# Multifractality and hyperuniformity in quasicrystalline Bose-Hubbard models
with and without disorder

Masahiro Hori1<EMAIL_ADDRESS>Takanori Sugimoto2 Yoichiro Hashizume3 and
Takami Tohyama1 1Department of Applied Physics1Department of Applied Physics
Tokyo University of Science Tokyo University of Science Katsushika Katsushika
Tokyo 125-8585 Tokyo 125-8585 Japan
2Center for Quantum Information and Quantum Biology Japan
2Center for Quantum Information and Quantum Biology Osaka University Osaka
University Osaka 560-0043 Osaka 560-0043 Japan
3Institute of Arts and Sciences Japan
3Institute of Arts and Sciences Oshamambe Division Oshamambe Division Tokyo
University of Science Tokyo University of Science Hokkaido 049-3514 Hokkaido
049-3514 Japan Japan

###### Abstract

Clarifying similarities and differences in physical properties between
crystalline and quasicrystalline systems is one of central issues in studying
quasicrystals. To contribute to this, we apply multifractal and hyperuniform
analyses to nonuniform spatial patterns in the Bose-Hubbard model on the
Penrose and Ammann-Beenker tilings. Based on the mean-field approximation, we
obtain real-space distributions of boson density and bosonic condensate. In
both Mott insulating and superfluid phases, the distributions are
hyperuniform. Analyzing the order metric that quantifies the complexity of
nonuniform spatial patterns, we find that both quasicrystals show a divergence
of the order metric at a phase boundary between the Mott insulating and
superfluid phases, in stark contrast to the case of a periodic square lattice.
Our results suggest that hyperuniformity is a useful concept to differentiate
between crystalline and quasicrystalline bosonic systems. Moreover, we
introduce on-site random potentials into these quasicrystalline Bose-Hubbard
models, leading to a Bose glass phase. Contrary to the Mott insulating and
superfluid phases, we find that the Bose glass phase is multifractal. The same
multifractality appears on a Bose glass phase in the periodic square lattice.
Therefore, multifractality is common in a Bose glass phase irrespective of the
periodicity of systems.

##  1 Introduction

A quasicrystal retains a structure that is not periodic. Thus spatial
distribution of physical quantities in quasicrystals is nonuniform unlike
periodic crystals. A quasicrystal also has a structure that is not random but
ordered, since the vertices of tiles forming a quasicrystalline pattern have a
long-range order with sharp Bragg peaks. As a result, the spatial pattern of
physical quantities in quasicrystals is neither random nor uniform [1, 2, 3,
4]. Multifractal and hyperuniform analyses have been proposed in
quantitatively characterizing the spatial distribution of vertices [5, 6, 7]
and nonuniform spatial patterns of physical quantities including electronic
states in quasicrystals and crystals [8, 9, 10, 11, 12, 13]. So far, it is
known that the spatial distribution of lattice points in crystals and vertices
in most quasicrystals is hyperuniform but that in random systems is
multifractal.

In contrast to states in fermion systems, multifractality and hyperuniformity
in boson systems have not been elucidated, although bosonic quasicrystalline
systems have been examined experimentally [14, 15, 16, 17] and theoretically
[18, 19, 20] by using ultra-cold atoms in optical quasicrystals. Such boson
systems can be described by a Bose-Hubbard model, which exhibits Mott
insulating phase (superfluid phase) when on-site Coulomb interaction (hopping
interaction) is dominant [18]. It has not been clarified whether spatial
distribution of physical quantities in those phases exhibits either
hyperuniform or multifractal.

Very recently, a Bose glass phase [21] has been observed in an optical
quasicrystal [15, 17]. In the Bose glass phase, spatial correlation in
physical quantities is weak and almost random. If randomness would determine
spatial properties, this Bose glass phase might show a multifractal feature
similar to a fermionic random system [22, 23]. If so, a transition between
states to be hyperuniform and multifractal may be possible by controlling
disorder in a bosonic quasicrystalline system.

In this paper, we apply the multifractal and hyperuniform analyses to physical
quantities in Bose-Hubbard models with and without disorder to clarify
similarities and differences between crystalline and quasicrystalline systems.
We find that the system without disorder is hyperuniform in both crystalline
and quasicrystalline systems. The order metric, which quantifies the
complexity of the system, becomes large near a phase boundary between the Mott
and superfluid phases only in quasicrystals. The origin of this enhancement in
quasicrystals is attributed to a unique feature of superfluid phases where
spatial patterns of physical quantities are governed by quasiperiodicity. With
disorder, we find a hyperuniform Mott insulating phase, a multifractal Bose
glass phase, and a hyperuniform superfluid phase. The presence of disorder
makes it difficult to clarify differences between crystalline and
quasicrystalline systems.

This paper is organized as follows. We introduce a mean-field Bose-Hubbard
model, perpendicular space, multifractality, hyperuniformity, replica overlap
for Bose glass, and the inverse participation ratio (IPR) in Sec. 2. In Sec.
3, we perform multifractal and hyperuniform analyses for physical quantities
without disorder. The effect of disorder is discussed in Sec. 4. Finally, we
give a summary of this work in Sec. 5.

##  2 Model and Method

###  2.1 Mean-field Bose-Hubbard model

We consider a Bose-Hubbard model with an on-site disorder potential. In
quasicrystals, we use a vertex model, where the links connecting vertices are
of the same length and hopping of bosons is along the links. A mean-field
Bose-Hubbard model reads [18]

$\displaystyle H_{\rm MF}=\sum_{i}H_{i}+E_{0},$ (1)

where a local Hamiltonian at an $i$th vertex is given by

$\displaystyle H_{i}=-J(\psi^{*}_{i}\hat{b}_{i}+{\rm
H.c.})-(\mu-\epsilon_{i})\hat{n}_{i}+\frac{U}{2}\hat{n}_{i}(\hat{n}_{i}-1),$
(2)

and a constant energy $E_{0}$ is given by

$\displaystyle E_{0}=J\sum_{i}\psi_{i}^{*}\braket{\hat{b}_{i}}.$ (3)

Here, $\hat{b}^{\dagger}_{i}$ ($\hat{b}_{i}$) is the creation (annihilation)
operator of a boson at the $i$th vertex, $\braket{\hat{b}_{i}}$ is the local
superfluid amplitude, $\psi_{i}=\sum_{j_{i}}\braket{\hat{b}_{j_{i}}}$ where
$j_{i}$ represents a vertex connected to the $i$th vertex by a link with
hopping $J$, $\mu$ is the chemical potential, $\epsilon_{i}$ represents the
vertex-dependent random on-site potential,
$\hat{n}_{i}=\hat{b}^{\dagger}_{i}\hat{b}_{i}$, and $U$ denotes an on-site
Coulomb interaction. In the following, we set $U$ as an energy unit, and we
take $\mu/U=0.5$ and 1.0.

We perform self-consistent calculations [24, 25, 18] for Eq. (2) to obtain
$\braket{\hat{b}_{i}}$ and $\braket{\hat{n}_{i}}$. Using an initial guess
$\psi_{i}$ and diagonalizing the local Hamiltonian (2), we obtain a wave
function of the ground state and calculate $\braket{\hat{n}_{i}}$ and
$\braket{\hat{b}_{i}}$ in the local Hilbert space that satisfies
$\braket{\hat{n}_{i}}\leq n_{b}$ bosons. We repeatedly calculate $\psi_{i}$ by
using the obtained $\braket{\hat{n}_{i}}$ and $\braket{\hat{b}_{i}}$ until the
calculation reaches its convergence. In the following, we set $n_{b}=10$.

Some of the present authors have shown that our model in Eq. (1) without
disorder, namely $\epsilon_{i}=0$, realizes the Mott insulating and superfluid
phases not only in crystals but also in quasicrystals [18]. We note that
$\braket{\hat{b}_{i}}$ is zero at all vertices (nonzero at some vertices) in
the Mott insulating (superfluid) phase.

###  2.2 Perpendicular space

We consider the Bose-Hubbard model (2) on a Penrose tiling that is a covering
of the plane by non-overlapping rhombi as shown in Fig. 1(a). Since a Penrose
tiling forms a quasicrystalline structure, we call structures formed by a
Penrose titling Penrose quasicrystals in this paper. The vertices of these
quasicrystals are obtained by projection from a five-dimensional hypercubic
lattice onto a two-dimensional plane. For the direction perpendicular to this
projection, we consider another projection from a five-dimensional hypercubic
lattice onto a three-dimensional plane, which results in the perpendicular-
space representation as shown in Fig. 1(b). Any vertex is located at one of
the four pentagonal planes at $\tilde{z}=0,$ $1/\sqrt{2},$ $2/\sqrt{2},$ and
$3/\sqrt{2}$. We show two of the four planes for $\tilde{z}=1/\sqrt{2}$ (left
panel) and 0 (right panel) in Fig. 1(c), where the colors distinguish a
coordination number $Z_{i}$ defined as the number of vertices that are
connected with the $i$th vertex along links. The black lines separate vertices
with different $Z_{i}$, resulting in the formation of star shapes in the
pentagonal planes. The average of $Z_{i}$ for the entire system is
$\overline{Z_{i}}=4$ in Penrose quasicrystals, which agrees with that of
square lattice. Note that the pentagonal planes at $\tilde{z}=3/\sqrt{2}$ and
$\tilde{z}=2/\sqrt{2}$ are identical to the planes at $\tilde{z}=0$ and
$\tilde{z}=1/\sqrt{2}$, respectively, with reversing them upside down. This
allows us to focus on only two planes at $\tilde{z}=0$ and $1/\sqrt{2}$.

We also consider the Bose-Hubbard model (2) on an Ammann-Beenker tiling as
shown in Fig. 1(d). We call structures with an Ammann-Beenker tiling Ammann-
Beenker quasicrystals, whose corresponding hypercubic lattice is in a four-
dimensional space. As shown in Fig. 1(e), its perpendicular-space
representation forms a two-dimensional octagon. Note that the coordination
number $Z_{i}$ increases with approaching the center of the octagonal plane.
The averaged coordination number is the same as Penrose quasicrystals, namely
$\overline{Z_{i}}=4$.

We can examine how quasiperiodicity affects the spatial distribution of a
physical quantity by plotting it in the perpendicular-space representation. If
the distribution is similar to Fig. 1(c) [1(e)] in Penrose [Ammann-Beenker]
quasicrystals, one can conclude that the physical quantity is governed by
local environment of vertices that produces quasiperiodicity. If not, one can
conclude that the physical quantity is not significantly affected by
quasiperiodicity.

In the following, we use the total number of vertices $N=75806$ (10th
generation) and $N=65281$ (6th generation) for Penrose and Ammann-Beenker
quasicrystals, respectively (see Appendix). For comparison, we also use square
lattice with $N=90000$. We use open boundary conditions.

Figure 1:  (a) The real-space representation of Penrose quasicrystals (3rd
generation, $N=86$). (b) The perpendicular-space representation of Penrose
quasicrystals. (c) Coordination number for $\tilde{z}=1/\sqrt{2}$ and 0. (d)
Same as (a) but for Ammann-Beenker quasicrystals (2nd generation, $N=65$). (e)
Same as (c) but for Ammann-Beenker quasicrystals.

###  2.3 Multifractality

For clarifying the effect of quasiperiodicity on physical quantities such as
$\braket{\hat{b}_{i}}$ or $\braket{\hat{n}_{i}}$, we employ the multifractal
analysis [5]. The multifractal analysis is one of the methods to quantify the
fractal property of a nonuniform spatial pattern, which has been employed also
in quasicrystals [26, 12].

The definition of multifractality depends on a spatial dimension. In this
paper, we concentrate on the multifractality in two dimensions. We introduce a
circular window of radius $R$ and consider a $R$-dependent quantity given by

$\displaystyle
D(\\{n_{i}\\},q,R)=\frac{1}{1-q}\frac{\ln{\frac{\sum_{i}n_{i}^{q}}{(\sum_{i}n_{i})^{q}}}}{\ln
N_{R}},$ (4)

where $\\{n_{i}\\}$ is the distribution of physical quantity $n_{i}$ and the
summation of $i$ is taken for all vertices within a circular window of radius
$R$, $N_{R}$ is the number of vertices within the circular window, and $q$ is
a real variable. By taking a limit $R\rightarrow\infty$ for Eq. (4), the
$q$-dependent multifractal dimension is defined as

$\displaystyle
D^{\infty}(\\{n_{i}\\},q)=\lim_{R\rightarrow\infty}D(\\{n_{i}\\},q,R).$ (5)

This dimension for a negatively (positively) large $q$ represents a fractal
property of larger (smaller) values of $n_{i}$.

A given system is multifractal in terms of $\\{n_{i}\\}$ if and only if
$D^{\infty}(\\{n_{i}\\},q)\neq 2$ for a certain $q$. In general,
$D^{\infty}(\\{n_{i}\\},q)$ is a decreasing function of $q$, and thus it is
enough to calculate $D^{\infty}(\\{n_{i}\\},q)$ with a large $q$ and a small
$q$ to judge if a system is multifractal or not. In the following, we
concentrate on $q=\pm 10$. We define multifractalness as

$\begin{split}&{\rm multifractalness~{}of~{}}\\{n_{i}\\}\\\
&=D^{\infty}(\\{n_{i}\\},q=-10)-D^{\infty}(\\{n_{i}\\},q=10).\end{split}$ (6)

This quantifies a tendency to be multifractal for a given system. That is,
when multifractalness is larger than 0, the system is multifractal.

###  2.4 Hyperuniformity

Hyperuniform analysis [6, 9, 11] is also one of the methods to quantify a
property of a nonuniform spatial pattern. Hyperuniform analysis has been used
for the distribution of physical quantities in quasicrystals [26]. In the
following, we explain a hyperuniformity in two-dimensional systems. First, the
sum of a physical quantity $n_{i}$ inside a circular window of radius $R$ is
given by

$\displaystyle N(R)=\sum_{i}n_{i}\Theta(R-|\bm{r}_{i}-\bm{r}_{c}|),$ (7)

where $\Theta(\bullet)$ is the Heaviside step function, $\bm{r}_{i}$ is a
position of the $i$th vertex, and $\bm{r}_{c}$ represents the center of the
circular window. The variance of $N(R)$ with respect to the random sampling of
$\bm{r}_{c}$ is given by

$\displaystyle\sigma(R)^{2}=\left[N(R)^{2}\right]_{\bm{r}_{c}}-\left(\left[N(R)\right]_{\bm{r}_{c}}\right)^{2},$
(8)

where $\left[\bullet\right]_{\bm{r}_{c}}$ denotes the average of $\bullet$
with respect to the random sampling of $\bm{r}_{c}$.

If the variance $\sigma(R)^{2}$ scales slower than the area of the window, a
given system is hyperuniform in terms of $\\{n_{i}\\}$: the condition to be
hyperuniform is that, defining

$\displaystyle A(R)=\frac{\sigma(R)^{2}}{R^{2}},$ (9)

$\lim_{R\rightarrow\infty}A(R)=0$. In general, there are three classes of
hyperuniformity [6] that are classified according to the $R$ dependence of
$\sigma(R)^{2}$ in two dimensions as

$\displaystyle\sigma(R)^{2}\sim\left\\{\begin{array}[]{lll}R&{\rm
Class~{}I}\\\ R\ln R&{\rm Class~{}II}\\\ R^{2-\alpha}(0<\alpha<1)&{\rm
Class~{}III}.\\\ \end{array}\right.$ (13)

For the point distribution where $n_{i}=1$, the system is Class-I hyperuniform
in all crystals and most quasicrystals [6].

When a system is Class-I hyperuniform, a proportionality coefficient in front
of $R$ in Eq. (13), called order metric, controls the complexity of the
distribution of $n_{i}$. Defining

$B(R)=\frac{\sigma(R)^{2}}{R},$ (14)

we numerically obtain the order metric as an average of $B(R)$ at large
distance. The order metrics of the point distribution in most quasicrystals
are larger than those in most crystals, due to the complex structures of
quasicrystals.

In most cases, a quasicrystalline system is either multifractal or
hyperuniform for each physical quantity. Note that multifractal systems are
never hyperuniform, and vice versa. Hence, as we confirm that a system is
multifractal (hyperuniform), we immediately conclude that the system is not
hyperuniform (multifractal).

###  2.5 Replica overlap for Bose glass

We consider an $L\times L$ matrix
$\mathbf{Q}=(q^{\alpha\beta})_{1\leq\alpha,\beta\leq L}$ whose
$(\alpha,\beta)$ component is given by

$\displaystyle
q^{\alpha\beta}=\frac{1}{N}\sum_{i=1}^{N}\kappa_{i}^{\alpha}\kappa_{i}^{\beta},$
(15)

where $N$ is the total number of vertices, $\alpha$ and $\beta$ represent
replica indices for a shot of randomness, $L$ is the number of kinds of
$\alpha$ and $\beta$, and a compressibility
$\kappa^{\alpha}_{i}=\braket{\hat{n}_{i}^{2}}-\braket{\hat{n}_{i}}^{2}$
denotes the fluctuation of vertex-dependent boson density
$\braket{\hat{n}_{i}}$ for a replica index $\alpha$. In the absence of
disorder, the superscript $\alpha$ denoting the replica number is omitted,
i.e., $\kappa_{i}$, in the following. The quantity $q^{\alpha\beta}$
represents the overlap of the fluctuations of $\braket{\hat{n}_{i}}$ between
the two shots of randomness $\alpha$ and $\beta$. Note that $q^{\alpha\beta}$
is similar to the Edwards-Anderson order parameter for spin glass [27, 28] and
Bose glass [29, 30].

We calculate the fluctuation of $q^{\alpha\beta}$ as an indicator for the Bose
glass phase. All of information on the overlaps of $\kappa_{i}^{\alpha}$ is
contained in a set of upper-triangular elements of $\mathbf{Q}$ excluding
diagonal elements. By using this set of $q^{\alpha\beta}$’s, we define a
replica overlap $q_{\rm BG}$ as

$\displaystyle q_{\rm
BG}=\sqrt{\frac{\frac{\sum_{\alpha<\beta}(q^{\alpha\beta})^{2}}{\frac{L(L-1)}{2}}-\left(\frac{\sum_{\alpha<\beta}q^{\alpha\beta}}{\frac{L(L-1)}{2}}\right)^{2}}{\left(\frac{\sum_{\alpha<\beta}q^{\alpha\beta}}{\frac{L(L-1)}{2}}\right)^{2}}},$
(16)

where we introduce a factor $L(L-1)/2$ to remove a trivial effect from the
number of types of randomness.

###  2.6 Inverse participation ratio (IPR)

IPR quantifies the degree of localization of a nonuniform spatial pattern,
which has been applied in quasicrystals [1, 31]. We define IPR for a set of
physical quantities $\\{n_{i}\\}_{i=1,2,\ldots,N}$ as

$\displaystyle{\rm IPR\hskip 2.84544ptof\hskip
2.84544pt}\\{n_{i}\\}=\sum_{i=1}^{N}\left(\frac{|n_{i}|}{\sum_{j=1}^{N}|n_{j}|}\right)^{2}.$
(17)

Since the quantity in the summation of $i$, $|n_{i}|/\sum_{j}|n_{j}|$ is
normalized, IPR does not depend on the magnitudes of $n_{i}$. We always obtain
$1/N\leq{\rm IPR}\leq 1$, and IPR is smaller (larger) when the physical
quantity is delocalized (localized) in a system.

##  3 Without disorder

In this section, we consider a model without disorder, $\epsilon_{i}=0$, for
$\mu/U=0.5$ and 1.0. Figure 2(a) shows a schematic phase diagram of our model
in Eq. (1) [18]. The horizontal axis represents a hopping amplitude
$\overline{Z_{i}}J/U$, and the upper (lower) panel shows a phase diagram for
$\mu/U=0.5$ (1.0), where MI and SF denote the Mott insulating phase and
superfluid phase, respectively. The system is in the Mott insulating phase
when $\braket{\hat{b}_{i}}=0$, $\braket{\hat{n}_{i}}$ is equal integer number,
and $\kappa_{i}=0$ for all vertices. On the other hand, the system is in the
superfluid phase when $\braket{\hat{b}_{i}}\neq 0$, $\braket{\hat{n}_{i}}\neq
1$, and $\kappa_{i}\neq 0$ for a certain vertex. When $\mu/U=0.5$, the system
undergoes a phase transition from the Mott insulating phase with
$\braket{\hat{n}_{i}}=1$ to the superfluid phase at the critical hopping
strength $J=J_{\rm c}$. The values of $\overline{Z_{i}}J_{\rm c}/U$ in both
Penrose and Ammann-Beenker quasicrystals are approximately 0.16, which is
smaller than that of square lattice being 0.17 [18]. When $\mu/U=1.0$, the
system is in the superfluid phase regardless of the strength of hopping $J$.

###  3.1 Hyperuniformity

By solving Eq. (1) self-consistently, we obtain the distribution of
$\braket{\hat{b}_{i}}$, $\braket{\hat{n}_{i}}$, and $\kappa_{i}$. To clarify
how quasiperiodicity of a system affects the distribution of physical
quantities, we apply a hyperuniform analysis to self-consistently obtained
physical quantities. We focus on hyperuniformity in $\braket{\hat{b}_{i}}$. In
the Mott insulating phase, the system is hyperuniform because
$\braket{\hat{b}_{i}}=0$. For the superfluid phase, we show $A(R)$ in Eq. (9)
normalized by $(\overline{|\braket{\hat{b}_{i}}|})^{2}$ as a function of $R/D$
for $\mu/U=0.5$ and $\overline{Z_{i}}J/U=0.19$ in Fig. 2(b), where $D$ is the
length of links. As indicated by a downward black arrow at the upper-right
corner in Fig. 2(a), the system is superfluid for this parameter set. The
blue, orange, and green lines represent the results for Penrose quasicrystal,
Ammann-Beenker quasicrystal, and square lattice, respectively. All systems are
hyperuniform because $A(R)$ approaches zero in the limit of
$R\rightarrow\infty$. We also find that the systems are hyperuniform
regardless of hopping amplitude (not shown). Figure 2(c) is the same as Fig.
2(b) but for $B(R)$ in Eq. (14). The blue, orange, and horizontal black-dashed
lines in Fig. 2(c) represent
$\overline{B}/(\overline{|\braket{\hat{b}_{i}}|})^{2}$ for Penrose
quasicrystal, Ammann-Beenker quasicrystal, and square lattice, respectively,
where $\overline{B}$ is the average of $B(R)$ within $30\leq R/D\leq 35$,
which we take as order metric. As shown in Fig. 2(c), $B(R)$ is fluctuating
around $\overline{B}$. This indicates Class I in hyperuniformity.

Figure 2:  (a) Schematic phase diagram of our model in Eq. (1). (b) $A(R)$ as
a function of $R/D$ for $\overline{Z_{i}}J/U=0.19$ and $\mu/U=0.5$. (c) Same
as (b) but for $B(R)$. (d) Same as (b) but for $\mu/U=1.0$. (e) Same as (c)
but for $\mu/U=1.0$. Horizontal colored lines in (c) and (e) represent an
average value of $B(R)$ for each system. (f) Order metric $\overline{B}$ as a
function of hopping amplitude for both $\mu/U=0.5$ and 1.0. The inset shows
behaviors near the phase boundary shown in (a). Sq represents square lattice
and Pen (AB) represents Penrose (Ammann-Beenker) quasicrystals, and MI (SF)
denotes the Mott insulating (superfluid) phase.

Figure 2(d) is the same as Fig. 2(b) but for $\mu/U=1.0$. The red, purple, and
brown lines represent the results for Penrose quasicrystal, Ammann-Beenker
quasicrystal, and square lattice, respectively. Figure 2(e) is the same as
Fig. 2(c) but for $\mu/U=1.0$. The yellow, cyan, and horizontal black-dashed
lines in Fig. 2(e) represent
$\overline{B}/(\overline{|\braket{\hat{b}_{i}}|})^{2}$ for Penrose
quasicrystal, Ammann-Beenker quasicrystal, and square lattice, respectively.
Similar to the case of $\mu/U=0.5$, Fig. 2(e) indicates Class I in
hyperuniformity for all systems.

Figure 2(f) shows the order metric $\overline{B}$ as a function of hopping
strength $J$. The inset is a magnified one near the phase boundary for
$\mu/U=0.5$. In the limit of large hopping amplitude $J\rightarrow\infty$, the
order metric approaches a value for point distribution [6, 7]. For
$\mu/U=0.5$, the order metric in quasicrystals (blue and orange dots)
significantly increases at $J\sim J_{\rm c}$. The origin of this significant
increase is attributed to a critical behavior near the phase transition [18].
With approaching a phase boundary, a typical correlation length increases and
thus the distribution of local superfluid amplitudes $\braket{\hat{b}_{i}}$
becomes wider [18], which leads to the enhancement of the order metric. On the
other hand, in square lattice, the order metric slightly increases with
approaching a phase boundary at $\overline{Z_{i}}J_{\mathrm{c}}/U\sim 0.17$
(green dots). For $\mu/U=1.0$, the order metric in all systems is almost
constant.

###  3.2 Near phase boundary

We focus on a phase boundary between the Mott insulating and superfluid
phases. At the vicinity of the phase boundary, the system may show an exotic
phase such as Bose glass [32, 33, 20]. The system is in the Bose glass phase
when superfluid vertices are randomly distributed. On the other hand, the
system is in the superfluid phase when the fraction of superfluid vertices is
large. We note that the Bose glass phase has been observed in a system of
ultra-cold atoms in optical quasicrystals [15, 17].

To clarify whether an exotic phase appears, we introduce a measure of
superfluidity at each vertex [34] as

$\displaystyle S_{i}=\left\\{\begin{array}[]{ll}0&{\rm if}\quad
1-\gamma_{n}\leq\braket{\hat{n}_{i}}\leq 1+\gamma_{n},\\\ 1&{\rm
else},\end{array}\right.$ (20)

where $\gamma_{n}$ is a small number. In the following, we set
$\gamma_{n}=5\times 10^{-3}$. We call the $i$th vertex with $S_{i}=1$ (0) a
superfluid (Mott insulating) vertex. Note that $S_{i}=1$ (0) for all vertices
in the superfluid (Mott insulating) phase. Since the Mott insulating phase
changes to the superfluid phase with increasing $J$, there would be some
signatures in the distribution of $S_{i}$ at a certain value of $J$ if an
exotic phase appears. To check this, we search a critical hopping amplitude
$J_{i}^{\rm c}$ for each vertex where $S_{i}$ changes from 0 (Mott insulating
vertex) to 1 (superfluid vertex).

####  3.2.1 Penrose quasicrystals

First, we discuss Penrose quasicrystals. After a self-consistent calculation
with the open boundary condition for $N=75806$, we remove vertices at the
edges in order to eliminate the effect of boundaries (see Appendix). We take
55651 vertices within $R/D\leq 120$ and calculate $J_{i}^{\rm c}$ for each
vertex. Then the values of $J_{i}^{\rm c}$ are grouped together by the
coordination number $Z_{i}$ and are plotted in Fig. 3(a) as colored symbols.
We find that, the larger $Z_{i}$ is, the smaller $J_{i}^{\rm c}$ is. For each
$Z_{i}$, they group around certain values of $J$. Each group is labeled by
symbols such as S3 and S4 [35, 36]. Such a grouping of $J_{i}^{\rm c}$ comes
from the difference of local environment beyond nearest-neighbor links.
Considering the perpendicular space of Penrose quasicrystals, we can map the
symbols onto the $\tilde{z}=0$ and $1/\sqrt{2}$ planes as shown in the inset
of Fig. 3(a).

Figure 3:  (a) The distribution of $J_{i}^{\rm c}$ for a given $Z_{i}$ as a
function $\overline{Z_{i}}J/U$ for a Penrose quasicrystal with $\mu/U=0.5$.
Symbols are assigned for each group of $J_{i}^{\rm c}$ and put into
corresponding regions in the $\tilde{z}=0$ and $1/\sqrt{2}$ perpendicular-
space planes in the inset. The distribution of $S_{i}$ in the real space and
perpendicular space for [(b) and (c)] the Mott insulating phase, [(d) and (e)]
$\overline{Z_{i}}J/U=0.1598$, [(f) and (g)] $\overline{Z_{i}}J/U=0.1625$, [(h)
and (i)] $\overline{Z_{i}}J/U=0.1634$, [(j) and (k)] the superfluid phase,
respectively.

In the Mott insulating phase where $\overline{Z_{i}}J/U\lesssim 0.159$, all
vertices are the Mott insulating vertex ($S_{i}=0$). We show the real-space
representation of $S_{i}=0$ in Fig. 3(b) by putting black dots on each vertex
around a center of a Penrose quasicrystal within $R/D\leq 10$. In the
perpendicular space, all vertices in the $\tilde{z}=0$ and $1/\sqrt{2}$ planes
are colored in black, i.e., $S_{i}=0$ as shown in the upper and lower panels
of Fig. 3(c), respectively. At $\overline{Z_{i}}J/U=0.1598$, some of the
$Z_{i}=7$ vertices satisfying $\overline{Z_{i}}J_{i}^{\rm c}/U<0.1598$ are the
superfluid vertex ($S_{i}=1$) as expected from Fig. 3(a). In this case, the
superfluid vertices (red dots) are regularly distributed in the real space as
shown in Fig. 3(d). In the perpendicular space, a part of the $1/\sqrt{2}$
plane is colored in red as seen in Fig. 3(e). With increasing
$\overline{Z_{i}}J_{i}^{\rm c}/U$ from 0.1625 to 0.1634, the number of red
dots increases keeping a regular pattern in Figs. 3(f) and 3(h). Accordingly,
a star shape colored in red appears in the $\tilde{z}=0$ and $1/\sqrt{2}$
planes as shown in Figs. 3(g) and 3(i). In the superfluid phase, all vertices
become the superfluid vertex (red dots) as seen in Figs. 3(j) and 3(k). Based
on these evolutions of $S_{i}$ with $J$ near the phase boundary, we conclude
that the distribution of the superfluid vertices is not random but regular and
thus it reflects the vertex structure of Penrose quasicrystals. This implies
that there is no Bose glass phase near the phase boundary. We will discuss the
Bose glass phase in the presence disorder in Sec. 4.

As shown in Fig. 2(a), we find a significant increase of order metric near the
phase boundary only in quasicrystals. Since order metric measures the
complexity of a given system, its increase may be attributed to the presence
of nonuniform superfluid phase where some of the vertices are not superfluid
vertices seen as discussed above. To examine whether the nonuniform superfluid
phase is caused by a finite-size effect or not, we show in Fig. 4(a) the
fraction of superfluid vertices as a function of $J$ near the phase boundary
for various system sizes $N_{\rm in}$, where $N_{\rm in}$ is the number of
vertices that exclude those around edge. The fraction is defined as the ratio
of superfluid vertices to $N_{\rm in}$. We find that the fraction is 0 (1) for
$J<J_{\rm c}$ ($J^{*}<J$), where $J_{\rm c}$ ($J^{*}$) depends on $N_{\rm
in}$, and $J^{*}$ refers to a characteristic $J$. The nonuniform superfluid
phase appears for $J_{\rm c}<J<J^{*}$. The difference $J^{*}-J_{\rm c}$ is
expected to remain finite in the limit of $N_{\rm in}\rightarrow\infty$ as
shown Fig. 4(b). Therefore, the presence of nonuniform superfluid phase is
intrinsic in Penrose quasicrystals. Accordingly, we conclude that the increase
of order metric near the phase boundary is also intrinsic.

Figure 4:  (a) The hopping amplitude $J$ dependence of the fraction of
superfluid vertices for various size $N_{\mathrm{in}}$ of Penrose
quasicrystals with $\mu/U=0.5$. (b) The critical hopping amplitude of $J_{\rm
c}$ (purple circles) and $J^{*}$ (red squares) for various $N_{\mathrm{in}}$.

####  3.2.2 Ammann-Beenker quasicrystals

Next, we examine behaviors near the phase boundary in Ammann-Beenker
quasicrystals. After a self-consistent calculation with the open boundary
condition for $N=65281$, we use 45873 vertices within $R/D\leq 110$ for the
evaluation of $S_{i}$ and $J_{i}^{\rm c}$. Figure 5(a) shows
$\overline{Z_{i}}J_{i}^{\rm c}/U$ for various $Z_{i}$. We find that
$J_{i}^{\rm c}$ decreases with increasing $Z_{i}$, which is the same as
Penrose quasicrystals. The inset represents an octagon in the perpendicular
space, where the number of $Z_{i}$ is assigned only in a one-eighth region of
the entire space.

Figure 5:  Same as Fig. 3 but for Ammann-Beenker quasicrystals with
$\mu/U=0.5$. (a) The distribution of $J_{i}^{\rm c}$ for a given $Z_{i}$ as a
function $\overline{Z_{i}}J/U$. The inset presents the entire perpendicular
space. The distribution of $S_{i}$ in the real space [(b) to (j)] and
perpendicular space [(c) to (k)]. We use [(d) and (e)]
$\overline{Z_{i}}J/U=0.161$, [(f) and (g)] $0.162$, and [(h) and (i)] $0.164$.

In Fig. 5(b), we show the distribution of $S_{i}$ around a center of the real
space $R/D\leq 10$ for the Mott insulating phase ($S_{i}=0$ represented by
black dots). Figure 5(c) is for the perpendicular space and all vertices are
Mott insulating vertices. Figures 5(d) [5(e)], 5(d) [5(e)], and 5(d) [5(e)]
are the same as Fig. 5(b) [5(c)] but for $\overline{Z_{i}}J/U=0.161$, 0.162,
and 0.164, respectively. All vertices for $Z_{i}$ more than 5, 4, and 3 are
superfluid vertices ($S_{i}=1$ represented by red dots) for
$\overline{Z_{i}}J/U=0.161$, 0.162, and 0.164, respectively. Figures 5(j) and
5(k) are for the superfluid phase where all the vertices are $S_{i}=1$.
Accordingly, a nonuniform superfluid phase reflecting a quasiperiodic
structure of the Ammann-Beenker quasicrystals also appears near the phase
boundary, which is similar to Penrose quasicrystals.

We show in Fig. 6(a) the fraction of superfluid vertices as a function of
$\overline{Z_{i}}J/U$ for various $N_{\rm in}$. We also show $J_{\rm c}$
(purple) and $J^{*}$ (red) in Fig. 6(b). In the limit of $N_{\rm
in}\rightarrow\infty$, $J^{*}-J_{\rm c}$ is expected to remain finite.
Accordingly, we conclude that the presence of nonuniform superfluid phase and
the resulting increase of order metric are not due to a finite-size effect but
intrinsic behaviors.

Figure 6:  Same as Fig. 4 but for Ammann-Beenker quasicrystals with
$\mu/U=0.5$.

####  3.2.3 Comparison between crystals and quasicrystals

To clarify the effect of quasiperiodicity on $|\braket{\hat{b}_{i}}|$ near a
phase transition point, we compare a critical behavior of
$|\braket{\hat{b}_{i}}|$ in crystals and quasicrystals for $\mu/U=0.5$. First,
we consider a square lattice. In Fig. 7(a), we show
$|\braket{\hat{b}_{i}}|^{2}$ as a function of $\overline{Z_{i}}J/U$ by a red
line. Since this red line is almost straight for larger $J$, we also plot a
linear fitting by a green-dashed line. By extrapolating this green-dashed line
to $|\braket{\hat{b}_{i}}|=0$, we obtain a critical hopping strength
$\overline{Z_{i}}J_{\rm c}/U\sim 0.1666$ for square lattice, as indicated by a
downward-black arrow. A critical exponent for $|\braket{\hat{b}_{i}}|$ is
$1/2$ [21, 37] and we have

$|\braket{\hat{b}_{i}}|=a\sqrt{\frac{J-J_{\rm c}}{J_{\rm c}}},$ (21)

where the constant $a$ is approximately $1.31$ for square lattice. Note that,
in square lattice, all vertices share the same $|\braket{\hat{b}_{i}}|$ and
thus a single $a$ is used for the entire system.

Next, we consider quasicrystals. Figure 7(b) shows
$|\braket{\hat{b}_{i}}|^{2}$ for various vertices in a Penrose quasicrystal,
where we select one vertex located close to the center of system in the real
system for each group shown in Fig. 3(a). As $J$ increases,
$|\braket{\hat{b}_{i}}|^{2}$ becomes finite for all vertices simultaneously at
$\overline{Z_{i}}J_{\rm c}/U\sim 0.1574$, and then linearly increases. The
slopes of these linear increases depend on types of vertices. Figure 7(c)
shows a result for an Ammann-Beenker quasicrystal, where we obtain
$\overline{Z_{i}}J_{\rm c}/U\sim 0.1579$. We also obtain a critical behavior
of $|\braket{\hat{b}_{i}}|$ following Eq. (21) regardless of a coordination
number $Z_{i}$.

Figure 7(d) shows a relationship between $a$ and $Z_{i}$ in a Penrose
quasicrystal. The blue circles represent mean values of $a$ with respect to
vertices with the same $Z_{i}$. As these mean values are almost on a straight
line, we show a linear fitting of them by a blue-dashed line. It implies that
$|\braket{\hat{b}_{i}}|$ near the phase transition point is almost
proportional to the number of linked vertices $Z_{i}$. This is reasonable
because the mean field should be larger for a vertex with larger $Z_{i}$ [see
Eq. (2)]. A value of $a$ for a square lattice is indicated by a green-dashed
line, which intersects with the blue-dashed line between $Z_{i}=3$ and 4.
Although the average value of $Z_{i}$ in Penrose quasicrystals is 4 which is
the same as in square lattice, we find that $a$ for a square lattice is larger
than $a$ for $Z_{i}=4$ in a Penrose quasicrystal. We show a result for Ammann-
Beenker quasicrystal in Fig. 7(e), where we also find a linear dependence of
$Z_{i}$ on $a$. The intersection of the blue-dashed and green-dashed lines is
between $Z_{i}=4$ and 5 in an Ammann-Beenkwe quasicrystal.

To clarify the effect of quasiperiodicity on $a$, we show the real-space
distribution of $a$ in a Penrose quasicrystal in Fig. 7(f). A colored-scale
bar is embedded at the far right, where a rightward-black arrow indicates $a$
for square lattice. We find an intricate pattern of $a$ in the real space,
however, a well-organized structure appears in the perpendicular space
representation [Fig. 7(g)], where a blue-star (red-star) shaped region in
$\tilde{z}=0$ ($1/\sqrt{2}$) is seen. Within each domain of the same type of
vertices, we find a non-uniform and ordered pattern. For example, we find a
yellow-triangular shaped region for D vertices (see the inset in Fig. 3).
Accordingly, we confirm that quasiperiodicity governs the critical behavior of
$|\braket{\hat{b}_{i}}|$ beyond the nearest-neighbor links. Figures 7(h) and
7(h) show the real-space and perpendicular-space distribution of $a$ in an
Ammann-Beenker quasicrystal, respectively. We also find that quasiperiodicity
affects the critical behavior.

According to the above discussion, $|\braket{\hat{b}_{i}}|$ becomes finite for
all vertices at the same hopping amplitude $J=J_{\rm c}$. For a certain
$(J_{\rm c}<)J$, $|\braket{\hat{b}_{i}}|$ for some vertices are larger than
others because $a$ is different among vertices [see Figs. 7(b) and 7(c)]. This
explains that $J^{\rm c}_{i}$ in Figs. 3(a) and 5(a) varies among vertices.
Also note that $J^{*}$ shown in Figs. 4(a) and 6(a) becomes $J_{\rm c}$ for
$\gamma_{n}\rightarrow 0$ as the critical hopping amplitude is shared by all
vertices. The origin of a large order metric at $J\sim J_{\rm c}$ in
quasicrystals is attributed to a wide variety of $a$ and hence
quasiperiodicity [see an inset of Fig. 2(f)].

Figure 7:  (a) $|\braket{\hat{b}_{i}}|^{2}$ as a function of
$\overline{Z_{i}}J/U$. (b) Same as (a) but for various types of vertices in a
Penrose quasicrystal. (c) Same as (b) but for an Ammann-Beenker quasicrystal.
(d) The relationship between $Z_{i}$ and $a$ in a Penrose quasicrystal. (e)
Same as (d) but for an Ammann-Beenker quasicrystal. The symbols in (b) and (d)
denote groups of $Z_{i}$ defined in Fig. 3(a). The vertical green-dotted line
in (d) and (e) represents the value of $a$ in a square lattice. (f) The real-
space distribution of $a$ in a Penrose quasicrystal. (g) The perpendicular-
space distribution of $a$ in a Penrose quasicrystal for $\tilde{z}=0$ (upper
panel) and $1/\sqrt{2}$ (lower panel). (h) Same as (f) but for an Ammann-
Beenker quasicrystal. (i) Same as (g) but for an Ammann-Beenker quasicrystal.
A common colored-scale bar for (f-i) is shown at the right side of (i).

##  4 With disorder

In this section, we consider a random on-site potential with the uniform
distribution $\epsilon_{i}\in[-\frac{\Delta}{2},\frac{\Delta}{2}]$, where
$\Delta$ represents a strength of disorder. With random on-site potentials, a
bosonic system shows the Bose glass phase for $J\sim J_{\rm c}$ both in
crystals and quasicrystals [33]. To clarify how randomness affects the
distribution of physical quantities, we apply multifractal analyses. We
calculate physical quantities for 30 shots of randomness, and we use the mean
values of them.

###  4.1 Penrose quasicrystals

First, we focus on Penrose quasicrystals. Figure 8(a) shows
$D(\\{\kappa_{i}^{\alpha}\\},q,R)$ in Eq. (4)
($\kappa^{\alpha}_{i}=\braket{\hat{n}_{i}^{2}}-\braket{\hat{n}_{i}}^{2}$) as a
function of $1/\ln(N_{R})$ for $q=-10$ (red circles) and $10$ (blue squares).
The parameter set used is $\mu/U=0.5,\Delta/U=0.6$, and
$\overline{Z_{i}}J/U=0.132$. By linearly fitting data points in Fig. 8(a), we
obtain extrapolated values of $D(\\{\kappa_{i}^{\alpha}\\},q,R)$ in the limit
of $N_{R}\rightarrow\infty$. The extrapolated
$D^{\infty}(\\{\kappa_{i}^{\alpha}\\},q=-10)$ and
$D^{\infty}(\\{\kappa_{i}^{\alpha}\\},q=10)$ deviate from 2.0, which indicates
that this system is multifractal. The inset of Fig. 8(a) shows the
distribution of $S_{i}$ [see Eq. (20)] in the perpendicular space for
$\tilde{z}=0$ (upper panel) and $1/\sqrt{2}$ (lower panel). We find that the
distribution of superfluid vertices ($S_{i}=1$ represented by red dots)
reflects quasiperiodic structure as a star-shaped region is relatively red in
color, especially for $\tilde{z}=1/\sqrt{2}$. However, this distribution does
not completely obey quasiperiodic structure because of the presence of on-site
random potentials. As a result, there are randomly distributed Mott insulating
vertices in the star-shaped region, whose distribution less reflects
quasiperiodic structure for larger $\Delta/U$. We refer to this phase as a
Bose glass phase, not a superfluid phase.

Figure 8:  (a) The $R$-dependent multifractal dimension
$D(\\{\kappa_{i}^{\alpha}\\},q=-10,R)$ and
$D(\\{\kappa_{i}^{\alpha}\\},q=10,R)$ as a function of $1/\ln(N_{R})$ for
$\Delta/U=0.6$, and $\overline{Z_{i}}J/U=0.132$ in a Penrose quasicrystal with
$\mu/U=0.5$. (b) Same as (a) but for $\overline{Z_{i}}J/U=0.151$. (c) The
$\overline{Z_{i}}J/U$ dependence of
$D^{\infty}(\\{\kappa_{i}^{\alpha}\\},q=-10)$ and
$D^{\infty}(\\{\kappa_{i}^{\alpha}\\},q=10)$. MI, BG, and SF represent Mott
insulating, Bose glass, and superfluid phases, respectively. (d) The fraction
of superfluid vertices as a function of $\overline{Z_{i}}J/U$. We use 30 shots
of randomness and average them, except for the insets in (a) and (b) where we
show results for a single shot of randomness.

Figure 8(b) is the same as 8(a) but for $\overline{Z_{i}}J/U=0.151$. Both
$D^{\infty}(\\{\kappa_{i}^{\alpha}\\},q=-10)$ and
$D^{\infty}(\\{\kappa_{i}^{\alpha}\\},q=10)$ are almost 2.0, which indicates a
nonmultifractal behavior. The inset clearly shows that this phase is a
superfluid phase. In Sec. 3, we have shown that the superfluid phase is
hyperuniform, which agrees with Fig. 8(b).

From the above discussion, the Bose glass phase is multifractal while the
superfluid phase is hyperuniform. Also, the Mott insulating phase is
hyperuniform as we have $\kappa_{i}^{\alpha}=0$ for all vertices. To examine
phase transition between these phases, we show the hopping amplitude
dependence of $D^{\infty}(\\{\kappa_{i}^{\alpha}\\},q=-10)$ (red circles) and
$D^{\infty}(\\{\kappa_{i}^{\alpha}\\},q=10)$ (blue squares) in Fig. 8(c) for
the same parameter set as Figs. 8(a) and 8(b). The two downward-black arrows
at the upper-right corner represent $\overline{Z_{i}}J/U$ used for Figs. 8(a)
and 8(b). We do not show data for $\overline{Z_{i}}J/U\lesssim 0.13$ because
$\kappa_{i}^{\alpha}$ is too small to calculate the multifractal dimension.
Because both $D^{\infty}(\\{\kappa_{i}^{\alpha}\\},q=-10)$ and
$D^{\infty}(\\{\kappa_{i}^{\alpha}\\},q=10)$ are away from 2.0, the system is
multifractal for $0.13\lesssim\overline{Z_{i}}J/U\lesssim$ 0.15. On the other
hand, the system is not multifractal for 0.15 $\lesssim\overline{Z_{i}}J/U$ as
both $D^{\infty}(\\{\kappa_{i}^{\alpha}\\},q=-10)$ and
$D^{\infty}(\\{\kappa_{i}^{\alpha}\\},q=10)$ are almost 2.0, and the system is
hyperuniform according to Sec. 3.

We note that it is difficult to determine the phase boundary precisely in the
mean field approximation. We roughly show the Mott insulating phase, the Bose
glass phase, and the superfluid phase in red, yellow, and blue-shaded regions,
respectively, in Fig. 8(c). To determine phase boundaries, we consider the
fraction of superfluid vertices as a function of $\overline{Z_{i}}J/U$ shown
in Fig. 8(d). For this calculation, we use 30 shots of randomness and average
them. The fraction is almost 1.0 for 0.15$\lesssim\overline{Z_{i}}J/U$, where
we assign a superfluid phase. Comparing Figs. 8(c) and 8(d), we conclude that
our system changes from a hyperuniform Mott insulating phase to a multifractal
Bose glass phase, and then to a hyperuniform superfluid phase with increasing
$\overline{Z_{i}}J/U$. The presence of phase transition among these phases has
been clarified for crystals in Refs. [38, 39, 40, 34] and quasicrystals in
Refs [19, 33, 20].

Figure 9 shows results for $\mu/U=1.0$. At $\overline{Z_{i}}J/U=0.034$, since
both $D^{\infty}(\\{\kappa_{i}^{\alpha}\\},q=-10)$ and
$D^{\infty}(\\{\kappa_{i}^{\alpha}\\},q=10)$ are away from 2.0, the system is
multifractal as shown in Fig. 9(a). The inset indicates that the system is a
Bose glass phase. At $\overline{Z_{i}}J/U=0.074$, both
$D^{\infty}(\\{\kappa_{i}^{\alpha}\\},q=-10)$ and
$D^{\infty}(\\{\kappa_{i}^{\alpha}\\},q=10)$ are almost 2.0, thus the system
is not multifractal [see Fig. 9(b)]. As shown in the inset, most vertices are
superfluid vertices and thus the system is a superfluid phase. Combining the
results of $D^{\infty}(\\{\kappa_{i}^{\alpha}\\},q)$ in Fig. 9(c) and the
fraction of superfluid vertices in Fig. 9(d), we approximately assign that the
system is (not) multifractal for $\overline{Z_{i}}J/U\lesssim 0.07$
($0.07\lesssim\overline{Z_{i}}J/U$). For $\mu/U=1.0$, we find a multifractal
Bose glass phase and a hyperuniform superfluid phase, but a Mott insulating
phase does not appear.

Figure 9:  Same as Fig. 8(a) but for $\mu/U=1.0$. We use (a)
$\overline{Z_{i}}J/U=0.034$ and (b) $0.074$.

Figure 10(a) shows multifractalness in $\\{\kappa_{i}^{\alpha}\\}$, namely
$D^{\infty}(\\{\kappa_{i}^{\alpha}\\},q=-10)-D^{\infty}(\\{\kappa_{i}^{\alpha}\\},q=10)$,
as a function of $\overline{Z_{i}}J/U$, for $\mu/U=0.5$ and $\Delta/U=0.2$
(blue circles), 0.6 (orange squares), 1.2 (green diamonds), and 2.0 (red
triangles). We find that multifractalness decreases with increasing
$\overline{Z_{i}}J/U$ regardless of $\Delta/U$. Note that, when
multifractalness is larger than 0, the system is multifractal. We confirm that
the system is hyperuniform for larger $\overline{Z_{i}}J/U$ irrespective of
$\Delta/U$.

Figure 10:  The hopping amplitude dependence of (a) the multifractalness of
$\\{\kappa_{i}^{\alpha}\\}$, (b) the fraction of superfluid vertices, (c) the
replica overlap for Bose glass, (d) IPR of $\\{\braket{\hat{b}_{i}}\\}$, (e)
IPR of $\\{\kappa_{i}^{\alpha}\\}$, and (f) IPR of $\\{S_{i}\\}$ for
$\mu/U=0.5$. For these calculations, we use 30 shots of randomness and average
them. We consider all vertices $N=75806$ including an edge for (d-f), and the
black-dashed lines indicate ${\rm IPR}=1/N$.

Figure 10(b) shows the fraction of superfluid vertices. The minimum hopping
amplitude for nonzero fraction at $\Delta/U=0.2$ and 0.6 is a critical hopping
amplitude at which a phase transition between Mott insulating and Bose glass
phases occurs. The fraction increases as the hopping amplitude increases. For
$\Delta/U=1.2$ and 2.0, the fraction at $J=0$ is larger than 0 and thus the
system is a Bose glass phase, which implies that a Mott insulating phase does
not appear for large $\Delta/U$. Then we have only one phase transition from a
multifractal Bose glass phase to a hyperuniform superfluid phase. Since
multifractalness is larger at $\Delta/U=2.0$ than 1.2 as shown in Fig. 10(a),
a multifractal phase, i.e., a Bose glass phase, becomes stable for large
$\Delta$, leading to disappearance of a Mott insulating phase.

Figure 10(c) shows a replica overlap for Bose glass $q_{\mathrm{BG}}$ in Eq.
(16). $q_{\mathrm{BG}}$ is nonzero in Bose glass phase, but almost zero in
superfluid phase. This indicates that, in Bose glass phase, the fluctuation of
$\braket{\hat{n}_{i}}$ overlaps significantly among different kinds of
randomness.

Figures 10(d), 10(e), and 10(f) show IPR of $\\{\braket{\hat{b}_{i}}\\}$,
$\\{\kappa_{i}^{\alpha}\\}$, and $\\{S_{i}\\}$, respectively. The black-dashed
lines indicate ${\rm IPR}=1/N$, which is the allowed minimum value of IPR. For
$J\rightarrow\infty$, all the vertices share almost the same value of physical
quantities, which makes IPR the minimum. The $J$ and $\Delta$ dependencies are
similar to those in Fig. 10(a), except for $\\{S_{i}\\}$. Since IPR is a
measure of localization for physical quantities, the enhancement of
multifractalness is closely related to a localized nature of the system.

Figure 11 is the same as Fig. 10 but for $\mu/U=1.0$. We find that
multifractalness in Fig. 11(a) decreases with increasing $\overline{Z_{i}}J/U$
and the system is multifractal (hyperuniform) for small (large)
$\overline{Z_{i}}J/U$ regardless of $\Delta/U$. We do not find the Mott
insulating phase in Fig. 11(b) as the fraction of superfluid vertices is
larger than zero. Comparing Figs. 11(a) and 11(b), we find that
multifractalness is (not) larger than zero when the system is a Bose glass
(superfluid) phase. The behavior of replica overlap in Fig. 11(c) is similar
to multifractalness in Fig. 11(a). The replica overlap is large (small) in
Bose glass phase (superfluid phase). IPR of $\\{\braket{\hat{b}_{i}}\\}$,
$\\{\kappa_{i}^{\alpha}\\}$, and $\\{S_{i}\\}$ in Figs. 11(d), 11(e), and
11(f) also show a $\Delta$ dependence similar to Fig. 11(a). Similar to the
$\mu/U=0.5$ case, physical quantities are localized when multifractalness is
enhanced.

Figure 11:  Same as Fig. 10 but for $\mu/U=1.0$.

###  4.2 Ammann-Beenker quasicrystals

Next, we focus on Ammann-Beenker quasicrystals. Figure 12(a) is the same as
Fig. 8(a) but for Ammann-Beenker quasicrystals with $\mu/U=0.5$ and
$\overline{Z_{i}}J/U=0.132$. As both
$D^{\infty}(\\{\kappa_{i}^{\alpha}\\},q=-10)$ and
$D^{\infty}(\\{\kappa_{i}^{\alpha}\\},q=10)$ deviate from 2.0, the system is
multifractal. The inset where red dots distribute implies that the system is a
Bose glass phase. Figure 12(b) is for $\overline{Z_{i}}J/U=0.151$. As both
$D^{\infty}(\\{\kappa_{i}^{\alpha}\\},q=-10)$ and
$D^{\infty}(\\{\kappa_{i}^{\alpha}\\},q=10)$ are almost 2.0, the system is not
multifractal. According to the inset where all vertices are superfluid vertex,
the system is a hyperuniform superfluid phase. From Figs. 12(c) and 12(d), we
find that the system changes from a hyperuniform Mott insulating phase to a
multifractal Bose glass phase, and then to a hyperuniform superfluid phase
with increasing $\overline{Z_{i}}J/U$.

Figure 12:  Same as Fig. 8 but for Ammann-Beenker quasicrystals with
$\mu/U=0.5$. We use (a) $\overline{Z_{i}}J/U=0.132$ and (b) 0.151.

Figure 13 is the same as Fig. 12 but for $\mu/U=1.0$. We find that the system
is a multifractal Bose glass (hyperuniform superfluid) phase for
$\overline{Z_{i}}J/U=0.034$ and 0.074 from Figs. 13(a) and 13(b). Figsures
13(c) and 13(d) show that the system changes from a multifractal Bose glass
phase to a hyperuniform superfluid phase with increasing
$\overline{Z_{i}}J/U$.

Figure 13:  Same as Fig. 12 but for $\mu/U=1.0$. We use (a)
$\overline{Z_{i}}J/U=0.034$ and (b) 0.074.

Figures 14 and 15 are the same as Figs. 10 and 11, respectively, but for
Ammann-Beenker quasicrystals. Both the $\mu/U=0.5$ and 1.0 exhibit almost the
same behavior between Penrose and Ammann-Beenker quasicrystals in several
quantities shown in the figures. This is a striking observation that
similarity between the two quasicrystals in various quantities is maintained
even in the presence of disorder.

Figure 14:  Same as Fig. 10 but for Ammann-Beenker quasicrystals with
$\mu/U=0.5$. We consider all vertices $N=65281$ including an edge for (d-f),
and the black-dashed lines indicate ${\rm IPR}=1/N$.  Figure 15:  Same as Fig.
14 but for $\mu/U=1.0$.

###  4.3 Comparison between quasicrystals and crystals

To clarify the effect of quasiperiodicity on the multifractal property of a
system, we compare multifractalness in quasicrystals and crystals. Without
disorder $\epsilon_{i}=0$, a critical hopping amplitude separating Mott
insulating and superfluid phases, namely $J_{\rm c}$, is different between
quasicrystals and crystals. To make effect of this difference smaller, we
normalize hopping amplitudes by $J_{\rm c}$ for both systems. We show in Fig.
16 multifractalness of $\\{\kappa_{i}^{\alpha}\\}$ as a function of $J/J_{\rm
c}$ in the presence of disorder $\Delta$ for Penrose quasicrystals, Ammann-
Beenker quasicrystals, and square lattice. Multifractalness does not show any
significant difference between quasicrystals and periodic lattice, for both
$\mu/U=0.5$ and 1.0.

Figure 16:  Scaled hopping amplitude dependence of multifractalness for various disorder $\Delta$. For $\mu/U=0.5$, (a) Penrose quasicrystals, (b) Ammann-Beenker quasicrystals, and (c) square lattice, and for $\mu/U=1.0$, (d) Penrose quasicrystals, (e) Ammann-Beenker quasicrystals, and (f) square lattice. For these calculations, we use 30 shots of randomness and average them.  Figure 17:  Inflation rule (a) [(f)], tiling of 1st (b) [(g)], 2nd (c) [(h)], 3rd (d) [(i)], and 4th (e) [(j)] generations for Penrose [Ammann-Beenker] tilings.  Table 1:  $N$,$N_{\rm in}$, and $R_{\rm in}/D$ for each generation in Penrose (Pen) and Ammann-Beenker (AB) quasicrystals.  | generation | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10
---|---|---|---|---|---|---|---|---|---|---|---
Pen | $N$ | 11 | 31 | 86 | 226 | 601 | 1591 | 4181 | 11006 | 28901 | 75806
| $N_{\rm in}$ |  | 16 | 51 | 141 | 401 | 861 | 2421 | 6171 | 18986 | 55651
| $R_{\rm in}/D$ |  | 2 | 4 | 6 | 10 | 15 | 25 | 40 | 70 | 120
AB | $N$ | 17 | 65 | 353 | 1969 | 11281 | 65281 |  |  |  |
| $N_{\rm in}$ |  | 65 | 193 | 1185 | 6825 | 42649 |  |  |  |
| $R_{\rm in}/D$ |  | 7$/\sqrt{2}$ | 10$/\sqrt{2}$ | 25$/\sqrt{2}$ | 60$/\sqrt{2}$ | 150$/\sqrt{2}$ |  |  |  |

##  5 Conclusion

In summary, we have investigated multifractal or hyperuniform properties in
the Bose-Hubbard model on the Penrose and Ammann-Beenker tilings with and
without disorder. We have found that, without disorder, the system is
hyperuniform in both Mott insulating and superfluid phases. From the
enhancement of order metric, we have found that the complexity of the system
increases near the phase boundary of the two phases. This is in contrast to a
constant order metric in square lattice. Examining a critical hopping strength
for generating superfluid vertex near the phase boundary, we have attributed
the origin of enhanced order metric to quasiperiodicity in the systems. This
contrasting behavior between crystalline and quasicrystalline systems can be
confirmed by using ultra-cold atoms if one measures the distribution of local
superfluid amplitude in optical lattice [14, 15, 16, 17].

With disorder, we have found that the system changes from a hyperuniform Mott
insulating phase to a multifractal Bose glass phase, and then to a
hyperuniform superfluid phase as hopping amplitude increases. We would like to
emphasize that, as far as we know, this is the first report for the phase
transition between hyperuniform and multifractal systems. We find no
significant differences between the multifractal properties of the Bose glass
phase in quasiperiodic and periodic systems. Accordingly, quasiperiodicity is
not essential for realizing multifractality in the presence of disorder.

In this study, underlying phase transitions are not quantitatively discussed
because we have simply applied the mean-field approximation. Future studies
could explore this issue further by using, for example, the cluster mean-field
approximation [41]. Also, future research on the phasons might extend the
explanations of the effect of quasiperiodicity on multifractality and
hyperuniformity [42]. The phasons are a kind of randomness, related to the
hyperspace of quasicrystals, and are realizable in quasicrystals only. The
Bose glass phase might be induced by the increases of the phasons in
quasicrystals.

## Acknowledgment

###### Acknowledgements.

This work was supported by JST SPRING (Grant No. JPMJSP2151) and the Japan
Society for the Promotion of Science, KAKENHI (Grant No. JP19H05821).

##  Appendix A Inflation rule

We generate Penrose and Ammann-Beenker tilings according to inflation method.
A inflation for a Penrose tiling is an operation of substitution illustrated
in Fig. 17(a). A Penrose tiling contains two types of rhombi: a “fat” rhombus
(left upper panel) and a “skinny” rhombus (right upper panel). Note that each
edge of these rhombi is distinguished by symbols of single and double arrows.
When a inflation operation acts on a tiling of $g$th generation, each
component of the tiling in upper panels is replaced with a set of tiles in
lower panels. We call the resulting tiling a tiling of $(g+1)$th generation.
The red edges of the tiles in lower panels are generated by inflations applied
to adjacent tiles.

We use a set of tiles in Fig. 17(b) as 1st generation of Penrose tiling. By
applying a inflation operation to Fig. 17(b), we obtain Fig. 17(c) as a tiling
of 2nd generation. A yellow fat rhombus in Fig. 17(b) results in a yellow-
shaded region in Fig. 17(c). Similarly, a blue skinny rhombus in Fig. 17(c)
results in a blue-shaded region in a tiling of 3rd generation illustrated in
Fig. 17(d).

We use open boundary condition. One can remove effects from edges by
considering the vertices inside a circular window near the center only. For
example, in Fig. 17(e), we show a red-dashed circular window with radius
$R_{\rm in}/D=6$ whose center is indicated by a red small circle. The number
of vertices inside the circular window is $N_{\rm in}=141$. Table 1 summarizes
$N$,$N_{\rm in}$, and $R_{\rm in}/D$ for each generation, which we use for
calculations in the main text. Figures 17(f-j) are the same as Figs. 17(a-e)
but for Ammann-Beenker quasicrystals. We have $N=1969$, $N_{\rm in}=1185$, and
$R_{\rm in}/D=25/\sqrt{2}$ in Fig. 17(j).

## References

  * [1] R. N. Araújo and E. C. Andrade: Phys. Rev. B 100, 014510 (2019).
  * [2] A. Koga: Phys. Rev. B 102, 115125 (2020).
  * [3] T. Fukushima, N. Takemori, S. Sakai, M. Ichioka, and A. Jagannathan: Phys. Rev. Res. 5, 043164 (2023).
  * [4] M. Hori, T. Sugimoto, T. Tohyama, and K. Tanaka: arXiv:2401.06355.
  * [5] T. C. Halsey, M. H. Jensen, L. P. Kadanoff, I. Procaccia, and B. I. Shraiman: Phys. Rev. A 33, 1141 (1986).
  * [6] S. Torquato, and F. H. Stillinger: Phys. Rev. E 68, 041113 (2003).
  * [7] A. Koga and S. Sakai: Phys. Rev. E 109, 044103 (2024).
  * [8] T. Tokihiro, T. Fujiwara, and M. Arai: Phys. Rev. B 38, 5981 (1988).
  * [9] S. Torquato: Phys. Rev. E 94, 022122 (2016).
  * [10] N. Macé, A. Jagannathan, P. Kalugin, R. Mosseri, and F. Piéchon: Phys. Rev. B 96, 045138 (2017).
  * [11] S. Torquato: Phys. Rep. 745, 0370 (2018).
  * [12] S. Sakai, R. Arita, and T. Ohtsuki: Phys. Rev. Res. 4, 033241 (2022).
  * [13] M. Sun, T. Čadež, I. Yurkevich, and A. Andreanov: Phys. Rev. B 109, 134504 (2024).
  * [14] M. Sbroscia, K. Viebahn, E. Carter, J.-C. Yu, A. Gaunt, and U. Schneider: Phys. Rev. Lett. 125, 200604 (2020).
  * [15] J.-C. Yu: Dr. Thesis, School of the Physical Sciences, University of Cambridge, Cambridge (2022).
  * [16] E. Gottlob and U. Schneider: Phys. Rev. B 107, 144202 (2023).
  * [17] J.-C. Yu, S. Bhave, L. Reeve, B. Song, and U. Schneider: arXiv:2303.00737.
  * [18] R. Ghadimi, T. Sugimoto, and T. Tohyama: Phys. Rev. B 102, 224201 (2020).
  * [19] R. Gautier, H. Yao, and L. Sanchez-Palencia: Phys. Rev. Lett. 126, 110401 (2021).
  * [20] M. Ciardi, A. Angelone, F. Mezzacapo, and F. Cinti: Phys. Rev. Lett. 131, 173402 (2023).
  * [21] M. P. A. Fisher, P. B. Weichman, G. Grinstein, and D. S. Fisher: Phys. Rev. B 40, 546 (1989).
  * [22] M. Puschmann, P. Cain, M. Schreiber, and T. Vojta: Eur. Phys. J. B 88, 314 (2015).
  * [23] B. Jäck, F. Zinser, E. J. König, S. N. P. Wissing, A. B. Schmidt, M. Donath, K. Kern, and C. R. Ast: Phys. Rev. Res. 3, 013022 (2021).
  * [24] R. V. Pai, J. M. Kurdestany, K. Sheshadri, and R. Pandit: Phys. Rev. B 85, 214524 (2012).
  * [25] J. M. Kurdestany, R. V. Pai, and R. Pandit: Ann. Phys. (Berlin) 524, 234 (2012).
  * [26] S. Sakai, R. Arita, and T. Ohtsuki: Phys. Rev. B 105, 054202 (2022).
  * [27] S. F. Edwards and P. W. Anderson: J. Phys. F: Metal Phys. 5, 965 (1975).
  * [28] G. Parisi: Phys. Rev. Lett. 50, 1946 (1983).
  * [29] T. Khellil and A. Pelster: J. Stat. Mech. 2016, 063301 (2016).
  * [30] S. J. Thomson, L. S. Walker, T. L. Harte, G. D. Bruce: Phys. Rev. A 94, 051601 (2016).
  * [31] R. Ghadimi, M. Hori, T. Sugimoto, and T. Tohyama: Phys. Rev. B 108, 125104 (2023).
  * [32] A. E. Niederle and H. Rieger: Phys. Rev. A 91, 043632 (2015).
  * [33] D. Johnstone, P. Öhberg, and C. W. Duncan: J. Phys. A: Math. Theor. 54, 395001 (2021).
  * [34] A. E. Niederle and H. Rieger: New J. Phys. 15, 075029 (2013).
  * [35] N.G. de Bruijn: Indag. Math. 84, 39 (1981).
  * [36] N.G. de Bruijn: Indag. Math. 84, 53 (1981).
  * [37] S. Sanders and M. Holthaus: New J. Phys. 19, 103036 (2017).
  * [38] P. Sengupta and S. Haas: Phys. Rev. Lett. 99, 050403 (2007).
  * [39] U. Bissbort and W. Hofstetter: Europhys. Lett. 86, 50007 (2009).
  * [40] U. Bissbort, R. Thomale, and W. Hofstetter: Phys. Rev. A 81, 063643 (2010).
  * [41] P. P. Gaude, A. Das, and R. V. Pai: J. Phys. A: Math. Theor. 55, 265004 (2022).
  * [42] S. V. Rajagopal, T. Shimasaki, P. Dotti, M. Račiūnas, R. Senaratne, E. Anisimovas, A. Eckardt, and D. M. André and Weld: Phys. Rev. Lett. 123, 223201 (2019).