# Unveiling unique properties of icosahedral magnetic quasicrystals -
Multipole physics and frustration

Junmo Jeon<EMAIL_ADDRESS>Korea Advanced Institute of Science and
Technology, Daejeon 34141, South Korea  SungBin Lee<EMAIL_ADDRESS>Korea
Advanced Institute of Science and Technology, Daejeon 34141, South Korea

###### Abstract

Multipolar degrees of freedom and their hidden orders have been widely
discussed in the context of heavy fermions, frustrated magnets and exotic
Kondo effects. Although there has been extensive search for multipolar degrees
of freedom in magnetic systems, there are few examples that allow pure
multipolar degrees of freedom in the absence of magnetic dipoles. In this
work, for the first time, we show that the magnetic behavior in an icosahedral
quasicrystal is generally described by multipolar degrees of freedom, and in a
specific case by the pure magnetic octupoles in the absence of dipoles,
resulting from the interplay of spin orbit coupling and crystal field
splitting. Importantly, we point out the non-crystallographic symmetries lead
to multipolar degrees of freedom, only allowed in quasicrystals but forbidden
in conventional crystals. For both Kramers and non-Kramers doublets, the
characteristics of multipoles are classified and the effective spin
Hamiltonian on symmetry grounds are derived. Based on the self-similar
triangular structure of the icosahedron, we argue the long-range frustration
in terms of the Ising model. We further classify the possible quantum phases
including quantum fluctuations, in terms of the instantaneous entanglement
generation of the ground state. Our study offers the magnetic icosahedral
quasicrystal as a new platform to search for the novel multipolar degrees of
freedom and their exotic phenomena.

Quasicrystal, Icosahedral symmetry, Multipoles, Octupoles, Magnetic
frustration

##  I Introduction

In condensed matter systems, there are several examples that cannot be easily
observed by conventional experimental techniques. Such hidden orders have been
in debate for several decades and have been waiting to be
discovered[buyers1996low, shah2000hidden, chakravarty2001hidden,
mydosh2011colloquium, mydosh2014hidden, doi:10.1021/jacs.1c09954]. In
particular, unusual higher rank multipole moments, beyond the conventional
electric and magnetic dipole moments, have been suggested as a key player to
exhibit various hidden orders[erkelens1987neutron, kitagawa1996possible,
shiina1997magnetic, lingg1999ultrasound, yamauchi1999antiferroquadrupolar,
tanaka1999evidence, kitagawa2000third, tayama2001antiferro, iwasa2002crystal,
matsuoka2001evidence, buschow2003handbook, hotta2005multipole, kiss2005group,
yanagisawa2005dilatometric, kiss2006scalar, santini2006hidden,
tanida2006possible, matsuoka2008simultaneous, carretta2010quadrupolar,
cricchio2009itinerant, kusunose2011hidden, ano2012quadrupole,
onimaru2011antiferroquadrupolar, goho2015electronic, onimaru2016exotic,
suzuki2018first]. Multipolar degrees of freedom are not only famous for their
ability to give rise to hidden orders, but also for their role in driving a
variety of other interesting and complex phenomena. For example, in heavy
fermion materials, multipolar degrees of freedom can lead to the emergence of
unconventional superconductivity and non-Fermi liquid behavior with exotic
Kondo physics[cox1987quadrupolar, tsvelik1993phenomenological, tou2005sb,
walker1993nature, bauer2002superconductivity, kohgi2003evidence,
suzuki2005quadrupolar, kotegawa2008anomalous, sakai2011kondo,
tsujimoto2014heavy, sumita2016superconductivity, onimaru2016quadrupole].
Moreover, beyond their hidden orders, magnetic frustration between multipole
moments can give rise to the emergence of exotic ground states, so called
multipolar quantum spin liquids[gao2019experimental, gaudet2019quantum,
sibille2020quantum, PhysRevB.61.476]. Hence, understanding the properties of
multipolar physics have been the focus of intense research with broad
implications and new insights with potential applications[totsuka1995matrix,
aoki2005novel, santini2009multipolar, tazai2019multipole, mydosh2020hidden,
patri2020theory]. However, there are limited examples of this kind and many of
the magnetic systems contain not only multipoles but also magnetic dipoles at
the same time.

Finding multipolar degrees of freedom in the magnetic systems requires a
delicate combination of spin-orbit couplings and crystalline electric field
(CEF) splitting based on the point group symmetries[cohen2016fundamentals,
suzuki2018first]. In conventional crystals, the point group symmetry, which
should be compatible with the translational symmetry, restricts searching pure
multipolar degrees of freedom in the magnetic systems[cohen2016fundamentals].
On the other hand, the quasicrystals, and their approximants could exhibit the
point group symmetries beyond the space group such as pentagonal rotational
symmetry because they are ordered without spatial
periodicity[kellendonk2015mathematics, suck2013quasicrystals,
labib2020magnetic]. Thus, the quasicrystalline materials would be a good
platform for finding multipolar degrees of freedom. Especially, several rare-
earth magnetic quasicrystals are present having icosahedral symmetry but have
never been explored in terms of their multipolar physics and related exotic
phenomena[deguchi2012quantum, suzuki2021magnetism, takeuchi2023high,
de2007lattice, judd1957crystal, kashimoto1998magnetic].

In this paper, we consider the noncrystallographic 5-fold rotational symmetry
and icosahedral symmetry with $f$-orbital electrons of the rare earth atoms.
First, we classify all possible multipole degrees of freedom present in rare-
earth magnetic quasicrystals in the presence of noncrystallographic 5-fold
symmetry. We note that it generally leads to the higher order multipoles of
the pseudospin $xy$ components, and magnetic dipoles of the Ising moments.
More interestingly, if magnetic quasicrystals with Yb3+ ions are in a perfect
icosahedral crystal field symmetry, they host the Kramers doublet that carries
pure magnetic octupole moments without magnetic dipole moments. On symmetry
grounds, we introduce the generic spin Hamiltonian for both Kramers doublet
and non-Kramers doublet. In the antiferromagnetic Ising limit, we first
discuss the degenerate ground state born of the geometrically frustrated
icosahedron structure. When the quantum fluctuation is introduced, we discuss
the unique ground state is stabilized having non-zero entanglement. In this
case, depending on (anti-) ferromagnetic $XY$ interaction, the specific linear
combination of the degenerate ground states found in the Ising limit becomes a
unique ground state. Our work provides a perspective for finding multipolar
degrees of freedom and their magnetic frustration originated from
noncrystallographic symmetries. Furthermore, it opens a new paradigm for
enriching hidden orders, spin liquids and novel Kondo effects in
quasicrystals.

##  II Results and discussion

###  II.1 Multipolar degrees of freedom in icosahedral magnetic quasicrystals

In this section, we first classify all possible multipolar degrees of freedom
for rare-earth magnetic systems in the presence of noncrystallographic 5-fold
symmetry and icosahedral symmetry. Then, we focus on a special case where pure
magnetic octupole moments exist in the absence of magnetic dipoles, and
discuss their characteristics.

The most general CEF Hamiltonian in a 5-fold rotational symmetry is given as
follows, using the Stevens operators,

$\displaystyle
H_{\mbox{CEF}}=B_{60}O_{6}^{0}+B_{65}O_{6}^{5}+B_{20}O_{2}^{0}+B_{40}O_{4}^{0}$
(1)

Here, $B_{nm}=-\gamma_{nm}qC_{nm}\braket{r^{n}}\theta_{n}$ are the Stevens
coefficients obtained by the radial integrals, where $\gamma_{nm}$ is a term
calculated from the ligand environment expressed in terms of tesseral
harmonics. $q$ is the charge of the central atom, $C_{nm}$ are the
normalization factors of the spherical harmonics, $r$ is the radial position,
$\theta_{n}$ are constants associated with electron orbitals of the magnetic
ion[Scheie:in5044]. Here, we assume the point charge model. Note that the
$z$-axis is a 5-fold rotational symmetry axis. The $O_{n}^{m}$ are Stevens
operators with respect to the total angular momentum operators (See
Supplementary Materials for the detailed forms of $O_{n}^{m}$.). When we
consider the full icosahedral symmetry, which has 5-fold rotational symmetry
plus mirror reflection symmetry, it leads to $B_{20}=B_{40}=0$, and
$B_{65}=-42B_{60}$. We consider both cases where a perfect icosahedral
symmetry is present or only 5-fold symmetry is present, applicable to
different situations respectively. The latter case can be taken into account
using the charge defects which breaks the icosahedral symmetry down to
$C_{5v}$. In this case, the charge defect on a ligand is placed on a $z$-axis
as $q\to q(1+\alpha)$. As a result, the Stevens coefficients,
$B_{65},B_{20},B_{40}$ are changing as a function of $\alpha$. Note that
$B_{nm}=-\alpha\gamma_{nm}^{\prime}qC_{nm}\braket{r^{n}}\theta_{n}$ for a
single point charge defect, $B_{60}$ is unchanged if the charge defect is
placed on the $z$-axis.

For both full icosahedral symmetry, $I_{h}$ and 5-fold rotational symmetry,
$C_{5v}$ with nonzero $\alpha$, we summarize the CEF states in Figs.1 and 2
for half-integer $J$ and integer $J$ values, respectively. For $\alpha=0$, the
point group symmetry restores the full icosahedral symmetry and multiple
degenerate levels appear. It is noteworthy that for $J=7/2$ which is the case
for Yb3+, the unique (Kramers) doublet exists under the perfect icosahedral
symmetry group as shown in red box of Fig.1 (a). Specifically, there are two
eigenspaces of $H_{\mbox{CEF}}$, the Kramers doublet and the
sextet[walter1987crystal]. On the other hand, for any given $\alpha\neq 0$,
the CEF states under 5-fold symmetry are split into doublets or singlets for
all given values of the total angular momentum $J$ of the rare earth atom. In
both Fig.1 and Fig.2, the tentative orders of the energy levels are given for
$\alpha=0.5$. Given $\alpha\neq 0$, every energy levels for half-integer $J$
are Kramers doublet due to the time-reversal symmetry. However, for integer
$J$, some singlets are allowed. Notably, for every case, the $z$-component of
the pseudospins of the doublets are magnetic dipoles, while the $x$ and
$y$-components of the pseudospins of the doublets are generally multipoles
such as quadrupoles, octupoles, and even higher ones. For half-integer values
of $J$, $\Sigma_{x(y)}$ could be dipole, octupole and dotriacontapole (see
Fig.1), whereas for integer values of $J$, the doublets carry either
quadrupole or hexadecapole $\Sigma_{x(y)}$ (see Fig.2). Again, it is because
of the time-reversal symmetry. It is interesting to note that for the low
lying Kramers doublet, $\Sigma_{x(y)}$ for $J=7/2$ always represents magnetic
octupoles originated from the unique Kramers doublet, whereas, $\Sigma_{x(y)}$
for $J=9/2$ and $J=15/2$ could take magnetic octupoles or dotriacontapoles. We
also note that for the low lying non-Kramers doublets, $\Sigma_{x(y)}$ for
$J=4$ and $J=6$ take either quadrupole or hexadecapole, while for $J=8$, it
could take only quadrupole (see Fig.2 (c)).

Figure 1: Summary of the CEF states and multipoles under noncrystallographic
point group symmetries for given half-integer values of $J$, (a) $J=7/2$
(Yb3+) (b) $J=9/2$ (Nd3+) and (c) $J=15/2$ (Dy3+) respectively. Under the
icosahedral symmetry $I_{h}$, the CEF states are split into several
multiplets. Particularly, note that $J=7/2$ which is the case for Yb3+ has a
unique Kramers doublet and it hosts pure magnetic octupoles without magnetic
dipoles or quadrupoles, as shown in the red box. Whereas, for $C_{5v}$ where
the mirror reflection is absent from $I_{h}$, all the multiplets are split
into either Kramers doublets. In this case, the $z$-components of the
pseudospin, $\Sigma_{z}$, are dipoles for every doublet state. But, the
$xy$-components, $\Sigma_{x,y}$, are not only dipoles but also octupoles and
dotriacontapoles. Depending on the energy scale of the breaking of mirror
symmetry, the order of energy could change. (d) Dipoles (black-copper),
octupoles (red-white) and dotriacontapoles (red-yellow). See the main text for
more details. Figure 2:  Summary of the CEF states and multipoles under
noncrystallographic point group symmetries for given integer values of $J$,
(a) $J=4$ (Pr3+) (b) $J=6$ (Tb3+) and (c) $J=8$ (Ho3+). Unlike the icosahedral
symmetry $I_{h}$ where the CEF states are split into several multiplets. the
five-fold symmetry $C_{5v}$ symmetry splits all the multiplets into either
non-Kramers doublets or singlets. Under $C_{5v}$, the $z$-components of the
pseudospin, $\Sigma_{z}$ are magnetic dipoles for every doublets. But, the
$xy$-components, $\Sigma_{x,y}$ represent either quadrupoles or hexadecapoles.
Depending on the energy scale of the breaking of mirror symmetry, the order of
energy could change. (d) Singlet (black), quadrupoles (blue-skyblue) and
hexadecapoles (violet-skyblue). See the main text for more details.

From now on, we focus on the special case where the lowest Kramers doublet is
represented by pure octupoles, i.e. $J=7/2$ in an icosahedral crystal field
symmetry and discuss their characteristics. Then, in the following
subsections, we discuss the generic spin Hamiltonian, geometrical frustration
and quantum fluctuation effects. It is important to note that all these
arguments are generally applicable to any cases of multipolar physics with $J$
listed in Fig.1 and Fig.2.

Let us define the Kramers doublet as $\left|{\pm}\right\rangle$, which are
written in terms of the eigenstates of the $J_{z}$ operator,

$\displaystyle\left|{+}\right\rangle=-\sqrt{\frac{7}{10}}\left|{J_{z}\\!\\!=\\!\\!-\frac{3}{2}}\right\rangle+\sqrt{\frac{3}{10}}\left|{+\frac{7}{2}}\right\rangle$
(2)
$\displaystyle\left|{-}\right\rangle=\sqrt{\frac{3}{10}}\left|{-\frac{7}{2}}\right\rangle+\sqrt{\frac{7}{10}}\left|{+\frac{3}{2}}\right\rangle.$

In case of Yb3+ ion, it is known that $B_{6}<0$, so the Kramers doublet is the
ground eigenspace of the CEF Hamiltonian, well separated from the sextet.
[lindgard1975tables, sandberg2021emergent]. The CEF energy gap is given by
$25200|B_{60}|\sim$ $\mathcal{O}$ (10meV), where detailed number depends on
icosahedral magnetic materials with Yb. Since the energy scale of spin
exchanges between rare-earth ions are generally much smaller than the crystal
field splittings, one can expect the magnetic properties at low temperature
are explained within this Kramers doublet. From Eq.(2), one can easily find
that $\braket{\pm}{J_{i}}{\mp}$ and $\braket{\pm}{J_{i}}{\pm}$ vanish where
$i=x,y,z$. Importantly, $J_{z}$ also vanishes due to the symmetric
coefficients of the states. This confirms that there is no magnetic dipole
moment. Thus, one should consider the multipolar degrees of freedom given by
the irreducible tensor operators. However, since $\left|{\pm}\right\rangle$
are the Kramers doublet, the time-reversal even operators such as quadrupole
moments vanish. Hence, we can expect the higher order degrees of freedom such
as octupoles, in the absence of any dipolar or quadrupolar degrees of freedom.

To show that the Kramers doublet, $\left|{\pm}\right\rangle$ in Eq.(2),
describes the octupolar degrees of freedom, let us define the pseudospin
ladder operators, $\Sigma^{\pm}$, as follows.

$\displaystyle\Sigma^{+}=\left|{+}\right\rangle\left\langle{-}\right|\ \ \ \ \
\Sigma^{-}=(\Sigma^{+})^{\dagger},$ (3)

and $\Sigma^{z}=[\Sigma^{+},\Sigma^{-}]/2$. Now define the octupolar operators
as the rank 3 spherical tensor operators, $T_{m}^{(3)}$ in terms of
$J_{+},J_{-}$ and $J_{z}$. Note that octupolar operators are time-reversal
odd, under the time-reversal transformation, $\mathcal{T}$, it satisfies
$\mathcal{T}\Sigma^{\pm}\mathcal{T}^{-1}\\!\\!=\\!\\!-\Sigma_{\mp}$,
$\mathcal{T}\Sigma^{z}\mathcal{T}^{-1}\\!\\!=\\!\\!-\Sigma^{z}$. As a result,
$\Sigma^{z}\sim T_{0}^{(3)}$ and $\Sigma^{\pm}\sim T_{m}^{(3)}$ for non-zero
$m$. However, $T_{1}^{(3)}$ and $T_{-1}^{(3)}$ vanish because $T_{\pm
1}^{(3)}\left|{\pm}\right\rangle$ is not in the doublet eigenspace. Note that
$T_{\pm 1}^{(3)}$ changes the eigenvalue of the $J_{z}$ operator by $\pm 1$.
Similarly, since $J_{\pm}^{2}\left|{\pm}\right\rangle$ and
$J_{\pm}^{3}\left|{\mp}\right\rangle$ are not in the doublet, the only non-
trivial matrix elements are $\braket{\pm}{T_{\pm 2}^{(3)}}{\mp}$ and
$\braket{\mp}{T_{\pm 3}^{(3)}}{\pm}$. This leads to $T_{\pm 2,\mp
3}^{(3)}\sim\Sigma^{\pm}$. In detail, one can represent the octupole
pseudospin operators in the doublet as,

$\displaystyle\Sigma^{z}\equiv T_{0}^{(3)},\ \ \ $
$\displaystyle\Sigma^{\pm}\equiv\frac{1}{2}\sqrt{\frac{2}{15}}T_{\pm
2}^{(3)}\mp\frac{1}{2}\sqrt{\frac{1}{5}}T_{\mp 3}^{(3)}.$ (4)

Specifically, $T_{\pm
2}^{(3)}=\frac{1}{4}\sqrt{\frac{105}{\pi}}\overline{J_{\pm}^{2}J_{z}}$,
$T_{\pm 3}^{(3)}=\mp\frac{1}{8}\sqrt{\frac{35}{\pi}}J_{\pm}^{3}$, and
$T_{0}^{(3)}=\frac{1}{4}\sqrt{\frac{7}{\pi}}(5J_{z}^{3}-3J_{z}J^{2})$, where
$\overline{\mathcal{O}}$ is the symmetrization of the operator
$\mathcal{O}$[sakurai1995modern]. Each pseudospin operator behaves like the
rank 3 tensors. From Eq.(4), one can write,

$\displaystyle\Sigma_{x(y)}\\!=\\!\frac{1}{4}\left[\sqrt{\frac{2}{15}}(T_{2}^{(3)}\pm
T_{-2}^{(3)})\pm\sqrt{\frac{1}{5}}(T_{3}^{(3)}-T_{-3}^{(3)})\right],$ (5)

where $x(y)$ takes $+(-)$ sign in Eq.(5), respectively.

###  II.2 Generic spin Hamiltonian on symmetry grounds

By applying the symmetry transformation of the icosahedron group ($I_{h}$) and
the time reversal symmetry transformation, one can find the generic spin
Hamiltonian of the nearest neighbor interactions. Let us define the local
$z$-axis pointing to the center of the icosahedron shell. Then, under the
5-fold rotational symmetries of $I_{h}$, we have $\Sigma^{\pm}_{i}\to e^{\mp
4i\pi/5}\Sigma^{\pm}_{j}$, where $i$ and $j$ are the nearest neighbor sites.
This leads to the bond dependent phase factors in the Hamiltonian, such as
$\Sigma^{+}_{i}\Sigma^{+}_{j}$ or $\Sigma^{+}_{i}\Sigma^{z}_{j}$. As a result,
the generic symmetry allowed Hamiltonian under the icosahedral point group
symmetry contains the four independent parameters, $J_{\pm\pm},J_{\pm
z},J_{\pm}$ and $J_{zz}$, and is given as,

$\displaystyle H$
$\displaystyle=\sum_{\braket{i,j}}\Bigl{[}J_{zz}\Sigma_{i}^{z}\Sigma_{j}^{z}+J_{\pm}\big{(}\Sigma_{i}^{+}\Sigma_{j}^{-}+\Sigma_{i}^{-}\Sigma_{j}^{+}\big{)}$
(6)
$\displaystyle+J_{\pm\pm}\big{(}\alpha_{ij}\Sigma_{i}^{+}\Sigma_{j}^{+}+\alpha_{ij}^{*}\Sigma_{i}^{-}\Sigma_{j}^{-}\big{)}$
$\displaystyle+J_{\pm
z}\Big{(}\Sigma_{i}^{z}\big{(}\beta_{ij}\Sigma_{j}^{+}+\beta_{ij}^{*}\Sigma_{j}^{-}\big{)}+i\leftrightarrow
j\Big{)}\Bigr{]}.$

Here, $\alpha_{ij}$ takes the values $1,e^{\pm i2\pi/5}$ and $e^{\pm i4\pi/5}$
depending on the bond orientation due to the 5-fold rotational symmetry, and
$\beta_{ij}=(\alpha_{ij}^{*})^{2}$. The Hamiltonian in Eq.(6) is written in
terms of local coordinate axes, where the local $z$-axis for each sites as
pointing in the icosahedron (See Supplementary Materials for detailed
derivation of the effective pseudospin Hamiltonian and local axes.). The
magnitudes of these spin exchange parameters would vary from case to case.
Here, for the simplest case, we first study the Ising limit with a finite
$J_{zz}$ and then consider the quantum fluctuations in the presence of
$J_{\pm}$.

Remarkably, we should emphasize that the spin Hamiltonian, $H$ in Eq.(6),
could be used to argue general characteristics of the multipolar pseudospin
models even for the $C_{5v}$ symmetry rather than $I_{h}$. The Hamiltonian in
Eq.(6) is applicable for any Kramers or non-Kramers doublets with half-integer
or integer values of $J$, in the case of $C_{5v}$ under small charge defect on
the ligand which has been discussed in terms of $\alpha$. The only difference
between Kramers and non-Kramers doublets is the presence and the absence of
the parameter $J_{\pm z}$. This is originated from the fact that for non-
Kramers doublet, $z$ components represent magnetic dipoles, whereas $x,y$
components represent quadrupoles or hexadecapoles, thus the coupling between
$\Sigma^{z}$ and $\Sigma^{\pm}$ is forbidden under the time reversal symmetry.

Figure 3: Icosahedral quasicrystal generated by the cut-and-project scheme
from 6-dimensional hyperspace. We draw the sites forming the icosahedron
shells only for visibility. The distances between neighboring icosahedron
shells vary. For the antiferromagnetic, $J_{zz}$, there is geometrical
frustration on each icosahedron.

###  II.3 Geometrical frustration

Let us first consider the Ising model, where only $J_{zz}$ is non-zero in
Eq.(6). Considering the icosahedral quasicrystal descended from 6-dimensional
hyperspace[kellendonk2015mathematics], Fig.3 represents the structure of
icosahedral quasicrystal. As shown in Fig.3, the distances between inter-shell
of icosahedron vary (See Supplementary Materials for detailed cut-and-project
scheme for the icosahedral quasicrystal.). In real materials, depending on the
structures of quasicrystals and approximants, the distances between the shells
could differ[suzuki2021magnetism, de2007lattice, takeuchi2023high,
steinhardt1987physics, deguchi2012quantum]. Furthermore, it is known that the
inter-shell distance can be also controlled in terms of the external
pressure[berns2013problem, deguchi2012quantum, watanuki2006pressure]. Thus,
for general argument, we mainly focus on the nearest neighboring sites in a
single icosahedron and discuss the magnetic states. Depending on the inter-
shell distances, one may consider the perturbative approach (See Supplementary
Materials for perturbative approach.). For ferromagnetic $J_{zz}$, it is
obvious there are only two degenerate ground states exist, where every
octupole points in local $+z$ and $-z$ directions, respectively.

For antiferromagnetic Ising, $J_{zz}>0$, the triangular faces of the
icosahedron cause geometric frustration. In this case, 72 degenerate states
exist and they are classified into two groups on symmetry grounds: (i) 60
degenerate states without 5-fold rotational symmetry, (ii) 12 degenerate
states with 5-fold rotational symmetry.

Figure 4: Two distinct octupolar ground states on an icosahedron for
antiferromagnetic $J_{zz}$ based on the symmetry ground. (a) 60 degenerate
states with no 5-fold rotational symmetry. (b) 12 degenerate states with
5-fold symmetry. $Z$ axis is the 5-fold rotational symmetry axis in (b). (c,d)
Top view along $Z$ axis in (a,b), respectively. (c) Mirror reflection with
respect to $\sigma$ plane containing $Z$ axis swaps the octupoles on the sites
Q and R that leads to another degenerate state without 5-fold symmetry. (d)
The state depicted in (b) possesses 5-fold rotational symmetry along $Z$ axis.

Fig.4 (a) show an example of the first group of ground states. Note the
octupolar moments arranged on the icosahedron in Fig.4 (a) do not have a
5-fold rotational symmetry axis. Since there are 6 independent choices of the
$Z$ axis, by applying 5-fold rotational transformation around each $Z$ axis,
we have 30 degenerate states. In addition, for each 30 degenerate states, the
energy is invariant under the swap of two octupoles on the sites Q and R in
Figs.4 (a) and (c). Hence $M_{\sigma}$, the spatial mirror reflection with
respect to the $\sigma$-plane depicted in Fig.4 (c) doubles the number of
degenerate states with no 5-fold rotational symmetry, resulting in total 60
degenerate states. (See Supplementary Materials for detailed discussion of the
symmetry argument.). Figs.4 (b) and (d) illustrate 5-fold rotational symmetric
ground state. There are 12 independent choice of the rotational symmetry axis,
$Z$ axis in Fig.4 (b) of the second group.

###  II.4 Quantum fluctuation

Now let us consider non-zero but small $J_{\pm}$ and study the effect of
quantum fluctuations. To study the fluctuation effects, we introduce three
subsets of the 72 degenerate states, $A,B$ and $C$ in terms of the orientation
preserving icosahedral rotation group, $I\trianglelefteq I_{h}$. Specifically,
the subsets $A$ and $C$ are generated by applying the spatial rotations in $I$
to the states in Figs.4 (a) and (b), respectively. While, the subset $B$ is
generated by applying the coset, $IM_{\sigma}=\\{gM_{\sigma}|g\in I\\}$ to the
state in Fig.4 (a). Thus, for
$H_{\pm}=J_{\pm}\sum_{\braket{i,j}}(\Sigma_{i}^{+}\Sigma_{j}^{-}+\Sigma_{i}^{-}\Sigma_{j}^{+})$,
two states in the same subset admit zero matrix element of $H_{\pm}$. One can
let $\left|{\psi_{A_{n}}}\right\rangle$, $\left|{\psi_{B_{l}}}\right\rangle$
and $\left|{\psi_{C_{r}}}\right\rangle$, where $1\leq n,l\leq 30$ and $1\leq
r\leq 12$ be the states in $A,B$ and $C$. Hence, in the sub-Hilbert space of
the 72 Ising ground states, $H_{\pm}$ has the matrix representation,
$[H_{\pm}]_{A,B,C}$, given by

$\displaystyle[H_{\pm}]_{A,B,C}=\begin{pmatrix}0&T_{AB}&T_{AC}\\\
T_{BA}&0&T_{BC}\\\ T_{CA}&T_{CB}&0\end{pmatrix}$ (7)

where $T_{BA}=T_{AB}^{\dagger}$ is a $30\times 30$ matrix, while
$T_{AC}=T_{CA}^{\dagger}$ and $T_{BC}=T_{CB}^{\dagger}$ are $30\times 12$
matrices. Here, each non-zero matrix element is $J_{\pm}$. On symmetry
grounds, we can write the general form of the ground state,
$\left|{GS}\right\rangle$, as,

$\displaystyle\left|{GS}\right\rangle=a\sum_{n=1}^{30}\left|{\psi_{A_{n}}}\right\rangle+b\sum_{l=1}^{30}\left|{\psi_{B_{l}}}\right\rangle+c\sum_{r=1}^{12}\left|{\psi_{C_{r}}}\right\rangle,$
(8)

where we have only three real coefficients, $a,b$ and $c$ for
$\left|{\psi_{A_{n}}}\right\rangle$, $\left|{\psi_{B_{l}}}\right\rangle$ and
$\left|{\psi_{C_{r}}}\right\rangle$, respectively(See Supplementary Materials
for detailed discussion for the perturbative method.). The energy correction
is $E(a,b,c)=\braket{GS}{H_{\pm}}{GS}$.

First, considering $J_{\pm}<0$, the Lagrange multiplier method leads to
$a=b=(1+\sqrt{6})c/5$ for the ground state. Next, if $J_{\pm}>0$, $E(a,b,c)$
is minimized when $a=-b$ and $c=0$. Remarkably, we have no degeneracy in
either cases. Thus, any small quantum fluctuation given by $H_{\pm}$ leads to
a unique ground state with particular superpositions of degenerate states (See
Supplementary Materials for detailed derivation).

To capture the entanglement, we compute the entanglement negativity of the
state defined by
$\mathcal{N}_{E}=\sum_{i}(\left|\lambda_{i}\right|-\lambda_{i})/2$, where
$\lambda_{i}$ are all eigenvalues of $\rho^{T_{A}}$, the partial transpose of
the density matrix of the ground state, $\rho$[horodecki2009quantum].
$\mathcal{N}_{E}=0$ if $\rho$ is separable, while $\mathcal{N}_{E}>0$ for an
entangled state. For the icosahedron shell, $\mathcal{N}_{E}$ is computed by
partitioning 12 vertices into two hemispherical region (one of them is
highlighted as the blue shaded region in the inset of Fig.5). Fig.5 (a)
illustrates the entanglement is instantaneously generated for non-zero
$J_{\pm}$.

Figure 5: Quantum fluctuation effect of nonzero $J_{\pm}$. (a) For nonzero
$J_{\pm}$, the entanglement negativity of the ground state, $\mathcal{N}_{E}$,
becomes finite. The sub-Hilbert space is defined by the blue shaded region of
the icosahedron as shown in the inset. Red arrows are drawn to emphasize the
local $z$ axes. At $J_{\pm}=0$ (marked by the triangle), the state is
separable. For $J_{\pm}>0$ (square, (b)), the unique ground state is comprised
of the states in the $A$ and $B$ subsets without 5-fold symmetry, with equal
probability but opposite sign. For $J_{\pm}<0$ (circle, (c)), all 72
degenerate states are superposed in the unique ground state as shown in (c).

##  III Discussion and Conclusion

In summary, we discover the magnetic quasicrystalline systems host multipolar
degrees of freedom, as a result of the interplay of spin-orbit coupling and
CEF splitting of the icosahedral point group symmetry. Despite intensive
previous work on magnetic properties in quasicrystals, the possibility of
multipolar degrees of freedom and their importance have never been pointed
out. In our work, for the first time, we have shown that the
noncrystallographic point group symmetry can generally host such
unconventional degrees of freedom and have studied their characteristics.
Depending on a perfect icosahedral symmetry $I_{h}$ or five-fold symmetry
$C_{5v}$, we classify the multipole characteristics for different values of
$J$. Strikingly, we have shown that pure octupolar degrees of freedom emerges
for $J=7/2$ under $I_{h}$ symmetry. In this case, the Kramers doublet has zero
expectation values for both magnetic dipoles and quadrupoles, however, the
rank 3 tensors which represent octupole degrees of freedom, have non-vanishing
expectation values. Based on the symmetry transformation, we further clarify
each component of the pseudospin-1/2 in terms of magnetic octupoles as
discussed in Eq.(4). On symmetry grounds, we also derive the spin exchange
Hamiltonian with four independent parameters. For antiferromagnetic Ising
model, magnetic frustration leads to 72 degenerate states for a single
icosahedron. For a small but finite $J_{\pm}$, quantum fluctuations make a
particular mixture of these degenerate states. It makes different but unique
ground states for (anti-) ferromagnetic $J_{\pm}$, producing a finite
entanglement even for arbitrary small $J_{\pm}$. Depending on the inter-shell
distances, possible macroscopic degeneracy and entanglement of octupoles would
be an interesting future work. Also, the studies in the presence of
$J_{\pm\pm}$ and $J_{\pm z}$, which do not preserve the total $\Sigma^{z}$,
can be explored which we leave as a future work.

Such octupolar degrees of freedom can be found in rare-earth based magnetic
quasicrystals and approximants such as Au-Al alloys, Cd-Mg alloys including
rare earth ions and etc[walter1987crystal, labib2020magnetic,
kashimoto1998magnetic, watanabe2021topological, suzuki2021magnetism,
deguchi2012quantum, watanuki2011pressure, sato2001magnetic,
imura2023variation, yamada2019formation, PhysRevB.61.476, PhysRevB.100.054417,
doi:10.1021/jacs.1c09954, hiroto2014sign]. However, many of the currently
present magnetic quasicrystals have issues of intermediate valence and some
mixed sites between non rare-earth atoms. [watanabe2021topological,
watanuki2012intermediate, watanuki2011pressure, lehr2013enhanced,
matsukawa2014valence, otsuki2016distributed, watanabe2016origin,
hayashi2017observation, watanabe2021magnetism, hiroto2014sign,
imura2023variation]. It makes imperfect symmetries, allowing small deviation
from perfect non-crystallographic point group symmetries. Nonetheless, one
expects that advancements in chemical synthesis techniques could make the
successful synthesis of finely controlled icosahedral
quasicrystals[watanuki2011pressure, lehr2013enhanced, yamada2019formation,
nagaoka2023quasicrystalline], and it may give us a chance to discover pure
magnetic octupoles or even higher order multipolar degrees of freedom and
their interesting physics.

Our work for the first time shows that the multipolar degrees of freedom
naturally emerge in icosahedral quasicrystals. It breaks new ground in the
magnetism of quasicrystals, and opens several interesting questions. One could
explore magnetic quasicrystals searching for hidden phases, magnetic
frustration induced long range entanglement such as spin liquids and non-Fermi
liquids due to exotic Kondo effect[kimura1986temperature, widom1988short,
goldman2014magnetism, andrade2015non, matsukawa2016pressure,
shi2018frustration, PhysRevLett.115.036403, tamura2021experimental]. Our study
motivates to experimentally find new rare-earth icosahedral quasicrystals
beyond conventional magnetism in periodic crystals. The field of magnetic
quasicrystals is an interesting research area, and continued advancements in
both experimental and theoretical studies lead us to discover new magnetic
phenomena in quasicrystals.

##  IV Methods

To construct the CEF Hamiltonian, we used the simplified point charge model,
where the point charge ligands are placed on each 12 verticies of the
icosahedron surrounding the rare earth atom. For the charge impurity model
where the icosahedral point group symmetry breaks down to the $C_{5v}$, we
place the charge impurity on the ligand site placed on the local $z$-axis. To
compute the Stevens parameter, we use the numerical values of the Stevens
factors and radial integrals of rare-earth ions in [EDVARDSSON1998230]. By
block diagonalizing the CEF Hamiltonians, we generally search the possible
doublet eigenspaces, and their possible multipolar degrees of freedom. We
examine the vanishing of the lower magnetic or electric multipole operators to
conclude the presence of the higher order multipolar degrees of freedoms. To
construct the symmetry allowed spin Hamiltonian on the icosahedral
quasicrystal, we apply the icosahedral point group symmetry, and time-reversal
symmetry to the octupolar pseudospin operators—that is described in the
Supplementary Materials in detail.

The icosahedral quasicrystal is constructed by the standard cut-and-project
scheme described in the Supplementary Materials in detail. Considering the
open boundary condition, we use the exact diagonalization method and symmetry
ground to find the ground states of the Ising model. With quantum fluctuation,
we analytically investigate the purely entangled ground state, and numerically
calculate its entanglement negativity emergent on the single icosahedron shell
by using the exact diagonalization method.

Acknowledgments

We thank Takanori Sugimoto and Taku J Sato for useful discussions. J.M.J and
S.B.L. are supported by National Research Foundation Grant (No.
2021R1A2C1093060)).

Competing interests The authors declare no conflict of interest.

Author contributions J.M.J and S.B.L develop main idea of the project. J.M.J
also produces the data and analyzes them. All authors contribute for writing
the manuscript.

Data availability The data that support the findings of this study are
available from the corresponding author upon request.

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## Supplementary Material

##  1 Crystal electric field Hamiltonian and Stevenes operator

The crystal field Hamiltonian under the icosahedral point group symmetry would
be written as $H_{CEF}=B_{6}(O_{6}^{0}-42O_{6}^{5})$. Here,
$B_{6}=A_{6}\gamma_{J}\braket{r^{6}}$ is the Stevens coefficient obtained by
the radial integral. $\gamma_{J}$ is the Stevens factor, and $r$ is the radial
position. Especially, $A_{6}=-\frac{33}{100}\frac{q_{0}|e|}{R_{0}^{7}}$ where
$q_{0}$ is the charge of the ligands and $R_{0}$ is the distance between
ligands and central rare earth atom[walter1987crystal, lindgard1975tables]. We
focus on the angular parts which are $O_{6}^{0}$ and $O_{6}^{5}$, Stevens
operators. They are given by[walter1987crystal]

$\displaystyle
O_{6}^{0}=231J_{z}^{6}-105(3J(J+1)-7)J_{z}^{4}+(105J^{2}(J+1)^{2}-525J(J+1)+294)J_{z}^{2}-5J^{3}(J+1)^{3}+40J^{2}(J+1)^{2}-60J(J+1)$
(S1) $\displaystyle
O_{6}^{5}=\frac{J_{z}(J_{+}^{5}+J_{-}^{5})+(J_{+}^{5}+J_{-}^{5})J_{z}}{4}$
(S2)

If the central rare earth atom is $\mbox{Yb}^{3+}$, the $4f$ electrons are the
valance electron. It is known that $q_{0}$ would be positive for some
materials such as Au-Al-Yb compounds. In such case, we have $A_{6}<0$.
Furthermore, $\mbox{Yb}^{3+}$ has the total angular momentum $J=7/2$ with
$\gamma_{J}=1.48\times 10^{-4}$, so we have $B_{6}<0$[walter1987crystal,
lindgard1975tables]. In this case the ground state sector of $H_{CEF}$, is
Kramer doublet.

Once the icosahedral point group symmetry is broken down to the $C_{5v}$
symmetry due to the single point charge impurity placed on the $z$-axis, the
additional diagonal Stevens operators, $O_{2}^{0}$ and $O_{4}^{0}$ could be
appeared. They are defined by

$\displaystyle O_{2}^{0}=3J_{z}^{2}-J(J+1)$ (S3) $\displaystyle
O_{4}^{0}=35J_{z}^{4}-(30J(J+1)-25)Jz^{2}+3J^{2}(J+1)^{2}-6J(J+1)$ (S4)

##  2 Local axes and symmetry allowed Hamiltonian

We consider an icosahedron whose one of the 5-fold rotationally symmetric axes
is the global $z$-axis. We define the local coordinate axes for the
icosahedron as follows.

$\displaystyle\vec{z}_{1}=(0,0,-1)$ (S5)
$\displaystyle\vec{z}_{2}=-\frac{1}{\sqrt{5}}(2,0,1)$
$\displaystyle\vec{z}_{3}=-\left(\frac{5-\sqrt{5}}{10},\sqrt{\frac{5+\sqrt{5}}{10}},\frac{1}{\sqrt{5}}\right)$
$\displaystyle\vec{z}_{4}=-\left(\frac{-5-\sqrt{5}}{10},\sqrt{\frac{5-\sqrt{5}}{10}},\frac{1}{\sqrt{5}}\right)$
$\displaystyle\vec{z}_{5}=-\left(\frac{-5-\sqrt{5}}{10},-\sqrt{\frac{5-\sqrt{5}}{10}},\frac{1}{\sqrt{5}}\right)$
$\displaystyle\vec{z}_{6}=-\left(\frac{5-\sqrt{5}}{10},-\sqrt{\frac{5+\sqrt{5}}{10}},\frac{1}{\sqrt{5}}\right)$

Note $\vec{x}_{i}=(0,1,0)\times\vec{z}_{i}$ and
$\vec{y}_{i}=\vec{z}_{i}\times\vec{x}_{i}$. The other six coordinates are
obtained by inversion symmetry. Specifically, the local $z$ and $x$ axes of
$i\leftrightarrow 13-i$ sites are related by the inversion transformation.
Note that $\vec{z}_{i}$ is the position vector of the site from the center of
the icosahedron shell.

The symmetry allowed spin Hamiltonian with respect to the local coordinate
axes is obtained by applying two mirror reflection symmetries, $M_{1}$ and
$M_{2}$ whose normal vectors of the mirror planes are $\vec{n}_{1}=(0,1,0)$
and $\vec{n}_{2}=(1,0,\varphi)/\sqrt{1+\varphi^{2}}$. Since $\Sigma^{\mu}$,
where $\mu=x,y,z$ are the pseudospin operators which are the rank 3 tensor
operators, the unitary representation of the reflection transformation is
given by $\pi$-rotation around these normal vectors of the mirror planes for
angular momentum $j=3$, octupoles. Considering $U(M_{1(2)})$ as the unitary
representation of the mirror reflection, $M_{1(2)}$, then the $T_{m}^{(3)}$
operators transform under the mirror reflections $M_{1(2)}$ as in Eq.(S6).

$\displaystyle T_{m}^{(3)}\to
U(M_{1(2)})T_{m}^{(3)}U^{\dagger}(M_{1(2)})=\sum_{m^{\prime}}D_{1(2),m^{\prime}m}^{(3)}T_{m^{\prime}}^{(3)}$
(S6)

where $D_{1(2),m^{\prime}m}$ are the Wigner D-matrix elements for the angular
momentum $j$. Each of the spherical tensor operator is represented as the
pseudospin operators in the Kramers doublet, as discussed in the main text. To
be more specific, the Wigner D-matrices are given by

$\displaystyle D_{1,m^{\prime}m}^{(3)}=\begin{pmatrix}0&0&0&0&0&0&1\\\
0&0&0&0&0&-1&0\\\ 0&0&0&0&1&0&0\\\ 0&0&0&-1&0&0&0\\\ 0&0&1&0&0&0&0\\\
0&-1&0&0&0&0&0\\\ 1&0&0&0&0&0&0\end{pmatrix}$ (S7) $\displaystyle
D_{2,m^{\prime}m}^{(3)}=\begin{pmatrix}\frac{-5-2\sqrt{5}}{25}&\frac{1}{5}\sqrt{\frac{3}{5}(7+3\sqrt{5})}&-\frac{1}{5}\sqrt{\frac{3}{2}(3+\sqrt{5})}&\frac{2}{5}&-\frac{1}{5}\sqrt{\frac{3}{2}(3-\sqrt{5})}&\frac{1}{5}\sqrt{\frac{3}{5}(7-3\sqrt{5})}&\frac{-5+2\sqrt{5}}{25}\\\
\frac{1}{5}\sqrt{\frac{3}{5}(7+3\sqrt{5})}&\frac{-15-\sqrt{5}}{50}&-\frac{1}{5}\sqrt{3-\sqrt{5}}&\frac{\sqrt{6}}{5}&-\frac{1}{5}\sqrt{3+\sqrt{5}}&\frac{15-\sqrt{5}}{50}&\frac{1}{25}\sqrt{\frac{3}{2}}(5-3\sqrt{5})\\\
-\frac{1}{5}\sqrt{\frac{3}{2}(3+\sqrt{5})}&-\frac{1}{5}\sqrt{3-\sqrt{5}}&\frac{1}{\sqrt{5}}&0&-\frac{1}{\sqrt{5}}&\frac{\sqrt{3+\sqrt{5}}}{5}&-\frac{1}{5}\sqrt{\frac{3}{2}(3-\sqrt{5})}\\\
\frac{2}{5}&\frac{\sqrt{6}}{5}&0&-\frac{1}{\sqrt{5}}&0&\frac{\sqrt{6}}{5}&-\frac{2}{5}\\\
-\frac{1}{5}\sqrt{\frac{3}{2}(3-\sqrt{5})}&-\frac{1}{5}\sqrt{3+\sqrt{5}}&-\frac{1}{\sqrt{5}}&0&\frac{1}{\sqrt{5}}&\frac{\sqrt{3-\sqrt{5}}}{5}&-\frac{1}{5}\sqrt{\frac{3}{2}(3+\sqrt{5})}\\\
\frac{1}{5}\sqrt{\frac{3}{5}(7-3\sqrt{5})}&\frac{15-\sqrt{5}}{50}&\frac{\sqrt{3+\sqrt{5}}}{5}&\frac{\sqrt{6}}{5}&\frac{\sqrt{3-\sqrt{5}}}{5}&\frac{1}{50}(-15-\sqrt{5})&-\frac{1}{5}\sqrt{\frac{3}{5}(7+3\sqrt{5})}\\\
\frac{1}{25}(-5+2\sqrt{5})&\frac{1}{25}\sqrt{\frac{3}{2}}(5-3\sqrt{5})&-\frac{1}{5}\sqrt{\frac{3}{2}(3-\sqrt{5})}&-\frac{2}{5}&-\frac{1}{5}\sqrt{\frac{3}{2}(3+\sqrt{5})}&-\frac{1}{5}\sqrt{\frac{3}{5}(7+3\sqrt{5})}&\frac{1}{25}(-5-2\sqrt{5})\end{pmatrix}$
(S8)

Note that the Hermitian and time-reversal invariant Hamiltonian possesses 5
independent parameters. Under the mirror reflection symmetry, $M_{1}$ given by
Eqs.(S6) and (S7), we have $\Sigma^{\pm}_{4(5)}\to-\Sigma^{\mp}_{5(4)}$ and
$\Sigma^{z}_{4(5)}\to-\Sigma^{z}_{5(4)}$, where the subscripts 4 and 5 are the
site indices (Refer to Fig.S1). Hence, there are only four independent
parameters, $J_{\pm\pm},J_{z\pm},J_{\pm}$ and $J_{zz}$ for the Hamiltonian
invariant under the mirror reflection $M_{1}$. On the other hand, we note that
Eq.(S8) does not give any further constraints on the parameters of the
Hamiltonian.

Under the 5-fold rotation, the pseudospin operators transform as
$\Sigma_{\pm}^{(i)}\to e^{\mp 4\pi/5}\Sigma_{\pm}^{(j)}$. Thus, the bond
dependent phase factors are added in order to make the Hamiltonian invariant
under the 5-fold rotations. Let $\alpha_{ij}$ and $\beta_{ij}$ be the matrix
of the additional phase factor for the interaction term between the $i$-th
site and the $j$-th site[gaudet2019quantum, patri2020theory] (See the main
text for the definition of $\alpha_{ij}$ and $\beta_{ij}$). Fig.S1 shows the
five types of nearest neighbor bond orientations, red, green, blue, cyan, and
black. Referring to the indices in Fig.S1, the bond-orientation dependent
phase factor are given as,

$\displaystyle\alpha_{ij}=1,$ $\displaystyle\mbox{ if }(i,j)\mbox{ or
}(j,i)\in\\{(1,2),(4,5),(3,7),(6,10),(8,9),(11,12)\\},$ (S9)
$\displaystyle\alpha_{ij}=e^{i2\pi/5},$ $\displaystyle\mbox{ if }(i,j)\mbox{
or }(j,i)\in\\{(2,3),(4,7),(1,5),(6.9),(10,11),(8,12)\\},$
$\displaystyle\alpha_{ij}=e^{i4\pi/5},$ $\displaystyle\mbox{ if }(i,j)\mbox{
or }(j,i)\in\\{(1,6),(3,4),(2,8),(5,11),(9,10),(7,12)\\},$
$\displaystyle\alpha_{ij}=e^{-i4\pi/5},$ $\displaystyle\mbox{ if }(i,j)\mbox{
or }(j,i)\in\\{(1,3),(5,6),(7,8),(4,11),(2,9),(10,12)\\},$
$\displaystyle\alpha_{ij}=e^{-i2\pi/5},$ $\displaystyle\mbox{ if }(i,j)\mbox{
or }(j,i)\in\\{(1,4),(3,8),(7,11),(9,12),(5,10),(2,6)\\}.$

For a given $\alpha_{ij}$, $\beta_{ij}=(\alpha_{ij}^{*})^{2}$.

Figure S1: Five different bond orientations allow different phase factors. The
red, green, blue, cyan and black colored nearest neighbor bonds have different
phase factor matrix elements, $\alpha_{ij}$. The numbering of the icosahedron
vertices indexes the sites shown as yellow spheres. Two mirror planes, $M_{1}$
and $M_{2}$, are shown as gray dashed line and orange plane, respectively.
Note that $M_{1}$ swaps the sites 4 and 5, which are invariant under the
$M_{2}$ reflection.

##  3 Icosahedron quasicrystal derived from cut-and-project scheme

The cut-and-project scheme for constructing the icosahedral quasicrystal is
introduced in this section[kellendonk2015mathematics, jeon2021discovery]. The
icosahedral point group symmetry is compatible in 6D space. Therefore, one can
construct the icosahedral quasicrystal in 3D space as the lattice points
descended from the 6D hypercubic lattice. In detail, let us consider the 6D
hypercubic lattice
$\mathcal{L}=\\{\bm{x}|\bm{x}=m_{i}\bm{e}_{i},m_{i}\in\mathbb{Z},1\leq i\leq
6\\}$, where $\bm{e}_{i}$ is a standard unit vector. The 6D space is
decomposed by two projection maps $\bm{\pi}$ and $\bm{\pi}^{\perp}$. Each of
them projects the lattice points in $\mathcal{L}$ onto the subspace of the
quasicrystal (physical space) and its orthogonal complement subspace
(perpendicular space), respectively. To produce the nontrivial
quasicrystalline pattern, the physical space should have an irrational angle
to the lattice surface. However, for such an irrational angle, the images of
the projection of whole lattice points, $\bm{\pi}(\mathcal{L})$, densely cover
the physical space. This violates the uniform discreteness of the definition
of quasicrystals. So one should choose a subset of $\mathcal{L}$, which is the
(relatively) compact subset of the perpendicular space $\bm{\pi}^{\perp}$ and
is often called the window, e.g. $K$. Only if the image of $\bm{\pi}^{\perp}$
belongs to the window $K$, we project the lattice points. The resulting
projection image in physical space is the discrete quasicrystalline lattice
structure. As a standard choice of the window,
$K=\bm{\pi}^{\perp}(\mathcal{W}(0))$, where $\mathcal{W}(0)$ is the Wigner-
Seitz cell of the origin.

$\displaystyle\bm{\pi}$ $\displaystyle=\begin{pmatrix}0&\tau&-1&0&1&-\tau\\\
\tau&-1&0&-\tau&0&-1\\\ 1&0&-\tau&1&-\tau&0\end{pmatrix}$ (S10)
$\displaystyle\bm{\pi}^{\perp}$
$\displaystyle=\frac{1}{\sqrt{2+\tau}}\begin{pmatrix}0&-1&-\tau&0&\tau&1\\\
1&\tau&0&-1&0&\tau\\\ \tau&0&1&\tau&1&0\end{pmatrix}.$ (S11)

Then, the window $K$ becomes an icosahedron whose vertices are
$\bm{\pi}^{\perp}$-projection images of the vertices of the Wigner-Seitz cell
(See Fig.S2 (a).). The icosahedral quasicrystal (See Fig.S2 (b)) is given by
the $\bm{\pi}$-projection of 6D hyper-cubic lattice points, whose image of
$\bm{\pi}^{\perp}$ belongs to the window $K$ in Fig.S2 (a).

Figure S2: (a) Icosahedron window, $K$ of the cut and project scheme. (b)
Icosahedral quasicrystal. The sites forming the icosahedron are shown only to
avoid confusion. Red balls indicate the lattice sites. Nearest neighboring
sites are connected by blue lines.

##  4 Possible intershell magnetic orderings and frustrations

This section shows an example of possible intershell magnetic states resulting
from the intershell interaction terms. For each pair of nearest neighboring
icosahedral shells, there are two pairs of nearest neighboring sites (see Fig.
S3). Fig.S3 shows an example of possible long-range intershell magnetic states
regarding the octupolar degrees of freedom. In particular, Fig.S3 (a) shows
the octupolar magnetic state when $J_{zz}$ is ferromagnetic for both the
intra- and the intershell sites. Every icosahedron is ordered as one of the
two ground states of the Ising model. On the other hand, Fig.S3 (b) shows the
octupolar magnetic state when $J_{zz}$ is ferromagnetic for the intrashell
sites but antiferromagnetic for the intershell sites. This gives rise to the
inter-shell geometric frustration. Note that the orange dashed lines
representing inter-shell interactions form the triangles leading to intershell
geometric frustration. Specifically, 12 icosahedron shells sit on the vertices
of the inflated icosahedron shape in the icosahedral quasicrystal. Therefore,
considering $\ket{FM_{\pm}}$ as two ferromagnetic Ising ground states of a
single icosahedron shell, the antiferromagnetic interaction between the shells
results in an antiferromagnetic Ising order on the inflated icosahedron in
terms of $\ket{FM_{\pm}}$. This is how the geometrical frustration is created
in the enlarged spatial scale in the icosahedral quasicrystal. Although we
have given the two particular examples, depending on the intra-shell distances
and their magnetic exchange couplings, it can stabilize different magnetic
ground state case by case.

Figure S3: Examples of octupolar magnetic states in the presence of intershell
interactions $J_{zz}$, where (a) both intra- and intershells are
ferromagnetic, and (b) the intra-shell is ferromagnetic but the intershell is
antiferromagnetic. The bonds between the nearest neighbors in the shell of an
icosahedron are shown as blue lines. In (b), the nearest neighbor bonds
between two adjacent icosahedron shells are shown as orange dashed lines. (c)
Orthographic projection of (b) to emphasize that three icosahedron shells
interact antiferromagnetically, leading to geometrical frustration.

##  5 Derivation of the unique ground state for $|J_{\pm}|\ll J_{zz}$

Here, we perturbatively derive the unique ground states emergent by small
$J_{\pm}$ compared to $J_{zz}$. We apply the degenerate perturbation theory
for 72-fold degenerate ground states for $J_{zz}>0$. Note that based on the
symmetry ground, we can classify the 72 states into three groups,
$\ket{\psi_{A_{n}}},\ket{\psi_{B_{l}}}$ and $\ket{\psi_{C_{r}}}$, where $1\leq
n,l\leq 30$, while $1\leq r\leq 12$. Figs.S4 (a,b,c) show the representative
states in each group, $A,B,C$, respectively. Each group is generated by
applying orientation-preserving icosahedral symmetry operations to these
states in Figs.S4 (a,b) and (c), respectively. Every orientation-preserving
operations are in the maximal normal subgroup of the full icosahedral symmetry
group, $I\trianglelefteq I_{h}$. Note $I$ is called as the icosahedral
rotation group of order 60[cornwell1997group].

Figure S4: Classification of the 72 degenerate ground states on an icosahedron
shell. (a,b) 60 degenerate states without 5-fold symmetry. The states admit
the 2-fold symmetry operation, which transforms the $Z$ axis into the
$Z^{\prime}$ axis. (c) 12 degenerate states with 5-fold rotational symmetry
around the $Z$ axis. The inset shows the view along the $Z$ axis to emphasise
the 5-fold symmetry. (d) Top view along the $Z$ axis of the state shown in
(a). There is a mirror plane, shown as $\sigma$, which includes the $Z$ axis
and the site P, and which interchanges the octupoles at the sites Q and R,
only under the mirror reflection of the site positions. As a result, the state
shown in (a) is transformed into (b) and vice versa.

First, the states shown in Figs.S4 (a) and (b) represent the states without
5-fold rotational symmetry. Instead, they have another orientation-preserving
icosahedral symmetry, which maps the $Z$ axis to the $Z^{\prime}$ axis in
Figs.S4 (a) and (b) respectively. Since the icosahedral rotation group has the
order, $|I|=60$, for each state in Figs.S4 (a) and (b), we have 30 degenerate
states generated by $I$, respectively. We denote them as $\ket{\psi_{A_{n}}}$
and $\ket{\psi_{B_{l}}}$, where $1\leq n,l\leq 30$. Note that the states in
Fig.S4 (a) and (b) are related by the mirror reflection of the sites with
respect to the $\sigma$ plane depicted in Fig.S4 (d). Here, $\sigma$ plane
contains the $Z$ axis and the site P. This mirror reflection, $M_{\sigma}$
with respect to the $\sigma$-plane only exchanges the octupoles on the sites Q
and R. Hence, the states, $\ket{\psi_{B_{l}}}$ are also obtained by applying
the coset $IM_{\sigma}=\\{gM_{\sigma}|g\in I\\}$ to the state in Fig.S4 (a).

On the other hand, $\ket{\psi_{C_{r}}}$ are 12 degenerate states with 5-fold
rotational symmetry (Refer to Fig.S4 (c) and the inset representing the
viewpoint along the rotational symmetry axis, $Z$ axis.). Since the state
itself has 5-fold rotational symmetry, we have only 12 distinct orientation
preserving transformations which give rise to the degenerate states in the
group $C$. Hence, we may let the states in the group $C$ as
$\ket{\psi_{C_{r}}}$, where $1\leq r\leq 12$.

Now, let us apply perturbative method to investigate the quantum fluctuation
based on above groups. Take

$\displaystyle
H_{\pm}=J_{\pm}\sum_{\braket{i,j}}(\Sigma_{i}^{+}\Sigma_{j}^{-}+\Sigma_{i}^{-}\Sigma_{j}^{+}).$
(S12)

Then, we have

$\displaystyle\braket{\psi_{A_{n}}}{H_{\pm}}{\psi_{A_{m}}}=\braket{\psi_{B_{l}}}{H_{\pm}}{\psi_{B_{k}}}=\braket{\psi_{C_{r}}}{H_{\pm}}{\psi_{C_{s}}}=0,$
(S13)

where $1\leq n,m,l,k\leq 30,1\leq r,s\leq 12$. This is because $H_{\pm}$ has
zero matrix element between two states related by the orientation preserving
transformation. Thus, one can write the block off-diagonal matrix
representation of $H_{\pm}$ for the 72-fold degenerate states, say
$[H_{\pm}]_{A,B,C}$ as

$\displaystyle[H_{\pm}]_{A,B,C}=\begin{pmatrix}0&T_{AB}&T_{AC}\\\
T_{BA}&0&T_{BC}\\\ T_{CA}&T_{CB}&0\end{pmatrix}$ (S14)

Here, the subscripts, $A,B$ and $C$ stand for the each groups.
$T_{BA}=T_{AB}^{\dagger}$ is a $30\times 30$ matrix, while
$T_{AC}=T_{CA}^{\dagger}$ and $T_{BC}=T_{CB}^{\dagger}$ are $30\times 12$
matrices. Here, each non-zero matrix element is $J_{\pm}$. We find that for
each state in $A(B)$, there are 4, and 2 different states in $B(A)$ and $C$,
respectively such that the matrix elements of $[H_{\pm}]_{A,B,C}$ is
$J_{\pm}$. On the other hand, for each state in $C$, there are 5 different
states in $A$ and $B$, respectively, and hence in total 10 states such that
the matrix elements of $[H_{\pm}]_{A,B,C}$ is $J_{\pm}$.

One can use the graph, $G$ in Fig.S5 to examine above facts. Here, the nodes
of the graph $G$ represents to each state (red circle, green pentagram, and
blue square represent the states in $A,B$ and $C$, respectively), and two
nodes are connected by an edge if they admit nonzero matrix element of
$[H_{\pm}]_{A,B,C}$. Thus, $[H_{\pm}]_{A,B,C}/J_{\pm}$ is the adjacency matrix
of the graph $G$[west2001introduction]. The graph $G$ is the decorated
pentakis icosidodecahedron (See Fig.S5.). The pentakis icosidodecahedron
(Fig.S5 (a)) has 42 vertices in two different types depending on the local
shape of the vertex, which are called pentagonal and hexagonal sites whose
connectivity is 5 and 6, respectively[bisztriczky2012polytopes]. Here, on the
other hand, we term by the decorated pentakis icosidodecahedron, whose
hexagonal sites are doubly occupied representing the states in $A$ (red
circles) and $B$ (green pentagram), while the pentagonal sites of the pentakis
icosidodecahedron are representing the states in $C$ (blue square) (See Fig.S5
(b).). Hence, in the graph $G$, each node for the states in $A(B)$ are
connected to the four nodes for the states in $B(A)$, and two nodes for the
states in $C$, while each node for the states in $C$ are connected to the five
nodes for the states in $A$ and $B$, respectively.

Figure S5: (a) Undecorated pentakis icosidodecahedron. There are 42 vertices,
120 edges, and 80 triangular faces. The 42 verticies are classified according
to their two kinds of connectivity, 5 and 6. Each vertex having 5 (6)
connectivity is called pentagonal (hexagonal) sites,
respectively[bisztriczky2012polytopes]. (b) The graph whose adjacency matrix
is $[H_{\pm}]_{A,B,C}/J_{\pm}$. We term this graph as decorated pentakis
icosidodecahedron comprised of 72 nodes which represent the 72-fold degenerate
antiferromagnetic ground states. The pentagonal vertices of the undecorated
pentakis icosidodecahedron are doubly occupied by the states in $A$ and $B$
groups. If two states admit nonzero matrix element of $H_{\pm}$, then we have
an edge depicted by the gray lines between two nodes representing each states.
Red square, green pentagram, blue squares which have 6,6 and 10
connectivities, respectively represent the states in $A,B$ and $C$. The
numberings of the nodes are the numberings of the state, where $1$ to $30$ are
the states in $A$, $31$ to $60$ are the states in $B$, and $61$ to $72$ are
the states in $C$.

It allows us to write the general form of the ground state, $\ket{GS}$, as

$\displaystyle\ket{GS}=a\sum_{n=1}^{30}\ket{\psi_{A_{n}}}+b\sum_{l=1}^{30}\ket{\psi_{B_{l}}}+c\sum_{r=1}^{12}\ket{\psi_{C_{r}}},$
(S15)

where we have three real coefficients, $a,b$ and $c$ for
$\ket{\psi_{A_{n}}},\ket{\psi_{B_{l}}}$ and $\ket{\psi_{C_{r}}}$,
respectively. Note that $[H_{\pm}]_{A,B,C}$ is a real symmetry matrix, so we
can assume without loss of generality that $a,b,c$ are real. Then we have,

$\displaystyle E(a,b,c)=\braket{GS}{H_{\pm}}{GS}=240ab+120ac+120bc.$ (S16)

In addition, we have the normalization condition
$N(a,b,c)=30a^{2}+30b^{2}+12c^{2}=1$.

The critical point is found by equating,

$\displaystyle\nabla E(a,b,c)=\lambda\nabla N(a,b,c),$ (S17)

where $\lambda$ is the Lagrange multiplier. For $a=b=(1+\sqrt{6})c/5$,
$E(a,b,c)$ is maximized, while for $a=-b$ and $c=0$, $E(a,b,c)$ is minimized.
Each case corresponds to the unique ground state for the ferromagnetic and
antiferromagnetic $J_{\pm}$. Remarkably, there is no degenerate ground state
in either case. Thus, any small quantum fluctuation given by $H_{\pm}$
completely eliminates the degeneracy, by superposing 72 ground states of the
antiferromagnetic Ising model.

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