#  Interplay between local and non-local frustration in the 1D ANNNI chain I
The even case

G. Torre  Institut Ruđer Bošković, Bijenička cesta 54, 10000 Zagreb, Croatia
A. G. Catalano  Institut Ruđer Bošković, Bijenička cesta 54, 10000 Zagreb,
Croatia  Université de Strasbourg, 4 Rue Blaise Pascal, 67081 Strasbourg,
France  S. B. Kožić  Institut Ruđer Bošković, Bijenička cesta 54, 10000
Zagreb, Croatia  F. Franchini<EMAIL_ADDRESS>Institut Ruđer Bošković, Bijenička
cesta 54, 10000 Zagreb, Croatia  S. M. Giampaolo<EMAIL_ADDRESS>Institut
Ruđer Bošković, Bijenička cesta 54, 10000 Zagreb, Croatia

###### Abstract

We consider the effects of the competition between different sources of
frustration in 1D spin chains through the analysis of the paradigmatic ANNNI
model, which possesses an extensive amount of frustration of local origin due
to the competition between nearest and next-to-nearest neighbor interactions.
An additional, non-extensive amount of topological frustration can be added by
applying suitable boundary conditions and we show that this seemingly
subdominant contribution significantly affects the model. Choosing periodic
boundary conditions with an even number of sites not divisible by $4$ and
using the entanglement entropy as a probe, we demonstrate that in one of the
model’s phases the ground state can be characterized as hosting two (almost)
independent excitations. Thus, not only we show an intriguing interplay
between different types of frustration, but also manage to propose a non-
trivial quasi-particle interpretation for it.

††preprint: RBI-ThPhys-2024-11

##  I Introduction

The field of quantum complex systems emerged as the natural extension of
classical statistical physics. Therefore, it is not surprising that the first
studies were focused on the analysis of quantities such as correlation
functions and local order parameters that, accordingly with Landau’s theory
[1], characterize the different phases of a physical system. This approach
achieved remarkable successes, but the inherent non-locality of quantum
mechanics gave rise to a much broader phenomenology than that of classical
systems, which was impossible to fully encompass within this framework. A
notable example of this fact is represented by topologically ordered phases
that are characterized by robust ground state degeneracies [2] and can be
unveiled by topological invariants that remain unaffected by continuous
deformations of the Hamiltonian parameters [3, 4, 5, 6]. In these phases, the
global structures of the ground states are connected to non-local correlations
whose presence can be highlighted by the entanglement properties [7, 12, 8, 9,
10, 11].

However, such order represents just one example, even if the most widely
known, of the different phenomenologies lying outside Landau’s theory which
are continually unveiled. Another one, which represents the topic of the
present work, is the so-called topological frustration (TF). The concept of
frustration in quantum mechanics must be handled with care. By definition, a
many-body system is frustrated when it is impossible to minimize
simultaneously all its local energy constrains. Due to the non-commuting
nature of quantum mechanics, with few exceptions such as the frustration free
models [13, 14] or systems at a factorization point [15, 16, 17], almost all
quantum systems include some amount of frustration [18, 19, 20, 21]. But, in
line with recent usage in the scientific community, in the following we
restrict the concept of frustration to effects induced by competing
interactions and/or geometrical properties, extending it directly from
classical systems [22, 23].

Returning to TF, initially it has been examined in models featuring
antiferromagnetic (AFM) nearest neighbor coupling and implemented on lattices
with frustrated boundary conditions (FBCs) [24]. FBCs refer to the coexistence
of periodic boundary conditions (PBCs) and structures comprised of an odd
number of spins. In the classical limit, where all terms in the Hamiltonian
commutes each other, these systems develop an extensive degeneracy of the
ground state which is typically lifted away when quantum terms are taken into
account [25, 26, 27]. As a result, a gapless band structure replaces the
gapped phase featured by the same systems in the same region of parameter
spaces but with different boundary conditions. FBC induce peculiar
phenomenologies, revealed, for instance, by an increment in complexity of the
ground state [28], the suppression of the usual AFM order [29, 30], a huge
sensitivity of the dynamical properties to small local perturbation [31], etc.
As for the topologically ordered phase, also the topological nature of such
phenomenology can be unveiled by the analysis of the entanglement entropy
[32], which clearly shows a topological contribution [33]. All these results
emerged in the past few years and have been quite unexpected, challenging the
prevailing belief that boundary conditions could not produce effects lasting
up to the thermodynamic limit.

However, the analysis carried out so far focused on one-dimensional spin
chains with just short-range (nearest-neighbor) interactions. These systems,
by their nature, could not show any source of local frustration, i.e.
generated by different interactions competing on the same sites. Therefore, TF
has been the only type of frustration and, being associated with boundary
conditions, provides a sub-extensive amount of frustration. With this work we
start extending the analysis to more complex systems, those that present also
local sources of frustration due to the interplay between neighboring and non-
neighboring spin pairs interactions, and investigate whether TF can influence
these systems as well. We consider the ANNNI model, being the simplest one in
this respect. It was initially introduced as a theoretical model to understand
numerous experimental observations [34, 35], and it is now regaining attention
as the right playground to test machine learning algorithms, due to its rich
phase diagram and the lack of a general analytical solution [36, 37].
Furthermore, it is amenable to experimental implementation with segmented ion
traps [38]. The Hamiltonian of the model for $L$ spins reads

$H=J_{1}\sum_{i=1}^{L}\sigma_{i}^{x}\sigma_{i+1}^{x}+J_{2}\sum_{i=1}^{L}\sigma_{i}^{x}\sigma_{i+2}^{x}+h\sum_{i=1}^{L}\sigma_{i}^{z},$
(1)

where $\sigma_{i}^{\alpha}$, with $\alpha=x,\,y,\,z$ are the Pauli operators
acting on the $i$-th spin, $J_{1}$ and $J_{2}$ are respectively the nearest
and the next-to-nearest neighbor interactions, and $h$ is a transverse
magnetic field. Unless stated otherwise, we assume periodic boundary
conditions, i.e. $\sigma_{i}^{\alpha}\equiv\sigma_{i+L}^{\alpha}$.

Indeed, an AFM next-to-nearest neighbors coupling ($J_{2}>0$) induces an
extensive frustration of local origin, regardless of the choice of the
boundary conditions and of the sign of $J_{1}$ and the different interaction
terms in eq. (1) result into a rich phase diagram (see Fig. 1), with four
different phases [35, 37], which we will present in the next section.

In this work we add TF, and we will show that this intensive source of
frustration affect the model, in particular in the so-called antiphase where
$J_{2}$ is the dominant interaction. To prove such a result, we focus on the
bipartite entanglement entropy (EE). In [32, 28] it was shown that in the
Ising chain (eq. (1) with $J_{2}=0$) TF, induced by FBC, adds an amount of EE
corresponding to the presence of a single delocalized excitation With a finite
$J_{2}>0$ we will prove that TF can be induced by choosing a chain with an
even number of sites non-divisible by $4$ and this results in a contribution
to the EE compatible with that of two delocalized excitations, obeying a Pauli
principle that prevents them from occupying the same momentum state. The case
with an odd number of site will be considered in the next article in this
series.

Figure 1: Phase diagram of the ANNNI model in the $(h/J_{1},\kappa)$ plane in
absence ($L=100$, left panel) and in presence ($L=102$, right panel) of
topological frustration, obtained by analyzing the second derivative of the
ground-state energy with respect to $h$. The solid orange lines represent the
quantum phase transitions detected by discontinuities in the energy
derivative. While the phase diagram, originally drawn in [35], is unaffected
by the different boundary conditions considered, we will show that the
antiphase has different properties for $L=4N$ and $L=4N+2$.

After discussing the phase diagram of the ANNNI model in Sec. II, we proceed
with our analysis of the EE. In Section III, with details provided in Appendix
A, we derive an analytical expression for the system’s unique ground state
close to the classical line $h=0$, by exploiting graph theory within a
perturbative approach. This result enables us to provide an analytical formula
for the EE in the thermodynamic limit, which can be cast as a sum of two
contributions: one coming from the double $\mathcal{Z}_{2}$ symmetry of
flipping each spin on each even/odd sub-lattice, and the other revealing the
presence of two delocalized excitations within the system’s ground state. We
obtained such an expression by exploiting the fact that for large systems the
elements of the reduced density matrix can be recasted in terms of Chebyshev
polynomials and thanks to this formulation he asymptotic eigenvalues can be
evaluated with the use of the so-called Cayley-Hamilton theorem [39].

Subsequently, we will move beyond the perturbative regime and consider the
whole phase diagram of the model using a DMRG numerical approach. In Section
IV we will present the value of the EE that we extracted for different system
sizes and Hamiltonian parameters and demonstrate that, in the thermodynamic
limit, it can be written as the sum of two contributions: one that coincides
with the amount of bipartite entanglement without TF, and one directly
associated with the presence of TF which matches the analytical expression we
obtained analytically close to the classical line. This result is consistent
with the same separation obtained for TF induced by FBCs for the EE in [32]
and for the non-stabilizerness entropy in [28]. Hence, it substantiates the
conjecture that every quantum resource of a TF system, in the thermodynamic
limit, can be decomposed as the sum of the value of the resource for the
corresponding unfrustrated model and the topological contribution measured in
proximity to the classical point.

At the end, in Section V we discuss the results and draw our conclusions,
while all the technical details are provided in the Appendices.

##  II The Phase Diagram of the ANNNI Model

In Fig. 1 we draw the phase diagram of the ANNNI chain, that we obtain by the
analysis of the energy discontinuities.

Without the external transverse field ($h=0$) every term of the Hamiltonian in
eq. (1) mutually commute and the model can be considered as classical.
Introduce the dimensionless coupling $\kappa\equiv J_{2}/|J_{1}|$, in such a
limit, we can identify two phases separated by a multicritical point located
at $\kappa=\kappa_{c}=1/2$. For $\kappa<\kappa_{c}$ we have a standard
ferromagnetic ($J_{1}<0$) or AFM ($J_{1}>0$) ordered phase, while for
$\kappa>k_{c}$ the system endeavors to arrange itself so that each spin has
one neighbor spin aligned and one anti-aligned (see also Section III). This
order is referred to as the antiphase [35].

Turning back on the magnetic field, three line of phase transition originate
from the multicritical point $(h,\kappa)=(0,1/2)$. One line extends up to
$(|h|,\kappa)=(|J_{1}|,0)$, and is a phase transition of the Ising-type
between an ordered and a paramagnetic disordered phase [40]. To the right of
this line, increasing $\kappa$ for $h\neq 0$ we encounter two more transition
lines. The first marks a Berezinskii–Kosterlitz–Thouless (BKT) transition
between the disordered and a floating phase [41, 42]. The latter is a gapless
phase described by a Luttinger liquid with algebraic incommensurate
correlations [43]. The second line marks a commensurate-to-incommensurate
transition between the floating and the antiphase. In this latter region the
AFM next-to-nearest interaction $J_{2}$ is the dominant one, and we will focus
on this antiphase to study the interplay between different kinds of
frustration. We will show that, in this phase TF can be induced not only when
the system is made of an odd number of spins, but also when the length of the
chain is equal to $L=4N+2$, for some $N\in\mathbb{N}$. For the sake of
clarity, in this paper we will focus on this even site configuration, leaving
the investigation of the case with an odd number of sites to the next work.

Figure 2: ANNNI chain for $L=10$ lattice sites viewed as two interacting
topologically frustrated Ising chains of length $L^{\prime}=L/2$, composed of
the odd and even spins, respectively. When $J_{1}=0$, the two rings are
disconnected.

##  III Analytical results close to the classical point

Among other properties, the phase transition across the multicritical point
$\kappa_{c}=0.5$ between the ordered phase and the antiphase at $h=0$
manifests itself through a change in the dimensionality of the ground state
manifold (GSM). For $\kappa<1/2$, the system shows a two-fold degenerate GSM
for any even $L$, generated by the two orthogonal Néel states. Instead, in the
antiphase ($\kappa>1/2$) two different situations arise, depending on the
chain lengths. When $L=4N$, the system is characterized by a four-fold GSM,
composed by periodic configurations with 4-sites periodicity
($\ket{\uparrow\uparrow\downarrow\downarrow\uparrow\uparrow\downarrow\downarrow\dots}$).
On the contrary, for $L=4N+2$, the GSM shows a massive ground state degeneracy
in which the number of elements scales quadratically with the system size. To
understand it better, let us start by considering the case $J_{1}=0$.

###  III.1 GSM degeneracy for $J_{1}=0$

In this limit the system decomposes exactly into two independent spin rings,
respectively made of the odd and the even lattice sites (see Fig. 2). Both the
two rings are made of an odd number of spins $L^{\prime}=L/2=2N+1$ and are
characterized by PBCs and AFM nearest neighbor interactions. Therefore, both
rings are geometrically frustrated and admit $2L^{\prime}=L$ independent
ground states each [29, 28]. These states differ from each other in the
position and in the orientation of a single ferromagnetic defect embedded in a
Neél AFM state. For each ring, they can be written as

$\ket{k,\pm}_{o,e}=T_{o,e}^{k-1}\bigotimes_{j=1}^{L^{\prime}-1}\sigma_{2j}^{z}\ket{\pm}_{o,e}^{\otimes
L^{\prime}},$ (2)

where $k$ runs from $1$ to $L^{\prime}$ and marks the position of the
ferromagnetic bond, $\ket{\pm}_{o,e}$ are the eigenstates of $\sigma^{x}_{i}$
on each odd/even ring, and $T_{o,e}$ is the translational operator acting on
each of the two rings. As long as we keep $J_{1}=0$, each state constructed as
the direct product of states in Eq. (2) is a ground state of the whole system,
resulting into a GSM with $L^{2}$ independent elements:

$\mathcal{G}_{0}=\\{\ket{k,\sigma}_{o}\ket{p,\sigma^{\prime}}_{e},k,p=1,\ldots
L^{\prime},\sigma,\sigma^{\prime}=\pm\\}.$ (3)

###  III.2 GSM degeneracy for $J_{1}>0$

For finite $J_{1}$, $\mathcal{G}_{0}$ splits into two subsets, each containing
$L^{2}/2$ elements. Which of the two has lower energy depends on the sign of
$J_{1}$, and thus for definiteness let us focus on the AFM next-neighbor case
($J_{1}>0$ ). As shown in Fig. 3, it is clear that once a ferromagnetic defect
is placed in one of the rings, the spin in the other ring lying between the
two aligned spins minimizes the $J_{1}$ interaction by pointing in the
opposite direction, thus halving the lowest energy configurations.

Therefore,for $J_{1}>0$, $h=0$ and $\kappa>1/2$ the GSM becomes

$\displaystyle\mathcal{G}$ $\displaystyle=$
$\displaystyle\\!\\!\left\\{\ket{\psi(k,p)}\equiv\ket{k,(-1)^{k}}_{o}\ket{p,(-1)^{p+1}}_{e},\right.$
(4)
$\displaystyle\left.k=1,\ldots,L,\;\;\;p=k,\ldots,\\!k\\!+\\!L^{\prime}\\!-\\!1\right\\},$

where we exploit the PBCs of each ring, i.e. $k,p\equiv k,p\mod L^{\prime}$,
and with a slight abuse of notation we identify the two eigenstates of
$\sigma^{x}$ with their eigenvalues, i.e. $\pm=\pm 1$. To make the
relationship between the elements of $\mathcal{G}_{0}$ and the ones of
$\mathcal{G}$ as clear as possible, in Fig. 4 we provide a simple pictorial
representation for the case $L=10$.

$\ldots$$+$$+$$-$$\ldots$kk+1kodd chaineven chain$J_{2}$$J_{1}$ Figure 3:
Specifying the configuration of the odd chain, namely fixing the ferromagnetic
defect position ($k$ in the figure), constrains the site on the even chain
coupled to $k$ and $k+1$ by $J_{1}$ to have the opposite orientation to lower
the state’s energy. The case of a state of type $\ket{k,+}_{o}$ is
illustrated. The overall effect of the $J_{1}$ interaction is then to split
the degeneracy and reduce the number of low energy states on the even chain
from $L^{2}$ to $L^{2}/2$.

###  III.3 Ground state near the classical point

A small transverse magnetic field $h$ further breaks the super-extensive
degeneracy left in $\mathcal{G}$, producing a band of closely lying states
with a unique ground-state $\ket{g}$, similarly to what occurs for 1D chains
with only next-neighbor interactions [25, 26, 27]. To extract the ground-state
of the system we employed lowest order degenerate perturbation theory. Despite
the intricate structure of the perturbation matrix resulting from the
interaction between the two TF rings, using graph theory (details for this
calculation are presented in Appendix A.1), we are able to recover the
analytic expression for the ground state in the antiphase with $L=4n+2$ sites
close to the classical line for $J_{1}>0$. It reads

$\ket{g}=\\!A\sum_{k=1}^{L}\sum_{p=k}^{k+L^{\prime}-1}\\!\\!\sin\left[\frac{(p-k+1)L\pi}{L+2}\right]\ket{\psi(k,p)},$
(5)

where $A=2/\sqrt{L\left(L+2\right)}$ is the normalization constant.

From Eq. (5) we can recover the physical quantities of interest, but we will
focus in particular on the bipartite Von Neumann Entanglement Entropy [44,
45].

###  III.4 EE near the classical point

Figure 4: Pictorial representation of $\mathcal{G}_{0}$ and $\mathcal{G}$ for
a system of length $L=10$. The states of the even and odd rings are
represented respectively on the horizontal and vertical axis. Every vertex in
the periodic grid is an element of the set $\mathcal{G}_{0}$. The effect of
the AFM interaction ($J_{1}>0$) is to select the subset compatible with the
constraint in Fig. 3, which are represented by the red vertices.

The EE for the ground state $\ket{g}$ in Eq. (5) with respect to a bipartition
of the chain into a subsystem $A$ made by $M$ contiguous spins and its
complement $\bar{A}$ is given by

$S_{M}(\rho_{A})=\mathrm{Tr}\left[\rho_{A}\log\rho_{A}\right].$ (6)

Here $\rho_{A}$ is the reduced density matrix obtained by tracing out from
$\ket{g}$ all degrees of freedom outside $A$, i.e.
$\rho_{A}=\Tr_{\bar{A}}(\ket{g}\bra{g})$. To evaluate the EE we have to
determine the spectrum of $\rho_{A}$. Let us a sketch of the calculation here,
while the details can be found in Appendix A.2.

We start noticing that, if both $A$ and $\bar{A}$ are made of more than four
sites, only 16 eigenvalues of $\rho_{A}$ are different from zero (and appear
in 4 degenerate multiplets with multiplicity four), while all the others
vanish identically. Taking into account the interaction graph showed in Fig.
4, it is possible to re-order the basis elements according to the position of
the ferromagnetic defects being inside or outside $A$, such that a block-
structure emerges in $\rho_{A}$. Although the resulting reduced density matrix
is not block-diagonal, it is possible to prove that the off-diagonal block
provide only subleading corrections to the matrix eigenvalues and thus in the
thermodynamic limit the eigenvalues of $\rho_{A}$ coincides with those of the
diagonal blocks, which are

$\displaystyle\lambda_{1}(x)$ $\displaystyle=$
$\displaystyle\dfrac{(1-x)^{2}}{4}-\dfrac{\sin^{2}\pi x}{4\pi^{2}},$
$\displaystyle\lambda_{2}(x)$ $\displaystyle=$
$\displaystyle\dfrac{x^{2}}{4}-\dfrac{\sin^{2}\pi x}{4\pi^{2}},$ (7)
$\displaystyle\lambda_{3,4}(x)$ $\displaystyle=$
$\displaystyle\dfrac{x(1-x)}{4}+\dfrac{\sin^{2}\pi
x}{4\pi^{2}}\pm\dfrac{\sin\pi x}{4\pi},$

where $x=M/L$ is the relative dimension of the partitions with respect to the
chain length. In Fig. 5 we can observe for $M=L/2$ how the eigenvalues of
$\rho_{A}$ obtained from numerical diagonalization tend to coincide with the
analytically determined values in the thermodynamic limit. Similar results can
be obtained for different values of $x$, hence proving the validity of Eq.
(III.4).

Figure 5: Absolute values of the difference of the numerical obtained $16$
non-zero eigenvalues of $\rho_{A}$ and the asymptotic values in Eq. (III.4) as
function of the inverse of the chain length $L$.

From Eq. (III.4) it is possible to recover the expression of the entanglement
entropy for diverging $L$ when $J_{1}\neq 0$. Remarkably, it can be put in the
form

$\displaystyle S_{M}(\rho_{A})$ $\displaystyle=$ $\displaystyle
2-y\log_{2}{y}-(1-y)\log_{2}{(1-y)}$ (8)
$\displaystyle-z\log_{2}{z}-(1-z)\log_{2}{(1-z)},$

where

$y=x-\frac{\sin\pi x}{\pi},\qquad{\rm and}\qquad z=x+\frac{\sin\pi x}{\pi}.$
(9)

The first term on the RHS of Eq. (8) stems from the four-fold degeneracy of
the eigenvalues and is due to the double $\mathcal{Z}_{2}$ symmetry of
flipping each spin on each even/odd sub-lattice. The other terms represent the
EE of two delocalized particles, each of them having probability $y$ ($z$) of
being in $A$. The factorization of the entropy contribution for each
excitation indicates their independence, however their coefficients do not
coincide with the geometrical probabilities $y=z=x=M/L$ (which emerge for
$J_{1}=0$): the corrections in eq. (9) indicate a correlation between these
excitation. Indeed, they are consistent with the result in Ref. [46], where
the EE was calculated in the case of few excitations over the vacuum of a
quadratic theory. In that case, each particle contributes with a probability
that displays a correction due to the relative distance in momentum space. Eq.
(9) fits the results in ref. [46] if the two excitations in the ANNNI chain
differ by $\Delta k=\frac{2\pi}{L}$, indicating that they are trying to
minimize their kinetic energy under the Pauli-like constraint of occupying
different momentum states.

Thus, quite remarkably, we found that the ground state EE of the TF ANNNI is
compatible with the existence of two excitations. While the effects of
geometrical frustration are commonly interpreted through a single particle
description, the existence of multiple proper quasi-particles due to more
complex frustration is quite unexpected. Based on previous results [26, 27,
29, 30], we expect that the emergence of this quasi-particle picture will have
consequences also on other observables of the system. This is the case, for
example, for the energy gap, which is expected to close as $L^{-2}$ in a TF
system, or for order parameters in the antiphase, which are expected to be
destroyed or become incommensurate. Since in this work we focus on the EE
properties, we leave a detailed analysis of these features to a future work.

##  IV Numerical analysis

Figure 6: Comparison of the EE at half chain as a function of $\kappa$ for
$h/J_{1}=0.3$, between the case with TF ($L=4N+2$ blue dots), and without TF
($L=4N$ red squares). The numerical data are obtained setting $N=25$.

The ANNNI model Eq. (1) is notoriously not analytically solvable: to obtain
results beyond the perturbative regime employed so far we use a density matrix
renormalization group algorithm (DMRG) based on tensor networks [47, 49, 50,
51, 48], in which the ground state is represented through a matrix product
state (MPS). In order to avoid ambiguities in the canonization of the MPSs, we
implemented the periodic boundary conditions directly in the matrix product
operator (MPO) encoding the system’s Hamiltonian [52] rather then using
periodic MPSs [53, 54]. Some details about our numerical approach can be found
in Appendix B. The MPS approximation is known to be efficient for states
possessing a finite amount of entanglement, as in the case of the ground-
states of one-dimensional systems satisfying the area law [55, 56, 57] and has
been already applied successfully for the ANNNI chain [40, 41]. From the
results of the previous section, and in analogy with the ones obtained in
other systems with TF [32], we expect that the amount of entanglement in the
ground state must stay finite inside the antiphase even in presence of TF.
Hence, the MPS representation of the ground state must be a faithful one. Such
a representation of the ground state is particularly useful to evaluate the EE
since the extraction of the partition related Schmidt coefficients [45] is
straightforward.

Let us start with an overview of the general behavior of the EE across the
phase diagram: we set the length of $A$ to a half chain ($M=L/2$) and we
evaluate the EE as a function of $\kappa$ for $h/J_{1}=0.3$. The results
obtained both for $L=4N$ and $L=4N+2$ with $N=25$, are plot in Fig. 6. In both
the ordered and disordered phases, the EE values are practically
indistinguishable from each other. Even in the floating phase, they tend to
coincide in the limit of large $N$, although the convergence is slower and for
$N=25$ the differences can still be observed. This slow convergence is not a
surprise, since the floating phase is known to be gapless and well
approximated by a conformal field theory and thus the EE at half chain keeps
growing logarithmically with the chain length and finite size effect are more
prominent [35].

Figure 7: Bipartite Entanglement Entropy at half chain as a function of the
length of the chain. The data are obtained using our DMRG algorithm for
$J_{2}=1$, $h=0.2$, $J_{1}=1$, for the case with TF ($L=4N+2$ blue dots), and
without TF ($L=4N$ red squares).

On the contrary, the behavior of the EE becomes different as soon as we enter
the antiphase, where the topological frustration drastically affects the
behavior of the entanglement. Such a difference cannot be explained as finite
size effects, as evidenced in Fig. 7, where the dependence of the EE at half
chain is analyzed by varying the chain length for a fixed set of the
Hamiltonian parameters inside the antiphase. While for systems whose length is
an integer multiple of four the EE is virtually independent from the size, TF
induces a dependence on $L$ in the EE that, however, remains finite also in
the thermodynamic limit. Fig. 8 highlights the different subsystem dependence
of the EE in the two cases, with the not TF case quickly saturating to the
(constant) area law. The inset further show that in the TF case the growth of
the EE with the subsystem size lies in between that expected for a single and
two delocalized, independent excitations.

Figure 8: Bipartite entanglement entropy as a function of the ratio $x=M/L$
between the subsystem length and the length of the chain. The data is obtained
using our DMRG algorithm for $J_{2}=1$, $h=0.2$, $J_{1}=1$, for the TF (blue
dots, green and orange triangles) and locally frustrated (red squares) ANNNI
model, with $L=102$, $78$, $54$ and $L=104$ respectively. Inset: EE of the TF
chain for $L=102$ compared to the EE for a state with one (cyan dashed line)
and two (purple dashed line) excitations (eq. (8) with $y=x$,$z=0$ and $y=z=x$
respectively).

From previous results on other TF models, a noteworthy observation emerged: in
the thermodynamic limit, the different contributions to quantum resources
decouple in the sum of two terms in which one coincide with the resource of
the unfrustrated counterpart and the other is purely due to TF [32, 28]. This
naturally leads to the hypothesis that a similar behavior occurs also in the
EE of the ANNNI model. However, providing evidences that support this
hypothesis proves to be more delicate than in the previous cases, since for
the ANNNI isolating the local contribution is challenging. In models with only
nearest neighbor coupling, it is enough to change the interaction sign, thus
removing TF. In the ANNNI model, reversing the signs of the interactions
eliminates both TF and also the local frustration. Therefore, to remove TF
while preserving the other local properties, we evaluate the EE for the same
set of the Hamiltonian parameters $\mathcal{J}\equiv(J_{1},J_{2},h)$ and with
the same length $L$ but applying open boundary conditions and considering the
subset $A$ in the middle of an open chain, to reduce boundary effects. We
compare the EE obtained in this way, which we denote as
$S_{M}^{o}(\mathcal{J},L)$, with the EE obtained assuming PBCs, namely
$S_{M}^{p}(\mathcal{J},L)$. If the hypothesis stands, the difference of these
two quantities must be equal to the topological contribution that is provided
by Eq. (8). In other words, if the hypothesis is verified the quantity
$R(\mathcal{J},x,L)$, defined as

$R(\mathcal{J},x,L)=\frac{S_{M}^{p}(\mathcal{J},L)-S_{M}^{o}(\mathcal{J},L)}{S_{M}(g)},$
(10)

when $L\rightarrow\infty$, shall tend to $R(\mathcal{J},x,L)\rightarrow 1$.
The data depicted in Fig. 9 clearly support this hypothesis. The analysis
carried out in the inset indicates a power-law convergence in the
thermodynamic limit with $\log(1-R)=-0.89\log L+1.47$.

It is worth mentioning here that, we numerically observe that the local
contribution to the EE in addition to the global $2$ factor is minimal
(approximately $10^{-3}$) within the bulk of the antiphase and increases only
close to the phase transition. Hence, even at finite $h$, the total
entanglement entropy of the frustrated system can be well approximated by the
perturbative formula obtained for $h\ll J_{1}$.

Figure 9: $R(h,L)$, defined in Eq. (10), as a function of the inverse system’s
size $L^{-1}$ for $h=0.2$, $\kappa=1$. The inset shows a plot in log-log scale
of $1-R$ vs $L$, together with a linear fit, hinting to a power-law
convergence in the thermodynamic limit, describe by the numerical law
$\log(1-R)=-0.89\log L+1.47$.

##  V Conclusion and outlook

In summary, we considered the effects of the interplay between local and non-
local sources of frustration in 1D spin chains through the analysis of the
entanglement entropy of the ANNNI model. The first type of frustration is due
to the interplay between the nearest and next-to nearest neighbor
interactions, while the non-local source (TF) is injected by a suitable choice
of boundary conditions. In this work our focus centered on the antiphase
region, where we revealed the presence of TF even in systems with an even
number of spins, when the number of sites is not divisible by four. In this
way, we showed that TF can emerge beyond the usual framework of geometrical
frustration induced by FBCs (odd number of sites), presenting both usual and
novel features.

The emergence of TF with an even number of sites can be traced back to the
fact that, in the limit in which the interaction with first neighbors becomes
negligible, the system decomposes into two independent topologically
frustrated quantum Ising models.

Starting from this consideration, exploiting an analytical, perturbative
approach valid close to the classical limit, we managed to obtain an
analytical expression of the unique ground state of the model. From it, we
were able to obtain the value of the EE for a generic bipartition composed of
connected subsets of contiguous spins in the thermodynamic limit. We proved
that the EE is decomposed into a contribution present also without TF and one
due to TF that indicates that the ground state hosts two excitations. The
latter terms are independent, except for the fact that both particles tend to
minimize their momenta, but cannot occupy the same momentum state and in this
way they develop a correlation, as often happens in 1D system, where
excitations typically acquire a fermionic nature.

To study the EE in the whole phase diagram of the model we employed a tensor
network based DMRG code and observed that for $L=4N+2$ the effect of TF
extends, and is limited, to the entire antiphase. There, we showed that the EE
can be decomposed again into a non TF contribution (obtained by applying open
boundary conditions to the same Hamiltonian) and that of two excitations, with
the same values obtained in the analytical perturbative regime. The importance
of this results stems from its consistency with what happens to every quantum
resource analyzed so far in models with only nearest-neighbor interaction. It
therefore supports the idea that this decomposition of quantum resources is a
general characteristic of topologically frustrated systems.

We would like to stress once more that the unveiled quasi-particle description
of the TF ANNNI chain is unexpected and required a meticulous effort to be
exposed. We expect that similar results may be unveiled also for other
frustrated systems. We plan to continue this investigation and in the next
work, we will consider the same ANNNI model, but with an odd number of sites.
There the interplay between local and topological frustrations is even more
intricate and affects the model beyond just the antiphase.

In general, we showed that the (extensive) frustration emerging from the
competition between local interactions and the intensive one of topological
origin contribute differently to the phenomenology of the model. In the future
we will consider other properties, such as order parameters, complexity,
quantum coherence, etc., but even more interesting will be to explore other
models with different types of frustration to map and understand the
phenomenology of the interplay between various source of frustration,
eventually moving to higher dimensions as well.

###### Acknowledgements.

AGC acknowledges support from the MOQS ITN programme, a European Union’s
Horizon 2020 research and innovation program under the Marie Skłodowska-Curie
grant agreement number 955479. SBK is supported by the Croatian Science
Foundation (HrZZ) Project DOK–2019–4–3321

##  Appendix A Perturbation theory close to the classical line

In this appendix we show the details of how to obtain the ground state of the
topologically frustrated ANNNI model near the classical line, i.e. in the
limit $h\rightarrow 0^{+}$, resorting to the lowest order perturbation theory
and how to use this result to extract its bipartite entanglement entropy.

###  A.1 Determination of the ground state

Let us divide the full Hamiltonian in eq. (1) as

$H=H_{0}+h\,\mathcal{H}^{\textrm{pert}},$ (11)

with

$H_{0}=J_{1}\sum_{i=1}^{L}\sigma_{i}^{x}\sigma_{i+1}^{x}+J_{2}\sum_{i=1}^{L}\sigma_{i}^{x}\sigma_{i+2}^{x},$
(12)

and

$\mathcal{H}^{\textrm{pert}}=\sum_{j=1}^{L}\sigma_{j}^{x}=\mathcal{H}^{\textrm{pert}}_{\textrm{o}}+\mathcal{H}^{\textrm{pert}}_{\textrm{e}},\quad\mathcal{H}^{\textrm{pert}}_{\textrm{o}/\textrm{e}}=\sum_{\begin{subarray}{c}j\in\\{o/e\\}\end{subarray}}\sigma_{j}^{x},$
(13)

where, keeping the notation introduced in the main text, the e (o) subscript
refer to the even (odd) sites subchain.

To find the analytical expression of the ground-state close to $h=0$ we thus
need to diagonalize the matrix $\mathcal{H}^{\textrm{pert}}$ over the
degenerate ground-state manifold $\mathcal{G}$ (Eq. (4) in the main text),
which reads

$\displaystyle\bra{\psi_{(k,p)}}\mathcal{H}^{\textrm{pert}}\ket{\psi_{(k^{\prime},p^{\prime})}}=f_{\textrm{o}}(k,p,k^{\prime},p^{\prime})+f_{\textrm{e}}(k,p,k^{\prime},p^{\prime}),$
(14) $\displaystyle
f_{\textrm{o}}(k,p,k^{\prime},p^{\prime})\equiv\left(\delta_{k,k^{\prime}-1}\delta_{(-1)^{k},(-1)^{k^{\prime}-1}}+\right.$
$\displaystyle\left.+\delta_{k,k^{\prime}+1}\delta_{(-1)^{k},(-1)^{k^{\prime}+1}}\right)\times\delta_{p,p^{\prime}}\delta_{(-1)^{p},(-1)^{p^{\prime}+1}},$
$\displaystyle
f_{\textrm{e}}(k,p,k^{\prime},p^{\prime})\equiv\delta_{k,k^{\prime}}\delta_{(-1)^{k},(-1)^{k^{\prime}}}\times$
$\displaystyle\times\left(\delta_{p,p^{\prime}+1}\delta_{(-1)^{p+1},(-1)^{p^{\prime}+2}}+\delta_{p,p^{\prime}-1}\delta_{(-1)^{p+1},(-1)^{p^{\prime}}}\right),$

where periodicity of the indices with respect to $L^{\prime}$ is intended. The
matrix plot and its associated graph are shown in Fig. 10.

Figure 10: Left: Matrix representation of the perturbation Eq. (13) over the
set $\mathcal{G}$ given in Eq. (4) for the ANNNI chain of length $L=14$.
Right: Graph representation of the matrix on the left. Figure 11: a) The Torus
Grid graph $T_{L,L}$ obtained as the Cartesian product $C_{L}\square C_{L}$
for $L=22$. b) The unfolded Torus with the two subsets corresponding to
$\mathcal{G}$ (red vertices) and $\mathcal{G}^{c}$ (gray vertices)
highlighted, and the corresponding disconnected sub-graphs. c) The same sub-
graphs can be also obtained as the tensor product of the Cycle graph of
cardinality $2L$ and the Path graph of length $L^{\prime}$.

Because of the complex structure of this matrix, its diagonalization is non
trivial. Nevertheless, a better insight can be obtained by looking first at
the simpler case in which $J_{1}=0$, corresponding to two non-interacting TF
Ising rings. In this case, it is easy to check that
$\mathcal{H}^{\textrm{pert}}$ can be written as the cartesian product (box
product) of two circulant matrices when evaluated over the manifold
$\mathcal{G}_{0}$ (see Eq. (3) of the main text), namely

$\mathcal{H}^{\textrm{pert}}=\mathcal{H}^{\textrm{pert}}_{\textrm{o}}\square\mathcal{H}^{\textrm{pert}}_{\textrm{e}}.$
(15)

While it is known that these matrices can be exactly diagonalized, it is worth
noting that all the entries of $\mathcal{H}^{\textrm{pert}}_{o/e}$ are either
zeros or one. Hence, they can be understood as adjacency matrices [58],
associated to two identical cycle graphs $C_{\frac{L}{2}}$ [59]. Remembering
that for any given two graphs $G_{1}$ and $G_{2}$ having adjacency matrix
$A(G_{i}),\,i=1,2$ respectively, $A(G_{1})\square A(G_{1})=A(G_{2}\square
G_{2})$ [60] (Fig, 11), we can conclude that the resulting graph associated
with the full matrix $\mathcal{H}^{\textrm{pert}}$ is the Torus Grid Graph
$T_{\frac{L}{2},\frac{L}{2}}=C_{\frac{L}{2}}\square C_{\frac{L}{2}}$ [59].

As discussed in the main text, setting $J_{1}\neq 0$ acts as a selection rule:
its effect being to split the set $\mathcal{G}_{0}$ in half according to the
orientation of the spin of the second chain that falls between the two spins
of the first ring where the magnetic defect is localized:

$\mathcal{G}_{0}=\mathcal{G}\cup\mathcal{G}^{c},$ (16)

where $\mathcal{G}^{c}$ is the complement set of $\mathcal{G}$. The graphs of
the two sets $\mathcal{G}$ and $\mathcal{G}^{c}$ are respectively the red and
gray part of the torus (see Fig. 11b for its unfolded representation and its
decomposition). The red one is also the graph obtained through Eq. (14), its
ground state being then the required solution. We can now observe that this
graph decomposition can be achieved through the Cartesian product of a Cycle
graph of length $2L$ and a Path graph of length $L^{\prime}$ (Fig. 11c).
Specifically, the $L$ kink states of the odd chain couple separately with half
of the kink states ($L^{\prime}$) that are compatible with the condition
$J_{1}>0$, and the other half with the condition $J_{1}<0$. The eigenvectors
of the adjacency matrix of the desired graph (the red ring in Fig. 11) can
then be obtained directly from the eigenvectors of the graphs involved in the
operation (see Tab. 1) through their tensor product, and finally projecting
into the subspace of the desired ring, since the two graphs are disconnected
(see Tab. 1).

Graph | A-eigenvalues | A-eigenvectors
---|---|---
$G_{1}$ | $\lambda_{i}$ | $X_{i}$
$G_{2}$ | $\mu_{i}$ | $Y_{j}$
$G_{1}\square\,G_{2}$ | $\lambda_{i}+\mu_{j}$ | $X_{i}\otimes Y_{j}$
$G_{1}\otimes\,G_{2}$ | $\lambda_{i}\mu_{j}$ | $X_{i}\otimes Y_{j}$
$G_{1}\cup G_{2}$ | $\lambda_{i}$ | $\left(\begin{array}[]{c}X_{i}\\\ 0\end{array}\right)$
$\mu_{j}$ | $\left(\begin{array}[]{c}0\\\ Y_{j}\end{array}\right)$
Table 1: Properties of the graph operations considered. First column: graphs and graph operations. Second column: eigenvalues of the corresponding adjacency matrix $A$. Third column: Eigenvector of the corresponding adjacency matrix $A$. Graph | A-eigenvalues | A-eigenvectors
---|---|---
Cycle | $\lambda_{l}=\sum_{k=0}^{m-1}c_{k}\omega^{kj}$ | $\ket{b_{l}}=\sum_{r=0}^{m-1}e^{i\frac{2\pi}{m}lr}{}\ket{r}$
$j=0,\ldots,m-1$
Path | $\mu_{k}=2\cos\left(\frac{k\pi}{m+1}\right)$ | $\ket{a_{k}}=\sum_{j=1}^{m}\sin\left(\frac{k\pi}{m+1}j\right)\ket{j}$
$k=1,2,...,m$
Table 2: Eigenvalues and eigenvectors of the adjacency matrix of the Cycle and
Path graphs having $m$ vertices. The coefficients $c_{k}$ is the matrix
element on the $k$-th row of the Circulant matrix associated to the Cycle
graph.

The general eigenvector will then be of the form
$\ket{b_{l}}\otimes\ket{a_{k}}$, that are respectively eigenvectors of the
Cycle and Path graphs, with eigenvalue $\lambda_{l}\,\mu_{k}$ (see Tab. 2).
The ground state will correspond to the $\\{l,k\\}$ values for which the
product is minimum. Since the solely effect of the $J_{1}$ interaction is to
select a sub-set of states, we expected for the odd chain the ground state of
the topologically frustrated Ising ring ($l=0$). As a consequence, the product
is minimized for $k=L^{\prime}$, and the ground states will be of the form
$\ket{b_{0}}\otimes\ket{a_{L^{\prime}}}$. The ground state will then be of the
form

$\ket{g}\\!=\\!\\!A\sum_{k=1}^{L}\\!\sum_{r=0}^{L^{\prime}-1}\sin\left[\alpha(r+1)\right]\ket{k,(-1)^{k}}\ket{k+r,(-1)^{k+r}}\\!,$
(17)

where $\alpha=L\pi(r+1)/(L+2)$, and where we disentangle the indexes in Eq.
(4), writing the general element as
$\ket{k,(-1)^{k}}\ket{k+r,(-1)^{k+r}},k=1,\ldots,L,r=0,\ldots,L^{\prime}-1$.
The periodicity of the indices with respect to $L^{\prime}$ is implicit.

To fix the normalization constant let us first introduce the following
quantity

$f_{d,b}^{m,n}(\alpha)=\sum_{r=m}^{n}\sin[\alpha(r+d+1)]\sin[\alpha(r+b+1)],$
(18)

with $\alpha\in\mathbb{R}$, and we write $f_{b,b}^{m,n}$ instead of
$f_{b,b}^{m,n}(\alpha)$. Eq. (18) admits a nice representation in terms of the
Chebyshev polynomials of first and second type, $T_{n}(\cos\alpha)$ and
$U_{n}(\cos\alpha)$ respectively, remembering that
$\sin[\alpha(n+1)]=\sin\alpha\;U_{n}(\cos\alpha)$ and resorting to the
relations [61]

$\displaystyle T_{a+b}(x)-T_{a-b}(x)=2(x^{2}-1)U_{a-1}(x)U_{b-1}(x),$ (19)
$\displaystyle T_{2(a-b)}(x)-1=2(x^{2}-1)U^{2}_{a-b-1}(x),$ (20)
$\displaystyle T_{a+b}(x)+T_{a-b}(x)=2T_{a}(x)T_{b}(x),$ (21) $\displaystyle
T_{2(a-b)(x)}+1=2T^{2}_{a-b}(x).$ (22)

We have

$\displaystyle f_{d,b}^{m,n}$ $\displaystyle=\frac{1}{2}(n-m+1)T_{b-d}+$
$\displaystyle-\dfrac{1}{4}\left(U_{-(2m+b+d+2)}+U_{2n+b+d+2}\right).$ (23)

Furthermore we also need the square of Eq. (18)

$\displaystyle(f_{d,b}^{m,n})^{2}$
$\displaystyle=\dfrac{(n-m+1)^{2}}{8}\left(1+T_{2(b-d)}\right)+$
$\displaystyle+\dfrac{1}{32\sin^{2}(\alpha)}\left(2-T_{2(2n+b+d+3)}-T_{2(2m+b+d+1)}+\right.$
$\displaystyle\left.T_{2(n+m+b+d+2)-T_{2(n-m+1)}}\right)+$
$\displaystyle-\dfrac{n-m+1}{16}\left(U_{2(n+b+1)}+U_{2(2n+d+1)}+\right.$
$\displaystyle\left.+U_{-2(n+d+1)}+U_{-2(n+b+1)}\right),$ (24)

and we evaluate the asymptotic expression

$\lim_{\begin{subarray}{c}L,M\to\infty\\\ M/L\to
C\end{subarray}}T_{aL+bM+c}(\cos\alpha)=(-1)^{b+c}\cos(2b\pi x),$ (25)

with $x=M/L$, $\alpha=L\pi/(L+2)$, $0<C<1$, and $a,b,c\in\mathbb{Z}$, which is
valid for even and odd $L,M\in\mathbb{N}$ respectively.

Simplifying the notation as $T_{n}(\cos\alpha)=T_{n}$ and
$U_{n}(\cos\alpha)=U_{n}$, the normalization constant can be then be
calculated resulting in

$\innerproduct{g}{g}=|A|^{2}L\,f_{0,0}^{0,\frac{L}{2}-1}=|A|^{2}\left(\dfrac{L^{2}}{4}+\dfrac{L}{4}-\dfrac{L}{4}U_{L}\right)=1.$
(26)

Furthermore, $U_{L}=-1$ for $L$ even. In fact, remembering that
$\sin[\alpha(n+1)]=\sin\alpha\;U_{n}(\cos\alpha)$ we can prove equivalently
that $\sin\left[\alpha(L+1)\right]=-\sin\left(\alpha\right),\forall\;L$ even,
condition that it is always satisfied for $L=4n+2,\forall n\in\mathbb{N}$.
From Eq. (26) it then follows that $A=2/\sqrt{L\left(L+2\right)}$. The
translational invariant form Eq. (5) is then obtained through the substitution
$p\rightarrow k+r$.

###  A.2 Asymptotic expression of the bipartite Entanglement Entropy close to
the classical line.

Having determined the expression for the ground state close to the classical
line, in this appendix we detail the analytic derivation of its bipartite
Entanglement Entropy. Without loos of generality, we consider a system sub-
partition $A$ made by an odd number $M$ of consecutive spins. The
corresponding spin number inside $A$ for the two sub-chains are then equal to
$M_{o}=(M+1)/2$ and $M_{e}=(M-1)/2$ respectively (see Fig. 12).

Figure 12: Partitioning of the ANNNI chain for the computation of the
Entanglement Entropy Eq. (6) near the classical point. The I (O) notation
indicates that the $(k,p):=k|p$ indexes are inside (outside) the partition
$A$.

As discussed in Sec. III, in analogy with the approaches used for the TF Ising
chain [33], it is useful to rewrite the state in Eq. (5) according to the kink
indexes $(k,p):=k|p$ being internal $(\textrm{I})$ or external $(\textrm{O})$
to the subsystem $A$

$\ket{g}=\ket{\textrm{I}|\textrm{I}}+\ket{\textrm{I}|\textrm{O}}+\ket{\textrm{O}|\textrm{I}}+\ket{\textrm{O}|\textrm{O}},$
(27)

where the vertical bar $(\textrm{|})$ separates the indexes corresponding to
the odd/even chains. We can then express the corresponding reduced density
matrix as a $16\times 16$ block matrix. Furthermore, we numerically observe
that only four blocks contribute to the spectrum in the thermodynamic limit as
shown in Fig. 5. We can then consider the following block-diagonal
approximated form for the reduced density matrix

$\rho_{A}=\left(\begin{array}[]{cccc}\rho^{\textrm{I}\textrm{I}|\textrm{I}\textrm{I}}&0&0&0\\\
0&\rho^{\textrm{I}\textrm{O}|\textrm{I}\textrm{O}}&0&0\\\
0&0&\rho^{\textrm{O}\textrm{I}|\textrm{O}\textrm{I}}&0\\\
0&0&0&\rho^{\textrm{O}\textrm{O}|\textrm{O}\textrm{O}}\end{array}\right).$
(28)

Let us now discuss each block matrix and derive their eigenvalues separately.

####  A.2.1 Asymptotic eigenvalues of the matrices
$\rho^{\textrm{I}\textrm{I}|\textrm{I}\textrm{I}}$ and
$\rho^{\textrm{O}\textrm{O}|\textrm{O}\textrm{O}}$

We start considering the matrix
$\rho^{\textrm{O}\textrm{O}|\textrm{O}\textrm{O}}$

$\rho^{\textrm{O}\textrm{O}|\textrm{O}\textrm{O}}=\dfrac{4}{2L+L^{2}}\sum_{n=0}^{\frac{L-M-1}{2}-1}f_{0,0}^{0,n}\mathbb{1}_{4\times
4},$ (29)

where we have used the property $c_{r}=c_{L/2-r}$. We have then one eigenvalue
with multiplicity four

$\displaystyle\lambda_{1}=\dfrac{4}{2L+L^{2}}\sum_{n=0}^{\frac{L-M-3}{2}}f_{0,0}^{0,n}\left(\alpha\right)=$
$\displaystyle=\dfrac{(L-M-1)(L-M+3)}{4L(L+2)}+$
$\displaystyle-\dfrac{1}{2L(L+2)\sin^{2}\alpha}\left(T_{2}-T_{L-M+1}\right).$
(30)

Its asymptotic expression is straightforward resorting to the limit Eq. (25)

$\lambda_{1}^{\mathrm{Asym}}(x)=\dfrac{(1-x)^{2}}{4}-\dfrac{\sin^{2}\pi
x}{4\pi^{2}}.$ (31)

The matrix $\rho^{\textrm{I}\textrm{I}|\textrm{I}\textrm{I}}$ is a $4\times 4$
diagonal block matrix with identical blocks once resorting to the property
$c_{r}^{2}=c_{L/2-r}^{2}$

$\rho^{\textrm{I}\textrm{I}|\textrm{I}\textrm{I}}=\bigoplus_{k=1}^{4}\tilde{\rho}^{\textrm{I}\textrm{I}|\textrm{I}\textrm{I}},$
(32)

with the $(M+1)/2\times(M+1)/2$ matrix
$\tilde{\rho}_{\textrm{I}\textrm{I}|\textrm{I}\textrm{I}}$ given by

$\tilde{\rho}^{\textrm{I}\textrm{I}|\textrm{I}\textrm{I}}=\left[v_{\frac{M-1}{2}},v_{\frac{M-3}{2}},\ldots,v_{0}\right]\otimes\left[v_{\frac{M-1}{2}},v_{\frac{M-3}{2}},\ldots,v_{0}\right],$
(33)

with $v_{i}=(c_{0},c_{1},\ldots,c_{i-1})$. Despite its complexity this matrix
has rank one. From the Cayley-Hamilton theorem then the only non-zero
eigenvalue with multiplicity four is given by its trace [39]

$\displaystyle\lambda_{2}$
$\displaystyle=\mathrm{Tr}\left[\tilde{\rho}^{\textrm{I}\textrm{I}|\textrm{I}\textrm{I}}\right]=\dfrac{4}{2L+L^{2}}\sum_{n=0}^{\frac{L-M-1}{2}-1}f_{0,0}^{0,n}=$
$\displaystyle=\dfrac{(M-1)(M+3)}{4L(L+2)}+$
$\displaystyle-\dfrac{1}{2L(L+2)\sin^{2}\alpha}\left(T_{2}-T_{M+1}\right).$
(34)

Its asymptotic limit is given by

$\lambda_{2}^{\mathrm{Asym}}(x)=\dfrac{x^{2}}{4}-\dfrac{\sin^{2}\pi
x}{4\pi^{2}}.$ (35)

####  A.2.2 Asymptotic eigenvalues of the matrices
$\rho^{\textrm{I}\textrm{O}|\textrm{I}\textrm{O}}$ and
$\rho^{\textrm{O}\textrm{I}|\textrm{O}\textrm{I}}$

The matrices $\rho^{\textrm{I}\textrm{O}|\textrm{I}\textrm{O}}$ and
$\rho^{\textrm{O}\textrm{I}|\textrm{O}\textrm{I}}$ are $2\times 2$ block
matrices with identical blocks of size respectively
$\frac{M+1}{2}\times\frac{M+1}{2}$ and $\frac{M-1}{2}\times\frac{M-1}{2}$

$\rho^{\textrm{I}\textrm{O}|\textrm{I}\textrm{O}}=\bigoplus_{k=1}^{2}\tilde{\rho}^{\textrm{I}\textrm{O}|\textrm{I}\textrm{O}},\quad\rho^{\textrm{O}\textrm{I}|\textrm{O}\textrm{I}}=\bigoplus_{k=1}^{2}\tilde{\rho}^{\textrm{O}\textrm{I}|\textrm{O}\textrm{I}},$
(36)

with

$\displaystyle\left(\tilde{\rho}^{\textrm{I}\textrm{O}|\textrm{I}\textrm{O}}\right)_{i,j}=f_{\frac{L-M-1}{2}-i,\frac{L-M-1}{2}-j}^{0,\frac{L-M-1}{2}},$
$\displaystyle\left(\tilde{\rho}^{\textrm{O}\textrm{I}|\textrm{O}\textrm{I}}\right)_{i,j}=f_{i,j}^{1,\frac{L-M-1}{2}},$
(37)

where we resort again to the property $c_{r}=c_{L/2-r}$.

The matrices Eqs. (A.2.2) has rank two. As a consequence, resorting to the
Cayley-Hamilton theorem, we can compute the non-zero eigenvalues solving the
following quadratic equation [39]

$\lambda^{2}-\mathrm{Tr}A\,\lambda-\dfrac{1}{2}\left[\mathrm{Tr}A^{2}-\left(\mathrm{Tr}A\right)^{2}\right]=0,$
(38)

with
$A=\tilde{\rho}^{\textrm{I}\textrm{O}|\textrm{I}\textrm{O}/\textrm{O}\textrm{I}|\textrm{O}\textrm{I}}$.
A straightforward computation thought Eq.s (A.1) and Eq. (A.1) shows that

$\displaystyle\mathrm{Tr}\tilde{\rho}_{\textrm{I}\textrm{O}|\textrm{I}\textrm{O}}$
$\displaystyle=\dfrac{4}{2L+L^{2}}\sum_{r=0}^{\frac{M-1}{2}}f_{0,0}^{r,\frac{L-M-1}{2}}=\dfrac{(M+1)(L-M+1)}{2L(L+2)}+\dfrac{1}{2L(L+2)\sin^{2}\alpha}\left(1-T_{M+1}-T_{L-M+1}+T_{L+2}\right).$
(39)
$\displaystyle\mathrm{Tr}(\tilde{\rho}^{\textrm{I}\textrm{O}|\textrm{I}\textrm{O}})^{2}$
$\displaystyle=\sum_{b,d=0}^{\frac{M-1}{2}}\left(f_{d,b}^{0,\frac{L-M-1}{2}}\right)^{2}=\dfrac{(L-M+1)^{2}(M+1)^{2}}{8L^{2}(L+2)^{2}}+\dfrac{(L-M+1)^{2}}{4L^{2}(L+2)^{2}\sin^{2}\alpha}(1-T_{M+1})+$
$\displaystyle-\dfrac{(L-M+1)(M+1)}{2L^{2}(L+2)^{2}\sin^{2}\alpha}(T_{L-M+1}-T_{L+2}+T_{M+1}-1)+\dfrac{(M+1)^{2}}{4L^{2}(L+2)^{2}\sin^{2}\alpha}(1-T_{L-M+1})+$
$\displaystyle+\dfrac{1}{4L^{2}(L+2)^{2}\sin^{4}\alpha}\left[2T_{L+2}-T_{L-M+3}-T_{L-M+1}-T_{2L-M+3}-T_{M+1}+\dfrac{1}{2}(T_{2(L-M+1)}+T_{2(L+2)}+T_{2(M+1)})\right],$
(40)
$\displaystyle\mathrm{Tr}\tilde{\rho}^{\textrm{O}\textrm{I}|\textrm{O}\textrm{I}}$
$\displaystyle=\dfrac{4}{2L+L^{2}}\sum_{r=1}^{\frac{M-1}{2}}f_{0,0}^{r,r+\frac{L-M-3}{2}}=\dfrac{(M-1)(L-M-1)}{2L(L+2)}+\dfrac{1}{2L(L+2)\sin^{2}\alpha}\left[T_{2}-T_{M+1}-T_{L-M+1}+T_{L}\right],$
(41)
$\displaystyle\mathrm{Tr}(\tilde{\rho}^{\textrm{O}\textrm{I}|\textrm{O}\textrm{I}})^{2}$
$\displaystyle=\sum_{b,d=0}^{\frac{M-3}{2}}\left(f_{d,b}^{1,\frac{L-M-1}{2}}\right)^{2}=\dfrac{(L-M-1)^{2}(M-1)^{2}}{8L^{2}(L+2)^{2}}+\dfrac{(L-M-1)^{2}}{4L^{2}(L+2)^{2}\sin^{2}\alpha}(1-T_{L-M+1})+$
$\displaystyle-\dfrac{(L-M-1)(M-1)}{2L^{2}(L+2)^{2}\sin^{2}\alpha}(T_{L-M+1}-T_{L}+T_{M+1}-T_{2})+\dfrac{(M-1)^{2}}{4L^{2}(L+2)^{2}\sin^{2}\alpha}(1-T_{L-M-1})+$
$\displaystyle+\dfrac{1}{4L^{2}(L+2)^{2}\sin^{4}\alpha}\left[2T_{L+2}-T_{L-M+3}-T_{L+M+1}-T_{2L-M+1}-T_{M+3}+\dfrac{1}{2}(T_{2(L-M+1)}+T_{2L}+T_{2(M+1)})+T_{4}\right].$
(42)

Through the limit Eq. (25) we obtain the same asymptotic expression for the
eigenvalues of the two matrices, with multiplicity four

$\displaystyle\lambda_{3,4}^{\mathrm{Asym}}(x)=\dfrac{x(1-x)}{4}+\dfrac{\sin^{2}\pi
x}{4\pi^{2}}\pm\dfrac{\sin\pi x}{4\pi}.$ (43)

####  A.2.3 Collecting the reduced density matrix eigenvalues

The asymptotic expression for the entanglement entropy Eq. (8) can be obtained
through the definition Eq. (6) and the asymptotic expression of the
eigenvalues Eq. (III.4), as well as with Eq.s (A.2.1), (35), and (43),
noticing that

$\displaystyle\lambda_{1}^{\mathrm{Asym}}=\dfrac{1}{4}(1-y)(1-z),\quad\lambda_{2}^{\mathrm{Asym}}=\dfrac{1}{4}yz,$
$\displaystyle\lambda_{3}^{\mathrm{Asym}}=\dfrac{1}{4}y(1-z),\quad\lambda_{4}^{\mathrm{Asym}}=\dfrac{1}{4}(1-y)z,$
(44)

with $y=x-\frac{\sin\pi x}{\pi}$ and $z=x+\frac{\sin\pi x}{\pi}$.

##  Appendix B Numerical analysis

Figure 13: Left: scaling of the infidelity between ground-states computed with
different bond dimension. Right: saturation of the entanglement entropy with
increasing bond dimension $D$.

We performed a numerical estimation of the ground-state of the Hamiltonian Eq.
(1) using the density matrix renormalization group (DMRG) technique. In
particular, we implemented a DMRG based on tensor networks, following the
approach of [48]. Despite the fact that in presence of topological frustration
the EE is expected to violate the area law, it is also known from [32] that it
will saturate to a finite value for large system’s sizes, as opposite to what
happens in critical systems where a logarithmic divergence of the EE is
expected in thermodynamic limit. Therefore, one could expect that an MPS
ansatz can still reproduce the ground-states of topologically frustrated
systems in an efficient way.

In order to benchmark the validity of the MPS ansatz we computed the fidelity
between ground-states corresponding to MPS with different bond-dimensions $D$

$\mathcal{F}=|\bra{GS(D+\Delta D)}\ket{GS(D)}|,$ (45)

and checked the saturation of the entanglement entropy as a function of the
bond dimension. The results are shown in Fig. 13. We observe that the
infidelity, i.e. $1-\mathcal{F}$, decays with increasing bond-dimension.
Moreover, also the EE entropy shows a saturation for increasing bond-
dimension, confirming the convergence of the DMRG algorithm. for sufficiently
large bond-dimension.

Based on the results of Fig. 13 we decided to use $D=70$ to collect the data
presented in Sec. IV.

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