columns and all elements equal to $1$, satisfying
$(\alpha\mathbb{I}-\vec{u}\vec{u}^{\mathrm{T}})^{-1}=(\mathbb{I}+\vec{u}\vec{u}^{\mathrm{T}})/\alpha$,
and we have defined the matrix

$B_{\ell,n}=\sum_{k=1}^{\alpha-1}f_{\frac{k}{\alpha}}\cos\left[2\pi\frac{k}{\alpha}(\ell-n)\right]\;.$
(7.6)

We can see that $\log(1-x)$ in Eq. (7.5) appears in the denominator, making it
complicated to conclude about the behavior of the theta function (2.6). We can
exploit the fact that it appears in the numerator in the inverse matrix. The
inverse, that follows from Neumann series formula, is given by

$\left(-i\Omega\right)^{-1}=\frac{1}{2\pi}|\log(1-x)|\left(\mathbb{I}+\vec{u}\vec{u}^{\mathrm{T}}\right)+\frac{1}{2\pi\alpha}\left(\mathbb{I}+\vec{u}\vec{u}^{\mathrm{T}}\right)B\left(\mathbb{I}+\vec{u}\vec{u}^{\mathrm{T}}\right)+O(1-x),\quad\textrm{as
}x\to 1\;.$ (7.7)

To express the theta function in terms of $\Omega^{-1}$ we exploit the formula

$\sum_{\vec{m}\in\mathbb{Z}^{\alpha-1}}e^{-\vec{m}\cdot
M\vec{m}}=\sqrt{\det(\pi
M^{-1})}\sum_{\vec{n}\in\mathbb{Z}^{\alpha-1}}e^{-\pi^{2}\vec{n}\cdot
M^{-1}\vec{n}}$ (7.8)

for $(\alpha-1)\times(\alpha-1)$ positive definite orthogonal matrices $M$.
For $\alpha=2$ the derivation of the formula is an application of the Poisson
summation formula, while for higher $\alpha$ it is a straightforward
generalization. Using formula (7.8) we get

$\frac{\Theta(0|4\eta\Omega)}{\Theta(0|\eta\Omega)}=\frac{1}{2^{\alpha-1}}\frac{\sum\limits_{\vec{n}\in\mathbb{Z}^{\alpha-1}}\exp\left[-\frac{1}{4\eta}\pi\vec{n}\cdot(-i\Omega)^{-1}\vec{n}\right]}{\sum\limits_{\vec{n}\in\mathbb{Z}^{\alpha-1}}\exp\left[-\frac{1}{\eta}\pi\vec{n}\cdot(-i\Omega)^{-1}\vec{n}\right]}$
(7.9)

and plugging the expansion (7.7) we get further

$\frac{\Theta(0|4\eta\Omega)}{\Theta(0|\eta\Omega)}=\frac{1}{2^{\alpha-1}}\frac{\sum\limits_{\vec{n}\in\mathbb{Z}^{\alpha-1}}(1-x)^{\frac{1}{8\eta}\vec{n}\cdot(\mathbb{I}+\vec{u}\vec{u}^{\mathrm{T}})\vec{n}}\exp\left\\{-\frac{1}{8\eta\alpha}\vec{n}\cdot\left[\left(\mathbb{I}+\vec{u}\vec{u}^{\mathrm{T}}\right)B\left(\mathbb{I}+\vec{u}\vec{u}^{\mathrm{T}}\right)+O(1-x)\right]\vec{n}\right\\}}{\sum\limits_{\vec{n}\in\mathbb{Z}^{\alpha-1}}(1-x)^{\frac{1}{2\eta}\vec{n}\cdot(\mathbb{I}+\vec{u}\vec{u}^{\mathrm{T}})\vec{n}}\exp\left\\{-\frac{1}{2\eta\alpha}\vec{n}\cdot\left[\left(\mathbb{I}+\vec{u}\vec{u}^{\mathrm{T}}\right)B\left(\mathbb{I}+\vec{u}\vec{u}^{\mathrm{T}}\right)+O(1-x)\right]\vec{n}\right\\}}\;.$
(7.10)

Both in the numerator and the denominator in Eq. (7.10) the leading term in
the expansion in $1-x$ as $x\to 1$ is a constant term obtained for
$\vec{n}=\vec{0}$. The first subleading term is given by those
$\vec{n}\neq\vec{0}$ that minimize

$\vec{n}\cdot(\mathbb{I}+\vec{u}\vec{u}^{\mathrm{T}})\vec{n}=\sum_{j=1}^{\alpha-1}n_{j}^{2}+\left(\sum_{j=1}^{\alpha-1}n_{j}\right)^{2}\;.$
(7.11)

By inspection, it is easy to see that this expression is minimized by
$\vec{n}=\pm\vec{e}_{j}$ for any $j$ and by
$\vec{n}=\vec{e}_{j}-\vec{e}_{j^{\prime}}$ for any $j\neq j^{\prime}$, where
we have denoted $(\vec{e}_{j})_{\ell}=\delta_{j\ell}$. In these cases the
expression in Eq. (7.11) is equal to $2$. If three $n_{j}$’s are nonzero we
cannot get anything smaller than $3$, so we can conclude that the previous
solutions complete the totality of the minima. Therefore, we find

$\begin{split}&\frac{\Theta(0|4\eta\Omega)}{\Theta(0|\eta\Omega)}\\\
&=\frac{1}{2^{\alpha-1}}\left\\{1+(1-x)^{\frac{1}{4\eta}}\left[2\sum_{j=1}^{\alpha-1}e^{-\frac{1}{8\eta\alpha}(\vec{e}_{j}+\vec{u})\cdot
B(\vec{e}_{j}+\vec{u})}+\sum_{j\neq
j^{\prime}=1}^{\alpha-1}e^{-\frac{1}{8\eta\alpha}(\vec{e}_{j}-\vec{e}_{j^{\prime}})\cdot
B(\vec{e}_{j}-\vec{e}_{j^{\prime}})}\right]+o\left((1-x)^{\frac{1}{4\eta}}\right)\right\\}\end{split}$
(7.12)

as $x\to 1$.

The coefficient of the subleading term in Eq. (7.12) can be simplified. From
the definition (7.6) of matrix $B$ we get

$(\vec{e}_{j}+\vec{u})\cdot
B(\vec{e}_{j}+\vec{u})=4\sum_{k=1}^{\alpha-1}f_{\frac{k}{\alpha}}\sin^{2}\left(\frac{\pi
kj}{\alpha}\right),\qquad(\vec{e}_{j}-\vec{e}_{j^{\prime}})\cdot
B(\vec{e}_{j}-\vec{e}_{j^{\prime}})=4\sum_{k=1}^{\alpha-1}f_{\frac{k}{\alpha}}\sin^{2}\left(\frac{\pi
k(j-j^{\prime})}{\alpha}\right)\;.$ (7.13)

Now, similarly to what was done in Ref. Calabrese2011Entanglement  for the
small $x$ expansion, using the definition of $f(k/\alpha)$ in Eq.
(LABEL:def_f_euler_gamma) and the integral representation of the polygamma
function

$\psi\left(\frac{k}{\alpha}\right)+\gamma_{E}=\int_{0}^{\infty}\frac{e^{-t}-e^{-\frac{k}{\alpha}t}}{1-e^{-t}}dt\;,$
(7.14)

exchanging the order of sum and integral and then performing the latter, after
simple algebra we find

$\sum_{k=1}^{\alpha-1}f_{\frac{k}{\alpha}}\sin^{2}\left(\frac{\pi
kj}{\alpha}\right)=-\alpha\log\alpha-\alpha\log\left[2\sin\left(\frac{\pi
j}{\alpha}\right)\right],\qquad j=1,2,\ldots,\alpha-1\;.$ (7.15)

It follows

$\frac{\Theta(0|4\eta\Omega)}{\Theta(0|\eta\Omega)}=\frac{1}{2^{\alpha-1}}\left[1+s(\alpha;\tfrac{1}{4\eta})\left(\frac{1-x}{4\alpha^{2}}\right)^{\frac{1}{4\eta}}+o\left((1-x)^{\frac{1}{4\eta}}\right)\right],$
(7.16)

where the function $s$ is defined in Eq. (4.8).

Finally, the large $x$ expansion of the tripartite information follows from
Eqs (7.2), (4.6) and (7.16),

$I_{3}^{(\alpha)}(x)=-\log 2\
+\frac{s(\alpha;\eta)}{\alpha-1}\left(\frac{1-x}{4\alpha^{2}}\right)^{\eta}+\frac{s(\alpha;\tfrac{1}{4\eta})}{\alpha-1}\left(\frac{1-x}{4\alpha^{2}}\right)^{\frac{1}{4\eta}}+o\left((1-x)^{\min(\eta,\tfrac{1}{4\eta})}\right),\quad\textrm{as
}x\to 1.$ (7.17)

The large $x$ expansion of the von Neumann tripartite information follows
immediately, using formula (4.9). The result is given in Eq. (2.22).

#### A physical argument.

Our predictions are based on the conjecture that the contributions from spin
structures associated with operators that have a fermionic string in the
central block are killed by the global quench. In addition, in the limit of
small quench the other contributions are not affected at all by the quench and
keep their ground state values. If that picture is correct, we can interpret
the residual tripartite information as a measure of the fraction of
correlations surviving the quench. Imagine indeed to focus on the contribution
from a correlation corresponding to a product of spins. For it to be nonzero,
the number of $\bm{\mathbf{\sigma}}^{x,y}$ in the first and third block should
be even (otherwise it would be represented with a string of JW fermions in the
block in between). On the other hand, in the ground state of the model, if
$|B|\ll|A|,|C|$, $\mathrm{tr}[\rho_{A\cup
C}^{\alpha}]\gg\mathrm{tr}[\rho_{(A\setminus A_{R})\cup(C\setminus
C_{L})}^{\alpha}]$, whenever $A_{R}\in A$ and $C_{L}\in C$, both adjacent to
$B$, have extent comparable with $A$ and $C$ (the mutual information diverges
in the limit of zero distance). Since $\mathrm{tr}[\rho_{A\cup C}^{\alpha}]$
and $\mathrm{tr}[\rho_{(A\setminus A_{R})\cup(C\setminus C_{L})}^{\alpha}]$
share the correlations in which there are no spins in $A_{R}\cup C_{L}$, it is
reasonable to assume that the typical correlation contributing to
$\mathrm{tr}[\rho_{A\cup C}^{\alpha}]$ should feature spin operators with a
distance from the edges of the central block that is negligible with respect
to the lengths of $A$ and $C$. We now argue that the value of a typical
correlation in the ground state should not be affected by a small displacement
(by a length comparable with $|B|$) of one of its spins. Let the latter be the
operator $\bm{\mathbf{\sigma}}^{x,y}$ closest to the edge with $B$. Moving it
to the other block is not expected to change its contribution to the ground
state entropy, but it changes the parity of the number of spins in the blocks.
Thus, such a small displacement maps a correlation that is relevant to the
quench tripartite information to an irrelevant one, both of them giving the
same contribution in the ground state of the model. This suggests that half of
the correlations contributing to the ground state tripartite information are
killed by the global quench, reducing the space of relevant correlations by
$\frac{1}{2}$. The logarithm of such ratio matches indeed our predictions for
the residual tripartite information.

##  8 Conclusions

We revisited the Rényi-$\alpha$ entropies of two disjoint blocks in the ground
state of the XXZ quantum spin chain, focusing on the universal contribution
that is captured by the tripartite information of the corresponding three
adjacent blocks. We have pointed out the connection between the tripartite
information and the partition functions of the underlying massless Thirring
quantum field theory with different spin structures. This has allowed us to
uncover the behaviour of the tripartite information in other models, e.g., in
the fermionic XXZ chain. For small cross ratio $x$, using the methods of Ref.
Calabrese2011Entanglement , we computed the limit $\alpha\to 1^{+}$ of the
analytic continuation, obtaining in turn also the small $x$ expansion of the
von Neumann tripartite information.

The results for the fermionic XXZ chain generalise the works of Refs
Casini2005 ; Casini2009reduced_density ; Casini2009Entanglement  on the
entanglement entropy of disjoint blocks in non-interacting hopping chains,
which are described by the free massless Dirac fermion. While for the latter
the tripartite information is zero for any $\alpha$, the tripartite
information in the fermionic XXZ chain, described by the massless Thirring QFT
with NS boundary conditions on all cycles, exhibits a more interesting
behavior. In particular, it can be both positive and negative.

Very recently Ref. Fujimura2023  studied the Rényi-$2$ entanglement entropy of
two disjoint intervals in the massless Thirring model, with the main focus on
the NS-NS spin structure of the torus. This problem is equivalent to the one
of the fermionic XXZ chain, studied in this work. They have obtained the
results by expressing the fermionic partition function in terms of bosonic
ones, in accordance with the duality discussed in Ref. Karch2019 . Our result
(2.4) for $\alpha=2$, while manifestly in a different form, is in agreement
with their result, and, moreover, the numerical checks we have performed
provide also a check for their formulas.

The identification of the contributions from different spin structures in the
XXZ spin chain has also allowed us to predict the behaviour of the tripartite
information in a family of models that we called “Jordan-Wigner-$\nu$
deformations”, the XXZ spin chain corresponding to the special case $\nu=1$.
Those are simple models with interactions beyond the two-body ones,
incorporating $\nu+1$ adjacent spins. Their low-energy properties are
described by a sum of massless Thirring models for $\nu$ different species of
fermions with correlated spin structures. The corresponding central charge is
equal to $\nu$. For $\nu=2$ we have obtained the curious result that the
tripartite information is exactly twice the one of the fermionic XXZ chain.
This provides an additional relation between the entanglement properties of
the fermionic XXZ model and those of a spin model. Furthermore, we have shown
that, for large enough $\nu$, the tripartite information in Jordan-
Wigner-$\nu$ deformations can become negative, generalizing the free case
addressed in Ref. Fagotti2012New .

Finally, the identification of the operators contributing to each spin
structure has allowed us to speculate about the stationary behavior of the
tripartite information at late times after a small quantum quench of the
anisotropy in the gapless XXZ spin chain. The resulting predictions depend on
the anisotropy and reduce to exact formulas analytically computed in Refs
Maric2022Universality ; Maric2023universality2 ; Maric2023Universality  for
the free case $\Delta=0$. We also extracted the limit $x\to 1^{-}$, which
corresponds to configurations in which the central subsystem is much smaller
than the others. We found that the tripartite information becomes equal to
$-\log 2$ irrespectively of the model parameters. Refs Maric2022Universality ;
Maric2023universality2 ; Maric2023Universality  uncovered the same universal
value in a variety of non-interacting quantum spin chains and termed it
“residual tripartite information”. The results of our work suggest that such
universality is likely to extend to interacting integrable models.

There are several open problems that our work brings to light, particularly in
the direction of quantum quenches. First, our conjecture about the asymptotic
behaviour of the tripartite information after a small quench in an interacting
integrable model should be confirmed and generalised to larger quenches by an
exact calculation of time evolution after the quench. Second, Ref.
Hayden2013Holographic  argued that a negative tripartite information should
suggest a domination of quantum entanglement over classical correlations.
Since we have confirmed that the residual tripartite information after global
quenches is negative, it is natural to wonder whether that is measuring a
quantum or classical feature (or both of them). To that aim it might be useful
to study some other quantity in which the distinction between classical and
quantum correlations is sharper, for example the entanglement negativity
Vidal2002 ; Calabrese2012negativityPRL . We also wonder whether the residual
tripartite information might be connected to some unusual property of the
entanglement Hamiltonian Dalmonte2022  of disjoint subsystems.

###### Acknowledgements.

VM is grateful to Shunsuke Furukawa, Tomotaka Kuwahara, Vincent Pasquier,
Tadashi Takayanagi and Tomonori Ugajin for stimulating discussions on topics
related to this work.

#### Funding information

This work was supported by the European Research Council under the Starting
Grant No. 805252 LoCoMacro. SB acknowledges the financial support from PEPR-Q
(QubitAF project).

##  Appendix A Spin structures and gapped spin chains

In this appendix we discuss how to show the formulas given in Sec. 3.3.

#### Disordered phases.

Let us consider the fermionic pseudo-RDMs (3.17) underlying the expansion in
the spin structures. After separating the contribution between even and odd
observables, clustering allows us for an immediate simplification

$\displaystyle\rho_{AC}^{+,s^{\prime}}\sim$
$\displaystyle\rho_{A}^{(e)}\otimes\rho_{C}^{(e)}+s^{\prime}\frac{1}{2^{|A|+|C|}}\sum_{\bm{\mathbf{O}}_{A}^{(o)},\bm{\mathbf{O}}_{C}^{(o)}}\braket{\bm{\mathbf{O}}_{A}^{(o)}\bm{\mathbf{T}}(v_{1})\bm{\mathbf{T}}(u_{2}-1)\bm{\mathbf{O}}_{C}^{(o)}}O_{A}O_{C}$
(A.1) $\displaystyle\rho_{AC}^{-,s^{\prime}}\sim$
$\displaystyle\frac{1}{2^{|A|+|C|}}\sum_{\bm{\mathbf{O}}_{A}^{(e)},\bm{\mathbf{O}}_{C}^{(e)}}\braket{\bm{\mathbf{O}}_{A}^{(e)}\bm{\mathbf{T}}(v_{1})\bm{\mathbf{T}}(u_{2}-1)\bm{\mathbf{O}}^{(e)}_{C}}O_{A}O_{C}+s^{\prime}\rho_{A}^{(o)}\otimes\rho_{C}^{(o)}$
$\displaystyle\qquad\qquad\qquad\overset{sp.flip}{=}\frac{1}{2^{|A|+|C|}}\sum_{\bm{\mathbf{O}}_{A}^{(e)},\bm{\mathbf{O}}_{C}^{(e)}}\braket{\bm{\mathbf{O}}_{A}^{(e)}\bm{\mathbf{T}}(v_{1})\bm{\mathbf{T}}(u_{2}-1)\bm{\mathbf{O}}^{(e)}_{C}}O_{A}O_{C}$
(A.2)

where the last identity follows from spin-flip symmetry, we introduced the
semilocal string $\bm{\mathbf{T}}(u)=(-1)^{\sum_{\ell\leq
u}\bm{\mathbf{c}}^{\dagger}_{\ell}\bm{\mathbf{c}}_{\ell}}$, and we used the
notations for the boundaries of the intervals $A=[u_{1},v_{1}]$ and
$C=[u_{2},v_{2}]$. These expressions can be simplified using clustering for
semilocal operators111Clustering for semilocal operators can be understood as
follows. In the product state with all spins aligned in the $z$ direction, the
second term of (A.1) vanishes and the first term matches
$\rho_{A}^{(e)}\otimes\rho_{C}^{(e)}$. We then observe that any spin-flip
symmetric state with a finite correlation length is mapped to that product
state by a spin-flip symmetric quasilocal unitary transformation. Rather than
transforming the product state, we can apply the inverse transformation to the
operators. Such a transformation would preserve the parity of the operators
and map the string $(-1)^{\bm{\mathbf{F}}_{B}}$ into itself, up to even
unitary operators
$\bm{\mathbf{Q}}_{\partial_{L}B}\bm{\mathbf{Q}}_{\partial_{R}B}$
quasilocalised around the boundaries $\partial_{L}B$ and $\partial_{R}B$ of
$B$:
$(-1)^{\bm{\mathbf{F}}_{B}}\rightarrow\bm{\mathbf{Q}}_{\partial_{L}B}\bm{\mathbf{Q}}_{\partial_{R}B}(-1)^{\bm{\mathbf{F}}_{B}}\bm{\mathbf{Q}}^{-1}_{\partial_{R}B}\bm{\mathbf{Q}}^{-1}_{\partial_{L}B}$.
Since the string acts as $1$ on the product state, this allows us to use
clustering also in this case., which gives

$\displaystyle\rho_{AC}^{+,s^{\prime}}\sim$
$\displaystyle\rho_{A}^{(e)}\otimes\rho_{C}^{(e)}$ (A.3)
$\displaystyle\rho_{AC}^{-,s^{\prime}}\sim$
$\displaystyle\left(\frac{1}{2^{|A|}}\sum_{\bm{\mathbf{O}}_{A}^{(e)}}\braket{\bm{\mathbf{O}}_{A}^{(e)}\bm{\mathbf{T}}(v_{1})}O_{A}\right)\otimes\left(\frac{1}{2^{|C|}}\sum_{\bm{\mathbf{O}}_{C}^{(e)}}\braket{\bm{\mathbf{T}}(u_{2}-1)\bm{\mathbf{O}}_{C}^{(e)}}O_{C}\right)$
(A.4)

We can manipulate the pseudo-RDMs in each block as follows

$\sum_{\bm{\mathbf{O}}_{A}^{(e)}}\braket{\bm{\mathbf{O}}_{A}^{(e)}\bm{\mathbf{T}}(v_{1})}\bm{\mathbf{O}}_{A}=\sum_{\bm{\mathbf{O}}_{A}^{(e)}}\braket{\bm{\mathbf{O}}_{A}^{(e)}\bm{\mathbf{T}}(x)}\bm{\mathbf{O}}_{A}\bm{\mathbf{T}}(x)\bm{\mathbf{T}}(v_{1})=\bm{\mathbf{R}}_{A}^{(e)}\bm{\mathbf{U}}_{x}^{\dagger}\bm{\mathbf{T}}(x)\bm{\mathbf{T}}(v_{1})\equiv\bm{\mathbf{T}}(v_{1})\bm{\mathbf{T}}(x)\bm{\mathbf{U}}_{x}\bm{\mathbf{R}}_{A}^{(e)}$
(A.5)

where $\bm{\mathbf{U}}_{x}$ is a unitary transformation quasilocalised around
$x$, which we assumed to be deep in the bulk of $A$, such that
$\bm{\mathbf{T}}(x)\ket{\Psi}=\bm{\mathbf{U}}_{x}\ket{\Psi}$. The last
identity in (A.5) comes from applying $\bm{\mathbf{T}}(x)$ to the bra instead
of the ket. Note that, even if the result is supposed to be independent of
$x$, we cannot simply replace
$\bm{\mathbf{U}}_{x}^{\dagger}\bm{\mathbf{T}}(x)$ by the identity. On the
other hand we have
$(\sum_{\bm{\mathbf{O}}_{A}^{(e)}}\braket{\bm{\mathbf{O}}_{A}^{(e)}\bm{\mathbf{T}}(v_{1})}O_{A})^{2}=[\rho_{A}^{(e)}]^{2}$
which implies that the pseudo-RDM is equal to $\rho_{A}^{(e)}$ up to changes
of signs in the eigenvalues. Since each term of the spin structure of the
tripartite information consists of an even number of
$\rho^{-,s^{\prime}}_{AC}$, this difference is irrelevant, so we can overlook
that subtlety and assume that all pseudo-RDMs are identical, which gives
(3.30).

#### Ordered phases.

Using clustering and the vanishing of the expectation values of strings of the
form $(-1)^{\bm{\mathbf{F}}_{X}}$ in the limit of large $|X|$ in $\ket{\psi}$
we find

$\displaystyle\rho_{AC}^{+,s^{\prime}}\sim\frac{1}{2^{|A|+|C|}}\sum_{\bm{\mathbf{O}}_{A}^{(e)},\bm{\mathbf{O}}_{C}^{(e)}}\braket{\psi}{\bm{\mathbf{O}}_{A}^{(e)}\bm{\mathbf{O}}_{C}^{(e)}}{\psi}O_{A}O_{C}\rightarrow\rho_{A;\ket{\psi}}^{+}\otimes\rho_{C;\ket{\psi}}^{+}$
(A.6) $\displaystyle\rho_{AC}^{-,s^{\prime}}\sim
s^{\prime}\frac{1}{2^{|A|+|C|}}\sum_{\bm{\mathbf{O}}_{A}^{(o)},\bm{\mathbf{O}}_{C}^{(o)}}\braket{\psi}{\bm{\mathbf{O}}_{A}^{(o)}\bm{\mathbf{O}}_{C}^{(o)}}{\psi}O_{A}O_{C}\rightarrow
s^{\prime}\rho_{A;\ket{\psi}}^{-}\otimes\rho_{C;\ket{\psi}}^{-}$ (A.7)

Thus, each term of the spin structure is proportional to the trace of a
product of $\rho_{A;\ket{\psi}}^{\pm}$ and $\rho_{C;\ket{\psi}}^{\pm}$. We
note

$\rho_{A;\ket{\psi}}^{\pm}=\frac{\rho_{A;\ket{\psi}}\pm(-1)^{\bm{\mathbf{F}}_{A}}\rho_{A;\ket{\psi}}(-1)^{\bm{\mathbf{F}}_{A}}}{2}=\frac{\rho_{A;\ket{\psi}}\pm\rho_{A;(-1)^{\bm{\mathbf{F}}}\ket{\psi}}}{2}\,.$
(A.8)

Since $\rho_{A;\ket{\psi}}$ satisfies the area law, it has only a finite
number of eigenvalues that are significantly different from zero. Let us call
them $\ket{\psi_{i},A}$ with eigenvalues $p_{i}$ such that $p_{i}\geq
p_{i+1}$. By assumption we have

$|\sum_{i}p_{i}\braket{\psi_{i},A}{(-1)^{\bm{\mathbf{F}}_{A}}}{\psi_{i},A}|=|\mathrm{tr}[(-1)^{\bm{\mathbf{F}}_{A}}\rho_{A;\ket{\psi}}]|\leq
e^{-\kappa|A|}$ (A.9)

for some $\kappa>0$ and it is reasonable to expect the inequality to hold term
by term, _i.e._ ,
$|\braket{\psi_{i}}{(-1)^{\bm{\mathbf{F}}_{A}}}{\psi_{i}}|\leq
e^{-\kappa|A|}$. This implies

$\mathrm{tr}[\rho_{A;\ket{\psi}}^{s_{1}}\cdots\rho_{A;\ket{\psi}}^{s_{n}}]\sim\frac{1}{2^{n-1}}\mathrm{tr}[\rho_{A;\ket{\psi}}^{n}]\,.$
(A.10)

Thus, in the spin structure the only information that matters is how many
$\rho_{AC}^{-,-}$ are present in a particular term, indeed each of them
carries a factor $-1$. This is exactly (3.32), which is therefore expected to
hold in general within this class of states.

##  Appendix B Reproducing the tripartite information of the XXZ chain

In this appendix we show how summing the massless Thirring model over all spin
structures reproduces the tripartite information of the XXZ chain (2.4),
obtained from the consideration of the free compactified boson
Calabrese2009Entanglement1 . We also derive some useful identities. The
tripartite information of the XXZ chain is obtained by summing over all spin
structures, as given by Eq. (3.18),

$I_{3}^{(\alpha)}(x)=\frac{1}{\alpha-1}\log\left(\frac{1}{2^{\alpha-1}}\frac{\sum\limits_{\vec{e}}\Theta\left[\begin{matrix}\vec{E}\\\
\vec{F}\end{matrix}\right]\left(\vec{0}|\tilde{\Omega}\right)}{\left[\Theta\left(\vec{0}|\Omega\right)\right]^{2}}\right)\;.$
(B.1)

We show that (B.1) reproduces the formula (4.5) obtained from the
considerations of the free compactified boson in Ref.
Calabrese2009Entanglement1 .

We remind the reader that the Riemannn-Siegel theta function with
characteristic is defined as follows

$\Theta\left[\begin{matrix}\vec{E}\\\
\vec{F}\end{matrix}\right]\left(\vec{0}|\tilde{\Omega}\right)=\sum_{\vec{M}\in\mathbb{Z}^{2(\alpha-1)}}e^{i\pi(\vec{M}+\vec{E})^{t}\tilde{\Omega}(\vec{M}+\vec{E})+2\pi
i\vec{M}\cdot\vec{F}}\,.$ (B.2)

Let us then split the sum over $\vec{M}$ into two sums

$\sum_{\vec{M}}\equiv\sum_{\vec{\mu}\in\\{0,\frac{1}{2}\\}^{\alpha-1}}\sum_{\vec{m},\vec{m}^{\prime}\in\mathbb{Z}^{\alpha-1}}\qquad\text{with}\quad\vec{M}=\begin{pmatrix}1\\\
1\end{pmatrix}\otimes(\vec{m}+\vec{\mu})+\begin{pmatrix}1\\\
-1\end{pmatrix}\otimes(\vec{m}^{\prime}+\vec{\mu})$ (B.3)

We then find

$\sum_{\vec{\mu}\in\\{0,\frac{1}{2}\\}^{\alpha-1}}(-1)^{4\vec{\mu}\cdot\vec{\delta}}\sum_{\vec{m},\vec{m}^{\prime}\in\mathbb{Z}^{\alpha-1}}\exp\left[i\pi
4\eta(\vec{m}^{\prime}+\vec{\varepsilon}+\vec{\mu})^{t}\Omega(\vec{m}^{\prime}+\vec{\varepsilon}+\vec{\mu})+i\pi\tfrac{1}{\eta}(\vec{m}+\vec{\mu})^{t}\Omega(\vec{m}+\vec{\mu})\right]\,,$
(B.4)

which can be compactly written as

$\Theta\left[\begin{matrix}\vec{E}\\\
\vec{F}\end{matrix}\right]\left(\vec{0}|\tilde{\Omega}\right)=\sum_{\vec{\mu}\in\\{0,\frac{1}{2}\\}^{\alpha-1}}(-1)^{4\vec{\mu}\cdot\vec{\delta}}\Theta\left[\begin{matrix}\vec{\mu}\\\
\vec{0}\end{matrix}\right]\left(\vec{0}|\tfrac{1}{\eta}\Omega\right)\Theta\left[\begin{matrix}\vec{\varepsilon}+\vec{\mu}\\\
\vec{0}\end{matrix}\right]\left(\vec{0}|4\eta\Omega\right)\,.$ (B.5)

Incidentally, we point out the following identity:

$\Theta\left[\begin{matrix}\vec{E}\\\
\vec{F}\end{matrix}\right]\left(\vec{0}|\tilde{\Omega}\right)\xrightarrow{\eta\rightarrow\frac{1}{4\eta}}(-1)^{4\vec{\varepsilon}\cdot\vec{\delta}}\Theta\left[\begin{matrix}\vec{E}\\\
\vec{F}\end{matrix}\right]\left(\vec{0}|\tilde{\Omega}\right)\;,$ (B.6)

obtained by shifting the summation index as
$\vec{\mu}\to\vec{\varepsilon}+\vec{\mu}$. This also gives the identity
(2.10).

Therefore we have

$I_{3}^{(\alpha)}(x)=\frac{1}{\alpha-1}\log\left(\frac{1}{2^{\alpha-1}}\frac{\sum\limits_{\vec{\varepsilon},\vec{\delta},\vec{\mu}}(-1)^{4\vec{\mu}\cdot\vec{\delta}}\Theta\left[\begin{smallmatrix}\vec{\mu}\\\
\vec{0}\end{smallmatrix}\right]\left(\vec{0}|\tfrac{1}{\eta}\Omega\right)\Theta\left[\begin{smallmatrix}\vec{\varepsilon}+\vec{\mu}\\\
\vec{0}\end{smallmatrix}\right]\left(\vec{0}|4\eta\Omega\right)}{\left|\Theta\left(\vec{0}|\Omega\right)\right|^{2}}\right)$
(B.7)

Let us work out the sums. If we sum over $\vec{\delta}$ we get

$\sum_{\vec{\delta}\in\\{0,\frac{1}{2}\\}^{\alpha-1}}\Theta\left[\begin{matrix}\vec{E}\\\
\vec{F}\end{matrix}\right]\left(\vec{0}|\tilde{\Omega}\right)=2^{\alpha-1}\Theta\left(\vec{0}|\tfrac{1}{\eta}\Omega\right)\Theta\left[\begin{matrix}\vec{\varepsilon}\\\
\vec{0}\end{matrix}\right]\left(\vec{0}|4\eta\Omega\right)\,.$ (B.8)

If we sum also over $\vec{\varepsilon}$ we find

$\sum_{\vec{\delta},\vec{\varepsilon}\in\\{0,\frac{1}{2}\\}^{\alpha-1}}\Theta\left[\begin{matrix}\vec{E}\\\
\vec{F}\end{matrix}\right]\left(\vec{0}|\tilde{\Omega}\right)=2^{\alpha-1}\Theta\left(\vec{0}|\tfrac{1}{\eta}\Omega\right)\sum_{\vec{\varepsilon}\in\\{0,\frac{1}{2}\\}^{\alpha-1}}\sum_{\vec{m}^{\prime}\in\mathbb{Z}^{\alpha-1}}\exp\left[i4\pi\eta(\vec{m}^{\prime}+\vec{\varepsilon})^{t}\Omega(\vec{m}^{\prime}+\vec{\varepsilon})\right]=\\\
2^{\alpha-1}\Theta\left(\vec{0}|\tfrac{1}{\eta}\Omega\right)\sum_{\vec{\varepsilon}\in\\{0,\frac{1}{2}\\}^{\alpha-1}}\sum_{\vec{m}^{\prime}\in\mathbb{Z}^{\alpha-1}}\exp\left[i\pi\eta(2\vec{m}^{\prime}+2\vec{\varepsilon})^{t}\Omega(2\vec{m}^{\prime}+2\vec{\varepsilon})\right]=2^{\alpha-1}\Theta\left(\vec{0}|\tfrac{1}{\eta}\Omega\right)\Theta\left(\vec{0}|\eta\Omega\right)\,,$
(B.9)

which is consistent with the result (4.5).

##  Appendix C Noninteracting systems: free fermion techniques

In non-interacting spin chains the computation of the entanglement entropies
is based on the matrix of fermionic two-point correlations and the Wick
theorem. The two-point correlation matrix reads
$\Gamma_{2\ell+i,2n+j}=\delta_{\ell
n}\delta_{ij}-\braket{\vec{\bm{\mathbf{a}}}_{\ell}\vec{\bm{\mathbf{a}}}_{n}^{\dagger}}_{ij}$,
where we have introduced the Majorana fermions
$\bm{\mathbf{a}}_{2\ell-1}=\bm{\mathbf{c}}_{\ell}+\bm{\mathbf{c}}_{\ell}^{\dagger}=\bm{\mathbf{\sigma}}_{\ell}^{x}\prod_{j<\ell}\bm{\mathbf{\sigma}}_{j}^{z},\
\bm{\mathbf{a}}_{2\ell}=i(\bm{\mathbf{c}}_{\ell}-\bm{\mathbf{c}}_{\ell}^{\dagger})=\bm{\mathbf{\sigma}}_{\ell}^{y}\prod_{j<\ell}\bm{\mathbf{\sigma}}_{j}^{z}$
and the shorthand vector notation
$\vec{\bm{\mathbf{a}}_{\ell}}=(\bm{\mathbf{a}}_{2\ell-1},\
\bm{\mathbf{a}}_{2\ell})^{\mathrm{T}}$. The Majorana fermions are self-adjoint
and satisfy the anticommutation relations
$\\{\bm{\mathbf{a}}_{\ell},\bm{\mathbf{a}}_{n}\\}=2\delta_{\ell n}$. We refer
the reader to Ref. Maric2023Universality  (Table 1) for a brief review how to
compute the correlation matrix in different settings. In particular, in the
ground state of the quantum XY chain

$H=\sum_{\ell}(1+\gamma)\bm{\mathbf{\sigma}}_{\ell}^{x}\bm{\mathbf{\sigma}}_{\ell+1}^{x}+(1-\gamma)\bm{\mathbf{\sigma}}_{\ell}^{y}\bm{\mathbf{\sigma}}_{\ell+1}^{y}+2h\bm{\mathbf{\sigma}}_{\ell}^{z}$
(C.1)

we have

$\begin{split}&\Gamma_{2j+1,2\ell+2}=-\Gamma_{2\ell+2,2j+1}=\int\limits_{-\pi}^{\pi}\frac{-\gamma\sin
k+i(h-\cos k)}{|-\gamma\sin k+i(h-\cos k)|}e^{ik(j-\ell)}\frac{dk}{2\pi}\,,\\\
&\Gamma_{2j+1,2\ell+1}=\Gamma_{2j+2,2\ell+2}=0\;.\end{split}$ (C.2)

In the case of a single block $A$, the entropy of spins is equal to the
entropy of the corresponding block of Jordan-Wigner fermions Jin2004 ;
Vidal2003 ; Peschel2003 . It can be written in terms of the correlation matrix
$\Gamma_{A}=(\Gamma_{j,\ell})_{j,\ell=1}^{2|A|}$,

$S_{\alpha}(A)=\tfrac{1}{1-\alpha}\tfrac{1}{2}\mathrm{tr}\Bigl{[}\log\Bigl{(}\bigl{(}\tfrac{\mathrm{I}+\Gamma_{A}}{2}\bigr{)}^{\alpha}+\bigl{(}\tfrac{\mathrm{I}-\Gamma_{A}}{2}\bigr{)}^{\alpha}\Bigr{)}\Bigr{]}\,.$
(C.3)

Such a simplicity is related to the property that the reduced density matrix
(3.2) is a fermionic Gaussian, i.e. an operator of the form
$\rho(\Gamma)\propto\exp{\left(\sum_{j,\ell}a_{j}^{\dagger}W_{j,l}a_{\ell}/4\right)}$,
where $W$ is an antisymmetric (in general complex) matrix. Clearly, the
moments appearing in Eq. (3.19) can be then computed using
$\mathrm{tr}\rho_{A}^{\alpha}=\exp[(1-\alpha)S_{\alpha}(A)]$.

In the case of disjoint blocks 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 Ref. Fagotti2010disjoint . The
main ingredient of this procedure is to notice that each pseudo-RDM (3.17) is
a fermionic Gaussian. Then the procedure is reduced to computing the trace of
products of different Gaussians.

The pseudo-RDMs $\rho_{AC}^{s,s^{\prime}}$ can be expressed in terms of four
correlation matrices: $\Gamma_{1}\equiv\Gamma^{+,+}\equiv\Gamma_{A\cup C,A\cup
C}$, $\Gamma_{2}\equiv\Gamma^{+,-}\equiv\mathrm{P}\Gamma_{1}\mathrm{P}$,
$\Gamma_{3}\equiv\Gamma^{-,+}\equiv\Gamma_{1}-\Gamma_{A\cup
C,B}\Gamma_{B,B}^{-1}\Gamma_{B,A\cup C}$ and
$\Gamma_{4}\equiv\Gamma^{-,-}\equiv P\Gamma_{3}P$, 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}$ (C.4)

and

$\mathrm{P}_{\ell n}\equiv\delta_{\ell n}\begin{cases}-1&1\leq\ell\leq 2|A|\\\
1&\text{otherwise}\end{cases}$ (C.5)

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

$\mathrm{tr}\left[\prod\limits_{k=1}^{\alpha}\rho_{AC}^{s_{k},s_{k}^{\prime}}\right]=|\det\Gamma_{B,B}|^{\frac{N^{-+}+N^{--}}{2}}\\{\Gamma^{s_{1},s_{1}^{\prime}},\Gamma^{s_{2},s_{2}^{\prime}},\ldots,\Gamma^{s_{\alpha},s_{\alpha}^{\prime}}\\}\,,$
(C.6)

where $N^{-+}$ and $N^{--}$ is the number of matrices $\Gamma^{-+}$ and
$\Gamma^{--}$, respectively, appearing in the expression.

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

$\rho(\Gamma)\rho(\Gamma^{\prime})=\\{\Gamma,\Gamma^{\prime}\\}\rho\left(\Gamma\times\Gamma^{\prime}\right),\quad\Gamma\times\Gamma^{\prime}\equiv\mathrm{I}-(\mathrm{I}-\Gamma^{\prime})(\mathrm{I}+\Gamma\Gamma^{\prime})^{-1}(\mathrm{I}-\Gamma)\,,$
(C.7)

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

$\\{\Gamma,\Gamma^{\prime}\\}^{2}=\mathrm{tr}[\rho(\Gamma)\rho(\Gamma^{\prime})]^{2}=\det\bigg{(}\frac{\mathrm{I}+\Gamma\Gamma^{\prime}}{2}\bigg{)}\;.$
(C.8)

Alternatively, (C.7) could be replaced by the formula for the trace of a
product of an arbitrary number of Gaussians reported in Ref. Klich2014 . Note
that (C.8) 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 eigenvalues with halved degeneracy Fagotti2010disjoint :

$\\{\Gamma,\Gamma^{\prime}\\}=\prod\limits_{\\{\lambda\\}/2}\lambda\,,$ (C.9)

where the product is over all eigenvalues $\lambda$ of the matrix
$(I+\Gamma\Gamma^{\prime})/2$ with degeneracy, that is always even, reduced by
half.

##  Appendix D About the effective parameters in the interleaved model

Figure 12: The same as in Fig. 4a, where open point markers correspond to data
obtained using the parameters $\gamma,h$ of the interleaved model (6.2) in Eq.
(6.4), whereas filled point markers correspond to renormalised data. The
latter data are obtained by choosing effective parameters in the XY subchain
so as to match the numerical expectation values of two local observables. The
R-R is not affected by the change, but the other spin structures are very
sensitive to it and only the renormalised data show good agreement with the
predictions. See the discussion around Eq. (6.5).

Here we comment a detail about the implementation of our numerical procedure.
Around eq. (6.5) we have discussed the importance of using the effective
parameters $\gamma,h$ in Eq. 6.4, obtained by considering local observables in
our tensor network simulations, instead of the ones we initially put in the
Hamiltonian. In Fig. 12 we explicitly compare the results of Fig. 4a, obtained
with the effective parameters, with the ones obtained using the initial ones.
There we can see the importance of using the effective values.

##  Appendix E From the tripartite information of Jordan-Wigner spin
deformations to XXZ spin structure

The tripartite information of the JW deformation has the form

$e^{(\alpha-1)[I_{3}^{(\alpha;\nu)}(x)+\log
2]}=\sum_{n=1}^{M_{\alpha}}g_{\alpha,n}\gamma_{\alpha,n}^{\nu}-\sum_{n=1}^{\tilde{M}_{\alpha}}\tilde{g}_{\alpha,n}(-\gamma_{\alpha,n})^{\nu}\,,$

where each $\gamma$ is a ratio of partition functions and $g_{\alpha,n}$ are
positive integers that represent the degeneracy of the terms. We can generally
express $\\{\gamma_{\alpha,n}\\}$ in terms of
$\\{e^{(\alpha-1)[I_{3}^{(\alpha;\nu)}(x)+\log 2]}\\}$ if we consider a large
enough number of deformations. If $g_{\alpha,n}=1$ and
$\tilde{g}_{\alpha,n}=0$, we can readily express $\\{\gamma_{\alpha,n}\\}$ as
the solutions to the polynomial

$x^{M_{\alpha}}+\sum_{n=1}^{M_{\alpha}}(-1)^{n}e_{n}x^{M_{\alpha}-n}=0\,,$
(E.1)

where $e_{n}$ are the elementary symmetric polynomials, which can be expressed
in terms of the power-sum symmetric polynomials through the Newton’s
identities. Specifically, we have

$e_{n}=\frac{1}{n!}\det\left|\begin{matrix}e^{(\alpha-1)[I_{3}^{(\alpha;1)}(x)+\log
2]}&1&0&\dots&\\\ e^{(\alpha-1)[I_{3}^{(\alpha;2)}(x)+\log
2]}&e^{(\alpha-1)[I_{3}^{(\alpha;1)}(x)+\log 2]}&2&0\dots\\\
\vdots&&\ddots&\ddots\\\ e^{(\alpha-1)[I_{3}^{(\alpha;n-1)}(x)+\log
2]}&e^{(\alpha-1)[I_{3}^{(\alpha;n-2)}(x)+\log
2]}&\dots&e^{(\alpha-1)[I_{3}^{(\alpha;1)}(x)+\log 2]}&n-1\\\
e^{(\alpha-1)[I_{3}^{(\alpha;n)}(x)+\log
2]}&e^{(\alpha-1)[I_{3}^{(\alpha;n-1)}(x)+\log
2]}&\dots&e^{(\alpha-1)[I_{3}^{(\alpha;2)}(x)+\log
2]}&e^{(\alpha-1)[I_{3}^{(\alpha;1)}(x)+\log 2]}\end{matrix}\right|\,.$

Since each tripartite information is symmetric under $x\rightarrow 1-x$, also
the roots of this polynomial are, a priori, symmetric. The symmetry is however
broken if we require the solutions to be smooth in $x\in(0,1)$. Those will be
the functions that should be identified with $\\{\gamma_{\alpha,n}\\}$.

In order to solve the inverse problem for given $\alpha$, we will have to
relax the constraints on $g_{\alpha,n}$ and $\tilde{g}_{\alpha,n}$ that we
imposed for the sake of exploration. Note however that the simplified solution
that we exhibited above will be useful to derive the exact expressions.

####  $2$-sheeted Riemann (torus, $g=1$).

In the $2$-sheeted Riemann surface, $M_{2}=3$, $\tilde{M}_{2}=1$, and the
coefficients are all equal to $1$. To solve this problem, it is convenient to
treat $\gamma_{2,4}$ as it were a correction “renormalizing” the tripartite
information (this abuse of terminology comes from the observation that, like
any other term multiplying the coefficients $\tilde{g}_{\alpha,n}$ in the
Riemann surfaces with genus $g$, $\gamma_{2,4}$ is zero in the noninteracting
case Coser2016Spin ). Let us then define

$e^{\tilde{I}_{3}^{(2;j)}(x)}=e^{I_{3}^{(2;j)}(x)}+(-1)^{j}\tfrac{\gamma_{2,4}^{j}(x)}{2}$
(E.2)

We then have

$\displaystyle e_{1}=$ $\displaystyle
2e^{\tilde{I}_{3}^{(2;1)}(x)}=2e^{I_{3}^{(2;1)}(x)}-\gamma_{2,4}(x)$ (E.3)
$\displaystyle e_{2}=$ $\displaystyle
2e^{2\tilde{I}_{3}^{(2;1)}(x)}-e^{\tilde{I}_{3}^{(2;2)}(x)}=2e^{2I_{3}^{(2;1)}(x)}-e^{I_{3}^{(2;2)}(x)}-2e^{I_{3}^{(2;1)}(x)}\gamma_{2,4}(x)$
$\displaystyle e_{3}=$
$\displaystyle\tfrac{4e^{3\tilde{I}_{3}^{(2;1)}(x)}+2e^{\tilde{I}_{3}^{(2;3)}(x)}-6e^{\tilde{I}_{3}^{(2;2)}(x)}e^{\tilde{I}_{3}^{(2;1)}(x)}}{3}=\tfrac{4e^{3I_{3}^{(2;1)}(x)}+2e^{I_{3}^{(2;3)}(x)}-6e^{I_{3}^{(2;2)}(x)}e^{I_{3}^{(2;1)}(x)}}{3}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad-(2e^{2I_{3}^{(2;1)}(x)}-e^{I_{3}^{(2;2)}(x)})\gamma_{2,4}(x)\,.$

Before discussing the roots of the corresponding polynomial—(E.1), let us
determine $\gamma_{2,4}(x)$. To that aim, it is convenient to rewrite (E.1)
with the unknown replaced by $\gamma_{2,j}(x)$, with $j\in\\{1,2,3\\}$,
multiply it by $\gamma_{2,j}(x)$ and finally sum over $j\in\\{1,2,3\\}$. This
gives a linear equation for $\gamma_{2,4}(x)$, which is solved by

$\gamma_{2,4}(x)=\frac{4e^{4I_{3}^{(2;1)}(x)}-12e^{2I_{3}^{(2;1)}(x)}e^{I_{3}^{(2;2)}(x)}+3e^{2I_{3}^{(2;2)}(x)}+8e^{I_{3}^{(2;1)}(x)}e^{I_{3}^{(2;3)}(x)}-3e^{I_{3}^{(2;4)}(x)}}{8e^{3I_{3}^{(2;1)}(x)}-12e^{I_{3}^{(2;1)}(x)}e^{I_{3}^{(2;2)}(x)}+4e^{I_{3}^{(2;3)}(x)}}\,.$
(E.4)

We point out that $\gamma_{2,4}(x)$ exhibits the symmetry $x\leftrightarrow
1-x$ and vanishes at $x=0$ (and $x=1$). We can now solve the cubic polynomial
in terms of $\tilde{I}_{3}^{(2,j)}(x)$ (they can be eventually expressed in
terms of $I_{3}^{(2,j)}(x)$ by using (E.4)). The roots of (E.1) read

$x_{k}=\frac{1}{3}\Bigl{(}e_{1}-e^{\frac{2\pi ik}{3}}C-e^{-\frac{2\pi
ik}{3}}\frac{\Delta_{0}}{C}\Bigr{)}\qquad k\in\\{-1,0,1\\}$

where

$\displaystyle C=$
$\displaystyle\sqrt[3]{\frac{\Delta_{1}+\sqrt{\Delta_{1}^{2}-4\Delta_{0}^{3}}}{2}}$
(E.5) $\displaystyle\Delta_{0}=$ $\displaystyle
e_{1}^{2}-3e_{2}=3e^{\tilde{I}_{3}^{(2;2)}(x)}-2e^{2\tilde{I}_{3}^{(2;1)}(x)}$
$\displaystyle\Delta_{1}=$
$\displaystyle-2e_{1}^{3}+9e_{1}e_{2}-27e_{3}=-16e^{3\tilde{I}_{3}^{(2;1)}(x)}+36e^{\tilde{I}_{3}^{(2;1)}(x)}e^{\tilde{I}_{3}^{(2;2)}(x)}-18e^{\tilde{I}_{3}^{(2;3)}(x)}$

For $x=0$ we can choose

$e_{1}=2\qquad\Delta_{0}=1\qquad\Delta_{1}=2\qquad C=1$

which gives

$x_{k}(0)=\frac{4}{3}\sin^{2}(\pi k/3)\,.$

It is not accidental that at $x=0$ all roots are either $0$ or $1$. This can
be seen as a consequence of clustering for all Jordan-Wigner deformations.
Importantly, the symmetry $x\leftrightarrow 1-x$ implies the same set of roots
at $x=1$.

As we commented before, the roots that we obtained using the standard solution
of a third order equation are not exactly our unknowns, as they’re generally
not smooth. In fact, the ratios of partition functions should be identified
with some root for given $x$, but the specific root could depend on $x$. They
can jump from one root to the other only if there is a multiple root, i.e., if
the discriminant vanishes

$\Delta_{1}^{2}=4\Delta_{0}^{3}\,.$

We’ve already seen that this happens at $x=0$ (and hence, at $x=1$). Since we
expect real roots, we are going to assume $\Delta_{1}^{2}(x)\leq
4\Delta_{0}^{3}(x)$ and $\Delta_{0}(x)\geq 0$ (the latter follows from the
former). Thus we have

$C=\sqrt[3]{\frac{\Delta_{1}+i\sqrt{4\Delta_{0}^{3}-\Delta_{1}^{2}}}{2}}=\sqrt{\Delta_{0}}e^{\frac{i}{3}\arctan\frac{\sqrt{4\Delta_{0}^{3}-\Delta_{1}^{2}}}{\Delta_{1}}+\frac{\pi}{3}\theta_{H}(-\Delta_{1})}\,,$

where the $\frac{\pi}{3}$ shift in the argument has been added to keep $C$
smooth even when $\Delta_{1}$ changes sign. The three solutions then read

$x_{k}=\frac{2}{3}\Bigl{(}e^{\tilde{I}_{3}^{(2;1)}(x)}-\sqrt{\Delta_{0}}\cos\Bigl{(}\tfrac{2\pi
k}{3}+\tfrac{1}{3}\arctan\tfrac{\sqrt{4\Delta_{0}^{3}-\Delta_{1}^{2}}}{\Delta_{1}}+\tfrac{\pi}{3}\theta_{H}(-\Delta_{1})\Bigr{)}\Bigr{)}$
(E.6)

Since $\Delta_{1}^{2}(x)\leq 4\Delta_{0}^{3}(x)$, when we expand
$4\Delta_{0}^{3}(x)-\Delta_{1}^{2}(x)$ about its zeroes $z_{j}$ we get
(assuming $\Delta_{1}(z_{j})\neq 0$)

$x_{k}(x)\xrightarrow{x\sim
z_{j}}\tfrac{2}{3}\Bigl{(}e^{\tilde{I}_{3}^{(2;1)}(x)}-\sqrt{\Delta_{0}}\cos\Bigl{(}\tfrac{2\pi
k}{3}\mathrm{sgn}(x-z_{j})+\tfrac{1}{3}\tfrac{(x-z_{j})\sqrt{\partial_{x}^{2}(4\Delta_{0}^{3}-\Delta_{1}^{2})|_{x=z_{j}}}}{\sqrt{2}\Delta_{1}(z_{j})}+\tfrac{\pi}{3}\theta_{H}(-\Delta_{1})\mathrm{sgn}(x-z_{j})\Bigr{)}\Bigr{)}$

which shows that $k=1$ matches $k=-1$ when $\Delta_{1}(z_{j})>0$ and $k=0$
does $k=-1$ when $\Delta_{1}(z_{j})<0$. As long as the tripartite information
is smooth around $z_{j}$, the jump will result in a smooth function. If
there’s an even number of zeroes of $4\Delta_{0}^{3}-\Delta_{1}^{2}$ with
$\Delta_{1}>0$, the solution corresponding to $k=1$ at $x=0$ corresponds to
$k=1$ at $x=1$.

Let us study the solutions depending on the number of zeroes of
$4\Delta_{0}^{3}-\Delta_{1}^{2}$ in $x\in(0,1)$, boundary excluded. If there’s
no zero, the roots are in a one-to-one correspondence with the partition
functions. If there’s a zero (hence it should be at $x=\frac{1}{2}$), there
are two possibilities. If $\Delta_{1}>0$ at the zero, then $k=1$ interchanges
with $k=-1$ at $x=\frac{1}{2}$, which we could indicate as $k=(1,-1)$ for one
solution and $k=(1,-1)$ for the other, whereas the root with $k=0$ remains
unchanged, i.e., $k=(0,0)$. This situation cannot really occur if we consider
that the fermionic tripartite information is supposed to satisfy clustering
(which means, the corresponding partition function should be $1$ at $x=0$) and
to be symmetric under $x\leftrightarrow 1-x$. The other possibility is
$\Delta_{1}(1/2)<0$, in which case the root $k=1$ remains unchanged,
$k=(1,1)$, and can be identified with the fermionic entropy, whereas $k=0$ and
$k=-1$ interchange at $x=\frac{1}{2}$, i.e., the other solutions are
$k=(0,-1)$ and $k=(-1,0)$. This situation occurs, in particular, in the
noninteracting case. If $4\Delta_{0}^{3}-\Delta_{1}^{2}$ has two zeroes
(clearly, symmetric around $x=\frac{1}{2}$), by symmetry $\Delta_{1}$ has the
same sign at both zeroes. If $\Delta_{1}>0$, the smooth solutions are
$k=(1,-1,1)$, $k=(-1,1,-1)$, and $k=(0,0,0)$. If $\Delta_{1}<0$, they are
$k=(1,1,1)$, $k=(-1,0,-1)$, and $k=(0,-1,0)$. This means that also the smooth
solutions are symmetric under $x\rightarrow 1-x$. If there are three zeroes,
one is necessarily at $x=1/2$. There are four possibilities for the sign of
$\Delta_{1}$. If it is always positive we have $k=(1,-1,1,-1)$,
$k=(-1,1,-1,1)$, and $k=(0,0,0,0)$, which cannot occur for the same reason why
we couldn’t have a single zero with $\Delta_{1}>0$. If $\Delta_{1}<0$ at each
zero we have $k=(1,1,1,1)$, $k=(0,-1,0,-1)$, and $k=(-1,0,-1,0)$. If
$\Delta_{1}(1/2)>0$ but $\Delta_{1}<0$ at the other zeroes, then
$k=(1,1,-1,0)$, $k=(0,-1,1,1)$, and $k=(-1,0,0,-1)$, where the latter
corresponds to the fermionic entropy. If $\Delta_{1}(1/2)<0$ but
$\Delta_{1}>0$ at the other zeroes, then $k=(1,-1,0,0)$, $k=(0,0,-1,1)$, and
$k=(-1,1,1,-1)$, where, again, the latter corresponds to the fermionic
entropy.

$\begin{array}[]{c||c|c|c|c|c|c}\mathrm{sgn}(\Delta_{1})&-&++&\--&+-+&-+-&\---\\\
\hline\cr\gamma_{2,1}&(1,1)&(\pm 1,\mp 1,\pm
1)&(1,1,1)\vee(-1,0,-1)&(-1,1,1,-1)&(-1,0,0,-1)&(1,1,1,1)\\\
\gamma_{2,2}&(-1,0)&(\mp 1,\pm 1,\mp
1)&(-1,0,-1)\vee(1,1,1)&(1,-1,0,0)&(1,1,-1,0)&(-1,0,-1,0)\\\
\gamma_{2,3}&(0,-1)&(0,0,0)&(0,-1,0)&(0,0,-1,1)&(0,-1,1,1)&(0,-1,0,-1)\\\
\hline\cr\text{RTI}&-\log 2&0&0&-\log 2&-\log 2&-\log 2\end{array}$ Table 1:
The smooth roots written in terms of the standard ones depending on the sign
of $\mathrm{sgn}(\Delta_{1})$ at the zeroes of
$4\Delta_{0}^{3}-\Delta_{1}^{2}$, for the smallest number of zeroes. In some
cases our qualitative considerations are not sufficient to unequivocally
identify all terms. The last line reports the expected residual tripartite
information after a small global quench (see Section 7).

The reader can easily generalise this discussion to potential cases with more
zeroes.

As a matter of fact, this qualitative analysis was more pedagogical than
necessary, indeed, using the explicit result for the spin structure of XXZ, we
find that, independently of $\Delta$, XXZ falls into the same category of the
noninteracting case, i.e., $4\Delta_{0}^{3}-\Delta_{1}^{2}$ has a single zero
at $x=\frac{1}{2}$ and $\mathrm{sgn}(\Delta_{1}(\frac{1}{2}))<0$. We can
therefore write

$\gamma_{2,1}(x)=\frac{2}{3}\Bigl{(}e^{\tilde{I}_{3}^{(2;1)}(x)}-\sqrt{\Delta_{0}(x)}\cos\Bigl{(}\frac{2\pi}{3}+\frac{1}{3}\arctan\frac{\sqrt{4\Delta_{0}^{3}(x)-\Delta_{1}^{2}(x)}}{\Delta_{1}(x)}+\frac{\pi}{3}\theta_{H}(-\Delta_{1}(x))\Bigr{)}\Bigr{)}\,.$
(E.7)
$\gamma_{2,2}(x)=\frac{2}{3}\left[e^{\tilde{I}_{3}^{(2;1)}(x)}-\sqrt{\Delta_{0}(x)}\cos\Bigl{(}\tfrac{\pi}{3}\mathrm{sgn}(x-\tfrac{1}{2})+\tfrac{1}{3}\arctan\tfrac{\sqrt{4\Delta_{0}^{3}(x)-\Delta_{1}^{2}(x)}}{\Delta_{1}(x)}-\tfrac{\pi}{3}\theta_{H}(\Delta_{1}(x))\Bigr{)}\right]$
(E.8)
$\gamma_{2,3}(x)=\frac{2}{3}\left[e^{\tilde{I}_{3}^{(2;1)}(x)}-\sqrt{\Delta_{0}(x)}\cos\Bigl{(}\tfrac{\pi}{3}\mathrm{sgn}(\tfrac{1}{2}-x)+\tfrac{1}{3}\arctan\tfrac{\sqrt{4\Delta_{0}^{3}(x)-\Delta_{1}^{2}(x)}}{\Delta_{1}(x)}-\tfrac{\pi}{3}\theta_{H}(\Delta_{1}(x))\Bigr{)}\right]$
(E.9)

which, together with (E.4) and (E.2), allows us to access the spin structure
in the torus through the Rényi-2 tripartite information of four local spin-
chain models. A similar procedure can be applied also to the Riemann surfaces
with higher genus, but the inversion becomes more complicated and the number
of models to consider increases. In particular, the $3$-sheeted Riemann
surface corresponding to the Rényi-$3$ tripartite information is worked out in
Appendix E.

####  $3$-sheeted Riemann surface ($g=2$)

For the Rényi-$3$ tripartite information, if we exploit some symmetries of the
partition functions, we obtain the system

$\gamma_{3,1}^{j}+3\gamma_{3,2}^{j}+3\gamma_{3,3}^{j}+3\gamma_{3,4}^{j}-6(-1)^{j}\gamma_{3,5}^{j}=4e^{2I_{3}^{(3;j)}(x)}\,,$
(E.10)

where $\gamma_{3,j}$ are the ratios of partition functions associated with
given boundary conditions and $I_{3}^{(3;j)}(x)$ is the Rényi-$3$ tripartite
information in the JW $j$-deformation of XXZ. Although we suspect that the
tripartite information of 5 deformations should be enough to invert that
system of equations, we propose a less refined approach that involves more
models but that can be easily generalised to surfaces with higher genus.

We start by giving up the information about the degeneracy of the terms,
splitting them in the corresponding number of unknowns, e.g.,
$3\gamma_{3,2}^{j}\rightarrow\gamma_{3,2_{1}}^{j}+\gamma_{3,2_{2}}^{j}+\gamma_{3,2_{3}}^{j}$.
We then use a similar trick to that employed for the torus. Specifically, we
write

$\gamma_{3,1}^{j}+\sum_{k=1}^{3}(\gamma_{3,2_{k}}^{j}+\gamma_{3,3_{k}}^{j}+\gamma_{3,4_{k}}^{j})=4e^{2I_{3}^{(3,j)}(x)}+\sum_{k=1}^{6}(-\gamma_{3,5_{k}})^{j}\,,$
(E.11)

where we have relaxed the symmetry constraints
$\gamma_{3,y_{k}}=\gamma_{3,y}$. Let us now work out the elementary symmetric
polynomial associated with the unknowns on the left hand side of the equation

$e_{n}[\\{\gamma_{3,1},\gamma_{3,2_{\bullet}},\gamma_{3,3_{\bullet}},\gamma_{3,4_{\bullet}}\\}]=\\\
\frac{4^{n}}{n!}\det\left|\begin{matrix}e^{2I_{3}^{(3;1)}}-\tfrac{\sum_{k}\gamma_{3,5_{k}}}{4}&\tfrac{1}{4}&0&\dots&\\\
e^{2I_{3}^{(3;2)}}+\tfrac{\sum_{k}\gamma_{3,5_{k}}^{2}}{4}&e^{2I_{3}^{(3;1)}}-\tfrac{\sum_{k}\gamma_{3,5_{k}}}{4}&\frac{2}{4}&0\dots\\\
\vdots&&\ddots&\ddots\\\
e^{2I_{3}^{(3;n-1)}}+\tfrac{\sum_{k}(-\gamma_{3,5_{k}})^{n-1}}{4}&e^{2I_{3}^{(3;n-2)}}+\tfrac{\sum_{k}(-\gamma_{3,5_{k}})^{n-2}}{4}&\dots&e^{2I_{3}^{(3;1)}}-\tfrac{\sum_{k}\gamma_{3,5_{k}}}{4}&\frac{n-1}{4}\\\
e^{2I_{3}^{(3;n)}}+\tfrac{\sum_{k}(-\gamma_{3,5_{k}})^{n}}{4}&e^{2I_{3}^{(3;n-1)}}+\tfrac{\sum_{k}(-\gamma_{3,5_{k}})^{n-1}}{4}&\dots&e^{2I_{3}^{(3;2)}}+\tfrac{\sum_{k}\gamma_{3,5_{k}}^{2}}{4}&e^{2I_{3}^{(3;1)}}-\tfrac{\sum_{k}\gamma_{3,5_{k}}}{4}\end{matrix}\right|\,.$
(E.12)

We construct the associated polynomial
$x^{10}+\sum_{n=1}^{10}(-1)^{n}e_{n}x^{10-n}$ and multiply it by $x^{m}$. We
then evaluate the polynomial at its roots and sum over the roots, using
(E.11). The result is

$4e^{2I_{3}^{(3;10+m)}}+\sum_{k=1}^{6}(-\gamma_{3,5_{k}})^{10+m}+\\\
\sum_{n=1}^{10}(-1)^{n}e_{n}[\\{\gamma_{3,1},\gamma_{3,2_{\bullet}},\gamma_{3,3_{\bullet}},\gamma_{3,4_{\bullet}}\\}](4e^{2I_{3}^{(3;10+m-n)}}+\sum_{k=1}^{6}(-\gamma_{3,5_{k}})^{10+m-n})=0\,.$
(E.13)

We restrict ourselves to $m\in\\{1,2,3,4,5,6\\}$; this turns out to be a
linear system of equations for the elementary symmetric polynomials associated
with the variables $\gamma_{3,5_{k}}$ and, in turn, can be inverted with
standard means. We can then express
$e_{n}[\\{\gamma_{3,5_{1}},\dots,\gamma_{3,5_{6}}\\}]$ as functions of
tripartite informations as follows

$e_{n}[\\{\gamma_{3,5_{1}},\dots,\gamma_{3,5_{6}}\\}]=\mathcal{E}_{n}[I_{3}^{(3;1)},\ldots,I_{3}^{(3;16)}]\qquad
n\in\\{1,2,3,4,5,6\\}\,.$

Imposing $\gamma_{3,5_{k}}=\gamma_{3,5}$ for every $k$ gives immediately
$\gamma_{3,5}$, for example using

$\gamma_{3,5}=\frac{1}{6}\mathcal{E}_{1}[I_{3}^{(3;1)},\ldots,I_{3}^{(3;16)}]\,.$

Since we are free to use any other $\mathcal{E}_{n}$, as a by-product we
obtain $5$ identities satisfied by
$\\{I_{3}^{(3;1)},\cdots,I_{3}^{(3;16)}\\}$. Similarly to the case $g=1$, we
are finally reduced to

$\gamma_{3,1}^{j}+\sum_{k=1}^{3}(\gamma_{3,2_{k}}^{j}+\gamma_{3,3_{k}}^{j}+\gamma_{3,4_{k}}^{j})=4e^{2\tilde{I}_{3}^{(3,j)}(x)}\,,$
(E.14)

where

$4e^{2\tilde{I}_{3}^{(3,j)}(x)}=4e^{2I_{3}^{(3,j)}(x)}+\sum_{k=1}^{6}(-\gamma_{3,5_{k}})^{j}$

are now known functions of the tripartite information. We now split variable
$\gamma_{3,1}$ in 5 unknowns that will be only eventually forced to match:

$\sum_{k=1}^{3}(\gamma_{3,1_{k}}^{j}+\gamma_{3,2_{k}}^{j}+\gamma_{3,3_{k}}^{j}+\gamma_{3,4_{k}}^{j})=4e^{2\tilde{I}_{3}^{(3,j)}(x)}+\gamma_{3,1_{4}}^{j}+\gamma_{3,1_{5}}^{j}\,.$
(E.15)

In this way we have increased the number of variables to $14$, but we will
momentarily treat $\gamma_{3,1_{4}}$ and $\gamma_{3,1_{5}}$ as if they were
known functions. As before, we construct the associated elementary symmetric
polynomials and perform the same manipulations. We end up with

$4e^{2\tilde{I}_{3}^{(3;12+m)}}+\sum_{k=4}^{5}\gamma_{3,1_{k}}^{12+m}+\sum_{n=1}^{10}(-1)^{n}\tilde{e}_{n}[\\{\gamma_{3,1_{\bullet}},\gamma_{3,2_{\bullet}},\gamma_{3,3_{\bullet}},\gamma_{3,4_{\bullet}}\\}](4e^{2\tilde{I}_{3}^{(3;12+m-n)}}+\sum_{k=4}^{5}\gamma_{3,1_{k}}^{12+m-n})=0$

where $\sim$ in $\tilde{e}_{n}$ is used to emphasise that $I_{3}^{(3,j)}$ is
now replaced by $\tilde{I}_{3}^{(3,j)}$. In this case we can restrict to
$m\in\\{1,2\\}$. As before, this is a linear system of equations for the
elementary symmetric polynomials associated with the variables
$\gamma_{3,1_{4}}$ and $\gamma_{3,1_{5}}$, which can be readily inverted. We
end up with

$\tilde{e}_{n}[\\{\gamma_{3,1_{4}},\gamma_{3,1_{5}}\\}]=\tilde{\mathcal{E}}_{n}[\tilde{I}_{3}^{(3;1)},\ldots,\tilde{I}_{3}^{(3;14)}]\qquad
n\in\\{1,2\\}\,.$

Imposing $\gamma_{3,1_{k}}=\gamma_{3,1}$ for every $k$ immediately gives
$\gamma_{1}$, for example using

$\gamma_{3,1}=\frac{1}{2}\tilde{\mathcal{E}}_{1}[\tilde{I}_{3}^{(3;1)},\ldots,\tilde{I}_{3}^{(3;14)}]\,.$

Since we could also use $\mathcal{E}_{2}$, we also get one identity satisfied
by $\\{\tilde{I}_{3}^{(3;1)},\cdots,\tilde{I}_{3}^{(3;14)}\\}$. Now that we
know $\gamma_{3,1}$, we can reduce the system of equations (E.15) into one of
the same form as for the torus:

$\gamma_{3,2}^{j}+\gamma_{3,3}^{j}+\gamma_{3,4}^{j}=\frac{4}{3}e^{2\tilde{I}_{3}^{(3,j)}(x)}-\frac{1}{3}\gamma_{3,1}^{j}(x)\,.$
(E.16)

This can be solved as we didfor the torus, and the entire discussion about the
roots holds true.

This procedure has allowed us to invert the problem using $16$ deformed
models, with a by-product of $6$ potentially independent identities satisfied
by $\\{I_{3}^{(3;1)},\ldots,I_{3}^{(3;16)}\\}$. In the best case scenario
those identities could be used to express
$\\{I_{3}^{(3;11)},\ldots,I_{3}^{(3;16)}\\}$ in terms of
$\\{I_{3}^{(3;1)},\ldots,I_{3}^{(3;10)}\\}$, realising a more refined
inversion based on just $10$ deformed models.

##  Appendix F Small $x$ expansion

In this section we derive the results (4.17), (4.19) and (4.20). The results
are obtained by studying the small $x$ expansion of

$\Theta\left[\begin{matrix}\vec{\mu}\\\
\vec{0}\end{matrix}\right]\left(\vec{0}|\kappa\Omega\right)$ (F.1)

for $\vec{\mu}\in\\{0,1/2\\}^{\alpha-1}$ and a generic parameter
$\kappa\in(0,\infty)$, and considering different contributions in Eq. (2.10) .

The small $x$ expansion relies heavily on the methods developed in Refs
Calabrese2009Entanglement1 ; Calabrese2011Entanglement , which start from the
expansion of $\beta_{k/\alpha}$, appearing in the period matrix (2.5), for
$x\to 0$,

$\beta_{\frac{k}{\alpha}}(x)=-\frac{\sin(\pi\frac{k}{\alpha})}{\pi}\left(\log
x+f_{\frac{k}{\alpha}}+\sum\limits_{l=1}^{\infty}p_{l}\left(\frac{k}{\alpha}\right)x^{l}\right),\quad
f_{\frac{k}{\alpha}}\equiv
2\gamma_{E}+\psi\left(\frac{k}{\alpha}\right)+\psi\left(1-\frac{k}{\alpha}\right),$
(F.2)

where $\gamma_{E}$ is the Euler gamma,
$\psi(z)\equiv\Gamma^{\prime}(z)/\Gamma(z)$ is the polygamma function and
$p_{l}(z)$ is a polynomial of degree 2$l$, whose explicit expression is not
needed. As in Ref. Calabrese2011Entanglement , we plug this expansion in the
theta function, getting

$\Theta\left[\begin{matrix}\vec{\mu}\\\
\vec{0}\end{matrix}\right]\left(\vec{0}|\kappa\Omega\right)=\sum\limits_{\vec{m}\in\mathbb{Z}^{\alpha-1}}x^{\kappa
a_{\vec{m}+\vec{\mu}}}e^{\kappa b_{\vec{m}+\vec{\mu}}}(1+O(x))\;,$ (F.3)

where the two vector-dependent coefficients are defined by

$\displaystyle
a_{\vec{v}}\equiv\frac{2}{\alpha}\sum\limits_{k=1}^{\alpha-1}\sin^{2}\left(\pi\frac{k}{\alpha}\right)\vec{v}\cdot
C_{k/\alpha}\vec{v}\;,$ (F.4) $\displaystyle
b_{\vec{v}}\equiv\frac{2}{\alpha}\sum\limits_{k=1}^{\alpha-1}\sin^{2}\left(\pi\frac{k}{\alpha}\right)f_{k/\alpha}\vec{v}\cdot
C_{k/\alpha}\vec{v}\;,$ (F.5)

and the matrix $C_{k/\alpha}$ has elements

$\left(C_{\frac{k}{\alpha}}\right)_{\ell,n}\equiv\cos\left[2\pi\frac{k}{\alpha}(\ell-n)\right]\;.$
(F.6)

Furthermore, for any vector $\vec{v}$ the coefficient $a_{\vec{v}}$ is equal
to Calabrese2011Entanglement

$a_{\vec{v}}=\sum_{\ell=1}^{\alpha-1}v_{\ell}^{2}-\sum_{\ell=1}^{\alpha-2}v_{\ell}v_{\ell+1}=\frac{v_{\ell}^{2}+v_{\alpha-1}^{2}}{2}+\frac{1}{2}\sum_{\ell=1}^{\alpha-2}(v_{\ell}-v_{\ell+1})^{2}\;.$
(F.7)

For $\vec{v}\in\mathbb{Z}^{\alpha-1}$ the first equality makes it apparent
that $a_{\vec{v}}\in\mathbb{Z}$, while from the second equality we see
$a_{\vec{v}}\geq 0$. It follows
$a_{\vec{m}+\vec{\mu}}=a_{2(\vec{m}+\vec{\mu})}/4\in\mathbb{N}_{0}/4$, where
the first equality follows from the definition (F.4).

The leading term in the expansion (F.3) is obtained by collecting all
$\vec{m}\in\mathbb{Z}^{\alpha-1}$ that minimize the coefficients
$a_{\vec{m}+\vec{\mu}}$. By inspection, $a_{\vec{m}+\vec{\mu}}=0$ is only
possible for $\vec{\mu}=0$ (for $\vec{m}=0$) and in that case the leading term
is a constant term. The case $\vec{\mu}=0$ has already been covered in Ref.
Calabrese2011Entanglement , which derived the first two subleading terms.
Focusing just on the first subleading term, the result reads

$\Theta(0|\kappa\Omega)=1+s(\alpha;\kappa)x^{\kappa}+o(x^{\kappa})\;,$ (F.8)

where $s(\kappa;\alpha)$ is defined in Eq. (4.8).

For $\vec{\mu}\neq 0$ we have $a_{\vec{m}+\vec{\mu}}\neq 0$ for all
$m\in\mathbb{Z}^{\alpha-1}$. Thus the leading term in (F.3) will have a non-
zero power. The smallest possible non-zero value of $a_{\vec{m}+\vec{\mu}}$ is
$1/4$, achieved when $\vec{\mu}$ contains two “domain walls”,

$\vec{\mu}^{\mathrm{T}}=\frac{1}{2}(\underbrace{0,0,\ldots,0}_{j_{1}},\underbrace{1,1,\ldots,1}_{j_{2}-j_{1}},0,0,\ldots,0)\;,\quad
0\leq j_{1}<j_{2}\leq\alpha-1\;,$ (F.9)

in combination with $\vec{m}=\vec{0}$ and $\vec{m}=-2\vec{\mu}$, which give
$\vec{m}+\vec{\mu}=\pm\vec{\mu}$ respectively. Thus for $\vec{\mu}$ of the
type (F.9)

$\Theta\left[\begin{matrix}\vec{\mu}\\\
\vec{0}\end{matrix}\right]\left(\vec{0}|\kappa\Omega\right)=x^{\kappa/4}\left(e^{\kappa
b_{\vec{\mu}}}+e^{\kappa b_{-\vec{\mu}}}\right)+o(x^{\kappa/4})\;.$ (F.10)

Now we resort to another formula derived in Ref. Calabrese2011Entanglement ,
about the coefficients $b_{\pm\vec{\mu}}$ for $\vec{\mu}$ given by (F.9),

$b_{\pm\vec{\mu}}=-\frac{1}{2}\log\left[2\alpha\sin\left(\pi\frac{j_{2}-j_{1}}{\alpha}\right)\right]\;.$
(F.11)

Using this formula we obtain

$\Theta\left[\begin{matrix}\vec{\mu}\\\
\vec{0}\end{matrix}\right]\left(\vec{0}|\kappa\Omega\right)=\frac{2}{\left[\sin\left(\pi\frac{j_{2}-j_{1}}{\alpha}\right)\right]^{\kappa/2}}\left(\frac{x}{4\alpha^{2}}\right)^{\kappa/4}+o(x^{\kappa/4})$
(F.12)

for $\vec{\mu}$ given by (F.9).

When $\vec{\mu}\neq 0$ and when $\vec{\mu}$ is not of the type (F.9), there
are more than two “domain walls” in $\vec{\mu}$, i.e. four or more. In this
case $a_{\vec{m}+\vec{\mu}}\geq 1/2$ for any $\vec{m}\in\mathbb{Z}^{\alpha-1}$
and, therefore,

$\Theta\left[\begin{matrix}\vec{\mu}\\\
\vec{0}\end{matrix}\right]\left(\vec{0}|\kappa\Omega\right)=O(x^{\kappa/2})\;.$
(F.13)

To derive the result (4.17), in (2.10) we separate the contributions with
respect to the number of domain walls in $\vec{\mu}$,

$\begin{split}&\mathcal{F}_{f}\left[\begin{matrix}\vec{0}\\\
\vec{\delta}\end{matrix}\right]=\frac{1}{\left[\Theta(0|\Omega)\right]^{2}}\Bigg{(}\Theta(0|\tfrac{1}{\eta}\Omega)\Theta\left(0|4\eta\Omega\right)\\\
&+\sum_{\vec{\mu}\textrm{ with two domain
walls}}(-1)^{4\vec{\mu}\cdot\vec{\delta}}\Theta\left[\begin{matrix}\vec{\mu}\\\
\vec{0}\end{matrix}\right]\left(\vec{0}|\tfrac{1}{\eta}\Omega\right)\Theta\left[\begin{matrix}\vec{\mu}\\\
\vec{0}\end{matrix}\right]\left(\vec{0}|4\eta\Omega\right)+\\\
&+\sum_{\vec{\mu}\textrm{ with more than two domain
walls}}(-1)^{4\vec{\mu}\cdot\vec{\delta}}\Theta\left[\begin{matrix}\vec{\mu}\\\
\vec{0}\end{matrix}\right]\left(\vec{0}|\tfrac{1}{\eta}\Omega\right)\Theta\left[\begin{matrix}\vec{\mu}\\\
\vec{0}\end{matrix}\right]\left(\vec{0}|4\eta\Omega\right)\Bigg{)}\end{split}$
(F.14)

Then we apply the expansion (F.8) to the denominator and the first term in the
brackets, the expansion (F.12) to the second term and the bound (F.13) to the
third term. Collecting the $\eta$-dependent leading terms we get the result
(4.17).

To derive the result (4.19), we rewrite (2.10) as

$\begin{split}\mathcal{F}_{f}\left[\begin{matrix}\vec{\varepsilon}\\\
\vec{\delta}\end{matrix}\right]=&\frac{1}{\left[\Theta(0|\Omega)\right]^{2}}\Bigg{(}\Theta\left[\begin{matrix}\vec{\varepsilon}\\\
\vec{0}\end{matrix}\right](0|\tfrac{1}{\eta}\Omega)\Theta\left(0|4\eta\Omega\right)+(-1)^{4\vec{\varepsilon}\cdot\vec{\delta}}\Theta(0|\tfrac{1}{\eta}\Omega)\Theta\left[\begin{matrix}\vec{\varepsilon}\\\
\vec{0}\end{matrix}\right]\left(0|4\eta\Omega\right)\\\
&+\sum_{\vec{\mu}\neq\vec{0},\vec{\varepsilon}}(-1)^{4\vec{\mu}\cdot\vec{\delta}}\Theta\left[\begin{matrix}\vec{\varepsilon}+\vec{\mu}\\\
\vec{0}\end{matrix}\right]\left(\vec{0}|\tfrac{1}{\eta}\Omega\right)\Theta\left[\begin{matrix}\vec{\mu}\\\
\vec{0}\end{matrix}\right]\left(\vec{0}|4\eta\Omega\right)\Bigg{)}\end{split}$
(F.15)

Here we have also used that the theta function is invariant under the shift of
a component of the characteristic by an integer, implying
$\Theta\left[\begin{matrix}2\vec{\varepsilon}\\\
\vec{0}\end{matrix}\right]=\Theta\left[\begin{matrix}\vec{0}\\\
\vec{0}\end{matrix}\right]$. Now, the expansion of the denominator in (F.15)
is given by (F.8). The first term in the brackets is a product of two theta
functions, one with zero characteristic and one with $\vec{\varepsilon}$ given
by the right hand side of (F.9). The leading term in the theta function with
zero characteristics is the constant term in (F.8), while the leading term for
the second theta function is described by (F.12). The second term in the
brackets is treated in an analogous way. The third term in the brackets is
subleading because both theta functions in the product have a non-zero
characteristic. Summing all these contributions gives the result (4.19).

To derive the bound (4.20) we recognize that when $\vec{\varepsilon}$ has more
than two domain walls then each term

$\Theta\left[\begin{matrix}\vec{\varepsilon}+\vec{\mu}\\\
\vec{0}\end{matrix}\right]\left(\vec{0}|\tfrac{1}{\eta}\Omega\right)\Theta\left[\begin{matrix}\vec{\mu}\\\
\vec{0}\end{matrix}\right]\left(\vec{0}|4\eta\Omega\right)$ (F.16)

in (2.10) has the following property. Either both characteristics, $\vec{\mu}$
and $\vec{\varepsilon}+\mu$, have two domain walls (e.g.
$\vec{\varepsilon}^{\mathrm{T}}=1/2(1,0,1),\
\vec{\mu}^{\mathrm{T}}=1/2(0,1,0)$ for $\alpha=4$), or one of the two
characteristics, $\vec{\mu}$ or $\vec{\varepsilon}+\mu$, has more than two
domain walls (e.g. $\vec{\varepsilon}^{\mathrm{T}}=1/2(1,0,1)$ with
$\vec{\mu}=0$ or $\vec{\mu}=\vec{\varepsilon}$ for $\alpha=4$). In the first
case we apply the expansion (F.12) and in the second case the bound (F.13),
giving us the result (4.20)

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# An upper limit to differential magnification effects in strongly
gravitationally lensed galaxies

Stephen Serjeant School of Physical Sciences
The Open University, Milton Keynes, MK7 6AA UK

###### Abstract

Differential magnification is now well-known to distort the spectral energy
distributions of strongly gravitationally lensed galaxies. However, that does
not mean that any distortions are possible. Here I prove an analytic upper
bound to differential magnification effects. For example, a thermal or sub-
thermal CO ladder cannot be made to appear super-thermal just from
gravitational lensing, and the Balmer decrement emission line ratio
H$\alpha$:H$\beta$ cannot reduce below the case B prediction just from
differential magnification. In general, if a physical model of a galaxy
predicts upper and/or lower bounds to an emission line ratio, then those
bounds also apply to the differentially magnified strongly gravitationally
lensed case. This applies not just for velocity-integrated emission lines, but
also for the line emission in any rest-frame velocity interval.

††journal: AAS Journals

[latex]

##  1 Introduction

The landmark discovery that bright submm and mm-wave galaxies are mainly
strongly gravitationally lensed (Negrello et al., 2010) has led to a glut of
strongly lensed systems (e.g. Negrello et al., 2017; Reuter et al., 2020;
Urquhart et al., 2022). Taken in conjunction with optical and near-infrared
strong lensing catalogues, there are now of the order a thousand confirmed or
candidate strong lensing systems, and future dark energy missions are expected
to increase these numbers by two orders of magnitude (e.g. Collett, 2015).
Many spectroscopic or continuum studies currently rely either on low angular
resolution data or integrated fluxes across the system (e.g. Urquhart et al.,
2022; Bendo et al., 2023; Hagimoto et al., 2023; Cox et al., 2023; Ismail et
al., 2023; Berta et al., 2023). Therefore studies have been made to assess the
effects of differential gravitational lensing magnification on the observed
integrated spectral energy distributions of strongly lensed galaxies (e.g.
Serjeant, 2012; Hezaveh et al., 2012). These effects are now widely cited and
recognised, but sometimes anomalies in observations are wrongly attributed to
differential magnification. This paper therefore sets out to address this by
proving an analytic bound on differential magnification effects.

##  2 Method

The formalism is a continuum generalisation of the discrete argument we
presented in Appendix E of Hagimoto et al. (2023). I consider two emission
lines, numbered 1 and 2, and line 2 is always brighter than line 1. For
example, line 2 could be the H$\alpha$ emission line, and line 1 could be
H$\beta$ under case B recombination with the additional possibility of dust
reddening reducing the Balmer decrement. Alternatively, both lines could be
$L^{\prime}$ luminosity measures of CO rotational transitions.

The line luminosities can be written as $L_{1}=kL_{2}$, with

$0\leq k\leq 1~{}.$ (1)

This line ratio $k$ can vary across a galaxy, $k=k(x,y)$ where $(x,y)$ are
Cartesian coordinates. It immediately follows that a luminosity increment
within $\mathrm{d}x\mathrm{d}y$ will satisfy

$k(x,y)L_{2}(x,y)\mathrm{d}x\mathrm{d}y\leq
L_{2}(x,y)\mathrm{d}x\mathrm{d}y~{}.$ (2)

Integrating over the $(x,y)$ positions in a galaxy image $G$ obtains

$\int_{G}k(x,y)L_{2}(x,y)\mathrm{d}x\mathrm{d}y\leq\int_{G}L_{2}(x,y)\mathrm{d}x\mathrm{d}y~{}.$
(3)

I define a luminosity-weighted observed mean $k$ as

$\langle
k\rangle_{\mathrm{unlensed}}=\frac{\int_{G}k(x,y)L_{2}(x,y)\mathrm{d}x\mathrm{d}y}{\int_{G}L_{2}(x,y)\mathrm{d}x\mathrm{d}y}\leq
1.$ (4)

In other words, integrating the (unlensed) light over a galaxy will not yield
a line ratio that exceeds the $k\leq 1$ bound that applies in any individual
region. For example, if one measures Balmer line luminosities relative to case
B predictions, then it follows that it is not possible to make the integrated
H$\beta$:H$\alpha$ luminosity ratio exceed the case B recombination limit, if
case B recombination applies throughout the galaxy.

This galaxy is then differentially magnified by a net factor $\mu(x,y)$, where
$\mu\geq 0$ (i.e. summing the modulus of the magnification factors of each
image, so images with negative parity contribute positively to the total
magnification of a differential region $\mathrm{d}x\mathrm{d}y$ in the
galaxy), and $(x,y)$ now measures positions in the source plane. Again, a
differential element in the source plane $\mathrm{d}x\mathrm{d}y$ will satisfy

$k(x,y)\mu(x,y)L_{2}(x,y)\mathrm{d}x\mathrm{d}y\leq\mu(x,y)L_{2}(x,y)\mathrm{d}x\mathrm{d}y$
(5)

(which follows immediately from Eqn. 1). Integrating over the $(x,y)$
positions in a galaxy image $G$ obtains

$\int_{G}k(x,y)\mu(x,y)L_{2}(x,y)\mathrm{d}x\mathrm{d}y\leq\int_{G}\mu(x,y)L_{2}(x,y)\mathrm{d}x\mathrm{d}y~{}.$
(6)

One can then define a luminosity-weighted observed mean $k$ in the
gravitationally-lensed case as

$\langle k\rangle_{\mathrm{lensed}}=\frac{\int_{G}k\mu
L_{2}\mathrm{d}x\mathrm{d}y}{\int_{G}\mu L_{2}\mathrm{d}x\mathrm{d}y}\leq
1~{}.$ (7)

##  3 Discussion

There is therefore a straightforward hard upper limit to the observed
integrated line ratios in a differentially magnified, strongly gravitationally
lensed galaxy. What can be learned about the underlying line ratios, in the
absence of a magnification map $\mu(x,y)$? Firstly, there is no guarantee that
any region within the galaxy has a line ratio exactly equalling the observed
one. This can be trivially shown to be true even in the absence of
differential magnification, i.e. $\mu=$ constant: if the galaxy is evenly
split between regions of $k=0$ and $k=1$, then the average observed $\langle
k\rangle=\frac{1}{2}$, even though no $(x,y)$ position will have that line
ratio.

Nevertheless, if the lensed line ratio exceeds $1$, then it must be the case
that the fundamental assumption in Eqn. 1 does not hold, i.e. $k(x,y)>1$ in at
least part of the galaxy. Therefore, if the integrated H$\beta$ emission line
exceeds the case B recombination prediction in a strongly lensed galaxy, then
at least part of the galaxy must somehow have a region exceeding that
prediction. Similarly, if the lensed CO ladder appears super-thermal, then
there must be at least one region within the galaxy that has super-thermal CO
transitions. An equivalent argument for line ratio lower bounds can trivially
be constructed by exchanging lines 1 and 2 in Section 1. Although this paper
describes galaxy lensing, the argument is independent of the structure of the
magnification map $\mu(x,y)$, so it applies in all gravitational lensing,
whether extragalactic or Galactic. In general, if a physical model predicts
lower or upper bounds to a line ratio, then those bounds also apply to the
differentially magnified strongly lensed case, so the model can in principle
be ruled out even without a magnification map.

There are examples of extreme, super-thermal CO line transitions in four
galaxies from the Bright Extragalactic ALMA Redshift Survey (BEARS, Hagimoto
et al., 2023). These line ratios can therefore not be attributed solely to
differential magnification. These observations are slightly complicated by the
fact that the data is from an interferometer, so differences in the Fourier
$uv$ coverage between emission lines could in principle lead to an anomalous
line ratio, but the $uv$ coverage is thorough in this case so this is not an
obvious solution. The argument in this paper holds not just for velocity-
integrated emission lines, but also for the line emission in any rest-frame
velocity interval; however the emission line profiles are comparable in most
of the targets. Deeper follow-up observations are recommended.

##  4 Conclusion

If a physical model of a galaxy predicts upper and/or lower bounds to an
emission line ratio, then those bounds also apply to the differentially
magnified strongly gravitationally lensed case. This applies not just for
velocity-integrated emission lines, but also for the line emission in any
rest-frame velocity interval.

##  5 Acknowledgements

SS acknowledges support from the ELSA project. “ELSA: Euclid Legacy Science
Advanced analysis tools” has received funding from the European Union’s
Horizon Europe research and innovation programme under Grant Agreement no.
101135203. UK participation is funded through the UK Horizon guarantee scheme
under Innovate UK grant 10093177. SS thanks Tom Bakx, Steve Eales and George
Bendo for constructive conversations on this topic during a workshop with
visitors’ funding from STFC under grant ST/W000830/1.

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