# Explainable Deep Reinforcement Learning
Using Introspection in a Non-episodic Task

Angel Ayala 1, Francisco Cruz 2,3, and Bruno Fernandes 1, and Richard Dezeley
2 1Escola Politécnica de Pernambuco, Universidade de Pernambuco, Recife,
Brasil
2School of Information Technology, Deakin University, Geelong, Australia
3Escuela de Ingeniería, Universidad Central de Chile, Santiago, Chile
Emails: {aaam<EMAIL_ADDRESS>{francisco.cruz,
<EMAIL_ADDRESS>
###### Abstract

Explainable reinforcement learning allows artificial agents to explain their
behavior in a human-like manner aiming at non-expert end-users. An efficient
alternative of creating explanations is to use an introspection-based method
that transforms Q-values into probabilities of success used as the base to
explain the agent’s decision-making process. This approach has been
effectively used in episodic and discrete scenarios, however, to compute the
probability of success in non-episodic and more complex environments has not
been addressed yet. In this work, we adapt the introspection method to be used
in a non-episodic task and try it in a continuous Atari game scenario solved
with the Rainbow algorithm. Our initial results show that the probability of
success can be computed directly from the Q-values for all possible actions.

##  I Introduction

Reinforcement learning (RL) [1] is a machine learning technique in which an
autonomous agent acquires a new skill by interacting with the environment.
Lately, the development of deep learning methods, especially for image
processing, has allowed the emergence of deep RL methods [2]. Deep RL combines
the use of deep neural networks and RL techniques [3]. In this approach, a
deep learning architecture may process raw images as inputs giving the value
function as output. Therefore, Q-values can be obtained directly from
unprocessed images that represent the state of the environment.

Empirical black-box models, as the case of deep RL, suffers the lack of
interpretation in terms of their parameters [4, 5]. Therefore, it is not easy
for a non-expert end-user to understand the behavior of a deep RL agent in
simple domain-based language. In this paper, we implement an introspection-
based explainable method [6]. The method is based on a mathematical
transformation of the internal Q-values into probabilities of completing the
intended task. Therefore, the deep RL agent needs to look into the internal
values, and no additional memory or processing is needed. We have tested the
introspection-based method in the Space Invaders environment, an Atari game
scenario.

##  II Deep introspection-based approach

Explanations from RL algorithms are useful to understand better why an action
was taken in a particular situation by the agent. Our researched approach
estimates the probability of success $\hat{P}_{s}$ directly from the Q-values
as the introspective knowledge of the agent’s self-motivation. This approach
is efficient due to the direct numerical transformation from the Q-values into
the probability of success and, therefore, no additional memory is needed to
compute $\hat{P}_{s}$. The Q-value in a given state-action pair for value-
based reinforcement learning algorithms indicates how much future reward the
agent may obtain. Hence, using the Q-values, we can compute the probability of
success $\hat{P}_{s}$ as follows:

$\hat{P}_{s}\approx\frac{1}{2}\cdot\log_{10}\frac{Q(s,a)}{R^{S}}+1,$ (1)

where $Q(s,a)$ is the Q-value and $R^{S}$ the maximal reward obtained in any
step. As a probability value, we limited $\hat{P}_{s}\in[0,1]$. In order to
compute the probability of success, in this work, we adapted Eq. (1) from [6]
to be used in non-episodic scenarios. Therefore, we use the maximal possible
reward obtained in any step $R^{S}$ instead of the maximal total reward
$R^{T}$ as per episodic tasks.

The introspection approach is used along with the Rainbow deep RL algorithm
[7], which integrates previous deep RL methods such as double Q-learning,
prioritized replay, dueling networks, multi-step learning, distributional RL,
and noisy nets including recommended parameters.

The network used as approximation function present a dueling network
architecture from [8] and adapted as [7] with a shared representation
$f_{\xi}(s)$, which is fed into a value stream $v_{\eta}$ with
$N_{\text{atoms}}$ and into and advantage stream $a_{\xi}$, formulated as
follows:

$p_{\theta}^{i}(s,a)=\frac{\exp(v_{\eta}^{i}(\phi)+a_{\psi}^{i}(\phi,a)-\bar{a}_{\psi}^{i}(s))}{\sum_{j}\exp(v_{\eta}^{j}(\phi)+a_{\psi}^{j}(\phi,a)-\bar{a}_{\psi}^{j}(s))},$
(2)

where $\phi=f_{\xi}(s)$ and
$\bar{a}_{\psi}^{i}=\frac{1}{N_{\text{actions}}}\sum_{a^{\prime}}a_{\psi}^{i}(\phi,a^{\prime})$.
Additionally, it replaces all the linear layers with noisy layers equivalents
as presented in [9] and adapted in [7].

##  III Experimental Scenario

Our approach has been tested in the Space Invaders Atari game environment. The
state space is composed of an RGB image with 210x160 pixels which is resized
to 84x84 and converted to grayscale. Additionally, 4 consecutive frames were
stacked to emulate a spatio-temporal analysis according to the suggested by a
DQN previous implementation [10]. The actions are in discrete domain,
configured by 6 possible choices $\in[0,5]$ with the following meaning, 0: do
nothing or no operation; 1: fire; 2: move to the right; 3: move to the left;
4: action 2 plus 1; and 5: action 3 plus 1.

In this scenario, the agent must destroy a swarm of alien ships with three
lives, i.e., it can receive a maximum of three alien bullets. The agent starts
at the bottom-left part of the frame as illustrated in Figure 1(a) and must
shoot to the alien swarm. The reward obtained depends on the alien ship
destroyed and its position in the swarm. It starts from 5 points for ships at
the bottom row with an increment of 5 points for ships each row up, until 30
points for ships at the top row. Additionally, an extra alien ship appears
randomly in the top part of the frame with a reward of 200 points, as shown in
Figure 1(b).

The agent was trained under 50 million steps in total using the Rainbow
algorithm [7] with all the recommended parameters. The agent was evaluated for
each 1 million steps to check learning performance, and, therefore, the
training process was split into 50 episodes. Episode 0 was used to fill the
prioritized replay memory with 80 thousand steps before starting the learning
process.

(a) Initial state.

(b) Extra alien ship.

Figure 1: Original observation for the Space Invaders arcade environment.

##  IV Discussion and Future Work

After 50 epochs of training for 9 agents, all the agents were able to fulfill
the task of destroying the swarm. In Figure 2(a) can be observed that the
agents during evaluation are able to get a high average reward, destroying at
least one time the alien swarm, which summarizes 630 points. In the last
episode (i.e., the last 1 million training steps), the average Q-values for
all action varied between 6.85 and 7.15, which is meaningless to explain the
agent behavior. Therefore, using the introspection-based method as shown in
Eq. (1), we have obtained the probabilities of success from the initial state.
Figure 2(b) shows the probability for each action during the last episode with
values from 0.678 to 0.688. As pointed out previously, these probabilities may
be used as the base to explain the agent behavior more naturally to non-
experts and provide a number easy to interpret than raw Q-values.

(a) Evaluation reward per episode.

(b) Probability of success.

Figure 2: Reward and probability of success from the initial state after
training.

However, the computed probabilities are not so different among them as
expected, and therefore, it may be difficult to use them to generate
counterfactual explanations. We hypothesize that this minor difference may be
came from that in this complex scenario, a sequence of actions to complete the
task may be elicited starting from any action.

Although the probability of success might provide a more understandable manner
to explain the agent behavior [6], it is still not clear to what extent.
Therefore, future work considers a user study to understand better if the
probability of success is a valid metric to use as a base for explainable
reinforcement learning. This is especially relevant in complex scenarios in
which the impact of an action, in terms of the intended goal, may not be clear
easily by non-expert end-users.

## Acknowledgment

This work has been financed in part by the Coordenação de Aperfeiçoamento de
Pessoal de Nível Superior - Brasil (CAPES) \- Finance Code 001, Fundação de
Amparo a Ciência e Tecnologia do Estado de Pernambuco (FACEPE), Conselho
Nacional de Desenvolvimento Científico e Tecnológico (CNPq) - Brazilian
research agencies, and Universidad Central de Chile under the research project
CIP2020013.

## References

  * [1] R. S. Sutton and A. G. Barto, Reinforcement learning: An introduction.  MIT press, 2018.
  * [2] N. Churamani, F. Cruz, S. Griffiths, and P. Barros, “iCub: Learning emotion expressions using human reward,” in Proceedings of the Workshop on Bio-inspired Social Robot Learning in Home Scenarios, IEEE/RSJ IROS, p. 2, 2016\.
  * [3] K. Arulkumaran, M. P. Deisenroth, M. Brundage, and A. A. Bharath, “Deep reinforcement learning: A brief survey,” IEEE Signal Processing Magazine, vol. 34, no. 6, pp. 26–38, 2017.
  * [4] F. Cruz, G. Acuña, F. Cubillos, V. Moreno, and D. Bassi, “Indirect training of grey-box models: application to a bioprocess,” in International Symposium on Neural Networks, pp. 391–397, Springer, 2007.
  * [5] F. C. Naranjo and G. A. Leiva, “Indirect training with error backpropagation in gray-box neural model: Application to a chemical process,” in 2010 XXIX International Conference of the Chilean Computer Science Society, pp. 265–269, 2010.
  * [6] F. Cruz, R. Dazeley, and P. Vamplew, “Explainable robotic systems: Understanding goal-driven actions in a reinforcement learning scenario,” arXiv e-prints, pp. arXiv–2006, 2020.
  * [7] M. Hessel, J. Modayil, H. Van Hasselt, T. Schaul, G. Ostrovski, W. Dabney, D. Horgan, B. Piot, M. Azar, and D. Silver, “Rainbow: Combining improvements in deep reinforcement learning,” in Thirty-second AAAI conference on artificial intelligence, 2018.
  * [8] Z. Wang, T. Schaul, M. Hessel, H. Hasselt, M. Lanctot, and N. Freitas, “Dueling network architectures for deep reinforcement learning,” in International conference on machine learning, pp. 1995–2003, PMLR, 2016.
  * [9] M. Fortunato, M. G. Azar, B. Piot, J. Menick, I. Osband, A. Graves, V. Mnih, R. Munos, D. Hassabis, O. Pietquin, et al., “Noisy networks for exploration,” arXiv preprint arXiv:1706.10295, 2017.
  * [10] V. Mnih, K. Kavukcuoglu, D. Silver, A. A. Rusu, J. Veness, M. G. Bellemare, A. Graves, M. Riedmiller, A. K. Fidjeland, G. Ostrovski, et al., “Human-level control through deep reinforcement learning,” nature, vol. 518, no. 7540, pp. 529–533, 2015.


# Hybrid skin-scale-free effect in non-Hermitian systems: A transfer matrix
approach

Yongxu Fu<EMAIL_ADDRESS>International Center for Quantum Materials,
School of Physics, Peking University, Beijing 100871, China  Yi Zhang
<EMAIL_ADDRESS>International Center for Quantum Materials, School of
Physics, Peking University, Beijing 100871, China

###### Abstract

Surpassing the individual buildings of the non-Hermitian skin effect (NHSE)
and the scale-free localization (SFL) observed lately, we systematically
exploit the exponential decay behavior of bulk eigenstates via the transfer
matrix approach in non-Hermitian systems. We concentrate on the one-
dimensional (1D) finite-size non-Hermitian systems with $2\times 2$ transfer
matrices in the participation of boundary impurity. We analytically unveil
that the unidirectional pure scale-free (UPSF) effect emerges with the
singular transfer matrices, while the hybrid skin-scale-free (HSSF) effect
emerges with the nonsingular transfer matrices even though impose open
boundary condition (OBC). The UPSF effect exceeds the scope of the SFL in
previous works, while the HSSF effect is a charming interplay between the
finite-size NHSE and SFL. Our results reveal that the NHSE under OBC prevails
in the blend with the SFL as the system tends to the thermodynamic limit. Our
approach paves an avenue to rigorously explore the finite-size display of the
NHSE and SFL in both Hermitian and non-Hermitian systems with generic boundary
conditions.

##  I Introduction

Exceeding the requirement of Hermitian operators for physical observables in
quantum mechanics, the non-Hermitian physics has been widely broadened in the
past few years [1, 2, 3, 4], which contains the basic energy band theory [5,
6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], the
higher-order topological phases [24, 25, 26, 27, 28, 29, 30, 31, 32], the
unique exceptional points (EPs) in non-Hermitian systems [33, 34, 35, 36, 37,
38, 39, 40, 41, 42, 43, 44, 45, 46], and other subjects in the scope of
condensed matter physics. The exploration of the NHSE in 1D non-Hermitian
systems, which supports the accumulation of extensive number of nominal bulk
eigenstates under OBC upon the single-particle level, is a milestone in the
avenue of the research on the non-Hermitian systems [9, 10, 11, 12, 13, 14,
15, 16, 17]. The non-Bloch band theory with the concept of generalized
Brillouin zone (GBZ) is fully explanatory for the NHSE [9, 11, 16], and more
recently, similar counterparts have paved our way towards higher dimensions
[47, 48, 49, 50].

A typical characteristic of the NHSE-eigenstates is their exponential decay in
the bulk region. The exponential factor, corresponding to the relevant point
on the GBZ, alters the wave vector in the traditional Bloch band theory from
real value to complex value, a core context of the non-Bloch band theory.
Strictly speaking, the complexity of the non-Hermitian systems does not
constrain the localized behavior of the bulk eigenstates in the NHSE. The SFL,
which suggests the existence of the eigenstates with size-dependent
localization length in 1D non-Hermitian systems, is explored lately [51, 52,
53, 54, 55, 56, 57]. Nevertheless, the scale-free behavior is model-dependent,
which is a seemingly accidental phenomenon closely related to the impurities
[58, 59, 60, 61, 62, 63, 64, 65], and lack a unified perspective in non-
Hermitian systems. Furthermore, the emergence of the interplay between NHSE
and SFL remains an unsurveyed subject in the field of non-Hermitian physics.

The transfer matrix approach is a powerful tool for elucidating the tight-
binding models for decades [66, 67, 68, 69, 70, 71, 72], which has a more
terse and compact form than the method of directly solving the eigenvalue
problem, and also applies to the non-Hermitian systems in recent years [73,
74, 75]. In this paper, we utilize the transfer matrix approach to establish a
unified depiction of the scale-free effect as well as the interplay between
NHSE and SFL. Without loss of generality, we concentrate on the 1D finite-size
non-Hermitian systems with $2\times 2$ transfer matrices, such as the Hatano-
Nelson (HN) model [76, 77, 78] and the non-Hermitian Su-Schrieffer-Heeger (NH-
SSH) model [5, 9] with boundary impurity. We unveil that the UPSF effect
emerges with the singular transfer matrices, while the HSSF effect appears
with the nonsingular transfer matrices, which shows the existence of the bulk
eigenstates possessing both NHSE and scale-free exponential decay factors. The
UPSF effect must accompany the boundary impurity, while the HSSF effect may
exist even under OBC. The localization length of the UPSF mode is generally
quasi-linearly dependent on the system size, which is the generalized
definition of the SFL in this paper - the results in previous works [53, 54,
55, 56] are equivalent to specific cases with linear dependence. The HSSF
effect implies that the NHSE under OBC blends with the emergent SFL in the
finite sizes yet prevails over the competition in the thermodynamic limit,
revealing the NHSE’s dominance. Once we turn on the boundary impurity, the
HSSF effect displays a finite-size dependence.

The rest of this paper is organized as follows. In Sec. II, we establish the
formalism for probing UPSF and HSSF effects from the transfer matrix
perspective, corresponding to the cases with singular and nonsingular transfer
matrices, respectively. In Secs. III.1 and IV.1, we analytically solve the
eigenstates as well as the energy spectrum with the emergence of the pure
scale-free (PSF) effect for the HN model and NH-SSH model with boundary
impurity. The HSSF effect is expounded for the HN model with boundary impurity
in Sec. III.2 and the NH-SSH model under OBC in Sec. IV.2. We give a
conclusion and further discussions in Sec. V.

##  II Probing UPSF and HSSF effect via transfer matrix approach

###  II.1 Review of transfer matrix approach for non-Hermitian tight-binding
models

Commonly, the real-space tight-binding Hamiltonian of a 1D non-interacting
non-Hermitian system reads

$\displaystyle\mathcal{H}$
$\displaystyle=\sum_{n}\sum_{l=-R}^{R}\sum_{\mu,\nu=1}^{q}t_{l,\mu\nu}c_{n,\mu}^{\dagger}c_{n+l,\nu}$
$\displaystyle=\sum_{n}\sum_{l=-R}^{R}\mathrm{c}_{n}^{\dagger}\mathrm{t}_{l}\mathrm{c}_{n+l},$
(1)

where $c_{n,\mu}^{\dagger}$ ($c_{n,\mu}$) is a creation (annihilation)
operator with the index of internal degrees of freedom $\mu$ in the $n$-th
unit cell, $\mathrm{c}_{n}^{\dagger}$ ($\mathrm{c}_{n}$) is a row (column)
vector containing $q$ $c_{n,\mu}^{\dagger}$ ($c_{n,\mu}$), and $t_{l,\mu\nu}$
($\mathrm{t}_{l}$) is a hopping amplitude (matrix) to the $l$-th nearest unit
cell.

Following the formalism established in Ref. [66, 72, 73], we bundle up at
least $R$ adjacent unit cells into a supercell, such that Eq. (II.1) reduces
to a nearest-neighbor tight-binding Hamiltonian

$\displaystyle\mathcal{H}=\sum_{n=1}^{N-1}\left[\boldsymbol{c}_{n}^{\dagger}\boldsymbol{J}_{L}\boldsymbol{c}_{n+1}+\boldsymbol{c}_{n}^{\dagger}\boldsymbol{M}\boldsymbol{c}_{n}+\boldsymbol{c}_{n+1}^{\dagger}\boldsymbol{J}_{R}^{\dagger}\boldsymbol{c}_{n}\right],$
(2)

with creation (annihilation) operator denoted as
$\boldsymbol{c}_{n}^{\dagger}$ ($\boldsymbol{c}_{n}$) and totally $N$
supercells. There are $\mathcal{N}\geq qR$ internal degrees of freedom in each
supercell, and $\boldsymbol{J}_{L,R}$ and $\boldsymbol{M}$ are hopping
matrices and onsite matrix, respectively. Without loss of generality, we
require $\boldsymbol{J}_{R}=\boldsymbol{J}_{L}\equiv\boldsymbol{J}$ with
$\boldsymbol{J}^{2}=0$ and impose non-Hermiticity merely on $\boldsymbol{M}$,
i.e., $\boldsymbol{M}\neq\boldsymbol{M}^{\dagger}$ 111This setup is adequate
for the core physics in the current paper; also, we can always ensure the
nilpotence of $\boldsymbol{J}$ by bundling a sufficient number of unit cells..
Consequently, the tight-binding Hamiltonian further reduces to

$\displaystyle\mathcal{H}=\sum_{n=1}^{N-1}\left[\boldsymbol{c}_{n}^{\dagger}\boldsymbol{J}\boldsymbol{c}_{n+1}+\boldsymbol{c}_{n}^{\dagger}\boldsymbol{M}\boldsymbol{c}_{n}+\boldsymbol{c}_{n+1}^{\dagger}\boldsymbol{J}^{\dagger}\boldsymbol{c}_{n}\right].$
(3)

Given an arbitrary single-particle state

$\displaystyle\ket{\Psi}=\sum_{n=1}^{N}\Psi_{n}\boldsymbol{c}_{n}^{\dagger}\ket{0},$
(4)

with $\Psi_{n}\in\mathbb{C}^{\mathcal{N}}$, the single-particle Schrödinger
equation $\mathcal{H}\ket{\Psi}=\varepsilon\ket{\Psi}$ reduces to the
recursion relation

$\displaystyle\boldsymbol{J}\Psi_{n+1}+\boldsymbol{J}^{\dagger}\Psi_{n-1}=\left(\varepsilon\mathbbm{1}-\boldsymbol{M}\right)\Psi_{n}.$
(5)

Next, we define
$\mathcal{G}=\left(\varepsilon\mathbbm{1}-\boldsymbol{M}\right)^{-1}$ as the
onsite Green’s function, which is nonsingular except when $\varepsilon$ is an
eigenvalue of $\boldsymbol{M}$. Performing the reduced singular value
decomposition (SVD) [80, 81], we obtain

$\displaystyle\boldsymbol{J}=V\Xi W^{\dagger},$ (6)

where $\Xi=\text{diag}\left\\{\xi_{1},\ldots,\xi_{r}\right\\}$ is a diagonal
matrix of singular value $\xi_{i}\in\mathbb{R}^{+},i=1,2,\ldots,r$ with
$r=\text{rank}\left(\boldsymbol{J}\right)$, and $V$ ($W^{\dagger}$) is the
matrix comprised of $r$ orthonormal bases $v_{i}$ ($w_{i}^{\dagger}$) in the
column (row) space of $\boldsymbol{J}$, that is

$\displaystyle V^{\dagger}V=W^{\dagger}W=\mathbbm{1},\qquad V^{\dagger}W=0.$
(7)

We focus on the $r=1$ case for simplicity; as a result,
$\Xi\equiv\xi\in\mathbb{R}^{+}$ and $\left\\{V\equiv v,W\equiv w\right\\}$
constitutes a set of orthonormal basis of $\mathbb{C}^{2}$. In this basis, we
expand $\Psi_{n}$ and $\mathcal{G}$ as

$\displaystyle\Psi_{n}=\alpha_{n}v+\beta_{n}w,\quad\alpha_{n}=v^{\dagger}\Psi_{n},\quad\beta_{n}=w^{\dagger}\Psi_{n},$
(8)

and

$\displaystyle\mathcal{G}_{AB}=B^{\dagger}\mathcal{G}A\in\mathbb{C},\qquad
A,B\in\left\\{v,w\right\\},$ (9)

respectively. Combining Eqs. (5)-(9), we obtain the propagating relation in
the bulk

$\displaystyle\Phi_{n+1}=T\Phi_{n},\qquad\Phi_{n}\equiv\left(\begin{matrix}\beta_{n}\\\
\alpha_{n-1}\end{matrix}\right),$ (10)

where $T$ is the $2\times 2$ transfer matrix [73]

$\displaystyle
T=\frac{1}{\xi\mathcal{G}_{vw}}\left(\begin{matrix}1&-\xi\mathcal{G}_{ww}\\\
\xi\mathcal{G}_{vv}&\xi^{2}\left(\mathcal{G}_{vw}\mathcal{G}_{wv}-\mathcal{G}_{vv}\mathcal{G}_{ww}\right)\end{matrix}\right).$
(11)

We define the trace and determinant of $T$ as

$\displaystyle\Delta=\text{tr}\left(T\right),\qquad\Gamma=\det{\left(T\right)}\equiv\frac{\mathcal{G}_{wv}}{\mathcal{G}_{vw}},$
(12)

which are rational functions of energy $\varepsilon$ in general, and for
simplicity, we suppress the explicit dependence on $\varepsilon$ hereafter. If
$T$ is singular ($\Gamma=0$), $T^{n}=\Delta^{n-1}T$; if $T$ is nonsingular
($\Gamma\neq 0$) [73],

$\displaystyle
T^{n}=\Gamma^{n/2}\left[\frac{U_{n-1}(z)}{\sqrt{\Gamma}}T-U_{n-2}(z)\mathbbm{1}\right],$
(13)

where

$\displaystyle
U_{n}(z)=\frac{\sin{\left(\left(n+1\right)\phi\right)}}{\sin{\phi}},$ (14)

is the Chebyshev polynomials of second kind [82, 83] and

$\displaystyle z\equiv
z(\varepsilon)=\frac{\Delta}{2\sqrt{\Gamma}}=\cos{\phi}\in\mathbb{C}.$ (15)

After reviewing the transfer matrix approach for non-Hermitian tight-binding
models, we shall utilize it for probing UPSF and HSSF effect.

Figure 1: (a) The propagating relation of $\Phi_{n}$ with the boundary
impurity in Eq. (16) and (b) the propagating relation of $\Phi_{n}$ under OBC
in the transfer matrix approach.

###  II.2 Transfer matrix approach for 1D tight-binding models with boundary
impurity

Without loss of generality, we consider the boundary impurity

$\displaystyle\boldsymbol{c}_{N}^{\dagger}\kappa_{L}\boldsymbol{c}_{1}+\boldsymbol{c}_{1}^{\dagger}\kappa_{R}\boldsymbol{c}_{N},$
(16)

with $\kappa_{L}=\gamma_{L}\boldsymbol{J}$,
$\kappa_{R}=\gamma_{R}\boldsymbol{J}^{\dagger}$, $\gamma_{L},\gamma_{R}\geq 0$
on top of the Hamiltonian Eq. (3), connecting the first and the last
supercells 222A more general impurity is still manageable with increased
complexity in the calculations.. Note that the periodic boundary condition
(PBC), as well as the Bloch band theory, is reproduced in the
$\gamma_{L}=\gamma_{R}=1$ case, while the OBC in the $\gamma_{L}=\gamma_{R}=0$
case. After some algebra, we obtain the equations on the boundaries (Appendix
A)

$\displaystyle\Phi_{1}=K_{L}T\Phi_{N},\qquad\Phi_{2}=TK_{R}\Phi_{1},$ (17)

where $K_{L}=\text{diag}\left\\{1/\gamma_{L},1\right\\}$ and
$K_{R}=\text{diag}\left\\{1,\gamma_{R}\right\\}$. Together with Eq. (10), we
illustrate the full propagating relation of $\Phi_{n}$ with the boundary
impurity in Fig. 1(a). Since $\Phi_{N}=T^{N-2}\Phi_{2}$, Eq. (17) gives

$\displaystyle\Phi_{1}=K_{L}T^{N}K_{R}\Phi_{1}.$ (18)

Defining $\varphi=K_{R}\Phi_{1}$ and
$K=K_{R}K_{L}=\text{diag}\left\\{1/\gamma_{L},\gamma_{R}\right\\}$, we final
obtain the compact boundary equation

$\displaystyle\varphi=KT^{N}\varphi.$ (19)

It implies that the legitimate $\varphi$ is the eigenvector of $KT^{N}$ with
respect to the eigenvalue $1$, thus the physical solution is
$\Phi_{1}=K_{R}^{-1}\varphi$ and $\Phi_{n}=T^{n-1}\varphi$ for
$n=2,3,\ldots,N$.

####  II.2.1 Pure scale-free effect

We first consider the cases with singular transfer matrices, which suggest the
occurrence of real-space EPs under OBC [73]. As we turn on the boundary
impurity, the determinant of $KT^{N}$ is zero due to $\Gamma=0$, which implies
$\text{tr}\left(KT^{N}\right)=1$ for satisfying Eq. (19). Consequently, we
obtain $\Delta^{N-1}\text{tr}\left(KT\right)=1$, namely
($\text{tr}\left(KT\right)\neq 0$)

$\displaystyle\Delta^{N}=\frac{\Delta}{\text{tr}\left(KT\right)}.$ (20)

Assuming $\Delta^{N}=c^{-1}$ with $c$ being an undetermined constant, we can
obtain the expression of $\varepsilon$ for $c$ through

$\displaystyle\Delta=c^{-\frac{1}{N}}e^{-i\frac{2\pi m}{N}},$ (21)

for each $m\in\left\\{1,2,\ldots,N\right\\}$, denoted as $\varepsilon_{m}$
(regardless of the possible band index). Substituting $\varepsilon_{m}$ into
Eq. (20), we figure out the constant $c$ as well as energies for each $m$.
Although multiple or null solutions of $c$ are perhaps obtained for some $m$
in general, we formally denote the solution of $c$ for each $m$ as $c_{m}$,
that is all of $c^{\prime}s$ generate the nominal bulk bands.

Consider a fixed $c_{m}$, the solution of eigenstate is given by

$\displaystyle\Phi_{1}^{m}$ $\displaystyle=K_{R}^{-1}\varphi_{m},$
$\displaystyle\Phi_{n}^{m}$
$\displaystyle=c_{m}^{-\frac{n-2}{N}}e^{-i\frac{2\pi m}{N}(n-2)}T\varphi_{m},$
(22)

where $n=2,3,\ldots,N$ and $KT^{N}\varphi_{m}=\varphi_{m}$. According to Eq.
(8), we obtain the eigenstate (up to the normalization coefficients hereafter)
concerning energy $\varepsilon_{m}$

$\displaystyle\Psi_{1}^{m}$
$\displaystyle=\left(T\varphi_{m}\right)_{2}v+\left(K_{R}^{-1}\varphi_{m}\right)_{1}w,$
(23) $\displaystyle\Psi_{n}^{m}$
$\displaystyle=c_{m}^{-\frac{n-1}{N}}e^{-i\frac{2\pi
m}{N}(n-1)}\psi_{m},\,\,n=2,3,\ldots,N-1,$ (24) $\displaystyle\Psi_{N}^{m}$
$\displaystyle=\left(K_{R}^{-1}\varphi_{m}\right)_{2}v+\left(c_{m}^{-\frac{N-2}{N}}e^{-i\frac{2\pi
m}{N}(N-2)}T\varphi_{m}\right)_{1}w,$ (25)

where
$\psi_{m}=\left(T\varphi_{m}\right)_{2}v+c_{m}^{\frac{1}{N}}e^{i\frac{2\pi
m}{N}}\left(T\varphi_{m}\right)_{1}w$, and $\left(\mathscr{V}\right)_{1,2}$
denote the first and second components of the column $2$-vector $\mathscr{V}$.

We specify the bulk region of the supercell systems Eq. (3) with boundary
impurity Eq. (16) as $\mathscr{B}=\left\\{2,3,\ldots,N-1\right\\}$, as well as
the boundary region $\mathscr{E}=\left\\{1,N\right\\}$. The bulk solution Eq.
(24) visibly displays a exponential decay

$\displaystyle\exp{\left\\{-\frac{\text{Re}(\log{c_{m}})}{N}(n-1)\right\\}},$
(26)

with the phase factor

$\displaystyle\exp{\left\\{-i\frac{\text{Im}(\log{c_{m}})+2\pi
m}{N}(n-1)\right\\}},$ (27)

which is a correction of the wave vector $2\pi m/N$. We define the occurrence
of such exponential-decay behavior of eigenstates in $\mathscr{B}$ as the UPSF
effect, which is a rigorous generalization of the SFL in previous work [51,
52, 53, 54, 55, 56]. Further, we denote the eigenstates like Eq. (24) in
$\mathscr{B}$ with $\text{Re}(\log{c_{m}})>0$ ($<0$) as the left
(right)-accumulation UPSF modes, and the localization lengths of these modes
are

$\displaystyle\xi_{m}=\frac{N}{|\text{Re}(\log{c_{m}})|}.$ (28)

Noteworthily, $c_{m}$ is generally dependent on system size $N$, thus we
define the localization length $\xi_{m}$ quasi-linear dependent on $N$, a
typical character of the UPSF effect.

####  II.2.2 Hybrid skin-scale-free effect

More imperceptible phenomena exceeding the NHSE emerge in cases of nonsingular
transfer matrices 333Here, we assume all physical energies corresponding to
$\Gamma\neq 0$, that is, neglect the extremely special situations where exist
some discrete energies leading to $\Gamma=0$.. When the system size is finite,
the NHSE, a rigorous terminology in the thermodynamic limit under OBC, shows
not only a finite-size extent but also a possible interplay with the SFL.

The propagating relation of $\Phi_{n}$ under OBC is illustrated in Fig. 1(b),
and accordingly, the physical condition (the constrain on $\phi$) is (Appendix
B)

$\displaystyle\frac{\sin{\left(N\phi\right)}}{\sin{\left[(N-1)\phi\right]}}=q,$
(29)

with $q=\xi\sqrt{\frac{\mathcal{G}_{wv}}{\mathcal{G}_{vw}}}\mathcal{G}_{vw}$.
Here, we have not simplified $q$ to avoid the branch problems of the square
root. Together with Eq. (15), we can, in principle, obtain the physical energy
$\varepsilon$ and $\phi$ simultaneously. If the energy $\varepsilon$ renders
$q$ real, the left-hand side of Eq. (29) must be real, thus leading to real
$\phi$. However, if the energy $\varepsilon$ renders $q$ complex, we obtain
complex $\phi=\phi_{R}+i\phi_{I}$ and $\phi_{I}\sim c/N$, which is quasi-
linearly dependent on $1/N$ (Appendix B). The eigenstate concerning energy
$\varepsilon$ reads

$\displaystyle\Psi_{n}=\Gamma^{n/2}\left[\mathscr{A}_{L}(\phi)e^{in\phi_{R}}e^{-n\phi_{I}}+\mathscr{A}_{R}(\phi)e^{-in\phi_{R}}e^{n\phi_{I}}\right],$
(30)

with $n=1,2,\ldots,N,$ and the coefficients
$\mathscr{A}_{L}(\phi),\mathscr{A}_{R}(\phi)$ given in Appendix B. The overall
exponential factor $\Gamma^{n/2}$ indicates the emergence of the NHSE for
$|\Gamma|\neq 1$, while it is the Hermitian cases or the non-Hermitian cases
with some special symmetries (such as parity-time (PT) symmetry) for
$|\Gamma|=1$. The exponential factors $e^{\pm n\phi_{I}}$ indicate the SFL
with bidirectionally accumulation for a set of $(N,\phi_{R})$. Therefore, we
define the occurrence of $|\Gamma|=1$ and $\phi_{I}\neq 0$ as the
superposition of bidirectional scale-free (SBSF) effect, while $|\Gamma|=1$
and $\phi_{I}=0$ suggests the extended eigenstate (ES). Noteworthily, the SBSF
effect reduces to the UPSF effect if one of $\mathscr{A}_{L,R}(\phi)$
vanishes, thus the PSF effect is defined as a union of the UPSF and SBSF
effects. Further, the occurrence of $|\Gamma|\neq 1$ and $\phi_{I}\neq 0$ is
defined as the HSSF effect, while the pure NHSE emerges with $|\Gamma|\neq 1$
and $\phi_{I}=0$. These correspondences are summarized in Table I.

$|\Gamma|=1$ $\phi_{I}=0$ | Yes | No
---|---|---
Yes | ES | PSF
No | NHSE | HSSF
Table 1: Summary of ES, NHSE, PSF, and HSSF concerning nonsingular transfer
matrices.

However, $n\phi_{I}\rightarrow 0$ with $n$ lying in the deep bulk as
$N\rightarrow+\infty$ under OBC, which means the exponential factor dominated
by $\Gamma^{n/2}$ in the deep bulk, namely, the exact context of NHSE in the
thermodynamic limit [23]. Hence, the emergent SFL participates in the NHSE
under finite size, and the NHSE gradually dominates with increasing system
size.

Next, we turn on the boundary impurity in Eq. (16), and the physical condition
is given by Eq. (19). Again, the complex $\phi=\phi_{R}+i\phi_{I}$ emerges
under finite size but the forms $\phi_{I}\sim c/N$ do not always exist when
$N\rightarrow+\infty$ (Appendix C). The general form of the eigenstate in
$\mathscr{B}$ with respect to energy $\varepsilon$ and complex $\phi$ reads

$\displaystyle\Psi_{n}^{imp}$
$\displaystyle=\Gamma^{n/2}\left[\mathcal{B}_{L}(\phi)e^{in\phi_{R}}e^{-n\phi_{I}}+\mathcal{B}_{R}(\phi)e^{-in\phi_{R}}e^{n\phi_{I}}\right],$
(31)

where the coefficients $\mathcal{B}_{L}(\phi)$, $\mathcal{B}_{R}(\phi)$ are
given in Appendix C. Similar to the OBC cases under finite sizes, the various
possible behaviors of eigenstates are also depicted in Table I. Nevertheless,
the thermodynamic limit may prevent both the NHSE and SFL effects with the
boundary impurity, an abrupt change from the OBC. In other words, the HSSF
effect in the presence of boundary impurity is sensitive to the system size.

##  III Hatano-Nelson model with boundary impurity

As a typical 1D non-Hermitian model, the HN model with boundary impurity reads

$\displaystyle\mathcal{H}_{hn}=\sum_{n=1}^{N-1}\left(t_{L}c_{n}^{\dagger}c_{n+1}+t_{R}c_{n+1}^{\dagger}c_{n}\right)+\gamma_{R}c_{1}^{\dagger}c_{N}+\gamma_{L}c_{N}^{\dagger}c_{1},$
(32)

where we assume $t_{L},t_{R},\gamma_{L},\gamma_{R}\geq 0$ for simplicity. The
model manifests an emergence of the SFL except for the NHSE [53, 56]. In this
section, we apply the transfer matrix approach to comprehensively study the
display and interplay of NHSE and scale-free effect in Eq. (32). Note the
single-band nature of the NH model offers an expedient expression of the
transfer matrix in the single-particle wave function space, denoted as
$\ket{\Psi}=\sum_{n=1}^{N}\psi_{n}c_{n}^{\dagger}\ket{0}$, instead of the
nominal $\left\\{v,w\right\\}$ space, after some algebra, we obtain the
propagating relation in the bulk (Appendix D)

$\displaystyle\left(\begin{matrix}\psi_{n+1}\\\
\psi_{n}\end{matrix}\right)=T\left(\begin{matrix}\psi_{n}\\\
\psi_{n-1}\end{matrix}\right),$ (33)

where the transfer matrix with respect to energy $\varepsilon$ is

$\displaystyle
T=\left(\begin{matrix}\frac{\varepsilon}{t_{L}}&-\frac{t_{R}}{t_{L}}\\\
1&0\end{matrix}\right),$ (34)

and we label

$\displaystyle\Delta$
$\displaystyle=\text{tr}\left(T\right)=\frac{\varepsilon}{t_{L}},$
$\displaystyle\Gamma$
$\displaystyle=\det{\left(T\right)}=\frac{t_{R}}{t_{L}}.$ (35)

###  III.1 Exact solutions of the PSF effect

We first consider the case with a singular transfer matrix, namely, $t_{R}=0$
leading to $\Gamma=0$. The eigenvectors and the corresponding energies are
(Appendix D.1)

$\displaystyle\varepsilon_{m}$
$\displaystyle=t_{L}c_{m}^{-\frac{1}{N-2}}e^{-i\frac{2\pi m}{N-2}},\quad
m=1,2,\ldots,N,$ $\displaystyle\psi_{n}^{m}$
$\displaystyle=c_{m}^{-\frac{n-2}{N-2}}e^{-i\frac{2\pi m}{N-2}(n-2)},\quad
n=3,4,\ldots,N,$ $\displaystyle\psi_{1}^{m}$
$\displaystyle=\frac{t_{L}}{\gamma_{L}}c_{m}^{-\frac{1}{N-2}}e^{-i\frac{2\pi
m}{N-2}}\psi_{N}^{m},$ (36)

with the physical condition

$\displaystyle\frac{t_{L}^{2}}{\gamma_{L}}c_{m}^{-\frac{2}{N-2}}e^{-i\frac{4\pi
m}{N-2}}=t_{L}c_{m}+\gamma_{R},$ (37)

where we have set $\psi_{2}^{m}=1$. Apparently, the exponential factor of
$\psi_{n}^{m}$ leads to the PSF effect, with the localization length

$\displaystyle\xi_{m}=\frac{N-2}{|\text{Re}\left(\log{c_{m}}\right)|},$ (38)

quasi-linearly dependent on the system size, and the left (right)-accumulation
UPSF modes correspond to $\text{Re}\left(\log{c_{m}}\right)>0$ ($<0$). The
phase diagrams of the UPSF modes in $\gamma_{L}$-$\gamma_{R}$ plane are
plotted in Figs. 2(A)(B) with system size $N=10,30$, where the green regions
label the existence of the UPSF modes with left-accumulation (LA), the cyan
regions label that with right-accumulation (RA), and the yellow regions label
that with both left- and right-accumulation (LRA) 444The LRA-region may
contain the extreme modes with $\text{Re}\left(\log{c_{m}}\right)=0$, which
are not our concerns in this paper.. In Fig. 2, we also plot the full single-
particle energy spectrum with the corresponding left-accumulation UPSF modes
for typical parameters [red stars in Figs. 2(A)(B)], where the analytic
results Eq. (III.1) (red dots and lines) are perfectly consistent with the
numerical results (blue dots) for various system sizes 555We uniformly label
the coordinate axis of the numerical energy spectrum (eigenstates) as
$(\text{Re}(\text{E}),\text{Im}(\text{E}))$ [($\text{site},|\psi|$)] for
simplicity throughout this paper.. In addition, the scale-free localized modes
given in Refs. [53, 56] are equivalent to the strong non-reciprocity or
extremely special $\gamma_{R}\rightarrow 0$ cases in our transfer matrix
formalism (Appendix D.1).

Figure 2: (A)(B): The phase diagrams of the UPSF modes in Eq. (32) with a
singular transfer matrix ($t_{L}=1.5$, $t_{R}=0$) in the
$\gamma_{L}$-$\gamma_{R}$ plane with $N=10$ and $N=30$, respectively.
(a1)(b1): The analytic (red dots) and numerical (blue dots) results of the
full energy spectrum concerning the parameters $\gamma_{L}=0.4$ and
$\gamma_{R}=2$, denoted as red stars in (A)(B), respectively. (a2)(b2): The
analytic (red lines) and numerical (blue dots) results of the normalized left-
accumulation UPSF modes corresponding to (a1)(b1), respectively.

Figure 3: (A)(B): The energy spectrum of Eq. (40) for $\gamma=4.2$ and
$\gamma=\sqrt{24}$, respectively. (a1)(a2): The analytic (red lines) and
numerical (blue dots) eigenstates concerning the blue points in (A). (b1)(b2):
The analytic (red lines) and numerical (blue dots) eigenstates concerning the
blue points in (B). We set $t=1$, $\delta=5$, and $N=20$.

Figure 4: (a) The LA component and (b) the RA component of the SBSF mode in
Fig. 3(a2).

###  III.2 The emergent HSSF effect with boundary impurity

We next consider the case with nonsingular transfer matrix, whose energy
expression is given by

$\displaystyle\varepsilon=2\sqrt{t_{L}t_{R}}\cos{\phi},\quad\phi=\phi_{R}+i\phi_{I}\in\mathbb{C},$
(39)

through Eq. (15). The solutions of $\phi$ lie in $\left[0,\pi\right]$ under
OBC, while lead to physical real or complex values with boundary impurity in
Eq. (32) (Appendix D.2). In order to observe the HSSF effect unambiguously, we
perform a generalized gauge transformation
$\mathcal{H}_{it}=S^{-1}\mathcal{H}_{hn}S$ to cancel out the more dominant
NHSE amplification factor (Appendix D.2), resulting in

$\displaystyle\mathcal{H}_{it}=\sum_{n=1}^{N-1}t\left(c_{n}^{\dagger}c_{n+1}+\text{h.c.}\right)+(\delta+\gamma)c_{1}^{\dagger}c_{N}+(\delta-\gamma)c_{N}^{\dagger}c_{1},$
(40)

where $S=\text{diag}\left\\{r,r^{2},\ldots,r^{N}\right\\}$ is the
transformation matrix, $t=\sqrt{t_{L}t_{R}}$, $r=\sqrt{t_{R}/t_{L}}$, and we
have set $r^{N-1}\gamma_{R}=\delta+\gamma$ and
$r^{-(N-1)}\gamma_{L}=\delta-\gamma$. Subsequently, the eigenstate
corresponding to energy $\varepsilon=2t\cos{\phi}$ of Eq. (40) reads

$\displaystyle\left(\begin{matrix}\psi_{n+1}\\\
\psi_{n}\end{matrix}\right)=\left[A_{L}(\phi)e^{i(n-1)\phi}+A_{R}(\phi)e^{-i(n-1)\phi}\right]\left(\begin{matrix}\psi_{2}\\\
\psi_{1}\end{matrix}\right),$ (41)

where the coefficients $A_{L,R}(\phi)$ are given in Appendix D.2. In Figs.
3(A)(B), we illustrate the energy spectrum for two typical points in the PT-
broken region $\gamma\in\left[|\delta-t|,\delta+t\right]$ (Appendix D.2). The
real (complex) energy (blue dots) in Fig. 3(A) corresponds to the ES (SBSF
mode), of which the match-perfectly analytic (red lines) and numerical (blue
dots) results are plotted in Figs. 3(a1)(a2). Further, the scale-free
distributions of the LA [$A_{R}(\phi)e^{-i(n-1)\phi}$] and RA
[$A_{L}(\phi)e^{i(n-1)\phi}$] components of the SBSF mode [Fig. 3(a2) with
$\phi_{I}<0$] are illustrated in Fig. 4. The two complex energies (blue dots)
in Fig. 3(B) correspond to the UPSF modes (vanishing RA term), of which the
match-perfectly analytic (red lines) and numerical (blue dots) results are
plotted in Figs. 3(b1)(b2). We remark that all solutions of $\phi$ for Fig.
3(B) are complex-valued, whose imaginary part is $\log{(\mu^{-1})}/N$ with
$\mu=\delta/t+\sqrt{(\delta/t)^{2}-1}$ (Appendix D.2). Therefore, we observe
the pure NHSE (HSSF effect) of $\mathcal{H}_{hn}$ ($t_{L}\neq t_{R}$)
corresponding to ESs (PSF modes) of $\mathcal{H}_{it}$ after the inverse gauge
transformation $S^{-1}$. We emphasize that the HSSF effect with boundary
impurity may be a finite-size phenomenon, since the thermodynamic limit may
prevent simultaneous NHSE and SFL.

##  IV The emergent UPSF and HSSF effects in the NH-SSH model

The well-known NH-SSH model is a typical $r=1$ Hamiltonian
($\mathcal{H}_{nssh}$) of Eq. (3) with

$\displaystyle\boldsymbol{M}=\left(\begin{matrix}0&t_{1}+\gamma\\\
t_{1}-\gamma&0\end{matrix}\right),\quad\boldsymbol{J}=\left(\begin{matrix}0&0\\\
t_{2}&0\end{matrix}\right),$ (42)

where we assume $t_{1},t_{2},\gamma\geq 0$ for simplicity. The transfer matrix
reads

$\displaystyle
T=\frac{1}{t_{2}(t_{1}+\gamma)}\left(\begin{matrix}\varepsilon^{2}-t_{1}^{2}+\gamma^{2}&-\varepsilon
t_{2}\\\ \varepsilon t_{2}&-t_{2}^{2}\end{matrix}\right),$ (43)

with

$\displaystyle\Delta$
$\displaystyle=\text{tr}\left(T\right)=\frac{\varepsilon^{2}-t_{1}^{2}+\gamma^{2}-t_{2}^{2}}{t_{2}(t_{1}+\gamma)},$
$\displaystyle\Gamma$
$\displaystyle=\det{\left(T\right)}=\frac{t_{1}-\gamma}{t_{1}+\gamma}.$ (44)

following the reduced SVD $\boldsymbol{J}=v\xi w^{\dagger}$ with $\xi=t_{2}$,
$v=(0,1)^{T}$, and $w=(1,0)^{T}$. We introduce the boundary impurity in Eq.
(16) into such an NH-SSH model and analyze its PSF or HSSF effect.

Figure 5: (A) The phase diagram of the UPSF modes of the Hamiltonian
$\mathcal{H}_{nssh}$ with a singular transfer matrix, where green, cyan, and
yellow regions label the LA, RA, and LRA, respectively. (a1)-(c1): The full
analytic (red dots) and numerical (blue dots) energy spectrum with
$(\gamma_{L},\gamma_{R})=(0.2,3)$, $(0.8,2)$, and $(3,0.2)$ corresponding to
the red stars in (A). (a2)-(c2): The analytic (red lines) and numerical (blue
dots) eigenstates corresponding to (a1)-(c1), respectively. We set
$t_{1}=\gamma=0.5$, $t_{2}=1$, and $N=16$.

###  IV.1 The PSF effect with boundary impurity

The transfer matrix at the critical point $t_{1}=\gamma$ of PT-symmetry
breaking is singular with $\Gamma=0$, where we can directly apply the analytic
results in Sec. II.2.1. The single-particle eigenstates with respect to the
two energy bands ($\pm$) are

$\displaystyle\varepsilon_{m}^{\pm}$
$\displaystyle=\pm\sqrt{t_{2}^{2}+2t_{2}\gamma
c_{m}^{-\frac{1}{N}}e^{-i\frac{2\pi m}{N}}},$ $\displaystyle\Psi_{n}^{\pm,m}$
$\displaystyle=\alpha_{n}^{\pm,m}v+\beta_{n}^{\pm,m}w,$ (45)

where $m=1,2,\ldots,N$,

$\displaystyle\alpha_{n}^{\pm,m}$
$\displaystyle=c_{m}^{-\frac{n}{N}}e^{-i\frac{2\pi
m}{N}n}\frac{c_{m}\gamma_{L}t_{2}}{\varepsilon_{m}^{\pm}},\quad
n=1,2,\ldots,N-1,$ $\displaystyle\beta_{n}^{\pm,m}$
$\displaystyle=c_{m}^{-\frac{n-1}{N}}e^{-i\frac{2\pi
m}{N}(n-1)}c_{m}\gamma_{L},\quad n=2,3,\ldots,N,$
$\displaystyle\alpha_{N}^{\pm,m}$
$\displaystyle=\frac{\varepsilon_{m}^{\pm\,2}-c_{m}\gamma_{L}(\varepsilon_{m}^{\pm\,2}-t_{2}^{2})}{\gamma_{R}\varepsilon_{m}^{\pm}t_{2}},$
$\displaystyle\beta_{1}^{\pm,m}$ $\displaystyle=1,$ (46)

and $c_{m}$ satisfies the physical condition

$\displaystyle t_{2}(1-\gamma_{L}\gamma_{R})=2\gamma
c_{m}^{-\frac{1}{N}}e^{-i\frac{2\pi m}{N}}(\gamma_{L}c_{m}-1).$ (47)

These results indicate the emergence of the UPSF effect with the localization
length $\xi_{m}=N/|\text{Re}(\log{c_{m}})|$, and the phase diagram of the UPSF
modes is illustrated in Fig. 5(A). Applying Eq. (IV.1) to the three selected
points (red stars) in Fig. 5(A), which correspond to the UPSF modes with LA,
RA, and LRA, respectively, we obtain full energy spectrum [red dots in Figs.
5(a1)-(c1)] and eigenstates [red lines in Figs. 5(a2)-(c2)] perfectly matching
the numerical results (blue dots).

Figure 6: (a) The full energy spectrum of $\bar{\mathcal{H}}$ with the
parameters $t_{1}=0.1$, $t_{2}=1$, $\gamma=0.5$, and $N=16$. (b) The solutions
of $\phi_{R}$ corresponding to (a), where the vertical axis denotes
$[2N\phi_{R}+\text{arg}(\mathscr{F})+2\pi p]/2\pi$ with $p=16$. All the
energies (orange, blue, and magenta dots), labeled by $j$, fall on the
horizontal gray lines, integer constants $m=1,2,\ldots,16$. (c) The solutions
of $2N\phi_{I}$ for each energy labeled as $j$ corresponding to (a) are shown
as orange, blue, and magenta dots, where the gray line labels
$\log{|\mathscr{F}|}$. (d)-(f): The analytic (red lines) and numerical (blue
dots) eigenstates with respect to the blue dots in (a)-(c) match consistently.
The magenta dots in (a)-(c) are related to the zero-energy modes. (g)-(i): The
distributions of the LA (cyan) and RA (purple) components corresponding to the
SBSF modes in (d)-(f), respectively.

###  IV.2 The HSSF effect under OBC

The most counter-intuitive phenomenon with the emergent HSSF effect appears in
the case with nonsingular transfer matrix under OBC. According to Appendix B,
the physical condition is given by

$\displaystyle\frac{\sin{\left(N\phi\right)}}{\sin{\left((N-1)\phi\right)}}=\frac{t_{2}\sqrt{(t_{1}+\gamma)(t_{1}-\gamma)}}{\varepsilon^{2}-t_{1}^{2}+\gamma^{2}}\equiv
q,$ (48)

which figures out the solutions of $\phi=\phi_{R}+i\phi_{I}$

$\displaystyle\phi_{I}$ $\displaystyle=\frac{\log{|\mathscr{F}|}}{2N},$ (49)
$\displaystyle\phi_{R}$ $\displaystyle=\frac{2\pi
m-\text{arg}(\mathscr{F})}{2N},$ (50) $\displaystyle\mathscr{F}$
$\displaystyle=\frac{1-qe^{-i\phi}}{1-qe^{i\phi}},$ (51)

with $m=1,2,\ldots,N$. In the region $t_{1}>\gamma$, real $\phi$ and OBC
energy spectrum exist simultaneously. However, in the region $t_{1}<\gamma$,
complex energies lead to the presence of imaginary part of $\phi$, namely the
quasi-$1/N$-proportion Eq. (49). Hence, the HSSF effect emerges according to
Eq. (30), which has never been uncovered in previous works.

As in Sec. III.2, we perform a generalized gauge transformation for
$\mathcal{H}_{nssh}$ to conceal the NHSE factor with
$S=\text{diag}\left\\{1,r,r,\ldots,r^{N-1},r^{N-1},r^{N}\right\\}$ and
$r=\sqrt{\Gamma}$, resulting in the Hamiltonian $\bar{\mathcal{H}}$ with

$\displaystyle\bar{M}=\left(\begin{matrix}0&\bar{t}_{1}\\\
\bar{t}_{1}&0\end{matrix}\right),\quad\bar{J}=\boldsymbol{J},\quad\bar{t}_{1}=\sqrt{(t_{1}+\gamma)(t_{1}-\gamma)}.$
(52)

Subsequently, we obtain
$\bar{\Delta}=(\varepsilon^{2}-\bar{t}_{1}^{2}-t_{2}^{2})/\bar{t}_{1}t_{2}$
and $\bar{\Gamma}=1$, which result the same physical condition for
$\mathcal{H}_{nssh}$ in Eq. (48), and according to Eq. (15), we obtain the
expression of the energy

$\displaystyle\varepsilon^{2}=t_{1}^{2}-\gamma^{2}+t_{2}^{2}+2t_{2}\bar{t}_{1}\cos{\phi},$
(53)

whose related eigenstates read

$\displaystyle\bar{\Psi}_{n}=\bar{\mathscr{A}}_{L}(\phi)e^{in\phi_{R}}e^{-n\phi_{I}}+\bar{\mathscr{A}}_{R}(\phi)e^{-in\phi_{R}}e^{n\phi_{I}},$
(54)

where the coefficients are given by

$\displaystyle\bar{\mathscr{A}}_{L}(\phi)$
$\displaystyle=\frac{1}{2iq\sin{\phi}}\left[t_{2}\bar{\mathcal{G}}_{vv}v+(e^{-i\phi}-qe^{-2i\phi})w\right],$
$\displaystyle\bar{\mathscr{A}}_{R}(\phi)$
$\displaystyle=\frac{1}{2iq\sin{\phi}}\left[-t_{2}\bar{\mathcal{G}}_{vv}v+(-e^{i\phi}+qe^{2i\phi})w\right],$
(55)

with
$\bar{\mathcal{G}}_{vv}=\varepsilon/(\varepsilon^{2}-t_{1}^{2}+\gamma^{2})$.
Utilizing the full energy spectrum in Fig. 6(a), we figure out the solutions
of $\phi$ for each energy labeled as $j$ in Figs. 6(b)(c), which are perfectly
consistent with the analytic formulas Eqs. (49)-(51). The analytic (red lines)
and numerical (blue dots) eigenstates concerning the three selected points
(blue dots) in Fig. 6(a) are plotted in Figs. 6(d)-(f), which match perfectly.
Further, the scale-free distributions of the LA
[$\bar{\mathscr{A}}_{L}(\phi)e^{in\phi_{R}}e^{-n\phi_{I}}$] and RA
[$\bar{\mathscr{A}}_{R}(\phi)e^{-in\phi_{R}}e^{n\phi_{I}}$] components of
these SBSF modes in Figs. 6(d)-(f) are illustrated in Figs. 6(g)-(i),
respectively. Therefore, the SBSF modes $\bar{\Psi}_{n}$ of
$\bar{\mathcal{H}}$, which resemble the ESs in appearance, indicating the
emergent HSSF modes of the original NH-SSH model under OBC after the inverse
gauge transformation.

As we turn on the boundary impurity, the physical condition is given by

$\displaystyle\left(\gamma_{L}\Gamma^{-\frac{N}{2}}+\gamma_{R}\Gamma^{\frac{N}{2}}\right)\sin{\phi}=\sin{\left((N+1)\phi\right)}$
$\displaystyle+\frac{t_{2}(1-\gamma_{L}\gamma_{R})}{\sqrt{(t_{1}+\gamma)(t_{1}-\gamma)}}\sin{(N\phi)}-\gamma_{L}\gamma_{R}\sin{\left((N-1)\phi\right)},$
(56)

according to Eq. (53) and Appendix C, which is equivalent to the result in
Ref. [64]. The HSSF modes in [Eq. (31)] appear with possible emergent complex
solutions of $\phi$, which match with the numerical results in Appendix E.
However, the HSSF effect with boundary impurity is usually limited to finite
sizes due to the restraints on the NHSE and SFL in large systems.

##  V Conclusions and discussions

We have developed a transfer-matrix perspective for a unified understanding of
the NHSE and the SFL in non-Hermitian systems with boundary impurity. We
derive the analytic expressions for the single-particle bulk eigenstates with
respect to the energy spectrum upon the 1D non-Hermitian systems with $2\times
2$ transfer matrices, which exhibit excellent consistency with numerical
results in the HN model and NH-SSH model. We find that the UPSF effect
accompanies singular transfer matrices, while the HSSF effect emerges with
nonsingular transfer matrices, even under OBC. The UPSF effect portrays a
localization length of the eigenstates quasi-linear with the system size,
while the HSSF effect shows an interplay between the finite-size NHSE and SFL.
We further reveal that the NHSE prevails over the SFL under OBC as the system
size increases, thus $\phi_{I}=0$ in the thermodynamic limit. Interestingly,
we may then generate the energy spectrum from
$\Delta=2\sqrt{\Gamma}\cos{\phi_{R}}$ [Eq. (15)]; correspondingly, the norm
$\sqrt{|\Gamma|}$ with corresponding $(\text{arg}(\Gamma),\phi_{R})$ outlines
the GBZ and re-invents the non-Bloch band theory through the transfer matrix
approach, open to further generalizations for future studies.

The constraints over the assumption $\boldsymbol{J}_{R}=\boldsymbol{J}_{L}$,
rank-$1$ of $\boldsymbol{J}$, namely $2\times 2$ transfer matrices, and the
boundary impurity in Eq. (16), are sufficiently general capture the core
mechanism and behaviors of the PSF and HSSF effects here. Nevertheless, there
exist more questions to explore in the future, such as global impurity and
disorder [56, 88, 89, 74, 90, 91], the role of various symmetries [1, 2],
realistic experiments and applications, and other subjects of the NHSE and SFL
in condensed matter physics.

## Acknowledgements

We acknowledge support from the National Key R&D Program of China
(No.2022YFA1403700) and the National Natural Science Foundation of China
(No.12174008 & No.92270102).

##  Appendix A The boundary equations with boundary impurity

Modifying Eq. (5) at the boundaries with the setting $\Psi_{0}\equiv\Psi_{N}$,
we obtain

$\displaystyle\Psi_{N}$
$\displaystyle=\mathcal{G}\boldsymbol{J}^{\dagger}\Psi_{N-1}+\gamma_{L}\mathcal{G}\boldsymbol{J}\Psi_{1}$
$\displaystyle\Psi_{1}$
$\displaystyle=\gamma_{R}\mathcal{G}\boldsymbol{J}^{\dagger}\Psi_{N}+\mathcal{G}\boldsymbol{J}\Psi_{2}.$
(57)

Combining with Eqs. (6)-(9), we obtain

$\displaystyle\alpha_{N}$
$\displaystyle=\mathcal{G}_{wv}\xi\alpha_{N-1}+\gamma_{L}\mathcal{G}_{vv}\xi\beta_{1},$
$\displaystyle\beta_{N}$
$\displaystyle=\mathcal{G}_{ww}\xi\alpha_{N-1}+\gamma_{L}\mathcal{G}_{vw}\xi\beta_{1},$
$\displaystyle\alpha_{1}$
$\displaystyle=\gamma_{R}\mathcal{G}_{wv}\xi\alpha_{N}+\mathcal{G}_{vv}\xi\beta_{2},$
$\displaystyle\beta_{1}$
$\displaystyle=\gamma_{R}\mathcal{G}_{ww}\xi\alpha_{N}+\mathcal{G}_{vw}\xi\beta_{2}.$
(58)

Further derivation gives

$\displaystyle\left(\begin{matrix}\beta_{1}\\\ \alpha_{N}\end{matrix}\right)$
$\displaystyle=\left(\begin{matrix}\frac{1}{\gamma_{L}}\xi^{-1}\mathcal{G}_{vw}^{-1}&-\frac{1}{\gamma_{L}}\xi^{-1}\mathcal{G}_{vw}^{-1}\mathcal{G}_{ww}\xi\\\
\mathcal{G}_{vv}\mathcal{G}_{vw}^{-1}&\left(\mathcal{G}_{wv}-\mathcal{G}_{vv}\mathcal{G}_{vw}^{-1}\mathcal{G}_{ww}\right)\xi\end{matrix}\right)\left(\begin{matrix}\beta_{N}\\\
\alpha_{N-1}\end{matrix}\right),$
$\displaystyle\left(\begin{matrix}\beta_{2}\\\ \alpha_{1}\end{matrix}\right)$
$\displaystyle=\left(\begin{matrix}\xi^{-1}\mathcal{G}_{vw}^{-1}&-\gamma_{R}\xi^{-1}\mathcal{G}_{vw}^{-1}\mathcal{G}_{ww}\xi\\\
\mathcal{G}_{vv}\mathcal{G}_{vw}^{-1}&\gamma_{R}\left(\mathcal{G}_{wv}-\mathcal{G}_{vv}\mathcal{G}_{vw}^{-1}\mathcal{G}_{ww}\right)\xi\end{matrix}\right)\left(\begin{matrix}\beta_{1}\\\
\alpha_{N}\end{matrix}\right),$ (59)

which are exactly the equations on the boundaries

$\displaystyle\Phi_{1}$
$\displaystyle=\left(\begin{matrix}\frac{1}{\gamma_{L}}&0\\\
0&1\end{matrix}\right)T\Phi_{N}=K_{L}T\Phi_{N},$ $\displaystyle\Phi_{2}$
$\displaystyle=T\left(\begin{matrix}1&0\\\
0&\gamma_{R}\end{matrix}\right)\Phi_{1}=TK_{R}\Phi_{1}.$ (60)

##  Appendix B The solutions of nonsingular cases under OBC

The OBC refers to the hard or Dirichlet boundary condition
$\Psi_{0}=\Psi_{N+1}=0$, implying $\alpha_{0}=\beta_{N+1}=0$ such that

$\displaystyle\Phi_{1}=\left(\begin{matrix}\beta_{1}\\\
0\end{matrix}\right),\quad\Phi_{N+1}=\left(\begin{matrix}0\\\
\alpha_{N}\end{matrix}\right).$ (61)

After normalization, we obtain the OBC equation

$\displaystyle T^{N}\left(\begin{matrix}1\\\
0\end{matrix}\right)=\left(\begin{matrix}0\\\ \tau\end{matrix}\right),$ (62)

with $\tau\in\mathbb{C}$. Accordingly, the propagating relation of $\Phi_{n}$
is illustrated in Fig. 1(b). Utilizing Eq. (13), we obtain the physical
condition under OBC [73]

$\displaystyle\frac{\sin{\left(N\phi\right)}}{\sin{\left((N-1)\phi\right)}}=q,$
(63)

with $q=\xi\sqrt{\frac{\mathcal{G}_{wv}}{\mathcal{G}_{vw}}}\mathcal{G}_{vw}$.
Further simplifying Eq. (63) with general complex $\phi=\phi_{R}+i\phi_{I}$
and $\phi_{R}\in[0,2\pi],\phi_{I}\in\mathbb{R}$, we obtain

$\displaystyle
e^{2N\phi_{I}}=\frac{1-qe^{-i\phi}}{1-qe^{i\phi}}e^{2iN\phi_{R}}.$ (64)

Hence, the solutions of $\phi$ reads

$\displaystyle\phi_{I}$ $\displaystyle=\frac{\log{|\mathscr{F}|}}{2N},$
$\displaystyle\phi_{R}$ $\displaystyle=\frac{2\pi
m-\text{arg}(\mathscr{F})}{2N},$ (65)

where $\mathscr{F}$ labels
$\left(1-qe^{-i\phi}\right)/\left(1-qe^{i\phi}\right)$ and $m=1,2,\ldots,N$.
Note that $|\mathscr{F}|=1$ leads to $\phi_{I}=0$ if $q$ is real. In general
complex-$q$ cases, the explicit $\phi_{I}$ and energy $\varepsilon$ are
dependent on $N$ and $\phi_{R}$ through the self-consistent transcendental
equations (15) and (B), that is we can denote $\phi_{I}=c/N$ with $c$ being a
finite-value function of $N$ and $\phi_{R}$, and thus $\phi_{I}$ is quasi-
linearly dependent on $1/N$.

The solution of an arbitrary eigenstate is given by
$\Phi_{n+1}=T^{n}\left(\begin{matrix}1\\\ 0\end{matrix}\right)$, resulting in

$\displaystyle\beta_{n}$
$\displaystyle=\frac{\Gamma^{(n-1)/2}}{q\sin{\phi}}\left[\sin{\left((n-1)\phi\right)}-q\sin{\left((n-2)\phi\right)}\right],$
$\displaystyle\alpha_{n}$
$\displaystyle=\frac{\Gamma^{n/2}}{q\sin{\phi}}\xi\mathcal{G}_{vv}\sin{(n\phi)},$
(66)

with $n=1,2,\ldots,N$. Note that $\beta_{N+1}=0$ is exactly the physical
condition Eq. (63) and $\alpha_{N}=\tau$. Consequently, an eigenstate
concerning energy $\varepsilon$ reads

$\displaystyle\Psi_{n}=\Gamma^{n/2}\left[\mathscr{A}_{L}(\phi)e^{in\phi_{R}}e^{-n\phi_{I}}+\mathscr{A}_{R}(\phi)e^{-in\phi_{R}}e^{n\phi_{I}}\right],$
(67)

where

$\displaystyle\mathscr{A}_{L}(\phi)$
$\displaystyle=\frac{1}{2iq\sin{\phi}}\left[\xi\mathcal{G}_{vv}v+\left(e^{-i\phi}-qe^{-2i\phi}\right)\frac{w}{\sqrt{\Gamma}}\right],$
$\displaystyle\mathscr{A}_{R}(\phi)$
$\displaystyle=\frac{1}{2iq\sin{\phi}}\left[-\xi\mathcal{G}_{vv}v+\left(-e^{i\phi}+qe^{2i\phi}\right)\frac{w}{\sqrt{\Gamma}}\right].$
(68)

##  Appendix C The solutions of nonsingular cases with boundary impurity

According to the boundary equation (19) with boundary impurity, the
eigenvalues of the $2\times 2$ matrix $KT^{N}$ must be $1$ and

$\displaystyle\det{\left(KT^{N}\right)}=\det{\left(K\right)}\left(\det{T}\right)^{N}=\frac{\gamma_{R}}{\gamma_{L}}\Gamma^{N},$
(69)

such that,

$\displaystyle\text{tr}\left(KT^{N}\right)=1+\frac{\gamma_{R}}{\gamma_{L}}\Gamma^{N}=\Gamma^{N/2}\left(\frac{\gamma_{R}}{{\gamma_{L}}}\right)^{1/2}\left[\left(\frac{\gamma_{R}}{{\gamma_{L}}}\Gamma^{N}\right)^{-1/2}+\left(\frac{\gamma_{R}}{{\gamma_{L}}}\Gamma^{N}\right)^{1/2}\right].$
(70)

On the other hand, utilizing the formula Eq. (13) of nonsingular $T$, we
obtain

$\displaystyle\text{tr}\left(KT^{N}\right)=\Gamma^{N/2}\left[\frac{U_{N-1}(z)}{\sqrt{\Gamma}}\text{tr}\left(KT\right)-U_{N-2}(z)\text{tr}\left(K\right)\right]=\Gamma^{N/2}\left[\frac{\text{tr}\left(KT\right)}{\sqrt{\Gamma}}\frac{\sin{\left(N\phi\right)}}{\sin{\phi}}-\text{tr}\left(K\right)\frac{\sin{\left((N-1)\phi\right)}}{\sin{\phi}}\right].$
(71)

Thus, we get the condition

$\displaystyle\left(\frac{\gamma_{R}}{{\gamma_{L}}}\right)^{1/2}\left[\left(\frac{\gamma_{R}}{{\gamma_{L}}}\Gamma^{N}\right)^{-1/2}+\left(\frac{\gamma_{R}}{{\gamma_{L}}}\Gamma^{N}\right)^{1/2}\right]\sin{\phi}=\frac{\text{tr}\left(KT\right)}{\sqrt{\Gamma}}\sin{\left(N\phi\right)}-\text{tr}\left(K\right)\sin{\left((N-1)\phi\right)}.$
(72)

Further simplifying with general complex $\phi=\phi_{R}+i\phi_{I}$ and
$\phi_{R}\in[0,2\pi],\phi_{I}\in\mathbb{R}$, we obtain

$\displaystyle
a(\phi)\left(e^{N\phi_{I}}\right)^{2}+b(\phi)e^{N\phi_{I}}+c(\phi)=0,$ (73)

where

$\displaystyle a(\phi)$
$\displaystyle=\frac{e^{-iN\phi_{R}}}{2i}\left[-\frac{\text{tr}\left(KT\right)}{\sqrt{\Gamma}}+\text{tr}\left(K\right)e^{i\phi}\right],$
$\displaystyle b(\phi)$
$\displaystyle=-\left[\left(\frac{\gamma_{R}}{{\gamma_{L}}}\Gamma^{N}\right)^{-1/2}+\left(\frac{\gamma_{R}}{{\gamma_{L}}}\Gamma^{N}\right)^{1/2}\right]\left(\frac{\gamma_{R}}{{\gamma_{L}}}\right)^{1/2}\sin{\phi},$
$\displaystyle c(\phi)$
$\displaystyle=\frac{e^{iN\phi_{R}}}{2i}\left[\frac{\text{tr}\left(KT\right)}{\sqrt{\Gamma}}-\text{tr}\left(K\right)e^{-i\phi}\right].$
(74)

The solution of $\phi_{I}$ is

$\displaystyle
e^{N\phi_{I}}=\frac{1}{2a(\phi)}\left[-b(\phi)\pm\sqrt{b^{2}(\phi)-4a(\phi)c(\phi)}\right],$
(75)

leading to physical real or complex $\phi$ in general. However, since
$a(\phi),b(\phi)$ are always finite and $b(\phi)$ contains the terms
$\Gamma^{\pm N/2}$, the emergent forms $\phi_{I}\sim c/N$ with finite size are
prevented by the large $N$.

The solution of an arbitrary eigenstate is given by
$\Phi_{n+1}=T^{n}\varphi,\,n=1,2,\ldots,N-1$ with $KT^{N}\varphi=\varphi$,
resulting in

$\displaystyle\Phi_{n+1}=\Gamma^{n/2}\left[\mathcal{A}_{L}(\phi)e^{in\phi_{R}}e^{-n\phi_{I}}+\mathcal{A}_{R}(\phi)e^{-in\phi_{R}}e^{n\phi_{I}}\right],$
(76)

where

$\displaystyle\mathcal{A}_{L}(\phi)$
$\displaystyle=\frac{1}{2i\sin{\phi}}\left(\frac{T}{\sqrt{\Gamma}}-e^{-i\phi}\mathbbm{1}\right)\varphi,$
$\displaystyle\mathcal{A}_{R}(\phi)$
$\displaystyle=\frac{-1}{2i\sin{\phi}}\left(\frac{T}{\sqrt{\Gamma}}-e^{i\phi}\mathbbm{1}\right)\varphi.$
(77)

Further,

$\displaystyle\beta_{1}$ $\displaystyle=\left(K_{R}^{-1}\varphi\right)_{1},$
$\displaystyle\alpha_{n}$
$\displaystyle=\Gamma^{n/2}\left[\mathcal{A}_{L,2}(\phi)e^{in\phi_{R}}e^{-n\phi_{I}}+\mathcal{A}_{R,2}(\phi)e^{-in\phi_{R}}e^{n\phi_{I}}\right],\quad
n=1,2,\ldots,N-1,$ $\displaystyle\beta_{n}$
$\displaystyle=\Gamma^{(n-1)/2}\left[\mathcal{A}_{L,1}(\phi)e^{i(n-1)\phi_{R}}e^{-(n-1)\phi_{I}}+\mathcal{A}_{R,1}(\phi)e^{-i(n-1)\phi_{R}}e^{(n-1)\phi_{I}}\right],\quad
n=2,3,\ldots,N,$ $\displaystyle\alpha_{N}$
$\displaystyle=\left(K_{R}^{-1}\varphi\right)_{2},$ (78)

where $\mathcal{A}_{L/R,i}(\phi),i=1,2$ denotes the $i$-th component of the
column vector $\mathcal{A}_{L/R}(\phi)$. Consequently, an eigenstate
concerning energy $\varepsilon$ reads

$\displaystyle\Psi_{1}^{imp}$
$\displaystyle=\Gamma^{1/2}\left[\mathcal{A}_{L,2}(\phi)e^{i\phi_{R}}e^{-\phi_{I}}+\mathcal{A}_{R,2}(\phi)e^{-i\phi_{R}}e^{\phi_{I}}\right]v+\left(K_{R}^{-1}\varphi\right)_{1}w,$
$\displaystyle\Psi_{n}^{imp}$
$\displaystyle=\Gamma^{n/2}\left[\mathcal{B}_{L}(\phi)e^{in\phi_{R}}e^{-n\phi_{I}}+\mathcal{B}_{R}(\phi)e^{-in\phi_{R}}e^{n\phi_{I}}\right],\quad
n=2,3,\ldots,N-1,$ $\displaystyle\Psi_{N}^{imp}$
$\displaystyle=\left(K_{R}^{-1}\varphi\right)_{2}v+\Gamma^{(N-1)/2}\left[\mathcal{A}_{L,1}(\phi)e^{i(N-1)\phi_{R}}e^{-(N-1)\phi_{I}}+\mathcal{A}_{R,1}(\phi)e^{-i(N-1)\phi_{R}}e^{(N-1)\phi_{I}}\right]w,$
(79)

where

$\displaystyle\mathcal{B}_{L}(\phi)$
$\displaystyle=\mathcal{A}_{L,2}(\phi)v+e^{-i\phi}\mathcal{A}_{L,1}(\phi)\frac{w}{\sqrt{\Gamma}},$
$\displaystyle\mathcal{B}_{R}(\phi)$
$\displaystyle=\mathcal{A}_{R,2}(\phi)v+e^{i\phi}\mathcal{A}_{R,1}(\phi)\frac{w}{\sqrt{\Gamma}}.$
(80)

##  Appendix D Details of the HN model with boundary impurity

Utilizing the single-particle Schrödinger equation of Eq. (32) with the
setting $\psi_{0}\equiv\psi_{N}$ , we obtain the bulk and boundary equations

$\displaystyle t_{R}\psi_{n-1}+t_{L}\psi_{n+1}$
$\displaystyle=\varepsilon\psi_{n},$ (81)
$\displaystyle\gamma_{R}\psi_{N}+t_{L}\psi_{2}$
$\displaystyle=\varepsilon\psi_{1},$ (82) $\displaystyle
t_{R}\psi_{N-1}+\gamma_{L}\psi_{1}$ $\displaystyle=\varepsilon\psi_{N},$ (83)

where $n\in\mathscr{B}$. The bulk equation (81) gives ($t_{L}\neq 0$)

$\displaystyle\psi_{n+1}=\frac{\varepsilon}{t_{L}}\psi_{n}-\frac{t_{R}}{t_{L}}\psi_{n-1},$
(84)

thus leading the propagating relation Eq. (33).

###  D.1 The case with singular transfer matrix

The real-space EP, or infernal point, emerges under OBC in the case with
singular transfer matrix ($t_{R}=0$), where all energies are degenerate at
zero but with only one eigenvector [73, 39, 40]. Combining the boundary
equations (82) and (83) for $t_{R}=0$, we can deduce the physical condition
with boundary impurity,

$\displaystyle\left(\begin{matrix}\psi_{2}\\\ \psi_{1}\end{matrix}\right)$
$\displaystyle=\left(\begin{matrix}\frac{\varepsilon}{t_{L}}&-\frac{\gamma_{R}}{t_{L}}\\\
1&0\end{matrix}\right)\left(\begin{matrix}\psi_{1}\\\
\psi_{N}\end{matrix}\right)$
$\displaystyle=\left(\begin{matrix}\frac{\varepsilon}{t_{L}}&-\frac{\gamma_{R}}{t_{L}}\\\
1&0\end{matrix}\right)\left(\begin{matrix}\frac{\varepsilon}{\gamma_{L}}&0\\\
1&0\end{matrix}\right)\left(\begin{matrix}\psi_{N}\\\
\psi_{N-1}\end{matrix}\right)$
$\displaystyle=\left(\begin{matrix}\frac{\varepsilon}{t_{L}}&-\frac{\gamma_{R}}{t_{L}}\\\
1&0\end{matrix}\right)\left(\begin{matrix}\frac{t_{L}}{\gamma_{L}}&0\\\
0&1\end{matrix}\right)\left(\begin{matrix}\frac{\varepsilon}{t_{L}}&0\\\
1&0\end{matrix}\right)\left(\begin{matrix}\psi_{N}\\\
\psi_{N-1}\end{matrix}\right)$
$\displaystyle=\left(\begin{matrix}\frac{\varepsilon}{t_{L}}&-\frac{\gamma_{R}}{t_{L}}\\\
1&0\end{matrix}\right)\left(\begin{matrix}\frac{t_{L}}{\gamma_{L}}&0\\\
0&1\end{matrix}\right)\left(\begin{matrix}\frac{\varepsilon}{t_{L}}&0\\\
1&0\end{matrix}\right)T^{N-2}\left(\begin{matrix}\psi_{2}\\\
\psi_{1}\end{matrix}\right)$
$\displaystyle=\left(\begin{matrix}\frac{\varepsilon}{\gamma_{L}}&-\frac{\gamma_{R}}{t_{L}}\\\
\frac{t_{L}}{\gamma_{L}}&0\end{matrix}\right)T^{N-1}\left(\begin{matrix}\psi_{2}\\\
\psi_{1}\end{matrix}\right)$ $\displaystyle\equiv
K_{d}T^{N-1}\left(\begin{matrix}\psi_{2}\\\ \psi_{1}\end{matrix}\right),$ (85)

where

$\displaystyle T=\left(\begin{matrix}\frac{\varepsilon}{t_{L}}&0\\\
1&0\end{matrix}\right),$ (86)

is the singular transfer matrix with
$\Delta=\text{tr}\left(T\right)=\varepsilon/t_{L}$. It implies that the
physical eigenvector of $K_{d}T^{N-1}$ concerns the eigenvalue $1$. Due to
$\det{\left(K_{d}T^{N-1}\right)}=\det{\left(K_{d}\right)}\Gamma^{N-1}=0$, we
obtain $\text{tr}\left(K_{d}T^{N-1}\right)=1$, thus

$\displaystyle\text{tr}\left(K_{d}\Delta^{N-2}T\right)=\Delta^{N-2}\text{tr}\left(K_{d}T\right)=\left(\frac{\varepsilon}{t_{L}}\right)^{N-2}\frac{\varepsilon^{2}-\gamma_{L}\gamma_{R}}{t_{L}\gamma_{L}}=1,$
(87)

which gives

$\displaystyle\frac{\varepsilon^{2}}{\gamma_{L}}-\gamma_{R}=t_{L}\left(\frac{t_{L}}{\varepsilon}\right)^{N-2}.$
(88)

Let $\left(t_{L}/\varepsilon\right)^{N-2}=c$ with $c$ being an undetermined
coefficient, and thus

$\displaystyle\varepsilon=t_{L}c^{-\frac{1}{N-2}}e^{-i\frac{2\pi
m}{N-2}},\quad m=1,2,\ldots,N-2.$ (89)

Substituting Eq. (89) into Eq. (88), we obtain the physical condition about
$c$

$\displaystyle\frac{t_{L}^{2}}{\gamma_{L}}c^{-\frac{2}{N-2}}e^{-i\frac{4\pi
m}{N-2}}=t_{L}c+\gamma_{R},$ (90)

where we will label $c_{m}$ as the solution concerning $m$. Applying the
propagating relation, we obtain

$\displaystyle\left(\begin{matrix}\psi_{n+1}\\\ \psi_{n}\end{matrix}\right)$
$\displaystyle=T^{n-1}\left(\begin{matrix}\psi_{2}\\\
\psi_{1}\end{matrix}\right)=\Delta^{n-2}T\left(\begin{matrix}\psi_{2}\\\
\psi_{1}\end{matrix}\right)=\left(\begin{matrix}\left(\frac{\varepsilon}{t_{L}}\right)^{n-1}\psi_{2}\\\
\left(\frac{\varepsilon}{t_{L}}\right)^{n-2}\psi_{2}\end{matrix}\right),$ (91)

with $n\in\mathscr{B}$. Together with the boundary equations (82) and (83) for
$t_{R}=0$, we finally obtain the eigenvectors with respect to energies

$\displaystyle\varepsilon_{m}$
$\displaystyle=t_{L}c_{m}^{-\frac{1}{N-2}}e^{-i\frac{2\pi m}{N-2}},\quad
m=1,2,\ldots,N,$ $\displaystyle\psi_{n}^{m}$
$\displaystyle=c_{m}^{-\frac{n-2}{N-2}}e^{-i\frac{2\pi
m}{N-2}(n-2)}\psi_{2}^{m},\quad n=3,4,\ldots,N,$ $\displaystyle\psi_{1}^{m}$
$\displaystyle=\frac{t_{L}}{\gamma_{L}}c_{m}^{-\frac{1}{N-2}}e^{-i\frac{2\pi
m}{N-2}}\psi_{N}^{m}.$ (92)

Furthermore, set $t_{L}=e^{\alpha},t_{R}=e^{-\alpha},\gamma_{L}=\mu
e^{\alpha},\gamma_{R}=\mu e^{-\alpha}$, then the strong non-reciprocity
$e^{\alpha}\gg e^{-\alpha}$ corresponds to the singular case
($t_{R}\rightarrow 0$). In the strong impurity case ($\mu\gg e^{\pm\alpha}$),
the physical condition Eq. (90) reduces to

$\displaystyle-\mu e^{-2\alpha}=c,$ (93)

which leading to the solution according to Eq. (D.1)

$\displaystyle\varepsilon_{m}$
$\displaystyle=e^{\alpha}e^{-\left(\kappa_{L}+ik_{m}\right)},$
$\displaystyle\psi_{n}^{(m)}$
$\displaystyle=e^{-\left(\kappa_{L}+ik_{m}\right)(n-2)}\psi_{2}^{(m)},$
$\displaystyle|\psi_{1}^{(m)}$
$\displaystyle|\sim|\frac{e^{2\alpha}}{\mu^{2}}|\ll 1,$ (94)

where $n=3,4,\ldots,N$, $\kappa_{L}=(\log{\mu}-2\alpha)/(N-2)$, and
$k_{m}=\left((2m+1)\pi\right)/(N-2)$. We next consider the limit case
$\gamma_{R}\rightarrow 0$, and the condition Eq. (88) reduces to

$\displaystyle\left(\frac{\varepsilon}{t_{L}}\right)^{N}=\frac{\gamma_{L}}{t_{L}}.$
(95)

Then, the eigenvectors with corresponding energies read

$\displaystyle\varepsilon_{m}$
$\displaystyle=t_{L}^{\frac{1}{N}(N-1)}\gamma_{L}^{\frac{1}{N}}e^{i\frac{2\pi
m}{N}},$ $\displaystyle\psi_{n}^{(m)}$
$\displaystyle=\left(\frac{\gamma_{L}}{t_{L}}\right)^{\frac{n-1}{N}}e^{i\frac{2\pi
m}{N}(n-1)}\psi_{1}^{(m)},$ (96)

with $m=1,2,\ldots,N$ and $n=2,3,\ldots,N$, which are similar to the results
of the $t_{L}=\gamma_{L}=0$ case in Ref. [56]. The weak impurity case ($\mu\ll
1$) corresponds the $\gamma_{R}\rightarrow 0$ limit, the solution Eq. (D.1)
becomes

$\displaystyle\varepsilon_{m}$
$\displaystyle=e^{\alpha}e^{i(k_{m}^{\prime}+i\kappa_{L}^{\prime})},$
$\displaystyle\psi_{n}^{(m)}$
$\displaystyle=e^{i(k_{m}^{\prime}+i\kappa_{L}^{\prime})(n-1)}\psi_{1}^{(m)},$
(97)

where $\kappa_{L}^{\prime}=-\log{\mu}/N$ and $k_{m}^{\prime}=2\pi m/N$.
Noteworthily, Eqs. (D.1) and (D.1) are exactly equivalent to the SFL solutions
in Ref. [53].

###  D.2 The case with nonsingular transfer matrix

Through the boundary equations (82)(83), we can imitate Eq. (D.1) to derive
the physical condition in current case

$\displaystyle\left(\begin{matrix}\psi_{2}\\\
\psi_{1}\end{matrix}\right)=KT^{N-1}\left(\begin{matrix}\psi_{2}\\\
\psi_{1}\end{matrix}\right),$ (98)

where

$\displaystyle
K=\left(\begin{matrix}\frac{\varepsilon}{\gamma_{L}}&-\frac{\gamma_{R}}{t_{L}}\\\
\frac{t_{L}}{\gamma_{L}}&0\end{matrix}\right).$ (99)

Consequently,

$\displaystyle\text{tr}\left(KT^{N-1}\right)$
$\displaystyle=1+\det{\left(KT^{N-1}\right)}=1+\frac{\gamma_{R}}{\gamma_{L}}\Gamma^{N-1}.$
(100)

On the other hand, utilizing the formula Eq. (13) of nonsingular $T$, we
obtain ($z=\cos{\phi}$)

$\displaystyle\text{tr}\left(KT^{N-1}\right)=\Gamma^{\frac{N-1}{2}}\left[\frac{U_{N-2}(z)}{\sqrt{\Gamma}}\text{tr}\left(KT\right)-U_{N-3}(z)\text{tr}\left(K\right)\right].$
(101)

Together with Eq. (39) and after some algebra, we obtain the final form of
physical condition

$\displaystyle\left(\frac{\gamma_{L}}{t_{L}}\Gamma^{-\frac{N}{2}}+\frac{\gamma_{R}}{t_{R}}\Gamma^{\frac{N}{2}}\right)\sin{\phi}=\sin{\left((N+1)\phi\right)}-\frac{\gamma_{L}\gamma_{R}}{t_{L}t_{R}}\sin{\left((N-1)\phi\right)},$
(102)

which is exactly the result in Ref. [64]. The solutions of $\phi$ in Eq. (102)
are real or complex values for different parameters according to Ref. [64],
and the imaginary part of the complex solution $\phi=\phi_{R}+i\phi_{I}$ can
possibly take the form $\phi_{I}=c/N$ with $c$ dependent on $N$ and $\phi_{R}$
as before.

The OBC condition corresponding to $\gamma_{L}=\gamma_{R}=0$ reduces Eq. (102)
to $\sin{\left((N+1)\phi\right)}=0$, which figures out the solutions
$\phi=l\pi/(N+1)\in\left[0,\pi\right]$ with $l=1,2,\ldots,N$. The eigenstate
concerning energy $\varepsilon=2\sqrt{t_{L}t_{R}}\cos{\phi}$ is deduced as

$\displaystyle\left(\begin{matrix}\psi_{n+1}\\\
\psi_{n}\end{matrix}\right)=T^{n}\left(\begin{matrix}1\\\
0\end{matrix}\right)=\frac{1}{\sin{\phi}}\left(\begin{matrix}\Gamma^{n/2}\sin{\left((n+1)\phi\right)}\\\
\Gamma^{(n-1)/2}\sin{\left(n\phi\right)}\end{matrix}\right),$ (103)

and normalized as $\psi_{n}=\mathcal{N}\Gamma^{n/2}\sin{\left(n\phi\right)}$
with $\mathcal{N}$ being the normalization coefficient, which is exact the
bulk eigenstate of pure finite-size NHSE.

We can conceal the NHSE factor $\Gamma^{n/2}$ through a generalized gauge
transformation and obtain the target Hamiltonian

$\displaystyle\mathcal{H}_{it}=S^{-1}\mathcal{H}_{hn}S=\sum_{n=1}^{N-1}t\left(c_{n}^{\dagger}c_{n+1}+c_{n+1}^{\dagger}c_{n}\right)+r^{N-1}\gamma_{R}c_{1}^{\dagger}c_{N}+\frac{\gamma_{L}}{r^{N-1}}c_{N}^{\dagger}c_{1},$
(104)

where $S=\text{diag}\left\\{r,r^{2},\ldots,r^{N}\right\\}$ is the
transformation matrix, $t=\sqrt{t_{L}t_{R}}$, and $r=\sqrt{t_{R}/t_{L}}$. We
identify that $r^{N-1}\gamma_{R}=\delta+\gamma$ and
$r^{-(N-1)}\gamma_{L}=\delta-\gamma$ for fixed $t_{L},t_{R}$, then Eq. (104)
is exactly the focused Hamiltonian in Ref. [54], which is a original Hermitian
Hamiltonian with non-Hermitian boundary impurity
$(\delta+\gamma)c_{1}^{\dagger}c_{N}+(\delta-\gamma)c_{N}^{\dagger}c_{1}$. The
transfer matrix of Eq. (104) becomes

$\displaystyle T_{it}=\left(\begin{matrix}\frac{\varepsilon}{t}&-1\\\
1&0\end{matrix}\right),$ (105)

with $\Gamma_{it}=1$ and $\Delta_{it}=\varepsilon/t$, and the physical
condition of $\mathcal{H}_{it}$ is also Eq. (102). In the PT-broken region
$\gamma\in\left[|\delta-t|,\delta+t\right]$, the solutions of Eq. (102) are
the coexistence of real and complex values, and specially at
$\gamma=\gamma_{a}=\sqrt{\delta^{2}-t^{2}}$, the complex solutions are
$\phi=2\pi m/N-i\log{\mu}/N$ with $m=1,2\ldots,N$ and
$\mu=\delta/t+\sqrt{(\delta/t)^{2}-1}$ [54]. The eigenstate concerning energy
$\varepsilon=2t\cos{\phi}$ of Eq. (104) is derived as

$\displaystyle\left(\begin{matrix}\psi_{n+1}\\\
\psi_{n}\end{matrix}\right)=\left[A_{L}(\phi)e^{i(n-1)\phi_{R}}e^{-(n-1)\phi_{I}}+A_{R}(\phi)e^{-i(n-1)\phi_{R}}e^{(n-1)\phi_{I}}\right]\left(\begin{matrix}\psi_{2}\\\
\psi_{1}\end{matrix}\right),\quad n=1,2,\ldots,N-1,$ (106)

where

$\displaystyle A_{L}(\phi)$
$\displaystyle=\frac{T_{it}-e^{-i\phi}\mathbbm{1}}{2i\sin{\phi}},$
$\displaystyle A_{R}(\phi)$
$\displaystyle=\frac{-T_{it}+e^{i\phi}\mathbbm{1}}{2i\sin{\phi}}.$ (107)

The real (complex) solutions of $\phi$ correspond to the ESs (PSF effect) of
Eq. (104), and accordingly correspond to the pure NHSE (HSSF effect) of
$\mathcal{H}_{hn}$ after the inverse of gauge transformation.

##  Appendix E The emergent HSSF modes of NH-SSH model with boundary impurity

Figure 7: (a) The energy spectrum of the Hamiltonian $\bar{\mathcal{H}}_{imp}$
for parameters
$t_{1}=0.1,t_{2}=1,\gamma=0.5,\gamma_{L}=0.3,\gamma_{R}=1.2,N=16$. (b)-(c):
The analytic (red line) and numerical (blue dots) results of the PSF modes
corresponding the blue dots in (a).

We again perform the generalized gauge transformation same as that in Sec.
IV.2 on $\mathcal{H}_{nssh}$ together with boundary impurity, and obtain the
target Hamiltonian $\bar{\mathcal{H}}_{imp}$, the Hamiltonian
$\bar{\mathcal{H}}$ in Sec. IV.2 together with boundary impurity strength
$\bar{\gamma}_{L}=r^{-N}\gamma_{L},\bar{\gamma}_{R}=r^{N}\gamma_{R}$. The
solutions of the emergent PSF modes are Eq. (C) with $\bar{\Gamma}=1$,
$\varphi=\bar{K}_{R}\Phi_{1}$, where

$\displaystyle\Phi_{1}=\left(\begin{matrix}\psi_{1}\\\
\psi_{2N}\end{matrix}\right),\quad\bar{K}_{R}=\left(\begin{matrix}1&0\\\
0&\bar{\gamma}_{R}\end{matrix}\right),$ (108)

and $\psi$ is the numerical $1\times 2N$ eigenvector. The energy spectrum and
the comparison between analytic and numerical results of the PSF modes are
illustrated in Fig. 7 for selected parameters, which expectedly match with
each other. Therefore, the PSF effect of $\bar{\mathcal{H}}_{imp}$ implies the
finite-size HSSF effect of $\mathcal{H}_{nssh}$ with boundary impurity.

## References

  * Gong  _et al._ [2018] Z. Gong, Y. Ashida, K. Kawabata, K. Takasan, S. Higashikawa, and M. Ueda, Topological phases of non-hermitian systems, Phys. Rev. X 8, 031079 (2018).
  * Kawabata  _et al._ [2019a] K. Kawabata, K. Shiozaki, M. Ueda, and M. Sato, Symmetry and topology in non-hermitian physics, Phys. Rev. X 9, 041015 (2019a).
  * Yuto  _et al._ [2020] A. Yuto, G. Zongping, and U. Masahito, Non-hermitian physics, Advances in Physics 69, 249 (2020).
  * Bergholtz  _et al._ [2021] E. J. Bergholtz, J. C. Budich, and F. K. Kunst, Exceptional topology of non-hermitian systems, Rev. Mod. Phys. 93, 015005 (2021).
  * Lee [2016] T. E. Lee, Anomalous edge state in a non-hermitian lattice, Phys. Rev. Lett. 116, 133903 (2016).
  * Leykam  _et al._ [2017] D. Leykam, K. Y. Bliokh, C. Huang, Y. D. Chong, and F. Nori, Edge modes, degeneracies, and topological numbers in non-hermitian systems, Phys. Rev. Lett. 118, 040401 (2017).
  * Shen  _et al._ [2018] H. Shen, B. Zhen, and L. Fu, Topological band theory for non-hermitian hamiltonians, Phys. Rev. Lett. 120, 146402 (2018).
  * Kunst  _et al._ [2018] F. K. Kunst, E. Edvardsson, J. C. Budich, and E. J. Bergholtz, Biorthogonal bulk-boundary correspondence in non-hermitian systems, Phys. Rev. Lett. 121, 026808 (2018).
  * Yao and Wang [2018] S. Yao and Z. Wang, Edge states and topological invariants of non-hermitian systems, Phys. Rev. Lett. 121, 086803 (2018).
  * Song  _et al._ [2019] F. Song, S. Yao, and Z. Wang, Non-hermitian skin effect and chiral damping in open quantum systems, Phys. Rev. Lett. 123, 170401 (2019).
  * Yokomizo and Murakami [2019] K. Yokomizo and S. Murakami, Non-bloch band theory of non-hermitian systems, Phys. Rev. Lett. 123, 066404 (2019).
  * Lee and Thomale [2019] C. H. Lee and R. Thomale, Anatomy of skin modes and topology in non-hermitian systems, Phys. Rev. B 99, 201103 (2019).
  * Longhi [2019] S. Longhi, Probing non-hermitian skin effect and non-bloch phase transitions, Phys. Rev. Research 1, 023013 (2019).
  * Zhang  _et al._ [2020a] K. Zhang, Z. Yang, and C. Fang, Correspondence between winding numbers and skin modes in non-hermitian systems, Phys. Rev. Lett. 125, 126402 (2020a).
  * Okuma  _et al._ [2020] N. Okuma, K. Kawabata, K. Shiozaki, and M. Sato, Topological origin of non-hermitian skin effects, Phys. Rev. Lett. 124, 086801 (2020).
  * Yang  _et al._ [2020a] Z. Yang, K. Zhang, C. Fang, and J. Hu, Non-hermitian bulk-boundary correspondence and auxiliary generalized brillouin zone theory, Phys. Rev. Lett. 125, 226402 (2020a).
  * Lin  _et al._ [2023] R. Lin, T. Tai, L. Li, and C. H. Lee, Topological non-hermitian skin effect, Frontiers of Physics 18, 53605 (2023).
  * Yao  _et al._ [2018] S. Yao, F. Song, and Z. Wang, Non-hermitian chern bands, Phys. Rev. Lett. 121, 136802 (2018).
  * Borgnia  _et al._ [2020] D. S. Borgnia, A. J. Kruchkov, and R.-J. Slager, Non-hermitian boundary modes and topology, Phys. Rev. Lett. 124, 056802 (2020).
  * Xue  _et al._ [2021] W.-T. Xue, M.-R. Li, Y.-M. Hu, F. Song, and Z. Wang, Simple formulas of directional amplification from non-bloch band theory, Phys. Rev. B 103, L241408 (2021).
  * Xue  _et al._ [2022] W.-T. Xue, Y.-M. Hu, F. Song, and Z. Wang, Non-hermitian edge burst, Phys. Rev. Lett. 128, 120401 (2022).
  * Longhi [2022] S. Longhi, Non-hermitian skin effect and self-acceleration, Phys. Rev. B 105, 245143 (2022).
  * Fu and Zhang [2023] Y. Fu and Y. Zhang, Anatomy of open-boundary bulk in multiband non-hermitian systems, Phys. Rev. B 107, 115412 (2023).
  * Liu  _et al._ [2019] T. Liu, Y.-R. Zhang, Q. Ai, Z. Gong, K. Kawabata, M. Ueda, and F. Nori, Second-order topological phases in non-hermitian systems, Phys. Rev. Lett. 122, 076801 (2019).
  * Lee  _et al._ [2019] C. H. Lee, L. Li, and J. Gong, Hybrid higher-order skin-topological modes in nonreciprocal systems, Phys. Rev. Lett. 123, 016805 (2019).
  * Edvardsson  _et al._ [2019] E. Edvardsson, F. K. Kunst, and E. J. Bergholtz, Non-hermitian extensions of higher-order topological phases and their biorthogonal bulk-boundary correspondence, Phys. Rev. B 99, 081302 (2019).
  * Kawabata  _et al._ [2020] K. Kawabata, M. Sato, and K. Shiozaki, Higher-order non-hermitian skin effect, Phys. Rev. B 102, 205118 (2020).
  * Okugawa  _et al._ [2020] R. Okugawa, R. Takahashi, and K. Yokomizo, Second-order topological non-hermitian skin effects, Phys. Rev. B 102, 241202 (2020).
  * Fu  _et al._ [2021] Y. Fu, J. Hu, and S. Wan, Non-hermitian second-order skin and topological modes, Phys. Rev. B 103, 045420 (2021).
  * Yu  _et al._ [2021] Y. Yu, M. Jung, and G. Shvets, Zero-energy corner states in a non-hermitian quadrupole insulator, Phys. Rev. B 103, L041102 (2021).
  * Palacios  _et al._ [2021] L. S. Palacios, S. Tchoumakov, M. Guix, I. Pagonabarraga, S. Sánchez, and A. G. Grushin, Guided accumulation of active particles by topological design of a second-order skin effect, Nature Communications 12, 4691 (2021).
  * Li  _et al._ [2022] Y. Li, C. Liang, C. Wang, C. Lu, and Y.-C. Liu, Gain-loss-induced hybrid skin-topological effect, Phys. Rev. Lett. 128, 223903 (2022).
  * Martinez Alvarez  _et al._ [2018] V. M. Martinez Alvarez, J. E. Barrios Vargas, and L. E. F. Foa Torres, Non-hermitian robust edge states in one dimension: Anomalous localization and eigenspace condensation at exceptional points, Phys. Rev. B 97, 121401 (2018).
  * Kawabata  _et al._ [2019b] K. Kawabata, T. Bessho, and M. Sato, Classification of exceptional points and non-hermitian topological semimetals, Phys. Rev. Lett. 123, 066405 (2019b).
  * Yokomizo and Murakami [2020] K. Yokomizo and S. Murakami, Topological semimetal phase with exceptional points in one-dimensional non-hermitian systems, Phys. Rev. Research 2, 043045 (2020).
  * Yang  _et al._ [2020b] Z. Yang, C.-K. Chiu, C. Fang, and J. Hu, Jones polynomial and knot transitions in hermitian and non-hermitian topological semimetals, Phys. Rev. Lett. 124, 186402 (2020b).
  * Zhang  _et al._ [2020b] Z. Zhang, Z. Yang, and J. Hu, Bulk-boundary correspondence in non-hermitian hopf-link exceptional line semimetals, Phys. Rev. B 102, 045412 (2020b).
  * Yang  _et al._ [2021] Z. Yang, A. P. Schnyder, J. Hu, and C.-K. Chiu, Fermion doubling theorems in two-dimensional non-hermitian systems for fermi points and exceptional points, Phys. Rev. Lett. 126, 086401 (2021).
  * Denner  _et al._ [2021] M. M. Denner, A. Skurativska, F. Schindler, M. H. Fischer, R. Thomale, T. Bzdušek, and T. Neupert, Exceptional topological insulators, Nature Communications 12, 5681 (2021).
  * Fu and Wan [2022] Y. Fu and S. Wan, Degeneracy and defectiveness in non-hermitian systems with open boundary, Phys. Rev. B 105, 075420 (2022).
  * Mandal and Bergholtz [2021] I. Mandal and E. J. Bergholtz, Symmetry and higher-order exceptional points, Phys. Rev. Lett. 127, 186601 (2021).
  * Delplace  _et al._ [2021] P. Delplace, T. Yoshida, and Y. Hatsugai, Symmetry-protected multifold exceptional points and their topological characterization, Phys. Rev. Lett. 127, 186602 (2021).
  * Liu  _et al._ [2021a] T. Liu, J. J. He, Z. Yang, and F. Nori, Higher-order weyl-exceptional-ring semimetals, Phys. Rev. Lett. 127, 196801 (2021a).
  * Stålhammar and Bergholtz [2021] M. Stålhammar and E. J. Bergholtz, Classification of exceptional nodal topologies protected by $\mathcal{PT}$ symmetry, Phys. Rev. B 104, L201104 (2021).
  * Ghorashi  _et al._ [2021a] S. A. A. Ghorashi, T. Li, M. Sato, and T. L. Hughes, Non-hermitian higher-order dirac semimetals, Phys. Rev. B 104, L161116 (2021a).
  * Ghorashi  _et al._ [2021b] S. A. A. Ghorashi, T. Li, and M. Sato, Non-hermitian higher-order weyl semimetals, Phys. Rev. B 104, L161117 (2021b).
  * Zhang  _et al._ [2022] K. Zhang, Z. Yang, and C. Fang, Universal non-hermitian skin effect in two and higher dimensions, Nature Communications 13, 2496 (2022).
  * Wang  _et al._ [2022] H.-Y. Wang, F. Song, and Z. Wang, Amoeba formulation of the non-hermitian skin effect in higher dimensions (2022), arXiv:2212.11743 [cond-mat.mes-hall] .
  * Yokomizo and Murakami [2023] K. Yokomizo and S. Murakami, Non-bloch bands in two-dimensional non-hermitian systems, Phys. Rev. B 107, 195112 (2023).
  * Hu [2023] H. Hu, Non-hermitian band theory in all dimensions: uniform spectra and skin effect (2023), arXiv:2306.12022 [cond-mat.mes-hall] .
  * Li  _et al._ [2020] L. Li, C. H. Lee, S. Mu, and J. Gong, Critical non-hermitian skin effect, Nature Communications 11, 5491 (2020).
  * Yokomizo and Murakami [2021] K. Yokomizo and S. Murakami, Scaling rule for the critical non-hermitian skin effect, Phys. Rev. B 104, 165117 (2021).
  * Li  _et al._ [2021] L. Li, C. H. Lee, and J. Gong, Impurity induced scale-free localization, Communications Physics 4, 42 (2021).
  * Guo  _et al._ [2023] C.-X. Guo, X. Wang, H. Hu, and S. Chen, Accumulation of scale-free localized states induced by local non-hermiticity, Phys. Rev. B 107, 134121 (2023).
  * Li  _et al._ [2023] B. Li, H.-R. Wang, F. Song, and Z. Wang, Scale-free localization and pt symmetry breaking from local non-hermiticity (2023), arXiv:2302.04256 [quant-ph] .
  * Molignini  _et al._ [2023] P. Molignini, O. Arandes, and E. J. Bergholtz, Anomalous skin effects in disordered systems with a single non-hermitian impurity (2023), arXiv:2302.09081 [cond-mat.dis-nn] .
  * Wang  _et al._ [2023] H.-R. Wang, B. Li, F. Song, and Z. Wang, Scale-free non-hermitian skin effect in a boundary-dissipated spin chain (2023), arXiv:2301.11896 [quant-ph] .
  * Zhu  _et al._ [2014] B. Zhu, R. Lü, and S. Chen, $\mathcal{PT}$ symmetry in the non-hermitian su-schrieffer-heeger model with complex boundary potentials, Phys. Rev. A 89, 062102 (2014).
  * Dangel  _et al._ [2018] F. Dangel, M. Wagner, H. Cartarius, J. Main, and G. Wunner, Topological invariants in dissipative extensions of the su-schrieffer-heeger model, Phys. Rev. A 98, 013628 (2018).
  * Liu and Chen [2020] Y. Liu and S. Chen, Diagnosis of bulk phase diagram of nonreciprocal topological lattices by impurity modes, Phys. Rev. B 102, 075404 (2020).
  * Yoshimura  _et al._ [2020] T. Yoshimura, K. Bidzhiev, and H. Saleur, Non-hermitian quantum impurity systems in and out of equilibrium: Noninteracting case, Phys. Rev. B 102, 125124 (2020).
  * Liu  _et al._ [2021b] Y. Liu, Y. Zeng, L. Li, and S. Chen, Exact solution of the single impurity problem in nonreciprocal lattices: Impurity-induced size-dependent non-hermitian skin effect, Phys. Rev. B 104, 085401 (2021b).
  * Koch and Budich [2020] R. Koch and J. C. Budich, Bulk-boundary correspondence in non-hermitian systems: stability analysis for generalized boundary conditions, The European Physical Journal D 74, 70 (2020).
  * Guo  _et al._ [2021] C.-X. Guo, C.-H. Liu, X.-M. Zhao, Y. Liu, and S. Chen, Exact solution of non-hermitian systems with generalized boundary conditions: Size-dependent boundary effect and fragility of the skin effect, Phys. Rev. Lett. 127, 116801 (2021).
  * Wu  _et al._ [2023] D. Wu, J. Chen, W. Su, R. Wang, B. Wang, and D. Y. Xing, Effective impurity behavior emergent from non-hermitian proximity effect, Communications Physics 6, 160 (2023).
  * Lee and Joannopoulos [1981] D. H. Lee and J. D. Joannopoulos, Simple scheme for surface-band calculations. i, Phys. Rev. B 23, 4988 (1981).
  * Pichard and Sarma [1981] J. L. Pichard and G. Sarma, Finite size scaling approach to anderson localisation, Journal of Physics C: Solid State Physics 14, L127 (1981).
  * Hatsugai [1993a] Y. Hatsugai, Chern number and edge states in the integer quantum hall effect, Phys. Rev. Lett. 71, 3697 (1993a).
  * Hatsugai [1993b] Y. Hatsugai, Edge states in the integer quantum hall effect and the riemann surface of the bloch function, Phys. Rev. B 48, 11851 (1993b).
  * Molinari [1998] L. Molinari, Transfer matrices, non-hermitian hamiltonians and resolvents: some spectral identities, Journal of Physics A: Mathematical and General 31, 8553 (1998).
  * Tauber and Delplace [2015] C. Tauber and P. Delplace, Topological edge states in two-gap unitary systems: a transfer matrix approach, New Journal of Physics 17, 115008 (2015).
  * Dwivedi and Chua [2016] V. Dwivedi and V. Chua, Of bulk and boundaries: Generalized transfer matrices for tight-binding models, Phys. Rev. B 93, 134304 (2016).
  * Kunst and Dwivedi [2019] F. K. Kunst and V. Dwivedi, Non-hermitian systems and topology: A transfer-matrix perspective, Phys. Rev. B 99, 245116 (2019).
  * Luo  _et al._ [2021] X. Luo, T. Ohtsuki, and R. Shindou, Transfer matrix study of the anderson transition in non-hermitian systems, Phys. Rev. B 104, 104203 (2021).
  * Ghaemi-Dizicheh [2023] H. Ghaemi-Dizicheh, Transport effects in non-hermitian nonreciprocal systems: General approach, Phys. Rev. B 107, 125155 (2023).
  * Hatano and Nelson [1996] N. Hatano and D. R. Nelson, Localization transitions in non-hermitian quantum mechanics, Phys. Rev. Lett. 77, 570 (1996).
  * Hatano and Nelson [1997] N. Hatano and D. R. Nelson, Vortex pinning and non-hermitian quantum mechanics, Phys. Rev. B 56, 8651 (1997).
  * Hatano and Nelson [1998] N. Hatano and D. R. Nelson, Non-hermitian delocalization and eigenfunctions, Phys. Rev. B 58, 8384 (1998).
  * Note [1] This setup is adequate for the core physics in the current paper; also, we can always ensure the nilpotence of $\boldsymbol{J}$ by bundling a sufficient number of unit cells.
  * Lay  _et al._ [2016] D. C. Lay, S. R. Lay, and J. J. McDonald,  _Linear algebra and its applications_ , 5th ed. (Pearson Education, Inc., United States of America, 2016).
  * Strang [2023] G. Strang,  _Introduction to Linear Algebra_ , 6th ed. (CUP, Cambridge, 2023).
  * Bateman [1953] H. Bateman,  _Higher transcendental functions: Vol 2_ , 381193 (McGraw-Hill, New York, 1953).
  * Abramowitz  _et al._ [1965] M. Abramowitz, I. A. Stegun, and D. M. Miller, Handbook of mathematical functions with formulas, graphs and mathematical tables (national bureau of standards applied mathematics series no. 55), Journal of Applied Mechanics 32, 239 (1965).
  * Note [2] A more general impurity is still manageable with increased complexity in the calculations.
  * Note [3] Here, we assume all physical energies corresponding to $\Gamma\neq 0$, that is, neglect the extremely special situations where exist some discrete energies leading to $\Gamma=0$.
  * Note [4] The LRA-region may contain the extreme modes with $\text{Re}\left(\log{c_{m}}\right)=0$, which are not our concerns in this paper.
  * Note [5] We uniformly label the coordinate axis of the numerical energy spectrum (eigenstates) as $(\text{Re}(\text{E}),\text{Im}(\text{E}))$ [($\text{site},|\psi|$)] for simplicity throughout this paper.
  * Longhi [2021] S. Longhi, Spectral deformations in non-hermitian lattices with disorder and skin effect: A solvable model, Phys. Rev. B 103, 144202 (2021).
  * Okuma and Sato [2021] N. Okuma and M. Sato, Non-hermitian skin effects in hermitian correlated or disordered systems: Quantities sensitive or insensitive to boundary effects and pseudo-quantum-number, Phys. Rev. Lett. 126, 176601 (2021).
  * Yuce and Ramezani [2022] C. Yuce and H. Ramezani, Coexistence of extended and localized states in the one-dimensional non-hermitian anderson model, Phys. Rev. B 106, 024202 (2022).
  * Sarkar  _et al._ [2022] R. Sarkar, S. S. Hegde, and A. Narayan, Interplay of disorder and point-gap topology: Chiral modes, localization, and non-hermitian anderson skin effect in one dimension, Phys. Rev. B 106, 014207 (2022).