# Higher-order topological Anderson insulators in quasicrystals

Tan Peng  Department of Physics, Hubei University, Wuhan 430062, China  Chun-
Bo Hua  Department of Physics, Hubei University, Wuhan 430062, China  Rui Chen
Shenzhen Institute for Quantum Science and Engineering and Department of
Physics, Southern University of Science and Technology (SUSTech), Shenzhen
518055, China  Zheng-Rong Liu  Department of Physics, Hubei University, Wuhan
430062, China  Dong-Hui Xu  Department of Physics, Hubei University, Wuhan
430062, China  Bin Zhou<EMAIL_ADDRESS>Department of Physics, Hubei
University, Wuhan 430062, China

###### Abstract

The disorder effects on higher-order topological phases in periodic systems
have attracted much attention. However, in aperiodic systems, such as
quasicrystalline systems, the interplay between disorder and higher-order
topology is still unclear. In this paper, we investigate the effects of
disorder on two types of second-order topological insulators, including a
quasicrystalline quadrupole insulator and a modified quantum spin Hall
insulator, in a two-dimensional Amman-Beenker tiling quasicrystalline lattice.
We demonstrate that the higher-order topological insulators are robust against
weak disorder in both models. More striking, the disorder-induced higher-order
topological insulators called higher-order topological Anderson insulators are
found at a certain region of disorder strength in both models. Our paper
extends the study of the interplay between disorder and higher-order topology
to quasicrystalline systems.

##  I Introduction

A higher-order topological insulator (HOTI), a generalization of conventional
topological insulator (TI), has been a hot point of research in condensed-
matter physics [1, 2, 3, 4, 5, 6, 7, 8]. Unlike the conventional TI, an $n$th-
order topological insulator that has $d$ dimensions will have gapless boundary
states in $d-n$ dimensions ($d\geq n$). For instance, a two-dimensional (2D)
second-order topological insulator (SOTI) has zero-dimensional (0D) corner
states localized at its boundary. Analogously, a three-dimensional (3D)
second- (third-) order topological insulator has one-dimensional (1D) (0D)
hinge (corner) states localized at its boundary. These novel bulk-boundary
correspondences, which are quite different from the conventional TIs, can be
described by the nested-Wilson-loop method [4, 5, 9] and the real-space
quadrupole moment [4, 7, 8, 6, 10, 5].

The HOTIs have been extensively studied in various systems [10, 11, 12, 13,
14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32,
33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51,
52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65]. Up to now, the great
majority of the previous works about the HOTIs were discussed in crystalline
systems. However, the aperiodic systems, especially the quasicrystalline
systems, which lack translational symmetry and possess forbidden symmetries in
crystals, such as the fivefold, eightfold, and twelvefold rotational
symmetries, have also been used to realize HOTIs [66, 67, 68, 69]. For
instance, Chen _et al_. proposed that two distinct types of SOTIs can be
realized in quasicrystalline lattices [67]. One is the quasicrystalline
quadrupole insulator which can be constructed in a modified Benlcazar-
Bernevig-Hughes model [4, 5], and this kind of SOTI is protected by chiral
symmetry. The other is the modified quantum spin Hall insulator which is
formed by a TI model with a mass term which gaps the counterpropagating edge
states and induces the appearance of topological corner states. They proved
that these types of the topological corner states are protected by combined
symmetries $C_{4}m_{z}$ and $C_{8}m_{z}$ with different boundary conditions.
Very recently, Lv _et al_. reported that the HOTI has been experimentally
implemented in a quasicrystalline lattice constructed by electrical circuits
[70].

Figure 1: (a) Schematic of the Ammann-Beenker tiling quasicrystal containing
$94$ cells. Each cell includes four sites, marked by orange. (b) An Ammann-
Beenker tiling quasicrystal containing $301$ vertices with the square boundary
condition. (c) An Ammann-Beenker tiling quasicrystal containing $297$ vertices
with the octagonal boundary condition. The first three nearest-neighbor
intercell bonds correspond to the short diagonal of the rhombus tile, the edge
of square and rhombus tiles, and the diagonal of the square tile,
respectively. The distance ratio of the three bonds is
$r_{0}:r_{1}:r_{2}=2\sin\frac{\pi}{8}:1:2\sin\frac{\pi}{4}$.

Another interesting topic is disorder-induced topological phase transition.
Generally, the topological phase is robust against weak disorder and
suppressed by strong disorder where the energy gap is closed and a topological
phase transition appears. Furthermore, a fascinating phenomenon is that
disorder can encourage the generation of a topological phase by adding a
certain strength of disorder to a topologically trivial phase. The disorder-
induced topological phase which is a so-called topological Anderson insulator
(TAI) was first proposed by Li _et al_. in 2009 [71]. Then, the TAIs have been
extensive studied in various systems [72, 73, 74, 75, 76, 77, 78, 79, 80, 81,
82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97] and realized
in several experiment platforms [98, 99, 100], such as the 1D-disordered
atomic wires [98] and the photonic lattices [99, 100]. TAI was implemented in
the crystalline systems, as well as in quasicrystalline systems. For instance,
TAI has been proposed in the Penrose tiling pentagonal quasicrystal [101] and
the Ammann-Beenker tiling octagonal quasicrystal [102, 103]. In addition, the
disorder-induced HOTI, dubbed the higher-order topological Anderson insulator
(HOTAI), has been studied in various condensed-matter systems, including
topological quadrupole insulators [104, 105], topological superconductors
[106], and topological Weyl semimetals [107]. Even more striking is that Zhang
_et al_. demonstrated that the HOTAI can appear in a modified Haldane model
and realize experimentally in the electric circuit setup [108]. However, the
investigation of the disorder-induced HOTI in quasicrystalline systems remains
lacking. An interesting question is whether the HOTAI can be realized in
quasicrystalline systems.

In this paper, we investigate the disorder-induced topological phase
transition in an Ammann-Beenker tiling octagonal quasicrystal with two types
of SOTI containing a quasicrystalline quadrupole insulator (named model I) and
a modified quantum spin Hall insulator with gapless corner states (named model
II) as mentioned in Ref. [67]. For model I, the lattice is cut into a square,
and each cell contains four sites as shown in Fig. 1(a). For model II, we
divide the discussion into two cases: a square boundary [shown in Fig. 1(b)]
and an octagonal boundary [shown in Fig. 1(c)]. By calculating the quadrupole
moment and the probability density of the in-gap eigenstates, we find that the
SOTI phases in the two models with the square boundary conditions are robust
against the weak disorder, and the HOTAI phase induced from an initial
topological trivial phase occurs at a certain area of disorder strength with
four localized gapless corner states characterized by a quantized quadrupole
moment ($q_{xy}=0.5$). More striking, a HOTAI phase with eight localized
gapless corner states is found in model II with an octagonal boundary, and
this HOTAI phase is a unique topological phase which cannot be realized in
crystalline systems.

The rest of the paper is organized as follows. In Sec. II, we introduce two
types of SOTIs with disorder in the 2D quasicrystalline lattice and give the
details of numerical methods. Then, we provide numerical results for studying
the topological phase transitions of the two models in Secs. III and IV,
respectively. Finally, we summarize our conclusions in Sec. V.

##  II Models and Method

We start with a tight-binding model of a quadrupole insulator with disorder in
an Ammann-Beenker tiling quasicrystalline lattice which has a square boundary
condition. Each vertex of the quasicrystalline lattice contains four sites as
shown in Fig. 1(a). In this section, we consider the first three nearest-
neighbor intercell hoppings and the nearest-neighbor intracell hopping. The
model Hamiltonian is given by [67]

$\displaystyle H_{1}$ $\displaystyle=$ $\displaystyle\lambda\sum_{m\neq
n}\frac{l(r_{mn})}{2}c_{m}^{\dagger}(\left|\cos\psi_{mn}\right|\Gamma_{4}-i\cos\psi_{mn}\Gamma_{3}$
(1)
$\displaystyle+\left|\sin\psi_{mn}\right|\Gamma_{2}-i\sin\psi_{mn}\Gamma_{1})c_{n}$
$\displaystyle+\sum_{m}c_{m}^{\dagger}[(\gamma_{x}+U_{m})\Gamma_{4}+\gamma_{y}\Gamma_{2}]c_{m},$

where
$c_{m}^{{\dagger}}=(c_{m1}^{{\dagger}},c_{m2}^{{\dagger}},c_{m3}^{{\dagger}},c_{m4}^{{\dagger}})$
is the creation operator in cell $m$. $\gamma_{x,y}$ are the intracell hopping
amplitudes along the $x$ axis and $y$ axis, respectively. $\lambda$ denotes
the intercell hopping amplitude. $U_{m}$ is the uniform random variable chosen
from $[-W/2,W/2]$, and $W$ is the disorder strength.
$\Gamma_{4}=\tau_{1}\tau_{0}$ and $\Gamma_{\mu}=-\tau_{2}\tau_{\mu}$ with
$\mu=1,2,3$. $\tau_{1-3}$ are the Pauli matrices acting on the sites in one
cell, and $\tau_{0}$ is the $2\times 2$ identity matrix. In polar coordinate
space, $\psi_{mn}$ is the polar angle between cells $m$ and $n$.
$l(r_{mn})=e^{1-r_{mn}/\zeta}$ is the spatial decay factor of hopping
amplitudes with the decay length $\zeta$ and $r_{mn}$ representing the spatial
distance of arbitrary two cells. Here, the spatial decay length $\zeta$ and
the side length of the rhombus and square $r_{1}$ are fixed as $1$, and the
energy unit is set as $\lambda=1$ for simplicity. The Hamiltonian $H_{1}$
respects time-reversal symmetry, particle-hole symmetry, and chiral symmetry
in the clean limit, i.e., $W=0$, and the time-reversal symmetry, particle-hole
symmetry, and chiral symmetry operators are $T=\tau_{0}\tau_{0}K$,
$P=\tau_{3}\tau_{0}K$ and $S=TP=\tau_{3}\tau_{0}$, respectively, where $K$ is
the complex conjugate operator. When the disorder strength is not zero, the
system also maintains these three symmetries. In fact, the Hamiltonian $H_{1}$
is a derivation of the Benalcazar-Bernevig-Hughes model [4, 5] in some sense.
In addition, we also will investigate the effects of disorder on the higher-
order topological phase of a modified Bernevig-Hughes-Zhang model in the
Ammann-Beenker tiling quasicrystalline lattice with the square boundary
condition and octagon boundary condition, respectively. As shown in Figs. 1(b)
and 1(c), each vertex of the quasicrystalline lattice is exactly one lattice
site. Only the first three nearest-neighbor hoppings are considered in our
computing. This model lattice can be described by a tight-binding Hamiltonian
with the form of [67]

$\displaystyle H_{2}$ $\displaystyle=$ $\displaystyle-\sum_{m\neq
n}\frac{l(r_{mn})}{2}c_{m}^{\dagger}[it_{1}(s_{3}\tau_{1}\cos\psi_{mn}+s_{0}\tau_{2}\sin\psi_{mn})$
(2)
$\displaystyle+t_{2}s_{0}\tau_{3}+t_{3}s_{1}\tau_{1}\cos(\xi\psi_{mn})]c_{n}$
$\displaystyle+\sum_{m}(M+2t_{2}+U_{m})c_{m}^{\dagger}s_{0}\tau_{3}c_{m},$

Table 1: Symmetries of the Hamiltonian $H_{2}$ on an Ammann-Beenker tiling quasicrystalline lattice with the square and octagonal boundaries without ($W=0$) and with ($W\neq 0$) disorder. $\sigma_{x,y,z}$ and $\tau_{x,y,z}$ are the Pauli matrices. $K$ is the complex conjugate operator, and $\mathcal{I}$ is the $N\times N$ unit matrix with the lattice number $N$. $\mathcal{M}_{x,y,}$ are orthogonal matrices permuting the sites of the tiling to flip the whole system vertically and horizontally. $\mathcal{R}_{4,8}$ are orthogonal matrix permuting the sites of the tiling to rotate the whole system by an angle of $\pi/2$ and $\pi/4$, respectively. Check mark indicates that the symmetry in this case is preserved, and a cross mark means the symmetry is absent. |  | Square | Octagon
---|---|---|---
|  |  $W=0$ |  $W\not=0$ |  $W=0$ |  $W\not=0$
$P=\sigma_{z}\tau_{x}\mathcal{I}K$ |  $PHP^{-1}=-H$ |  ✓ |  $\times$ |  ✓ |  $\times$
$T=i\sigma_{y}\tau_{0}\mathcal{I}K$ |  $THT^{-1}=H$ |  $\times$ |  $\times$ |  $\times$ |  $\times$
$S=PT$ |  $SHS^{-1}=-H$ |  $\times$ |  $\times$ |  $\times$ |  $\times$
$m_{x}=\sigma_{x}\tau_{0}\mathcal{M}_{x}$ |  $m_{x}Hm_{x}^{-1}=H$ |  ✓ |  $\times$ |  ✓ |  $\times$
$m_{y}=\sigma_{y}\tau_{z}\mathcal{M}_{y}$ |  $m_{y}Hm_{y}^{-1}=H$ |  ✓ |  $\times$ |  ✓ |  $\times$
$m_{z}=\sigma_{z}\tau_{0}\mathcal{I}$ |  $m_{z}Hm_{z}^{-1}=H$ |  $\times$ |  $\times$ |  $\times$ |  $\times$
|  |  |  |  |
$C_{4}=e^{-i\frac{\pi}{4}\sigma_{z}\tau_{z}}\mathcal{R}_{4}$ |  $C_{4}HC_{4}^{-1}=H$ |  $\times$ |  $\times$ |  $\times$ |  $\times$
$C_{4}T$ |  $C_{4}TH(C_{4}T)^{-1}=H$ |  ✓ |  $\times$ |  $\times$ |  $\times$
$C_{4}m_{x}$ |  $C_{4}m_{x}H(C_{4}m_{x})^{-1}=H$ |  $\times$ |  $\times$ |  ✓ |  $\times$
$C_{4}m_{y}$ |  $C_{4}m_{y}H(C_{4}m_{y})^{-1}=H$ |  $\times$ |  $\times$ |  ✓ |  $\times$
$C_{4}m_{z}$ |  $C_{4}m_{z}H(C_{4}m_{z})^{-1}=H$ |  ✓ |  $\times$ |  $\times$ |  $\times$
|  |  |  |  |
$C_{8}=e^{-i\frac{\pi}{8}\sigma_{z}\tau_{z}}\mathcal{R}_{8}$ |  $C_{8}HC_{8}^{-1}=H$ |  $\times$ |  $\times$ |  $\times$ |  $\times$
$C_{8}T$ |  $C_{8}TH(C_{8}T)^{-1}=H$ |  $\times$ |  $\times$ |  ✓ |  $\times$
$C_{8}m_{x}$ |  $C_{8}m_{x}H(C_{8}m_{x})^{-1}=H$ |  $\times$ |  $\times$ |  $\times$ |  $\times$
$C_{8}m_{y}$ |  $C_{8}m_{y}H(C_{8}m_{y})^{-1}=H$ |  $\times$ |  $\times$ |  $\times$ |  $\times$
$C_{8}m_{z}$ |  $C_{8}m_{z}H(C_{8}m_{z})^{-1}=H$ |  $\times$ |  $\times$ |  ✓ |  $\times$

where
$c_{m}^{{\dagger}}=(c_{m\alpha\uparrow}^{{\dagger}},c_{m\alpha\downarrow}^{{\dagger}},c_{m\beta\uparrow}^{{\dagger}},c_{m\beta\downarrow}^{{\dagger}})$
represents the creation operator of an electron on a site $m$. In each site,
$\alpha$ ($\beta$) is the index of orbitals, and $\uparrow$ ($\downarrow$)
represents the spin direction. $s_{1-3}$ and $\tau_{1-3}$ are the Pauli
matrices acting on the spin and orbital degree of freedom, respectively.
$s_{0}$ and $\tau_{0}$ are the $2\times 2$ identity matrices. $t_{1-3}$ are
the hopping strength, and $M$ is the Dirac mass. The term containing $t_{3}$
is actually equivalent to a mass term that destroys the time-reversal symmetry
of the system so that the original helical boundary state of the system opens
the energy gap and evolves into a higher-order corner state [67, 68]. $\xi$ is
the varying period of the mass term, and $\xi=2$ (4) for square (octagonal)
samples. In the clean limit, the Hamiltonian $H_{2}$ respects particle-hole
symmetry, mirror symmetry $m_{x,y}$, and some combined symmetries, such as
$C_{4}T$, $C_{4}m_{z}$ with the square boundary condition and $C_{4}m_{x}$,
$C_{4}m_{y}$, $C_{8}T$, $C_{8}m_{z}$ with the octagonal boundary condition as
shown in Table 1. In fact, it has been demonstrated that the higher-order
corner state is protected by the combined symmetry $C_{4}m_{z}$ ($C_{8}m_{z}$)
with the square (octagonal) boundary condition by employing some uniform
perturbations to test the stability of the corner states [67]. However, all
symmetries are broken when the disorder is introduced and whether the higher-
order corner states induced by the disorder can appear is unclear. Without
loss of generality, we will set $t_{1}=t_{2}=1$.

The nested-Wilson-loop method [4, 9, 5] in the momentum space is efficient to
characterize the topological phase of an electric quadrupole insulator.
However, the quasicrystalline lattice is the lack of the translation
invariance, thus, the topological invariant defined in the momentum space is
no longer applicable for our models of quasicrystals. Therefore, we employ a
real-space quadrupole moment to characterize the topological phases of the
quasicrystalline lattice with disorder. The real-space quadrupole moment is
given by [109, 110, 111, 104, 112]

$q_{xy}=\frac{1}{2\pi}{\rm{Im}}\ln[\det(\Psi_{occ}^{\dagger}\hat{U}\Psi_{occ})\sqrt{\det(\hat{U}^{\dagger})}],$
(3)

where $\Psi_{occ}$ is the eigenvector of occupied states.
$\hat{U}\equiv\exp[i2\pi\hat{X}\hat{Y}/N]$ where $\hat{X}$ and $\hat{Y}$ are
the position operators, and $N$ represents the total number of the lattice
sites. If $q_{xy}=0.5$, the system is a SOTI phase with topological corner
states. Besides, $q_{xy}=0$ indicates a trivial phase. Note that the
subsequent calculations of $q_{xy}$ in this paper are based on the periodic
boundary condition. It is also noted that the validity of the formulation of
the bulk quadrupole moment proposed by two previous works [109, 110] is still
controversial. Ono _et al_. presented that the proposed definition of the bulk
quadrupole moment fails even for a simple noninteracting example [113]. Thus,
a satisfactory formulation of the bulk quadrupole moment should be worthy of
further study in the future works.

##  III Model I: Chiral symmetry-protected higher-order topological insulator

In this section, we focus on the disorder-induced topological phase transition
with chiral symmetry in an Ammann-Beenker tiling quasicrystal with the square
boundary condition. The disorder is of the $U_{m}\Gamma_{4}$ type, which does
not destroy the chiral symmetry.

Figure 2 shows the real-space quadrupole moment as a function of disorder
strength $W$ and intracell hopping amplitude along the $y$-axis $t_{y}$ with
fixed $t_{x}$. The color map shows the magnitude of the real-space quadrupole
moment. It is found that when $W=0$, that is, in the clean limit, the system
is in a SOTI phase with $q_{xy}=0.5$ if $\gamma_{y}$ satisfies
$-1.8<\gamma_{y}<-0.9$. However, with the gradual increase in the disorder
strength, the SOTI phase will transform to the Anderson localized state phase
with $q_{xy}$ changing from $0.5$ to $0$. There are a series of critical
maximum disorder strengths increased monotonically with the increase in
$\gamma_{y}$. Meanwhile, we also find a phase which is a disorder-induced SOTI
phase in the region where $\gamma_{y}>-0.9$. However, in our calculation,
$q_{xy}$ is not a quantum number strictly equal to $0.5$ in the phase region.
We believe that this is due to the finite-size effect of the system and will
be discussed in the follow-up.

Figure 2: Topological phase diagram of the Ammann-Beenker tiling quasicrystal
in ($W,\gamma_{y}$) space obtained by calculating the real-space topological
invariant quadrupole moment $q_{xy}$ with $\gamma_{x}=-1.5$. The system is cut
to a square sample containing $1257$ cells with periodical boundary
conditions. Some $100$ random configuration averages are taken in our
computing.

In order to explore the role of the disorder effect in the quasicrytalline
lattice with chiral symmetry in more depth, we take two specific parameter
values of $\gamma_{y}$ and plot the variation of $q_{xy}$ with respect to the
strength of disorder as shown in Figs. 3(a) and 3(b). For the case of
$\gamma_{y}=-1.5$, the second-order phase remains stable in a weakly
disordered situation ($W<2.5$) with a quantized quadrupole moment plateau and
is destroyed in the strongly disordered situation ($W>7$) where $q_{xy}=0$. On
the other hand, when $\gamma_{y}=-0.75$, the system hosts a trivial phase with
$q_{xy}=0$ in the clean limit. As the strength of disorder increasing,
$q_{xy}$ gradually increases from $0$ and approaches $0.5$, indicating that
the system has undergone a phase transition from a trivial phase to a
topological nontrivial phase. Actually, there is not a quantized quadrupole
moment plateau in Fig. 3(b), and we attribute this to the finite-size effect.
Therefore, we plot $q_{xy}$ versus system size $N$ when $W=5.5$ with $500$
disorder configurations in the inset in Fig. 3(b). It is found that $q_{xy}$
approaches $0.5$ with a large system size ($N=16437$). To further certify the
existence of SOTI phases, we set some specific values of $W$ in Figs. 3(a) and
3(b) to give the energy spectrum and wave-function diagram of the system. It
is found that the SOTI phase is robust against the weak disorder [$W=1.5$ in
Figs. 3(c) and 3(e)] since the system hosts four zero-energy modes which are
localized at the four corners of the lattice. This property is similar to the
first-order topological state. Similarly, when $W=4$, four zero-energy modes
appear at the four corners of the lattice which indicate the presence of the
disorder-induced SOTI phases. The corner states are protected by the chiral
symmetry which is quite similar to the corner states that appeared in
crystalline systems in some previous works [104, 105].

Figure 3: The real-space quadrupole moment $q_{xy}$ versus disorder strength
$W$ with different initial states including (a) a higher-order topological
phase with $\gamma_{x}=-1.5,\gamma_{y}=-1.5$ and (b) a topological trivial
phase with $\gamma_{x}=-1.5,\gamma_{y}=-0.75$. The periodic boundary condition
is taken, and 500 disorder configurations are performed. The inset shows the
quadrupole moments $q_{xy}$ versus $N$ when $W=5.5$. $N$ is the total number
of the cells. The energy modes near the zero energy for (c) a higher-order
topological initial phase with $\gamma_{y}=-1.5$, $W=1.5$ and (d) a trivial
initial phase with $\gamma_{y}=-0.75$, $W=4$, respectively. (e) and (f) The
wave-function distribution of the zero modes corresponds to (c) and (d),
respectively. The system contains $1257$ cells, and the open boundary
condition is taken in (c)-(f).

##  IV Model II: Combined symmetry-protected higher-order topological
insulator

In this section, we concentrate on the effects of disorder on the combined
symmetry-protected higher-order topological phase in an Ammann-Beenker tiling
quasicrystal with square and octagonal boundary conditions, respectively. All
calculations are based on the Hamiltonian $H_{2}$. In the clean limit, the
HOTI phase is protected by combined symmetry $C_{4}m_{z}$ and $C_{8}m_{z}$ for
different boundary conditions. So far, it was revealed that the HOTI phase is
protected by the symmetries, such as chiral, partial-hole, $C_{4}T$,
$C_{4}m_{z}$ and $C_{8}m_{z}$ symmetries. According to the previous work
[104], the values of $q_{xy}$ can be quantized to $0$ or $1/2$ only if the
system has chiral or partial-hole symmetry. However, all of these symmetries
are destroyed when the disorder is introduced in the Hamiltonian $H_{2}$ (see
Table I). Hence, the real-space quadrupole moment discussed in Sec. III may
not be appropriate for model II with the square boundary condition. In
addition, there is no well-defined topological invariant for a lattice with an
octagonal boundary condition. One appropriate way to characterize the higher-
order topological phase is to adopt the existence of the corner states as a
working definition [114, 108]. Thus, we calculate the energy spectrum and
wave-function distribution of the system to determine whether the corner
states exist. To reveal HOTAI in the quasicrystal described by model II, in
the following calculations, we will not only perform disorder configuration
average by many enough times, but also try to ensure that the size of the
samples is large enough.

###  IV.1 Square boundary condition

In Figs. 4(a) and 4(b), we plot the eigenspectrum of the open lattice as the
function of disorders with different $M$. For the case of $M=-1$, the
probability density of the four in-gap eigenstates near zero energy in the
clean limit presents a picture with four corner states localized at the four
corner of the lattice [see Fig. 7(a) in the Appendix], indicating that the
system hosts a SOTI phase. Upon introducing the disorder and increasing its
strength, in Fig. 4(a), it is shown that the midgap modes remain stable until
$W\approx 5.5$, beyond which the bulk gap disappears, and the system is
converted to an Anderson localized state phase. To further illustrate the
stability of the SOTI phase, Figs. 4(c) and 4(e) display the eigenspectrum and
probability density of the in-gap eigenstates with $W=1.5$. It is found that
the four corner states are stable under weak disorder, indicating that the
second-order phase is robust against the weak disorder. For another case of
$M=1.6$, the system is a normal insulator phase in the clean limit due to the
fact that the middle four eigenstates near the zero energy are localized in
the bulk [see Fig. 7(b) in the Appendix]. With the increase in $W$, two
topological phase transitions occur in Fig. 4(b). First, in the region
$4<W<8$, the four middle eigenvalues gradually tend to be degenerate near the
zero energy, and midgap modes are generated, indicating that a phase
transition from normal insulator phase to the HOTI phase occurs. Figures 4(d)
and 4(f) show the energy spectrum and probability density of the in-gap
eigenstates of $H_{2}$ under the open boundary condition with $M=1.6$ and
$W=6.6$. It is found that there are fourfold energy degenerate in-gap states
under the condition of this set of parameters. What is more interesting is
that the wave functions corresponding to these degenerate energies are all
localized at the four corners of the lattice, which are the so-called corner
states as shown in Fig. 4(f). The corner states, induced by disorder, is
strong evidence for the emergence of the HOTAI. Then, with the increase in the
disorder strength, the higher-order phase converts to an Anderson insulator
phase at $W\approx 8$ with the energy gap closure and all eigenstates being
localized. Based on our calculations, we can draw two conclusions: first, the
higher-order topological phase is relatively stable under weak disorder;
second, disorder can also induce the higher-order topological phase in model
II.

Figure 4: The eigenspectrum versus disorder strength $W$ with different
initial states including: (a) a higher-order topological phase with $M=-1$ and
(b) a topological trivial phase with $M=1.6$. Some $200$ disorder
configurations are performed with a square sample containing $4061$ sites. The
energy modes near the zero energy for (c) a higher-order topological initial
phase with $M=-1$, $W=1.5$ and (d) a trivial initial phase with $M=1.6$,
$W=6.6$, respectively. (e) and (f) The wave-function distribution of the four
in-gap states corresponds to (c) and (d), respectively. All calculations are
based on the open boundary condition.

As mentioned above, since both the chiral and the partial-hole symmetries are
broken in model II with disorder, the necessary condition for the application
of the real-space formula of the quadrupole moment is not be satisfied, and
the real-space quadrupole moment should not be applied to characterize the
higher-order topological phase in model II with the square sample. However,
here we also try to calculate the real-space quadrupole moment $q_{xy}$ versus
disorder strength with different $M$ as shown in Fig. 5. Strikingly, it is
found that the values of $q_{xy}$ are also quantized $1/2$ in certain disorder
strength regions. For the case of $M=-1$ [see Fig. 5(a)], the system hosts a
SOTI phase with $q_{xy}=1/2$ in the clean limit, and a typical plateau is
accompanied by quantized $q_{xy}$ until the strength of $W$ reaches a certain
value ($W\approx 5.5$), indicating that the SOTI phase is robust against the
weak disorder. However, the SOTI phase is eventually destroyed by strong
disorder. For another case of $M=1.6$ [see Fig. 5(b)], the system is a normal
insulator phase with $q_{xy}=0$ in the clean limit. With the increase in $W$,
two topological phase transitions occur, accompanied by $q_{xy}$ changing from
$0$ to $0.5$ at $W\approx 4$ and returning to $q_{xy}=0$ at $W\approx 8$. In
the region $4<W<8$, a remarkable plateau of quantized $q_{xy}=0.5$ appears,
which indicates a SOTI phase induced by disorder. Thus, it is shown that the
results given by $q_{xy}$ match well with the energy spectrum [comparing Figs.
4(a) and 4(b) with Figs. 5(a) and 5(b)]. It is implied that the validity of
the operator-based formulation of the bulk quadruploe moment proposed by two
pervious works [109, 110] is still an open issue. An intriguing question is
whether this real-space quadrupole moment can still characterize the topology
of the system in these situations without any symmetry constraint, and it will
be further investigated in future work.

Figure 5: The real-space quadrupole moment $q_{xy}$ versus disorder strength
$W$ with different initial states including (a) a higher-order topological
phase with $M=-1$ and (b) a topological trivial phase with $M=1.6$. The
periodic boundary condition is taken, and $500$ disorder configurations are
performed. The system is cut to a square sample which contains $1257$ sites.

###  IV.2 Octagonal boundary condition

In Fig. 6(a), we plot the eigenspectrum of the open lattice as the function of
disorders with $M=-1$. The probability density of the eight in-gap eigenstates
near zero energy in the clean limit presents a picture with eight corner
states localized at the eight corners of the lattice [see Fig. 7(c) in the
Appendix], indicating that the system hosts a SOTI phase. Upon introducing the
disorder and increasing its strength, the midgap modes remain stable until
$W\approx 4$, beyond which the bulk gap disappears, and the system is
converted to an Anderson localized state phase. To further illustrate the
stability of the SOTI phase, Figs. 6(c) and 6(e) display the eigenspectrum and
probability density of the in-gap eigenstates with $W=2$. It is found that the
eight corner states are stable under weak disorder.

In Fig. 6(b), we plot the eigenspectrum of the open lattice as the function of
disorders with $M=1.6$. The probability density of the middle eight
eigenstates near the zero energy in the clean limit are localized in the bulk
[see Fig. 7(d) in the Appendix], indicating that the system hosts a trivial
phase. Upon introducing the disorder and increasing its strength, a series of
interesting changes occur in the energy spectrum. First, in the region
$0<W<5.5$, the eight middle eigenvalues gradually tend to be degenerate near
the zero energy, and midgap modes are generated, indicating that a phase
transition from a normal insulator phase to a HOTI phase may occur. To verify
this conclusion, we plot the eigenspectrum and probability density of the
midgap eigenstates at $W=6.6$ as shown in Figs. 6(d) and 6(f). It is shown
that the eight midgap states are localized at the eight corners of the
lattice, and these corner states are the powerful proof of disorder-induced
HOTI. Then, with the increase in the disorder strength, the higher-order phase
converts to an Anderson insulator phase at $W\approx 8$ with the energy gap
closure and all eigenstates being localized.

Figure 6: The eigenspectrum versus disorder strength $W$ with different
initial states including (a) a higher-order topological phase with $M=-1$ and
(b) a topological trivial phase with $M=1.6$. Some $200$ disorder
configurations are performed with a octagonal sample containing $13289$ sites.
The energy modes near the zero energy for (c) a higher-order topological
initial phase with $M=-1$, $W=2$ and (d) a trivial initial phase with $M=1.6$,
$W=6.6$, respectively. (e) and (f) The wave-function distribution of the eight
in-gap states corresponds to (c) and (d), respectively. All calculations are
based on the open boundary condition.

##  V Conclusions and discussions

In this paper, we investigate the disorder-induced higher-order topological
phase transition in an Ammann-Beenker tiling quasicrystal. Two types of SOTI
phases are considered: One is the quasicrystalline quadrupole insulator (model
I), and the other is a quantum spin Hall insulator with a mass term which
gapped the edge states and the topological corner states emerge (model II).
Without disorder, model I (II) in the SOTI phase hosts gapless topological
corner states protected by chiral ($C_{4}m_{z}$ or $C_{8}m_{z}$) symmetry and
localized at the lattice corners. Based on calculating the quadrupole moment
and the probability density of the middle gap eigenstates, it is found that in
both models, the SOTI phases stay stable with weak disorder and are destroyed
by strong disorder. More interesting is that the chiral symmetry-protected
disorder-induced HOTAI is found by adding a certain strength of disorder to a
topological trivial phase in model I. Meanwhile, a topological phase
transition from a topological trivial phase to a HOTAI phase with topological
corner states is also found in model II.

Based on the self-consistent Born approximation (SCBA), the disorder-induced
topological phase transition forms a topological trivial phase to a
topological nontrivial phase is attributed to the disorder that renormalizes
the system parameters, such as the mass term, hopping term, and chemical
potential [78, 115, 87, 75, 89, 81, 90, 91, 104]. However, the SCBA theory is
invalid for the aperiodic systems, such as amorphous and quasicrystalline
lattices which are the lack of translation symmetry. Up to now, there is not a
well-defined theory to reveal the generating mechanism of TAI in aperiodic
systems, and it is will be studied in the future work. Nevertheless, analogous
to the generating mechanism of the TAI in a periodic system, we suppose that
the generation of the TAI or HOTAI in the quasicrystalline system is also due
to the renormalization of the parameters caused by disorder, and the initial
trivial phase is converted to the HOTAI phase. In addition, disorder in model
I does not destroy the chiral symmetry, and this symmetry also protects the
topology of the system [104]. In model II, the introducing of disorder has
caused all of the symmetries of the system to be broken. It seems difficult to
find a symmetry to guarantee the topology of the system, however, our
calculations show that the HOTAI phases can be also induced by disorder in
model II. We note that the quadrupole moment is easy to get a quantized value
in model II. This may be caused by the following two points. One is that model
I is more sensitive to the finite-size effect. Two is that the wave function
of HOTI in model II is more local than model I. In previous work, Fu _et al_.
have proposed that, when the disorder is strong, the topological surface
states exist, due to symmetries that are destroyed by disorder but remain
unbroken on average [116]. Here, two key points are employed to guarantee that
the averaged symmetries exist. One is that enough disorder configurations are
needed for the average. The other one is the size of the systems should be
large enough. Under these conditions, we suppose that the combined symmetries,
such as $C_{4}m_{z}$ and $C_{8}m_{z}$ which are broken by random disorder will
recovered statistically by taking an ensemble average [117] and the HOTAI
phases are protected by the average combined symmetries. More details about
average symmetry which can keep the HOTI phases with disorder in
quasicrystalline systems will be further investigated in the future work.

Recently, the HOTAI has been successfully implemented in a modified Haldane
model based on electric circuits system [108]. Moreover, the quasicrystalline
quadrupole topological insulatiors has been experimentally realized in
electrical circuits [70]. Therefore, we propose an experimental setup to
construct the quasicrystalline lattice in electronic circuits and realize the
introduction of random disorder by changing the magnitude of the inductors and
the capacitors. By this way, we believe that the HOTAI phase in the
quasicrystalline system can be observed.

## Acknowledgments

B.Z. was supported by the NSFC (under Grant No. 12074107), and the program of
outstanding young and middle-aged scientific and technological innovation team
of colleges and universities in Hubei Province (under Grant No. T2020001).
D.-H.X. was supported by the NSFC (under Grant No. 12074108). D.-H.X. also
acknowledges financial support from the Chutian Scholars Program in Hubei
Province.

## Appendix: Wave function with square and octagonal boundary condition in the
clean limit

In this Appendix, we plot the probability density of the four (eight)
eigenstates which are nearest to zero energy in the clean limit with different
Dirac mass $M$ to identify the initial phase of the system. All calculations
are based on $H_{2}$. As shown in Figs. 7(a) and 7(c), four and eight in-gap
states symmetrically distributed at the corners of a quasicrystal octagon,
indicating that the system is a HOTI phase at $M=-1$ in the clean limit.
Meanwhile, when $M=1.6$, the system is in a topological trivial phase as shown
in Fig. 7(b) and 7(d).

Figure 7: Probability density of the eigenstates in the clean limit with
different Dirac mass (a) and (c)$M=-1$ and (b) and (d) $M=1.6$. The system is
cut to a square sample which contains $4061$ sites for (a) and (b). For (c)
and (d), the system is cut into a octagonal sample with $13289$ sites.

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