# Higher-order topological insulators in hyperbolic lattices

Zheng-Rong Liu  Department of Physics, Hubei University, Wuhan 430062, China
Chun-Bo Hua<EMAIL_ADDRESS>School of Electronic and Information
Engineering, Hubei University of Science and Technology, Xianning 437100,
China  Tan Peng  School of Mathematics, Physics and Optoelectronic
Engineering, Hubei University of Automotive Technology, Shiyan 442002, China
Rui Chen  Department of Physics, The University of Hong Kong, Pokfulam Road,
Hong Kong, China  Bin Zhou<EMAIL_ADDRESS>Department of Physics, Hubei
University, Wuhan 430062, China

###### Abstract

To explore the non-Euclidean generalization of higher-order topological
phenomena, we construct a higher-order topological insulator model in
hyperbolic lattices by breaking the time-reversal symmetry (TRS) of quantum
spin Hall insulators. We investigate three kinds of hyperbolic lattices, i.e.,
hyperbolic $\\{4,5\\}$, $\\{8,3\\}$ and $\\{12,3\\}$ lattices, respectively.
The non-Euclidean higher-order topological behavior is characterized by zero-
energy effective corner states appearing in hyperbolic lattices. By adjusting
the variation period of TRS breaking term, we obtain four, eight and twelve
zero-energy effective corner states in these three different hyperbolic
lattices, respectively. It is found that the number of zero-energy effective
corner states of a hyperbolic lattice depends on the variation period of TRS
breaking term. Via symmetry analysis, it is confirmed that the hyperbolic
zero-energy effective corner states are protected by both the particle-hole
symmetry $P$ and the effective chiral symmetry $S_{\rm eff}$, where $S_{\rm
eff}$ is a combination of chiral symmetry $S$ and mirror symmetry $m_{z}$. The
hyperbolic zero-energy effective corner states remain stable unless both the
particle-hole symmetry and the effective chiral symmetry are broken
simultaneously. The topological nature of hyperbolic zero-energy effective
corner states is further confirmed by checking the robustness of the zero
modes in the hyperbolic lattices in the presence of disorder. Our work
provides a route for the research on hyperbolic higher-order topological
insulator in the non-Euclidean geometric systems.

##  I Introduction

The study of topological insulators is an important field in condensed matter
physics [1, 2, 3, 4, 5, 6]. Topological insulators in the $d$-dimensional
geometry host a $d$-dimensional gapped bulk and $(d-1)$-dimensional gapless
boundary states [7, 8, 9, 10, 11]. These boundary states are protected by the
topological properties of the bulk, characterized by a quantized number called
the topological invariant [12, 13, 9, 1, 14]. Recently, the notion of
topological insulators has been generalized, and a new class of topological
materials called higher-order topological insulators (HOTIs) has attracted
increasing research interest [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].
In comparison with conventional first-order topological insulators, the HOTIs
possess topologically localized states that are at least two dimensions lower
than the bulk. Namely, a $n$th-order topological insulator is characterized by
$(d-n)$-dimensional gapless boundary states for $d$-dimensional bulk system.
The HOTIs can be realized in a wide variety of systems, such as bismuth [19],
phononic [39], photonic [40], microwave [41], acoustic metamaterial [42, 43],
and electrical circuit systems [44]. In two-dimensional (2D) geometry, the
HOTIs refer to second-order topological insulators. A second-order topological
insulator in 2D geometry has a gapped 2D bulk state and gapped one-dimensional
(1D) boundary state, but has topologically protected gapless zero-dimensional
(0D) corner states localized at the intersections between 1D boundaries [22,
23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41,
42, 43, 44].

In addition to Euclidean geometry, non-Euclidean geometry also possesses
abundant novel physical phenomena. Hyperbolic lattices with constant negative
curvature are promising platforms for fault-tolerant quantum computing [46,
47, 48, 49, 50]. Inspired by the recent experimental realizations of
hyperbolic lattices in circuit quantum electrodynamics [51], a considerable
number of theoretical studies and experimental observations of hyperbolic
lattices have been reported [52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63,
64, 65, 66, 67, 68, 69, 70, 71, 72]. Based on the Riemann surface theory and
algebraic geometry framework, a hyperbolic lattice generalization of Bloch
band theory and crystallography has been proposed [52, 53, 54, 56, 57, 55].
Recently, the non-Euclidean generalization of topological phenomena has also
received enthusiastic attention [58, 59, 60, 61, 62, 63, 64, 65]. Yu
$et~{}al$. proposed a non-Euclidean analog of the quantum spin Hall effect in
hyperbolic lattices [58]. Subsequently, the topological phase transition and
transport properties of Chern insulator in a hyperbolic lattice were explored
[60]. Urwyler $et~{}al$. investigated the topological aspects arising in
hyperbolic band theory based on the hyperbolic Haldane model and the
hyperbolic Kane-Mele model [61]. It is noted that the first experimental
demonstration of topological hyperbolic lattices on an electric circuit
network has been reported by Zhang $et~{}al$ [59]. In their work, in addition
to the theoretical proposal and experimental simulation of the hyperbolic
Haldane model, they also showed that the fractal-like midgap higher-order zero
modes appear in deformed hyperbolic lattices, in which a pair of distinct
coupling strengths is introduced [59]. Motivated by the recent theoretical
progress and experimental realization in topological hyperbolic lattices,
further exploring the non-Euclidean generalization of higher-order topological
phenomena is a fascinating issue. In this work, we aim to investigate a non-
Euclidean analog of HOTIs. We construct a HOTI model in hyperbolic lattices by
introducing a time-reversal symmetry (TRS) breaking Wilson mass term into the
quantum spin Hall insulator. For concreteness, the hyperbolic $\\{4,5\\}$,
$\\{8,3\\}$ and $\\{12,3\\}$ lattices are considered, respectively. The non-
Euclidean higher-order topological behavior is characterized by zero-energy
effective corner states appearing in hyperbolic lattices. The symmetry
analysis and robustness against weak disorder further confirm the topological
nature of hyperbolic zero-energy effective corner states.

The rest of the paper is organized as follows. In Sec. II, we introduce a HOTI
model in real space into hyperbolic $\\{4,5\\}$, $\\{8,3\\}$, and $\\{12,3\\}$
lattices, and calculate the energy spectrum of the system and the probability
distribution of the (near) zero-energy eigenstates when the Wilson mass term
is turned off or on. In Sec. III.1, we separately add sixteen perturbations to
the hyperbolic HOTI system to examine the stability of zero-energy effective
corner states. In Sec. III.2, we demonstrate the robustness of zero-energy
effective corner states against symmetry-preserving weak disorder. Finally, we
summarize our conclusions in Sec. IV.

##  II Zero-energy effective corner states in hyperbolic lattices

In this paper, we adopt Schläfli symbol $\\{p,q\\}$ to label the hyperbolic
lattice in the Poincaré disk, which is obtained by projecting a hyperbolic
plane with constant negative curvature into the unit circle in Euclidean plane
[73, 74, 75]. $p$ represents the number of vertices of regular polygons that
are uniformly tessellated in hyperbolic space, and $q$ means the number of
polygons adjacent to the same vertex. In Euclidean geometry, a regular polygon
uniformly tessellated in a plane must satisfy the relation $(p-2)(q-2)=4$.
This means that only three regular polygons $\\{3,6\\}$, $\\{4,4\\}$ and
$\\{6,3\\}$ can be uniformly tessellated in the Euclidean plane. However,
there are infinite kinds of regular polygons in the hyperbolic plane, which is
guaranteed by the relation $(p-2)(q-2)>4$. It provides support for the noval
physical phenomena in hyperbolic lattices.

Quantum spin Hall insulators are a promising platform for realizing HOTIs [15,
16, 32, 35, 36, 38]. First, we introduce Bernevig-Hughes-Zhang (BHZ) model
[11, 10] of quantum spin Hall insulators to hyperbolic lattice in the real
space, which can be described by the following tight-binding model
Hamiltonian:

$\displaystyle
H_{0}=-\sum_{\left<j,k\right>}c_{j}^{\dagger}it_{1}\left[s_{z}\tau_{x}\cos(\theta_{jk})+s_{0}\tau_{y}\sin(\theta_{jk})\right]c_{k}$
$\displaystyle-\sum_{\left<j,k\right>}c_{j}^{\dagger}t_{2}s_{0}\tau_{z}c_{k}+\sum_{j}(M+2t_{2})c_{j}^{\dagger}s_{0}\tau_{z}c_{j},$
(1)

where $c_{j}^{\dagger}$ and $c_{j}$ are the creation and annihilation
operators of electrons on site $j$. $\theta_{jk}$ represents the polar angle
of the vector from the site $k$ to the site $j$ in the Poincaré disk. $s_{0}$
is the identity matrix, $s_{x,y,z}$ and $\tau_{x,y,z}$ are the Pauli matrices
representing spin and orbital, respectively. $M$ denotes the Dirac mass,
$t_{1}$ is the spin-orbit coupling strength, and $t_{2}$ is the hopping
amplitude.

Before constructing higher-order topological states, we discuss the symmetries
of the Hamiltonian $H_{0}$. Similar to that in the Euclidean geometric system,
the Hamiltonian $H_{0}$ in the hyperbolic lattice possesses TRS
$TH_{0}T^{-1}=H_{0}$ with TRS operator $T=is_{y}\tau_{0}\mathcal{K}$ (where
$\mathcal{K}$ is the complex conjugation), particle-hole symmetry (PHS)
$PH_{0}P^{-1}=-H_{0}$ with PHS operator $P=s_{z}\tau_{x}\mathcal{K}$, and
chiral symmetry $SH_{0}S^{-1}=-H_{0}$ with chiral symmetry operator $S=PT$. In
addition to the fundamental symmetries, the Hamiltonian $H_{0}$ in the
hyperbolic lattice also satisfies some spatial symmetries. For the hyperbolic
$\\{p,q\\}$ lattice, $H_{0}$ has mirror symmetry $\left[H_{0},m_{z}\right]=0$
with mirror symmetry operator $m_{z}=s_{z}\tau_{0}$ and $p$-fold rotational
symmetry $\left[H_{0},C_{p}\right]=0$. The $p$-fold rotational symmetry
operator is $C_{p}=e^{-i\frac{\pi}{p}s_{z}\tau_{z}}\mathcal{R}_{p}$, where
$\mathcal{R}_{p}$ is an orthogonal matrix permuting the vertices of the
hyperbolic $\\{p,q\\}$ lattice to rotate the whole system by an angle of
$2\pi/p$.

Figure 1: Schematic illustration of the hyperbolic (a) $\\{4,5\\}$, (b)
$\\{8,3\\}$ and (c) $\\{12,3\\}$ lattice. In the Poincaré disk, the solid
black line means the distance between the nearest neighbors of two lattice
sites, and any solid black line in the same disk represents the same distance.
Energy spectrum of the Hamiltonian $H_{0}$ on the hyperbolic (d) $\\{4,5\\}$,
(e) $\\{8,3\\}$ and (f) $\\{12,3\\}$ lattice. In (d), (e) and (f), the
energies marked by red dots correspond to states localized at the edge. (g)
The probability distribution of four eigenstates near zero energy
($E_{n}=-0.0091,-0.0091,0.0091,0.0091$) marked by the red arrow in (d). The
parameters are $M=-1$ and the lattice site number $N=436$ in (d) and (g). (h)
The probability distribution of four eigenstates near zero energy
($E_{n}=-0.0012,-0.0012,0.0012,0.0012$) marked by the red arrow in (e). The
parameters are $M=-1.2$ and $N=2888$ in (e) and (h). (i) The probability
distribution of four eigenstates near zero energy
($E_{n}=-0.0005,-0.0005,0.0005,0.0005$) marked by the red arrow in (f). The
parameters are $M=-1$ and $N=7680$ in (f) and (i). Here, we take the
parameters $t_{1}=t_{2}=1$.

Here, for concreteness, the hyperbolic $\\{4,5\\}$, $\\{8,3\\}$ and
$\\{12,3\\}$ lattices are constructed in the Poincaré disk as shown in Figs.
1(a), 1(b) and 1(c), respectively. Energy spectra of the Hamiltonian $H_{0}$
are plotted in Figs. 1(d), 1(e) and 1(f). In Ref. [60], the gapless chiral
edge states were presented when the Chern insulator model was applied to a
hyperbolic lattice. Similarly, we plot the probability distribution of the
edge states of the BHZ model $H_{0}$ in the hyperbolic $\\{4,5\\}$,
$\\{8,3\\}$ and $\\{12,3\\}$ lattices, as shown in Figs. 1(g), 1(h) and 1(i).
It can be found that gapless edge states exist in three different hyperbolic
lattices, and the double degeneracy of the edge states is guaranteed by TRS.

One of the ways to realize HOTIs is to construct new mass domain walls on the
boundaries of first-order topological insulators [32, 23, 31, 15, 30, 16, 29,
21, 35, 36]. The energy gap of the first-order topological insulator can be
opened by breaking the TRS, therefore, we construct the following Hamiltonian
$H_{1}$,

$\displaystyle H_{1}$ $\displaystyle=H_{0}+H_{g},$ $\displaystyle H_{g}$
$\displaystyle=\frac{g}{2}\sum_{j\neq
k}c_{j}^{\dagger}\cos(\eta\theta_{jk})s_{x}\tau_{x}c_{k},$ (2)

where $H_{g}$ is a Wilson mass term, $g$ is the magnitude of the Wilson mass.
$\eta$ can only take even numbers, which is used to adjust the variation
period of the Wilson mass. The TRS breaking term $H_{g}$ gaps out 1D edge
states in the first-order topological insulator. In addition, since
$\cos(\eta\theta_{jk})$ in $H_{g}$ is alternating positive and negative,
$H_{g}$ can construct new mass domain walls on the edge.

Figure 2: (a) Energy spectrum of the Hamiltonian $H_{1}$ in the hyperbolic
$\\{4,5\\}$ lattice. (b) The probability distribution of the four zero-energy
eigenstates marked with red dots in (a). The parameters are $M=-1$, $g=0.5$,
$\eta=2$, and $N=436$ in (a) and (b). (c) Energy spectrum of the Hamiltonian
$H_{1}$ in the hyperbolic $\\{8,3\\}$ lattice. (d) The probability
distribution of the eight zero-energy eigenstates marked with red dots in (c).
The parameters are $M=-1.2$, $g=0.5$, $\eta=4$, and $N=2888$ in (c) and (d).
(e) Energy spectrum of the Hamiltonian $H_{1}$ in the hyperbolic $\\{12,3\\}$
lattice. (f) The probability distribution of the twelve zero-energy
eigenstates marked with red dots in (e). The parameters are $M=-1$, $g=0.5$,
$\eta=6$, and $N=7680$ in (e) and (f). Here, we take the parameters
$t_{1}=t_{2}=1$.

Next, we turn on the $H_{g}$ term in order to explore higher-order topological
states. Here we first set $\eta=2$ and use the Jackiw-Rebbi mechanism to
analyze the newly generated Dirac-mass domain walls [76]. The sign change of
$H_{g}$ is modulated by $\cos(\eta\theta_{jk})$, and when $\eta=2$, $H_{g}$ is
divided into four parts with alternating positive and negative signs in the
whole plane by $\cos(\eta\theta_{jk})$. Then, the boundary between the
positive and negative regions of Dirac mass is the domain wall with the Dirac
mass equal to zero. When $\eta=2$, there are four such domain walls. As shown
in Fig. 2(a), there are four zero energies marked with red dots in the energy
gap opened by the $H_{g}$ term. In Fig. 2(b), we show the probability
distribution of these four zero-energy eigenstates and find that they are
localized exactly at the new domain walls constructed by $H_{g}$. This is
consistent with our analysis results. These four zero modes are called as the
hyperbolic effective corner states, since there are not the real geometric
corners (intersections between 1D boundaries) in the Poincaré disk
corresponding to a hyperbolic lattice [59]. Then, we apply the Hamiltonian
$H_{1}$ with $\eta=4$ and $\eta=6$ in the hyperbolic $\\{8,3\\}$ and
$\\{12,3\\}$ lattices, respectively. By diagonalizing the Hamiltonian $H_{1}$
in two different hyperbolic lattices, we obtain partial energies near zero
energy, as shown in Figs. 2(c) and 2(e). It can be found that there are eight
and twelve in-gap zero energies, respectively. Similarly, we calculate the
probability distributions of these eight (twelve) zero-energy eigenstates and
find that they all exhibit a $0$D form localized at the edge of the hyperbolic
lattice. The eight and twelve zero-energy effective corner states in the
hyperbolic $\\{8,3\\}$ and $\\{12,3\\}$ lattices are plotted in Figs. 2(d) and
2(f), respectively. Numerical calculations show that the number of zero-energy
effective corner states of a hyperbolic lattice can be tuned by changing the
value of $\eta$. For instance, there can also exit twelve zero-energy
effective corner states in the hyperbolic $\\{8,3\\}$ lattice when the value
of $\eta$ in the Hamiltonian $H_{1}$ is taken as $\eta=6$ for the hyperbolic
$\\{8,3\\}$ lattice.

##  III Stability of zero-energy effective corner states

In this section, we will verify that zero-energy effective corner states are
topologically nontrivial. First, we examine whether zero-energy effective
corner states in hyperbolic lattices are protected by specific symmetries
through symmetry analysis and the responses of zero-energy effective corner
states to different perturbations. Then, we present the robustness of the zero
modes in the hyperbolic lattices against weak disorder.

###  III.1 Symmetry analysis

Figure 3: Energy spectrum of $H_{1}+\Delta_{mn}$ in the hyperbolic $\\{8,3\\}$
lattice, where $m,n=0,x,y,z$. Here, we take the parameters $M=-1.2$,
$t_{1}=t_{2}=1$, $g=0.5$, $\eta=4$, the perturbation strength $U=0.01$, and
$N=2888$. Table 1: Symmetry analysis of the Hamiltonian $H_{1}$ without
($U=0$) and with ($U\neq 0$) the perturbation terms $\Delta_{mn}$ on the
hyperbolic $\\{4,5\\}$, $\\{8,3\\}$, and $\\{12,3\\}$ lattices. Here, for the
hyperbolic $\\{4,5\\}$, $\\{8,3\\}$, and $\\{12,3\\}$ lattices, $C_{p}$ refers
to $4$-fold, $8$-fold, and $12$-fold rotational symmetries, respectively.
Check mark indicates that the symmetry in this case is preserved, and a cross
mark means the symmetry is absent.

| $U=0$ | $U\not=0$
---|---|---
$00$ | $0x$ | $0y$ | $0z$ | $x0$ | $xx$ | $xy$ | $xz$ | $y0$ | $yx$ | $yy$ | $yz$ | $z0$ | $zx$ | $zy$ | $zz$
$P$ | ✓ | $\times$ | $\times$ | $\times$ | ✓ | ✓ | ✓ | ✓ | $\times$ | $\times$ | $\times$ | $\times$ | ✓ | $\times$ | $\times$ | $\times$ | ✓
$T$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$
$S$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$
$m_{x}$ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$
$m_{y}$ | ✓ | ✓ | $\times$ | $\times$ | ✓ | $\times$ | ✓ | ✓ | $\times$ | ✓ | $\times$ | $\times$ | ✓ | $\times$ | ✓ | ✓ | $\times$
$m_{z}$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$
$Sm_{z}$ | $\checkmark$ | $\times$ | $\times$ | $\checkmark$ | $\checkmark$ | $\checkmark$ | $\checkmark$ | $\times$ | $\times$ | $\times$ | $\times$ | $\checkmark$ | $\checkmark$ | ✓ | $\checkmark$ | $\times$ | $\times$
$C_{p}$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$
$C_{p}T$ | ✓ | ✓ | $\times$ | $\times$ | ✓ | $\times$ | $\times$ | ✓ | $\times$ | $\times$ | $\times$ | ✓ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$
$C_{p}m_{x}$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$
$C_{p}m_{y}$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$
$C_{p}m_{z}$ | ✓ | ✓ | $\times$ | $\times$ | ✓ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | ✓ | $\times$ | $\times$ | ✓

So far, there is not yet a suitable topological invariant for characterizing
higher-order topological states in hyperbolic lattices. It is known that, by
adding perturbations to the system, the gapless topological boundary states
remain stable as long as the symmetry protecting the topological system is
preserved. In this subsection, we add a uniform perturbation
$\Delta_{mn}=U\sum_{j}c_{j}^{\dagger}s_{m}\tau_{n}c_{j}$ to the Hamiltonian
$H_{1}$ to examine the stability of zero-energy effective corner states. $U$
is the strength of the perturbation, and $m,n=0,x,y,z$. Obviously, there are
sixteen kinds of perturbations in total. Here, for concreteness, we take the
$\\{8,3\\}$ lattice as an example to show the changes of the near-zero energy
spectrum under the influence of different perturbations. The changes of the
systems in the $\\{4,5\\}$ and $\\{12,3\\}$ lattices under the influence of
perturbations are similar to those in the $\\{8,3\\}$ lattices. As shown in
Fig. 3, the zero modes remain stable in the presence of ten kinds of uniform
perturbations, which are $\Delta_{0y}$ [Fig. 3(c)], $\Delta_{0z}$ [Fig. 3(d)],
$\Delta_{x0}$ [Fig. 3(e)], $\Delta_{xx}$ [Fig. 3(f)], $\Delta_{xy}$ [Fig.
3(g)], $\Delta_{yy}$ [Fig. 3(k)], $\Delta_{yz}$ [Fig. 3(l)], $\Delta_{z0}$
[Fig. 3(m)], $\Delta_{zx}$ [Fig. 3(n)], and $\Delta_{zz}$ [Fig. 3(p)],
respectively. The other six kinds of perturbations can destroy the zero modes.
Like conventional topological insulators, topologically localized states in
HOTIs are likewise protected by specific symmetries. Next, we explore the
symmetries that protect zero-energy effective corner states in hyperbolic
lattices by analyzing the changes in symmetry under perturbation.

The results of symmetry analysis are presented in Table 1. In the absence of
perturbation $\Delta_{mn}$, the Hamiltonian $H_{1}$ satisfies PHS, mirror
symmetries $m_{x}$ and $m_{y}$, and combined symmetries $Sm_{z}$, $C_{p}T$,
$C_{p}m_{z}$. Comparing with Fig. 3, we find that the zero modes remain stable
when PHS $P$ or effective chiral symmetry $S_{\rm eff}=Sm_{z}$ is preserved,
while the zero modes are broken in the absence of both $P$ and $S_{\rm eff}$.
Therefore, the zero-energy effective corner states in the hyperbolic
$\\{4,5\\}$, $\\{8,3\\}$, and $\\{12,3\\}$ lattices are indeed higher-order
topological states, which are protected by both PHS and the effective chiral
symmetry.

Figure 4: (a) Eigenenergy as a function of disorder strength $W$ in the
hyperbolic $\\{4,5\\}$ lattice. (b) Energy spectrum of the Hamiltonian $H_{\rm
dis}$ in the hyperbolic $\\{4,5\\}$ lattice when $W=0.5$. (c) The probability
distribution of the four zero modes marked with red dots in (b). The
parameters are $M=-1$, $g=0.5$, $\eta=2$, and $N=436$ in (a), (b), and (c).
(d) Eigenenergy as a function of disorder strength $W$ in the hyperbolic
$\\{8,3\\}$ lattice. (e) Energy spectrum of the Hamiltonian $H_{\rm dis}$ in
the hyperbolic $\\{8,3\\}$ lattice when $W=0.5$. (f) The probability
distribution of the eight zero modes marked with red dots in (e). The
parameters are $M=-1.2$, $g=0.5$, $\eta=4$, and $N=2888$ in (d), (e), and (f).
(g) Eigenenergy as a function of disorder strength $W$ in the hyperbolic
$\\{12,3\\}$ lattice. (h) Energy spectrum of the Hamiltonian $H_{\rm dis}$ in
the hyperbolic $\\{12,3\\}$ lattice when $W=0.2$. (i) The probability
distribution of the twelve zero modes marked with red dots in (h). The
parameters are $M=-1$, $g=0.5$, $\eta=6$, and $N=7680$ in (g), (h), and (i).
In (a), (d), and (g), the vertical axis represents the average energy of $100$
samples. Here, we take the parameters $t_{1}=t_{2}=1$.

###  III.2 Robustness of the zero modes against disorder

Now, we focus on the robustness of zero-energy effective corner states in
hyperbolic lattices with disorder. The above subsection has mentioned that
some specific uniform perturbations can break PHS and effective chiral
symmetry, that is, can break the zero-energy effective corner states.
Therefore, here we only consider a disorder that does not break PHS and
effective chiral symmetry. The Hamiltonian of system with disorder is written
as

$\displaystyle H_{\rm
dis}=H_{1}+W\sum_{j}c_{j}^{\dagger}\omega_{j}s_{0}\tau_{z}c_{j},$ (3)

where $W$ is the disorder strength, and $\omega_{j}$ is a uniformly
distributed variable within $[-0.5,0.5]$. Due to the lack of topological
invariants, we use the multi-degenerate zero-energy stability as an indicator
to verify whether the system is robust against weak disorder. Similar to the
clean system, we first substitute the Hamiltonian $H_{\rm dis}$ with $\eta=2$
into the hyperbolic $\\{4,5\\}$ lattice and calculate the energy as a function
of the disorder strength $W$. As shown in Fig. 4(a), the degenerate four zero-
energy states remain stable in weak disorder until the disorder strength
increases to a certain value. In Fig. 4(b), we plot the energy spectrum of the
Hamiltonian $H_{\rm dis}$ in the hyperbolic $\\{4,5\\}$ lattice when the
disorder strength $W=0.5$, and the four zero energies are marked with red
dots. Figure 4(c) shows the probability distribution of the four zero-energy
eigenstates. It is obvious that the localized four zero-energy states remain
stable in the presence of weak disorder. Subsequently, we investigate the
response of systems in the hyperbolic $\\{8,3\\}$ and $\\{12,3\\}$ lattices to
disorder, respectively. As shown in Figs. 4(d) and 4(g), the degenerate eight
and twelve zero-energy states are robust against weak disorder. According to
the above results, it can be known that the zero-energy effective corner
states in the hyperbolic $\\{4,5\\}$, $\\{8,3\\}$, and $\\{12,3\\}$ lattices,
which are jointly protected by PHS and the effective chiral symmetry, remain
stable when the disorder strength is weak.

##  IV Conclusion

In this work, we investigate higher-order topological behavior of the
hyperbolic lattices by introducing a TRS breaking Wilson mass term into the
quantum spin Hall insulator. Three hyperbolic $\\{4,5\\}$, $\\{8,3\\}$ and
$\\{12,3\\}$ lattices are considered, respectively. By adjusting the variation
period of TRS breaking term, different numbers zero-energy effective corner
states can appear in the hyperbolic lattices. It is found that the hyperbolic
zero-energy effective corner states are protected by both the particle-hole
symmetry and the effective chiral symmetry. The robustness of the zero modes
in the hyperbolic lattices against weak disorder further confirms the
topological nature of hyperbolic zero-energy effective corner states. Our
results present a non-Euclidean analog of HOTI.

## 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).

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