$x_{4}=1$. Since we have Calabrese2011Entanglement

$\tfrac{\Theta(0,0|\hat{\tau}_{x_{4}}^{(3)})}{\Theta(0,\frac{1}{2}|\hat{\tau}_{x_{4}}^{(3)})}\sim\frac{3^{3/4}}{2}(1-x_{4})^{-\frac{1}{4}}$
(7.49)

we find

$\tilde{G}_{3}(x_{4})+\log
2\sim\tfrac{3}{2}(\tfrac{27}{16})^{-\frac{729}{64}\sum_{k_{F}}\beta_{1}^{2}(\gamma_{\pm}^{(k_{F})})}(1-x_{4})^{-2\sum_{k_{F}}\beta_{1}(\gamma_{\pm}^{(k_{F})})-\frac{135}{64}\sum_{k_{F}}\beta_{1}^{2}(\gamma_{\pm}^{(k_{F})})}\,$
(7.50)

where we used that, for any $\gamma_{\pm}\neq 0$, the exponent of $(1-x_{4})$
is strictly positive ($-\frac{128}{135}<\beta_{1}(\gamma_{\pm})<0$). We point
out that such an estimate is perturbative and, in principle, is subject to
corrections coming from higher order contributions.

##  8 Conclusion

In this work we have investigated the tripartite information in the stationary
state emerging after global quenches from two types of initial states: ground
states of critical systems and inhomogeneous states obtained by joining two
macroscopically different parts, e.g., a domain wall. We have identified the
quantum field theories capturing the limit of large subsystem’s lengths and
derived analytic expressions for the Rényi-$\alpha$ tripartite information.
Specifically, using a mapping to a Riemann-Hilbert problem with a piecewise
constant matrix for a doubly connected domain, we have proved the conjecture
of Ref. Maric2022Universality  for the asymptotic behaviour of the Rényi-$2$
tripartite information. We have also worked out an implicit expression for the
Rényi-$\alpha$ tripartite information, eq. (7.21), which we have then
unravelled within a rapidly convergent perturbation theory. By keeping just
the first three orders (where, in fact, the second order vanishes) of the
expansion, we have obtained predictions that, in all quench settings
investigated, approximate the Rényi-$\alpha$ tripartite information with an
accuracy that easily surpasses the limitations of our numerical simulations.

Our findings have confirmed the conjectures of Ref Maric2022Universality :
first, the Rényi-$\alpha$ tripartite information approaches a function of the
cross ratio, and, second, the limit in which the length of the second block
$B$ becomes negligible with respect to that of $A$ and $C$ (but still much
larger than the lattice spacing) is $-\log 2$ for any Rényi index $\alpha$,
confirming the emergence of a negative residual tripartite information. The
vanishing of the residual tripartite information that is expected in thermal
states (cf. sec. 2) suggests that a nonzero value is always associated with
non-thermal stationary properties. In translationally invariant global
quenches, this is generally expected only in systems with infinitely many
conservation laws, which are usually integrable. Indeed, in one dimension,
only with infinitely many local integrals of motion the system can retain
memory of algebraically decaying correlations. In bipartitioning protocols, we
do not have a comparably strong argument to exclude the emergence of a nonzero
residual entropy in generic systems.

This work leaves several open questions that range from improving our results
to generalising them. If we overlook the powerful perturbation theory that we
extracted from it, the exact implicit expression that we derived by formally
solving the Riemann-Hilbert problem (for $\alpha>2$) is unsatisfactory: it
does not make it manifest neither the dependency on the cross ratio nor the
limit of the residual tripartite information. An explicit solution could then
be desirable, both for confirming that our conjectures are exact and not just
excellent approximations and for establishing an exact connection with the
known CFT results.

Second, we have assessed the universality of the residual tripartite
information only in noninteracting models, and it is not even clear to us
whether or not the residual tripartite information could take only discrete
values (in our systems, $0$ or $-\log 2$). In this respect, generalising our
findings to interacting integrable spin chains is not straightforward, but
perhaps not impossible. For example, bosonisation could help in mapping the
problem to a noninteracting one, in which the entropies are expressed in terms
of just a correlation operator; carefully taking into account the nonlocality
of the Jordan-Wigner transformation might then allow one to carry out the
analogous calculation.

In addition, we have investigated only models that do not allow for string
order at infinite times after the quench. There are however spin-chain systems
in which symmetry protected topological order can emerge even after a global
quench Fagotti2022Global ; FagottiMaricZadnik2022 . Considering that, in 2D,
the tripartite information is sensitive to topological order, the question of
whether the tripartite information could display some peculiar features in
such systems is of central importance and still open.

Another point that we overlooked is the actual relaxation dynamics: by
assuming a description through a stationary state we have ignored that finite-
time effects could be significant. With Ref. Bocini2022Connected  in mind, we
wonder whether some of those corrections could be included by taking into
account how correlations change when the lengths of the subsystems are large
enough not to allow us to ignore the inhomogeneity of the state.

Finally, two natural generalisations of our work are the analysis of the
tripartite information of detached subsystems, as well as of entanglement
measures such as the negativity, which could also help clarify whether the
residual tripartite information is a merely classical quantity or if instead
it witnesses some exceptional entanglement structure.

##  Appendix A Generalized XY model

A generalisation of the quantum XY model Lieb1961  was proposed in Ref.
Suzuki1971The  and is described by the Hamiltonian

$\bm{\mathbf{H}}=\sum_{\ell}\left(\sum_{\gamma_{1},\gamma_{2}\in\\{x,y\\}}\sum_{n=1}^{\infty}J_{\gamma_{1},\gamma_{2}}^{(n)}\bm{\mathbf{\sigma}}^{\gamma_{1}}_{\ell}\left(\bm{\mathbf{\prod}}_{j=1}^{n-1}\bm{\mathbf{\sigma}}_{\ell+j}^{z}\right)\bm{\mathbf{\sigma}}_{\ell+n}^{\gamma_{2}}+h\bm{\mathbf{\sigma}}_{\ell}^{z}\right)\,.$
(A.1)

The common denominator of those models is that a Jordan-Wigner transformation
maps the Hamiltonian into a quadratic form of fermions. The XY model
considered in (3.3) corresponds to the special case
$J_{\gamma_{1},\gamma_{2}}^{(k)}=\delta_{k,1}\delta_{\gamma_{1},\gamma_{2}}\left(\delta_{\gamma_{1},x}J_{x}+\delta_{\gamma_{1},y}J_{y}\right)$.
In this work we consider only local Hamiltonians, so we assume
$J^{(n)}_{\gamma_{1},\gamma_{2}}=0$ for any $n$ larger than some given
integer. The symbol of (A.1), as defined in (3.4), reads

$\displaystyle\mathcal{H}(k)=f_{0}(k)I+f_{1}(k)\sigma^{x}+f_{2}(k)\sigma^{y}+f_{3}(k)\sigma^{z},$
(A.2) $\displaystyle
f_{0}(k)=2\sum_{n=1}^{\infty}\left(J^{(n)}_{x,y}-J^{(n)}_{y,x}\right)\sin(nk),$
(A.3) $\displaystyle
f_{1}(k)=2\sum_{n=1}^{\infty}\left(J^{(n)}_{x,x}-J^{(n)}_{y,y}\right)\sin(nk),$
(A.4) $\displaystyle
f_{2}(k)=2h-2\sum_{n=1}^{\infty}\left(J^{(n)}_{x,x}+J^{(n)}_{y,y}\right)\cos(nk),$
(A.5) $\displaystyle
f_{3}(k)=-2\sum_{n=1}^{\infty}\left(J^{(n)}_{x,y}+J^{(n)}_{y,x}\right)\sin(nk).$
(A.6)

The excitation energies $\varepsilon(k)$ can be identified with the
eigenvalues of $\mathcal{H}(k)$, i.e.,
$\varepsilon(k)=f_{0}(k)+\sqrt{\sum_{j=1}^{3}f_{j}^{2}(k)}$. We note that
reflection symmetric Hamiltonians satisfy $f_{0}(k)=0$.

##  Appendix B Determinant representation

### Integrating the fields out

The partition function corresponding to the action (5.30) with the boundary
conditions (5.37) can be rewritten as a partition function in which the fields
are continuous at the boundaries provided that the following term is added to
the action

$\delta
S_{E}^{(\alpha)}(k_{F};\Sigma)=\int_{A}\mathrm{d}x\sum_{\ell,n}^{\alpha}(\delta_{\ell
n}-[\Sigma T\Sigma]_{\ell n})\vec{\psi}^{(\ell){\dagger}}(x,0^{+})\vec{u}\
\vec{u}^{\dagger}\vec{\psi}^{(n)}(x,0^{-})+\\\
\int_{C}\mathrm{d}x\sum_{\ell,n}^{\alpha}(\delta_{\ell n}-T_{\ell
n})\vec{\psi}^{(\ell){\dagger}}(x,0^{+})\vec{u}\
\vec{u}^{\dagger}\vec{\psi}^{(n)}(x,0^{-})\,.$ (B.1)

Here $\vec{u}$ is the normalised vector defined in (5.28). Indeed, this new
action leads to the same classical solutions and boundary conditions as the
original problem. Being now the dependency on $\Sigma$ explicit, we have eased
the notations and replaced $\vec{\psi}^{(n)}(x,\tau;\Sigma)$ by
$\vec{\psi}^{(n)}(x,\tau)$.

We now linearise $\delta S_{E}^{(\alpha)}(k_{F};\Sigma)$. As shown below, this
can be done by introducing an appropriate number of auxiliary Grassmann
fields. Before doing it, however, we should take care of a subtlety. Since the
eigenvalues $-z$ of $T$ satisfy $z^{\alpha}=-1$ (cf. (5.33)), the kernel of
$\mathrm{I}-T$ is nonempty when $\alpha$ is odd. For $\alpha$ even we could
linearise $\delta S_{E}^{(\alpha)}(k_{F};\Sigma)$ by introducing $2\alpha$
auxiliary fields, half of them defined in $A$ and the rest in $C$ (see below);
for $\alpha$ odd, however, we would need only $2\alpha-2$ auxiliary fields. We
choose here to overcome this discrepancy between even and odd $\alpha$ by
introducing a small regulator $\epsilon$ and replacing $I-T$ by
$(1+\epsilon)I-T$. In this way the kernel is non-empty even when $\alpha$ is
odd and we can treat the even and odd case on the same footing. The regulator
$\epsilon$ will be sent to zero at the end of the calculation. For the
reader’s information, we mention that the calculation could be effectively
carried out also without introducing $\epsilon$, with the advantage of
slightly reducing the size of the matrices appearing in the final result for
$\alpha$ odd, at the cost however of having to make a distinction between even
and odd $\alpha$. That said, it is convenient to rewrite $\delta
S_{E}^{(\alpha)}(k_{F};\Sigma)$ in a basis diagonalising $T$:

$\int_{A}\mathrm{d}x\sum_{z|z^{\alpha}=-1}\Bigl{[}\frac{1}{\sqrt{\alpha}}\sum_{\ell=1}^{\alpha}\Sigma_{\ell\ell}(-z)^{\ell}\vec{\psi}^{(\ell){\dagger}}(x,0^{+})\vec{u}\Bigr{]}(1+\epsilon+z)\Bigl{[}\frac{1}{\sqrt{\alpha}}\sum_{n=1}^{\alpha}\Sigma_{nn}(-1/z)^{n}\vec{u}^{\dagger}\vec{\psi}^{(n)}(x,0^{-})\Bigr{]}\\\
+\int_{C}\mathrm{d}x\sum_{z|z^{\alpha}=-1}\Bigl{[}\frac{1}{\sqrt{\alpha}}\sum_{\ell=1}^{\alpha}(-z)^{\ell}\vec{\psi}^{(\ell){\dagger}}(x,0^{+})\vec{u}\Bigr{]}(1+\epsilon+z)\Bigl{[}\frac{1}{\sqrt{\alpha}}\sum_{n=1}^{\alpha}(-1/z)^{n}\vec{u}^{\dagger}\vec{\psi}^{(n)}(x,0^{-})\Bigr{]}$
(B.2)

where we used

$\bigl{[}\mathrm{(}1+\epsilon)I-T[\\{0,\ldots,0\\}]\bigr{]}_{\ell
n}=\frac{1}{\alpha}\sum_{z|z^{\alpha}=-1}(1+\epsilon+z)(-z)^{\ell-n}\,.$ (B.3)

Let us then introduce the fields $\mathcal{A}(x;z)$ and $\mathcal{C}(x;z)$
with action

$\delta
S_{E}^{(\alpha)}[\mathcal{A},\mathcal{C}]=-f_{\alpha}\sum_{z|z^{\alpha}=-1}\left[\int_{A}\mathrm{d}x\mathcal{A}^{\dagger}(x;z)\frac{1}{1+\epsilon+z}\mathcal{A}(x;z)+\int_{C}\mathrm{d}x\mathcal{C}^{\dagger}(x;z)\frac{1}{1+\epsilon+z}\mathcal{C}(x;z)\right]$
(B.4)

where the constant

$f_{\alpha}\equiv\Big{[}\prod_{z|z^{\alpha}=-1}(1+\epsilon+z)\Big{]}^{1/\alpha}$
(B.5)

has been chosen in such a way that the total partition function is not
affected by the addition of the new fields, indeed

$\idotsint\mathcal{D}\mathcal{A}^{\dagger}\mathcal{D}\mathcal{A}\mathcal{D}\mathcal{C}^{\dagger}\mathcal{D}\mathcal{C}e^{-\delta
S_{E}^{(\alpha)}[\mathcal{A},\mathcal{C}]}=1\,.$ (B.6)

The term $\delta S_{E}^{(\alpha)}(k_{F};\Sigma)$ incorporating the boundary
conditions can now be linearised by shifting the auxiliary fields
$\mathcal{A}$ and $\mathcal{C}$. This results in the comprehensive action

$S_{E}^{(\alpha)}=\int\mathrm{d}\tau\int\mathrm{d}x\
\sum_{n=1}^{\alpha}\Bigl{[}\vec{\psi}^{(n){\dagger}}(x,\tau)\begin{pmatrix}\partial_{\tau}-i\partial_{x}&0&0\\\
0&\partial_{\tau}+i\partial_{x}&0\\\
0&0&\partial_{\tau}+\sigma\end{pmatrix}\vec{\psi}^{(n)}(x,\tau)\\\
-f_{\alpha}\sum_{z|z^{\alpha}=-1}\left[\int_{A}\mathrm{d}x\mathcal{A}^{\dagger}(x;z)\frac{1}{1+\epsilon+z}\mathcal{A}(x;z)+\int_{C}\mathrm{d}x\mathcal{C}^{\dagger}(x;z)\frac{1}{1+\epsilon+z}\mathcal{C}(x;z)\right]\\\
+\sqrt{\frac{f_{\alpha}}{\alpha}}\sum_{z|z^{\alpha}=-1}\int_{A}\mathrm{d}x\sum_{n=1}^{\alpha}\Bigl{[}\mathcal{A}^{\dagger}(x;z)\Sigma_{nn}(-\tfrac{1}{z})^{n}\vec{u}^{\dagger}\vec{\psi}^{(n)}(x,0^{-})+\vec{\psi}^{(n){\dagger}}(x,0^{+})\vec{u}\Sigma_{nn}(-z)^{n}\mathcal{A}(x;z)\Bigr{]}\\\
+\sqrt{\frac{f_{\alpha}}{\alpha}}\sum_{z|z^{\alpha}=-1}\int_{C}\mathrm{d}x\sum_{n=1}^{\alpha}\Bigl{[}\mathcal{C}^{\dagger}(x;z)(-\tfrac{1}{z})^{n}\vec{u}^{\dagger}\vec{\psi}^{(n)}(x,0^{-})+\vec{\psi}^{(n){\dagger}}(x,0^{+})\vec{u}(-z)^{n}\mathcal{C}(x;z)\Bigr{]}$
(B.7)

The third step is to integrate over $\vec{\psi}^{(n){\dagger}}$ and
$\vec{\psi}^{(n)}$. This produces the normalization factor associated with the
path integral representation of $\mathrm{tr}[\rho_{A\cup C}]^{\alpha}$
multiplied by the partition function of the theory with Euclidean action

$S_{E}=-f_{\alpha}\sum_{z|z^{\alpha}=-1}\left[\int_{A}\mathrm{d}x\mathcal{A}^{\dagger}(x;z)\frac{1}{1+\epsilon+z}\mathcal{A}(x;z)+\int_{C}\mathrm{d}x\mathcal{C}^{\dagger}(x;z)\frac{1}{1+\epsilon+z}\mathcal{C}(x;z)\right]\\\
+\frac{f_{\alpha}}{\alpha}\int\mathrm{d}x\sum_{n=1}^{\alpha}\sum_{z|z^{\alpha}=-1}\Bigl{\\{}\int_{A}\mathrm{d}x^{\prime}\mathcal{A}^{\dagger}(x^{\prime},z)\Sigma_{nn}(-\tfrac{1}{z})^{n}+\int_{C}\mathrm{d}x^{\prime}\mathcal{C}^{\dagger}(x^{\prime},z)(-\tfrac{1}{z})^{n}\Bigr{\\}}\vec{u}^{\dagger}\braket{\vec{\psi}^{(n)}(x^{\prime})\vec{\psi}^{(n)\dagger}(x)}\vec{u}\\\
\sum_{w|w^{\alpha}=-1}\Bigl{\\{}\chi_{A}(x)\Sigma_{nn}(-w)^{n}\mathcal{A}(x;w)+\chi_{C}(x)(-w)^{n}\mathcal{C}(x;w)\Bigr{\\}}$
(B.8)

The correlation functions
$\braket{\vec{\psi}^{(n)}(x)\vec{\psi}^{(n)\dagger}(y)}$ are independent of
$n$ and from (5.28) it follows

$\vec{u}^{\dagger}\braket{\vec{\psi}^{(n)}(x)\vec{\psi}^{(n)\dagger}(y)}\vec{u}=\braket{\bm{\mathbf{\psi}}(x)\bm{\mathbf{\psi}}^{\dagger}(y)},$
(B.9)

where the right hand side is given by (5.9).

Finally, the partition function (3.12) is the determinant of the operator
$\bm{\mathbf{L}}_{\underline{\sigma}}$ coupling the fields in $S_{E}$, i.e.

$\\{\Gamma_{j_{1}},\ldots,\Gamma_{j_{\alpha}}\\}=\det\bm{\mathbf{L}}_{\underline{\sigma}}.$
(B.10)

Defining the operator $\bm{\mathbf{C}}$ by

$\bm{\mathbf{C}}(x,y)=\braket{\bm{\mathbf{\psi}}(x)\bm{\mathbf{\psi}}^{\dagger}(y)},\quad
x,y\in A\cup C,$ (B.11)

where the right-hand side is given by (5.9), gives

$\displaystyle\bm{\mathbf{L}}_{\underline{\sigma}}(z,x;w,y)\sim$
$\displaystyle\begin{cases}f_{\alpha}\delta_{zw}\Bigl{[}\frac{\delta(x-y)}{1+\epsilon+z}-\bm{\mathbf{C}}(x,y)\Bigr{]}&x,y\in
A\bigvee x,y\in C\\\
-\frac{f_{\alpha}}{\alpha}\sum_{n=1}^{\alpha}\sigma_{n}(\tfrac{w}{z})^{n}\bm{\mathbf{C}}(x,y)&x\in
A,y\in C\bigvee x\in C,y\in A\end{cases}$ (B.12) $\displaystyle
z^{\alpha}=w^{\alpha}=-1.$

Here we used the symbol $\sim$ rather than $=$ because
$\bm{\mathbf{L}}_{\underline{\sigma}}(z,x;w,y)$ can be replaced by any other
operator with the same determinant. More compactly, we have

$\bm{\mathbf{L}}_{\underline{\sigma}}\sim
f_{\alpha}\left(\mathrm{D}\otimes\bm{\mathbf{1}}-\mathrm{I}\otimes\bm{\mathbf{C}}+\frac{\mathrm{I}-\mathrm{M}_{\underline{\sigma}}}{2}\otimes(\bm{\mathbf{C}}-\bm{\mathbf{P}}\bm{\mathbf{C}}\bm{\mathbf{P}})\right)$
(B.13)

where

$\displaystyle{}_{zw}=$ $\displaystyle\frac{1}{1+\epsilon+z}\delta_{zw}$
(B.14) $\displaystyle[\mathrm{M}_{\underline{\sigma}}]_{zw}=$
$\displaystyle\frac{1}{\alpha}\sum_{n=1}^{\alpha}\sigma_{n}(\tfrac{w}{z})^{n}$

and $\bm{\mathbf{P}}$ is defined in (5.34).

### Standard form

To relate the calculation of the Rényi entropies to the solution of a Riemann-
Hilbert problem with a piecewise constant matrix it is convenient to represent
$\bm{\mathbf{L}}_{\underline{\sigma}}$ by an operator with a more standard
form, specifically, an operator in which $\bm{\mathbf{P}}$ acts only on the
left of $\bm{\mathbf{C}}$. As shown before long, bringing $\bm{\mathbf{P}}$ to
the left of $\bm{\mathbf{C}}$ can be readily achieved by increasing the size
of the representation of $\bm{\mathbf{L}}_{\underline{\sigma}}$. Since,
however, we would also like to eventually reduce the space back to (almost)
its original size, it is firstly convenient to diagonalise
$\mathrm{M}_{\underline{\sigma}}$. This is straightforward, indeed, the matrix
is circulant and the eigenvalues are equal to $\sigma_{j}$. We call
$U_{\underline{\sigma}}$ a unitary matrix that diagonalises
$\mathrm{M}_{\underline{\sigma}}$. Specifically, we choose the unitary
transformation $\mathrm{U}_{\underline{\sigma}}$ that brings
$\mathrm{M}_{\underline{\sigma}}$ into $\Sigma$:

$\mathrm{M}_{\underline{\sigma}}=\mathrm{U}_{\underline{\sigma}}\Sigma\mathrm{U}^{\dagger}_{\underline{\sigma}}$
(B.15)

where $j_{n}\in\\{1,\ldots,\alpha\\}$. The change of basis results in

$\bm{\mathbf{L}}_{\underline{\sigma}}\sim
f_{\alpha}\left(\mathrm{U}^{\dagger}_{\underline{\sigma}}\mathrm{D}\mathrm{U}_{\underline{\sigma}}\otimes\bm{\mathbf{1}}-\mathrm{I}\otimes\bm{\mathbf{C}}+\frac{\mathrm{I}-\Sigma}{2}\otimes(\bm{\mathbf{C}}-\bm{\mathbf{P}}\bm{\mathbf{C}}\bm{\mathbf{P}})\right)\,.$
(B.16)

We now use the identity

$\det\begin{pmatrix}A_{11}&A_{12}\\\ A_{21}&A_{22}\end{pmatrix}=\det
A_{11}\det(A_{22}-A_{21}A_{11}^{-1}A_{12})$ (B.17)

with

$\displaystyle A_{11}=$ $\displaystyle\mathrm{I}\otimes\bm{\mathbf{1}}$ (B.18)
$\displaystyle A_{12}=$ $\displaystyle
f_{\alpha}\mathrm{I}\otimes\bm{\mathbf{P}}$ $\displaystyle A_{21}=$
$\displaystyle\frac{\mathrm{I}-\mathrm{\Sigma}}{2}\otimes\bm{\mathbf{P}}\Bigl{(}\bm{\mathbf{C}}-\tfrac{\bm{\mathbf{1}}}{2}\Bigr{)}$
$\displaystyle A_{22}=$ $\displaystyle
f_{\alpha}\mathrm{U}^{\dagger}_{\underline{\sigma}}(\mathrm{D}-\tfrac{\mathrm{I}}{2})\mathrm{U}_{\underline{\sigma}}\otimes\bm{\mathbf{1}}-f_{\alpha}\frac{\mathrm{I}+\Sigma}{2}\otimes\Bigl{(}\bm{\mathbf{C}}-\tfrac{\bm{\mathbf{1}}}{2}\Bigr{)}$

Thus we have

$\det\bm{\mathbf{L}}_{\underline{\sigma}}=\det\left|\begin{pmatrix}\mathrm{I}\otimes\bm{\mathbf{1}}&f_{\alpha}\mathrm{I}\otimes\bm{\mathbf{P}}\\\
0&f_{\alpha}\mathrm{U}^{\dagger}_{\underline{\sigma}}(\mathrm{D}-\tfrac{\mathrm{I}}{2})\mathrm{U}_{\underline{\sigma}}\otimes\bm{\mathbf{1}}\end{pmatrix}+\right.\\\
\left.\begin{pmatrix}0&0\\\
\frac{\mathrm{I}-\mathrm{\Sigma}}{2}\otimes\bm{\mathbf{P}}(\bm{\mathbf{C}}-\tfrac{\bm{\mathbf{1}}}{2}\Bigr{)}&-f_{\alpha}\frac{\mathrm{I}+\Sigma}{2}\otimes\Bigl{(}\bm{\mathbf{C}}-\tfrac{\bm{\mathbf{1}}}{2}\Bigr{)}\end{pmatrix}\right|\,,$
(B.19)

in which, as desired, $\bm{\mathbf{P}}$ does not appear to the right of
$\bm{\mathbf{C}}-\frac{\bm{\mathbf{1}}}{2}$. We now simplify the diagonal part
in this representation of $\bm{\mathbf{L}}_{\underline{\sigma}}$. This can
done by multiplying from the left by the operator

$\kappa_{\alpha}\begin{pmatrix}\mathrm{I}\otimes\bm{\mathbf{1}}&f_{\alpha}\mathrm{I}\otimes\bm{\mathbf{P}}\\\
0&f_{\alpha}\mathrm{U}^{\dagger}_{\underline{\sigma}}(\mathrm{D}-\frac{\mathrm{I}}{2})\mathrm{U}_{\underline{\sigma}}\otimes\bm{\mathbf{1}}\end{pmatrix}^{-1}=\kappa_{\alpha}\begin{pmatrix}\mathrm{I}\otimes\bm{\mathbf{1}}&-\mathrm{U}^{\dagger}_{\underline{\sigma}}(\mathrm{D}-\tfrac{\mathrm{I}}{2})^{-1}\mathrm{U}_{\underline{\sigma}}\otimes\bm{\mathbf{P}}\\\
0&f_{\alpha}^{-1}\mathrm{U}^{\dagger}_{\underline{\sigma}}(\mathrm{D}-\tfrac{\mathrm{I}}{2})^{-1}\mathrm{U}_{\underline{\sigma}}\otimes\bm{\mathbf{1}}\end{pmatrix}$
(B.20)

where $\kappa_{\alpha}$ ensures the transformation to have unitary determinant
and is given by

$\kappa_{\alpha}=\frac{1}{\sqrt{2}}\bigg{[}\prod_{z|z^{\alpha}=-1}(1-\epsilon-z)\bigg{]}^{\frac{1}{2\alpha}}\xrightarrow[\epsilon\to
0]{}2^{\frac{1-\alpha}{2\alpha}}$ (B.21)

Thus we have

$\bm{\mathbf{L}}_{\underline{\sigma}}\sim\kappa_{\alpha}\begin{pmatrix}\mathrm{I}\otimes\bm{\mathbf{1}}&0\\\
0&\mathrm{I}\otimes\bm{\mathbf{1}}\end{pmatrix}+\\\
\kappa_{\alpha}\begin{pmatrix}-\mathrm{U}^{\dagger}_{\underline{\sigma}}(\mathrm{D}-\tfrac{\mathrm{I}}{2})^{-1}\mathrm{U}_{\underline{\sigma}}\frac{\mathrm{I}-\mathrm{\Sigma}}{2}\otimes\bm{\mathbf{1}}&f_{\alpha}\mathrm{U}^{\dagger}_{\underline{\sigma}}(\mathrm{D}-\tfrac{\mathrm{I}}{2})^{-1}\mathrm{U}_{\underline{\sigma}}\frac{\mathrm{I}+\mathrm{\Sigma}}{2}\otimes\bm{\mathbf{P}}\\\
f_{\alpha}^{-1}\mathrm{U}^{\dagger}_{\underline{\sigma}}(\mathrm{D}-\tfrac{\mathrm{I}}{2})^{-1}\mathrm{U}_{\underline{\sigma}}\frac{\mathrm{I}-\mathrm{\Sigma}}{2}\otimes\bm{\mathbf{P}}&-\mathrm{U}^{\dagger}_{\underline{\sigma}}(\mathrm{D}-\tfrac{\mathrm{I}}{2})^{-1}\mathrm{U}_{\underline{\sigma}}\frac{\mathrm{I}+\mathrm{\Sigma}}{2}\otimes\bm{\mathbf{1}}\end{pmatrix}\mathrm{I}\otimes\Bigl{(}\bm{\mathbf{C}}-\tfrac{\bm{\mathbf{1}}}{2}\Bigr{)}\,.$
(B.22)

We can get rid of $f_{\alpha}$ by applying

$\begin{pmatrix}f_{\alpha}^{-1}\mathrm{I}&0\\\
0&\mathrm{I}\end{pmatrix}\otimes\bm{\mathbf{1}}\qquad\text{and}\qquad\begin{pmatrix}f_{\alpha}\mathrm{I}&0\\\
0&\mathrm{I}\end{pmatrix}\otimes\bm{\mathbf{1}}$ (B.23)

to the left and to the right, respectively. Making the dependency on the
spatial coordinates explicit, we then find

$\bm{\mathbf{L}}_{\underline{\sigma}}(x;y)\sim\kappa_{\alpha}\delta(x-y)\begin{pmatrix}\mathrm{I}&0\\\
0&\mathrm{I}\end{pmatrix}-\kappa_{\alpha}\Bigl{(}\bm{\mathbf{C}}(x,y)-\tfrac{1}{2}\delta(x-y)\Bigr{)}\\\
\times\left[\chi_{A}(x)\begin{pmatrix}\mathrm{U}^{\dagger}_{\underline{\sigma}}(\mathrm{D}-\tfrac{\mathrm{I}}{2})^{-1}\mathrm{U}_{\underline{\sigma}}\frac{\mathrm{I}-\mathrm{\Sigma}}{2}&\mathrm{U}^{\dagger}_{\underline{\sigma}}(\mathrm{D}-\tfrac{\mathrm{I}}{2})^{-1}\mathrm{U}_{\underline{\sigma}}\frac{\mathrm{I}+\mathrm{\Sigma}}{2}\\\
\mathrm{U}^{\dagger}_{\underline{\sigma}}(\mathrm{D}-\tfrac{\mathrm{I}}{2})^{-1}\mathrm{U}_{\underline{\sigma}}\frac{\mathrm{I}-\mathrm{\Sigma}}{2}&\mathrm{U}^{\dagger}_{\underline{\sigma}}(\mathrm{D}-\tfrac{\mathrm{I}}{2})^{-1}\mathrm{U}_{\underline{\sigma}}\frac{\mathrm{I}+\mathrm{\Sigma}}{2}\end{pmatrix}\right.\\\
\left.+\chi_{C}(x)\begin{pmatrix}\mathrm{U}^{\dagger}_{\underline{\sigma}}(\mathrm{D}-\tfrac{\mathrm{I}}{2})^{-1}\mathrm{U}_{\underline{\sigma}}\frac{\mathrm{I}-\mathrm{\Sigma}}{2}&-\mathrm{U}^{\dagger}_{\underline{\sigma}}(\mathrm{D}-\tfrac{\mathrm{I}}{2})^{-1}\mathrm{U}_{\underline{\sigma}}\frac{\mathrm{I}+\mathrm{\Sigma}}{2}\\\
-\mathrm{U}^{\dagger}_{\underline{\sigma}}(\mathrm{D}-\tfrac{\mathrm{I}}{2})^{-1}\mathrm{U}_{\underline{\sigma}}\frac{\mathrm{I}-\mathrm{\Sigma}}{2}&\mathrm{U}^{\dagger}_{\underline{\sigma}}(\mathrm{D}-\tfrac{\mathrm{I}}{2})^{-1}\mathrm{U}_{\underline{\sigma}}\frac{\mathrm{I}+\mathrm{\Sigma}}{2}\end{pmatrix}\right]$
(B.24)

The matrices appearing in the last two lines have $\alpha$ columns of zeroes,
indeed, for any $\sigma_{j}$, either $\frac{1-\sigma_{j}}{2}=0$ or
$\frac{1+\sigma_{j}}{2}=0$. Moving these columns to the left through a unitary
transformation allows us to reduce the size of the matrix by means of the
determinant identity

$\det\left|\begin{pmatrix}\lambda\mathrm{I}^{[d_{0}\times
d_{0}]}\otimes\bm{\mathbf{1}}&\lambda B^{[d_{0}\times
d]}\otimes\bm{\mathbf{B}}\\\ 0^{[d\times d_{0}]}\otimes\bm{\mathbf{0}}&\lambda
A^{[d\times
d]}\otimes\bm{\mathbf{D}}\end{pmatrix}\right|=\det[\lambda^{1+\frac{d_{0}}{d}}A^{[d\times
d]}\otimes\bm{\mathbf{D}}]\,,$ (B.25)

where we have specified the size of the matrices as superscripts. The
following idempotent transformation does the job

$\mathrm{V}_{\underline{\sigma}}=\begin{pmatrix}\frac{\mathrm{I}+\Sigma}{2}&\frac{\mathrm{I}-\Sigma}{2}\\\
\frac{\mathrm{I}-\Sigma}{2}&\frac{\mathrm{I}+\Sigma}{2}\end{pmatrix}\otimes\bm{\mathbf{I}}\,.$
(B.26)

Finally, the operator is reduced to the following

$\bm{\mathbf{L}}_{\underline{\sigma}}(x;y)\sim
2^{\frac{1-\alpha}{\alpha}}\delta(x-y)\mathrm{I}-i2^{\frac{1-\alpha}{\alpha}}\Bigl{(}2\bm{\mathbf{C}}(x,y)-\delta(x-y)\Bigr{)}\left[\chi_{A}(x)\mathrm{M}+\chi_{C}(x)\Sigma\mathrm{M}\Sigma\right]\,.$
(B.27)

where $\mathrm{M}$ is the Hermitian circulant matrix

${}[\mathrm{M}]_{\ell
n}=-i[e^{i\pi\frac{(n-\ell\mod\alpha)}{\alpha}}-\delta_{\ell
n}]=\frac{1}{\alpha}\sum_{j=1}^{\alpha}\cot(\tfrac{\pi(j+\frac{1}{2})}{\alpha})e^{2\pi
ij\frac{\ell-n}{\alpha}}\qquad\ell,n\in\\{1,\ldots,\alpha\\}$ (B.28)

Thus, we have shown that the operator $\bm{\mathbf{L}}_{\underline{\sigma}}$
is equivalent to the following

$\bm{\mathbf{L}}_{\underline{\sigma}}\sim
2^{\frac{1-\alpha}{\alpha}}\mathrm{I}\otimes\bm{\mathbf{1}}-i2^{\frac{1-\alpha}{\alpha}}\left[\mathrm{M}\otimes\frac{\bm{\mathbf{1}}-\bm{\mathbf{P}}}{2}+\Sigma\mathrm{M}\Sigma\otimes\frac{\bm{\mathbf{1}}+\bm{\mathbf{P}}}{2}\right]\mathrm{I}\otimes\Bigl{(}2\bm{\mathbf{C}}-\bm{\mathbf{1}}\Bigr{)}\,,$
(B.29)

##### Example: $\alpha=2$.

Besides the “fermionic” configuration $\underline{\sigma}=(1,1)$, there is
only one nontrivial configuration, $\underline{\sigma}=(-1,1)$, whose
corresponding operator reads

$\bm{\mathbf{L}}_{(-1,1)}\sim\frac{1}{\sqrt{2}}\begin{pmatrix}\bm{\mathbf{1}}&0\\\
0&\bm{\mathbf{1}}\end{pmatrix}+\frac{i}{\sqrt{2}}\begin{pmatrix}0&\bm{\mathbf{P}}\\\
\bm{\mathbf{P}}&0\end{pmatrix}(2\bm{\mathbf{C}}-\bm{\mathbf{1}})$ (B.30)

This is a very special case where we can reduce the problem even further (as
it is always the case for the fermionic configuration). The matrices indeed
commute and we can block-diagonalise the operator, which results in

$\det\bm{\mathbf{L}}_{(-1,1)}=\det\left|\frac{1}{\sqrt{2}}\bm{\mathbf{1}}+\frac{i}{\sqrt{2}}\bm{\mathbf{P}}(2\bm{\mathbf{C}}-1)\right|\det\left|\frac{1}{\sqrt{2}}\bm{\mathbf{1}}-\frac{i}{\sqrt{2}}\bm{\mathbf{P}}(2\bm{\mathbf{C}}-1)\right|\,.$
(B.31)

##### Example: $\alpha=3$.

Besides the fermionic configuration, there is only one additional
nonequivalent configuration, $\underline{\sigma}=(-1,1,1)$, whose
corresponding operator can be represented as follows

$\bm{\mathbf{L}}_{(-1,1,1)}\sim
2^{-\frac{2}{3}}\left[\begin{pmatrix}\bm{\mathbf{1}}&0&0\\\
0&\bm{\mathbf{1}}&0\\\
0&0&\bm{\mathbf{1}}\end{pmatrix}+\begin{pmatrix}0&e^{i\frac{\pi}{3}}\bm{\mathbf{P}}&e^{i\frac{2\pi}{3}}\bm{\mathbf{P}}\\\
e^{i\frac{2\pi}{3}}\bm{\mathbf{P}}&0&-e^{i\frac{\pi}{3}}\bm{\mathbf{1}}\\\
e^{i\frac{\pi}{3}}\bm{\mathbf{P}}&-e^{i\frac{2\pi}{3}}\bm{\mathbf{1}}&0\end{pmatrix}(2\bm{\mathbf{C}}-\bm{\mathbf{1}})\right]$
(B.32)

##### Example: $\alpha=4$ and $\underline{\sigma}=(-1,-1,1,1)$.

We can exploit the simple structure of $\Sigma$ to halve the size of the
auxiliary space. We find

$\det\bm{\mathbf{L}}_{(-1,-1,1,1)}=\prod_{s=\pm
1}\det\left|2^{-\frac{3}{4}}\begin{pmatrix}\bm{\mathbf{1}}&0\\\
0&\bm{\mathbf{1}}\end{pmatrix}-i2^{-\frac{3}{4}}\begin{pmatrix}s\bm{\mathbf{P}}&e^{-i\frac{\pi}{4}}\bm{\mathbf{1}}+se^{i\frac{\pi}{4}}\bm{\mathbf{P}}\\\
e^{i\frac{\pi}{4}}\bm{\mathbf{1}}+se^{-i\frac{\pi}{4}}\bm{\mathbf{P}}&s\bm{\mathbf{P}}\end{pmatrix}\Bigl{(}2\bm{\mathbf{C}}-1\Bigr{)}\right|$
(B.33)

###  B.1 Back to the chain

We have derived (5.40) starting from a specific class of correlation
operators. We claim here that the final result is formally correct also
relaxing those assumptions, and even reintroducing the chain. Following Ref.
Fagotti2010disjoint  (see also Sec. 3.1.2 of FagottiPhDThesis ), such a
generalisation can be done by the formal replacement of
$2\bm{\mathbf{C}}-\bm{\mathbf{1}}$ by the correlation matrix $\Gamma$, and of
the determinant by its square root (in our specific case it is reasonable to
expect that the sign is always positive, but in general the sign ambiguity can
be lifted by computing the product of the eigenvalues with halved degeneracy).
This prescription results in the formula (4.6). This representation is more
useful than the one proposed in Ref. Fagotti2010disjoint : on the one hand,
the matrix in the determinant is linear in $\Gamma_{A\cup C}$ 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 show the correctness of this representation for $\alpha=2$ and $\alpha=3$,
which are the commonest cases investigated.

Case $\alpha=2$ is actually trivial: by applying the prescription to (B.31) we
immediately find

$\\{\Gamma_{1},\Gamma_{2}\\}^{2}=\det\left|\frac{1}{\sqrt{2}}\mathrm{I}_{A\cup
C}+\frac{i}{\sqrt{2}}P\Gamma_{A\cup
C}\right|\det\left|\frac{1}{\sqrt{2}}\mathrm{I}_{A\cup
C}-\frac{i}{\sqrt{2}}P\Gamma_{A\cup
C}\right|=\det\left|\frac{\mathrm{I}_{A\cup C}+P\Gamma_{A\cup C}P\Gamma_{A\cup
C}}{2}\right|\,,$ (B.34)

which is the definition of $\\{\Gamma_{1},\Gamma_{2}\\}$ in Ref.
Fagotti2010disjoint .

Case $\alpha=3$ is more interesting. We start applying the prescription to
(B.32)

$\\{\Gamma_{2},\Gamma_{1},\Gamma_{1}\\}^{2}=\det\left|2^{-\frac{2}{3}}\begin{pmatrix}\mathrm{I}_{A\cup
C}&e^{i\frac{\pi}{3}}P\Gamma_{A\cup C}&e^{i\frac{2\pi}{3}}P\Gamma_{A\cup C}\\\
e^{i\frac{2\pi}{3}}P\Gamma_{A\cup C}&\mathrm{I}_{A\cup
C}&-e^{i\frac{\pi}{3}}\Gamma_{A\cup C}\\\ e^{i\frac{\pi}{3}}P\Gamma_{A\cup
C}&-e^{i\frac{2\pi}{3}}\Gamma_{A\cup C}&\mathrm{I}_{A\cup
C}\end{pmatrix}\right|\,,$ (B.35)

express the determinant of the block matrix in terms of the blocks

$\\{\Gamma_{2},\Gamma_{1},\Gamma_{1}\\}^{2}=\det\left|2^{-\frac{2}{3}}\mathrm{I}_{A\cup
C}\right|\det\left|2^{-\frac{2}{3}}\begin{pmatrix}\Gamma_{A\cup
C}+P\Gamma_{A\cup C}P\Gamma_{A\cup
C}&-e^{i\frac{\pi}{3}}\left(\mathrm{I}_{A\cup C}+P\Gamma_{A\cup
C}P\Gamma_{A\cup C}\right)\\\ -e^{i\frac{2\pi}{3}}\left(\mathrm{I}_{A\cup
C}-P\Gamma_{A\cup C}P\Gamma_{A\cup C}\right)&\Gamma_{A\cup C}+P\Gamma_{A\cup
C}P\Gamma_{A\cup C}\end{pmatrix}\right|\,,$ (B.36)

recognize $\Gamma_{2}=P\Gamma_{A\cup C}P$, express the second determinant in
terms of blocks, and compare the obtained expression with

$\\{\Gamma_{2},\Gamma_{1},\Gamma_{1}\\}^{2}=\\{\Gamma_{2}\times\Gamma_{1},\Gamma_{1}\\}^{2}\\{\Gamma_{2},\Gamma_{1}\\}^{2},$
(B.37)

where the latter is evaluated using formulas (3.16) and (3.17).

##  Appendix C Abelian approximation in the pseudo-conformal limit

In this Appendix we work out the Abelian approximation when
$\gamma_{+}=-\gamma_{-}\rightarrow\pm 1$ at each Fermi point.

##### Rényi-3 tripartite information.

There is a single nontrivial contribution associated with
$\underline{\sigma}=(1,1,-1)$. By enforcing (7.3) we find

$\left.\tfrac{\det\bm{\mathbf{L}}_{(1,1,-1)}^{(k_{F})}}{\det\bm{\mathbf{L}}_{(1,1,1)}^{(k_{F})}}\right|_{Abelian}=(1-x_{4})^{\frac{8}{27}}\,,$
(C.1)

which results in the following Abelian approximation

$G_{3}^{Abelian}(x_{4})=-\log
2+\frac{1}{2}\log\Bigl{(}1+3(1-x_{4})^{\frac{8\nu}{27}}\Bigr{)}\,.$ (C.2)

where $\nu$ is the number of Fermi points.

##### Rényi-4 tripartite information.

There are three nontrivial contributions. By enforcing (7.3) we find

$\begin{gathered}\left.\tfrac{\det\bm{\mathbf{L}}_{(1,1,1,-1)}^{(k_{F})}}{\det\bm{\mathbf{L}}_{(1,1,1,1)}^{(k_{F})}}\right|_{Abelian}=(1-x_{4})^{\frac{5}{16}}\qquad\left.\tfrac{\det\bm{\mathbf{L}}_{(1,1,-1,-1)}^{(k_{F})}}{\det\bm{\mathbf{L}}_{(1,1,1,1)}^{(k_{F})}}\right|_{Abelian}=(1-x_{4})^{\frac{3}{8}}\\\
\tfrac{\det\bm{\mathbf{L}}_{(1,-1,1,-1)}^{(k_{F})}}{\det\bm{\mathbf{L}}_{(1,1,1,1)}^{(k_{F})}}=\left.\tfrac{\det\bm{\mathbf{L}}_{(1,-1,1,-1)}^{(k_{F})}}{\det\bm{\mathbf{L}}_{(1,1,1,1)}^{(k_{F})}}\right|_{Abelian}=(1-x_{4})^{\frac{1}{2}}\end{gathered}$
(C.3)

This results in the following Abelian approximation

$G_{4}^{Abelian}(x_{4})=-\log
2+\frac{1}{3}\log\Bigl{(}1+4(1-x_{4})^{\frac{5\nu}{16}}+2(1-x_{4})^{\frac{3\nu}{8}}+(1-x_{4})^{\frac{\nu}{2}}\Bigr{)}\,.$
(C.4)

##### Rényi-5 tripartite information.

There are three nontrivial contributions. By enforcing (7.3) we find

$\begin{gathered}\left.\tfrac{\det\bm{\mathbf{L}}_{(1,1,1,1,-1)}^{(k_{F})}}{\det\bm{\mathbf{L}}_{(1,1,1,1,1)}^{(k_{F})}}\right|_{Abelian}=(1-x_{4})^{\frac{8}{25}}\qquad\left.\tfrac{\det\bm{\mathbf{L}}_{(1,1,1,-1,-1)}^{(k_{F})}}{\det\bm{\mathbf{L}}_{(1,1,1,1,1)}^{(k_{F})}}\right|_{Abelian}=(1-x_{4})^{\frac{4}{125}(15-\sqrt{5})}\\\
\left.\tfrac{\det\bm{\mathbf{L}}_{(1,1,-1,1,-1)}^{(k_{F})}}{\det\bm{\mathbf{L}}_{(1,1,1,1,1)}^{(k_{F})}}\right|_{Abelian}=(1-x_{4})^{\frac{4}{125}(15+\sqrt{5})}\,.\end{gathered}$
(C.5)

This results in the following Abelian approximation

$G_{5}^{Abelian}(x_{4})=-\log
2+\frac{1}{4}\log\Bigl{(}1+5(1-x_{4})^{\frac{8\nu}{25}}+5(1-x_{4})^{\frac{4\nu(15-\sqrt{5})}{125}}+5(1-x_{4})^{\frac{4\nu(15+\sqrt{5})}{125}}\Bigr{)}\,.$
(C.6)

##### Rényi-6 tripartite information.

There are seven nontrivial contributions. By enforcing (7.3) we find

$\begin{gathered}\left.\tfrac{\det\bm{\mathbf{L}}_{(1,1,1,1,1,-1)}^{(k_{F})}}{\det\bm{\mathbf{L}}_{(1,1,1,1,1,1)}^{(k_{F})}}\right|_{Abelian}=(1-x_{4})^{\frac{35}{108}}\qquad\left.\tfrac{\det\bm{\mathbf{L}}_{(1,1,1,1,-1,-1)}^{(k_{F})}}{\det\bm{\mathbf{L}}_{(1,1,1,1,1,1)}^{(k_{F})}}\right|_{Abelian}=(1-x_{4})^{\frac{23}{54}}\\\
\left.\tfrac{\det\bm{\mathbf{L}}_{(1,1,1,-1,1,-1)}^{(k_{F})}}{\det\bm{\mathbf{L}}_{(1,1,1,1,1,1)}^{(k_{F})}}\right|_{Abelian}=(1-x_{4})^{\frac{31}{54}}\qquad\left.\tfrac{\det\bm{\mathbf{L}}_{(1,1,-1,1,1,-1)}^{(k_{F})}}{\det\bm{\mathbf{L}}_{(1,1,1,1,1,1)}^{(k_{F})}}\right|_{Abelian}=(1-x_{4})^{\frac{16}{27}}\\\
\left.\tfrac{\det\bm{\mathbf{L}}_{(1,1,1,-1,-1,-1)}^{(k_{F})}}{\det\bm{\mathbf{L}}_{(1,1,1,1,1,1)}^{(k_{F})}}\right|_{Abelian}=(1-x_{4})^{\frac{49}{108}}\qquad\left.\tfrac{\det\bm{\mathbf{L}}_{(1,1,-1,1,-1,-1)}^{(k_{F})}}{\det\bm{\mathbf{L}}_{(1,1,1,1,1,1)}^{(k_{F})}}\right|_{Abelian}=(1-x_{4})^{\frac{67}{108}}\\\
\tfrac{\det\bm{\mathbf{L}}_{(1,-1,1,-1,1,-1)}^{(k_{F})}}{\det\bm{\mathbf{L}}_{(1,1,1,1,1,1)}^{(k_{F})}}=\left.\tfrac{\det\bm{\mathbf{L}}_{(1,-1,1,-1,1,-1)}^{(k_{F})}}{\det\bm{\mathbf{L}}_{(1,1,1,1,1,1)}^{(k_{F})}}\right|_{Abelian}=(1-x_{4})^{\frac{3}{4}}\end{gathered}$
(C.7)

This results in the following Abelian approximation

$G_{6}^{Abelian}(x_{4})=-\log
2+\frac{1}{5}\log\Bigl{(}1+6(1-x_{4})^{\frac{35\nu}{108}}+6(1-x_{4})^{\frac{23\nu}{54}}+6(1-x_{4})^{\frac{31\nu}{54}}+6(1-x_{4})^{\frac{16\nu}{27}}\\\
+3(1-x_{4})^{\frac{49\nu}{108}}+3(1-x_{4})^{\frac{67\nu}{108}}+(1-x_{4})^{\frac{3\nu}{4}}\Bigr{)}\,.$
(C.8)

###### Acknowledgements.

##### Funding information

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

## References

  * (1) A. Polkovnikov, K. Sengupta, A. Silva and M. Vengalattore,  _Colloquium: Nonequilibrium dynamics of closed interacting quantum systems_ ,  Rev. Mod. Phys. 83, 863 (2011),  doi:10.1103/RevModPhys.83.863.
  * (2) J. Eisert, M. Friesdorf and C. Gogolin,  _Quantum many-body systems out of equilibrium_ ,  Nature Phys 11, 124 (2015),  doi:10.1038/nphys3215.
  * (3) C. Gogolin and J. Eisert,  _Equilibration, thermalisation, and the emergence of statistical mechanics in closed quantum systems_ ,  Rep. Prog. Phys. 79(5), 056001 (2016),  doi:10.1088/0034-4885/79/5/056001.
  * (4) M. Rigol, V. Dunjko and M. Olshanii,  _Thermalization and its mechanism for generic isolated quantum systems_ ,  Nature 452(7189), 854 (2008),  doi:10.1038/nature06838.
  * (5) J. M. Deutsch,  _Quantum statistical mechanics in a closed system_ ,  Phys. Rev. A 43, 2046 (1991),  doi:10.1103/PhysRevA.43.2046.
  * (6) M. Srednicki,  _Chaos and quantum thermalization_ ,  Phys. Rev. E 50, 888 (1994),  doi:10.1103/PhysRevE.50.888.
  * (7) J. M. Deutsch,  _Eigenstate thermalization hypothesis_ ,  Reports on Progress in Physics 81(8), 082001 (2018),  doi:10.1088/1361-6633/aac9f1.
  * (8) T. Kinoshita, T. Wenger and D. S. Weiss,  _A quantum newton’s cradle_ ,  Nature 440(7086), 900 (2006),  doi:10.1038/nature04693.
  * (9) M. Rigol, V. Dunjko, V. Yurovsky and M. Olshanii,  _Relaxation in a Completely Integrable Many-Body Quantum System: An Ab Initio Study of the Dynamics of the Highly Excited States of 1D Lattice Hard-Core Bosons_ ,  Phys. Rev. Lett. 98, 050405 (2007),  doi:10.1103/PhysRevLett.98.050405.
  * (10) L. Vidmar and M. Rigol,  _Generalized Gibbs ensemble in integrable lattice models_ ,  J. Stat. Mech. 2016(6), 064007 (2016),  doi:10.1088/1742-5468/2016/06/064007.
  * (11) J. Eisert, M. Cramer and M. B. Plenio,  _Colloquium: Area laws for the entanglement entropy_ ,  Rev. Mod. Phys. 82, 277 (2010),  doi:10.1103/RevModPhys.82.277.
  * (12) V. Marić and M. Fagotti,  _Universality in the tripartite information after global quenches_ ,  doi:10.48550/ARXIV.2209.14253,  URL https://arxiv.org/abs/2209.14253 (2022), 2209.14253.
  * (13) L. Amico, R. Fazio, A. Osterloh and V. Vedral,  _Entanglement in many-body systems_ ,  Rev. Mod. Phys. 80, 517 (2008),  doi:10.1103/RevModPhys.80.517.
  * (14) R. Horodecki, P. Horodecki, M. Horodecki and K. Horodecki,  _Quantum entanglement_ ,  Rev. Mod. Phys. 81, 865 (2009),  doi:10.1103/RevModPhys.81.865.
  * (15) A. M. Kaufman, M. E. Tai, A. Lukin, M. Rispoli, R. Schittko, P. M. Preiss and M. Greiner,  _Quantum thermalization through entanglement in an isolated many-body system_ ,  Science 353(6301), 794 (2016),  doi:10.1126/science.aaf6725,  https://www.science.org/doi/pdf/10.1126/science.aaf6725.
  * (16) P. Calabrese and J. Cardy,  _Entanglement entropy and quantum field theory_ ,  Journal of Statistical Mechanics: Theory and Experiment 2004(06), P06002 (2004),  doi:10.1088/1742-5468/2004/06/p06002.
  * (17) P. Calabrese and J. Cardy,  _Entanglement entropy and conformal field theory_ ,  Journal of Physics A: Mathematical and Theoretical 42(50), 504005 (2009),  doi:10.1088/1751-8113/42/50/504005.
  * (18) C. Holzhey, F. Larsen and F. Wilczek,  _Geometric and renormalized entropy in conformal field theory_ ,  Nuclear Physics B 424(3), 443 (1994),  doi:https://doi.org/10.1016/0550-3213(94)90402-2.
  * (19) V. E. Korepin,  _Universality of entropy scaling in one dimensional gapless models_ ,  Phys. Rev. Lett. 92, 096402 (2004),  doi:10.1103/PhysRevLett.92.096402.
  * (20) A. Kitaev and J. Preskill,  _Topological Entanglement Entropy_ ,  Phys. Rev. Lett. 96, 110404 (2006),  doi:10.1103/PhysRevLett.96.110404.
  * (21) N. J. Cerf and C. Adami,  _Information theory of quantum entanglement and measurement_ ,  Physica D: Nonlinear Phenomena 120(1), 62 (1998),  doi:https://doi.org/10.1016/S0167-2789(98)00045-1,  Proceedings of the Fourth Workshop on Physics and Consumption.
  * (22) P. Calabrese and J. Cardy,  _Evolution of entanglement entropy in one-dimensional systems_ ,  J. Stat. Mech. 2005(04), P04010 (2005),  doi:10.1088/1742-5468/2005/04/p04010.
  * (23) S. Sotiriadis and J. Cardy,  _Inhomogeneous quantum quenches_ ,  Journal of Statistical Mechanics: Theory and Experiment 2008(11), P11003 (2008),  doi:10.1088/1742-5468/2008/11/P11003.
  * (24) A. Bastianello and P. Calabrese,  _Spreading of entanglement and correlations after a quench with intertwined quasiparticles_ ,  SciPost Phys. 5, 033 (2018),  doi:10.21468/SciPostPhys.5.4.033.
  * (25) B. Bertini, K. Klobas, V. Alba, G. Lagnese and P. Calabrese,  _Growth of rényi entropies in interacting integrable models and the breakdown of the quasiparticle picture_ ,  Phys. Rev. X 12, 031016 (2022),  doi:10.1103/PhysRevX.12.031016.
  * (26) V. Alba and P. Calabrese,  _Entanglement and thermodynamics after a quantum quench in integrable systems_ ,  Proceedings of the National Academy of Sciences 114(30), 7947 (2017),  doi:10.1073/pnas.1703516114,  https://www.pnas.org/doi/pdf/10.1073/pnas.1703516114.
  * (27) V. Alba and P. Calabrese,  _Entanglement dynamics after quantum quenches in generic integrable systems_ ,  SciPost Phys. 4, 017 (2018),  doi:10.21468/SciPostPhys.4.3.017.
  * (28) H. Casini, H. Liu and M. Mezei,  _Spread of entanglement and causality_ ,  Journal of High Energy Physics 2016(7), 77 (2016),  doi:10.1007/JHEP07(2016)077.
  * (29) H. Liu and S. J. Suh,  _Entanglement growth during thermalization in holographic systems_ ,  Phys. Rev. D 89, 066012 (2014),  doi:10.1103/PhysRevD.89.066012.
  * (30) B. Skinner, J. Ruhman and A. Nahum,  _Measurement-induced phase transitions in the dynamics of entanglement_ ,  Physical Review X 9(3) (2019),  doi:10.1103/physrevx.9.031009.
  * (31) V. Alba and P. Calabrese,  _Quantum information scrambling after a quantum quench_ ,  Phys. Rev. B 100, 115150 (2019),  doi:10.1103/PhysRevB.100.115150.
  * (32) B. Bertini, K. Klobas and T.-C. Lu,  _Entanglement negativity and mutual information after a quantum quench: Exact link from space-time duality_ ,  Phys. Rev. Lett. 129, 140503 (2022),  doi:10.1103/PhysRevLett.129.140503.
  * (33) V. Eisler and Z. Zimborás,  _Area-law violation for the mutual information in a nonequilibrium steady state_ ,  Phys. Rev. A 89, 032321 (2014),  doi:10.1103/PhysRevA.89.032321.
  * (34) G. Parez and R. Bonsignori,  _Analytical results for the entanglement dynamics of disjoint blocks in the xy spin chain_ ,  Journal of Physics A: Mathematical and Theoretical 55(50), 505005 (2023),  doi:10.1088/1751-8121/acb097.
  * (35) P. Calabrese, F. H. L. Essler and M. Fagotti,  _Quantum Quench in the Transverse-Field Ising Chain_ ,  Phys. Rev. Lett. 106, 227203 (2011),  doi:10.1103/PhysRevLett.106.227203.
  * (36) P. Calabrese, F. H. L. Essler and M. Fagotti,  _Quantum quenches in the transverse field Ising chain: II. Stationary state properties_ ,  Journal of Statistical Mechanics: Theory and Experiment 2012(07), P07022 (2012),  doi:10.1088/1742-5468/2012/07/p07022.
  * (37) H. Casini and M. Huerta,  _Reduced density matrix and internal dynamics for multicomponent regions_ ,  Classical and Quantum Gravity 26(18), 185005 (2009),  doi:10.1088/0264-9381/26/18/185005.
  * (38) H. Casini and M. Huerta,  _Entanglement entropy in free quantum field theory_ ,  Journal of Physics A: Mathematical and Theoretical 42(50), 504007 (2009),  doi:10.1088/1751-8113/42/50/504007.
  * (39) P. Fries and I. A. Reyes,  _Entanglement and relative entropy of a chiral fermion on the torus_ ,  Phys. Rev. D 100, 105015 (2019),  doi:10.1103/PhysRevD.100.105015.
  * (40) D. Blanco, T. Ferreira Chase, J. Laurnagaray and G. Pérez-Nadal,  _Rényi entropies of the massless dirac field on the torus_ ,  Phys. Rev. D 105, 045014 (2022),  doi:10.1103/PhysRevD.105.045014.
  * (41) F. Iglói and I. Peschel,  _On reduced density matrices for disjoint subsystems_ ,  Europhysics Letters 89(4), 40001 (2010),  doi:10.1209/0295-5075/89/40001.
  * (42) M. Fagotti and P. Calabrese,  _Entanglement entropy of two disjoint blocks in XY chains_ ,  Journal of Statistical Mechanics: Theory and Experiment 2010(04), P04016 (2010),  doi:10.1088/1742-5468/2010/04/p04016.
  * (43) M. M. Wolf, F. Verstraete, M. B. Hastings and J. I. Cirac,  _Area laws in quantum systems: Mutual information and correlations_ ,  Phys. Rev. Lett. 100, 070502 (2008),  doi:10.1103/PhysRevLett.100.070502.
  * (44) T. Kuwahara, A. M. Alhambra and A. Anshu,  _Improved thermal area law and quasilinear time algorithm for quantum gibbs states_ ,  Phys. Rev. X 11, 011047 (2021),  doi:10.1103/PhysRevX.11.011047.
  * (45) H. Bernigau, M. J. Kastoryano and J. Eisert,  _Mutual information area laws for thermal free fermions_ ,  Journal of Statistical Mechanics: Theory and Experiment 2015(2), P02008 (2015),  doi:10.1088/1742-5468/2015/02/P02008.
  * (46) M. Lemm and O. Siebert,  _Thermal area law for lattice bosons_ ,  doi:10.48550/ARXIV.2207.07760,  URL https://arxiv.org/abs/2207.07760 (2022).
  * (47) A. M. Alhambra,  _Quantum many-body systems in thermal equilibrium_ ,  doi:10.48550/ARXIV.2204.08349,  URL https://arxiv.org/abs/2204.08349 (2022).
  * (48) H. Casini, C. D. Fosco and M. Huerta,  _Entanglement and alpha entropies for a massive dirac field in two dimensions_ ,  Journal of Statistical Mechanics: Theory and Experiment 2005(07), P07007 (2005),  doi:10.1088/1742-5468/2005/07/P07007.
  * (49) P. Calabrese, J. Cardy and E. Tonni,  _Entanglement entropy of two disjoint intervals in conformal field theory_ ,  Journal of Statistical Mechanics: Theory and Experiment 2009(04), P11001 (2009),  doi:10.1088/1742-5468/2009/11/P11001.
  * (50) H. Casini,  _Mutual information challenges entropy bounds_ ,  Classical and Quantum Gravity 24(5), 1293 (2007),  doi:10.1088/0264-9381/24/5/013.
  * (51) S. Furukawa, V. Pasquier and J. Shiraishi,  _Mutual Information and Boson Radius in a $c=1$ Critical System in One Dimension_,  Phys. Rev. Lett. 102, 170602 (2009),  doi:10.1103/PhysRevLett.102.170602.
  * (52) M. Headrick,  _Entanglement rényi entropies in holographic theories_ ,  Phys. Rev. D 82, 126010 (2010),  doi:10.1103/PhysRevD.82.126010.
  * (53) B. Chen and J. ju Zhang,  _On short interval expansion of rényi entropy_ ,  Journal of High Energy Physics 2013(11) (2013),  doi:10.1007/jhep11(2013)164.
  * (54) B. Chen, F.-y. Song and J.-j. Zhang,  _Holographic rényi entropy in ads3/lcft2 correspondence_ ,  Journal of High Energy Physics 2014(3), 137 (2014),  doi:10.1007/JHEP03(2014)137.
  * (55) M. Headrick, A. Maloney, E. Perlmutter and I. G. Zadeh,  _Rényi entropies, the analytic bootstrap, and 3d quantum gravity at higher genus_ ,  Journal of High Energy Physics 2015(7), 59 (2015),  doi:10.1007/JHEP07(2015)059.
  * (56) H. Casini, M. Huerta, R. C. Myers and A. Yale,  _Mutual information and the f-theorem_ ,  Journal of High Energy Physics 2015(10), 3 (2015),  doi:10.1007/JHEP10(2015)003.
  * (57) D. D. Blanco and H. Casini,  _Entanglement entropy for non-coplanar regions in quantum field theory_ ,  Classical and Quantum Gravity 28(21), 215015 (2011),  doi:10.1088/0264-9381/28/21/215015.
  * (58) R. E. Arias, H. Casini, M. Huerta and D. Pontello,  _Entropy and modular hamiltonian for a free chiral scalar in two intervals_ ,  Phys. Rev. D 98, 125008 (2018),  doi:10.1103/PhysRevD.98.125008.
  * (59) D. Blanco, A. Garbarz and G. Pérez-Nadal,  _Entanglement of a chiral fermion on the torus_ ,  Journal of High Energy Physics 2019(9), 76 (2019),  doi:10.1007/JHEP09(2019)076.
  * (60) F. Ares, P. Calabrese, G. Di Giulio and S. Murciano,  _Multi-charged moments of two intervals in conformal field theory_ ,  Journal of High Energy Physics 2022(9), 51 (2022),  doi:10.1007/JHEP09(2022)051.
  * (61) C. T. Asplund and A. Bernamonti,  _Mutual information after a local quench in conformal field theory_ ,  Phys. Rev. D 89, 066015 (2014),  doi:10.1103/PhysRevD.89.066015.
  * (62) J. Molina-Vilaplana and P. Sodano,  _Holographic view on quantum correlations and mutual information between disjoint blocks of a quantum critical system_ ,  Journal of High Energy Physics 2011(10), 11 (2011),  doi:10.1007/JHEP10(2011)011.
  * (63) L. Lepori, S. Paganelli, F. Franchini and A. Trombettoni,  _Mutual information for fermionic systems_ ,  Phys. Rev. Res. 4, 033212 (2022),  doi:10.1103/PhysRevResearch.4.033212.
  * (64) H. Casini and M. Huerta,  _Remarks on the entanglement entropy for disconnected regions_ ,  Journal of High Energy Physics 2009(03), 048 (2009),  doi:10.1088/1126-6708/2009/03/048.
  * (65) N. Shiba,  _Entanglement entropy of two spheres_ ,  Journal of High Energy Physics 2012(7), 100 (2012),  doi:10.1007/JHEP07(2012)100.
  * (66) J. Cardy,  _Some results on the mutual information of disjoint regions in higher dimensions_ ,  Journal of Physics A: Mathematical and Theoretical 46(28), 285402 (2013),  doi:10.1088/1751-8113/46/28/285402.
  * (67) A. Allais and E. Tonni,  _Holographic evolution of the mutual information_ ,  Journal of High Energy Physics 2012(1), 102 (2012),  doi:10.1007/JHEP01(2012)102.
  * (68) C. A. Agón and T. Faulkner,  _Quantum corrections to holographic mutual information_ ,  Journal of High Energy Physics 2016(8), 118 (2016),  doi:10.1007/JHEP08(2016)118.
  * (69) C. A. Agón, I. Cohen-Abbo and H. J. Schnitzer,  _Large distance expansion of mutual information for disjoint disks in a free scalar theory_ ,  Journal of High Energy Physics 2016(11), 73 (2016),  doi:10.1007/JHEP11(2016)073.
  * (70) B. Chen and J. Long,  _Rényi mutual information for a free scalar field in even dimensions_ ,  Phys. Rev. D 96, 045006 (2017),  doi:10.1103/PhysRevD.96.045006.
  * (71) H. Casini, F. D. Mazzitelli and E. Testé,  _Area terms in entanglement entropy_ ,  Phys. Rev. D 91, 104035 (2015),  doi:10.1103/PhysRevD.91.104035.
  * (72) E. Tonni,  _Holographic entanglement entropy: near horizon geometry and disconnected regions_ ,  Journal of High Energy Physics 2011(5), 4 (2011),  doi:10.1007/JHEP05(2011)004.
  * (73) B. Groisman, S. Popescu and A. Winter,  _Quantum, classical, and total amount of correlations in a quantum state_ ,  Phys. Rev. A 72, 032317 (2005),  doi:10.1103/PhysRevA.72.032317.
  * (74) M. Caraglio and F. Gliozzi,  _Entanglement entropy and twist fields_ ,  Journal of High Energy Physics 2008(11), 076 (2008),  doi:10.1088/1126-6708/2008/11/076.
  * (75) M. A. Rajabpour and F. Gliozzi,  _Entanglement entropy of two disjoint intervals from fusion algebra of twist fields_ ,  Journal of Statistical Mechanics: Theory and Experiment 2012(02), P02016 (2012),  doi:10.1088/1742-5468/2012/02/P02016.
  * (76) P. Calabrese, J. Cardy and E. Tonni,  _Entanglement entropy of two disjoint intervals in conformal field theory: II_ ,  Journal of Statistical Mechanics: Theory and Experiment 2011(01), P01021 (2011),  doi:10.1088/1742-5468/2011/01/p01021.
  * (77) A. Coser, L. Tagliacozzo and E. Tonni,  _On rényi entropies of disjoint intervals in conformal field theory_ ,  Journal of Statistical Mechanics: Theory and Experiment 2014(1), P01008 (2014),  doi:10.1088/1742-5468/2014/01/P01008.
  * (78) P. Ruggiero, E. Tonni and P. Calabrese,  _Entanglement entropy of two disjoint intervals and the recursion formula for conformal blocks_ ,  Journal of Statistical Mechanics: Theory and Experiment 2018(11), 113101 (2018),  doi:10.1088/1742-5468/aae5a8.
  * (79) V. Alba, L. Tagliacozzo and P. Calabrese,  _Entanglement entropy of two disjoint blocks in critical ising models_ ,  Phys. Rev. B 81, 060411 (2010),  doi:10.1103/PhysRevB.81.060411.
  * (80) V. Alba, L. Tagliacozzo and P. Calabrese,  _Entanglement entropy of two disjoint intervals in c=1 theories_ ,  Journal of Statistical Mechanics: Theory and Experiment 2011(06), P06012 (2011),  doi:10.1088/1742-5468/2011/06/p06012.
  * (81) M. Fagotti,  _New insights into the entanglement of disjoint blocks_ ,  EPL (Europhysics Letters) 97(1), 17007 (2012),  doi:10.1209/0295-5075/97/17007.
  * (82) V. Balasubramanian, A. Bernamonti, N. Copland, B. Craps and F. Galli,  _Thermalization of mutual and tripartite information in strongly coupled two dimensional conformal field theories_ ,  Phys. Rev. D 84, 105017 (2011),  doi:10.1103/PhysRevD.84.105017.
  * (83) T. Grava, A. P. Kels and E. Tonni,  _Entanglement of two disjoint intervals in conformal field theory and the 2d coulomb gas on a lattice_ ,  Phys. Rev. Lett. 127, 141605 (2021),  doi:10.1103/PhysRevLett.127.141605.
  * (84) F. Ares, R. Santachiara and J. Viti,  _Crossing-symmetric twist field correlators and entanglement negativity in minimal CFTs_ ,  Journal of High Energy Physics 2021(10), 175 (2021),  doi:10.1007/JHEP10(2021)175.
  * (85) C. A. Agón, P. Bueno and H. Casini,  _Tripartite information at long distances_ ,  SciPost Phys. 12, 153 (2022),  doi:10.21468/SciPostPhys.12.5.153.
  * (86) M. Ali-Akbari, M. Rahimi and M. Asadi,  _Holographic mutual and tripartite information in a non-conformal background_ ,  Nuclear Physics B 964, 115329 (2021),  doi:https://doi.org/10.1016/j.nuclphysb.2021.115329.
  * (87) P. Hayden, M. Headrick and A. Maloney,  _Holographic mutual information is monogamous_ ,  Phys. Rev. D 87, 046003 (2013),  doi:10.1103/PhysRevD.87.046003.
  * (88) F. Carollo and V. Alba,  _Entangled multiplets and spreading of quantum correlations in a continuously monitored tight-binding chain_ ,  Phys. Rev. B 106, L220304 (2022),  doi:10.1103/PhysRevB.106.L220304.
  * (89) G. Parez, P.-A. Bernard, N. Crampé and L. Vinet,  _Multipartite information of free fermions on hamming graphs_ ,  Nuclear Physics B 990, 116157 (2023),  doi:https://doi.org/10.1016/j.nuclphysb.2023.116157.
  * (90) P. Hosur, X.-L. Qi, D. A. Roberts and B. Yoshida,  _Chaos in quantum channels_ ,  Journal of High Energy Physics 2016(2), 4 (2016),  doi:10.1007/JHEP02(2016)004.
  * (91) O. Schnaack, N. Bölter, S. Paeckel, S. R. Manmana, S. Kehrein and M. Schmitt,  _Tripartite information, scrambling, and the role of Hilbert space partitioning in quantum lattice models_ ,  Phys. Rev. B 100, 224302 (2019),  doi:10.1103/PhysRevB.100.224302.
  * (92) C. Sünderhauf, L. Piroli, X.-L. Qi, N. Schuch and J. I. Cirac,  _Quantum chaos in the Brownian SYK model with large finite N : OTOCs and tripartite information_ ,  Journal of High Energy Physics 2019(11), 38 (2019),  doi:10.1007/JHEP11(2019)038.
  * (93) Y. Kuno, T. Orito and I. Ichinose,  _Purification and scrambling in a chaotic hamiltonian dynamics with measurements_ ,  Phys. Rev. B 106, 214304 (2022),  doi:10.1103/PhysRevB.106.214304.
  * (94) M. B. Hastings,  _An area law for one-dimensional quantum systems_ ,  Journal of Statistical Mechanics: Theory and Experiment 2007(08), P08024 (2007),  doi:10.1088/1742-5468/2007/08/P08024.
  * (95) B.-Q. Jin and V. E. Korepin,  _Quantum Spin Chain, Toeplitz Determinants and the Fisher—Hartwig Conjecture_ ,  Journal of Statistical Physics 116(1), 79 (2004),  doi:10.1023/B:JOSS.0000037230.37166.42.
  * (96) S. Fraenkel and M. Goldstein,  _Entanglement measures in a nonequilibrium steady state: Exact results in one dimension_ ,  SciPost Phys. 11, 085 (2021),  doi:10.21468/SciPostPhys.11.4.085.
  * (97) M. Fagotti, V. Marić and L. Zadnik,  _Nonequilibrium symmetry-protected topological order: emergence of semilocal gibbs ensembles_ ,  doi:10.48550/ARXIV.2205.02221,  URL https://arxiv.org/abs/2205.02221 (2022), 2205.02221.
  * (98) V. Alba, M. Fagotti and P. Calabrese,  _Entanglement entropy of excited states_ ,  Journal of Statistical Mechanics: Theory and Experiment 2009(10), P10020 (2009),  doi:10.1088/1742-5468/2009/10/p10020.
  * (99) F. Ares, J. G. Esteve, F. Falceto and E. Sánchez-Burillo,  _Excited state entanglement in homogeneous fermionic chains_ ,  Journal of Physics A: Mathematical and Theoretical 47(24), 245301 (2014),  doi:10.1088/1751-8113/47/24/245301.
  * (100) A. Bluhm, Á. Capel and A. Pérez-Hernández,  _Exponential decay of mutual information for Gibbs states of local Hamiltonians_ ,  Quantum 6, 650 (2022),  doi:10.22331/q-2022-02-10-650.
  * (101) M. B. Hastings and T. Koma,  _Spectral gap and exponential decay of correlations_ ,  Communications in Mathematical Physics 265(3), 781 (2006),  doi:10.1007/s00220-006-0030-4.
  * (102) M. Suzuki,  _The dimer problem and the generalized X-model_ ,  Physics Letters A 34(6), 338 (1971),  doi:https://doi.org/10.1016/0375-9601(71)90901-7.
  * (103) E. Lieb, T. Schultz and D. Mattis,  _Two soluble models of an antiferromagnetic chain_ ,  Annals of Physics 16(3), 407 (1961),  doi:https://doi.org/10.1016/0003-4916(61)90115-4.
  * (104) M. Fagotti,  _Charges and currents in quantum spin chains: late-time dynamics and spontaneous currents_ ,  J. Phys. A: Math. Theor. 50(3), 034005 (2016),  doi:10.1088/1751-8121/50/3/034005.
  * (105) P. Calabrese and J. Cardy,  _Quantum quenches in 1+1 dimensional conformal field theories_ ,  J. Stat. Mech. 2016(6), 064003 (2016),  doi:10.1088/1742-5468/2016/06/064003.
  * (106) F. H. L. Essler and M. Fagotti,  _Quench dynamics and relaxation in isolated integrable quantum spin chains_ ,  J. Stat. Mech. 2016(6), 064002 (2016),  doi:10.1088/1742-5468/2016/06/064002.
  * (107) J.-S. Caux,  _The Quench Action_ ,  Journal of Statistical Mechanics: Theory and Experiment 2016(6), 064006 (2016),  doi:10.1088/1742-5468/2016/06/064006.
  * (108) T. Sabetta and G. Misguich,  _Nonequilibrium steady states in the quantum xxz spin chain_ ,  Phys. Rev. B 88, 245114 (2013),  doi:10.1103/PhysRevB.88.245114.
  * (109) V. Eisler and Z. Zimborás,  _Entanglement negativity in the harmonic chain out of equilibrium_ ,  New Journal of Physics 16(12), 123020 (2014),  doi:10.1088/1367-2630/16/12/123020.
  * (110) L. Mazza, J. Viti, M. Carrega, D. Rossini and A. De Luca,  _Energy transport in an integrable parafermionic chain via generalized hydrodynamics_ ,  Phys. Rev. B 98, 075421 (2018),  doi:10.1103/PhysRevB.98.075421.
  * (111) B. Bertini, M. Fagotti, L. Piroli and P. Calabrese,  _Entanglement evolution and generalised hydrodynamics: noninteracting systems_ ,  Journal of Physics A: Mathematical and Theoretical 51(39), 39LT01 (2018),  doi:10.1088/1751-8121/aad82e.
  * (112) V. Alba, B. Bertini and M. Fagotti,  _Entanglement evolution and generalised hydrodynamics: interacting integrable systems_ ,  SciPost Phys. 7, 5 (2019),  doi:10.21468/SciPostPhys.7.1.005.
  * (113) M. Gruber and V. Eisler,  _Magnetization and entanglement after a geometric quench in the XXZ chain_ ,  Phys. Rev. B 99, 174403 (2019),  doi:10.1103/PhysRevB.99.174403.
  * (114) M. Collura, A. De Luca, P. Calabrese and J. Dubail,  _Domain wall melting in the spin- ${\frac{1}{2}}$ XXZ spin chain: Emergent Luttinger liquid with a fractal quasiparticle charge_,  Phys. Rev. B 102, 180409 (2020),  doi:10.1103/PhysRevB.102.180409.
  * (115) A. Bastianello, B. Bertini, B. Doyon and R. Vasseur,  _Introduction to the special issue on emergent hydrodynamics in integrable many-body systems_ ,  Journal of Statistical Mechanics: Theory and Experiment 2022(1), 014001 (2022),  doi:10.1088/1742-5468/ac3e6a.
  * (116) V. Alba, B. Bertini, M. Fagotti, L. Piroli and P. Ruggiero,  _Generalized-hydrodynamic approach to inhomogeneous quenches: correlations, entanglement and quantum effects_ ,  Journal of Statistical Mechanics: Theory and Experiment 2021(11), 114004 (2021),  doi:10.1088/1742-5468/ac257d.
  * (117) B. Bertini and M. Fagotti,  _Determination of the Nonequilibrium Steady State Emerging from a Defect_ ,  Phys. Rev. Lett. 117, 130402 (2016),  doi:10.1103/PhysRevLett.117.130402.
  * (118) B. Bertini, M. Collura, J. De Nardis and M. Fagotti,  _Transport in Out-of-Equilibrium ${XXZ}$ Chains: Exact Profiles of Charges and Currents_,  Phys. Rev. Lett. 117, 207201 (2016),  doi:10.1103/PhysRevLett.117.207201.
  * (119) O. A. Castro-Alvaredo, B. Doyon and T. Yoshimura,  _Emergent Hydrodynamics in Integrable Quantum Systems Out of Equilibrium_ ,  Phys. Rev. X 6, 041065 (2016),  doi:10.1103/PhysRevX.6.041065.
  * (120) J.-S. Caux and F. H. L. Essler,  _Time Evolution of Local Observables After Quenching to an Integrable Model_ ,  Phys. Rev. Lett. 110, 257203 (2013),  doi:10.1103/PhysRevLett.110.257203.
  * (121) M. Fagotti,  _Locally quasi-stationary states in noninteracting spin chains_ ,  SciPost Phys. 8, 48 (2020),  doi:10.21468/SciPostPhys.8.3.048.
  * (122) G. Vidal, J. I. Latorre, E. Rico and A. Kitaev,  _Entanglement in Quantum Critical Phenomena_ ,  Phys. Rev. Lett. 90, 227902 (2003),  doi:10.1103/PhysRevLett.90.227902.
  * (123) I. Klich,  _A note on the full counting statistics of paired fermions_ ,  Journal of Statistical Mechanics: Theory and Experiment 2014(11), P11006 (2014),  doi:10.1088/1742-5468/2014/11/p11006.
  * (124) I. Affleck,  _Field Theory and Quantum Critical Phenomena_ ,  In _Les Houches Summer School in Theoretical Physics: Fields, Strings, Critical Phenomena_ (1988).
  * (125) L. P. Kadanoff,  _Scaling laws for ising models near ${T}_{c}$_,  Physics Physique Fizika 2, 263 (1966),  doi:10.1103/PhysicsPhysiqueFizika.2.263.
  * (126) K. G. Wilson,  _Renormalization group and critical phenomena. i. renormalization group and the kadanoff scaling picture_ ,  Phys. Rev. B 4, 3174 (1971),  doi:10.1103/PhysRevB.4.3174.
  * (127) J. L. Cardy, O. A. Castro-Alvaredo and B. Doyon,  _Form factors of branch-point twist fields in quantum integrable models and entanglement entropy_ ,  Journal of Statistical Physics 130(1), 129 (2008),  doi:10.1007/s10955-007-9422-x.
  * (128) H. Kleinert,  _Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets_ ,  World Scientific, 5th edn. (2009).
  * (129) A. Coser, E. Tonni and P. Calabrese,  _Spin structures and entanglement of two disjoint intervals in conformal field theories_ ,  Journal of Statistical Mechanics: Theory and Experiment 2016(5), 053109 (2016),  doi:10.1088/1742-5468/2016/05/053109.
  * (130) X. Chen and E. Fradkin,  _Quantum entanglement and thermal reduced density matrices in fermion and spin systems on ladders_ ,  Journal of Statistical Mechanics: Theory and Experiment 2013(08), P08013 (2013),  doi:10.1088/1742-5468/2013/08/P08013.
  * (131) A. Mollabashi, N. Shiba and T. Takayanagi,  _Entanglement between two interacting cfts and generalized holographic entanglement entropy_ ,  Journal of High Energy Physics 2014(4), 185 (2014),  doi:10.1007/JHEP04(2014)185.
  * (132) S. Furukawa and Y. B. Kim,  _Entanglement entropy between two coupled tomonaga-luttinger liquids_ ,  Phys. Rev. B 83, 085112 (2011),  doi:10.1103/PhysRevB.83.085112.
  * (133) C. Xu,  _Entanglement entropy of coupled conformal field theories and fermi liquids_ ,  Phys. Rev. B 84, 125119 (2011),  doi:10.1103/PhysRevB.84.125119.
  * (134) N. I. Muskhelishvili,  _Singular Integral Equations_ ,  P. Noordhoff N.V. - Groningen-Holland (1953).
  * (135) A. Its,  _The Riemann-Hilbert Problem and Integrable Systems_ ,  Notices of the American Mathematical Society 50(11) (2003).
  * (136) M. Fagotti,  _Global Quenches after Localized Perturbations_ ,  Phys. Rev. Lett. 128, 110602 (2022),  doi:10.1103/PhysRevLett.128.110602.
  * (137) S. Bocini,  _Connected correlations in partitioning protocols: a case study and beyond_ ,  doi:10.48550/ARXIV.2212.07151,  URL https://arxiv.org/abs/2212.07151 (2022).
  * (138) M. Fagotti,  _Entanglement & Correlations in exactly solvable models_,  Ph.D. thesis (2011).