# Non-Hermitian total-loss high-order topological insulator based on 1D Su-
Schrieffer-Heeger (SSH)

Hui-Chang Li  Jing-Wei Xu  Chen Luo  Tai-Lin Zhang  Jian-Wei Xu  Xiang Zhou
Yun Shen<EMAIL_ADDRESS>Department of Physics, Nanchang University,
Nanchang 330031, China  Xiao-Hua Deng<EMAIL_ADDRESS>Institute of
Space Science and Technology, Nanchang University, Nanchang 330031, China

###### Abstract

Non-Hermiticity alters topology with the presence of non-Hermitian factors in
topological systems. Most existing non-Hermitian topological systems derive
their topological phases from Hermitian components, that is, the gain and loss
of the system are considered simultaneously. In this work, we reveal two-
dimensional non-Hermitian high-order topological insulator based on one-
dimensional SSH chain, the nontrivial topology of which induced by total-loss.
By introducing the loss of a specific configuration, we get a band gap with
corner and edge states whose topology is characterized by the gapped wannier
band. In addition, we demonstrate the existence of ’real-energy’ edge states
in pseudo-PT symmetric domain wall system. These results can be easily
implemented in experiments, and promote the research of topological
transmission of lossy systems in the real world.

##  I Introduction

Hermiticity of system is the basic assumption of quantum mechanics, which
plays an important role in the study of topological states and allows the
existence of real eigenvalues and orthogonal eigenstates which help to define
the classification of topological phase 1-1 ; 1-2 ; 1-3  and various
topological invariants 2-1 ; 2-2 ; 2-3 ; 2-4 ; 2-5 ; 2-6 ; 2-7 . Nowadays, the
research of topological states 3-1 ; 3-3  has been extended to various fields,
such as superconductor 4-1 ; 4-2 ; 4-3 , cold atom 5-1 ; 5-2 ; 5-3 ; 5-4 ; 5-5
; 5-6 , solid states 6-1 ; 6-2 , etc. More recently, topological crystalline
insulators (TCIs) hosting higher-order topological (HOT) states have drawn
extensive attention 7-1 ; 7-2 ; 7-3 ; 7-4 . Directly from the physical
phenomena, for example, a two-dimensional (2D) second-order TCI can exhibit
both one-dimensional (1D) gapped edge states and zero-dimensional (0D) corner
states at mid-gap.

However, hermiticity is only an ideal assumption of quantum mechanics.
Recently, the study of topological states has gradually turned to open quantum
system with non-Hermiticity 9-1 , which is closer to the real system. Many
research have confirmed the topology of non-Hermitian systems, such as the
photon and phono platform 10-1 ; 10-2 ; 10-3 ; 10-4 ; 10-5 ; 10-6 ; 10-7 ;
10-8 ; 10-9 ; 10-10 , the cold atomic systems with gain and loss acting as
non-Hermitian factor 11-1 ; 11-2 ; 11-3 ; 11-4 ; 11-5 ; 11-6 , as well as the
periodic driven Floquet non-Hermitian systems 12-1 ; 12-2 ; 12-3 .

Generally, in non-Hermitian systems, the real eigenvalues are not obtained,
and the eigen wavefunctions are no longer orthogonal. Instead, the left and
right eigen wavefunctions with biorthogonality are needed, and some unique
properties will appear, such as skin effect, exceptional point/ring, bulk
Fermi arcs and so on 13-1 ; 13-2 ; 13-3 . In addition, for any total-loss non-
Hermitian Hamiltonian, the gain-loss balanced part can be separated, whereby
the PT symmetry of non-Hermitian total-loss system can be discussed.

For the higher-order topology in non-Hermitian systems, it has been proved
that the generalized bulk-corner correspondence can work for a wide class of
non-Hermitian high-order topological systems with reflection or chiral
symmetries. However, the existing non-Hermitian high-order topological
insulators (NH-HOTIs) can not meet the diverse requirements of the real world,
so it is of great significance to develop new NH-HOTIs. In addition, most of
the existing studies consider the system with balanced gain and loss. But in
reality, the laborious gain realization is especially unfavorable to the
practical experimental research and application preparation. In this paper, we
propose a NH-HOTI, the energy gap of which is opened by configuring the
losses, and the edge and corner states in the energy gap are observed. Then we
demonstrate the ’real-energy’ edge states of pseudo-PT symmetry in total-loss
non-Hermitian systems, and realize this pseudo-PT symmetry in NH-HOTIs domain
wall system.

##  II NH-HOTI

In this section, we first briefly analyze the topological phase transition in
two cases of loss configuration of 1D non-Hermitian SSH chain, then propose a
NH-HOTI.

We focus on non-Hermitian system of total-loss, and start with a regular 1D
SSH chain consisting of N unit cells as depicted in Fig. 1. Here, this 1D SSH
chain consist of four uniformly coupled cavities, with low-loss $\gamma$ (only
background loss) introduced into two cavities and high-loss $\gamma_{1}$
(background loss $\gamma$ \+ additional loss $\gamma^{\prime}$) introduced
into the other two. The Hamiltonian of this chain based on the tight-binding
(TB) model can be written as

$\left(\begin{array}[]{cccc}-i\gamma&t&0&t\cdot e^{-iak}\cr
t&-i(\gamma+\frac{\gamma^{\prime}-\gamma^{\prime\prime}}{2})&t&0\cr
0&t&-i(\gamma+\gamma^{\prime})&t\cr t\cdot
e^{iak}&0&t&-i(\gamma+\frac{\gamma^{\prime}+\gamma^{\prime\prime}}{2})\end{array}\right)$
(1)

where $a$ is the lattice constant and $k$ is the Bloch wave number. $t$ is the
coupling strength between cavities.
$\gamma^{\prime}$and$\gamma^{\prime\prime}$ mark the loss and satisfy:
$\gamma^{\prime\prime}=\pm\gamma^{\prime}$. The imaginary onsite potential
becomes
$[-i\gamma,-i(\gamma+\gamma^{\prime}),-i(\gamma+\gamma^{\prime}),-i\gamma]$
where $\gamma^{\prime}=-\gamma^{\prime\prime}$, which is called case 1,
corresponding to Fig. 1(a). The imaginary onsite potential becomes
$[-i\gamma,-i\gamma,-i(\gamma+\gamma^{\prime}),-i(\gamma+\gamma^{\prime})]$
where $\gamma^{\prime}=\gamma^{\prime\prime}$, which is called case 2,
corresponding to Fig. 1(b).The nontrivial topology of this system can be
captured by a biorthogonal polarization 10-10 ; 14-2 ; 14-3

$P=\frac{i}{2\pi}\int_{0}^{2\pi}Tr[A(k)]\,dk$ (2)

where
$A(k)=\left\langle\phi_{\omega}(k)|\partial_{k}|\varphi_{\upsilon}(k^{\prime})\right\rangle$
is the biorthogonal non-Ablien Berry connection.
$|\phi_{\omega}(k)\rangle$,$|\varphi_{\upsilon}(k^{\prime})\rangle$ represent
left and right eigenvectors, and $\omega$ and $\upsilon$ denote the lower two
occupied bands. The normalization condition
$\left\langle\phi_{\omega}(k)|\varphi_{\upsilon}(k^{\prime})\right\rangle=\delta_{\omega\upsilon}\delta_{kk^{\prime}}$
is guaranteed. In general, the biorthogonal polarization is complex. Thus, the
real-valued polarization can be defined as $P=P^{LR}+P^{RL}$. The biorthogonal
polarization can be calculated:$P=0.5$ in case 1 and $P=0$ in case 2,
represent trivial topology and nontrivial topology, respectively.

Figure 1: Schematic of 1D non-Hermitian SSH chain. (a) In case 1, the low-loss
is on both sides, and the high-loss is added in the middle two cavities.(b) In
case 2, there are two continuous background loss and additional loss in the
unit cell. $N=20$. The coupling strength between resonators is $t$. low-loss
$\gamma$ and high-loss $\gamma_{1}$ are shown in red and blue, respectively.
Energy spectrum of the model. The energy spectra of case 1 and case 2 are (c)
and (d) respectively. The parameters of the both case:
$t=1$,$\gamma=1$,$\gamma^{\prime}=|\gamma^{\prime\prime}|=5$.

The topological property of this model can also be reflected in the energy
band which opens a gap by the loss $\gamma^{\prime}$ and
$\gamma^{\prime\prime}$, and the energy spectrum are shown in Fig. 1(c)(d). We
numerically calculate the eigenfrequency of the finite chain with open
boundary on both sides, and get the result in Fig. 2. In case 1, we obtain the
inner-gap edge states with zero-energy. The states here are very well
localized on both sides. In case 2, there is no inner-gap states, but two
kinds of inner-band edge states appear, corresponding to low loss and high
loss respectively. This shows that the two different loss configurations are
trivial and nontrivial topology.

Next, we will demonstrate the edge and corner states in a total-loss NH-HOTI.
Here, we only consider the nearest-neighbor hopping, and neglect the real part
of the onsite energy. The Hamiltonian of the corresponding TB model then can
be given as:

$H=\left(\begin{array}[]{cc}H_{1}&H_{2}\cr H_{3}&H_{4}\end{array}\right)$ (3)

where

$H_{1}=\left(\begin{array}[]{cccccc}0&0&p_{11}&0&p_{21}&0\cr
0&0&0&p_{21}&0&p_{31}\cr p_{12}&0&0&0&p_{31}&0\cr 0&p_{22}&0&0&0&p_{21}\cr
p_{22}&0&p_{32}&0&0&0\cr
0&p_{32}&0&p_{22}&0&0\end{array}\right)-\gamma\cdot\left(\begin{array}[]{cccccc}1&0&0&0&0&0\cr
0&1&0&0&0&0\cr 0&0&1&0&0&0\cr 0&0&0&1&0&0\cr 0&0&0&0&1&0\cr
0&0&0&0&0&1\end{array}\right);H_{2}=t\cdot\left(\begin{array}[]{cccccc}1&0&0&0&0&1\cr
1&1&0&0&0&0\cr 0&1&1&0&0&0\cr 0&0&1&1&0&0\cr 0&0&0&1&1&0\cr
0&0&0&0&1&1\end{array}\right)$
$H_{3}=t\cdot\left(\begin{array}[]{cccccc}1&1&0&0&0&0\cr 0&1&1&0&0&0\cr
0&0&1&1&0&0\cr 0&0&0&1&1&0\cr 0&0&0&0&1&1\cr
1&0&0&0&0&1\end{array}\right);H_{4}=t\cdot\left(\begin{array}[]{cccccc}0&1&0&0&0&1\cr
0&0&1&0&0&0\cr 0&0&0&1&0&0\cr 0&0&0&0&1&0\cr 0&0&0&0&0&1\cr
0&0&0&0&0&0\end{array}\right)-(\gamma+\gamma^{\prime})\cdot\left(\begin{array}[]{cccccc}1&0&0&0&0&0\cr
0&1&0&0&0&0\cr 0&0&1&0&0&0\cr 0&0&0&1&0&0\cr 0&0&0&0&1&0\cr
0&0&0&0&0&1\end{array}\right)$

Figure 2: The bulk states (black), zero-energy edge states (red), low-loss
edge states (magenta) and high-loss edge states (green) are shown in (a)(b)
(case 1) and (c)(d) (case 2), which is the eigenfrequency diagram of finite
model with $N=20$, and the distribution of eigenmode is shown in the
illustration. All parameters are consistent with Fig. 1.

where $p_{11}=t\cdot
exp(i\cdot(-\frac{1}{2}k_{x}+\frac{\sqrt{3}}{2}k_{y}))$;$p_{21}=t\cdot
exp(i\cdot(\frac{1}{2}k_{x}+\frac{\sqrt{3}}{2}k_{y}))$;$p_{31}=t\cdot
exp(i\cdot
k_{x})$;$p_{12}=p^{\ast}_{11}$;$p_{22}=p^{\ast}_{21}$;$p_{32}=p^{\ast}_{31}$
are the hopping unit vectors. $\ast$ denotes complex conjugate. The value of
$t$, $\gamma$ and $\gamma^{\prime}$ is consistent with the previous. As is
shown in Fig. 3(a), the unit cell is composed of six internal (Armchair) and
external (Zigzag) resonant cavities. The six external cavities only have low-
loss $\gamma$, and the six internal cavities have high-loss(case 3). This case
can be regarded as case 1 of 1D SSH chain in the first section in six
directions and demonstrated in Sec III.

Figure 3: (a) 12 resonant cavities and the loss configuration(case 3). Red and
blue represent two different lattice points of low-loss and high-loss
respectively, $a$ is the lattice constant of the unit cell. Red arrows mark
nontrivial topology in six directions. The illustration is the first Brillouin
zone. As the model satisfies the C6 symmetry, the simplified Brillouin zone is
described by the red triangle. (b) is a finite model based on (a).

Then we calculate the band structure along the $\Gamma-M-K-\Gamma$ path. The
cases of lossless and loss-considering are shown in Fig. 4(a) and (b),
respectively. Three bandgaps emerge after two different losses are introduced.
In addition, we numerically calculate the eigenfrequency of the finite 2D
modelFig. 4(c). In addition to the bulk states, we also find the inner-gap
edge states and corner states which are the only two discussed states here. We
plot the sum of probabilities (using right eigenvectors) of the bulk, edge,
and corner states in Fig. 4(d–f), respectively, further identifying their
existence. We note, as can be seen from Fig. 4(b–d), the states distribute all
over the bulk, edges, and corners, so this system does not feature non-
Hermitian skin effect, but belongs to higher-order topological system.

Here, we calculate the wannier band in case 3, to characterize its topology.
First, we use $H|\varphi\rangle=E_{1}|\varphi\rangle$,
$H^{\dagger}|\phi\rangle=E_{2}|\phi\rangle$ to define the right
$|\varphi\rangle$ and left $|\phi\rangle$ eigenstate. The Brillouin zone is
discretized into $M\times N$ k-points along $b_{1}$ and $b_{2}$ direction
respectively SM (we use $M=N=100$ in all calculations). Then we define Wilson
loop element:
$\Re_{\omega,\upsilon}(k)=\left\langle\phi_{\omega}(k)|\partial_{k}|\varphi_{\upsilon}(k+\triangle
k)\right\rangle$ in which $\triangle k=\frac{2\pi}{N}$. Now, the wilson loop
can be defined:

$W_{\omega,\upsilon}=F_{\omega,\upsilon}(k)\cdot
F_{\omega,\upsilon}(k+\triangle k)\cdot\cdot\cdot
F_{\omega,\upsilon}(k+(N-1)\triangle k)$ (4)

where

$F_{\omega,\upsilon}(k)=\left(\begin{array}[]{cccc}\Re_{1,1}(k)&\Re_{1,2}(k)&\cdots&\Re_{1,\upsilon}(k)\cr\Re_{2,1}(k)&\Re_{2,2}(k)&\cdots&\Re_{2,\upsilon}(k)\cr\vdots&\vdots&\ddots&\vdots\cr\Re_{\omega,1}(k)&\Re_{\omega,2}(k)&\cdots&\Re_{\omega,\upsilon}(k))\end{array}\right)$
(5)

We then diagonalize the Wilson loop operator
$W_{\omega,\upsilon}|\chi\rangle=exp(i\cdot 2\pi N_{1}(b_{2}))|\chi\rangle$,
where $|\chi\rangle$ is the eigenvector which depends on the wilson loop, and
the phase $N_{1}(b_{2})$ is the wannier center. The wannier band is
calculated(Fig. 5(a)) by using five bands under the second band gap, that is,
$\omega$ and $\upsilon$ are equal to 5. It can be seen that the calculated
Wannier band is gapped, which is the reason why edge states and corner states
appear in the second band gap 14-2 .

Figure 4: (a) Energy band of lossless system($\gamma=1,\gamma_{1}=6$). (b)
Energy band with two kinds of loss. The red dotted line depicts the
topological band gap. (c) Eigenfrequency spectrum of finite lattice model in
Fig. 3(b). Abscissa and ordinate represent the real and imaginary part
respectively. The black dot, green dot and red dot represent the bulk states,
edge states and corner states, respectively. Eigenmodes (right eigenvector) of
finite lattice: (d) corner states (e) edge states (f) bulk states. Figure 5:
(a) Wannier band in case 3(red, blue and black are used to mark wannier center
$>0$, $<0$ and $=0$, respectively). (b) Calculated eigenfrequencies for Fig.
3(b) as a function of the $\gamma^{\prime}$(case 3). The red dashed line
indicates the parameter value used in Fig. 4.

Next, we realize the NH-HOTI through the photonic crystal platform. The energy
band structure and the unit cell are shown in Fig. 6(a)(b). Here, the loss is
introduced through the imaginary part of the non-zero imaginary part of
relative permittivity. We simulate the supercell with finite $y$ direction and
obtain the energy band structure of the supercell with edge states (as shown
in Fig. 6(c)). Without the domain wall, supercell has edge states as shown in
Fig. 6(d), with two edge states on two boundaries.

Figure 6: Band structure and schematics of non-Hermitian optical photonic
structure. (a) and (c) are band structures of unit cell and supercell,
respectively. The red dotted line marks the topological energy gap, and the
magenta solid line is the edge states. (b) The lattice constant $a=800um$, the
lattice radius $r=0.12a$, the real part of the relative permittivity of the
material $\epsilon_{Re}=11.7$, the imaginary part of the background loss
$\epsilon_{Im}=1$, and after adding loss $\epsilon_{Im}=15$, and (d) is the
eigenmodes of the simulated supercell edge states.

Then we simulate the eigenfrequency of the finite hexagonal lattice model, as
is shown in Fig. 7(a). The corresponding edge states appear at the frequency
corresponding to the supercell band structure (shown by the green and red dots
in the figure).

Figure 7: Simulation results of finite lattice model. (a) is the
eigenfrequency of hexagonal finite lattice model. (b) (c) are the edge and
corner mode field distributions of the green and red dots in (a).

##  III Pseudo-PT symmetric in total-loss non-Hermitian system

In this section, we will demonstrate that there exist ’real-energy’ edge
states in total-loss non-Hermitian pseudo-PT symmetric systems, which are
implemented in 1D SSH chain and NH-HOTIS syetems. Then, the flexible
correspondence between 1D non-Hermitian SSH chain and NH-HOTI is discussed.

Generally, in a non-Hermitian system with gain and loss, a domain wall system
satisfying PT symmetry can be constructed by configuring the size of loss and
gain 14-1 . This system grows ’real-energy’ edge states embedded in the bulk
states. Here we find that there also exist ’real-energy’ edge states embedded
in the bulk states in the total-loss non-Hermitian system.

For the total-loss non-Hermitian pseudo-PT symmetric domain wall system, we
define: the whole domain wall system can balance the gain and loss around a
’real-energy’. In addition, the generalized PT part can be separated from the
pseudo-PT symmetric system based on the ’real-energy’. Also satisfy
$PTH_{PT}^{L}=H_{PT}^{R}$, where $H_{PT}^{L},H_{PT}^{R}$ are the Hamiltonian
of left and right sides of the domain wall which separated from the pseudo-PT
symmetric system. Here, we use case 1 in Section II to construct a pseudo-PT
domain wall system for example.

For any total-loss non-Hermitian system, the PT symmetric part can be
separated. In case 1, the Hamiltonian can be written as $H=H_{PT}+H^{\prime}=$

$\left(\begin{array}[]{cccc}i\frac{\gamma^{\prime}}{2}&t&0&e^{-iak}\cr
1&-i\frac{\gamma^{\prime}}{2}&1&0\cr 0&1&-i\frac{\gamma^{\prime}}{2}&1\cr
e^{iak}&0&1&i\frac{\gamma^{\prime}}{2}\end{array}\right)-i\gamma_{Re}\cdot\left(\begin{array}[]{cccc}1&0&0&0\cr
0&1&0&0\cr 0&0&1&0\cr 0&0&0&1\end{array}\right)$ (6)

where $-i\gamma_{Re}=-i(\gamma+\frac{\gamma^{\prime}}{2})$ is the ’real-
energy’.

According to this concept, we can construct pseudo-PT symmetric domain wall
system in 1D SSH chain. First, we show a pseudo-PT symmetric system without a
’real-energy’ edge states. Based on the non-Hermitian model of case 1 and case
2, we construct it by creating domain wall in the middle of the whole model,
as shown in Fig. 8(a). The numerical results (Fig. 8(b)(c)) show that there
are zero-energy edge states at the domain wall. The appearance of edge states
is a simple combination of two loss configurations, means that the imaginary
part of the frequency of edge states at the domain wall corresponds to the
size of loss at the domain wall.

Figure 8: The Schematic, eigenfrequencies and eigenmodes of pseudo-PT
symmetric domain wall system. (a) is the finite model of 1D SSH chain in case
1 with $N=20$. At both sides of the domain wall, the positions of the
imaginary parts $\gamma$ and $\gamma_{1}$ of the potential energy of the
cavity are reversed. (b) is the eigenfrequencies distribution. (c) is the
eigenmodes distribution. It can be seen that the edge states appear at both
the domain wall and the side in the same way as the case without domain wall,
that is, only corresponding to the loss. The meaning representative of colors
and the selection of parameters are consistent with the Fig. 2.

According to Eq. (6), in the pseudo-PT symmetric domain wall system with
’real-energy’ edge states, the Hamiltonian can have a ’real-energy’ part in
addition to the PT symmetric part. We construct the domain wall according to
this ’real-energy’, and the imaginary parts of the potential energy on both
sides of the domain wall are $[-i\gamma_{Re},-i\gamma,-i\gamma,-i\gamma_{Re}]$
and $[-i\gamma_{Re},-i\gamma_{1},-i\gamma_{1},-i\gamma_{Re}]$. In the
numerical calculation of the domain wall system with $\gamma_{Re}=3.5$(for
$\gamma=1$ and $\gamma^{\prime}=5$), it is obtained that the ’real-energy’
edge states in the bulk states appears at the imaginary part of the
eigenfrequency of -3.5 (Fig. 9(d)). In addition, the domain wall system also
has the edge states corresponding to the size of loss at both sides. In Fig.
9(e), it can be seen that the imaginary part of the characteristic frequency
is linear with $\gamma^{\prime}$.

Figure 9: The schematic, eigenfrequencies and eigenmodes of pseudo-PT
symmetric domain wall system. (a) is the finite model of SSH chain with
$N=20$. Representative of low-loss and high-loss is same as before, and white
dots indicate loss of $\gamma_{Re}$. (b) and (c) are the eigenfrequencies and
eigenmodes of this pseudo-PT symmetric domain wall system, respectively. (d)
and (e) are the real part and imaginary part of eigenfrequencies as a function
of the $\gamma^{\prime}$. The meaning representative of colors and the
selection of parameters are consistent with the Fig. 2.

Then, we use this NH-HOTI which can be regarded as 1D non-Hermitian SSH chain
in six directions to fabricate a 2D pseudo-PT symmetric domain wall system,
and obtain the ’real-energy’ edge states at the domain wall.

In order to verify that NH-HOTI can be regarded as a simple superposition of
1D SSH chain in Sec II in six different directions, we use a loss
configuration different from Sec II: add low-loss at ’2 3 5 6 8 11’ and add
high-loss at other positions(case 4). In this instance, it is a nontrivial
topology only in the positive and negative directions of x, and we construct a
finite model(Fig. 10(a)) based on this loss configuration. The eigenmodes of
three edge states show that the required states only appears in the positive
and negative directions of x, which is exactly what we want and need. Wannier
bands of three bands with edge states are shown in the appendix SM .

Next, wo add the loss as $i\gamma_{Re}$ at ’2 3 5 6 8 11’ and add low-loss and
high-loss on two sides at other positions respectively. In this way, we obtain
the loss configuration shown in Fig. 11(a), which is the case as Fig. 9(a) in
$x$ direction. Then we construct a finite domain wall system of this model,
which is composed of 4 * 10 unit cells. This domain wall system only has edge
states at both ends of the $x$ direction of the whole system, except for
forming ’real-energy’ boundary states at the domain wall, as is shown in Fig.
11(b).

So far, we have demonstrated that 1D SSH chain with non-trivial or trivial
topology can be constructed in different directions, and various high-order
topology models that meet different needs can be obtained flexibly.

Figure 10: (a) Finite model based on the case: add low-loss at ’2 3 5 6 8 11’
and add high-loss at other positions($\gamma=1$, $\gamma_{1}=11$). Its
eigenfrequency spectrum is (b) in which edge state I(red dots), edge state
II(green dots), edge state III(magenta dots) exist in three band gaps(marked
with dashed lines of three corresponding colors) respectively, and the loss
configuration is called case 4. (c) is the eigenmodes of (a), and the unit
cell is shown in the illustration in which red and blue arrows mark nontrivial
and trivial topology in different directions. (d) is the sum of the square of
right eigenvertors module corresponding to edge states I, II and III in (b).
Figure 11: The schematic, eigenfrequencies and eigenmodes of pseudo-PT
symmetric domain wall system. (a) is the finite model constructed of 4*10 unit
cells. The imaginary onsite potential of red, white and blue: $\gamma=1$,
$\gamma_{Re}=6$, $\gamma_{1}=11$, respectively. The black solid line marks the
unit cell, and the cyan and red dotted lines mark the domain wall and both end
walls. (b) is the eigenfrequencies of this system, in which, red and cyan dots
are the edge states at the end wall and the ’real-energy’ edge states at the
domain wall, and black dots are states other than the edge states marked in
cyan and red. (c) is the sum of the square of right eigenvertors module
corresponding to edge states and ’real-energy’ edge states in (b).

##  IV CONCLUSION

In summary, we propose a total-loss NH-HOTI, which realizes the existence of
edge states and corner states in the energy band gap by adjusting the loss
configuration, and can obtain edge states in different directions by adjusting
the loss configuration flexibly. The topology of this NH-HOTI can be
characterized by the gapped wannier band. In addition, we discuss the
existence of ’real-energy’ edge states in pseudo-PT symmetric domain wall
system, and demonstrate it theoretically in the domain wall system of 1D SSH
chain and NH-HOTI. Finally, we demonstrate the flexible correspondence between
1D non-Hermitian SSH chain and NH-HOTI. These results are expected to be
useful for exploring high-order non-Hermitian models, and provide a flexible
and useful scheme for finding topological states of real systems considering
loss only.

###### Acknowledgements.

Acknowledgments: This work was supported by the National Natural Science
Foundation of China (Grant numbers 61865009, 61927813).

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