# Higher-order topology in Fibonacci quasicrystals

Chaozhi Ouyang1,2 Qinghua He1,2 Dong-Hui Xu3<EMAIL_ADDRESS>Feng
Liu1,2,4<EMAIL_ADDRESS>1Institute of High Pressure Physics, Ningbo
University, Ningbo, 315-211, China  2School of Physical Science and
Technology, Ningbo University, Ningbo, 315-211, China  3Deparment of Physcis &
Chongqing Key Laboratory for Strongly Coupled Physics, Chongqing University,
Chongqing 400044, China  4Department of Nanotechnology for Sustainable Energy,
School of Science and Technology, Kwansei Gakuin University, Gakuen 2-1, Sanda
669-1337, Japan

###### Abstract

In crystalline systems, higher-order topology, characterized by topological
states of codimension greater than one, typically arises from the mismatch
between Wannier centers and atomic sites, leading to filling anomalies.
However, this phenomenon is less understood in aperiodic systems, such as
quasicrystals, where Wannier centers are not well defined. In this study, we
examine Fibonacci chains and squares, a quintessential type of quasicrystal,
to investigate their higher-order topological properties. We discover that
topological interfacial states, including corner states, can be inherited from
their higher-dimensional periodic counterparts, such as the two-dimensional
Su-Schrieffer-Heeger model. This finding is validated through numerical
simulations of both phononic and photonic Fibonacci quasicrystals by the
finite element method, revealing the emergence of topological edge and corner
states at interfaces between Fibonacci quasicrystals with differing topologies
inherited from their parent systems. Our results not only provide insight into
the higher-order topology of quasicrystals but also open avenues for exploring
novel topological phases in aperiodic structures.

##  I Introduction

Over the past two decades, the concept of topology has become a cornerstone in
classifying solid-state materials, revealing essential differences even for
materials with seemingly identical energy-band structures. This paradigm
shift, heralded by the topological band theory, has profoundly reshaped our
understanding of solid-state physics [1, 2, 3]. One of the most striking
properties of topological band theory is the bulk-edge correspondence,
connecting the bulk topological invariant in crystalline systems to the
emergence of robust interfacial states in finite, topologically distinct
samples [4, 5, 6, 7, 8]. In this evolving landscape, the exploration of
higher-order topological phases, which deviate from the conventional
topological insulators by exhibiting bulk-corner correspondence, has gained
prominence in recent years. These higher-order topological phases allow for
topological states of codimension greater than one, such as corner states,
with potential applications in topological lasers and quantum computing [9,
10, 11, 12, 13, 14, 15, 16, 17, 18, 19].

Significant progress has been made in understanding higher-order topological
states in crystalline systems, as evidenced by a wealth of studies. These
studies could be exemplified through several models based on filling anomalies
induced by the mismatches between Wannier centers and atomic sites [20, 21,
22, 23, 24]. Protected by various symmetries [25, 26, 27, 28], such as point
group symmetry [29, 30, 31], inversion symmetry [32], and chiral symmetry [33,
34], higher-order topological states are observed not only in solid-state
materials [35, 36, 37, 38, 39, 40, 41], but also in metamaterials, such as
photonic [42, 43, 44, 45, 46, 47], sonic crystals [48, 49, 50, 51, 52, 53,
54], and electric circuit arrays [55, 56, 57].

A less explored but equally interesting direction is the exploration of these
states in systems lacking transnational symmetry, such as quasicrystals. These
ideas have recently been explored in two-dimensional (2D) quasicystals [58,
59, 60, 61, 62, 63], which requires a fine-tuned mass term to generate a
higher-order topology. While these initial studies have shed some light on the
interplay between aperiodicity and high-order topology, the fine-tuned mass
may hinder the realization of higher-order topology in quasicrystals. A common
approach to probing the topology in quasicrystals involves examining their
higher-dimensional parent systems. For instance, the topology of a one-
dimensional (1D) Fibonacci quasicrystal can be elucidated through its mapping
to a 2D periodic ancestor in the Harper model, which can be characterized by
the Chern number [64, 65, 66].

Drawing inspiration from these foundational studies, our research shifts focus
to a modified Fibonacci quasicrystal, whose higher-dimensional ancestor is the
2D Su-Schrieffer-Heeger (SSH) model. The 2D SSH model has several distinct
topological phases characterized by the vector Zak phase or equivalently the
Wannier centers, depending on the ratio between intra-cell and inter-cell
hopping. In the nontrivial topological phase of the 2D SSH model, filling
anomalies give rise to topological corner states in addition to edge states.
By extending the Fibonacci chains to Fibonacci squares, it is possible to
explore higher-order topological interfacial states and investigate the
preservation of topological properties from high-dimensional parent systems to
their lower-dimensional quasicrystal counterparts. Such preservation of
topological property cross dimensions without translational symmetry can be
verified using artificial structures such as the phononic and photonic
quasicrystals, which may offer a hint peeping at the topology of various
quasicrystals.

In this work, we employ the cut-and-project method to construct modified
Fibonacci chains and squares from the 2D SSH model. We introduce topological
interfaces in these structures, derived from parent systems belonging to
different topological classes, and investigate the emergence of topological
edge and corner states. These states are observed in phononic and photonic
quasicrystals through numerical simulations using the finite element method by
COMSOL. Our findings suggest that lower-dimensional systems, even in the
absence of translational symmetry, can exhibit topological states influenced
by the distinct topological classifications of their higher-dimensional
periodic ancestors. This research contributes to a deeper understanding of
topology in aperiodic systems and may offer new insights into the design and
application of topological materials.

The structure of the remaining parts is organized as follows. In Sec. II, we
introduce the modified Fibonacci chains and the corresponding photonic and
phononic quasicrystals whose parent system is the 2D SSH model using the cut-
and-project method. In Sec. III, we discuss the topological interfacial states
formed in photonic and phononic Fibonacci chains. In Sec. IV, we extend the
modified Fibonacci chains to Fibonacci squares and explore higher-order
topological phases hosting corner states in the phononic and photonic
Fibonacci squares. Finally, the discussion and summary are given in Sec. V.

##  II Construction of modified Fibonacci chains

The Fibonacci chain, a 1D quasicrystal, is constructed using a recursive
relation, $F_{n}=F_{n-1}+F_{n-2}$. Starting with $F_{1}=F_{2}=1$,
$F_{n-1}/F_{n-2}$ asymptotically approaches the golden ratio
$\tau=(1+\sqrt{5})/2$ as $n$ becomes large. In our model, these elements
$F_{1}$ and $F_{2}$ represent two different types of atomic sites, denoted as
letters “A” and “B”. These sites differ in terms of spacing distances and
resulting hopping amplitudes, similar to the SSH model [67]. Applying the
Fibonacci sequence, we can construct a simple quasicrystal exemplified by an
atomic chain such as ”ABBABABBABBA$\cdots$”.

The tight-binding Hamiltonian of the Fibonacci chain can be written as

$\mathcal{H}=\sum_{n}t_{n}c^{\dagger}_{n+1}c_{n}+\text{H.c.},$ (1)

where $t_{n}=(t+\lambda V_{n})$ denotes the hopping amplitude between site $n$
and $n+1$, $V_{n}=2[\lfloor(n+2)/\tau\rfloor)-\lfloor(n+1)/\tau\rfloor)]-1$ is
the characteristic function with $\lfloor x\rfloor$ the floor function, and
$\lambda$ is the hopping difference between A and B sites. It is noted that
the characteristic function can be equivalently obtained by the cut-and-
project method, where the higher-dimensional parent system is a simple 2D
square lattice for the conventional Fibonacci chain [65].

In the following, we extend the conventional Fibonacci chain to a modified one
through the cut-and-project method, whose higher-dimensional parent system is
the 2D SSH model, which has several distinct topological phases identified by
the Wannier centers. We expect these modified Fibonacci chains to inherit
topological properties including higher-order ones from their parent systems,
and we verify this observation through numerical simulations of phononic and
photonic quasicrystals.

###  II.1 cut-and-project method

Figure 1: (a) Schematic of the cut-and-project for the modified Fibonacci
chain. The higher-dimensional parent system is the 2D SSH model, which has
four sites in one unit cell. When the projection angle is $\tan\theta=1/\tau$,
Fibonacci sequence forms with parts A and B differed by the distances, i.e.,
$(l+d)\cos\theta$ and $(l+d)\sin\theta$. (b) Parts A and B of the modified
Fibonacci chain and the corresponding phononic quasicrystal structure. The
phononic modified Fibonacci chain is constructed by mapping the distance
between sites to the width of coupling waveguide as $D(L)=1/L^{2}$. (c)
Illustration of the modified phononic Fibonacci chain.

Construction of the Fibonacci chain using the cut-and-project method provides
an intuitive approach to understanding its structure. As illustrated in Fig.
1(a), this method starts with a parent 2D square lattice. The Fibonacci
quasicrystal is then generated by projecting these lattice sites onto an axis
labeled $x^{\prime}$. The orientation of this axis is critical, defined by an
angle $\theta$ such that $\tan\theta=\omega$, where $\omega=1/\tau$. The
projection strip used for this operation has a width equal to one unit cell of
the lattice. As a result, the spacing between the projected centers of the
unit cells can take on two distinct values: $\cos\theta=1/\sqrt{1+\omega^{2}}$
and $\sin\theta=\omega/\sqrt{1+\omega^{2}}$. These spacing values correspond
to the A and B sites of the Fibonacci chain, as previously mentioned.

Incorporating the 2D SSH model into the cut-and-project method, we obtain a
modified Fibonacci chain. The topological phases of the 2D SSH model are
distinguished by two tpes of hopping: inter-cell hopping
$\gamma_{\text{inter}}$ and intra-cell hopping $\gamma_{\text{intra}}$. As
illustrated in Fig. 1(a), these hoppings are represented with varying
distances and line thicknesses. Crucially, the 2D SSH model exhibits four
distinct topological phases, which can be characterized by the vector Zak
phase, or equivalently, by the Wannier center coordinates $(\nu_{x},\nu_{y})$,
where $\nu_{i}$ can be either 0 or 1/2 in terms of the lattice constant along
the $i$-direction. For the symmetric case under the point group $C_{4}$ that
we focus on, $\nu_{x}=\nu_{y}=0$ for
$\gamma_{\text{intra}}>\gamma_{\text{inter}}$, and $\nu_{x}=\nu_{y}=1/2$ for
$\gamma_{\text{intra}}<\gamma_{\text{inter}}$. The interplay between these
hopping parameters in the 2D SSH model directly influences the topological
characteristics of the resulting modified Fibonacci chain, providing a rich
framework for exploring topological phenomena in quasicrystalline structures.

In the 2D SSH model, each unit cell contains four sublattices. By projecting
these sublattices onto the $x^{\prime}$ axis, as shown by the red dots in Fig.
1(a), we derive the modified Fibonacci chain. This projection results in two
distinct types of tiling within the chain, labeled as parts A and B, similar
to the conventional Fibonacci chain. However, a key difference in the modified
version is that each of these parts encompasses four sites. The positions of
these four sites along the $x^{\prime}$ axis are determined by the following
euqations:

$\begin{split}x^{\prime}_{a}=(nl+\frac{l}{2})\sin\theta+(ml-\frac{l}{2})\cos\theta+md\cos\theta+nd\sin\theta\\\
x^{\prime}_{b}=(nl+\frac{l}{2})\sin\theta+(ml+\frac{l}{2})\cos\theta+md\cos\theta+nd\sin\theta\\\
x^{\prime}_{c}=(nl-\frac{l}{2})\sin\theta+(ml+\frac{l}{2})\cos\theta+md\cos\theta+nd\sin\theta\\\
x^{\prime}_{d}=(nl-\frac{l}{2})\sin\theta+(ml-\frac{l}{2})\cos\theta+md\cos\theta+nd\sin\theta,\end{split}$
(2)

where $(m,n)$ represents the index of the unit cell in the 2D SSH model,
$a,b,c,d$ are the labels of the sublattices as indicated in Fig. 1(a), $l$ is
the distance between sites within a unit cell, and $d$ is the distance between
adjacent unit cells. The values of $(m,n)$ are constrained by the relation:

$\frac{n-1}{m+1}<\tan\theta\leq\frac{n}{m}.$ (3)

It is important to note that for $d>l$, there is a possibility for the sites
of neighboring parts A and B to overlap, a factor that must be considered in
analyzing the structure of the modified Fibonacci chain.

###  II.2 modified phononic and photonic Fibonacci quasicrystals

We translate the theoretical framework of the cut-and-project method from the
parent 2D SSH model into tangible constructs of phononic and photonic
Fibonacci chains. Starting with the photonic quasicrystal, we follow a
straightforward method: artificial atoms, such as dielectric rods, are
positioned in accordance with the projected sites on the $x^{\prime}$ from the
2D SSH model as indicated by Eq. (2). This arrangement is depicted in the left
panel of Fig. 1(b). In our numerical simulations, we set the radius of the
dielectric rod at $0.2~{}\text{[mm]}$ and the dielectric constant at 100 on an
air background. We employ a perfect electric conductor as the boundary
condition in the simulation of photonic quasicrystals. Notably, because of the
absence of a characteristic length in Maxwell’s equations, the size of the
photonic quasicrystal can be adjusted without affecting its fundamental
properties. The parameters selected here serve illustrative purposes only.
This approach effectively replicates the modified Fibonacci chain,
encapsulating its long-range interactions, and allowing us to explore its
topological properties in photonic analogues.

For the phononic quasicrystal, we use square resonators connected by
waveguides. The coupling between each resonator is proportional to the width
of the waveguides, and we map the distance between the sites to the width of
the waveguide as $D=1/L^{n}$ with $n=2$. It should be noted that for other
mapping functions, the simulation results remain qualitatively similar. For
our simulations, we set the dimensions of each square resonator to 30 [mm] in
length, with the waveguides also at a length of 30 [mm]. The uniformity of
waveguide length ensures consistent propagation characteristic throughout the
whole phononic quasicrsytals. The boundary condition applied in the simulation
of phononic quasicrystals is the hard boundary condition, which ensures that
the sound pressure is zero at the boundaries, mimicking an isolated system.
Figure 1(c) illustrates an example of the modified phononic Fibonacci
quasicrystal, where the various shades are used to denote parts A and B.

##  III topological interfacial states

In this section, we delve into the phenomenon of topological interfacial
states, which arise at the junction of two topological distinct systems. Our
focus is on the demonstration of the emergence of these topological
interfacial states in modified Fibonacci chains derived from phononic and
photonic quasicrystals, each with distinct topological characteristics
inherited from parent systems. This demonstration is intriguing because, in
quasicrystals, the concept of a Wannier center is not well defined because of
the absence of translational symmetry, unlike the case of the SSH model.

In addition to demonstration of the emergence of topological interfacial
states, symmetry protection of topological states in quasicrystals is also
important. Like in the SSH model, chiral symmetry pins the energy of edge
states at zero. The role of chiral symmetry in these quasicrystalline systems
presents a rich area for exploration, potentially revealing new insights into
symmetry protection in topological states. An in-depth investigation of
chiral-symmetry protection in the topological interfacial states of
quasicrystals is reserved for future work.

Figure 2: (a) Topological interface formed by the two types of modified
phononic Fibonacci chains, whose higher-dimensional parent systems belonging
two topological classes of the 2D SSH model. The coupling between two chains
is denoted by $t_{\tau}$ and controlled by the width of coupling waveguide.
(b) Eigenspectra of the two types of modified phononic Fibonacci chains in
(a), as displayed by left and right panels, respectively. (c) Eigenspectra of
the combined structure of the two types of modified phononic Fibonacci chains.
(d) Wave profile of the topological interfacial state as sound pressure in the
setup of (a).

###  III.1 phononic topological interfacial states

We construct a topological interface by joining two phononic Fibonacci chains,
each originating from the 2D SSH model but with distinct topological
characteristics. This interface, as illustrated in Fig. 2(a), servers as a
physical realization of the boundary between different topological phases of
the quasicrystals. The left phononic chain is adopted from the 2D SSH model
with $l=1\text{[mm]}$ and $d=3~{}\text{[mm]}$, effectively resulting in a
scenario of $\gamma_{\text{intra}}>\gamma_{\text{inter}}$. On the contrary,
the right phononic chain adopts the parameters $l=3~{}\text{[mm]}$ and
$d=~{}1\text{[mm]}$, which correspond to
$\gamma_{\text{intra}}<\gamma_{\text{inter}}$.

In Fig. 2(b), we display the eigenspectra of the two topologically distinct
types of phononic Fibonacci chains derived from the 2D SSH model. There appear
multiple gaps in the eigenspectra similar to the conventional Fibonacci chain
defined in Eq. (1). The contrast in the eigenspectra of the modified Fibonacci
chain is the four subbands in the middle, which reflect the underlying
structure of the four sublattices in the 2D SSH model, as displayed by the
left panel of Fig. 2(b). In the 2D SSH model, it is observed that the two
different topological phases can be exchanged by redefining the unit cell
following a half-period shift. This shift results in a pattern mapping between
the two topologically distinct Fibonacci chains, where part A is transformed
into part B and part B becomes BA. However, despite this transformation, parts
A and B in the two chains remain nonequivalent due to the different values of
$l$ and $d$, which do not result in the closure of the band gap in their
parent systems.

As displayed in Fig. 2(c), we observe the emergence of a topological
interfacial state within the shared gap of the two phononic Fibonacci chains.
Up to the frequency that we solve, there is only one topological interfacial
state. By adjusting the coupling strength, denoted as $t_{\tau}$, between the
two chains, we can effectively modulate the frequency of this state within a
certain range. This tunability offers a versatile tool for manipulating the
properties of the interfacial state for potential applications.

The wave profile of the topological interfacial state, as shown in Fig. 2(d),
reveals another critical feature: the state exhibits expected exponential
decay along both directions from the interface. This decay is a hallmark of
topological interfacial states, indicating the localized nature of these
states at the interface of the two distinct topological phases.

The total number of sites in parts A and B of the simulation is 48, which
corresponds to the Fibonacci sequence “ABBABABBABBA”. For a larger number of
sites, the eigenspectra remain qualitatively the same, and the band gaps do
not close.

Figure 3:  (a) Topological interface formed by the two types of modified
photonic Fibonacci chains, whose higher-dimensional parent systems belonging
two topological classes of the 2D SSH model. The coupling between two chains
is denoted by $t_{\tau}$ and controlled by the distance between two chains.
(b) Eigenspectra of the two types of modified photonic Fibonacci chains in
(a), as displayed by left and right panels, respectively. (c) Eigenspectra of
the combined structure of the two types of modified photonic Fibonacci chains.
(d) Wave profile of the topological interfacial state as electric field in the
setup of (a).

###  III.2 photonic topological interfacial states

In addition to the phononic Fibonacci chains, it is equally compelling to
investigate the topological interfacial states in photonic quasicrystals. One
notable aspect of the photonic case is the non-negligible impact of long-range
interactions.

Figure 3(a) depicts the topological interface between the two distinct types
of photonic Fibonacci chains, represented by green dots indicating the
placement of dielectric rods. Upon examination of the eigenspectra of these
photonic chains, as shown in Fig. 3(b), we find a similarity to the phononic
case, suggesting similar topological behaviors. The eigenspectra of these two
types of photonic Fibonacci chains are displayed in Fig. 3(b), which looks
similar to the phononic case. The eigenspectra of the combined structure of
these photonic Fibonacci chains, as displayed in Fig. 3(c), reveal a
topological interfacial state, marked by a red dot. Moreover, the wave profile
of this topological interfacial state, represented as an electric field in
photonic Fibonacci chains, is displayed in Fig. 3(d).

Figure 4: Schematic of the extended cut-and-project method for the modified
Fibonacci square. There are two projecting axes $x^{\prime}$ and $y^{\prime}$,
whose projection angles are complementary angles. The diagonal sites of the 2D
SSH model are projected to the $x^{\prime}$ axis, and the off-diagonal sites
are projected to the $y^{\prime}$ axis.  Figure 5: (a) Schematics of the two
types of modified phononic Fibonacci squares, which belongs two topologically
distinct classes of the 2D SSH model, constructed by the extended cut-and-
project method. (b) Eigenspectra of the two types of modified phononic
Fibonacci squares. Left panel is for the case of
$\gamma_{\text{intra}}>\gamma_{\text{inter}}$, and right panel is for
$\gamma_{\text{intra}}<\gamma_{\text{inter}}$. (c) Similar to (a) for the
modified photonic Fibonacci squares. (d) Similar to (b) for the modified
photonic Fibonacci squares. Figure 6: (a) Topological interface with a corner
structure formed by two types of modified phononic Fibonacci squares. (b)
Topological interface with a corner structure formed by two types of modified
photonic Fibonacci squares. (c) Eigenspectra for the combined structure in
(a). (d) Eigenspectra for the combined structure in (b). (e) Wave profiles as
sound pressure of the topological corner states in (c). (f) Wave profiles as
electric field of the topological corner states in (d).

##  IV higher-order topological interfacial states

In Sec. II(A), we introduce the cut-and-project method for constructing
modified Fibonacci chains from the 2D SSH model. Building upon this method, we
extend its application from constructing Fibonacci chains to Fibonacci squares
by delving into the study of higher-order topological interfacial states in
both phononic and photonic quasicrystals.

###  IV.1 Construction of modified Fibonacci squares

In our quest to construct the Fibonacci square, the traditional chain product
method proves to be somewhat cumbersome, resulting in a complex structure with
16 sites in each of parts A and B. To streamline this process, we have adapted
the cut-and-project method to project sites on both the $x^{\prime}$ and
$y^{\prime}$ axes. This adaptation involves projecting diagonal sites of the
2D SSH model, such as $a$ and $c$ on the $x^{\prime}$ axis, while aligning the
off-diagonal sites, namely $b$ and $d$ on the $y^{\prime}$ axis. The schematic
of the extended cut-and-project method for modified Fibonacci squares is
depicted in Fig. 4, where there are two projection axes, namely $x^{\prime}$
and $y^{\prime}$. A key aspect of this method is the relationship between the
projection angles that satisfies
$\theta_{x^{\prime}}+\theta_{y^{\prime}}=\pi/2$. This angular arrangement is
critical, as it imparts symmetry of the $C_{4}$ point group to the resulting
modified Fibonacci square, which simplifies the construction of corresponding
phononic and photonic Fibonacci squares.

###  IV.2 phononic and photonic modified Fibonacci squares

Upon determining the projected positions of the sites on the $x^{\prime}$ and
$y^{\prime}$ axes, we construct the corresponding modified phononic and
photonic Fibonacci squares. Figure 5(a) displays the structures for the two
different topological phases in the phononic case. It is seen that in modified
phononic Fibonacci squares, Fibonacci sequences appear along the $x-$ and $y-$
directions. In the nontrivial phase that
$\gamma_{\text{intra}}<\gamma_{\text{inter}}$, we notice a mix of sites for
parts A and B similar to the modified Fibonacci chain. The length of the
squares and the coupling waveguide is $5~{}\text{[mm]}$. The mapping function
between the distance and the width of the coupling waveguide is $D=1/L^{2}$.

Figure 5(b) displays the eigenspectra for the modified phononic Fibonacci
squares. There is a common band gap for the two topologically distinct
modified Fibonacci squares around $10^{4}~{}\text{[Hz]}$, allowing us to
explore the higher-order topological states.

Turning to the photonic counterpart, Fig. 5(c) illustrates the modified
photonic Fibonacci squares, with the left panel representing the topologically
trivial case and the right panel depicting the nontrivial case. The radius of
the dielectric robs is $0.2~{}\text{[mm]}$, and the dielectric constant is
100. The remaining part is air. The eigenspectra of the modified photonic
Fibonacci squares is displayed in Fig. 5(d). Similarly to the phononic case,
we observe a common band gap for both types of squares. It is important to
note that in the nontrivial photonic Fibonacci squares, boundary effects lead
to the emergence of localized states within the band gap.

###  IV.3 corner states in modified Fibonacci squares

We demonstrate that topological corner states emerge when two distinct types
of modified Fibonacci squares are combined. The setup of combined modified
phononic and photonic Fibonacci squares are displayed in Figs. 6(a) and 6(b),
respectively. The different shades indicate their topological classifications.
In each setup, a corner structure is constructed in the combined phononic and
photonic Fibonacci squares.

The eigenspectra of these combined structures for both phononic and photonic
cases are displayed in Figs. 5(c) and 5(d). A notable observation is the
appearance of corner states within the common band gaps of these
configurations. Interestingly, these corner states exhibit double degeneracy,
distinguishing them from the topological interfacial states observed in the
modified Fibonacci chain. This characteristic is further elucidated in the
wave profiles of the corner states for both the phononic and photonic cases,
shown in Figs. 5(e) and 5(f). Here, we observe that, unlike the corner states
typically seen in the 2D SSH model, which are localized at the corner sites,
the corner states in our modified Fibonacci square model manifest around the
corner sites rather than directly at them. This unique localization pattern of
the corner states in our model provides new insights into the behavior of
topological states in quasicrystalline systems and contributes to our
understanding of their topological characteristics.

##  V Summary

In summary, we have proposed a modified Fibonacci quasicrystal whose higher-
dimensional parent system is the two-dimensional Su-Schrieffer-Heeger model.
By constructing the corresponding phononic and photonic quasicrsytals of the
modified Fibonacci chains and squares, we observe that topological interfacial
states including higher-order one such as corner states emerge in the finite-
element simulations. Our results suggest that, in addition to the Chern-
insulator states, higher-order topological states induced by filling anomalies
can also be inherited from the parent systems in quasicrystals. In particular,
we observe the double degeneracy of the corner states. The discussed phononic
and photonic quasicrytals of modified Fibonacci quasicrystals offer a platform
for further study of the higher-order topological properties in quasicrystals.

## Acknowledgments

This work is supported by the NSFC (under No. 12074205, No. 12074108, and No.
12347101), NSFZP Grant No. LQ21A040004, the Natural Science Foundation of
Chongqing (Grant No. CSTB2022NSCQ-MSX0568) and the Fundamental Research Funds
for the Central Universities (Grant No. 2023CDJXY-048).

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