††thanks: These authors contributed equally to this work††thanks: These
authors contributed equally to this work

# Constructing higher-order topological states in higher dimension

Yao Wang  Center for Integrated Quantum Information Technologies, School of
Physics and Astronomy, State Key Laboratory of Advanced Optical Communication
Systems and Networks, Shanghai Jiao Tong University, Shanghai 200240, China
Synergetic Innovation Center of Quantum Information and Quantum Physics,
University of Science and Technology of China, Hefei, Anhui 230026, China
Yongguan Ke  Guangdong Provincial Key Laboratory of Quantum Metrology and
Sensing & School of Physics and Astronomy,
Sun Yat-Sen University (Zhuhai Campus), Zhuhai 519082, China  Nonlinear
Physics Centre, Research School of Physics and Engineering,
Australian National University, Canberra ACT 2601, Australia  Yi-Jun Chang
Center for Integrated Quantum Information Technologies, School of Physics and
Astronomy, State Key Laboratory of Advanced Optical Communication Systems and
Networks, Shanghai Jiao Tong University, Shanghai 200240, China  Synergetic
Innovation Center of Quantum Information and Quantum Physics, University of
Science and Technology of China, Hefei, Anhui 230026, China  Yong-Heng Lu
Center for Integrated Quantum Information Technologies, School of Physics and
Astronomy, State Key Laboratory of Advanced Optical Communication Systems and
Networks, Shanghai Jiao Tong University, Shanghai 200240, China  Synergetic
Innovation Center of Quantum Information and Quantum Physics, University of
Science and Technology of China, Hefei, Anhui 230026, China  Jun Gao  Center
for Integrated Quantum Information Technologies, School of Physics and
Astronomy, State Key Laboratory of Advanced Optical Communication Systems and
Networks, Shanghai Jiao Tong University, Shanghai 200240, China  Synergetic
Innovation Center of Quantum Information and Quantum Physics, University of
Science and Technology of China, Hefei, Anhui 230026, China  Chaohong Lee
<EMAIL_ADDRESS>Guangdong Provincial Key Laboratory of Quantum
Metrology and Sensing & School of Physics and Astronomy,
Sun Yat-Sen University (Zhuhai Campus), Zhuhai 519082, China  State Key
Laboratory of Optoelectronic Materials and Technologies, Sun Yat-Sen
University (Guangzhou Campus), Guangzhou 510275, China  Xian-Min Jin
<EMAIL_ADDRESS>Center for Integrated Quantum Information
Technologies, School of Physics and Astronomy, State Key Laboratory of
Advanced Optical Communication Systems and Networks, Shanghai Jiao Tong
University, Shanghai 200240, China  Synergetic Innovation Center of Quantum
Information and Quantum Physics, University of Science and Technology of
China, Hefei, Anhui 230026, China

Higher-order topological phase as a generalization of Berry phase attracts an
enormous amount of research. The current theoretical models supporting higher-
order topological phases, however, cannot give the connection between lower
and higher-order topological phases when extending the lattice from lower to
higher dimensions. Here, we theoretically propose and experimentally
demonstrate a topological corner state constructed from the edge states in one
dimensional lattice. The two-dimensional square lattice owns independent
spatial modulation of coupling in each direction, and the combination of edge
states in each direction come up to the higher-order topological corner state
in two-dimensional lattice, revealing the connection of topological phase in
lower and higher dimensional lattices. Moreover, the topological corner states
in two-dimensional lattice can also be viewed as the dimension-reduction from
a four-dimensional topological phase characterized by vector Chern number,
considering two modulation phases as synthetic dimensions in Aubry-André-
Harper model discussed as example here. Our work deeps the understanding to
topological phases breaking through the lattice dimension, and provides a
promising tool constructing higher topological phases in higher dimensional
structures.

##  I Introduction

The higher-order topological phases introduced in higher-dimensional lattices
recently extend the conventional understanding on the topological nontrivial
materials, where the $d$-dimensional lattice owns not only the first-order
($d-1$)-dimensional edge states but also the $n$-order ($d-n$)-dimensional
edge states T20171 ; T20172 ; T20181 ; T20190 ; T20191 ; T20192 ; T20201 ;
T20202 . The second-order corner states in two-dimensional (2D) lattices are
widely investigated since 2019 in sonic Es1 ; Es2 ; Es3 ; Es4 ; Es5 ; Es6 ,
ring resonator Er , waveguide Ew1 ; Ew2 ; Ew3 , cavity Ec1 ; Ec2 , and cold
atom Ecold  systems. Recently, the higher-order topological states in three-
dimensional lattices are also reported E3D1 ; E3D2 . The investigations on
higher-order topological phases in both theories and experiments promote and
extend the development of topological photonics Topo_review_1 ; Topo_review_2
.

The current principles to seek high-order topological states are mainly based
on analyzing spatial or (and) nonspatial symmetries T20171 ; T20172 ; TPRL118
; T20181 ; Langbehn2017 ; Song2017 ; Linhu2018 ; Max2018 . In spatial-
symmetric (such as inversion- or rotational-symmetric) systems, high-order
topological states may originate from quantized dipole polarization TPRL118 ;
T20181  or multipole moments Langbehn2017 ; Song2017 . In nonspatial-symmetric
(such as chiral-symmetric) systems, corner states may arise due to nontrivial
edge winding numbers Linhu2018 . By combining nonspatial and spatial
symmetries, second-order topological insulator and superconductors have been
partially classified Max2018 . The existing schemes requiring delicate designs
of overall symmetry, as a top-to-bottom approach, cannot provide insight of
the connection between lower-order and higher-order topological states. Since
lower-order edge states are well-known, we may wonder whether it is possible
to use lower-order topological states as building blocks assembling to higher-
order topological states. If possible, what are their topological
correspondence to the bulk?

Here, we theoretically propose and experimentally demonstrate a bottom-to-top
scheme for constructing topological corner states by using topological edge
states as building blocks. In each direction the topological edge states are
snapshot states in a topological pumping by means of a changing one-
dimensional dipole moment which is related to Chern number. Such scheme
naturally extends Chern number to vector Chern number with individual
components separately defined in each direction, from lower- to higher-
dimensional lattices. The hierarchical relation between two-dimensional Zak
phase TPRL118 ; T20181  and vector Chern number can be understood as a varying
of two-dimensional dipole polarization generates quantized charge pumping in
two directions. The fact that corner states are guaranteed by nontrivial
vector Chern number can be termed as _bulk-corner correspondence_ , and they
inherit the topological origin of edge states as a dimension reduction of
quantum Hall phase. We have to emphasize that such corner states do not
require any fine tuning of spatial or nonspatial symmetries.

Taking the off-diagonal Aubry-André-Harper (AAH) model for example, we
theoretically analyze the topological origin when extending the lattice from
one dimension to two dimensions. We construct the two-dimensional photonic
topological lattice and successfully observe the higher-order topological
corner states predicted in theory. Our model gives an intuitive understanding
on the raising of higher-order topological phases in higher dimensional
lattice, which connects the topological phases in different dimensional
lattices and provides a convenient tool for constructing higher-order
topological phases in higher dimensional lattices.

Figure 1: Schematic of constructing corner states. (a) Three types of edge
states in one-dimensional lattices. The edge states in one-dimensional lattice
can be regarded as the building block for the higher-order topological states.
(b-c) Corner states in two- and three-dimensional lattices built by edge
states in one-dimensional lattices. We can find the connection between the
topological states in different dimensional lattice by projecting the higher-
dimensional lattice into one-dimensional lattice in direction of x, y and z
axis respectively.

##  II Vector Chern number and corner states

We explore a systematical method to construct corner states in a two-
dimensional square lattice. Consider that the position of a particle is
denoted by $(i,j)$, where $i$ and $j$ are the site indices of $x$ and $y$
directions respectively. The coupling between $(i,j)$th and $(k,l)$th lattice
sites has the form, $H_{x}(i,k)\delta_{j,l}+\delta_{i,k}H_{y}(j,l)$, where
$H_{x}(i,k)$ is the coupling matrix along $x$ direction irrelevant to
positions in $y$ direction, and vice versa. The motion of a particle hopping
in such lattice is governed by the Hamiltonian,

$H=H_{x}\otimes I_{y}+I_{x}\otimes H_{y},$ (1)

where $I_{s}$ is an identical matrix in $s$ direction with $s\in x,\ y$. Once
we obtain the eigenvalues $E_{p}^{s}$ and eigenstates $|\psi_{p}^{s}\rangle$
corresponding to $H_{s}$ where $p$ are quantum numbers, we can immediately
prove that

$H|\psi_{m}^{x}\rangle\otimes|\psi_{n}^{y}\rangle=(E_{m}^{x}+E_{n}^{y})|\psi_{m}^{x}\rangle\otimes|\psi_{n}^{y}\rangle,$
(2)

that is, $E_{m}^{x}+E_{n}^{y}$ and
$|\psi_{m}^{x}\rangle\otimes|\psi_{n}^{y}\rangle$ are the eigenvalues and
eigenstates corresponding to $H$, respectively. If $|\psi_{m}^{x}\rangle$ and
$|\psi_{n}^{y}\rangle$ are topological edge states, the product of these edge
states becomes a topological corner state in two dimensions. Hence the seeking
for topological corner states is transformed to separately design the coupling
matrix in each direction that supports topological edge states.

Consider that the coupling matrix $H_{s}$ is controlled by two parameters
$\phi^{s}$ and $\theta^{s}$, which satisfied
$H_{s}(\phi^{s},\theta^{s})=H_{s}(\phi^{s}+2\pi,\theta^{s}+2\pi)$. In
practice, $\phi^{s}$ is a modulated phase, and $\theta^{s}$ is a twisted phase
when a particle hopping across the boundary in $s$ direction by imposing
twisted boundary condition. To characterize the bulk topology, we can define
vector Chern number $(C_{\mathbf{u}}^{x};C_{\mathbf{v}}^{y})$ with individual
components given by

$C_{\mathbf{w}}^{s}=\frac{1}{{2\pi
i}}\int_{BZ}d\theta^{s}{d\phi^{s}\det[\mathcal{F}({\phi^{s},\theta^{s}})]}.$
(3)

Here,
$[\mathcal{F}({\phi^{s},\theta^{s}})]^{m,n}=\partial_{\phi^{s}}A_{\theta^{s}}^{m,n}-\partial_{\theta^{s}}A_{\phi^{s}}^{m,n}+i[A_{\phi^{s}},A_{\theta^{s}}]^{m,n}$
are elements of the non-Abelian Berry curvature with elements of Berry
connection
$(A_{\mu})^{m,n}=\langle\psi_{m}^{s}({\phi^{s},\theta^{s}})|\nabla_{\mu}|\psi_{n}^{s}({\phi^{s},\theta^{s}})\rangle$.
$m,n\in\mathbf{w}$ which is a subset of near degenerate bands. If both the two
components of the vector Chern number
$(C_{\mathbf{u}}^{x};C_{\mathbf{v}}^{y})$ are integer, in the open boundary
condition there exists at least one topological corner state at some fixed
modulated phases $(\phi^{x},\phi^{y})$. We term this relation as _bulk-corner
correspondence_ , which gives a clear topological origin of corner states,
i.e., a dimensional reduction of a four-dimensional topological space
characterized by vector Chern number.

Figure 2: The density of states for (a) one-dimensional AAH lattice and (b)
two-dimensional AAH lattices. The finite energy gaps between corner states and
extended states inherit from those between edge states and extended states.
The corner states are constructed from the edge states, the former has twice
the energy of the latter. The local density of states are shown in the insets,
the bulk state and edge state in one-dimensional AAH lattice are shown in (i)
and (ii) respectively, the edge states and corner states in two-dimensional
AAH lattice are shown in (iii, v) and (iv, vi) respectively. The parameters in
simulation are adopted as: $t_{x(y)}=0.5$, $\lambda_{x(y)}=0.95$,
$b_{x(y)}=(\sqrt{5}+1)/2$, and there are 15 sites for one-dimensional lattice
and 15$\times$15 sites for two-dimensional lattice. DOS: density of states.
Figure 3: Corner states. (a) Schematic of the fabricated photonic
quasicrystal. Nine input sites are set in the lattice, four waveguides marked
as C1, C2, C3 and C4 are designed to observe the corner states. Four
waveguides marked as B1, B2, B3 and B4 are designed to observe the edge
states. The waveguide marked as T is designed to observe bulk state. (b)
Spectrum of one-dimensional off-diagonal AAH lattice. For 16-sited lattice,
two boundary modes (green lines) cross in the band gap, for 15-sited lattice,
only one boundary mode connects the bands separated by the gap. The red dash
lines give the $\phi$ adopted in experiment. (c-d) Measured corner states. The
photons are confined in the corners when we excite the lattice corner sites
(c). The white dotted squares point out the ranges of corner states. The
quantified results confirm the theoretically predicted topological corner
states arising by extending topological lattice with edge state from one
dimension to two dimensions. Figure 4: Edge states. (a) The measured result of
edge states. We inject the photons into the lattices from the input sites B1,
B2, B3, and B4 respectively, the photons are localized in the edges of the
lattices. (b) The trivial cases. The photon cannot be confined in the edges of
the lattices. The white dotted squares point out the ranges of edge states.
(c) The quantified results. The edge states can be conveniently extended from
lower dimension to higher dimension.

For explicitness, we focus on a two-dimensional off-diagonal AAH lattice with
hopping strengths varying in space,

$\displaystyle H=$ $\displaystyle\sum_{i,j}t_{x}[1+\lambda_{x}\cos(2\pi
b_{x}i+\phi^{x})]\hat{a}_{i,j}\hat{a}_{i+1,j}^{\dagger}$
$\displaystyle+t_{y}[1+\lambda_{y}\cos(2\pi
b_{y}j+\phi^{y})]\hat{a}_{i,j}\hat{a}_{i,j+1}^{\dagger}+H.c.,$ (4)

where $\hat{a}_{i,j}^{\dagger}$ ($\hat{a}_{i,j}$) is the creation
(annihilation) operator at site ($i,j$), $t_{x(y)}$ is the average coupling
strengths, $\lambda_{x(y)}$, $b_{x(y)}$, and $\phi^{x(y)}$ are modulated
strengths, frequencies and phases, respectively. In numerical calculations of
vector Chern number, we choose $t_{x(y)}=0.5$, $\lambda_{x(y)}=0.95$,
$b_{x(y)}=(\sqrt{5}+1)/2$, and the total sites are $15\times 15$ for two-
dimensional lattice. There are three subsets of bands in each direction, and
the vector Chern number takes values of $(1,-2,1;1,-2,1)$, indicating the
existence of topological corner states.

In each direction, there are three types of one-dimensional topological edge
states, one localized at both edges, one localized at the left edge and the
last one localized at the right edge; see Fig. 1(a). These edge states
constitute basic building blocks to construct topological corner states in
higher dimension. As shown in Fig. 1(b), we can construct corner states by
using the edge states in both $x$ and $y$ directions (see Supplemental
Materials for more details SM ). Since the couplings along $x$ and $y$
directions are independent, the robustness of corner states inherits that of
one-dimensional edge states in each dimension. Taking edge states in $x$
direction for example, there are energy gaps between edge states and extended
states; see Fig. 2(a). These topological edge states are robust to
considerable disorder, perturbation and long-range couplings that mix the two
dimensions provided energy gaps keep open. Hence, the constructing corner
states also share similar topological protection to the one dimensional edge
states, where there are finite energy gaps between corner states and extended
states; see Fig. 2(b). Apart from corner states, products of edge state in one
direction and extended state in other direction form a continuum subset which
is also separated from extended states. Such approach can be naturally
generalized to construct corner in three dimensional periodically-modulated
lattice, see Fig. 1 (c), along with hinge and surface states (see Supplemental
Materials for more details SM ).

What’s more, when $b_{x}=b_{y}=1/2$, the above model reduces to a two-
dimensional Su-Schrieffer-Heeger (SSH) lattice, where coupling matrices are
changed in a stagger way in both $x$ and $y$ directions TPRL118 . Indeed,
topological corner states are predicted with an alternative theory of two-
dimensional polarization T20201 ; T20181  and observed in experiment of
photonic crystal slabs Es1 ; Ew2 ; Ew3 . The varying of two-dimensional
polarization TPRL118  could give rise to topologial charge pumping in two
dimensions which is characterized by vector Chern number. However, on one
hand, these corner states can be more easily and naturally understood in our
theoretical framework, that is, they are the product of two edge states in
both $x$ and $y$ directions. On the other hand, in contrast to two-dimensional
polarization which relies on spatial symmetries, our theory of vector Chern
number can also predict corner states without requiring any fine-tuning of
symmetries.

##  III Experimental implement

In experiment, we first realize the Hamiltonian (II) in a two-dimensional
array of waveguides with modulated spacing. The site number is 15 or 16 in
both $x$ and $y$ directions, the average coupling strength $t_{x}=t_{y}=t$ is
adopted as 0.3 for photon with wavelength of 810 nm, the modulating amplitude
$\lambda_{x}=\lambda_{y}=\lambda$ is set as 0.5, the periodic parameter
$b_{x}=b_{y}=b$ is $(\sqrt{5}+1)/2$, and the initial phases $\phi^{x}$ and
$\phi^{y}$ are set as the same value. We fabricate the photonic waveguide
lattices according the Hamiltonian using the femtosecond laser direct-writing
technique fabri_1 ; fabri_2 ; fabri_3 ; fabri_4 ; PIT_Gap . As shown in Fig.
3(a), the propagation direction of the waveguide lattice maps the evolution
time, hence the fabricated tree dimensional waveguide lattice realizes the
designed two-dimensional off-diagonal AAH lattice. We further set nine sites
for injecting the photons, including four corner sites labeled as from C1 to
C4, four boundary sites labeled as from B1 to B4, and one site in the lattice
center labeled as T.

According to the prediction in theory, the corner state will appear when we
extend the one-dimensional topological lattice with edge state to a two-
dimensional lattice, and the corresponding topological origin is also extended
to higher dimensions. As shown in Fig. 3(b), there are edge states in both two
ends of lattice for 16-sited one-dimensional off-diagonal AAH lattice with
initial phase $\phi$ = 0.14$\pi$. We fabricate the two-dimensional off-
diagonal AAH lattice with initial phase $\phi^{x}=\phi^{y}$ = 0.14$\pi$ to
demonstrate the predicted corner states. We inject the photons with wavelength
of 810 nm in to the lattice from four corners respectively, the photons will
be confined in the excited lattice corners if there are higher-order corner
states in theoretical prediction. As shown in Fig. 3(c), the photon output
distributions after 40 mm evolution distance are localized in the white dotted
squares, which give the ranges of corner states.

In Fig. 3(d), we give the quantified results for measured corner states, which
is calculated by

$\xi_{p,q}=\sum_{i,j}\hat{a}_{i,j}^{\dagger}\hat{a}_{i,j}\quad(|i-p|\leq
l,|j-q|\leq l),$ (5)

where $(p,q)$ presents the excited site indices, and $l$ describes the range
of corner states adopted as 3. Compared with the 16$\times$16-sited lattice
with $\phi=0.14\pi$, there is no corner state for the case of $\phi=0.75\pi$,
and the photons flow out of the corner state range. This is because there is
no edge state in the one-dimensional AAH lattice for the case of
$\phi=0.75\pi$. Furthermore, we fabricate three two-dimensional
15$\times$15-sited lattices with phase $\phi=0.14\pi$, 0.75$\pi$, and
1.25$\pi$ respectively. There is only left (right) edge state for one-
dimensional lattice with $\phi=0.14\pi$ (1.25$\pi$), therefore the corner
state can only be observed by exciting the lattice from input C2 (C4). Similar
to 16$\times$16-sited lattice, there is no corner state for the case of
$\phi=$ 0.75$\pi$. We excite the lattices from input C2 and C4 respectively
and measure the photon output distributions, the quantified results, together
with the results of 16$\times$16-sited lattices, confirm theoretical
predictions on topological corner states arising by extending topological
lattice with edge state from one dimension to two dimension.

The corner states appearing in two-dimensional lattice require combination of
the one-dimensional lattices owning edge states in both the $x$ and $y$
directions. Differently, The edge states in higher dimensional lattices, as a
product of edge state in one direction and extended state in other direction,
can be naturally extended from the edge states in lower dimensional lattices.
As shown in Fig. 4(a), we inject the photons into the 16$\times$16-sited
lattice with $\phi$ = 0.14$\pi$ from the input B1, B2, B3 and B4 respectively,
the photons are confined in the boundaries. For the case of $\phi$ =
0.75$\pi$, there is no edge state that can be extended, so we can find that
the photons flow out the boundary ranges, as shown in Fig. 4(b) taking the
cases of B2 and B3 for example. The quantified results in Fig. 4(c) show the
observed edge states extended from one-dimensional lattices. As intuitive
understanding, edge states are extended from dots to lines when the
topological lattices are extended from one dimension to two dimensions.

In conclusion, we present a theoretical explanation on topological origin of
higher-order topological phase in higher dimensional lattices, which is
connected to the topological phase in lower dimensional lattices. We
experimentally observe the theoretically predicted higher-order topological
corner states in two-dimensional off-diagonal AAH lattices. Our model
intuitively explains the connection of topological phases in different
dimensional lattices, which is universal to various models and is promising to
be a convenient and practical tool for constructing higher-order topological
phases in higher-order lattices.

###### Acknowledgements.

The authors thank Yidong Chong and Jian-Wei Pan for helpful discussions.
X.-M.J. is supported by National Key R&D Program of China (2019YFA0308700,
2017YFA0303700), National Natural Science Foundation of China (11761141014,
61734005, 11690033), Science and Technology Commission of Shanghai
Municipality (17JC1400403), Shanghai Municipal Education Commission
(2017-01-07-00-02-E00049). X.-M.J. acknowledges additional support from a
Shanghai talent program. C. Lee is supported by the Key-Area Research and
Development Program of GuangDong Province (2019B030330001), the National
Natural Science Foundation of China (11874434, 11574405), and the Science and
Technology Program of Guangzhou (201904020024). Y. K. is partially supported
by the Office of China Postdoctoral Council (20180052), the National Natural
Science Foundation of China (11904419), and the Australian Research Council
(DP200101168).

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##  IV Supplementary Materials:
Constructing higher-order topological states in higher dimension

###  IV.1 Constructing corner states with off-diagonal AAH model

Figure S1: Appearance of edge modes and their topological origin. (a) Energy
bands in ($k,\phi^{s}$) space. The corresponding Chern numbers for three bands
are $(-1,2,-1)$. (b) Energy spectrum as a function of $\phi^{s}$ under open
boundary condition. (c) Asymmetry edge state and (d) Symmetry edge state for
$\phi^{s}=0(2\pi)$. The parameters are chosen as $t_{s}=1$, $\lambda_{s}=0.5$,
$b_{s}=1/3$, and the system size is $30$.

The Hamiltonian (4) in the main text can be rewritten as $H=H_{x}\otimes
I_{y}+I_{x}\otimes H_{y}$ with

$H_{s}=\sum_{j}\\{t_{s}[1+\lambda_{s}\cos(2\pi
b_{s}j+\phi_{s})]\hat{a}_{j}\hat{a}_{j+1}^{\dagger}+H.c.\\}.$ (S1)

Before proceeding to corner states, we first study the topological properties
of the Hamiltonian (S1) with rational modulated frequency $b_{s}=\mu/\nu$,
where $\mu$ and $\nu$ are coprime numbers. In this case, we can calculate the
vector Chern number in an alternative way. When the twisted angle is $0$
(i.e., in periodic boundary condition), the system is invariant when
translating a particle from the $j$th to $(j+\nu)$th sites, that is, the
system has translational symmetry. According to Bloch theorem, the eigenstates
are Bloch functions with quasi-momentum $k$, and the eigenvalues form $\nu$
Bloch bands. Here, the quasi-momentum is a good quantum number and provides an
intrinsic parameter to define topological invariant. Instead of integral over
twisted angle, each component of vector Chern number can be defined in the
space formed by quasi-momentum and modulated phase,

${C_{n}^{s}}=\frac{1}{{2\pi
i}}\int_{BZ}dk{d\phi^{s}\left({\langle{\partial_{\phi^{s}}}\psi_{n}^{s}|{\partial_{k}}\psi_{n}^{s}\rangle}\right)-\langle{\partial_{k}}\psi_{n}^{s}|{\partial_{\phi^{s}}}\psi_{n}^{s}\rangle}.$
(S2)

Such definition is consistent with Eq. (3) in the main text. According to
bulk-boundary correspondence, when the sum of all the Chern numbers below a
gap is non-zero, edge modes should appear in this gap. To show this, in the
periodic boundary condition we calculate the energy bands in the
($k,\phi^{s}$) parameter space and their Chern numbers, and in the open
boundary condition we calculate energy spectrum as a function of modulated
phase $\phi^{s}$; see Fig. S1. The parameters are chosen as $t_{s}=1$,
$\lambda_{s}=0.5$, $b_{s}=1/3$, and the system size is $30$. There are three
bands in each direction and the corresponding vector Chern number is
$(-1,2,-1;-1,2,-1)$. Indeed, in each direction, there are isolated edge modes
in each energy gap. When $\phi^{s}=0(2\pi)$, the two edge states in the first
gap become degenerate and their wave-functions are asymmetric [Fig. S1(c)] and
symmetric [Fig. S1(d)]. Here, we have clearly shown that the appearance of the
edge states results from dimension reduction of topological phases in the two-
dimensional parameter space $(k,\phi^{s})$.

Figure S2: Constructing corner states with edge states in $x$ and $y$
directions. (a) and (b): Energy spectral for off-diagonal AAH models in $y$
and $x$ directions, respectively. (c) corner states obtained by combining
symmetric and asymmetric edge states in $x$ and $y$ directions. (d) The corner
states obtained by diagonalizing Hamiltonian (4) in the main text with the
same energies as those in the top panel. The parameters are chosen as
$t_{x}=t_{y}=1$, $\lambda_{x}=0.4$, $\lambda_{y}=0.5$, $b_{x}=b_{y}=1/3$,
$\phi^{x}=\phi^{y}=0$ and the numbers of lattices in both $x$ and $y$
directions are $30$.

With the edge states at hand, we can construct the corner states with the
method depicted in the main text. For a set of degenerate eigenstates, the
superposition of these eigenstates is also the eigenstates. To avoid
unnecessary degeneracy, we consider the parameters for off-diagonal AAH model
in the $x$ direction are slightly different from those in the $y$ dimension.
The parameters are chosen as $t_{x}=t_{y}=1$, $\lambda_{x}=0.4$,
$\lambda_{y}=0.5$, $b_{x}=b_{y}=1/3$, $\phi^{x}=\phi^{y}=0$ and the numbers of
lattices in both $y$ and $x$ directions are $30$. The eigenvalues of the off-
diagonal AAH models in $y$ and $x$ directions are shown in Fig. S2(a) and (b),
respectively. Since the parameters are slightly different, the energy spectral
in $x$ and $y$ directions are quite similar. The symmetric and asymmetric edge
modes are marked by ‘S’ and ‘A’ in the spectral. With the edge modes in the
first gap, we can construct four corner states (namely, $SS$, $SA$, $AS$ and
$AA$) by any combination of the symmetric and asymmetric edge states in two
dimensions; see Fig. S2(c) respectively. For comparison, we show the corner
states obtained by directly diagonalizing Hamiltonian (4) in the main text;
see Fig. S2(d). The eigenvalues and spatial distributions of the corner states
in the bottom panel are exactly the same as those in the top panel. It means
that the constructed corner states are indeed the eigenstates of the two-
dimensional off-diagonal AAH model.

For irrational modulated frequency, translational symmetry is broken and
quasi-momentum is no-longer a good quantum number to define topological
invariant. However, vector Chern number for the irrational off-diagonal AAH
model can still be defined by introducing twisted angles, as shown in the main
text. The vector Chern number well characterizes the topology of such
irrational off-diagonal AAH model. Topological corner states also appear in
the energy gap as a consequence of such nontrival vector Chern number. We show
experimental observation of corner states based on the irrational off-diagonal
AAH model in the main text.

###  IV.2 Generally constructing corner states in two-dimensional lattice

Figure S3: Constructing corner states in two-dimensional lattice. The second-
order states can be assembled using the edge states in one-dimensional
lattices.

As we have shown in main text and last section by taking the examples, one is
able to freely construct the higher-order topological phase in two-dimensional
lattice by assembling edge states in one-dimensional lattices. In this way,
the higher-order states can be assembled using the edge states in lower-
dimensional lattices, similar to playing LEGO game. In Fig. S3, we show the
combination way to constructing corner states in two-dimensional lattice using
edge states in $x$ and $y$ directional one-dimensional lattices. For example,
we can choose the modulation of the lattice owning both left and right edge
states in $x$ direction and just owning left edge state in $y$ direction to
construct the corner states in top left and top right corners.

###  IV.3 Constructing corner states in three dimensional lattice

Here we extend our approach to a three-dimensional cubic lattice model. The
position of a particle is denoted by $(i,j,k)$ which are site indices of $x$,
$y$ and $z$ directions respectively. The coupling between $(i,j,k)$th and
$(l,m,n)$th lattice sites has the form,
$H_{x}(i,l)\delta_{j,m}\delta_{k,n}+\delta_{i,l}H_{y}(j,m)\delta_{k,n}+\delta_{i,l}\delta_{j,m}H_{z}(k,n)$,
where $H_{x}(i,k)$ is the coupling matrix along $x$ direction irrelevant to
positions in $y$ and $z$ directions, and vice versa. The motion of a particle
hopping in such lattice is governed by the Hamiltonian,

$H=H_{x}\otimes I_{y}\otimes I_{z}+I_{x}\otimes H_{y}\otimes
I_{z}+I_{x}\otimes I_{y}\otimes H_{z},$ (S3)

where $I_{\\{x,y,z\\}}$ are identical matrices in $\\{x,y,z\\}$ directions.
Solving the eigenproblems,

$\displaystyle H_{x}|\psi_{p}^{x}\rangle$ $\displaystyle=$ $\displaystyle
E_{p}^{x}|\psi_{p}^{x}\rangle,$ $\displaystyle H_{y}|\psi_{q}^{y}\rangle$
$\displaystyle=$ $\displaystyle E_{q}^{y}|\psi_{q}^{y}\rangle,$ $\displaystyle
H_{z}|\psi_{l}^{z}\rangle$ $\displaystyle=$ $\displaystyle
E_{l}^{z}|\psi_{l}^{z}\rangle,$ (S4)

we can obtain the eigenvalues $\\{E_{p}^{x},E_{q}^{y},E_{l}^{z}\\}$ and
eigenstates
$\\{|\psi_{p}^{x}\rangle,|\psi_{q}^{y}\rangle,|\psi_{l}^{z}\rangle\\}$
corresponding to $\\{H_{x},H_{y},H_{z}\\}$. We can immediately prove that

$\displaystyle
H|\psi_{p}^{x}\rangle\otimes|\psi_{q}^{y}\rangle\otimes|\psi_{l}^{z}\rangle$
(S5) $\displaystyle=$
$\displaystyle(E_{p}^{x}+E_{q}^{y}+E_{l}^{z})|\psi_{p}^{x}\rangle\otimes|\psi_{q}^{y}\rangle\otimes|\psi_{l}^{z}\rangle,$

that is, $E_{p}^{x}+E_{q}^{y}+E_{l}^{z}$ and
$|\psi_{p}^{x}\rangle\otimes|\psi_{q}^{y}\rangle\otimes|\psi_{l}^{z}\rangle$
are the eigenvalues and eigenstates corresponding to $H$, respectively. If
$\\{|\psi_{p}^{x}\rangle,|\psi_{q}^{y}\rangle,|\psi_{l}^{z}\rangle\\}$ are the
edge states, the product of the three edge states becomes a corner state in
three dimensions.

Similarly, we can also construct hinge state by selecting edge states in any
two dimensions and extended state in the other dimension, and the surface
states by constructing extended states in any two dimensions and edge state in
the other dimension.

Figure S4: Constructing (a) corner state, (b) Hinge state and (c) Surface
state in a three-dimensional lattice. The parameters are chosen as
$t_{x}=t_{y}=t_{z}=1$, $\lambda_{x}=\lambda_{y}=\lambda_{z}=0.5$,
$b_{x}=b_{y}=b_{z}=1/3$, $\phi^{x}=\phi^{y}=\phi_{z}=0$ and the numbers of
lattices in $x$, $y$ and $z$ directions are $30$.  Figure S5: Constructing
higher-order topological states in three-dimensional lattice. Higher-order
topological states are observed in quasi-crystal lattice (a) and SSH lattice
(b) in simulation. The line width and color depth of markers show the output
probabilities. The red arrows point out the exciting position in simulation.

In Fig. S4, we show the construction of corner, hinge and surface states in a
three-dimensional lattice with $H_{s}$ given by Eq. (S1). The parameters are
chosen as $t_{x}=t_{y}=t_{z}=1$, $\lambda_{x}=\lambda_{y}=\lambda_{z}=0.5$,
$b_{x}=b_{y}=b_{z}=1/3$, $\phi^{x}=\phi^{y}=\phi_{z}=0$ and the numbers of
lattices in $x$, $y$ and $z$ directions are $30$. The corner state is the
product of the first symmetric edge states [as shown in Fig. S2(d)] in $x$,
$y$ and $z$ directions. The Hinge state is the product of the extended state
with the highest energy of the first band in $z$ direction and the first
symmetric edge states in $x$ and $y$ directions. The Surface state is the
product of the extended states with the highest energy of the first band in
$x$ and $z$ directions and the first symmetric edge state in $y$ direction.

To simulate the observation of higher-order topological states in three-
dimensional lattices, in the Fig. S5, we show the evolution results of photons
in assembled higher-order topological three-dimensional lattices. In the first
example, the parameters are chosen as $t_{x(y,z)}=0.5$,
$\lambda_{x(y,z)}=0.5$, $\phi_{x(y,z)}=0.15$, $b_{x(y,z)}=(\sqrt{5}+1)/2$, the
site numbers of lattice in both $x$ and $y$ directions are $16$, and the site
number of lattice in $z$ direction is $15$. This is a three-dimensional quasi-
crystal lattice based on AAH model, and the simulated results is shown in Fig.
S5(a). In the second example, the parameters are chosen as $t_{x(y,z)}=0.5$,
$\lambda_{x(y,z)}=0.5$, $\phi_{x(y,z)}=0$, $b_{x(y,z)}=1/2$, the site numbers
of lattice in both $x$ and $y$ directions are $16$, and the site number of
lattice in $z$ direction is $15$. This is a three-dimensional SSH lattice, and
the simulated results is shown in Fig. S5(b).