# Exploring quantum entanglement in chiral symmetry partial restoration with
1+1 string model

Wei Kou<EMAIL_ADDRESS>Institute of Modern Physics, Chinese Academy of
Sciences, Lanzhou 730000, China  University of Chinese Academy of Sciences,
Beijing 100049, China  Xurong Chen<EMAIL_ADDRESS>(Corresponding author)
Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou 730000,
China  University of Chinese Academy of Sciences, Beijing 100049, China

###### Abstract

Within the confining color flux tube picture, we assess the color electric
field generated by quark-antiquark pairs within the framework of the Schwinger
model and estimate its impact on the chiral condensate. We observe differences
in the distribution of the color flux tube generated by quark-antiquark pairs
at different separation distances, leading to discrepancies in the partial
restoration of chiral symmetry. Furthermore, we suggest incorporating the
color magnetic field in the calculation of chiral condensate, leading to
quantum entanglement effects, and proceed to compute the entanglement entropy.
We observe that the entanglement entropy increases with the distance of the
color source (the separation distance between quark and anti-quark), and the
magnitude of the color electric field and chiral condensate restoration after
spatial integration also increases with the distance of the color source.
Then, we try to provide a qualitative explanation for the existence of these
phenomena.

###### pacs:

24.85.+p, 13.60.Hb, 13.85.Qk

##  I Introduction

The Schwinger model Schwinger (1962a, b) with massless fermions is exactly
solvable in (1+1) dimensional spacetime. From this classical solvable model,
one can gain insight into some properties of Quantum Chromodynamics (QCD),
such as chiral symmetry breaking and confinement (nonzero fermion mass). These
unresolved issues motivate physicists to deeply explore QCD, and much related
works have been done by using the equivalence of the massless fermion case of
the Schwinger model to the theory of free massive boson fields Brown (1963);
Zumino (1964); Lowenstein and Swieca (1971); Casher  _et al._ (1974); Coleman
_et al._ (1975); Kharzeev and Loshaj (2014); Tomiya (2022); Florio  _et al._
(2023).

As one of the unsolved mysteries in QCD, the quark confinement problem has
consistently troubled many researchers. In recent years, researchers have made
significant progress in confinement studies by using the description of the
Schwinger model in holographic QCD to establish a connection between the
quantum entanglement of quark-antiquark pairs accelerating away from each
other in the vacuum and the Einstein-Rosen bridge, and incorporating
confinement potential into the analysis Xiao (2008); Semenoff and Zarembo
(2011); Chernicoff  _et al._ (2013); Jensen and Karch (2013); Lewkowycz and
Maldacena (2014); Sonner (2013); Hubeny and Semenoff (2014); Jensen  _et al._
(2014); Ghodrati (2015); Grieninger  _et al._ (2023a, b). These works inspire
the question: Is there a duality between wormholes in the bulk and quantum
entanglement?

In general, the interpretation of the confinement effect in the language of
string theory can be simplified as the breaking of strings due to the
separation of quark-antiquark pairs, leading to the excitation of new quark-
antiquark pairs from the vacuum. The newly generated quark-antiquark pairs can
affect the distribution of color-electric fields in the vacuum, thus
influencing the vacuum dynamics of QCD. Discussions of such phenomena have
been reflected in recent simulation calculations Florio  _et al._ (2023);
Ikeda  _et al._ (2023), still based on the Schwinger model.

Another feature of QCD is the chiral symmetry breaking. As massless quarks
move in the QCD vacuum, they acquire dynamical mass, resulting in the
emergence of a non-zero quark condensate $\langle q\bar{q}\rangle_{0}$. In
other words, chiral symmetry breaking does not occur in the presence of chiral
symmetry. This property is referred to as dynamical chiral symmetry
spontaneous breaking (DCSB) Nambu and Jona-Lasinio (1961a, b). It is
meaningful to calculate the vacuum chiral condensate using an appropriate
model. Ref. Hamer  _et al._ (1982) provides a method for computing the
condensate considering the massive Schwinger model. Assuming the change in
vacuum condensate behavior under the influence of background fields is
interesting, which has prompted the development of a theoretical framework to
describe non-Abelian background fields in QCD. The description of
(3+1)-dimensional non-Abelian gauge fields is complex. In order to simplify
the framework, a quasi-Abelian picture based on the dual Meissner effect has
been proposed, providing support for many lattice QCD studies Singh  _et al._
(1993); Schilling  _et al._ (1999); Chernodub  _et al._ (2001, 2005); Suzuki
_et al._ (2009); Cea  _et al._ (2012). This is also referred to as the
“maximum Abelian projection.” Thees arguments are summarized and reviewed in
Ref. Kharzeev and Loshaj (2014), where the relationship between confinement
and vacuum magnetic monopole condense is discussed. In a simple string model,
confined quarks are connected by Abelian color-electric flux tubes. These flux
tubes have thin transverse size, and are dual to Abrikosov-Nielsen-Olesen
(ANO) vortices Abrikosov (1957); Nielsen and Olesen (1973); Cea  _et al._
(2012) in type-II superconductors. This implies that charged massless fermions
can be localized to the core of ANO vortices containing a magnetic field, and
their dynamical behavior can be described by (1+1) dimensional effective
theory Witten (1985).

The difference between vacuum chiral condensate and chiral condensate in a
background field is intriguing, and how background fields affect the vacuum
properties of QCD is a worthwhile question to explore. We assume that the
chiral condensate, generated by chiral symmetry spontaneous breaking in
vacuum, can be analogized as a Hilbert space quantum pure state, and its value
can be computed using specific models. When a background field is introduced,
whether it is a non-Abelian gauge field or a quasi-Abelian gauge field
(currently only dealing with the Abelian case), it breaks the “stable” vacuum
condensate and causes a decrease in the condensate value, known as the chiral
symmetry partial restoration Kharzeev and Loshaj (2014); Kaneko  _et al._
(2014); Iritani  _et al._ (2013, 2014a, 2014b, 2015, 2016). The changes
induced by the background field on chiral condensates may be reflected in
quantum entanglement, which should be sourced by the background field (quark-
antiquark pairs). In this work, we employ the (1+1)-dimensional Schwinger
model to construct the background field and calculate the variation of the
vacuum chiral condensate under the background field. Additionally, we
introduce two subspaces of the total Hilbert space after the direct product
decomposition: (i) the chiral condensate space without the background field
and (ii) the chiral condensate space affected by the background field. Then we
calculate the entanglement entropy. We change the distance between the quark-
antiquark pairs, affecting the distribution of the background field, and find
that the entanglement entropy increases with the increased distance between
the quark-antiquark pairs, similar to recent quantum simulation results. We
argue that in the framework of the (1+1)-dimensional Schwinger model, the
entanglement entropy between the quark-antiquark pairs will increase with
their distance, and as the source of the background field, the entanglement
entropy between them will be reflected in the vacuum chiral condensate they
affect.

The sections organization of this work are as follows. In section II, we
briefly review some concepts related to the partial restoration of chiral
symmetry, followed by the introduction of entanglement entropy. In section
III, we present the computational results and discuss possible correspondences
between several physical quantities. In the final section, we provide the
conclusion and outlook for the entire paper.

##  II Formalism

###  II.1 1+1 Schwinger model and chiral condensate

This section begins with the construction of the chiral condensate using the
bosonization of the Schwinger model. The Lagrangian of the Abelian gauge
theory in (1+1) dimensions for the case of massless fermions is given by

$\mathcal{L}=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+\bar{\psi}(i\gamma^{\mu}\partial_{\mu}-g\gamma^{\mu}A_{\mu})\psi,$
(1)

where $g$ is the coupling constant with the dimension of mass, $F_{\mu\nu}$ is
gauge strength, $\psi$ denotes the fermion field. As mentioned earlier, this
theory is exactly solvable and processes properties such as confinement,
chiral symmetry breaking, axial anomaly, and periodic $\theta$-vacuum
Schwinger (1962b); Lowenstein and Swieca (1971); Coleman  _et al._ (1975).
Additionally, it is used to calculate the chiral condensate depended on the
electric field Hamer  _et al._ (1982). In other words, the presence of a
background electric field suppresses the magnitude of the chiral condensate,
with its periodicity depending on the $\theta$-vacuum of the theoretical
model.

In one-dimensional space, fermionic degrees of freedom can be precisely
expressed in terms of bosonic degrees of freedom. This correspondence is
typically manifested as duality between fermionic bilinears and the real
scalar fields. This scheme is known as bosonization. In the case of an Abelian
gauge, the correspondence is as follows Coleman (1975); Mandelstam (1975)

$\displaystyle\bar{\psi}i\gamma^{\mu}\partial_{\mu}\psi\to\frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi,$
(2)
$\displaystyle\bar{\psi}\gamma^{\mu}\psi\to-\frac{1}{\sqrt{\pi}}\epsilon^{\mu\nu}\partial_{\nu}\phi,$
$\displaystyle\bar{\psi}\psi\to-c\frac{g}{\sqrt{\pi}}\cos(2\sqrt{\pi}\phi),$

where $c=e^{\gamma_{E}}/2\pi$ with the Euler constant $\gamma_{E}\simeq
0.5772$. Note that the scalar field has mass dimension $[\phi]=0$. Introducing
the effective mass of scalar field, the Lagrangian (1) is equivalent to

$\mathcal{L}=\frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi-\frac{1}{2}\frac{g^{2}}{\pi}\phi^{2}.$
(3)

From Eq. (2), one can express the chiral condensate in terms of the scalar
filed is
$\bar{\psi}\psi=-\frac{ge^{\gamma_{E}}}{2\pi^{3/2}}\cos(2\sqrt{\pi}\phi)$.
Thus the chiral condensate is evaluated through the Feynman-Hellman theorem by
differentiating the energy of the ground state in the presence of an electric
filed $E^{1+1}$ with respect to the fermion mass $m$, in the chiral limit
$m\to 0$ Hamer  _et al._ (1982):

$\langle\bar{\psi}\psi\rangle_{E_{1+1}}=-\frac{ge^{\gamma_{E}}}{2\pi^{3/2}}\cos\theta,$
(4)

with $\theta=2\pi E_{1+1}/g$. Here, $E_{1+1}$ is the electric field along the
string flux tube. Formally, one can see that the value of the condensate
becomes a constant once the background electric field is specified. If the
electric field $E_{1+1}=0$, the condensate must be constant
$\langle\bar{\psi}\psi\rangle_{0}=-\frac{ge^{\gamma_{E}}}{2\pi^{3/2}}$ with
the coupling $g$.

###  II.2 3+1 dimensional physical electric field and string fluctuation

The actual physical presence of a (3+1)-dimensional electric field
distribution, with the “thin” (1+1)-dimensional string corresponding to the
electric field along the flux tube direction ($x_{l}$ direction), see Figure
1. The electric field present in reality should have complete physical
significance, denoted as $E_{3+1}(x_{t})$, where $x_{t}$ is the transverse
coordinate in the plane perpendicular to the flux tube. In this scenario, it
has been demonstrated that the color fields of a static quark-antiquark pair
are almost entirely described by the longitudinal chromoelectric field, which
in turn is approximately constant along the flux tube Cea and Cosmai (1995);
Cardaci  _et al._ (2011). Therefore, one can assume that $E_{1+1}$ and
$E_{3+1}(x_{t})$ have the same string tension in the longitudinal flux
direction, i.e., the energy per unit length of the string should be equal Cea
_et al._ (2012); Kharzeev and Loshaj (2014):

$\sigma_{E}\simeq\frac{1}{2}\int
d^{2}x_{t}E_{3+1}^{2}(x_{t})=\frac{1}{2}(E_{1+1})^{2}.$ (5)

Clearly, the above equation allows one to introduce a quantity $P(x_{t})$ to
represent the probability of finding fluctuations of the string at the
transverse plane position $x_{t}$, and the probability distribution is
constrained as:

$(E_{3+1}(x_{t}))^{2}=(E_{1+1})^{2}P(x_{t}),$ (6)

since $\int d^{2}x_{t}P(x_{t})=1$ condition is satisfied.

Figure 1: The schematic diagram of the electric field string flux tube.
Positive and negative static fermions (separated by a longitudinal distance
$d$) are connected by a thin color flux tube with width $a$, the string
fluctuating in the transverse space ($x_{t}$ direction). The string area is
simplified to $\pi a^{2}$.

Figure 1 depicts the transverse plane position fluctuations of the
(1+1)-dimensional string. Although it is not possible to accurately describe
this fluctuation process, the information of the fluctuations is encoded into
the introduced probability $P(x_{t})$. This is reasonable under the assumption
that Eq. (5) holds.

###  II.3 Partial restoration of chiral symmetry

Let us now consider how the vacuum chiral condensate will change under the
influence of the background field. Firstly, a natural way to describe the
variation of chiral symmetry in the background field is the ratio of the
condensate with and without the background field. It is expected that the
ratio should be 1.0 when there is no background electric field, which is
obvious. When the background field is present, as described in the previous
subsection, the non-zero $E_{1+1}$ should be included in the calculation of
the condensate (4). Here we need to consider the property of the transverse
plane fluctuations of the string described in Figure 1, and take into account
the size $a$ of the string and its corresponding effective area $\pi a^{2}$.
Therefore, the probability of finding the string (flux) at a given transverse
position is $\pi a^{2}P(x_{t})$, indicating that if we want to find the
electric field flux tube at a certain $x_{t}$, the condensate will incorporate
$x_{t}$-dependent information Kharzeev and Loshaj (2014):

$\displaystyle\langle\bar{\psi}\psi(x_{t})\rangle$ $\displaystyle=(1-\pi
a^{2}P(x_{t}))\langle\bar{\psi}\psi\rangle_{0}$ (7) $\displaystyle+\pi
a^{2}P(x_{t})\langle\bar{\psi}\psi\rangle_{E^{1+1}}.$

In Kharzeev and Loshaj (2014), the authors showed that transverse size of
string $a$ appears comparable to the lattice spacing.

Quantitatively calculating the impact of confining flux tube on chiral
condensate has been the subject of numerous lattice studies Kaneko  _et al._
(2014); Iritani  _et al._ (2013, 2014a, 2014b, 2015, 2016). It is worth noting
that in the Ref Iritani  _et al._ (2015), the authors extended the confining
string formation to two quarks, introducing the influence of a third quark.
Based on the lattice definition, the effect of confining flux tube on the
chiral condensate is written as

$r(x_{t})=\frac{\langle\bar{q}q(x_{t})W\rangle}{\langle\bar{q}q\rangle\langle
W\rangle},$ (8)

where $W$ is the Wilson loop of the static quarks. In this study, this
quantity is given by

$r(x_{t})=\frac{\langle\bar{\psi}\psi(x_{t})\rangle}{\langle\bar{\psi}\psi\rangle_{0}},$
(9)

with the condensate (7). For specific details on the restoration of partial
chiral symmetry, please refer to Refs. Kharzeev and Loshaj (2014); Kaneko  _et
al._ (2014); Iritani  _et al._ (2013, 2014a, 2014b, 2015, 2016).

###  II.4 Entanglement entropy

The above discussions are all based on the influence of background fields on
vacuum chiral condensate. We now consider what specific changes should occur
in the Hilbert space of vacuum condensate after the introduction of background
fields. When there are no confining flux tubes present, the information of the
vacuum chiral condensate is completely knowable, depending only on the model’s
coupling constant $g$ and some mathematical parameters. At this point, we can
speculate that it is in a quantum pure state, as it satisfies the conditions
of a pure state. However, when the background field is introduced, with the
partial restoration of chiral symmetry, it can be understood as the energy
condensed in the vacuum appearing as a“depression” in the transverse plane,
which is caused by the confining flux tube Iritani  _et al._ (2016).
Throughout the entire $x_{t}$ plane, the states space is decomposed into (i)
the chiral condensate space ($A$) without the background field and (ii) the
chiral condensate space ($B$) affected by the background field. The physical
states inside the region $A$ “probed” as the chiral condensate states without
background color field in Hilbert space $\mathcal{H}_{A}$, and states is
affected by background field $\mathcal{H}_{B}$.

The composite system in $A\cap B$ (the entire condensate) is then described by
the vector $\Psi_{AB}$ in the space $\mathcal{H}_{A}\otimes\mathcal{H}_{B}$
that is a tensor product of the two space:

$|\Psi_{AB}\rangle=\sum_{i,j}c_{ij}|\varphi_{i}^{A}\rangle\otimes|\varphi_{j}^{B}\rangle,$
(10)

where $c_{i,j}$ are the elements of the matrix $C$ that has the dimension
$n_{A}\times n_{B}$. If the sum (10) contains only one term, then the state
$|\Psi_{AB}\rangle$ is separable, or a product state. Otherwise the state
$|\Psi_{AB}\rangle$ is entangled. The corresponding density matrix is
$\rho_{AB}=|\Psi_{AB}\rangle\langle\Psi_{AB}|$.

The Schmidt decomposition theorem Schmidt (1907); Peres (1995) allows the pure
state $|\Psi_{AB}\rangle$ to be expanded as a single sum

$|\Psi_{AB}\rangle=\sum_{n}\alpha_{n}|\Psi_{n}^{A}\rangle|\Psi_{n}^{B}\rangle$
(11)

for a suitably chosen orthonormal sets of $|\Psi_{n}^{A}\rangle$ and
$|\Psi_{n}^{B}\rangle$ localized in the domains $A$ and $B$, respectively,
where $\alpha_{n}$ are positive and real numbers that are the square roots of
the eigenvalues of matrix $CC^{\dagger}$. Thus one can write down the density
matrix $\rho_{AB}$ of the mixed state probed in region $A$ as

$\hat{\rho}_{A}=\mathrm{tr}_{B}\rho_{AB}=\sum_{n}\alpha_{n}^{2}|\Psi_{n}^{A}\rangle\langle\Psi_{n}^{A}|,$
(12)

where $\alpha_{n}^{2}\equiv P_{n}$ is the probability of a state
$|\Psi_{n}^{A}\rangle$. Now let’s discuss the states $|\Psi_{n}^{A}\rangle$
and $|\Psi_{n}^{B}\rangle$ mentioned above. In fact, we do not have a
straightforward method to construct the specific form of the wave function
corresponding to a state after the Schmidt decomposition. However, under the
assumptions we made earlier, we can acknowledge that the two states in Eq.
(11) respectively represent the information of the two subspaces obtained by
the direct product decomposition of the Hilbert space. Based on this, we can
define the von Neumann entropy

$S_{EE}=-\sum_{n}P_{n}\ln P_{n}.$ (13)

In our model, the probabilities $P_{n}$ only have two terms: $P_{1}=\pi
a^{2}P(x_{t})$ and $P_{2}=1-\pi a^{2}P(x_{t})$ correspond to the probabilities
of finding or not finding the flux at a given transverse position $x_{t}$.

##  III Numerical results and discussions

Before presenting our numerical results, we now briefly introduce how the
background field $E_{3+1}(x_{t})$, which is used to obtain $E_{1+1}$ and
$P(x_{t})$, is chosen. We utilize lattice calculation results from Cea  _et
al._ (2012); Baker  _et al._ (2020) as inputs for the distribution of color
flux tubes, emphasizing that the distribution of color flux tubes is
essentially independent of the longitudinal position of the probe, as
explained in Cea  _et al._ (2012); Kharzeev and Loshaj (2014); Baker  _et al._
(2020); Cardaci  _et al._ (2011); Cea and Cosmai (1995). The measured color
electric field as a function of the transverse coordinate can be well
described by the following parameterization Clem (1975)

$E_{3+1}(x_{t})=\frac{\varphi}{2\pi}\frac{\mu^{2}}{\alpha}\frac{K_{0}[(\mu^{2}x_{t}^{2}+\alpha^{2})^{1/2}]}{K_{1}[\alpha]},$
(14)

where $\varphi$, $\alpha$ and $\mu$ are fitting parameters. Eq. (14) is a
simple analytical expression derived from Ampère’s law and the Ginzburg-Landau
equation. In particular, it can be simplified to the London model outside the
vortex core in the presence of a transverse magnetic field distribution. It
reflects the compatibility of color flux tubes with dual superconducting
theories.

In this work, we adopt the lattice results from Ref. Baker  _et al._ (2020),
which include the distribution of color flux tubes obtained as input from
distances between different color charges. The relevant parameters are listed
in Table 1. The lattice coupling constant $\beta=2N_{c}/g^{2}$ with color
number $N_{c}$.

Table 1: The Clem parameters describing the non-perturbative filed transverse section going through the midpoint between the quark and antiquark positions Baker  _et al._ (2020). Sets | $\beta$ |  $d$ (fm) | $\varphi$ |  $\mu$ (fm-1) | $\alpha$
---|---|---|---|---|---
1 | 6.475 | 0.37 | 3.474(4) | 4.999(9) | 1.192(4)
2 | 6.333 | 0.45 | 3.83(3) | 5.30(6) | 1.55(3)
3 | 6.240 | 0.51 | 4.028(11) | 6.039(26) | 2.141(20)
4 | 6.500 | 0.54 | 4.370(15) | 5.71(4) | 2.02(3)
5 | 6.539 | 0.69 | 4.50(7) | 6.25(20) | 2.47(16)
6 | 6.370 | 0.85 | 5.40(25) | 6.7(9) | 4.0(1.1)
7 | 6.299 | 0.94 | 5.2(4) | 7.8(1.9) | 5.5(2.8)
8 | 6.240 | 1.02 | 8.0(7) | 4.4(8) | 2.4(9)
9 | 6.218 | 1.06 | 6.6(7) | 6.0(1.8) | 4.0(2.4)

We chose the parameter sets of Set 1, 5, and 9 to plot the profile of the
color electric field $E_{3+1}(x_{t})$ as a function of the transverse
coordinate, formed by color charges at different distances. It can be seen
that the profiles at different $d$ are distinct. Additionally, we provide the
plots of $P(x_{t})$ and $S_{EE}(x_{t})$ by Eqs. (5, 13), see Figure 2. We
observe that the color field distribution, the probability distribution, and
the entanglement entropy distribution all resemble “Gaussian” distributions.
This is based on the results of the lattice simulation and fitting using the
Clem parameterized model Clem (1975). This indicates that the rise and fall of
a color flux tube excited between two static color charges in the transverse
plane also exhibits Gaussian-like behavior. Furthermore, it suggests that the
probability of finding the tube is highest at the origin of the polar
coordinates (assuming the effective radius of the flux, $\pi a^{2}$, is
known).

Figure 2: The transverse distribution of the color electric field existing
between two static color charges (top left), along with the corresponding
transverse probability (top right) and entanglement entropy (bottom)
distributions. The 1st, 5th, and 9th parameter sets from Table 1 are chosen,
with the parameters from the Ref. Baker  _et al._ (2020).

We now investigate the variation of the chiral condensate with respect to the
background field. Initially, we provide the square root of the string tension
corresponding to the static color charge from the position separation by Eq.
(5). Following this, we integrate the chiral condensate under the influence of
the background field, which represents the total condensate contribution. We
anticipate that the total condensate value will differ for various color field
backgrounds. For convenience, we include the integral of the vacuum condensate
over the entire space, denoted as
$\langle\bar{\psi}\psi\rangle^{int}=\frac{1}{g}\int d^{2}x_{t}\pi
a^{2}P(x_{t})\langle\bar{\psi}\psi\rangle_{E^{1+1}}-C_{0}$ with
$C_{0}=\frac{1}{g}\int d^{2}x_{t}\langle\bar{\psi}\psi\rangle_{0}$. Finally,
we examine the integration results for the entanglement entropy distribution.
All the numerical results are presented in Table 2, corresponding to the
parameter selections listed in Table 1, respectively. We emphasize that all
numerical integration rests on the rotational symmetry of the field
distribution.

Table 2: The numerical results corresponding to different sets of parameters in Table 1, including $\sigma_{E}$, $\langle\bar{\psi}\psi\rangle^{int}$, and $S_{EE}^{int}$.  Sets | $\beta$ |  $d$ (fm) |  $\sqrt{\sigma_{E}}$ (GeV) |  $\langle\bar{\psi}\psi\rangle^{int}$ (fm2) |  $S_{EE}^{int}$ (GeV-2)
---|---|---|---|---|---
1 | 6.475 | 0.37 | 0.4659(3) | 1.9657(2) | 0.8361(4)
2 | 6.333 | 0.45 | 0.5045(20) | 1.973(13) | 0.8478(22)
3 | 6.240 | 0.51 | 0.5430(10) | 1.9814(6) | 0.8469(11)
4 | 6.500 | 0.54 | 0.5687(10) | 1.9929(10) | 0.8567(18)
5 | 6.539 | 0.69 | 0.596(5) | 2.0018(48) | 0.8543(82)
6 | 6.370 | 0.85 | 0.640(17) | 2.011(23) | 0.8962(369)
7 | 6.299 | 0.94 | 0.620(24) | 2.006(42) | 0.8938(703)
8 | 6.240 | 1.02 | 0.77(4) | 2.044(37) | 0.9563(469)
9 | 6.218 | 1.06 | 0.62(3) | 2.026(56) | 0.9294(804)

Our numerical results are also displayed in Figure 3. The uncertainties arise
from the calculation of the uncertainty covariance matrix for the parameters
in Table 1. It can be observed that the results for all three sets of physical
quantities increase as the color charge distance $d$ increases. However, it is
important to clarify the presence of large uncertainties in the current
results.

Figure 3: Numerical results of string tensions, integrated chiral condensates,
and integrated entanglement entropy at different color-charge distances $d$,
with uncertainties from a Clem parameterized model Clem (1975) fit to the
lattice simulation Baker  _et al._ (2020).

The variations of physical quantities with $d$ in Figure 3 are clearly
attributed to the distribution of color flux tubes between positive and
negative color charges at different distances. The string tension defines the
energy contained in a unit length of the color flux tube, and its increase
with $d$ is a result obtained from lattice simulations. Based on this premise,
the two remaining physical quantities are both associated with the probability
distribution $P(x_{t})$ introduced by the conversion of $E_{3+1}$ to
$E_{1+1}$, and after integration over the transverse space, they both
degenerate to the $E_{1+1}$ dependence, similar to the cosine oscillatory
behavior of chiral condensate under the influence of the background field (4).
As the separation of color sources increases, the thickness of the flux may be
scaled by a logarithmic function with respect to its length Hasenfratz  _et
al._ (1981); Luscher  _et al._ (1981), and this information is encoded in the
energy per unit length, i.e., the string tension. Such behavior has indeed
been studied in quenched lattice QCD calculations Cardoso  _et al._ (2013) as
well as in finite temperature Bakry  _et al._ (2012). This directly affects
the change in vacuum chiral condensate, and precisely reduces the amount of
condensate, which corresponds to a partial chiral symmetry restoration Iritani
_et al._ (2015).

The integrated value of entanglement entropy also increases with the color
source separation $d$, which can be interpreted as an increase in the effect
of string tension on the chiral condensate when the color source flux
thickness is increased. In other words, the vacuum chiral condensate
decreases, the degrees of freedom in $A$ space increase, and the entanglement
between $A$ and $B$ (the complement of $A$) is increased. We can conjecture
that this process will be similar to the entanglement entropy change process
of two quantum subsystems into which other similar pure state systems have
been separated. When the chiral condensate of the whole space completely
disappears after the influence of the flux with infinite thickness, then the
chiral symmetry is completely recovered and the entanglement entropy goes to
zero.

##  IV Summary and outlook

In this work, we mainly provide three discussions: (i) We reviewed the method
of calculating the chiral condensate using the Schwinger model and introduced
the influence of the vacuum color flux tube on the vacuum chiral condensate.
(ii) We utilized the color electric field distribution obtained from recent
lattice calculations Baker  _et al._ (2020), corresponding to different color
source separations. We found that as the color source separation increases,
the corresponding thickness of the color flux tube increases, which affects
the transverse distribution shape of the color field, thereby influencing the
string tension. (iii) The string tension increases with the increase of color
source separation and affects the amount of chiral condensate recovery. We
observed that the recovery amount increases with the increase of color source
separation. Additionally, we hypothesize the influence of color sources as the
basis for the direct product decomposition of the entire system’s Hilbert
space and provide the change in the system’s entanglement entropy. We found
that the entanglement entropy also increases with the increase of color source
separation. The entanglement entropy corresponding to the chiral condensate
may be induced by the entanglement between two color sources, and the
information encoded in the entanglement among two color sources may be
discovered from the observation of the chiral condensate.

The current numerical results depend on the uncertainties of the parameters
corresponding to the color electric field distribution obtained from lattice
calculations Baker  _et al._ (2020). It can be observed that parameters at
larger separations of color charges exist higher fitting uncertainties. This
is related to the lattice configurations in lattice QCD calculations, where it
is difficult to satisfy the lattice spacing limit at the edges of the lattice.
Meanwhile, it is also interesting to investigate the effects of introducing
the third color source on the color flux tube Iritani  _et al._ (2015, 2016).
It is worth considering how the influence caused by the third color source is
reflected in the entanglement entropy. Additionally, vacuum polarization of
oppositely moving fermion pairs and string breaking-induced vacuum
fluctuations also increase the entanglement entropy between color sources and
the vacuum fermion condensate. This has been discussed in recent works in
detail Florio  _et al._ (2023); Ikeda  _et al._ (2023). Whether there are
connections between our present study and the recent works Florio  _et al._
(2023); Ikeda  _et al._ (2023) could be a direction for us to explore in the
future.

###### Acknowledgements.

This work is supported by the Strategic Priority Research Program of Chinese
Academy of Sciences under the Grant NO. XDB34030301.

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# Spin Dynamics, Loop Formation and Cooperative Reversal in Artificial
Quasicrystals with Tailored Exchange Coupling

Vinayak Shantaram Bhat<EMAIL_ADDRESS>Laboratory of Nanoscale
Magnetic Materials and Magnonics, Institute of Materials (IMX), École
Polytechnique Fédérale de Lausanne (EPFL), 1015 Lausanne, Switzerland
International Research Centre MagTop, Institute of Physics, Polish Academy of
Sciences, 02668 Warsaw, Poland  Sho Watanabe  Laboratory of Nanoscale Magnetic
Materials and Magnonics, Institute of Materials (IMX), École Polytechnique
Fédérale de Lausanne (EPFL), 1015 Lausanne, Switzerland  Florian Kronast
Helmholtz-Zentrum Berlin für Materialien und Energie, Albert-Einstein-Strasse
15, 12489, Berlin, Germany  Korbinian Baumgaertl  Laboratory of Nanoscale
Magnetic Materials and Magnonics, Institute of Materials (IMX), École
Polytechnique Fédérale de Lausanne (EPFL), 1015 Lausanne, Switzerland  Dirk
Grundler<EMAIL_ADDRESS>Laboratory of Nanoscale Magnetic Materials and
Magnonics, Institute of Materials (IMX), École Polytechnique Fédérale de
Lausanne (EPFL), 1015 Lausanne, Switzerland  Institute of Electrical and Micro
Engineering (IEM), École Polytechnique Fédérale de Lausanne (EPFL), 1015
Lausanne, Switzerland

###### Abstract

Aperiodicity and un-conventional rotational symmetries allow quasicrystalline
structures to exhibit unprecedented physical and functional properties. In
magnetism, artificial ferromagnetic quasicrystals exhibited knee anomalies
suggesting reprogrammable magnetic properties via non-stochastic switching.
However, the decisive roles of short-range exchange and long-range dipolar
interactions have not yet been clarified for optimized reconfigurable
functionality. We report broadband spin-wave spectroscopy and X-ray
photoemission electron microscopy on different quasicrystal lattices
consisting of ferromagnetic Ni81Fe19 nanobars arranged on aperiodic Penrose
and Ammann tilings with different exchange and dipolar interactions. We imaged
the magnetic states of partially reversed quasicrystals and analyzed their
configurations in terms of the charge model, geometrical frustration and the
formation of flux-closure loops. Only the exchange-coupled lattices are found
to show aperiodicity-specific collective phenomena and non-stochastic
switching. Both, exchange and dipolarly coupled quasicrystals show magnonic
excitations with narrow linewidths in minor loop measurements. Thereby
reconfigurable functionalities in spintronics and magnonics become realistic.

## Introduction

Quasicrystals exhibit aperiodic long-range order and unconventional rotational
symmetry, but no translational invariance. Since their discovery [1, 2] the
impact of aperiodicity on fundamental physical phenomena is pursued with great
interest [2]. One powerful avenue to gain novel insight resides in the
materials-by-design approach making use of nanofabrication and imaging
techniques [3]. Thereby microscopic understanding of e.g. geometrical
frustration in spin ice systems was achieved. The artificial spin ices (ASIs)
consisted of either disconnected or interconnected ferromagnetic nanobars that
were arranged on strictly periodic lattices with translational invariance [4,
5, 6]. In both types of ASIs cooperative phenomena were found and analyzed via
spin ice rules, charge model and energy minimization through clockwise (CW)
and counter-clockwise (CCW) flux-closure loops (FCLs) [7, 8]. Vortex-like
microstates, i.e., flux-closure loops in a lattice (which are sometimes called
microvortices), indicated energy minimization due to magnetic coupling [9].
Tailored dipolar interaction was named key for novel devices [10] based on
e.g. reproducible microstates upon cycling an applied magnetic field [11, 12].
The spin dynamics in periodic ASIs with reconfigurable magnetic configurations
and intentionally introduced magnetic defects have already generated enormous
interest [12, 13, 14, 15, 16, 17, 18]. Still Iacocca et al. pointed out very
recently that dipolar coupling in real ASI might not be sufficient for
reconfigurable magnon waveguides [19] consistent with earlier experiments
[20]. For artificial magnetic quasicrystals (AMQs) unconventional magnetic
properties and non-stochastic switching were reported for both interconnected
lattices with exchange coupling and lattices with edge-to-edge separations of
up to about 150 nm between nanomagnets exhibiting dipolar interaction only [3,
21, 22, 23, 24]. Following the orthodox understanding [25] and recently
performed micromagnetic simulations one anticipates aperiodicity-induced
phenomena for edge-to-edge separations even wider than 150 nm [24].
Particularly, aperiodic quasicrystals promise a plethora of reconfigurable
magnetic configurations due to non-stochastic switching in a global magnetic
field, while periodic lattices would require the serial writing process based
on a magnetic force microscope [25]. Domains and domain walls between
differently oriented lattice segments of ASIs have been foreseen already as
conduits which steer magnons in a reconfigurable manner inside the magnetic
lattice [26] or in an underlayer [19]. However, the following questions are
unanswered: (1) How do the cooperative phenomena show up in real samples when
one systematically varies the type and relative strength of coupling among
aperiodic nanobars? (2) What is the origin of the knee-like anomalies which
were reported for the magnetic hysteresis of quasicrystalline Penrose P2
tilings? [3] (3) How do functional properties of ferromagnetic Penrose P2 and
P3 tilings compare? They belong to the same class of ten-fold rotationally
symmetric quasicrystal lattices but consist of different geometric prototiles
[27]. In this article, we report on dynamic and quasi-static investigations
based on broadband spectroscopy and magnetic imaging (Fig. 1), respectively,
performed on ten-fold rotationally symmetric Penrose P2, P3 and the eight-fold
rotationally symmetric Ammann tilings. We address the open questions by
magnetic resonance spectra taken in the major loop and the hysteretic regime
as well as X-ray photo-electron emission microscopy (XPEEM) using x-ray
circular dichroism (XMCD) with high spatial resolution (Methods).
For our studies we prepared artificial magnetic quasicrystals in the form of
nanobars (Fig. 1) out of the magnetically isotropic alloy Ni81Fe19 (Py) on
Penrose P3 [Fig. 1(b) to (d)], P2 [Fig. 1(h)], and Ammann (Fig. 2)
quasicrystal lattices using nanofabrication techniques.

Figure 1: (a) Sketch of broadband spin wave spectroscopy. Scanning electron
microscopy (SEM) images showing 3rd generation Penrose P3 lattices: (b) P3A
for which nanobars are fully connected, (c) P3B with partially connected
nanobars, and (d) P3C with disconnected nanobars. The scale bars correspond to
1 $\mu$m. Gray-scale spin wave spectra obtained on 8th generation (e) P3A, (f)
P3B, and (g) P3C samples. The magnetic field was applied along the horizontal
direction of graphs (b) to (d) and varied from +90 mT to - 90 mT in a step-
wise manner. In the field regime between the blue arrows [Fig. (e)] the
reversal of the AMQ takes place. The magenta color arrows mark $H=0$ in (e).
The green arrow in (g) marks the high-frequency mode in the reversal regime.
The yellow arrow highlights the branch attributed to nanomagnets being almost
perpendicular to the applied field. (h) Sketch of the XPEEM imaging experiment
performed on ferromagnetic quasicrystals. Here a Penrose P2 tiling is shown as
an SEM image. D and w represent the length and width of a nanobar,
respectively. (i) (Back row) PEEM topography images of nanobars arranged on
Penrose P2 lattices P2A (left), P2B (center), and P2C (right). (Center row)
XPEEM images taken on P2A, P2B, and P2C (from left to right). (Front row)
Analysis of the magnetic configurations of vertices based on the charge ($Q$)
model. The magnitude of $Q$ is given by diameter of circles (legend). Orange
(green) color indicates negative (positive) charge. The broken lines guide the
eye for the allocation of charges to a specific vertex. For the configurations
shown the maximum evaluated $|Q|$ amounted to 3. The XPEEM images were taken
at $\mu_{0}H=0~{}$mT after applying $\mu_{0}H=-52~{}$mT such that $\mathbf{H}$
had pointed in $-x$-direction. The bright (dark) regions in (h) represent Py
(Si substrate).

The width $w$ of nanobars, their thickness and intervertex spacing were kept
at 120 nm, 25 nm, and 810 nm, respectively. We varied the lengths $D$ of
nanobars from sample to sample between 810 nm and 408 nm. Thereby we created
quasicrystals which consisted of interconnected (A), partially connected (B)
and fully separated nanobars (C), respectively [compare Fig. 1(b) to 1(d)].
The interconnected nanobars joining in the vertices of an AMQ of kind A were
both exchange and dipolarly coupled. Separated nanobars in kind B and C were
dipolarly coupled only. In sample B (sample C) opposing nanobars exhibited
edge-to-edge separations of up to about 200 nm (400 nm). The imaging presented
here show that for non-stochastic switching and cooperative reversal in
Penrose and Ammann tilings the recently explored dipolar coupling [24] is not
sufficient. We observe significant domain formation only when quasicrystalline
Penrose P3 and Ammann tilings are exchange-coupled. Both exchange- and
dipolarly coupled lattices show reprogrammable magnonic excitations of narrow
linewidth. Our findings are key when designing quasicrystals for field-
controlled functionalities exploiting return-point memory and reproducible
magnetic states [11, 26, 28].

## Results and Discussion

### Broadband spectroscopy in the major loop

For the presentation of the results we decompose the Penrose lattices reported
in Fig. 1 into three types of nanobars, i.e., Type I, Type II, and Type III
based on the angle $\phi$ that the nanobars take with respect to the direction
of magnetic field H used in Fig. 1: Type I exhibits $\phi$ = 0o, Type II
$\phi$ = $\pm$ 36o, and Type III $\phi$ = $\pm$ 72o. In the Ammann tilings
there exist nanobars with $\phi$ = 90o. Considering Ref. [29] we calculated
shape anisotropy fields $\mu_{0}$Hani for individual nanobars of lengths 810
nm, 609 nm, and 408 nm and obtained 147 mT, 140 mT, and 117 mT, respectively.
Note that the maximum field $\mu_{0}$$|$Hmax$|$ = 90 mT that was available in
the broadband spin-wave spectroscopy setup was smaller than the calculated
fields $\mu_{0}$Hani. As a consequence, nanobars perpendicular to the applied
field could not be saturated. The estimated nucleation field for incoherent
reversal in an ideal isolated nanobar with $D=810~{}$nm via curling amounted
to 70 mT [30]. The minimum reversal field for coherent rotation was about the
same value. Considering these values, we expected the field regime ranging
from $+90$ mT$\leq\mu_{0}H\leq-90~{}$mT to be large enough to reverse the
magnetization of Type I and II nanobars but not of individual nanobars of Type
III exhibiting an angle $|\phi|$ of 72o.

In the following we present and discuss broadband spectroscopy data obtained
on P3 lattices and Ammann tilings for which we have observed large domain
formation in the XPEEM imaging experiments. In Fig. 1(e) to (g) we display
spectra taken on Penrose P3 tilings when varying the applied magnetic field
from +90 mT to -90 mT. For all three samples we see two strong branches at
large absolute field values consistent with the interconnected AMQs reported
earlier in Ref. [31]. We attribute the branches to resonances in Type I
(highest frequency) and Type II (second highest frequency) nanobars.
Considering a field of 90 mT, the branch frequencies decrease from (e) to (g),
i.e., for the interconnected nanobars (P3A) the two prominent branches reside
at overall larger frequencies than for the disconnected nanobars (P3C). From
sample P3A to P3C the lengths of nanobars reduce and the demagnetization
effect enhances, thereby reducing the internal fields and consequently the
resonance frequencies [32]. Below the prominent branches manifolds of further
resonances are found which are attributed partly to standing spin waves
confined along the nanobars. Due to the corresponding backward volume
magnetostatic spin wave configuration their resonance frequencies reside at
small values. We consider the field regime between the blue arrows in Fig.
1(e) to be the regime in which nanobars of sample P3A reverse. The switching
field regime extends from about -30 mT to -65 mT, very similar to a nominally
identical interconnected AMQ investigated in Ref. [31]. Before we report
imaging of magnetic configurations in the reversal regime it is instructive to
discuss further details of the spin dynamics in the different quasicrystals.
In Fig. 1(e) and (f) the magenta arrows highlight the branches of Type I and
Type II nanobars at zero field. In Fig. 1(e) the two branches of P3A are
clearly split at $H=0$. This is not the case in P3B. Here the two branches are
degenerate at $H=0$, indicating that Type I and Type II experience the same
internal magnetic field. The same degeneracy is observed for P3C at $H=0$ in
Fig. 1(g). We attribute the frequency difference observed in Fig. 1(e) to the
coupling between spin-wave modes in Type I and Type II nanobars leading to an
avoided crossing. Nanobars in the interconnected AMQ P3A hence interact. The
yellow arrow in Fig. 1(g) highlights a faint branch which exhibits an agility
$df/dH>0$ for $H<0$ and, at $H=0$, approaches the degenerate frequencies of
Type I and Type II nanobars. The characteristics of this branch are consistent
with the field-dependent resonance frequency of Type III nanobars which are at
$\pm 72$ deg. For them, $H$ is applied almost along their hard-axis direction.
A similar faint branch is seen for P3B in Fig. 1(f). In P3C we resolve a
specific high-frequency mode in the reversal regime (green arrow) which is not
observed in P3A and P3B and will be discussed after the presentation of the
magnetic imaging.
In Fig. 2(d) to (f) we display spectra of interconnected (ATA), partially
connected (ATB) and disconnected nanobars (ATC). For the Ammann tilings,
nanobars of Type I, II and III make angles $\phi$ = 0o, $\pm$ 45o, and $\pm$
90o, respectively. The faint branches highlighted by yellow arrows in Fig.
2(d), (e) and (f) are found in all three Amman tilings ATA, ATB and ATC,
respectively. These branches originate from the Type III nanobars which are at
an angle of 90 deg with respect to the applied field $H$. $H$ thus points
along their hard axis. Again at large absolute fields, two prominent branches
are found which are consistent with Type I (high frequency) and Type II
(second highest frequency) nanobars. Only for the interconnected AMQ ATA we
observe a frequency splitting between Type I and Type II nanobars (magenta
arrows) near $H=0$. For AMQs ATB and ATC (disconnected nanobars) the frequency
degeneracy occurs. The reversal field regime for ATA [blue arrows in Fig.
2(d)] is found to be narrow compared to P3A [Fig. 1(e)]. The interconnected
nanobars reverse close to -30 mT without a significant switching field
distribution. The distribution widens for the other (partly) disconnected
AMQs. A detailed analysis of spectra of ATC provides again a specific high-
frequency mode [green arrow in Fig. 2(f)] in the reversal regime, similar to
P3C.

Figure 2: SEM images showing 1st generation Ammann tilings (a) ATA for which
nanobars are fully connected, (b) ATB with partially connected nanobars and
(c) ATC with disconnected nanobars. The black color scale bar represents 1
micrometer. Gray-scale spin wave spectra obtained on 4th generation (d) ATA,
(e) ATB and (f) ATC. The magnetic field was applied along the horizontal
direction of graphs (a) to (c) and varied from +90 mT to - 90 mT in a step-
wise manner. In a relatively small field regime near the blue arrow in (d) to
(f) the reversal of AT takes place. The magenta color arrows mark $H=0$. The
yellow arrows in (d) to (f) highlight the branches attributed to nanomagnets
being perpendicular to the applied field. The green arrow in (f) marks the
high-frequency mode in the reversal regime of ATC.

Spin-wave spectra obtained in minor loops and the reconfigurable
characteristics of AMQs are discussed after presenting XPEEM experiments by
which we image magnetic states in the hysteretic regime.

### Magnetic imaging of partly reversed quasicrystals

Figure 3:  (a) - (f) XPEEM topography images for six different quasicrystals
as labelled in the graphs. Bright (dark) regions correspond to magnetic (non-
magnetic) material. The scale bar given in (a) corresponds to 1 $\mu$m. The
arrow in (a) represents the X-ray direction and the magnetic field direction.
Selected magnetic images of remnant states using XPEEM after applying
different field values $\mu_{o}H$ (given in parenthesis) for (g) P2A (32.5
mT), (h) P2B (41.6 mT), (i) P2C (42.64 mT), (j) P3A (36.4 mT), (k) P3B (42.9
mT), and (l) P3C (48.1 mT) representing the switching of 44%, 51%, 46%, 56%,
44%, and 55% type I nanobars, respectively. Blue (red) colors represent
magnetization parallel (opposite) to the X-ray direction. Blue color indicates
a reversed nanobar. The images represent the states attained when about 50% of
Type I nanobars switched. Notice the weak contrast in case of nanobars which
do not point along the X-ray direction. (m) – (r) CM analysis of the XMCD
experimental data shown in (g) – (l). The green and orange filled circles at
the vertices represent positive and negative vertex charges, respectively.
Here the circles with smallest and largest diameter represent charge $|Q|$ = 0
and $|Q|$ = 5, respectively. When quasicrystals consist of disconnected
nanobars a large charge of up to $|Q|$ = 5 can be found. The magnetic
configurations of P2 and P3 tilings in the as-grown state are displayed in
Supplementary Fig. 1.

In the following we discuss magnetic imaging [Fig. 1(h)] of partly reversed
AMQs. We have studied the three designs of AMQs introduced previously [31],
i.e., Penrose P2 and P3 lattices as well as Ammann tilings. In the XPEEM
microscope the maximum field $\mu_{0}$$|$Hmax$|$ that was available to
magnetize the samples in initial configurations [Fig. 1(i) of P2A and
Supplementary Fig. 3] amounted to 52 mT. This value was smaller than the
calculated fields $\mu_{0}$Hani but larger than the fields that initiated
reversal for Type I and Type II nanobars in the broadband spectroscopy
experiments of Figs. 1 and 2. Still the field was not large enough to reverse
the magnetization of an individual Type III nanobar ($\phi$ = $\pm$72o
(Penrose) or 90o (Ammann)) if it did not interact with nanobars of Type I and
II.
In order to evaluate magnetic states in the saturated and partly reversed
quasicrystals, we considered the shape-anisotropy induced bistability (Ising
nature) of nanobars and exploited the so-called charge model (CM) (see
Methods). We first present experimental data obtained on the Penrose lattices
[Fig. 3(a) to 3(f)]. In Fig. 3(g) to 3(l) we show XPEEM images taken on
magnetic states of P2 and P3 AMQs after initiating partial reversal in a minor
loop (as described in Methods). Each image displays a remnant state in zero
field after application of a specific field. Images of P2A in Fig. 3(g) and
P3A in Fig. 3(j) show that these AMQs with interconnected nanobars contain
domains or chains of reversed nanobars (blue) next to domains in which
nanobars remained in the original orientation (red). In particular P3A
incorporates a large-area domain of reversed nanobars. The reversed domains
and chains are found to include reversed nanobars of Type III though their
anisotropy field was estimated to be larger than the applied reversal field.
The detection of reversed Type III nanobars reflects a cooperative phenomenon
in the magnetic hysteresis of a quasicrystal. Their reversal is not triggered
by the relatively weak external field alone. The reversal is attributed to the
influence of neighboring Type I and Type II nanobars which exhibit $|\phi|$
$\leq$ 36o. The analysis in terms of the CM (see Supplementary Fig. 4)
displayed in Fig. 3(m) and 3(p) shows that vertices formed by the
interconnected nanobars exhibit overall small values of Q in their remnant
states [the maximum value amounts to Q = 3 in Fig. 3(m) and 3(p)].
AMQs from partially and fully separated nanobars, imaged as P2B [Fig. 3(h)],
P2C [Fig. 3(i)], P3B [Fig. 3(k)], and P3C [Fig. 3(l)], show numerous reversed
magnets that are (more) randomly distributed over the whole AMQ compared to
the two AMQs with completely interconnected nanobars. From XPEEM images taken
for many different magnetic histories (not shown) we found that switching
events in the form of avalanches were seen in the interconnected (exchange-
coupled) P2A and P3A samples but were not seen in the disconnected P2C and P3C
samples. In the latter samples, opposing nanobars formed fragmented domains
which might not be functional for the proposed magnon steering.
In Fig. 4 we analyze the reversal in the AMQs in detail.

Figure 4: (a) - (f) Field-dependent reversal of Type I (red circle), Type II
(blue square) and Type III (green triangle) in AMQs P2A, P2B, P2C, P3A, P3B,
and P3C, respectively. (g) - (l) Total charge Qtot for P2A, P2B, P2C, P3A,
P3B, and P3C, respectively. (m)-(r) Number of FCLs [see the insets in (m) and
(p) for their definition] for P2A, P2B, P2C, P3A, P3B, and P3C, respectively.
(s) - (x) Evaluation of relative amounts of maximum possible individual vertex
charges Q in P2A, P2B, P2C, P3A, P3B, and P3C, respectively. The purple, dark
yellow, and dark cyan colored symbols represent charges 3 (for N = 3), 4 (for
N = 4), and 5 (for N = 5), respectively. The values in (s) - (w) were
multiplied by 3 for better visualization using a unique y scale for all
graphs. Notice the presence of charges $|Q|$ = 5 in P2C for N = 5 vertices.

In Fig. 4(a) to 4(f) the relative numbers of reversed nanobars are depicted as
extracted from a series of XPEEM microscopy images taken at remnance after
applying different magnetic fields H. For P2A and P2B we find reversals to
start from Type I nanobars [red symbols in Fig. 4(a)]. For the other AMQs P2C,
P3A, and P3B, Type II nanobars switch first. Once Type I or Type II nanobars
reverse in the interconnected P2A or P3A, the reversal of their Type III
nanobars follows. A less pronounced successive reversal of Type III nanobars
is found for P2B and P3B. Here, the number of reversed Type III nanobars is
much smaller compared to P2A and P3A. In P2C and P3C consisting of completely
disconnected nanobars we do not find the reversal of Type III nanobars in the
accessible field regime. Here, the interaction between nanomagnets is too
small.
In Fig. 4(g) to 4(l) we show the total charge Qtot extracted from XPEEM
images. In AMQs P2A, P2B and P2C the onset of reversal is accompanied by a
small global minimum in Qtot. In case of P2A and P2B Qtot regains a value
close to the initial state at large H. This is different for P2C for which
Qtot grows with increasing reversal field. For the Penrose P3 tiling
consisting of interconnected nanobars (P3A), Qtot stays small at large
$\mu_{0}$H (vertex charges analyzed for +52 mT are displayed in Supplementary
Fig. 4). A growth of Qtot is observed for P3B and P3C. We attribute the growth
in Qtot to the fact that disconnected nanobars of Type III do not experience
cooperative reversal.
In Fig. 4(m) to 4(r) we summarize the number of flux-closure loops (FCLs)
present in the XPEEM images. For Penrose P2 tilings the minimum in Qtot is
accompanied by a maximum of FCLs. The maximum in FCLs roughly occurs when
about 50 % of Type I and Type II nanobars have undergone switching. For P3B a
pronounced maximum in FCLs is found as well. For P3A and P3C the corresponding
variation is small. Note that P2B, P2C and P3B exhibit a large number of FCLs
already before pronounced switching has taken place. We attribute this
observation to the limited field strength (52 mT) that was available to define
the initial magnetic states (Supplementary Fig. 3). The maximum field was not
large enough to saturate Type III nanobars that were not connected to
neighboring nanobars.
In Fig. 4(s) to 4(x) we depict the number of individual vertices that exhibit
the maximum charge Q = 3, Q = 4 and Q = 5 for neighbor numbers N = 3, N = 4
and N = 5, respectively. For interconnected lattices P2A and P2B the number of
vertices with Q = 4 and Q = 5 is negligible. Strikingly, the number of
vertices with Q = 3 takes its lowest value when the maximum number of FCLs is
reached in P2A. A similar behavior is observed for P2B and P3A. P2C, P3B, and
P3C behave differently in that the number of vertices with Q = 3 does not go
through a global minimum as a function of reversal field. In case of P2C and
P3C the number grows with H. A considerable number of vertices with Q = 4 (Q =
5) is found only in P3B and P3C (P2C). Large individual vertex charges hence
occur in Penrose P2 and P3 tilings with disconnected nanobars, but not for P2
and P3 tilings with interconnected ones. In the latter cases, vertex
configurations with large Q [as stabilized in micromagnetic simulations for
interconnected nanobars in Fig. 6(a) of the supplementary information] have
thus not been observed in the reversal regime. The real AMQ lattices made from
interconnected nanobars avoided these high-energy configurations and formed
FCLs instead. We argue that the exchange interaction in the vertices of the
interconnected P2 and P3 tilings plays the major role for the observed
cooperative magnetization reversal which included Type III nanobars.
Precursors for the cooperative reversal are the Type I and Type II nanobars
that meet at a vertex exhibiting a relatively large charge (e.g. Q = 2 for N =
4). The occurrence of large Q indicates the violation of the local ice rule.
Figure 4(s) to 4(x) reveals that ice rule violations are pronounced in P2C,
P3B and P3C with disconnected nanobars. Our data suggest that weakly
interacting nanobars in AMQs provoke ice rule violations. Note that P3B and
P3C did not show the frequency splitting between Type I and Type II nanobar
resonances near $H=0$ [Fig. 1(f) and (g)]. The absence of splitting is
consistent with weakly interacting nanobars.
To gain further insight into the violation of spin ice rules we have simulated
low- and high-energy vertex states for connected and disconnected nanobars
[Supplementary Figs. 6(b) to 6(i)]. We see that for interconnected nanobars
found e.g. in P2A, the total vertex energy increases by about 80 % from
Supplementary Fig. 6(b) (Q = -1) to Fig. 6(c) (Q = -5), respectively,
considering a vertex with N = 5. For the disconnected nanobars relevant e.g.
in P2C, a violation of the spin ice rule [Q = -5 in Supplementary Fig. 6(e)]
causes an energy higher by only 7 % compared to Fig. 6(d) (Q = -1). Consistent
energy variations with Q are found for vertices with N = 4 in Supplementary
Fig. 6(f) to 6(i). The large energy cost for a spin ice rule violation found
in simulations for fully interconnected nanobars favors low vertex charges
which agree with the spin ice rule. Indeed we experimentally detected mainly
low-Q states in the interconnected lattices of P2A, P3A and ATA in the
reversal regime. If present, high-Q (high energy) vertices in interconnected
AMQs act as nucleation sites for the cooperative reversal. They give rise to
non-stochastic switching [23], presumably causing the staircase-like jumps or
knee anomalies [3]. Vertices containing disconnected nanobars are found to
accommodate particularly large Q and violate the spin ice rule. However these
charges do not trigger cooperative reversal which we attribute to the
relatively weak dipolar coupling in our AMQs. Still our results indicate that
dipolar interaction is enough to provoke energy minimization on a local scale
between vertices via flux-closure loops. We find an increase in the number of
flux-closure loops in almost any of the investigated quasicrystals in the
reversal regime. The dipolar interaction and loop formation are however not
sufficient for creating extended domains.
In Fig. 5 we summarize the experimental data obtained on Amman tilings (AT)
depicted in Fig. 5(a), (b) and (c). The initial magnetic configurations are
displayed in Supplementary Fig. 5.

Figure 5: XMCD-PEEM topography image for Ammann tilings (a) ATA ($\mu_{0}H$ =
34 mT), (b) ATB (at 47 mT), and (c) ATC (at 47 mT) representing the switching
of 47%, 47%, and 53% Type I nanobars. Bright (dark) regions correspond to
magnetic (non-magnetic) regions. The scale bar corresponds to 1 $\mu$m. The
arrow represents the field direction. The X-ray direction was misaligned by
22o to optimize the detection of nanobars of all the different orientations.
(d) - (f) XMCD data of topography images shown in (a) – (f). Blue (red) colors
represent magnetization parallel (opposite) to the X-ray direction. Blue color
indicates a reversed nanobar. Notice the weak contrast of nanobars whose
orientation deviates from the X-ray direction. (g) – (i) CM analysis of XMCD
experimental data shown in (d) – (f). The green and orange filled circles at
the vertices represent positive and negative vertex charges, respectively.
Here the circles with smallest and largest diameter represent charge $|Q|$ = 0
and $|Q|$ = 5, respectively. (j) - (l) Classification of reversal in terms of
switching of Type I and II of nanobars for ATA, ATB, and ATC, respectively.
Legends I and II represent nanobars of Types I and II. (m) - (o) Total charge
count for ATA, ATB, and ATC, respectively. (p) - (r) Number of FCLs [see the
inset in (p) for their definition] for ATA, ATB, and ATC, respectively. (s) -
(u) Charge depiction for ATA, ATB, and ATC, respectively. The purple, dark
yellow, and dark cyan colored symbols represent charges 3 (N = 3), 4 (N = 4),
and 5 (N = 5), respectively.

The XPEEM experiments are shown in Fig. 5(d) to 5(f). For interconnected
nanobars we detect a large central domain of reversed nanobars. For the
partially and fully disconnected nanobars the reversed nanobars are more
distributed over the AMQs. In analogy to disconnected Penrose tilings the
analysis based on the CM provides a tendency towards slightly larger
individual vertex charges when going from ATA to ATC. Still this tendency is
less significant in the graphs of Fig. 5(g) to 5(i) compared to the Penrose
tilings. Analyzing all our XPEEM datasets we find that almost all of Type III
nanobars with $\phi$ = $\pm$ 90o do not undergo switching in Ammann tilings
(compare also data taken at +52 mT and shown in Supplementary Fig. 5).
Studying the reversal of Type I and Type II nanobars in detail [Fig. 5(j) to
5(l)] ATA shows a narrower switching field distribution for these nanobars
compared to P2A and P3A. About 40% of Type I and Type II nanobars of ATA
undergo reversal within a span of 7.5 mT [Fig. 5(j)], i.e., within a field
regime smaller than for P2A and P3A. Full reversal of the Type I and Type II
nanobars is seen at about 34 mT. The corresponding nanobars that are
disconnected require a field of up to about 52 mT for full reversal. These
observations are qualitatively consistent with the field dependencies of spin
wave resonances presented in Fig. 2. We suppose that the narrow field
distribution for switching in ATA reflects a cooperative reversal phenomenon
like an avalanche [compare the blue chain-like domain in Fig. 5(d)].
The analysis of total charges Qtot is depicted in Fig. 5(m) to 5(o). In all
three Ammann tilings we observe a minimum in Qtot when Type I and Type II
nanobars undergo the reversal process. The drop in Qtot is accompanied by an
increased number of FCLs [Fig. 5(p) to 5(r)]. In the magnetization reversal of
ATB [Fig. 5(k)] the switching seems to take place in two separate steps [see
the increases in the number of reversed nanobars at 28 mT and 42 mT in Fig.
5(k)]. Consistent with these features, there are local minima (a shoulder and
a maximum) in Qtot (FCLs). We also see that the maximum in FCLs coincides with
the minimum in the sum of charges. The slope of reversal in ATC [Fig. 5(l)]
resembles the high-field slope of ATB in Fig. 5(k) and the ones seen in P2B,
P2C and P3C. We argue that the high-field reversal processes which are
detected over a broad field regime reflect the disconnected nanobars. In this
field regime we find the spin ice rule violations for ATB and ATC [Fig. 5(t)
and 5(u), respectively] in that the maximum vertex charge Q = 3 is present in
case of N = 3. This large vertex charge is not observed in the reversal of ATA
[Fig. 5(s)].
The data shown in Fig. 4(d) are consistent with the evolution of spin-wave
branches. Reversal of P3A starts near 30 mT via switching of Type I and Type
II nanobars. The switching of Type III nanobars occurs at higher fields. At 52
mT, i.e., the maximum field in the XPEEM microscoype, less than 50 % have been
switched. Consistently, in our spectroscopy data we need to apply an opposing
field with $\mu_{0}|H|=70$ mT to obtain fully developed high-frequency spin-
wave branches in Fig. 1(e). In P3B [Fig. 4(e)] and P3C [Fig. 4(f)] we do not
find reversed Type III nanobars up to 52 mT in XPEEM, which explains the faint
monotonous spin-wave signals marked by a yellow arrow in Fig. 1(g). The
evolution of spin-wave branches measured on ATA [Fig. 2(d)] is also consistent
with the XPEEM imaging [Fig. 5(j)]: a large number of Type I and Type II
nanobars reverse in a narrow regime between 30 and 34 mT in both experiments.
Type III nanobars do not reverse in Fig. 5(j) explaining the monotonous
variation of the branch highlighted by yellow arrows in Fig. 2(d).

### Broadband spectroscopy in the minor loop: reconfigurable magnon
excitations

In the following we discuss the magnonic excitations which are detected in
minor loop measurements starting from intermediate negative fields after
magnetizing ATA, ATB and ATC at $+90~{}$mT. In Fig. 6 we show color-coded
spin-wave spectra taken on partially reversed quasicrystals. In each case the
depicted spectra belong to a minor loop which starts near the magnetic field
value for which the XPEEM data suggest a minimum in the sum of vertex charges
and a large number of flux-closure loops for the Ammann tilings. XPEEM images
for ATA, ATB, and ATC were shown Fig. 4(d) - (f), respectively. For these
three samples the branches in the minor loop [Fig. 6(d) - (f)] are more
complex compared to major loop measurements. They exhibit both $df/dH>0$ and
$df/dH<0$ for $H<0$ reflecting non-reversed and reversed nanobars,
respectively. The white arrows highlight branches which were not observed in
Fig. 2(d) - (f). The number of flux-closure loops does not seem to be large
enough to resolve additional x-shaped magnon branches near $H=0$ which are
characteristic for spin dynamics in magnetic vortex configurations [33, 34].
All the reprogrammed branches in Fig. 6 exhibit small linewidths which are
promising in view of magnonic functionalities like reprogrammable frequency
filters.

Figure 6: SEM images showing 1st generation Ammann tilings (a) ATA for which
nanobars are fully connected, (b) ATB with partially connected nanobars and
(c) ATC with disconnected nanobars. The black color scale bar represents 1
micrometer. Gray-scale spin wave spectra for large arrays measured on 4th
generation (d) ATA, (e) ATB, and (f) ATC, respectively. The minor loop
magnetic field protocol implemented was (d) +90 mT $\rightarrow$ -34 mT
$\rightarrow$ 90 mT, (e) +90 mT $\rightarrow$ -46 mT $\rightarrow$ 90 mT, and
(f) +90 mT $\rightarrow$ -46 mT $\rightarrow$ 90 mT, respectively.

## Conclusions

To conclude, we varied the exchange and dipolar interactions in Penrose P2,
P3, and Ammann quasicrystal lattices. We explored magnonic excitations and
imaged ferromagnetic reversal in these exotic ASI structures using XPEEM. Our
data show compliance with ice rules in the exchange coupled nanobars. Ice rule
violation occurs prominently in tilings without exchange coupling among
nanobars. Owing to the asymmetric and aperiodic arrangements around each
vertex, nanobars taking the same angle with the applied field have
significantly different switching behavior depending on their local
environment. We observe the narrowest distribution of reversal fields in an
exchange-coupled Ammann tiling, followed by the exchange-coupled Penrose P2
and P3 tilings. Here reversal is triggered by vertices whose charge Q deviates
from the ground state. We demonstrated that spin-wave resonances remain sharp
for the partially reversed quasicrystals. The exchange-coupled Penrose P3 and
Ammann tilings show the formation of extended domains via non-stochastic
cooperative reversal which might be functionalized for reprogrammable magnon
steering.

## Methods

### Sample Fabrication

A bilayer PMMA/MMA resist was spin-coated on a silicon substrate, and exposed
via Raith electron beam lithography system at 100 KeV. After development of
the resist a 25 nm thick Py film was deposited using ebeam evaporator.
Subsequently, ultrasonic assisted lift-off was performed in N-methyl
Pyrilidone solution.

### XPEEM Measurements

XPEEM imaging was done at the SPEEM station located at the UE49/PGMa beamline
at BESSY-II (Helmholtz Zentrum, Berlin). The samples were mounted on a sample
holder which allowed us to apply magnetic field to the sample in-situ. The
Penrose P2 and P3 AMQs were patterned on the same silicon substrate, whereas
Ammann AMQs were fabricated on another identical silicon substrate. The
Magnetic images were obtained by performing XMCD at the Fe L3-edge. The
obtained contrast is a measure of the projection of the magnetization on the
X-ray polarization vector. Thus nanobars with a magnetization parallel or
antiparallel to the X-ray polarization either appear red or blue. The sample
orientation was optimized to maximize the contrast.

### Broadband spin-wave spectroscopy

To detect the resonances, we connected two ports of a vector network analyzer
to both ends of the CPW using microwave probes and coaxial cables.
Subsequently, we applied a constant global magnetic field (from $\pm$ 90 mT to
$\mp$90 mT in steps of $\pm$1 mT) at a given in-plane angle $\phi$ and
performed frequency sweeps from 1 GHz up to 20 GHz using a vector network
analyzer. We collect the S-parameters from the vector network analyzer as a
function of frequency at the constant applied magnetic field, $\mu_{o}H$, and
angle $\phi$, and this corresponds to single spin-wave spectra at one
$\mu_{o}H$ and $\phi$ value. We then subtracted spin-wave spectra taken at
successive fields $\mu_{o}H$; that is, we obtain $\Delta$ S (i) = S [H (i+1),
$\phi$] - S [H (i), $\phi$].

### Micromagnetic Simulations

Simulations were performed using the OOMMF code [35], and the Py parameters
used in simulations were as follows: Exchange constant A = 1.3 $\times$ 10 -11
J m-1, saturation magnetization MS = 8 $\times$ 105 A m-1, magnetocrystalline
anisotropy constant K = 0, gyromagnetic ratio $\gamma$ = 2.211 $\times$ 105 m
A-1 s-1, and dimensionless damping coefficient $\alpha$ = 0.01. Different
magnetic configurations were explored by initializing magnetization vectors of
individual segments and relaxing the spin system at the given magnetic field.
For this, we first created a colored bitmap (on a grid of 5 nm$\times$ 5 nm
$\times$ 25 nm) where each segment was assigned a color corresponding to its
assumed magnetization orientation. We then imported this colored bitmap into
OOMMF and equilibrated it in the presence of a field that resided within the
experimental switching regime.

### Charge model

The charge model (CM) assigns a magnetic charge to each of the vertices for
evaluating the magnetic energy [7]. For this one assumes each nanobar to be a
dumbbell of length l with two equal charges of opposite polarity, $\pm$q =
$\pm$ m/l = $\pm$ Mtwl/l = $\pm$ Mtw, at the dumbbell’s ends. m, M, t, and w
represent the magnetic dipole moment, saturation magnetization, thickness, and
width of a segment, respectively. A vertex with a coordination number (CN) N
can acquire a charge Q = $\sum$ q = +Nq $\dots$ -Nq which is the sum of
individual charges q. For a given CN a vertex acquires the lowest possible
charge Q in the ground state to minimize the total energy.

## Data Availability

Requests concerning data should be addressed to D.G. The datasets analysed in
the current study are available in the Zenodo repository, ….

## Code Availability

The code used for micromagnetic simulations is found in Ref. [35] and input
scripts are available in the Zenodo repository, ….

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## Acknowledgements

The research was supported by the Swiss National Science Foundation via Grant
No. 163016. We thank HZB for the allocation of synchrotron radiation beamtime.
V.S. Bhat acknowledges support from the foundation for Polish Science through
the IRA Programme financed by EU within SG OP Programme.

## Author contributions statement

V.B. and D.G. conceived the project. V.B. fabricated the samples and performed
the micromagnetic simulations. V.B., S.W., F.K, and K.B. conducted the XPEEM
experiments. V.B., S.W., and D.G performed the data analysis and wrote the
manuscript on which all authors commented.

## Competing interests

The authors declare no competing interests.

## Materials & Correspondence

Correspondence and material requests should be addressed to D.G.