#  Quantum bridge states and sub-dimensional transports in quasicrystals

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

Exotic tiling patterns of quasicrystals have gotten a lot of attention for
unique quantum phenomena such as critical state and multifractality. In this
regard, finding new quasi-periodic tiling patterns and the relevant quantum
states is one of the main interests in quasicrystals. Here, we focus on new
types of quasicrystals described by coexisting phason flips and discuss their
quantum transports with distinct local tiling patterns. It turns out that such
new tilings are uniquely descended from the translation of high-dimensional
lattices and their projection, in particular, a four-dimensional hypercubic
lattice. Such unconventional tiling gives rise to sub-dimensionally confined
states forming a bridge, so-called quantum bridge states. It turns out that
electrons in such states could be strongly localized by applying a magnetic
field at a particular time but are transported to the sites which share the
same local patterns along the bridge with time evolution. The speed of
transport is controlled via magnetic field strength, and one can even expect
electron transportation between multiple bridges where their numbers are
determined by the translation vectors of a four-dimensional hypercubic
lattice. Our work paves a way to discover anomalous quantum states and their
transports uniquely present in quasicrystals with new tiling structures.

##  I Introduction

Quasicrystal (QC), an ordered but not periodic structure, is famous for exotic
tiling patterns with fractality[1, 2, 3, 4, 5, 6, 7, 8, 9, 10]. They are
represented by self-similar structures as a compensation of the absence of
translational symmetry[11, 12, 13, 10, 14, 15, 8, 16, 11]. The tiling patterns
of quasicrystals are important because they give rise to novel quantum states,
for instance, a critical wave function, a power-law decaying wave packet,
which has been discovered in many distinct quasicrystals[17, 18, 19, 20, 21,
18, 19, 20, 17, 22, 20]. Beyond the critical states, another quantum states in
quasicrystals have been recently discussed under magnetic field[23]. In this
case, electrons are strictly localized forming each island with particular
length scales, similar to the Aharonov-Bohm-like cage, and such islands are
extensively distributed and interfere with each other under field control[24,
23]. Such states are uniquely present depending on quasi-periodic pattern.
Therefore, discovery of exotic tiling patterns and understanding their quantum
phenomena are important tasks.

In this regard, we focus on quasicrystals with new local tiling patterns,
which can be understood by cut-and-project scheme to construct
quasicrystalline structures from the high-dimensional lattice[4, 8, 25, 5].
According to the translations in high-dimensional lattice, the distinct local
tiling patterns emerge from the atomic rearrangement, which is also known as a
phason flip[12, 26, 27, 28, 29, 30, 31]. In particular, it is well known that
the Ammann-Beenker tiling results from a four-dimensional (4D) hypercubic
lattice and its projection. Interestingly, with particular lattice
translations in 4D hypercubic lattice and its projection, the QC structures
result in the phason flip and hence the atomic rearrangement gives rise to new
local tiling patterns distinct from the well-known Ammann-Beenker tiling[15,
32]. In such case, one could ask how electronic wave functions and their
transports are characterized.

In this paper, we consider the QC structures with local tiling patterns
descended from 4D hypercubic lattice translations, and study their unique
quantum states and transports emerging from new tiling patterns. In detail, we
consider two dimensional quasicrystals projected from 4D hypercubic lattice.
Distinct from the Ammann-Beenker tiling, certain translations in 4D lattice
result in new local tiling patterns forming one dimensional bridges when it is
projected. Considering such quasicrystal structures, we demonstrate the sub-
dimensionally localized eigenstates forming a bridge itself, so called a
quantum bridge state in the presence and absence of the magnetic field.
Interestingly, under magnetic field, the quantum bridge states give rise to an
interesting time-flying of the wave function that is not only confined on the
bridges but also jumping between the sites on the bridges which share the same
local pattern, leading to the sub-dimensional quantum transport. Moreover, we
show that the flux-control could manipulate both the localization of the
quantum bridge states and the mean current of such sub-dimensional quantum
transports. Hence, our finding clarifies the field-controlled sub-dimensional
quantum states and dynamics, purely originated from the novel quasicrystalline
tiling patterns.

##  II Quantum bridge states

In this section, we focus on quantum states in new tiling patterns, originated
from translations of a 4D hypercubic lattice. In the cut-and-project scheme,
the quasicrystal (QC) is descended from high-dimensional lattice points
belonging to the compact subspace[25, 8]. Hence, exotic tiling patterns of the
QCs depend on the transformation in the high-dimensional lattice[33, 32].
Particularly, the atomic positions are rearranged by the high-dimensional
lattice translations, which is known as a phason flip[26, 27, 28, 29, 30, 12].
In general, the phason flips do not significantly change the given tiling
pattern due to indistinguishable local tiling patterns[27]. However,
particular lattice translations in high-dimensional lattice allow an
interesting situation where two independent atomic arrangement related by
phason flip coexists. For instance, certain translation vectors in four-
dimensional hypercubic lattice make the two atoms (colored in blue and red at
the center in Fig.1(c)) coexist, where these two atomic arrangement is related
by the phason flips. In such a way, simple four-dimensional hypercubic lattice
shifts could alter the orders of rotational symmetry, and create a distinct
sub-dimensional quasiperiodic tiling pattern shown in Fig.1 (a) compared to
the eight-fold rotational symmetric Ammann-Beenker tiling[15, 32]. Depending
on the hypercubic lattice translations, the sub-dimensional tiling pattern as
described in a boxed region in Fig.1 (a) appears along at most four different
directions[32]. We name this sub-dimensional quasi-periodic tiling pattern as
bridges. See Supplemental Material for detailed information of the bridges
pattern. In Fig.1(a), we exemplify the QC with a bridge along the $x$-axis
resulting from a particular translation vector $a(1/2,0,0,0)$ in 4D hypercubic
lattice with a lattice constant $a$.

Figure 1:  (a) An example of QC descended from the four-dimensional lattice,
different from the Ammann-Beenker tiling. It is due to the particular
translation, $a(1/2,0,0,0)$, in four-dimensional hypercubic lattice with a
lattice constant $a$. The magnified image below shows the unique sub-
dimensional local tiling pattern, bridge distinct from the Ammann-Beenker
tiling. (b) The spatial distribution of a quantum bridge state without
magnetic field. The calculation is performed with the number of sites 13694.
(c) Typical example of the phason flip from the original atomic arrangement
(red color) to the new atomic arrangement (blue color). Outer six atoms are
not affected by this phason flip. For the hypercubic lattice translations,
blue and red atoms coherently coexist. This gives rise to a distinct sub-
dimensional tiling pattern, bridge from the Ammann-Beenker tiling. See the
main text for details.

In the QC shown in Fig.1(a), let us consider the tight-binding Hamiltonian
$H=t\sum_{\left\langle i,j\right\rangle}c_{j}^{\dagger}c_{i}+\mbox{h.c.}$
where $t$ is the uniform hopping magnitude and $i$ and $j$ sites are al length
$l=1$ apart from each other, where $l$ is the length of an edge of the
rhombus. Since the tiling pattern of the QC in Fig.1 (a) has the local
patterns of the Ammann-Beenker tiling except the bridges, they share the
similar eigenstate characters as the Ammann-Beenker tiling case. The important
question that could be asked here is the emergent states forming the sub-
dimensional localization on the bridge, so called quantum bridge states and
the influence of an external fields.

Fig.1 (b) illustrates a quantum bridge state when the magnetic field is
absent. Note that the wave function is localized on the bridge along the
$x$-axis. Let us now consider applying external magnetic field. It turns out
that the external magnetic field could enhance the localization due to
destructive interference forming Aharonov-Bohm like cages. In detail, the
tight-binding Hamiltonian under the magnetic field becomes
$H=\sum_{\left\langle
i,j\right\rangle}t_{ij}c_{j}^{\dagger}c_{i}+\mbox{h.c.}$, where the hopping
parameter from the $i$-site to the $j$-site becomes[34],

$\displaystyle
t_{ij}=t\exp{\left(i\int_{\vec{r}_{i}}^{\vec{r}_{j}}\vec{A}(\vec{r})\cdot
d\vec{r}\right)}.$ (1)

Here, $\vec{A}$ is the magnetic vector potential, $\vec{r}_{i}$ and
$\vec{r}_{j}$ are the position vectors of the $i$ site and the $j$ site,
respectively. Throughout the paper, we choose the Hartree atomic units,
$\hbar=e=1$. For Landau gauge, $\vec{A}(\vec{r})=Bx\hat{y}$, where $B$ is the
strength of the magnetic field along $z$-axis, let us parametrize the edge
from $\vec{r}_{i}=(x_{i},y_{i})$ to $\vec{r}_{j}=(x_{j},y_{j})$ as
$\vec{r}(s)=(1-s)\vec{r}_{i}+s\vec{r}_{j}$ for $0\leq s\leq 1$. Then, the
hopping parameter with Peierls phase is given by,

$\displaystyle
t_{ij}=t\exp{\left(-i\pi\frac{\Phi(x_{i}+x_{j})(y_{i}-y_{j})}{2\phi_{0}}\right)},$
(2)

where $\Phi$ is the flux through the unit square and $\phi_{0}$ is a flux
quanta. Then, the Peierls phases in Eq.(2) can give rise to the destructive
interference due to the exotic local tiling pattern of QCs, resulting in the
enhancement of localization[23].

Figure 2:  A quantum bridge state under the magnetic field,
$\Phi=1.35\phi_{0}$. (a) The sub-dimensional quantum bridge state along the
$x$-axis. Its probability distributions ($|\psi_{i}|^{2}$) are illustrated as
functions of the $y$-position (b) or the $x$-position (c). (d) Fidelity of
each eigenstate and $\ket{\psi_{0}}$ as a function of IDOS, where
$\ket{\psi_{0}}$ is a $\delta$-function-like initial wave function placed on a
particular $A$-site in the inset of (a). The quantum bridge state illustrated
in (a) has the highest fidelity. Each marker indicates the fidelity with each
eigenstate. The calculation is performed with the number of the sites, 13694.
See the main text for details.

Fig.2 (a) shows the typical example of the quantum bridge state with
$\Phi=1.35\phi_{0}$. Figs.2 (b,c) illustrate the probability distributions of
the quantum bridge state as the functions of the $y$ and $x$ positions,
respectively. Fig.2 (b) shows that the quantum bridge state is more strongly
localized by the magnetic field compared to the case without the magnetic
field. This is because the Peierls phase in Eq.(2) induces the destructive
interference of wave function dependent on the local tiling pattern.
Furthermore, the probability distribution along the bridge illustrated in
Fig.2 (c) also shows that the wave function is localized along the bridges
under the magnetic field. Specifically, the quantum bridge state is highly
localized along the sites that share the same local pattern of the $A$-site
shown in the inset of Fig.2 (a). We emphasize that the local pattern of the
$A$-site results from the existence of atoms at both positions related by
phason flips and these sites are uniquely emergent due to the special
hypercubic translations. Such sites are extensively present along the bridge
since these sites repeatedly appear along the bridge.

Now let us consider the time evolution of such localized electron wave
function. Let $\ket{\psi_{0}}$ be the initial state that is the
$\delta$-function-like wave function placed on a particular $A$-site as shown
in the inset of Fig.2 (a). For a given set of eigenstates
$\\{\ket{\phi_{n}}\\}$, where $n$ is the index for $n$-th energy, the initial
state would be represented by the linear combination of the eigenstates. Thus,
with the coefficients $c_{n}=\braket{\phi_{n}}{\psi_{0}}$,

$\displaystyle\ket{\psi_{0}}=\sum_{n}c_{n}\ket{\phi_{n}}.$ (3)

Its dynamics is given by,

$\displaystyle\ket{\psi(\tau)}=\sum_{n}c_{n}\exp{(-i\tau
E_{n})}\ket{\phi_{n}}.$ (4)

Here, $E_{n}$ is the $n$-th eigenvalue, and $\tau$ is the time. When almost
all of the $c_{n}$ are negligible except for a few $n$, the dynamics originate
from such a few dominant eigenstates. Fig.2 (d) shows the fidelity,
$\left|c_{n}\right|^{2}$ of each $\ket{\phi_{n}}$ and $\ket{\psi_{0}}$ as a
function of the integrated density of state (IDOS) in the presence of magnetic
field($\Phi=1.35\phi_{0}$). The eigenstates having a significant fidelity are
the quantum bridge states. Thus, under the magnetic field, the time-flying of
the state $\ket{\psi_{0}}$ leads to the sub-dimensional transport along the
bridges that is jumping between the sites as we demonstrate in the next
section.

##  III Transports along the bridges

Now we discuss the sub-dimensional quantum transport on the bridges and its
controllability under the magnetic field. Here, the external magnetic field
would play a role of the adjuster that controls the transport current. For a
concrete argument, we demonstrate the sub-dimensionally confined quantum
transports along the bridge on the $x$-axis.

Initially, let us consider the $\delta$-function-like wave function placed on
the $A$-site shown in the inset of Fig.2 (a). Fig.3 (a) illustrates its time-
flying. The top panel of Fig.3 (a) shows the initial state $\ket{\psi_{0}}$.
With the time evolution $\tau$, the wave function jumps to the sites having
the same local pattern on the bridge (See the middle and bottom panels of
Fig.3 (a) for $\tau=1500t^{-1}$ and $\tau=3000t^{-1}$.). This results in sub-
dimensional transport along the bridges.

Figure 3:  Quantum transport demonstrated on a bridge along $x$-axis for
$\Phi=1.35\phi_{0}$. (a) Top panel shows the initial $\delta$-function like
wave function at time $\tau=0$. At time $\tau=1500t^{-1}$ and $3000t^{-1}$,
the electron wave functions evolve to the sites having the same local pattern
of the initial site as shown in the middle and bottom panels, respectively.
(b) The time-averaged mean current of the transport, $\bar{J}_{x}$, as a
functions of flux $\Phi$. Current drastically changes with flux. Time-averaged
$\bar{J}_{x}$ is calculated during the time $\tau=3000t^{-1}$. See the main
text for details.

One could ask how such transport could be controlled via magnetic flux change.
For quantitative argument, we consider the mean current defined by,

$\displaystyle\bar{J}_{x}=-e\frac{i}{\tau}\int_{0}^{\tau}d\tau^{\prime}\left\langle[\hat{x},H]\right\rangle(\tau^{\prime}).$
(5)

Here, $\tau$ is the time for averaging. As shown in Fig.3 (b), $\bar{J}_{x}$
is indeed dependent on the magnetic flux. This originates from the magnetic
flux dependence of localization of the quantum bridge states.

One could generalize above argument related to the sub-dimensional transport
for the case of more number of bridges. Note that the number of bridges depend
on the translation of the four-dimensional hypercubic lattice. We demonstrate
the quantum transports with two and four bridges in Supplemental Materials. As
the number of bridges increases, the wave function can be transmitted in at
most eight different directions. We would suggest the angle-dependent current
measurement controlled by magnetic flux along the bridges.

##  IV Conclusion

In summary, we discussed the new quantum phenomena relevant to the exotic
tiling pattern in quasicrystals. Particular translations in four dimensional
hypercubic lattice, give rise to the atomic rearrangement where two distinct
phason flips coexist and thus result in new local tiling patterns. In such
system, we discovered the sub-dimensionally localized quantum bridge states
and their relevant transports. In the presence of particular magnetic field,
the wave function tends to be more strongly localized on the bridges forming
the Aharonov-Bohm like cages. It allows the electron jumps from one site to
others that share the same local pattern with time evolution. Interestingly,
the amount of current can be also controlled in terms of magnetic flux.
Furthermore, the number of bridges which are the paths of the sub-dimensional
transport, depends on the four-dimensional hypercubic lattice translations.
Our study paves a way for finding unique quantum transports and their magnetic
field control in quasicrystals with distinct local tiling patterns.

## Acknowledgements

This work is supported by National Research Foundation Grant
(NRF-2020R1A4A3079707, NRF grand 2021R1A2C1093060).

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Supplementary Materials of Quantum bridge states and
sub-dimensional transports in quasicrystals

Junmo Jeon, SungBin Lee

##  Appendix A Review: New tiling patterns — bridges

Here, we briefly review new tiling pattern, say bridges pattern derived from
some higher-dimensional translation in the cut-and-project scheme. Starting
from a standard example of the quasicrystal, the Ammann-Beenker tiling
descended from four-dimensional hypercubic lattice and its projection, we have
found that the simple four-dimensional hypercubic lattice shifts could alter
the order of rotational symmetry, and create a new quasicrystalline tiling
pattern.

Let us briefly review the cut-and-project scheme (CPS). The CPS is a way to
understand non-crystallographic symmetry of the low-dimensional quasicrystals
in terms of the high-dimensional lattices[25]. Let high-dimensional cubic
lattice $\mathcal{L}=\\{\bm{x}|\bm{x}=m_{i}\bm{e}_{i},m_{i}\in\mathbb{Z},1\leq
i\leq D\\}$, where $D$ is the dimension of the lattice and $\bm{e}_{i}$ is a
standard unit vector. The $D$-dimensional space is decomposed into 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(QC)
(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 entire lattice
points, $\bm{\pi}(\mathcal{L})$, densely cover the physical space. Thus, one
should select a subset of $\mathcal{L}$, which is the compact subset of the
perpendicular space $\bm{\pi}^{\perp}$ that is closed and bounded. This
compact subset is often termed as the window, say $K$. Now, we project the
lattice points only when those image of $\bm{\pi}^{\perp}$ belongs to the
window $K$. Then, the resultant projection image onto the physical space
emerges 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[25, 35, 36].

To illustrate the details, let us consider the example of the CPS, the Ammann-
Beenker tiling. The Ammann-Beenker tiling is generated by the CPS with four-
dimensional hyper-cubic lattice with $D=4$. The Wigner-Seitz cell is given by
the polytope whose 16 verticies are
$(\pm\frac{1}{2},\pm\frac{1}{2},\pm\frac{1}{2},\pm\frac{1}{2})$ where the
signs of each component are independent. The $\bm{\pi}$ and $\bm{\pi}^{\perp}$
projections are given by,

$\displaystyle\bm{\pi}$
$\displaystyle=\begin{pmatrix}1&\frac{1}{\sqrt{2}}&0&-\frac{1}{\sqrt{2}}\\\
0&\frac{1}{\sqrt{2}}&1&\frac{1}{\sqrt{2}}\end{pmatrix}$ (S1)
$\displaystyle\bm{\pi}^{\perp}$
$\displaystyle=\begin{pmatrix}1&-\frac{1}{\sqrt{2}}&0&\frac{1}{\sqrt{2}}\\\
0&\frac{1}{\sqrt{2}}&-1&\frac{1}{\sqrt{2}}\end{pmatrix}.$ (S2)

Then, the window $K$ becomes a tesseract graph whose vertices are
$\bm{\pi}^{\perp}$-projection images of the vertices of the Wigner-Seitz cell
(See Fig.S1 (a).)[8]. The Ammann-Beenker tiling (See Fig.S1 (c)) is given by
the $\bm{\pi}$-projection of four-dimensional hyper-cubic lattice points,
whose image of $\bm{\pi}^{\perp}$ belongs to the window $K$ in Fig.S1 (a).

Figure S1:  Schematic illustration of formation of the quasicrystals with
coexisting phason flips from a translation operation in the high-dimensional
lattice. The unit cell of four-dimensional hypercubic lattice – a tesseract,
is drawn at the center. (The faint image in the back is a tesseract before
translation with a shifting vector $\gamma$.) (a) The compact window formed by
the $\bm{\pi}^{\perp}$-projection of the Wigner-Seitz cell. The special set of
translations given by the shift vector $\bm{\gamma}$ (green arrow) of the
high-dimensional lattice generates quasicrystals with distinct tiling pattern
with coexisting atomic positions which are related by the phason flips. Before
translation, the $\bm{\pi}$-projected quasicrystal is the 8-fold symmetric
Ammann-Beenker tiling. (c) With translation by
$\bm{\gamma}=(x,y,z,w)=\frac{1}{2}(1,0,1,0)$, the $\bm{\pi}$-projection
results in a 4-fold symmetric QC with new local tiling pattern. In both (b)
and (c), the local 8-fold symmetry is shown as marked in orange circle. (d)
Enlarged image of the new sub-dimensional tiling pattern shown as a blue
circle in (c), which is absent in the pure Ammann-Beenker tiling (b).

In general, the higher-dimensional lattice translation in CPS gives unique
degrees of freedom of quasicrystalline structure that is the rearrangement of
atoms called by the phason flips. According to the phason flips, the tiling
pattern of the quasicrystal could be locally deformed. Two or more atomic
arrangement would locally compete depending on the higher-dimensional lattice
translation as the phason flips. For the case that one of the competing atomic
arrangement is favored, the kind of local tiling patterns would not be
drastically changed, and hence the tilings are locally isomorphic. However,
for some special higher-dimensional lattice shifts, the locally competing
atomic arrangements would be coexisting, thus, the distinct local tiling
patterns would be appeared.

As a specific example, let us consider the CPS for the Ammann-Beenker tiling
illustrated in Fig.S1 again. Generally, for a hypercubic lattice shift,
$\bm{\gamma}$, the tiling pattern could be locally altered by selecting one of
two competing local atomic arrangements. However, for a special
$\bm{\gamma}=\frac{1}{2}(1,0,1,0)$, the compact (closed and bounded) window
accepts both competing local atomic positions related by the phason flilp.
FigS1 (d) shows the local tiling pattern comprised of many such coexisting
atomic arrangements related by the phason flips. It turns out that the
distinct quasicrystalline tiling pattern in Fig.S1 (c) from the pure Ammann-
Beenker tiling is formed at such special hypercubic lattice translation
$\bm{\gamma}$.

The new tiling pattern called by the bridge illustrated in Fig.S1 (d) is sub-
dimensionally extending. The directions of the extension of the bridges are
depending on the hypercubic lattice shift $\bm{\gamma}$, and they could be
1,2, and 4 different directions depending on the $\bm{\gamma}$[32]. See Sec.C
for the demonstration of the sub-dimensional quantum transport along the
bridges with two and four different directions.

Although the randomized phason flips just give defect, here, the coexisting
phason flips appear in a quasi-periodically ordered way. Hence, the tiling
pattern in Fig.S1 (d) is ordered quasicrystalline structure rather than random
defects.

##  Appendix B Irregular diffusion on the bulk

For comparison, we investigate a typical dynamics in the new quasicrystalline
tiling starting from the initial $\delta$-function-like wave function placed
on a random bulk site out of the bridge pattern (bllue shaded region in
Fig.S2). In detail, Fig S2 illustrates the probability distribution (colored)
of the wave function evolved from the initial position $X_{0}$ out of the
bridge pattern after time $\tau=3000t^{-1}$. Note that the quasicrystal has
the bridge pattern along the $x$-axis (blue shaded region). Fig.S2
demonstrates that the dynamics of the wave function becomes the irregular
diffusion in the bulk if the initial position is out of the bridges pattern.
Here, the flux is given by $1.35\phi_{0}$ where $\phi_{0}$ is the flux quanta.
Hence, we can conclude that the electronic transport along the bridges
discussed in the main text is indeed an unique phenomenon dependent on the
initial position.

Figure S2:  Time evolution after time $\tau=3000t^{-1}$ of the
$\delta$-function-like wave function initially placed on general bulk site,
$X_{0}$ out of the bridges represented by the blue shaded region. Here, $t$ is
the hopping magnitude in the tight-binding Hamiltonian. See the main text for
detailed infromation of the Hamiltonian. The magnetic flux is $1.35\phi_{0}$.
The quasicrystal is two-fold symmetric with the bridges pattern along
$x$-axis.

##  Appendix C Transports with more number of the bridges

The number of new tiling pattern, bridges could be two and four depending on
the hypercubic lattice shifts[32]. Thus, for completeness of our discussion,
we illustrate similar quantum transports with two and four numbers of bridges.
Figs.S3 (a,b) and (c,d) demonstrate the quantum transports along with the two
and four numbers of bridges, respectively. The $A$ in Figs.S3 (a,c) indicate
the initial positions whose local patterns are the same as the local pattern
of the $A$ site discussed in the main text.

To be more specific, Figs.S3 (a,c) and (b,d) show the probability
distributions after time $\tau=1000t^{-1}$ and $\tau=3000t^{-1}$,
respectively. Here, the fluxes are set to be $1.35\phi_{0}$ where $\phi_{0}$
is the flux quanta. Note that the quantum transports are both sub-dimensional
along the bridges patterns. Importantly, as the numbers of the bridges pattern
increases, the sub-dimensional wave function transports can occur in various
directions. It turns out that the angle-dependent sub-dimensional transports
can emerge with different numbers of bridges.

Figure S3:  New quasicrystals descended from the four-dimensional lattice
translations (a,b) $1/2(1,0.1,0)$ and (c,d) $1/2(1,1,1,1)$ where the lattice
constant of the hypercubic lattice is unity. Each new quasicrystal has four-
fold (a,b) and eight-fold (c,d) symmetric bridges patterns. Dashed lines are
drawn to emphasize the bridges tiling pattern region. Under the flux
$\Phi=1.35\phi_{0}$, (a,c) and (b,d) show the probability distributions after
time $\tau=1000t^{-1}$ and $\tau=3000t^{-1}$, respectively. The initial
positions of the $\delta$-function-like wave function are represented by $A$
in (a,c) whose local patterns are the same as the local pattern of the
$A$-site discussed in the main text.