\vartheta(k)+\vartheta(-k)-1]\sigma^{y}\Bigr{\\}}e^{-iW(k)}\,,$
(5.10)

where $W(k)=W^{\dagger}(k)$ and $W^{t}(-k)=-W(k)$. This is a general way to
parametrise the famous Bogoliubov transformation required to diagonalise
systems such as the transverse-field Ising model. In fact, $W(k)$ is nothing
but the symbol of the quadratic operator that generates the transformation
mapping the post-quench Hamiltonian $\bm{\mathbf{H}}$ to an operator commuting
with the Hamiltonian of the XX model. Such a transformation exists because all
quadratic systems have an eigenbasis of Slater determinants. In addition, if
$\bm{\mathbf{H}}$ has a conservation law with a non-critical ground state, the
generator $\bm{\mathbf{W}}$ can be chosen in such a way that $e^{iW(k)}$ is
smooth. Since such a quasilocal transformation does not affect the notion of
space in the continuum limit, it can be simply incorporated in the definition
of the fields; specifically, we can redefine the $\bm{\mathbf{c}}$-fermions as
follows

$\bm{\mathbf{c}}_{\ell}=\sum_{n=-\infty}^{\infty}\frac{1}{2}\begin{pmatrix}1&i\end{pmatrix}\int_{-\pi}^{\pi}\frac{\mathrm{d}k}{2\pi}e^{ik(\ell-n)}e^{iW(k)}\begin{pmatrix}\bm{\mathbf{a}}_{2n-1}\\\
\bm{\mathbf{a}}_{2n}\end{pmatrix}$ (5.11)

and leave the relation between the $\bm{\mathbf{d}}$-fermions and the
$\bm{\mathbf{c}}$ ones unchanged. Mapping (5.11) manifests the quasilocality
of the transformation: while $\bm{\mathbf{c}}_{\ell}$ is a superposition of
fermions at different sites, we are still labelling it using a single index
$\ell$, generating in turn a spatial indetermination

$\Delta
x\sim\frac{a}{-\lim_{r\rightarrow\infty}\frac{1}{r}\log\left|\int_{-\pi}^{\pi}\frac{\mathrm{d}k}{2\pi}e^{ikr}e^{iW(k)}\right|}$
(5.12)

which however approaches zero in the continuum limit. As a final remark we
point out that even with a non-smooth $W(k)$ one would generally obtain an
expression similar to (5.8), though with a different parametrisation. Since
$W(k)$ is smooth in all the quenches that we consider, we do not develop
further for that special situation; we just mention that a non-smooth $W(k)$
can be obtained by quenching to the critical Ising (or XY) model.

###  5.2 Tripartite information

Plugging the asymptotic behaviour of the late-time correlation functions into
the formulas reviewed in Section 3.2 would be sufficient to obtain the
asymptotic behaviour of the entanglement entropies of the subsystems entering
in the tripartite information. The nonlinear way in which the correlation
matrix appears in the determinants describing the various terms of the
expansion of the entropy of disjoint blocks is however not ideal. By deriving
again the formulas within a quantum field theory, we have found a more
convenient representation, cf. (4.6); we sketch in this section the method
employed.

####  5.2.1 Entanglement entropies in a QFT at equilibrium: a brief overview

In a quantum field theory at equilibrium there is a standard formalism to
express the Rényi entropies in terms of partition functions Cardy2008Form ;
Calabrese2009Entanglement ; Casini2009Entanglement , which starts with
interpreting the density matrix of a thermal state at inverse temperature
$\beta$ as a path integral on the imaginary time interval $(0,\beta)$. Here we
briefly review the formalism, focusing on fermionic fields (for bosons see
e.g. Calabrese2009Entanglement ; Casini2009Entanglement ) in the ground state
$\ket{0}$ ($\beta=\infty$), which is the relevant case for this work.

For the sake of simplicity we assume that the system is described by a single
fermionic field $\bm{\mathbf{\psi}}(x)$, satisfying the standard
anticommutation relations
$\\{\bm{\mathbf{\psi}}(x),\bm{\mathbf{\psi}}^{\dagger}(y)\\}=\delta(x-y)$. We
employ the fermionic coherent state path integral formalism based on Grassmann
variables Kleinert2009 . In this formalism the brakets $\braket{\psi}{0}$ and
$\braket{0}{\psi^{\prime}}$ can be represented as path integrals in the lower
and upper half plane respectively. If $L=\cup_{i}^{n}(u_{i},v_{i})$ is a
subsystem (which we have assumed to be a union of several intervals
$(u_{i},v_{i})$) the reduced density matrix of
$\bm{\mathbf{\rho}}=\ket{0}\bra{0}$ is obtained by tracing out the degrees of
freedom in the complement of $L$, that we denote by $\overline{L}$,

$\rho_{L}(\psi_{L},\psi^{\prime}_{L})=\int\mathcal{D}\psi^{\dagger}_{\overline{L}}\mathcal{D}\psi_{\overline{L}}\bra{-\psi_{\overline{L}},\psi_{L}}\bm{\mathbf{\rho}}\ket{\psi_{\overline{L}},\psi_{L}^{\prime}},$
(5.13)

where a minus sign appears, that is a well-known peculiarity of the
representation of trace with Grassmann variables. After an integral
transformation $\psi\to-\psi$ in the lower half plane, the reduced density
matrix reads

$\rho_{L}(\psi_{L};\psi_{L}^{\prime})=\frac{1}{Z_{1}(L)}\int_{\psi(x,0^{-})=\psi_{L}(x),\;x\in
L}^{\psi(x,0^{+})=-\psi_{L}^{\prime}(x),\;x\in
L}\mathcal{D}\psi^{\dagger}\mathcal{D}\psi\
e^{-S_{E}[\psi^{\dagger},\psi]}\,,$ (5.14)

where $S_{E}[\psi^{\dagger},\psi]$ is the Euclidean action and $\psi(x)$,
$\psi^{\dagger}(x)$ are Grassmann fields. It corresponds to gluing the upper
and lower half plane on $\overline{L}$, leaving a cut on $L$. The partition
function $Z_{1}(L)$, which ensures the normalization
$\mathrm{tr}[\rho_{L}]=1$, reads

$Z_{1}(L)=\int\mathcal{D}\psi_{L}^{\dagger}\mathcal{D}\psi_{L}\
\rho_{L}(\psi_{L};-\psi_{L}).$ (5.15)

The Rényi entropies with integral index can be obtained by cyclically gluing
together a given number of copies of the cylinder, so that one obtains

$\mathrm{tr}[\rho_{L}^{\alpha}]=\frac{\mathrm{Z}_{\alpha}(L)}{Z_{1}^{\alpha}(L)}$
(5.16)

with $\mathrm{Z}_{\alpha}(L)$ the partition function on an $\alpha$-sheeted
Riemann surface with branch points at $u_{j}$ and $v_{j}$. A convenient
alternative representation introduces $\alpha$ copies of the field,
$\psi^{(i)\dagger}(x)$, $\psi^{(i)}(x)$, for $i=1,2,\ldots,\alpha$, described
by decoupled equivalent actions (so that the total action is the sum of the
actions of the copies), moving the original complication of having to deal
with a nontrivial Riemann surface to the boundary conditions satisfied by the
copies. Specifically, in the simple case that we are describing, subsequent
copies are identified at the branch cuts, that is to say,

$\displaystyle\psi^{(i)}(x,0^{-})=\psi^{(i+1)}(x,0^{+}),\quad
i=1,2,\ldots,\alpha-1$ (5.17)
$\displaystyle\psi^{(\alpha)}(x,0^{-})=(-1)^{\alpha+1}\psi^{(1)}(x,0^{+})$

for $x\in L$, and $\psi^{(i)}(x,0^{-})=\psi^{(i)}(x,0^{+})$ for $x\notin L$,
for any $i$. Here, we have changed the integration variables (redefining them
with appropriate minus signs) so as to glue the copies with no additional
signs except possibly for the last one. The factor $(-1)^{\alpha+1}$ comes
from collecting one minus sign for each product in $\rho_{L}^{\alpha}$ and one
from the final trace. For future convenience, we define the (anti)circulant
matrix

$T_{\ell,n}=\delta_{\ell+1,n}+(-1)^{\alpha+1}\delta_{\ell,\alpha}\delta_{n,1}\,,\quad\ell,n=1,2,\ldots,\alpha\,,$
(5.18)

in terms of which the boundary conditions read

$\psi^{(i)}(x,0^{-})=\sum_{j=1}^{\alpha}T_{ij}\psi^{(j)}(x,0^{+}).$ (5.19)

The advantage of this formulation stands in the fact that
$\mathrm{tr}[\rho_{L}^{\alpha}]$ can be interpreted as a $2n$-points
correlation function of special fields, called “twist fields”, where $n$ is
the number of disjoint intervals. Importantly, boundary conditions (5.17)
ensure the locality of the twist fields Cardy2008Form . In the presence of
conformal symmetry such a locality condition allows one to use the power of
conformal field theory to predict the asymptotic behaviour of
$\mathrm{tr}[\rho_{L}^{\alpha}]$.

As presented so far, this approach is effective to compute the Rényi entropies
of fermionic subsystems. In order to compute the entropies of spin blocks in
spin chains, however, this picture should be slightly generalised to take into
account the nonlocality of the Jordan-Wigner transformation. In particular,
also the trace of products of different density matrices appears in the Rényi
entropies of disjoint blocks — cf. (3.12). A case that is relevant to our
specific problem is when the reduced density matrices to multiply are
equivalent to one another through a local unitary that, in each block of the
subsystem, acts as the transformation associated with some global symmetry.
One can then attach the corresponding unitary transformation to the boundary
conditions and still describe the system as consisting of $\alpha$ copies of
the original one. For example, instead of (5.19) one could have

$\psi^{(i)}(x,0^{-})=U_{i}(x)\sum_{j=1}^{\alpha}T_{ij}\psi^{(j)}(x,0^{+}),\quad
x\in L,$ (5.20)

where the factors $U_{i}(x)$ depend on the sequence of reduced density
matrices in the product.

The reader is referred to Ref. Coser2016Spin  for the application of this
approach to the calculation of the Rényi-$\alpha$ tripartite information in
the conformal critical ground state of a non-interacting spin chain. For the
analogous calculation after a global quench we face the additional problem
that the formulation in terms of twist fields is not local (the energy density
within $L$ does not commute with the twist fields), so we are not going to use
the interpretation of the Rényi entropies as correlation functions.

####  5.2.2 Popping down to a field theory on a ladder

In view of the results reviewed in the previous section, we start by
reinterpreting the late-time correlations calculated in Section 5.1 as the
correlations in the ground state of a field theory. This could sound
impossible at first sight: the stationary state has extensive entropies, so it
should not be possible to reinterpret it as a ground state of a local theory.
This obstacle can be however overcome by immersing the system in a bigger one
with unphysical degrees of freedom. Specifically, we propose to envelop the
chain in a 3-legs ladder connected with one another through local
interactions. For this task we introduce three decoupled free fields,
$\bm{\mathbf{\psi}}_{j}(x)$, with $j=1,2,3$, characterised by the correlation
functions

$\displaystyle\braket{\bm{\mathbf{\psi}}_{j}(x)\bm{\mathbf{\psi}}_{j}^{\dagger}(y)}=\frac{1}{2}\delta(x-y)-\frac{(-1)^{j}}{2\pi
i(x-y)},\quad j=1,2$ (5.21)
$\displaystyle\braket{\bm{\mathbf{\psi}}_{3}(x)\bm{\mathbf{\psi}}^{\dagger}_{3}(y)}=\frac{1+\sigma}{2}\delta(x-y),$
(5.22)

where $\sigma=\mathrm{sgn}[\gamma_{+}+\gamma_{-}]$. These correlations are
found in the ground state of the Hamiltonian

$\bm{\mathbf{H}}=\int
dx\left[-i\bm{\mathbf{\psi}}_{1}^{\dagger}(x)\partial_{x}\bm{\mathbf{\psi}}_{1}(x)+i\bm{\mathbf{\psi}}_{2}^{\dagger}(x)\partial_{x}\bm{\mathbf{\psi}}_{2}(x)+\sigma\bm{\mathbf{\psi}}_{3}^{\dagger}(x)\bm{\mathbf{\psi}}_{3}(x)\right].$
(5.23)

Let us then introduce a $3\times 3$ unitary matrix $V$ with the following
elements of the first column

$(V_{11},V_{21},V_{31})=\begin{cases}\left(\sqrt{\frac{1+\gamma_{+}}{2}},\sqrt{\frac{1+\gamma_{-}}{2}},\sqrt{-\frac{\gamma_{+}+\gamma_{-}}{2}}\right),&\sigma<0\\\
\left(\sqrt{\frac{1-\gamma_{-}}{2}},\sqrt{\frac{1-\gamma_{+}}{2}},\sqrt{\frac{\gamma_{+}+\gamma_{-}}{2}}\right),&\sigma\geq
0\end{cases}$ (5.24)

and define the fields

$\begin{pmatrix}\bm{\mathbf{\psi}}(x)\\\ \bm{\mathbf{\psi}}^{\prime}(x)\\\
\bm{\mathbf{\psi}}^{\prime\prime}(x)\end{pmatrix}=V^{\dagger}\begin{pmatrix}\bm{\mathbf{\psi}}_{1}(x)\\\
\bm{\mathbf{\psi}}_{2}(x)\\\ \bm{\mathbf{\psi}}_{3}(x)\end{pmatrix}\,.$ (5.25)

The first field $\bm{\mathbf{\psi}}(x)$, given by

$\bm{\mathbf{\psi}}(x)=V_{11}\bm{\mathbf{\psi}}_{1}(x)+V_{21}\bm{\mathbf{\psi}}_{2}(x)+V_{31}\bm{\mathbf{\psi}}_{3}(x)\,,$
(5.26)

has the desired correlation functions (5.9). The remaining two components,
$\bm{\mathbf{\psi}}^{\prime}(x)$ and $\bm{\mathbf{\psi}}^{\prime\prime}(x)$,
define two unphysical fields. Inverting the relation (5.25) enables us to
interpret (5.23) as a Hamiltonian on a three-leg ladder in which the physical
field lives on the first leg and interacts locally with two unphysical fields
living on the remaining two legs. In this interpretation, we are studying the
entanglement of a subsystem immersed in one, infinite, leg of the ladder with
the remaining part of the ladder. Note that this is a different problem from
studying the entanglement between one whole leg of the ladder with the
remaining legs, i.e. of one field with the other fields, that has been
addressed in Refs. Chen2013 ; Mollabashi2014 ; Furukawa2011 ; Xu2011 .

For future convenience, let us also introduce the following compact notation
for the decoupled fields

$\vec{\bm{\mathbf{\psi}}}\equiv\begin{pmatrix}\bm{\mathbf{\psi}}_{1}\\\
\bm{\mathbf{\psi}}_{2}\\\ \bm{\mathbf{\psi}}_{3}\end{pmatrix}\;.$ (5.27)

Thus we have

$\braket{\bm{\mathbf{\psi}}(x)\bm{\mathbf{\psi}}^{\dagger}(y)}=\vec{u}^{\dagger}\braket{\vec{\bm{\mathbf{\psi}}}(x)\vec{\bm{\mathbf{\psi}}}^{\dagger}(y)}\vec{u},\qquad\vec{u}\equiv
V\begin{pmatrix}1\\\ 0\\\ 0\end{pmatrix}=\begin{pmatrix}V_{11}\\\ V_{21}\\\
V_{31}\end{pmatrix}.$ (5.28)

In order to study the tripartite information at late time after a global
quench we need to compute the entropy of a block of spins and also the one of
two disjoint blocks — cf. (2.5). The latter case requires the evaluation of
(3.12) with $j_{1},\ldots,j_{\alpha}\in\\{1,2\\}$. To that aim, we build on
the procedure used in Casini2009reduced_density ; Casini2009Entanglement ;
Fries2019 ; Blanco2022  for computing the fermionic entropies. As reviewed in
the previous section, the reduced density matrix can be expressed as a path
integral, with boundary conditions corresponding to gluing the space on the
complement of the subsystem of interest. The trace of a power of the reduced
density matrix is computed by introducing the appropriate number of copies of
the fields, related by a suitable boundary conditions on the subsystem. Our
case is peculiar in that the boundary conditions are imposed only on the
physical field, i.e., on one leg of the ladder, and should also account for
the operator $\eqref{eq:operator P definition}$, which appears for every index
$j_{\ell}=2$ in (3.12).

Since the fields associated with different Fermi momenta are decoupled, their
contribution in (3.12) is factorized. In addition, the remaining fields that
are not associated with Fermi momenta give contributions independent of
$j_{\ell}$, which therefore cancel out in the ratios appearing in (3.18).
Without loss of generality we can then focus on a single field
$\vec{\bm{\mathbf{\psi}}}$, corresponding to a particular Fermi point, with
correlation functions (5.9). We introduce $\alpha$ copies of the field
$\vec{\bm{\mathbf{\psi}}}$, which we denote by
$\vec{\bm{\mathbf{\psi}}}^{(1)},\ldots,\vec{\bm{\mathbf{\psi}}}^{(\alpha)}$,
and the corresponding Grassmann fields
$\vec{\psi}^{(1)},\ldots,\vec{\psi}^{(\alpha)}$. According to the
representation reviewed in the previous section, the trace of a product of
reduced density matrices is given by the path integral over those fields

$\\{\Gamma_{j_{1}},\Gamma_{j_{2}},\ldots,\Gamma_{j_{\alpha}}\\}=\mathrm{tr}\left[\rho(\Gamma_{j_{1}})\rho(\Gamma_{j_{2}})\ldots\rho(\Gamma_{j_{\alpha}})\right]\sim\int\mathcal{D}\psi^{\dagger}\mathcal{D}\psi\exp{\left(-S_{E}^{(\alpha)}(k_{F};\Sigma)\right)},$
(5.29)

where the Euclidean action follows from (5.23) and reads

$S_{E}^{(\alpha)}(k_{F};\Sigma)=\int\mathrm{d}\tau\int\mathrm{d}x\
\sum_{n=1}^{\alpha}\Bigl{[}\vec{\psi}^{(n){\dagger}}(x,\tau;\Sigma)\begin{pmatrix}\partial_{\tau}-i\partial_{x}&0&0\\\
0&\partial_{\tau}+i\partial_{x}&0\\\
0&0&\partial_{\tau}+\sigma\end{pmatrix}\vec{\psi}^{(n)}(x,\tau;\Sigma)\,.$
(5.30)

The boundary conditions read

$\displaystyle\psi^{(\ell)}(x,0^{-})=\begin{cases}\sum_{n=1}^{\alpha}T_{\ell
n}\psi^{(n)}(x,0^{+}),&x\in A\cup C\\\ \psi^{(\ell)}(x,0^{+}),&x\notin A\cup
C\end{cases}$ (5.31)

for the copies of the physical field, with $T$ defined in (5.18), and

$\displaystyle\psi^{\prime(\ell)}(x,0^{-})=$
$\displaystyle\psi^{\prime(\ell)}(x,0^{+})$ (5.32)
$\displaystyle\psi^{\prime\prime(\ell)}(x,0^{-})=$
$\displaystyle\psi^{\prime\prime(\ell)}(x,0^{+})\,,$

for the copies of the unphysical ones. We remind the reader that these
boundary conditions and the factor $(-1)^{\alpha+1}$ come from the
representation of the trace in the Grassmann formalism. Note that $T$ is an
(anti-)circulant matrix that can also be represented as follows

$T_{\ell n}=\frac{1}{\alpha}\sum_{z|z^{\alpha}=-1}(-z)^{\ell-n+1}\,.$ (5.33)

We now come to the main complication of working with spins, i.e., $j_{n}$ not
being generally equal to $1$. For each index $j_{1},\ldots,j_{\alpha}$ equal
to $2$ there is an additional minus sign in the boundary conditions of the
copies of the physical field on $x\in A$; this comes from matrix $\mathrm{P}$
— (3.10) — which in the continuum limit reads

$\bm{\mathbf{P}}(x,y)=[-\chi_{A}(x)+\chi_{C}(x)]\delta(x-y)\,.$ (5.34)

We can then write the boundary conditions for the copies of the physical field
as follows

$\psi^{(\ell)}(x,0^{-};\Sigma)=\begin{cases}\sum_{n=1}^{\alpha}(\Sigma
T\Sigma)_{\ell n}\psi^{(n)}(x,0^{+};\Sigma),&x\in A\\\
\sum_{n=1}^{\alpha}T_{\ell n}\psi^{(n)}(x,0^{+};\Sigma),&x\in C\\\
\psi^{(\ell)}(x,0^{+};\Sigma),&x\notin A\cup C\end{cases},$ (5.35)

where

$\Sigma_{\ell n}\equiv\delta_{\ell n}\sigma_{n}=-\delta_{\ell n}e^{i\pi
j_{\ell}}\,.$ (5.36)

Using (5.25) the boundary conditions for the decoupled fields follow
immediately

$\begin{split}&\vec{\psi}^{(\ell)}(x,0^{+};\Sigma)=\\\
&V\sum_{n}\begin{pmatrix}\chi_{A}(x)[\Sigma T\Sigma]_{\ell
n}+\chi_{C}(x)T_{\ell n}+\chi_{\overline{A\cup C}}(x)\delta_{\ell n}&0&0\\\
0&\delta_{\ell n}&0\\\ 0&0&\delta_{\ell
n}\end{pmatrix}V^{\dagger}\vec{\psi}^{(n)}(x,0^{-};\Sigma)\,.\end{split}$
(5.37)

The partition function for this system can be worked out using the method of
Ref. Casini2009Entanglement  and is detailed in Appendix B. The result of the
calculation can be expressed as follows

$\\{\Gamma_{j_{1}},\Gamma_{j_{2}},\ldots,\Gamma_{j_{\alpha}}\\}\sim\prod_{k_{F}}D^{(k_{F})}_{j_{1},j_{2},\ldots,j_{\alpha}},$
(5.38)

where we have written $\sim$ instead of the equality sign because the members
of the equation are equal up to a multiplicative constant, which is
independent of $j_{1},j_{2},\ldots,j_{\alpha}$ and hence does not contribute
to the tripartite information. The factor
$D^{(k_{F})}_{j_{1},\ldots,j_{\alpha}}$ reads

$D^{(k_{F})}_{j_{1},\ldots,j_{\alpha}}=\det\bm{\mathbf{L}}_{\underline{\sigma}}^{(k_{F})},$
(5.39)

where the operator $\bm{\mathbf{L}}_{\underline{\sigma}}^{(k_{F})}$ is given
by

$\bm{\mathbf{L}}_{\underline{\sigma}}^{(k_{F})}=2^{\frac{1-\alpha}{\alpha}}\mathrm{I}\otimes\bm{\mathbf{1}}-i2^{\frac{1-\alpha}{\alpha}}\left[\mathrm{M}\otimes\frac{\bm{\mathbf{1}}-\bm{\mathbf{P}}}{2}+\Sigma\mathrm{M}\Sigma\otimes\frac{\bm{\mathbf{1}}+\bm{\mathbf{P}}}{2}\right]\mathrm{I}\otimes\Bigl{(}2\bm{\mathbf{C}}^{(k_{F})}-\bm{\mathbf{1}}\Bigr{)}\,.$
(5.40)

Here we have denoted by $\bm{\mathbf{C}}^{(k_{F})}$ the operator with the
elements equal to the correlator (5.9) on $A\cup C$, i.e.

$\bm{\mathbf{C}}^{(k_{F})}(x,y)=\braket{\bm{\mathbf{\psi}}(x)\bm{\mathbf{\psi}}^{\dagger}(y)},\qquad
x,y\in A\cup C\,.$ (5.41)

Matrix $\mathrm{M}$ is given by

${}[\mathrm{M}]_{\ell
n}=-i[e^{i\pi\frac{(n-\ell\mod\alpha)}{\alpha}}-\delta_{\ell
n}]=\frac{1}{\alpha}\sum_{j=1}^{\alpha}\cot(\tfrac{\pi(j+\frac{1}{2})}{\alpha})e^{2\pi
ij\frac{\ell-n}{\alpha}}\qquad\ell,n\in\\{1,\ldots,\alpha\\}$ (5.42)

As discussed in Appendix B.1, this formula can be expressed in terms of the
correlation matrix $\Gamma$ so as to become exact also on the lattice. This
provides an indirect confirmation of the correctness of our procedure.

Equations (5.38), (5.39), and (5.40) are the main results of this section and
the very reason why we worked out a quantum field theory description. On the
one hand, a numerical implementation of Eq. (5.39) gives an efficient way to
compute the tripartite information. On the other hand, the correlator appears
linearly in Eq. (5.40), and this will allow us to map the calculation into a
Riemann-Hilbert problem, which is the subject of the next section.

##  6 Mapping to a Riemann-Hilbert problem

In this section we connect (5.39) to a Riemann-Hilbert problem with piece-wise
constant matrix for a doubly connected domain. The starting point is to
represent the determinant in terms of the resolvent
$(\lambda\bm{\mathbf{1}}-\bm{\mathbf{L}}_{\underline{\sigma}}^{(k_{F})})^{-1}$
as follows:

$\det\bm{\mathbf{L}}_{\underline{\sigma}}^{(k_{F})}=\exp\mathrm{tr}[\log\bm{\mathbf{L}}_{\underline{\sigma}}^{(k_{F})}]=\exp\mathrm{tr}\Bigl{[}\int_{-\infty}^{0}\mathrm{d}\lambda\Bigl{(}\frac{\bm{\mathbf{1}}}{1-\lambda}+\frac{\bm{\mathbf{1}}}{\lambda\bm{\mathbf{1}}-\bm{\mathbf{L}}_{\underline{\sigma}}^{(k_{F})}}\Bigr{)}\Bigr{]}\,,$
(6.1)

where we have implicitly assumed that the eigenvalues of
$\bm{\mathbf{L}}_{\underline{\sigma}}$ do not fall on the negative real axis;
we will comment later on the validity of this assumption. This representation
traces the problem back to inverting a singular integral equation of the form
$F(x)=\int\mathrm{d}yK(x,y)\phi(y)$, with (cf. (5.40) and (5.9))

$\bm{\mathbf{K}}(x,y)=\mathrm{A}(x)\delta(x-y)+\mathrm{B}(x)\frac{1}{\pi
i(y-x)},$ (6.2)

where the coordinates $x,y$ belong to $A\cup C$ and, in our specific case,
$\mathrm{A}(x)$ and $\mathrm{B}(x)$ are the $\alpha\times\alpha$ matrices

$\displaystyle A(x)=$
$\displaystyle(\lambda-2^{\frac{1-\alpha}{\alpha}})\mathrm{I}+i2^{\frac{1-\alpha}{\alpha}}\frac{\gamma_{+}+\gamma_{-}}{2}\left[\mathrm{M}\chi_{A}(x)+\Sigma\mathrm{M}\Sigma\chi_{C}(x)\right]$
(6.3) $\displaystyle B(x)=$ $\displaystyle
i2^{\frac{1-\alpha}{\alpha}}\frac{\gamma_{-}-\gamma_{+}}{2}\left[\mathrm{M}\chi_{A}(x)+\Sigma\mathrm{M}\Sigma\chi_{C}(x)\right]\,,$

where $\chi_{A}(x)$ and $\chi_{C}(x)$ are the indicator functions of $A$ and
$C$, respectively. Note that $A(x)$ and $B(x)$ are constant in the blocks,
i.e. they are independent of $x$ in each block, and
$\mathrm{A}(x)\pm\mathrm{B}(x)$ are invertible for $\lambda\leq 0$. In what
follows the region associated with $A\cup C$ will be denoted by $L$.

The inverse $\bm{\mathbf{K}}^{-1}$ can be recognised by solving the singular
equation

$\mathrm{A}(x)\phi(x)+\frac{\mathrm{B}(x)}{\pi
i}\int_{L}\frac{\phi(y)}{y-x}dy=\mathrm{F}(x),\quad x\in L,$ (6.4)

for an arbitrary matrix function $\mathrm{F}$. We indeed have

$\phi(x)=\int_{L}\bm{\mathbf{K}}^{-1}(x,y)\mathrm{F}(y)dy\,.$ (6.5)

The integrals here and in the following are understood as Cauchy principal
values. In the remainder of the section we show, following Muskhelishvili1953
, how the singular equation is mapped to a Riemann-Hilbert problem and the
inverse $\bm{\mathbf{K}}^{-1}$ expressed in terms of its solution.

We define the sectionally holomorphic (holomorphic outside $L$, in which the
limit to $L$ from up/down exists) matrix function

$\Phi(z)=\frac{1}{2\pi i}\int_{L}\frac{\phi(y)}{y-z}dy\qquad z\in\mathbb{C}$
(6.6)

and its limits to $L$ from up and down $\Phi^{\pm}(x)=\lim_{\varepsilon\to
0^{+}}\Phi(x\pm i\varepsilon)$. From now on, if not explicitly written
otherwise, we will use $z$ for complex variables and $x$ for real ones. By
virtue of the Plemelj formulae

$\Phi^{\pm}(x)=\pm\frac{1}{2}\phi(x)+\frac{1}{2\pi
i}\int_{L}\frac{\phi(y)}{y-x}dy,$ (6.7)

we can recast the singular equation (6.4) into the Riemann-Hilbert problem of
finding the sectionally holomorphic matrix function $\Phi(z)$ that approaches
the identity at infinity and satisfies on $L$ the equation

$\Phi^{+}(x)=\mathrm{G}(x)\Phi^{-}(x)+[\mathrm{A}(x)+\mathrm{B}(x)]^{-1}\mathrm{F}(x)\qquad
x\in L\,,$ (6.8)

where

$\mathrm{G}(x)=[\mathrm{A}(x)+\mathrm{B}(x)]^{-1}[\mathrm{A}(x)-\mathrm{B}(x)]\,.$
(6.9)

Let $\mathrm{X}(z)$ be the solution to the homogeneous Riemann-Hilbert
problem, i.e., the sectionally holomorphic matrix function that on $L$
satisfies

$\mathrm{X}^{+}(x)=\mathrm{G}(x)\mathrm{X}^{-}(x)\qquad x\in L$ (6.10)

and approaches the identity in the limit $|z|\rightarrow+\infty$. The Plemelj
formula allows us to express the solution to the non-homogeneous problem (6.8)
as follows

$\Phi(z)=\frac{\mathrm{X}(z)}{2\pi
i}\int_{L}\mathrm{d}x\frac{[\mathrm{X}^{+}(x)]^{-1}[\mathrm{A}(x)+\mathrm{B}(x)]^{-1}}{x-z}\mathrm{F}(x)\,.$
(6.11)

Plugging this solution into (6.7) and applying again the Plemelj formulae
gives

$\phi(x)=\left[\tfrac{[\mathrm{A}(x)+\mathrm{B}(x)]^{-1}}{2}+\tfrac{[\mathrm{A}(x)-\mathrm{B}(x)]^{-1}}{2}\right]\mathrm{F}(x)+\tfrac{\mathrm{X}^{+}(x)}{2\pi
i}\smallint_{L}\mathrm{d}y\tfrac{[\mathrm{X}^{+}(y)]^{-1}[\mathrm{A}(y)+\mathrm{B}(y)]^{-1}}{y-x}\mathrm{F}(y)\\\
-\tfrac{[\mathrm{A}(x)-\mathrm{B}(x)]^{-1}[\mathrm{A}(x)+\mathrm{B}(x)]\mathrm{X}^{+}(x)}{2\pi
i}\smallint_{L}\mathrm{d}y\tfrac{[\mathrm{X}^{+}(y)]^{-1}[\mathrm{A}(y)+\mathrm{B}(y)]^{-1}}{y-x}\mathrm{F}(y)$
(6.12)

By comparison with (6.5) we identify the inverse of $\bm{\mathbf{K}}$

$\bm{\mathbf{K}}^{-1}(x,y)=\left[\tfrac{[\mathrm{A}(x)+\mathrm{B}(x)]^{-1}}{2}+\tfrac{[\mathrm{A}(x)-\mathrm{B}(x)]^{-1}}{2}\right]\delta(x-y)\\\
+\left[\tfrac{[\mathrm{A}(x)+\mathrm{B}(x)]^{-1}}{2}-\tfrac{[\mathrm{A}(x)-\mathrm{B}(x)]^{-1}}{2}\right]\tfrac{[\mathrm{A}(x)+\mathrm{B}(x)]\mathrm{X}^{+}(x)[\mathrm{X}^{+}(y)]^{-1}[\mathrm{A}(y)+\mathrm{B}(y)]^{-1}}{\pi
i(y-x)}\,.$ (6.13)

In our specific problem,
$\bm{\mathbf{K}}=\lambda\bm{\mathbf{1}}-\bm{\mathbf{L}}_{\sigma}^{(k_{F})}$
and we only need the trace of $\bm{\mathbf{K}}^{-1}$ — cf. (6.1). Therefore,
we can expand $\bm{\mathbf{K}}(x,y)$ for $y$ close to $x$ (a similar trick was
used in Ref. Blanco2022 ); we obtain

$\bm{\mathbf{K}}^{-1}(x,y)\sim\left[\tfrac{[\mathrm{A}(x)+\mathrm{B}(x)]^{-1}}{2}+\tfrac{[\mathrm{A}(x)-\mathrm{B}(x)]^{-1}}{2}\right]\delta(x-y)+\left[\tfrac{[\mathrm{A}(x)+\mathrm{B}(x)]^{-1}}{2}-\tfrac{[\mathrm{A}(x)-\mathrm{B}(x)]^{-1}}{2}\right]\frac{1}{\pi
i(y-x)}\\\
+\left[\tfrac{[\mathrm{A}(x)-\mathrm{B}(x)]^{-1}}{2}-\tfrac{[\mathrm{A}(x)+\mathrm{B}(x)]^{-1}}{2}\right]\tfrac{[\mathrm{A}(x)+\mathrm{B}(x)]\partial_{x}\mathrm{X}^{+}(x)[\mathrm{X}^{+}(x)]^{-1}[\mathrm{A}(x)+\mathrm{B}(x)]^{-1}}{\pi
i}\,,$ (6.14)

where we used that our specific operator is such that $\mathrm{A}(x)$ and
$\mathrm{B}(x)$ are constant in $A$ and $C$ and therefore their derivative is
zero within $L$. Plugging this into (6.1) gives

$\det\bm{\mathbf{L}}_{\underline{\sigma}}^{(k_{F})}=\exp\Bigl{(}\int_{L}\frac{\mathrm{d}x}{a}\frac{1}{2}\mathrm{tr}[\log(\tilde{\mathrm{A}}^{2}(x)-\mathrm{B}^{2}(x))]\Bigr{)}\\\
\times\exp\left(\int_{-\infty}^{0}\frac{\mathrm{d}\lambda}{2\pi
i}\int_{L}\mathrm{d}x\mathrm{tr}\Bigl{[}[\partial_{\lambda}\log\mathrm{G}(x)][\partial_{x}\mathrm{X}^{+}(x)][\mathrm{X}^{+}(x)]^{-1}\Bigr{]}\right)\,,$
(6.15)

where we made it explicit the dependency on $\lambda$ by defining the
$\lambda$-independent matrix function
$\tilde{\mathrm{A}}(x)=\mathrm{A}(x)-\lambda$ — cf. (6.3) — and used that, in
our specific case, $[\tilde{\mathrm{A}}(x),\mathrm{B}(x)]=0$. We also
regularised $\mathrm{tr}[\bm{\mathbf{1}}]$ by introducing a cut-off
$a^{-1}\sim\delta(0)$, which can be interpreted as the inverse lattice
spacing. The first exponential represents the leading extensive contribution
to the entropy, which is not universal and therefore is not supposed to
exactly match the lattice result. We note however that, as it should be, it is
independent of $\underline{\sigma}$. The second exponential captures the
entire universal part of the entropy of disjoint blocks and it is expressed in
terms of the solution to the homogeneous Riemann-Hilbert problem (6.10) with
$\log\mathrm{G}(x)$ the piecewise constant circulant matrix explicitly given
by

$\log\mathrm{G}(x)=\chi_{A}(x)2\pi\mathrm{R}+\chi_{C}(x)2\pi\Sigma\mathrm{R}\Sigma$
(6.16)

where (cf. (6.3) and (5.42))

$\displaystyle\mathrm{R}_{\ell n}=$ $\displaystyle[\mathrm{R}_{-}]_{\ell n}$
(6.17) $\displaystyle[\mathrm{R}_{s}]_{\ell n}=$
$\displaystyle\frac{1}{2\pi\alpha}\sum_{j=1}^{\alpha}\log\Bigl{(}\tfrac{2^{1-\frac{1}{\alpha}}\lambda-1-is\gamma_{-s}\cot\bigl{(}\frac{\pi(j+\frac{1}{2})}{\alpha}\bigr{)}}{2^{1-\frac{1}{\alpha}}\lambda-1-is\gamma_{s}\cot\bigl{(}\frac{\pi(j+\frac{1}{2})}{\alpha}\bigr{)}}\Bigr{)}e^{2\pi
ij\frac{\ell-n}{\alpha}}\,.$

We note $[\partial_{\lambda}\mathrm{R},\mathrm{R}]=0$ and
$\mathrm{R}_{s}^{\dagger}=-\mathrm{R}_{-s}$.

The $\alpha$-Rényi entropies of the disjoint blocks $A\cup C$ are finally
given by

$S_{\alpha}[A\cup
C]=\tfrac{1}{1-\alpha}\Bigl{(}\smallint_{L}\tfrac{\mathrm{d}x}{a}\sum_{k_{F}}\frac{1}{2}\mathrm{tr}[\log(\tilde{\mathrm{A}}^{2}(x)-\mathrm{B}^{2}(x))]\Bigr{)}\\\
+\tfrac{1}{1-\alpha}\log\Bigl{[}\tfrac{1}{2^{\alpha}}\sum_{\Sigma}\exp\Bigl{(}\sum_{k_{F}}\smallint_{-\infty}^{0}\tfrac{\mathrm{d}\lambda}{2\pi
i}\smallint_{L}\mathrm{d}x\mathrm{tr}\Bigl{[}[\partial_{\lambda}\log\mathrm{G}(x)][\partial_{x}\mathrm{X}^{+}(x)][\mathrm{X}^{+}(x)]^{-1}\Bigr{]}\Bigr{)}\Bigr{]}\,.$
(6.18)

The tripartite information is obtained by evaluating the corresponding linear
combination of entropies or using the shortcut (3.18) based on the fact that
replacing $\Sigma$ by $\mathrm{I}$ corresponds to computing the fermionic
entropies, resulting in turn in a vanishing tripartite information.

The next sections will be devoted to solve the homogeneous problem (6.10) and
finally compute the tripartite information.

##  7 Solution to the Riemann-Hilbert problem

###  7.1 Simple cases and Abelian approximation

We start with the simple case in which

$[\mathrm{R},\Sigma\mathrm{R}\Sigma]=0\,.$ (7.1)

This happens, in particular, when $\Sigma=\mathrm{I}$ or the diagonal elements
of $\Sigma$ have alternating sign. The solution to this Abelian homogeneous
Riemann-Hilbert problem readily follows from the Plemelj formulae
Muskhelishvili1953 ; Its2003TheRP

$X^{+}(x)=\exp\Bigl{(}i\smallint_{0}^{1}\mathrm{d}t\tfrac{\mathrm{R}}{t-\frac{v_{1}-x-i\epsilon}{v_{1}-u_{1}}}-\tfrac{\Sigma\mathrm{R}\Sigma}{t-\frac{x+i\epsilon-
u_{2}}{v_{2}-u_{2}}}\Bigr{)}=\exp\left[iR\log\Bigl{(}\tfrac{x-u_{1}+i\epsilon}{x-v_{1}+i\epsilon}\Bigr{)}-i\Sigma
R\Sigma\log\Bigl{(}\tfrac{v_{2}-x-i\epsilon}{u_{2}-x-i\epsilon}\Bigr{)}\right]\,.$
(7.2)

The universal contributions to the entropy can then be expressed as follows

$\exp\Bigl{(}\smallint_{-\infty}^{0}\tfrac{\mathrm{d}\lambda}{2\pi
i}\smallint_{L}\mathrm{d}x\mathrm{tr}\Bigl{[}[\partial_{\lambda}\log\mathrm{G}(x)][\partial_{x}\mathrm{X}^{+}(x)][\mathrm{X}^{+}(x)]^{-1}\Bigr{]}\Bigr{)}=\\\
\exp\left(\mathrm{tr}[R_{0}^{2}]\log\Bigl{(}\tfrac{(v_{1}-u_{1})(v_{2}-u_{2})}{\epsilon^{2}}\Bigr{)}+\mathrm{tr}[(R_{0}\Sigma)^{2}]\log\Bigl{(}\tfrac{(v_{2}-u_{1})(u_{2}-v_{1})}{(v_{2}-v_{1})(u_{2}-u_{1})}\Bigr{)}\right)$
(7.3)

where $\mathrm{R}_{0}$ is $\mathrm{R}$ at $\lambda=0$, that is to say,

${}[R_{0}]_{\ell
n}=\tfrac{1}{2\pi\alpha}\sum_{j=1}^{\alpha}\log\Bigl{(}\tfrac{1-i\gamma_{+}\cot\bigl{(}\frac{\pi(j+\frac{1}{2})}{\alpha}\bigr{)}}{1-i\gamma_{-}\cot\bigl{(}\frac{\pi(j+\frac{1}{2})}{\alpha}\bigr{)}}\Bigr{)}e^{2\pi
ij\frac{\ell-n}{\alpha}}\,.$ (7.4)

Eq.(7.3) describes the contribution to the entropy from terms satisfying
(7.1). Since such a condition is always satisfied for $\alpha=2$, we are in a
position to compute the Rényi-$2$ tripartite information

$I_{3}^{(2)}(A,B,C)=G_{2}(x_{4})=-\log
2+\log\left[1+(1-x_{4})^{\sum_{k_{F}}\frac{[\arctan(\gamma_{+})-\arctan(\gamma_{-})]^{2}}{\pi^{2}}}\right]\,,$
(7.5)

where, to avoid confusion with the position variable $x$, we have renamed the
cross ratio $x_{4}$, i.e.,

$x_{4}=\frac{(v_{1}-u_{1})(v_{2}-u_{2})}{(u_{2}-u_{1})(v_{2}-v_{1})}\,.$ (7.6)

This was first conjectured in Ref. Maric2022Universality  and we have
therefore proved its validity.

If we enforce (7.3) also for the terms that do not satisfy (7.1), we get what
we call “Abelian approximation”, which implicitly neglects any potential
contribution coming from the commutator $[\mathrm{R},\Sigma\mathrm{R}\Sigma]$.
The Abelian approximation of the tripartite information is then given by

$G^{Abelian}_{\alpha}(x_{4})=-\log
2+\tfrac{1}{\alpha-1}\log\Bigl{[}\tfrac{1}{2}\sum_{\Sigma}(1-x_{4})^{\sum_{k_{F}}\mathrm{tr}[(R_{0}\Sigma)^{2}-R_{0}^{2}]}\Bigr{]}\,.$
(7.7)

At first glance this might appear as a very crude simplification. Remarkably,
instead, it turns out to be an excellent approximation whatever $\alpha$ and
$\gamma_{\pm}$. We will come back to this point in Section 7.2, where we will
also work out the non-Abelian contributions. In the following we focus instead
on the limit $\gamma_{+}=-\gamma_{-}\rightarrow\pm 1$, in which we can compare
the Abelian approximation with exact analytic results.

##### Pseudo-conformal limit.

Figure 11: Abelian approximation of the Rényi-$\alpha$ tripartite information
against the exact CFT prediction in a state with $2$ Fermi points (a standard
Fermi sea) as a function of the cross ratio. The agreement is far beyond
expectations.

With “pseudo-conformal limit” we mean the limit in which the Gaussian
contributions (3.12) to the tripartite information after a global quench match
a subset of the terms characterising the tripartite information in the ground
state of a conformal field theory. Strictly speaking, indeed, the state at
late times after the quench does not exhibit conformal symmetry, but when
$\gamma_{+}=-\gamma_{-}\rightarrow\pm 1$ the effect of such a symmetry
breaking is completely captured by our preliminary selection of the Gaussian
terms which are supposed to contribute at infinite time after the quench —
compare (3.15) with (3.11).

Specifically, from the exact analytic results of Refs Coser2016Spin  and as
pointed out in Ref. Maric2022Universality , assuming
$\gamma_{+}=-\gamma_{-}\rightarrow\pm 1$ we have (for each Fermi point)

$\frac{\det\bm{\mathbf{L}}_{\underline{\sigma}}^{(k_{F})}}{\det\bm{\mathbf{L}}_{\underline{1}}^{(k_{F})}}=\frac{\Theta(\vec{\delta}|\hat{\tau}^{(\alpha)}_{x_{4}})}{\Theta(\vec{0}|\hat{\tau}^{(\alpha)}_{x_{4}})}\,,\qquad[\vec{\delta}]_{j}=\frac{1-\sigma_{j}\sigma_{j+1}}{4}\,,$
(7.8)

where
$\Theta(\vec{z},M)=\sum_{\vec{m}\in\mathbb{Z}^{\alpha-1}}e^{i\pi\vec{m}^{t}M\vec{m}+2\pi
i\vec{m}\cdot\vec{\delta}}$ is the Siegel theta function and $\hat{\tau}_{x}$
is the $(\alpha-1)\times(\alpha-1)$ period matrix of the Riemann surface
$\mathcal{R}_{\alpha}$ with elements

$[\hat{\tau}^{(\alpha)}_{x}]_{\ell
n}=\frac{2i}{\alpha}\sum_{k=1}^{\alpha-1}\sin(\tfrac{\pi
k}{\alpha})\cos(\tfrac{2\pi
k(\ell-n)}{\alpha})\tfrac{P_{(k/\alpha)-1}(2x-1)}{P_{(k/\alpha)-1}(1-2x)}\,,$
(7.9)

and $P_{\mu}(z)$ are the Legendre polynomials. Fig. 11 shows a comparison
between the exact CFT predictions and the corresponding Abelian
approximations, which are explicitly worked out for a few values of $\alpha$
in Appendix C. The agreement is excellent for small values of $\alpha$ and
remains very good also increasing $\alpha$. It is nevertheless clear that the
Abelian approximation is not exact.

###  7.2 Beyond the Abelian approximation

The goodness of the Abelian approximation is striking, so one could wonder why
is that the case. In addition, the Abelian approximation of the tripartite
information is a function of solely the cross ratio, which is not a property
of the tripartite information that we can accept without skepticism: in our
state there are also finite correlation lengths so, strictly speaking, there
is no global conformal symmetry. In order to shed light on these questions, we
work out the solution to the Riemann-Hilbert problem even when
$[\mathrm{R},\Sigma\mathrm{R}\Sigma]\neq 0$ (we refer the reader to Ref.
Its2003TheRP  for a general review on similar Riemann-Hilbert problems).

In such a non-Abelian case, (7.2) does not solve (6.10) because the matrices
do not appear in the correct order. Order of the operators apart, however, the
structure of the solution should remain the same. Let us then start with the
following Ansatz

$X(z)=\hat{T}^{-1}\exp\Bigl{[}i\smallint_{0}^{1}\mathrm{d}t\tfrac{\mathrm{R}_{-}(t)}{t-\frac{v_{1}-z}{v_{1}-u_{1}}}+\tfrac{\Sigma\mathrm{R}^{\dagger}_{+}(t)\Sigma}{t-\frac{z-u_{2}}{v_{2}-u_{2}}}\Bigr{]}\qquad
z\in\mathbb{C}\setminus L\,,$ (7.10)

where $\hat{T}$ ($\hat{T}^{-1}$) denotes the (anti) time ordering operator.
The matrix function $X^{+}(x)$ is defined as the limit $\epsilon\rightarrow
0^{+}$ for $z=x+i\epsilon$ and $x\in A\cup C$, that is to say,

$X^{+}(x)=\lim_{\epsilon\rightarrow
0^{+}}\hat{T}^{-1}\exp\Bigl{[}i\smallint_{0}^{1}\mathrm{d}t\tfrac{\mathrm{R}_{-}(t)}{t-\frac{v_{1}-x-i\epsilon}{v_{1}-u_{1}}}+\tfrac{\Sigma\mathrm{R}_{+}^{\dagger}(t)\Sigma}{t-\frac{x+i\epsilon-
u_{2}}{v_{2}-u_{2}}}\Bigr{]}\qquad x\in L\,.$ (7.11)

Plugging this into (6.10) results in the consistency conditions

$\mathrm{R}_{s}(\tau)=\lim_{\epsilon\rightarrow
0^{+}}[\mathscr{L}_{s}(\tau-i\tfrac{\epsilon}{\ell_{s}},\tau)]^{-1}\mathrm{R}_{s}\mathscr{L}_{s}(\tau-i\tfrac{\epsilon}{\ell_{s}},\tau)$
(7.12)

where we introduced the shorthand

$\ell_{-}=v_{1}-u_{1}\qquad d=u_{2}-v_{1}\qquad\ell_{+}=v_{2}-u_{2}$ (7.13)

and

$\mathscr{L}_{s}(z,\tau^{\prime})=\hat{T}^{-1}\exp\Bigl{[}i\smallint_{0}^{\tau^{\prime}}\mathrm{d}t\tfrac{\mathrm{R}_{s}(t)}{t-z}+\tfrac{\Sigma\mathrm{R}_{-s}^{\dagger}(t)\Sigma}{t+\frac{\ell_{s}}{\ell_{-s}}z+\frac{d}{\ell_{-s}}}\Bigr{]}$
(7.14)

This implicitly gives $X^{+}(x)$ as a function of $\mathrm{R}_{s}$ (note, in
particular, that $\mathrm{R}_{s}(\tau)$ could be constructed starting from
$\mathrm{R}_{s}(0)=\mathrm{R}_{s}$). En passant, we can already argue that our
preliminary assumption about the eigenvalues of
$\bm{\mathbf{L}}_{\underline{\sigma}}^{(k_{F})}$ not falling on the negative
real axis is satisfied: the inverse (6.13) of
$\lambda\bm{\mathbf{1}}-\bm{\mathbf{L}}_{\underline{\sigma}}^{(k_{F})}$
depends on $\lambda$ only through $[A(x)\pm B(x)]^{-1}$ and $R_{\pm}$, which
are smooth matrix functions of $\lambda$ on the real line, justifying in turn
representation (6.1).

We report two identities that are particularly useful for the upcoming
calculation:

$\Sigma[\mathscr{L}_{s}(-\tfrac{d}{\ell_{s}}-\tfrac{\ell_{-s}}{\ell_{s}}z,\tau^{\prime})]^{\dagger}\Sigma=[\mathscr{L}_{-s}(z^{\ast},\tau^{\prime})]^{-1}$
(7.15)

and

${}[\partial_{z}\mathscr{L}_{s}(z,\tau^{\prime})][\mathscr{L}_{s}(z,\tau^{\prime})]^{-1}=i\smallint_{0}^{\tau^{\prime}}\mathrm{d}\tau\Bigl{[}\tfrac{\mathrm{R}_{s}(z,\tau)}{(\tau-z)^{2}}-\tfrac{\ell_{s}}{\ell_{-s}}\tfrac{\Sigma[\mathrm{R}_{-s}(-\frac{d}{\ell_{-s}}-\frac{\ell_{s}}{\ell_{-s}}z^{\ast},\tau)]^{{\dagger}}\Sigma}{(\tau+\frac{\ell_{s}}{\ell_{-s}}z+\frac{d}{\ell_{-s}})^{2}}\Bigr{]}\,.$
(7.16)

Here we introduced what will become the main ingredient of our analytical
analysis, i.e., the auxiliary matrix function

$R_{s}(z,\tau^{\prime})=\mathscr{L}_{s}(z,\tau^{\prime})\mathrm{R}_{s}(\tau^{\prime})[\mathscr{L}_{s}(z,\tau^{\prime})]^{-1}\,,$
(7.17)

which satisfies the following integro-differential equation

$\displaystyle\partial_{\tau_{1}}\mathrm{R}_{s}(\tau_{1}-i\tfrac{\epsilon}{\ell_{s}},\tau_{2})=$
(7.18) $\displaystyle\qquad
i\left[\smallint_{0}^{\tau_{2}}\mathrm{d}\tau_{2}^{\prime}\Bigl{[}\tfrac{\mathrm{R}_{s}(\tau_{1}-i\tfrac{\epsilon}{\ell_{s}},\tau_{2}^{\prime})}{(\tau_{2}^{\prime}-\tau_{1}+i\tfrac{\epsilon}{\ell_{s}})^{2}}-\tfrac{\ell_{s}}{\ell_{-s}}\tfrac{\Sigma[\mathrm{R}_{-s}(-\frac{d}{\ell_{-s}}-\frac{\ell_{s}}{\ell_{-s}}\tau_{1}-i\frac{\epsilon}{\ell_{-s}},\tau_{2}^{\prime})]^{{\dagger}}\Sigma}{(\tau_{2}^{\prime}+\frac{\ell_{s}}{\ell_{-s}}\tau_{1}-i\tfrac{\epsilon}{\ell_{-s}}+\frac{d}{\ell_{-s}})^{2}}\Bigr{]},\mathrm{R}_{s}(\tau_{1}-i\tfrac{\epsilon}{\ell_{s}},\tau_{2})\right]$
$\displaystyle\lim_{\epsilon\rightarrow
0^{+}}\mathrm{R}_{s}(\tau-i\tfrac{\epsilon}{\ell_{s}},\tau)=\lim_{\epsilon\rightarrow
0^{+}}\mathrm{R}_{s}(\tau-i\tfrac{\epsilon}{\ell_{s}},0)=\mathrm{R}_{s}\,.$

We stress that, even if not manifest in our notations,
$\mathrm{R}_{s}(\tau_{1},\tau_{2})$ depends on $\ell_{\pm}$ and $d$
independently of its arguments. Since
$X^{+}(x)=\mathscr{L}_{-}(\frac{v_{1}-x-i\epsilon}{\ell_{-}},1)$, we have

${}[\partial_{x}X^{+}(x)][X^{+}(x)]^{-1}=i\smallint_{0}^{1}\mathrm{d}\tau^{\prime}\Bigl{[}\tfrac{1}{\ell_{+}}\tfrac{\Sigma[\mathrm{R}_{+}(\frac{x-u_{2}-i\epsilon}{\ell_{+}},\tau^{\prime})]^{\dagger}\Sigma}{(\tau^{\prime}-\frac{x+i\epsilon-
u_{2}}{\ell_{+}})^{2}}-\tfrac{1}{\ell_{-}}\tfrac{\mathrm{R}_{-}(\frac{v_{1}-x-i\epsilon}{\ell_{-}},\tau^{\prime})}{(\tau^{\prime}-\frac{v_{1}-x-i\epsilon}{\ell_{-}})^{2}}\Bigr{]}\,,$
(7.19)

which finally gives

$\int_{L}\tfrac{\mathrm{d}x}{2\pi i}\mathrm{tr}[\partial_{\lambda}\log
G(x)\partial_{x}X^{+}(x)][X^{+}(x)]^{-1}]=\\\
\iint_{0}^{1}\mathrm{d}^{2}\tau\mathrm{tr}\left[(\partial_{\lambda}\mathrm{R}_{-})\Bigl{[}\tfrac{\ell_{-}}{\ell_{+}}\tfrac{\Sigma[\mathrm{R}_{+}(-\frac{\ell_{-}}{\ell_{+}}\tau_{1}-\frac{d}{\ell_{+}}-i\frac{\epsilon}{\ell_{+}},\tau_{2})]^{\dagger}\Sigma}{(\tau_{2}+\frac{\ell_{-}}{\ell_{+}}\tau_{1}+\frac{d}{\ell_{+}}-i\frac{\epsilon}{\ell_{+}})^{2}}-\tfrac{\mathrm{R}_{-}(\tau_{1}-i\frac{\epsilon}{\ell_{-}},\tau_{2})}{(\tau_{2}-\tau_{1}+i\frac{\epsilon}{\ell_{-}})^{2}}\Bigr{]}\right]+\\\
\iint_{0}^{1}\mathrm{d}^{2}\tau\left(\mathrm{tr}\left[(\partial_{\lambda}\mathrm{R}_{+})\Bigl{[}\tfrac{\ell_{+}}{\ell_{-}}\tfrac{\Sigma[\mathrm{R}_{-}(-\frac{\ell_{+}}{\ell_{-}}\tau_{1}-\frac{d}{\ell_{-}}-i\frac{\epsilon}{\ell_{-}},\tau_{2})]^{\dagger}\Sigma}{(\tau_{2}+\frac{\ell_{+}}{\ell_{-}}\tau_{1}+\frac{d}{\ell_{-}}-i\frac{\epsilon}{\ell_{-}})^{2}}-\tfrac{\mathrm{R}_{+}(\tau_{1}-i\frac{\epsilon}{\ell_{+}},\tau_{2})}{(\tau_{2}-\tau_{1}+i\frac{\epsilon}{\ell_{+}})^{2}}\Bigr{]}\right]\right)^{\ast}$
(7.20)

Plugging this into (6.18) gives the entropy of the disjoint blocks, from which
we can finally obtain the tripartite information of adjacent blocks $A,B,C$

$e^{(\alpha-1)I_{3}^{(\alpha)}(A,B,C)}=\\\
\tfrac{1}{2^{\alpha}}\sum_{\Sigma}\exp\Bigl{[}\iint_{0}^{1}\\!\\!\\!\mathrm{d}^{2}\tau\int_{C_{[0,\mathrm{R}_{0}]}}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\mathrm{tr}\Bigl{[}\mathrm{d}\mathrm{R}_{-}\
\Bigl{(}\tfrac{\ell_{-}}{\ell_{+}}\tfrac{\Sigma(\mathrm{R}_{+}(-\frac{\ell_{-}}{\ell_{+}}\tau_{1}-\frac{d}{\ell_{+}}-i\frac{\epsilon}{\ell_{+}},\tau_{2}))^{\dagger}\Sigma+\mathrm{R}_{-}}{(\tau_{2}+\frac{\ell_{-}}{\ell_{+}}\tau_{1}+\frac{d}{\ell_{+}}-i\frac{\epsilon}{\ell_{+}})^{2}}-\tfrac{\mathrm{R}_{-}(\tau_{1}-i\frac{\epsilon}{\ell_{-}},\tau_{2})-\mathrm{R}_{-}}{(\tau_{1}-\tau_{2}-i\frac{\epsilon}{\ell_{-}})^{2}}\Bigr{)}\Bigr{]}\Bigr{]}\\\
\exp\Bigl{[}\iint_{0}^{1}\\!\\!\\!\mathrm{d}^{2}\tau\Bigl{(}\int_{C_{[0,-\mathrm{R}_{0}^{\dagger}]}}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\mathrm{tr}\Bigl{[}\mathrm{d}\mathrm{R}_{+}\
\Bigl{(}\tfrac{\ell_{+}}{\ell_{-}}\tfrac{\Sigma(\mathrm{R}_{-}(-\frac{\ell_{+}}{\ell_{-}}\tau_{1}-\frac{d}{\ell_{-}}-i\frac{\epsilon}{\ell_{-}},\tau_{2}))^{\dagger}\Sigma+\mathrm{R}_{+}}{(\tau_{2}+\frac{\ell_{+}}{\ell_{-}}\tau_{1}+\frac{d}{\ell_{-}}-i\frac{\epsilon}{\ell_{-}})^{2}}-\tfrac{\mathrm{R}_{+}(\tau_{1}-i\frac{\epsilon}{\ell_{+}},\tau_{2})-\mathrm{R}_{+}}{(\tau_{1}-\tau_{2}-i\frac{\epsilon}{\ell_{+}})^{2}}\Bigr{)}\Bigr{]}\Bigr{)}^{\ast}\Bigr{]}\,.$
(7.21)

##### Logarithmic (divergent) contributions.

Since $R_{\pm}(z,\tau_{2})$ is bounded (it is equivalent to $R_{\pm}$), for
generic values of $\ell_{\pm}$ and $d$, the divergent contribution in (7.20)
comes from the neighbourhoods of $\tau_{1}=\tau_{2}$. By expanding about that
line we identify the following asymptotic logarithmic divergency

$\int_{-\infty}^{0}\mathrm{d}\lambda\int_{L}\frac{\mathrm{d}x}{2\pi
i}\mathrm{tr}[\partial_{\lambda}\log
G(x)\partial_{x}X^{+}(x)][X^{+}(x)]^{-1}]\sim\\\
-\int_{-\infty}^{0}\mathrm{d}\lambda\left[\iint_{0}^{1}\mathrm{d}^{2}\tau\tfrac{\mathrm{tr}[(\partial_{\lambda}\mathrm{R}_{-})\mathrm{R}_{-}]}{(\tau_{2}-\tau_{1}+i\frac{\epsilon}{\ell_{-}})^{2}}+\iint_{0}^{1}\mathrm{d}^{2}\tau\left(\tfrac{\mathrm{tr}(\partial_{\lambda}\mathrm{R}_{+})\mathrm{R}_{+}]}{(\tau_{2}-\tau_{1}+i\frac{\epsilon}{\ell_{+}})^{2}}\right)^{\ast}\right]=\mathrm{tr}[\mathrm{R}_{0}^{2}]\log\tfrac{\ell_{+}\ell_{-}}{\epsilon^{2}}\,.$
(7.22)

This is completely captured by the Abelian approximation and, as expected, is
independent of the particular value of $\Sigma$. In the chain it results in
the asymptotic behaviour

$S_{A\cup C}\sim
O(\ell_{+}+\ell_{-})+\frac{1}{1-\alpha}\mathrm{tr}[\mathrm{R}_{0}^{2}]\log(\ell_{+}\ell_{-})+O(1)$
(7.23)

where the leading non-universal extensive behaviour $O(\ell_{+}+\ell_{-})$ is
partially captured by the first line of (6.18).

##### Residual tripartite information.

Ref. Maric2022Universality  introduced the concept of “residual tripartite
information”, which quantifies the non-commutativity of limits when
$(a\ll)d\ll\ell_{\pm}$. Since we have already taken the continuum limit
$a\rightarrow 0$, we can simply set $d=0$ in (7.21), obtaining

$\frac{1}{2^{\alpha}}\sum_{\Sigma}\exp\Bigl{[}\iint_{0}^{1}\mathrm{d}^{2}\tau\int_{C_{[0,\mathrm{R}_{0}]}}\\!\\!\\!\\!\\!\\!\mathrm{tr}\Bigl{[}\mathrm{d}\mathrm{R}_{-}\
\Bigl{(}\tfrac{\ell_{-}}{\ell_{+}}\tfrac{\Sigma(\mathrm{R}_{+}(-\frac{\ell_{-}}{\ell_{+}}\tau_{1}-i\frac{\epsilon}{\ell_{+}},\tau_{2}))^{\dagger}\Sigma+\mathrm{R}_{-}}{(\tau_{2}+\frac{\ell_{-}}{\ell_{+}}\tau_{1}-i\frac{\epsilon}{\ell_{+}})^{2}}-\tfrac{\mathrm{R}_{-}(\tau_{1}-i\frac{\epsilon}{\ell_{-}},\tau_{2})-\mathrm{R}_{-}}{(\tau_{1}-\tau_{2}-i\frac{\epsilon}{\ell_{-}})^{2}}\Bigr{)}\Bigr{]}\Bigr{]}\\\
\exp\Bigl{[}\iint_{0}^{1}\mathrm{d}^{2}\tau\Bigl{(}\int_{C_{[0,-\mathrm{R}_{0}^{\dagger}]}}\\!\\!\\!\\!\\!\\!\mathrm{tr}\Bigl{[}\mathrm{d}\mathrm{R}_{+}\
\Bigl{(}\tfrac{\ell_{+}}{\ell_{-}}\tfrac{\Sigma(\mathrm{R}_{-}(-\frac{\ell_{+}}{\ell_{-}}\tau_{1}-i\frac{\epsilon}{\ell_{-}},\tau_{2}))^{\dagger}\Sigma+\mathrm{R}_{+}}{(\tau_{2}+\frac{\ell_{+}}{\ell_{-}}\tau_{1}-i\frac{\epsilon}{\ell_{-}})^{2}}-\tfrac{\mathrm{R}_{+}(\tau_{1}-i\frac{\epsilon}{\ell_{+}},\tau_{2})-\mathrm{R}_{+}}{(\tau_{1}-\tau_{2}-i\frac{\epsilon}{\ell_{+}})^{2}}\Bigr{)}\Bigr{]}\Bigr{)}^{\ast}\Bigr{]}$
(7.24)

with

$\mathrm{R}_{s}(\tau_{1}-i\tfrac{\epsilon}{\ell_{s}},\tau_{2})=\mathrm{R}_{s}+\\\
\qquad
i\int_{\tau_{2}}^{\tau_{1}}\mathrm{d}\tau_{1}^{\prime}\left[\int_{0}^{\tau_{2}}\mathrm{d}\tau_{2}^{\prime}\Bigl{[}\tfrac{\mathrm{R}_{s}(\tau_{1}^{\prime}-i\frac{\epsilon}{\ell_{s}},\tau_{2}^{\prime})}{(\tau_{2}^{\prime}-\tau_{1}^{\prime}+i\frac{\epsilon}{\ell_{s}})^{2}}-\tfrac{\ell_{s}}{\ell_{-s}}\tfrac{\Sigma[\mathrm{R}_{-s}(-\frac{\ell_{s}}{\ell_{-s}}\tau_{1}^{\prime}-i\frac{\epsilon}{\ell_{-s}},\tau_{2}^{\prime})]^{{\dagger}}\Sigma}{(\tau_{2}^{\prime}+\frac{\ell_{s}}{\ell_{-s}}\tau_{1}^{\prime}-i\frac{\epsilon}{\ell_{-s}})^{2}}\Bigr{]},\mathrm{R}_{s}(\tau_{1}^{\prime}-i\tfrac{\epsilon}{\ell_{s}},\tau_{2})\right]$
(7.25)

This produces a new singularity in the integrals at $\tau_{2}\sim\tau_{1}\sim
0$ for any $\Sigma\neq\mathrm{I}$, which appears already in the Abelian
approximation. Contrary to the logarithmic divergency of the entropy of
disjoint blocks, however, this singularity has also non-Abelian contributions.
This can be seen by comparing the asymptotics (in the limit $d\rightarrow 0$)
of the Abelian approximations in the pseudo-conformal limit, as worked out in
the previous section, with the exact CFT predictions. Despite the Abelian
approximation not capturing the singular behaviour exactly, we find that non-
Abelian contributions are not strong enough to remove the divergency, with the
remarkable consequence that the Rényi-$\alpha$ tripartite information
approaches the finite value $-\log 2$ in the limit $d\rightarrow 0$, for any
$\alpha$ and any $\gamma_{\pm}\neq 0$.

###  7.3 Perturbation theory for the tripartite information

Representation (7.21) for the tripartite information is implicit, indeed the
matrix functions $\mathrm{R}_{s}(z,\tau)$ are defined as the solutions to
(7.18). We develop in this section a perturbation theory to overcome this
problem. Specifically, we carry out a formal expansion of
$R_{s}(\tau_{1},\tau_{2})$ in $R_{s}$.

First, we rewrite (7.18) in integral form

$\mathrm{R}_{s}(\tau_{1}-i\tfrac{\epsilon}{\ell_{s}},\tau_{2})=\mathrm{R}_{s}+\\\
i\smallint_{\tau_{2}}^{\tau_{1}}\mathrm{d}\tau_{1}^{\prime}\Bigl{[}\smallint_{0}^{\tau_{2}}\mathrm{d}\tau_{2}^{\prime}\Bigl{[}\tfrac{\mathrm{R}_{s}(\tau_{1}^{\prime}-i\frac{\epsilon}{\ell_{s}},\tau_{2}^{\prime})}{(\tau_{2}^{\prime}-\tau_{1}^{\prime}+i\frac{\epsilon}{\ell_{s}})^{2}}-\tfrac{\ell_{s}}{\ell_{-s}}\tfrac{\Sigma\bigl{(}\mathrm{R}_{-s}(-\frac{d}{\ell_{-s}}-\frac{\ell_{s}}{\ell_{-s}}\tau_{1}^{\prime}-i\frac{\epsilon}{\ell_{-s}},\tau_{2}^{\prime})\bigr{)}^{\dagger}\Sigma}{(\tau_{2}^{\prime}+\frac{\ell_{s}}{\ell_{-s}}\tau_{1}^{\prime}-i\frac{\epsilon}{\ell_{-s}}+\frac{d}{\ell_{-s}})^{2}}\Bigr{]},\mathrm{R}_{s}(\tau_{1}^{\prime}-i\tfrac{\epsilon}{\ell_{s}},\tau_{2})\Bigr{]}\,.$
(7.26)

It is clear that such a formal expansion is a triangularization of the
integral equation, indeed the terms with a given number of $R_{s}$ matrices
are quadratic functionals of the terms with a strictly lower number of
$\mathrm{R}_{s}$ matrices. In particular, the linear term of the expansion is
trivially equal to $\mathrm{R}$. The quadratic terms can only come from the
commutators of the linear term, and so on and so forth. From now on we assume
$\tau_{2}\in\mathbb{R}_{+}$, $\tau_{1}\in\mathbb{R}$, and expand
$\mathrm{R}_{s}(\tau_{1}-i\frac{\epsilon}{\ell_{s}},\tau_{2})$ as follows:

$\mathrm{R}_{s}(\tau_{1}-i\tfrac{\epsilon}{\ell_{s}},\tau_{2})=\sum_{n=1}^{\infty}\mathrm{R}^{(n)}_{s}(\tau_{1}-i\tfrac{\epsilon}{\ell_{s}},\tau_{2})\,,$
(7.27)

where $\mathrm{R}_{s}^{(n)}(\tau_{1}-i\tfrac{\epsilon}{\ell_{s}},\tau_{2})$ is
written in terms of products of matrices in which matrix $\mathrm{R}_{s}$
appears $n$ times. Thus we have

$\mathrm{R}_{s}^{(n)}(\tau_{1}-i\tfrac{\epsilon}{\ell_{s}},\tau_{2})=\delta_{n1}\mathrm{R}_{s}-i\sum_{j=1}^{n-1}\int_{\tau_{2}}^{\tau_{1}}\mathrm{d}\tau_{1}^{\prime}\Bigl{[}\int_{0}^{\tau_{2}}\mathrm{d}\tau_{2}^{\prime}\Bigl{[}\mathrm{R}_{s}^{(n-j)}(\tau_{1}^{\prime}-i\tfrac{\epsilon}{\ell_{s}},\tau_{2}),\\\
\tfrac{\mathrm{R}_{s}^{(j)}(\tau_{1}^{\prime}-i\tfrac{\epsilon}{\ell_{s}},\tau_{2}^{\prime})}{(\tau_{2}^{\prime}-\tau_{1}^{\prime}+i\tfrac{\epsilon}{\ell_{s}})^{2}}-\tfrac{\ell_{s}}{\ell_{-s}}\tfrac{\Sigma\bigl{(}\mathrm{R}_{-s}^{(j)}(-\tfrac{d}{\ell_{-s}}-\tfrac{\ell_{s}}{\ell_{-s}}\tau_{1}^{\prime}-i\tfrac{\epsilon}{\ell_{-s}},\tau_{2}^{\prime})\bigr{)}^{\dagger}\Sigma}{(\tau_{2}^{\prime}+\frac{d}{\ell_{-s}}+\frac{\ell_{s}}{\ell_{-s}}\tau_{1}^{\prime}-i\tfrac{\epsilon}{\ell_{-s}})^{2}}\Bigr{]}\,.$
(7.28)

which also implies

$-\Sigma(\mathrm{R}_{-s}^{(n)}(-\tfrac{d}{\ell_{-s}}-\tfrac{\ell_{s}}{\ell_{-s}}\tau_{1}-i\tfrac{\epsilon}{\ell_{-s}},\tau_{2}))^{\dagger}\Sigma=\delta_{n1}\Sigma\mathrm{R}_{s}\Sigma+\\\
i\sum_{j=1}^{n-1}\int_{\tau_{1}}^{-\tfrac{d}{\ell_{s}}-\tfrac{\ell_{-s}}{\ell_{s}}\tau_{2}}\mathrm{d}\tau_{1}^{\prime}\Bigl{[}\int_{0}^{\tau_{2}}\mathrm{d}\tau_{2}^{\prime}\Bigl{[}\tfrac{\mathrm{R}_{s}^{(j)}(\tau_{1}^{\prime}-i\tfrac{\epsilon}{\ell_{s}},\tau_{2}^{\prime})}{(\tau_{2}^{\prime}-\tau_{1}^{\prime}+i\tfrac{\epsilon}{\ell_{s}})^{2}}-\tfrac{\ell_{s}}{\ell_{-s}}\tfrac{\Sigma(\mathrm{R}_{-s}^{(j)}(-\tfrac{d}{\ell_{-s}}-\tfrac{\ell_{s}}{\ell_{-s}}\tau_{1}^{\prime}-i\tfrac{\epsilon}{\ell_{-s}},\tau_{2}^{\prime}))^{\dagger}\Sigma}{(\tau_{2}^{\prime}+\tfrac{d}{\ell_{-s}}+\tfrac{\ell_{s}}{\ell_{-s}}\tau_{1}^{\prime}-i\tfrac{\epsilon}{\ell_{-s}})^{2}}\Bigr{]},\\\
\Sigma(\mathrm{R}_{-s}^{(n-j)}(-\tfrac{d}{\ell_{-s}}-\tfrac{\ell_{s}}{\ell_{-s}}\tau_{1}^{\prime}-i\tfrac{\epsilon}{\ell_{-s}},\tau_{2})^{\dagger}\Sigma\Bigr{]}\,.$
(7.29)

This expansion can be used to set up a perturbation theory for the tripartite
information

$\begin{gathered}e^{(\alpha-1)I_{3}^{(\alpha)}}=\frac{1}{2^{\alpha}}\sum_{\Sigma}\prod_{n=1}^{\infty}K^{(n,\Sigma)}_{\alpha}\,,\qquad\text{with}\\\
K^{(n,\Sigma)}_{\alpha}=\exp\Bigl{[}\iint_{0}^{1}\\!\\!\\!\mathrm{d}^{2}\tau\\!\\!\\!\int_{C_{[0,\mathrm{R}_{0}]}}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\mathrm{tr}\Bigl{[}\mathrm{d}\mathrm{R}_{-}\Bigl{(}\tfrac{\Sigma(\mathrm{R}_{+}^{(n)}(-\frac{\ell_{-}}{\ell_{+}}\tau_{1}-\frac{d}{\ell_{+}}-i\frac{\epsilon}{\ell_{+}},\tau_{2}))^{\dagger}\Sigma+\mathrm{R}_{-}}{\frac{\ell_{+}}{\ell_{-}}(\tau_{2}+\frac{\ell_{-}}{\ell_{+}}\tau_{1}+\frac{d}{\ell_{+}}-i\frac{\epsilon}{\ell_{+}})^{2}}-\tfrac{\mathrm{R}^{(n)}_{-}(\tau_{1}-i\frac{\epsilon}{\ell_{-}},\tau_{2})-\mathrm{R}_{-}}{(\tau_{1}-\tau_{2}-i\frac{\epsilon}{\ell_{-}})^{2}}\Bigr{)}\Bigr{]}\Bigr{]}\\\
\exp\Bigl{[}\iint_{0}^{1}\mathrm{d}^{2}\tau\Bigl{(}\int_{C_{[0,-\mathrm{R}_{0}^{\dagger}]}}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\mathrm{tr}\Bigl{[}\mathrm{d}\mathrm{R}_{+}\Bigl{(}\tfrac{\Sigma(\mathrm{R}_{-}^{(n)}(-\frac{\ell_{+}}{\ell_{-}}\tau_{1}-\frac{d}{\ell_{-}}-i\frac{\epsilon}{\ell_{-}},\tau_{2}))^{\dagger}\Sigma+\mathrm{R}_{+}}{\frac{\ell_{-}}{\ell_{+}}(\tau_{2}+\frac{\ell_{+}}{\ell_{-}}\tau_{1}+\frac{d}{\ell_{-}}-i\frac{\epsilon}{\ell_{-}})^{2}}-\tfrac{\mathrm{R}^{(n)}_{+}(\tau_{1}-i\frac{\epsilon}{\ell_{+}},\tau_{2})-\mathrm{R}_{+}}{(\tau_{1}-\tau_{2}-i\frac{\epsilon}{\ell_{+}})^{2}}\Bigr{)}\Bigr{]}\Bigr{)}^{\ast}\Bigr{]}\,.\end{gathered}$
(7.30)

Since the dependency on $\lambda$ is just through $\mathrm{R}_{\pm}$, the
variables $\tau_{1}$ and $\tau_{2}$ turn out to be completely decoupled from
$\lambda$, and we can work out the respective integrals separately. The
integral in $\lambda$ encodes all the system details, whereas the integrals in
$\tau_{1}$ and $\tau_{2}$ depend only on the configuration of the subsystems,
i.e., on $\ell_{\pm}$ and $d$. For this reason, we call the former the
“physical part” and the latter the “configurational part”. A priori one could
expect this expansion to be effective only in the limit of small $\gamma$,
indeed $\mathrm{R}_{s}\rightarrow 0$ as $\gamma\rightarrow 0$ — cf. (6.17). It
turns out, instead, that such a perturbation theory is quickly convergent for
any value of $\gamma$, including the limit $\gamma\rightarrow 1$. In the
latter case, as discussed in the previous section, the tripartite information
is a function of the cross ratio $x_{4}$, therefore we have

$e^{(\alpha-1)G^{\gamma\rightarrow
1}_{\alpha}(x_{4})}=\frac{1}{2^{\alpha}}\sum_{\Sigma}\prod_{n=1}^{\infty}K^{(n,\Sigma)}_{\alpha}(\tfrac{\ell_{-}}{d},\tfrac{\ell_{+}}{d};\gamma\rightarrow
1)$ (7.31)

and we can decompose
$K^{(n,\Sigma)}_{\alpha}(\tfrac{\ell_{-}}{d},\tfrac{\ell_{+}}{d};\gamma\rightarrow
1)$ as the product of a finite numbers of terms, let us say $M_{n}$ (at the
first orders of the expansion, $M_{n}=1$), corresponding to independent
configurational parts, as follows

$K^{(n,\Sigma)}_{\alpha}(\tfrac{\ell_{-}}{d},\tfrac{\ell_{+}}{d};\gamma\rightarrow
1)=\prod_{j=1}^{M_{n}}[k_{n,j}(\tfrac{\ell_{-}}{d},\tfrac{\ell_{+}}{d})]^{\beta_{n,j}^{(\alpha)}(\Sigma)}\,.$
(7.32)

Here $\beta_{n,j}^{(\alpha)}$ are exponents characterising the physical part
and $k_{n,j}(\ell_{-},d,\ell_{+})$ the corresponding functions associated with
the configurational part. Since the exponents can be changed by changing
$\alpha$ and $\Sigma$, it is reasonable to expect that the dependency on $x$
is not peculiar to the limit $\gamma\rightarrow 1$, whereas we have

$\textsc{Conjecture:}\quad
k_{n,j}(\tfrac{\ell_{-}}{d},\tfrac{\ell_{+}}{d})\equiv k_{n,j}(x_{4})\,.$
(7.33)

The configurational part is however independent of the physical details, i.e.,
it does not change when $\gamma_{\pm}$ are finite. Consequently, the
tripartite information is expected to be a function of the cross ratio alone
in all the quench protocols investigated. In the next section we will check
the validity of this conjecture at the first orders of the perturbative
expansion.

##### First order (Abelian approximation).

At the first order we can neglect the commutator on the right hand side of
(7.26). Thus we have

$\displaystyle R^{(1)}_{s}(\tau_{1}-i\tfrac{\epsilon}{\ell_{s}},\tau_{2})=$
$\displaystyle\mathrm{R}_{s}$ (7.34)
$\displaystyle-\Sigma(\mathrm{R}_{-s}^{(1)}(-\tfrac{d}{\ell_{-s}}-\tfrac{\ell_{s}}{\ell_{-s}}\tau_{1}-i\tfrac{\epsilon}{\ell_{-s}},\tau_{2}))^{\dagger}\Sigma=$
$\displaystyle\Sigma\mathrm{R}_{s}\Sigma\,.$

The physical part of $K^{(1,\Sigma)}_{\alpha}$ is then captured by the
following (identical) integrals ($M_{1}=1$)

$\beta_{1,1}^{(\alpha)}(\Sigma)=\int^{\mathrm{R}_{0}}_{0}\mathrm{tr}[\mathrm{d}\mathrm{R}_{-}(\mathrm{R}_{-}-\Sigma\mathrm{R}_{-}\Sigma)]=\Bigl{(}\int^{-\mathrm{R}_{0}^{\dagger}}_{0}\mathrm{tr}[\mathrm{d}\mathrm{R}_{+}(\mathrm{R}_{+}-\Sigma\mathrm{R}_{+}\Sigma)]\Bigr{)}^{\ast}=\tfrac{1}{2}\mathrm{tr}[\mathrm{R}_{0}^{2}-(\Sigma\mathrm{R}_{0})^{2}]\,,$
(7.35)

where we didn’t specify the paths $C_{[0,\mathrm{R}_{0}]}$ and
$C_{[0,-\mathrm{R}_{0}^{\dagger}]}$ as in (7.30) because the integrals are
independent of the particular path.

Given that the two integrals of the physical part match each other, the
configurational part requires just the evaluation of the following integral

$\log
k_{1,1}=\iint_{0}^{1}\mathrm{d}^{2}\tau\tfrac{\ell_{-}}{\ell_{+}}\tfrac{1}{(\tau_{2}+\frac{\ell_{-}}{\ell_{+}}\tau_{1}+\frac{d}{\ell_{+}}-i\frac{\epsilon}{\ell_{+}})^{2}}+\tfrac{\ell_{+}}{\ell_{-}}\tfrac{1}{(\tau_{2}+\frac{\ell_{+}}{\ell_{-}}\tau_{1}+\frac{d}{\ell_{-}}+i\frac{\epsilon}{\ell_{-}})^{2}}=-2\log(1-x_{4})\,.$
(7.36)

Putting all together we recover the “Abelian approximation” that we worked out
in the previous section:

$e^{(\alpha-1)G_{\alpha}(x)}\approx\frac{1}{2^{\alpha}}\sum_{\Sigma}(1-x_{4})^{\mathrm{tr}[(\Sigma\mathrm{R}_{0})^{2}-\mathrm{R}_{0}^{2}]}\,.$
(7.37)

Identifying the Abelian approximation as the leading order of a perturbation
theory helps understand why such a simple approximation turns out to be so
good when compared with exact results, for example, in the CFT limit.

##### Second and third order.

At the second order in $\mathrm{R}$ we have

$\displaystyle\mathrm{R}_{s}^{(2)}(\tau_{1}-i\tfrac{\epsilon}{\ell_{s}},\tau_{2})=$
$\displaystyle\chi^{(s)}_{1}(\tau_{1},\tau_{2})i[\Sigma\mathrm{R}_{s}\Sigma,\mathrm{R}_{s}]$
(7.38)
$\displaystyle-\Sigma(\mathrm{R}_{-s}^{(2)}(-\tfrac{d}{\ell_{-s}}-\tfrac{\ell_{s}}{\ell_{-s}}\tau_{1}-i\tfrac{\epsilon}{\ell_{-s}},\tau_{2}))^{\dagger}\Sigma=$
$\displaystyle\chi^{(-s)\ast}_{1}(-\tfrac{d}{\ell_{-s}}-\tfrac{\ell_{s}}{\ell_{-s}}\tau_{1},\tau_{2})i[\Sigma\mathrm{R}_{s}\Sigma,\mathrm{R}_{s}]]$

where

$\chi_{1}^{(s)}(\tau_{1},\tau_{2})=\frac{\ell_{s}}{\ell_{-s}}\int_{\tau_{2}}^{\tau_{1}}\int_{0}^{\tau_{2}}\tfrac{\mathrm{d}\tau_{1}^{\prime}\mathrm{d}\tau_{2}^{\prime}}{(\tau_{2}^{\prime}+\frac{d}{\ell_{-s}}+\frac{\ell_{s}}{\ell_{-s}}\tau_{1}^{\prime}-i\tfrac{\epsilon}{\ell_{-s}})^{2}}=\\\
\log(d+\ell_{s}\tau_{1}-i\epsilon)+\log(d+(\ell_{s}+\ell_{-s})\tau_{2}-i\epsilon)-\log(d+\ell_{s}\tau_{2}-i\epsilon)-\log(d+\ell_{s}\tau_{1}+\ell_{-s}\tau_{2}-i\epsilon)\,.$
(7.39)

Since $\mathrm{tr}[\mathrm{d}\mathrm{R}[\mathrm{R},\mathrm{Y}]]=0$ for any
$\mathrm{Y}$, however, the physical part of $K^{(2,\Sigma)}_{\alpha}$ vanishes
($M_{2}=0$), therefore the leading correction to (7.37) is higher order. Let
us then consider the third order. We have

$\mathrm{R}_{s}^{(3)}(\tau_{1}-i\tfrac{\epsilon}{\ell},\tau_{2})=-\chi_{21}^{(s)}(\tau_{1},\tau_{2})[\Sigma\mathrm{R}_{s}\Sigma,[\Sigma\mathrm{R}_{s}\Sigma,\mathrm{R}_{s}]]-\chi_{22}^{(s)}(\tau_{1},\tau_{2})[\mathrm{R}_{s},[\mathrm{R}_{s},\Sigma\mathrm{R}_{s}\Sigma]]$
(7.40)
$-\Sigma(\mathrm{R}_{-s}^{(3)}(-\tfrac{d}{\ell_{-s}}-\tfrac{\ell_{s}}{\ell_{-s}}\tau_{1}-i\tfrac{\epsilon}{\ell_{-s}},\tau_{2}))^{\dagger}\Sigma=-\chi^{(-s)\ast}_{21}(-\tfrac{d}{\ell_{-s}}-\tfrac{\ell_{s}}{\ell_{-s}}\tau_{1},\tau_{2})[\mathrm{R}_{s},[\mathrm{R}_{s},\Sigma\mathrm{R}_{s}\Sigma]]\\\
-\chi^{(-s)\ast}_{22}(-\tfrac{d}{\ell_{-s}}-\tfrac{\ell_{s}}{\ell_{-s}}\tau_{1},\tau_{2})[\Sigma\mathrm{R}_{s}\Sigma,[\Sigma\mathrm{R}_{s}\Sigma,\mathrm{R}_{s}]]\,,$
(7.41)

where

$\displaystyle\chi_{21}^{(s)}(\tau_{1},\tau_{2})=$
$\displaystyle\tfrac{\ell_{s}}{\ell_{-s}}\int_{\tau_{2}}^{\tau_{1}}\mathrm{d}\tau_{1}^{\prime}\int_{0}^{\tau_{2}}\mathrm{d}\tau_{2}^{\prime}\tfrac{\chi^{(s)}_{1}(\tau_{1}^{\prime},\tau_{2})}{(\tau_{2}^{\prime}+\frac{d}{\ell_{-s}}+\frac{\ell_{s}}{\ell_{-s}}\tau_{1}^{\prime}-i\frac{\epsilon}{\ell_{-s}})^{2}}$
(7.42) $\displaystyle\chi_{22}^{(s)}(\tau_{1},\tau_{2})=$
$\displaystyle\int_{\tau_{2}}^{\tau_{1}}\mathrm{d}\tau_{1}^{\prime}\int_{0}^{\tau_{2}}\mathrm{d}\tau_{2}^{\prime}\Bigl{[}\tfrac{\chi^{(s)}_{1}(\tau_{1}^{\prime},\tau_{2}^{\prime})-\chi^{(s)}_{1}(\tau_{1}^{\prime},\tau_{2})}{(\tau_{2}^{\prime}-\tau_{1}^{\prime}+i\frac{\epsilon}{\ell_{s}})^{2}}+\tfrac{\ell_{s}}{\ell_{-s}}\tfrac{\chi^{(-s)\ast}_{1}(-\frac{d}{\ell_{-s}}-\frac{\ell_{s}}{\ell_{-s}}\tau_{1}^{\prime},\tau_{2}^{\prime})}{(\tau_{2}^{\prime}+\frac{d}{\ell_{-s}}+\frac{\ell_{s}}{\ell_{-s}}\tau_{1}^{\prime}-i\frac{\epsilon}{\ell_{-s}})^{2}}\Bigr{]}$

Concerning the physical part of $K^{(3,\Sigma)}_{\alpha}(x_{4})$, this is
captured again by two identical integrals ($M_{3}=1$):

$\beta_{3,1}^{(\alpha)}(\Sigma)=-\int_{0}^{R_{0}}\mathrm{tr}[\mathrm{d}\mathrm{R}_{-}[\Sigma\mathrm{R}_{-}\Sigma,[\Sigma\mathrm{R}_{-}\Sigma,\mathrm{R}_{-}]]]=-\Bigl{(}\int_{0}^{-R_{0}^{\dagger}}\mathrm{tr}[\mathrm{d}\mathrm{R}_{+}[\Sigma\mathrm{R}_{+}\Sigma,[\Sigma\mathrm{R}_{+}\Sigma,\mathrm{R}_{+}]]]\Bigr{)}^{\ast}\\\
-\frac{1}{2}\mathrm{tr}[(\Sigma R_{0}^{2})^{2}-(\Sigma
R_{0})^{4}]\equiv\frac{1}{4}\mathrm{tr}[([\Sigma\mathrm{R}_{0}\Sigma,\mathrm{R}_{0}])^{2}]\,,$
(7.43)

where, as before, we omitted the paths because the integrals do not depend on
it. As manifest in the rightmost expression, this terms capture a purely non-
Abelian contribution. As before, given that the two integrals of the physical
part match each other, the configurational part is characterised by a single
integral

$\log
k_{3,1}=-\iint_{0}^{1}\mathrm{d}^{2}\tau\tfrac{\ell_{-}}{\ell_{+}}\tfrac{\chi^{(+)\ast}_{22}(-\frac{d}{\ell_{+}}-\frac{\ell_{-}}{\ell_{+}}\tau_{1},\tau_{2})}{(\tau_{2}+\frac{\ell_{-}}{\ell_{+}}\tau_{1}+\frac{d}{\ell_{+}}-i\frac{\epsilon}{\ell_{+}})^{2}}+\tfrac{\chi_{21}^{(-)}(\tau_{1},\tau_{2})}{(\tau_{1}-\tau_{2}-i\frac{\epsilon}{\ell_{-}})^{2}}+\Bigl{(}\tfrac{\ell_{+}}{\ell_{-}}\tfrac{\chi^{(-)\ast}_{22}(-\frac{d}{\ell_{-}}-\frac{\ell_{+}}{\ell_{-}}\tau_{1},\tau_{2})}{(\tau_{2}+\frac{\ell_{+}}{\ell_{-}}\tau_{1}+\frac{d}{\ell_{-}}-i\frac{\epsilon}{\ell_{-}})^{2}}+\tfrac{\chi_{21}^{(+)}(\tau_{1},\tau_{2})}{(\tau_{1}-\tau_{2}-i\frac{\epsilon}{\ell_{+}})^{2}}\Bigr{)}^{\ast}\,.$
(7.44)

Despite this could be calculated analytically, the result is cumbersome and we
have not been able to simplify it so as to exhibit it here. We have not even
found a satisfactory way to make it manifest that it depends only on the cross
ratio. We have however made some simplifications that allow for an efficient
numerical evaluation. Specifically, the integrals above can be reduced to the
following:

$\iint_{0}^{1}\mathrm{d}^{2}\tau\Bigl{(}\tfrac{\chi_{21}^{(s)}(\tau_{1},\tau_{2})}{(\tau_{1}-\tau_{2}-i\frac{\epsilon}{\ell_{s}})^{2}}+\tfrac{\ell_{s}}{\ell_{-s}}\tfrac{\chi^{(-s)\ast}_{22}(-\frac{d}{\ell_{-s}}-\frac{\ell_{s}}{\ell_{-s}}\tau_{1},\tau_{2})}{(\tau_{2}+\frac{\ell_{s}}{\ell_{-s}}\tau_{1}+\frac{d}{\ell_{-s}}-i\frac{\epsilon}{\ell_{-s}})^{2}}\Bigr{)}\xrightarrow{\epsilon\rightarrow
0^{+}}\\\
\tfrac{\ell_{s}}{\ell_{-s}}\iint_{0}^{1}\mathrm{d}^{2}\tau\int_{\tau_{2}}^{\tau_{1}}\mathrm{d}\tau_{1}^{\prime}\int_{0}^{\tau_{2}}\mathrm{d}\tau_{2}^{\prime}\tfrac{\log\left(\frac{(d+\ell_{s}\tau_{1}^{\prime})(d+(\ell_{s}+\ell_{-s})\tau_{2})}{(d+\ell_{s}\tau_{2})(d+\ell_{s}\tau_{1}^{\prime}+\ell_{-s}\tau_{2})}\right)}{(\tau_{1}-\tau_{2})^{2}(\tau_{2}^{\prime}+\frac{d}{\ell_{-s}}+\frac{\ell_{s}}{\ell_{-s}}\tau_{1}^{\prime})^{2}}\\\
-\int_{0}^{\ell_{-s}}\mathrm{d}\tau_{2}^{\prime}\int_{-d-\ell_{s}}^{\ell_{-s}}\mathrm{d}\tau_{1}^{\prime}\tfrac{\left[\log\Bigl{(}\frac{d+(\frac{\ell_{s}}{\ell_{-s}}+1)\tau_{2}^{\prime}}{d+\tau_{2}^{\prime}}\Bigr{)}-\log(d+\tau_{1}^{\prime}+\frac{\ell_{s}}{\ell_{-s}}\tau_{2}^{\prime}+i\epsilon)\right]\log\left(\frac{(\max(0,-d-\tau_{1}^{\prime})+d+\ell_{-s})(\max(\tau_{1}^{\prime},\tau_{2}^{\prime})+d+\ell_{s})}{(d+\ell_{-s}+\ell_{s})(\max(0,-d-\tau_{1}^{\prime})+\max(\tau_{1}^{\prime},\tau_{2}^{\prime})+d)}\right)}{(\tau_{2}^{\prime}-\tau_{1}^{\prime}-i\epsilon)^{2}}\\\
-\int_{0}^{\ell_{s}}\mathrm{d}\tau_{1}\int_{0}^{\ell_{-s}}\mathrm{d}\tau_{2}\tfrac{\log\Bigl{(}\frac{(d+\tau_{2})}{(d+(\frac{\ell_{s}}{\ell_{-s}}+1)\tau_{2})}\Bigr{)}\Bigl{[}\log(i\epsilon)-\log(\tau_{2}+i\epsilon)+\log\Bigl{(}\frac{d+\tau_{1}}{d+\tau_{1}+\tau_{2}}\Bigr{)}\Bigr{]}}{(\tau_{2}+\tau_{1}+d)^{2}}\\\
-\int_{0}^{\ell_{-s}}\mathrm{d}\tau_{2}\int_{-d-\ell_{s}}^{\tau_{2}}\mathrm{d}\tau_{1}^{\prime}\tfrac{\tau_{2}(\ell_{s}-\max(0,-d-\tau_{1}^{\prime}))\log\left(d+\tau_{1}^{\prime}+\frac{\ell_{s}}{\ell_{-s}}\tau_{2}+i\epsilon\right)}{(\tau_{1}^{\prime}+i\epsilon)(\tau_{1}^{\prime}-\tau_{2}+i\epsilon)(d+\ell_{s}+\tau_{2})(\max(0,-d-\tau_{1}^{\prime})+d+\tau_{2})}\\\
-\tfrac{\ell_{s}}{\ell_{-s}}\int_{0}^{\ell_{-s}}\\!\\!\\!\\!\\!\\!\mathrm{d}\tau_{2}^{\prime}\int_{-d-\ell_{s}}^{\ell_{-s}}\\!\\!\\!\\!\\!\\!\mathrm{d}\tau_{1}^{\prime}\tfrac{\Bigl{[}\log(\tau_{1}^{\prime}+i\epsilon)-\log(\tau_{1}^{\prime}-\tau_{2}^{\prime}+i\epsilon)+\log\Bigl{(}\frac{d+(\frac{\ell_{s}}{\ell_{-s}}+1)\tau_{2}^{\prime}}{d+\frac{\ell_{s}}{\ell_{-s}}\tau_{2}^{\prime}}\Bigr{)}\Bigr{]}\log\Bigl{(}\frac{(\max(0,-d-\tau_{1}^{\prime})+d+\ell_{-s})(\max(\tau_{1}^{\prime},\tau_{2}^{\prime})+d+\ell_{s})}{(d+\ell_{-s}+\ell_{s})(\max(0,-d-\tau_{1}^{\prime})+\max(\tau_{1}^{\prime},\tau_{2}^{\prime})+d)}\Bigr{)}}{(\frac{\ell_{s}}{\ell_{-s}}\tau_{2}^{\prime}+d+\tau_{1}^{\prime}+i\epsilon)^{2}}$
(7.45)

The numerical evaluation of the integrals above is consistent with $\log
k_{3,1}$ being a function of just the cross ratio $x_{4}$, i.e.,
$k_{3,1}=k_{3,1}(x_{4})$. Figure 5 shows a plot of $k_{3,1}(x_{4})$.
Importantly and contrary to $k_{1,1}(x_{4})$, $k_{3,1}(x_{4})$ is smaller than
$1$ and approaches zero as $x_{4}\rightarrow 1$. Since also
$\beta_{3,1}^{(\alpha)}(\Sigma)$ is negative, the third order correction tends
to increase the tripartite information. Nevertheless, such a correction is not
strong enough to affect the residual tripartite information, which remains
$-\log 2$ also at this perturbative level.

##### Inclusion of the leading non-Abelian correction.

The leading non-Abelian correction is characterised by the function
$k_{3,1}(x_{4})$ defined in (7.44) and the corresponding exponent (7.43). We
remedy not having provided an explicit expression for $k_{3,1}(x_{4})$ by
replacing $k_{3,1}(x_{4})$ with a function $\tilde{k}_{3,1}(x_{4})$ that is
equivalent to it up to higher order corrections. We can indeed use the freedom
in the choice of $\tilde{k}_{3,1}(x_{4})$ to exhibit an explicit expression.
To that aim, we exploit the knowledge of the exact result in the limit
$|\gamma_{\pm}|\rightarrow 1$ with the observation that truncating the
perturbative series at the third order provides an excellent approximation.
The function that replaces $k_{3,1}(x_{4})$ so as to make the pseudo-conformal
limit $|\gamma_{\pm}|\rightarrow 1$ exact is indeed a captivating candidate
for $\tilde{k}_{3,1}(x_{4})$. We stress that, in this, we give up the
independence of $k_{3,1}(x_{4})$ from $\Sigma$ and $\alpha$. For $\alpha=3$,
this results in the following replacement

$k_{3,1}(x_{4})\rightarrow\tilde{k}_{3,1}(x_{4})=(1-x_{4})^{54}\Bigl{[}\tfrac{\Theta(0,0|\hat{\tau}_{x_{4}}^{(3)})}{\Theta(0,\frac{1}{2}|\hat{\tau}_{x_{4}}^{(3)})}\Bigr{]}^{\frac{729}{4}}\,,$
(7.46)

The corresponding estimation of the Rényi-$3$ tripartite information reads

$\tilde{G}_{3}(x_{4})=-\log
2+\tfrac{1}{2}\log\Bigl{(}1+3(1-x_{4})^{-2\sum_{k_{F}}\beta_{1}(\gamma_{\pm}^{(k_{F})})+54\sum_{k_{F}}\beta_{3}(\gamma_{\pm}^{(k_{F})})}\Bigl{[}\tfrac{\Theta(0,0|\hat{\tau}_{x_{4}}^{(3)})}{\Theta(0,\frac{1}{2}|\hat{\tau}_{x_{4}}^{(3)})}\Bigr{]}^{\frac{729}{4}\sum_{k_{F}}\beta_{3}(\gamma_{\pm}^{(k_{F})})}\Bigr{)}\,,$
(7.47)

where the exponents are explicitly given by — cf. Eqs (7.35) and (7.44)

$\displaystyle\beta_{1}(\gamma_{\pm})=$
$\displaystyle\tfrac{\log^{2}\left(\frac{\sqrt{3}\gamma_{+}-i}{\sqrt{3}\gamma_{-}-i}\right)+\log^{2}\left(\frac{\sqrt{3}\gamma_{+}+i}{\sqrt{3}\gamma_{-}+i}\right)-\log\left(\frac{\sqrt{3}\gamma_{+}+i}{\sqrt{3}\gamma_{-}+i}\right)\log\left(\frac{\sqrt{3}\gamma_{+}-i}{\sqrt{3}\gamma_{-}-i}\right)}{9\pi^{2}}$
(7.48) $\displaystyle\beta_{3}(\gamma_{\pm})=$
$\displaystyle-\tfrac{1}{4}\beta_{1}^{2}(\gamma_{\pm})\,.$

Note that, having used the result for $\gamma_{\pm}\rightarrow 1$, this
approximation is rougher for small $\gamma_{\pm}$, where it is expected to
overestimate the correction to the Abelian approximation.

We can now check that the residual tripartite information does not depend on
$\gamma_{\pm}$. To that aim, we should expand the prediction above around