# Enhancing Markov and Chebyshev’s inequalities

Joan del Castillo  Departament de Matemàtiques, Universitat Autònoma de
Barcelona, Cerdanyola del Vallès, Spain<EMAIL_ADDRESS>
###### Abstract.

The idea of the restricted mean has been used to establish a significantly
improved version of Markov’s inequality that does not require any new
assumptions. The result immediately extends on Chebyshev’s inequalities and
Chernoff’s bound. The improved Markov inequality yields a bound that is
hundreds or thousands of times more accurate than the original Markov bound
for high quantiles in the most prevalent and diverse situations.

The Markov inequality benefits from being model-independent, and the long-
standing issue of its imprecision is solved. Practically speaking, avoidance
of model risk is decisive when multiple competing models are present in a
real-world situation.

###### Key words and phrases:

Restricted mean, Inequalities in Probability, Concentration Inequalities.

##  1\. Introduction

Historically, Markov and Chebyshev’s inequalities, which provide proof of the
weak law of large numbers, date back to the articles by Chebyshev (1867) and
Markov (1884). Today’s most basic version of this inequality carries Markov’s
name. Chebyshev was undoubtedly aware of it as well, and Markov was a student
of his. These inequalities have survived to our day in most undergraduate
probability textbooks, for instance, Ross (2014) and Casella and Berger
(2002).

Markov’s inequality has the benefit of being model-independent from a
practical standpoint. When numerous competing models are provided in relation
to a real-world problem, the elimination of model risk is crucial. The first
problem with inequality is its lack of accuracy. Recently, several researchers
have taken an interest in the idea of updating these classic inequalities in
relation to current problems, see Satyanarayana (2014), Cohen (2015), Huber
(2019), Ogasawara (2020), Ren et al. (2021), Solari and Djordjilović (2022),
Bhat and Kosuru (2022), Choi and Kim (2023).

Theorem 1 provides an enhanced Markov inequality that is hundreds or thousands
of times more accurate than the original for high quantiles and does not
require any new assumptions, see Figure 3.1 and Table 1. The result
immediately extends Chebyshev’s inequalities and Chernoff’s bound (Chernoff,
1952), see Corollary 2. The proof is practically as simple as that of the
traditional Markov inequality, using the concept of restricted expectation.
This idea is used, for example, in the Lorenz curve (Lorenz, 1905), which
measures how wealth is distributed in a society, see Arnold (2015), and in
medicine as the restricted mean survival time, see Royston and Parmar (2013).

##  2\. Improving the accuracy of Markov’s inequality

According to Markov’s inequality, see Ross (2014), the probability of a
nonnegative random variable $X$ being at least $\nu>0$ is at most the
expectation of $X$ divided by $\nu$. That is

(2.1) $\Pr\left\\{X>\nu\right\\}\leq E\left(X\right)/\nu.$

Chebyshev’s inequality, an extended version for a nondecreasing nonnegative
function, $\varphi$, is as follows

$\Pr\left\\{X>\nu\right\\}\leq
E(\varphi\left(X\right))/\varphi\left(\nu\right),$

The proof of this extended version is as simple as applying Markov (2.1) to
the second part of the equality:

(2.2)
$\Pr\left\\{X>\nu\right\\}=\Pr\left\\{\varphi\left(X\right)>\varphi\left(\nu\right)\right\\}.$

The proof of the weak law of large numbers follows from Chebyshev’s inequality
for the variable$\left|X-E\left(X\right)\right|$ and
$\varphi\left(x\right)=x^{2}$, when $X$ has a finite variance, see Casella and
Berger (2002). Other immediate inequalities follow using higher moments,
$\varphi\left(X\right)=X^{k}$, and using the moment-generating function,
$\varphi\left(X\right)=exp\left(tX\right)$, which led to Chernoff’s bound
(1952).

A typical proof of Markov’s inequality makes use of the indicator random
variable of the event $\left\\{X>\nu\right\\}$

$1_{\left\\{X>\nu\right\\}}=\left\\{\begin{array}[]{cc}0,\;when&X\leq\nu\\\
1,\;when&X>\nu\end{array}\right.$

and the inequality

(2.3) $\nu 1_{\left\\{X>\nu\right\\}}\leq X$

which is clear if we consider the two possible values of _X._ If $0\leq
X\leq\nu$, $\nu 1_{\left\\{X>\nu\right\\}}=0$ and if $X>\nu$, $\nu
1_{\left\\{X>\nu\right\\}}=\nu$. Then, taking the expectation of both sides of
the inequality (2.3), Markov inequality (2.1) follows, since
$E\left(1_{\left\\{X>\nu\right\\}}\right)=\Pr\left\\{X>\nu\right\\}$ and
expectation is a monotonically increasing operator.

Let $X$ be a nonnegative random variable with cumulative distribution function
$F\left(x\right)$. For clarity, it is now assumed that $X$ has a probability
density function $f\left(x\right)$, so its expectation is expressed as

$E\left(X\right)=\int_{0}^{\infty}x\,f(x)\,dx,$

For $\nu\geq 0$, the _restricted expectation_ of $X$ over $\nu$ is introduced
as follows

(2.4)
$E_{\nu}\left(X\right)=E\left(X1_{\left\\{X>\nu\right\\}}\right)=\int_{\nu}^{\infty}x\,f(x)\,dx.$

Naturally, if $\nu=0$, $E_{0}\left(X\right)=E\left(X\right)$. This concept is
employed, for instance, in the Lorenz curve (Arnold, 2015), and in medicine as
the restricted mean survival time (Royston and Parmar, 2013).

The following is the paper’s primary finding.

######  Theorem 1.

If $X$ is a nonnegative random variable and $\nu>0$, then the probability of
$X$ being at least $\nu$ is at most the restricted expectation of X over $\nu$
divided by $\nu$. This improves on Markov’s inequality.

$\Pr\left\\{X>\nu\right\\}\leq E_{\nu}\left(X\right)/\nu\leq
E\left(X\right)/\nu.$

###### Proof.

Multiplying both sides of the inequality (2.3) by the nonnegative random
variable $1_{\left\\{X>\nu\right\\}}$, bearing in mind that
$\left(1_{\left\\{X>\nu\right\\}}\right)^{2}=1_{\left\\{X>\nu\right\\}}$, the
following inequality follows

(2.5) $\nu 1_{\left\\{X>\nu\right\\}}\leq X1_{\left\\{X>\nu\right\\}},$

Then taking the expectation of both sides of the above inequality, the main
part of the result follows. Moreover, it is clear that
$X1_{\left\\{X>\nu\right\\}}\leq X$ and taking the expectation on both sides
it is demonstrated that the new inequality improves on Markov’s inequality.

Note that with a measure-theoretic definition of expectation, the probability
density function is not necessary and the result is as general as traditional
Markov inequality. In particular for discrete random variables, this
conclusion is true. ∎

The three terms of the inequalities of Theorem 1 will henceforth be called the
tail function _,_
$\Pr\left\\{X>\nu\right\\}=(1-F(x))=\overline{F}\left(x\right)$_,_ traditional
Markov bound, $E\left(X\right)/\nu$, and enhanced Markov bound,
$E_{\nu}\left(X\right)/\nu$. Enhanced Markov inequality will from now on be
used to refer to the first inequality of Theorem 1.

Theorem 1 for Markov’s inequality instantly generalizes Chebyshev and
Chernoff’s inequalities under the restricted expectation, as the following
result shows.

######  Corollary 2.

If $X$ is a nonnegative random variable and $\nu>0$, then for a nondecreasing
nonnegative function $\varphi$

$\Pr\left\\{X>\nu\right\\}\leq
E_{\nu}(\varphi\left(X\right))/\varphi\left(\nu\right)\leq
E(\varphi\left(X\right))/\varphi\left(\nu\right),$

###### Proof.

We simply need to apply Theorem 1 to the variable $\varphi\left(X\right)$ in
the second part of equality (2.2). ∎

For instance, using higher moments, with $\varphi\left(X\right)=X^{k}$, the
previous Corollary leads to the following conclusion.

(2.6) $\Pr\left\\{X>\nu\right\\}\leq E_{\nu}(X^{k})/\nu^{k}\leq
E(X^{k})/\nu^{k},$

Using the moment generating function, with
$\varphi\left(X\right)=exp\left(tX\right)$, the Chernoff bound is also
improved

(2.7) $\Pr\left\\{X>\nu\right\\}\leq E_{\nu}(exp\left(tX\right))e^{-t\nu}\leq
E(exp\left(tX\right))e^{-t\nu}$

Below, the results of Theorem 1 will be checked in different examples to show
the extraordinary improvement they represent. It will also be shown how the
restricted expectation can be computed just as easily as the general
expectation. We believe that the large increase in accuracy will lead to
significant practical applications.

##  3\. Examples

In order to confirm the assertion expressed in the Introduction, we offer some
examples to go along with the Section 2 results. For the half-normal and
exponential distributions across high quantiles, we will compare the enhanced
and traditional Markov bounds.

In general, for a sample of size $n$, we shall take into account the $(1-1/n)$
quantile, which is typically a little below the sample maximum. The improved
Markov bound is often about $n$ times more accurate than the traditional bound
when we consider samples larger than $100$.

Figure 3.1 shows a preliminary idea of how the enhanced Markov bound and the
traditional Markov bound approximate the tail function in the half-normal and
exponential distributions with expected value 1.

Figure 3.1. The tail function (thick), the enhanced Markov bound (thin) for
half-normal (blue) and exponential (red) distributions, both with expected
value 1. The traditional Markov bound (dashed, blue) is the same in both
cases.

######  Example 3.

The probability density function of the standard exponential distribution,
$X$, is

$f\left(x\right)=e^{-x},\;x\geq 0.$

The cumulative distribution function is given by

$F\left(x\right)=1-e^{-x},\;x\geq 0.$

The restricted expectation is straightforward to compute using integration by
parts

$E_{\nu}\left(X\right)=\int_{\nu}^{\infty}x\,e^{-x}\,dx=e^{-\nu}\left(1+\nu\right),$

and if $\nu=0$, the expectation is $E\left(X\right)=1$.

The $0.99$ and $0.999$ quantiles for $X$ are $4.605$ and $6.908$. The three
terms of the inequalities of Theorem 1 for $\nu=6.908$ are: the tail function
$1.00\,10^{-3}$, the enhanced __ Markov bound $1.14\,10^{-3}$ and the
traditional Markov bound $0.145$. Simply by looking at the orders of
magnitude, the difference is obvious, for it is about a thousand times better.
See also Table 1.

Table 1. Comparison between the adjustments of the tail function by the enhanced Markov bound, $E_{\nu}\left(X\right)/\nu$, and traditional Markov bound, $E\left(X\right)/\nu$, for the half-normal and exponential distributions with expected value $1$. | Half normal | Exponential
---|---|---
$\nu$ | $1-F(\nu)$ | $E_{\nu}\left(X\right)/\nu$ | $E\left(X\right)/\nu$ | $1-F(\nu)$ | $E_{\nu}\left(X\right)/\nu$ | $E\left(X\right)/\nu$
1 | 0.425 | 0.727 | 1.000 | 0.368 | 0.736 | 1.000
2 | 0.111 | 0.140 | 0.500 | 0.135 | 0.203 | 0.500
3 | 0.017 | 0.019 | 0.333 | 0.050 | 0.066 | 0.333
4 | 1.4E-03 | 1.5E-03 | 0.250 | 0.018 | 0.023 | 0.250
5 | 6.6E-05 | 7.0E-05 | 0.200 | 6.7E-03 | 8.1E-03 | 0.200
6 | 1.7E-06 | 1.8E-06 | 0.167 | 2.5E-03 | 2.9E-03 | 0.167
7 | 2.3E-08 | 2.4E-08 | 0.143 | 9.1E-04 | 1.0E-03 | 0.143
8 | 1.7E-10 | 1.8E-10 | 0.125 | 3.4E-04 | 3.8E-04 | 0.125

If $X$ is normally distributed with zero mean, $N\left(0,\sigma^{2}\right)$,
then its absolute value$\left|X\right|$ follows a _half-normal_ distribution
with a probability density function

$f_{H}\left(x;\sigma\right)=\frac{2}{\sqrt{2\pi}\sigma}exp\left(-\frac{x^{2}}{2\sigma^{2}}\right),\;x\geq
0.$

The distribution is supported on the interval $[0,\infty)$. If
$\Phi\left(x\right)$ is the cumulative distribution function of the standard
normal distribution, then the cumulative distribution function of the _half-
normal_ distribution is

$F_{H}\left(x;\sigma\right)=2\Phi\left(x/\sigma\right)-1,\;x\geq 0.$

######  Example 4.

The restricted expectation of the half-normal distribution is given by

$E_{\nu}\left(\left|X\right|\right)=\frac{\sqrt{2}}{\sqrt{\pi}\sigma}\int_{\nu}^{\infty}x\,exp\left(-\frac{x^{2}}{2\sigma^{2}}\right)\,dx,$

since

$-\sigma^{2}\frac{d}{dx}\left\\{exp\left(-\frac{x^{2}}{2\sigma^{2}}\right)\right\\}=x\,exp\left(-\frac{x^{2}}{2\sigma^{2}}\right),$

then

$E_{\nu}\left(\left|X\right|\right)=\sigma\sqrt{2/\pi}exp\left(-\frac{\nu^{2}}{2\sigma^{2}}\right),$

in particular $E\left(\left|X\right|\right)=\sigma\sqrt{2/\pi}$.

Then, a half-normal distribution $Z$ where $\sigma=\sqrt{\pi/2}$ has unit
expectation, $E\left(Z\right)=1$, and its restricted expectation is

$E_{\nu}\left(Z\right)=exp\left(-\nu^{2}/\pi\right),$

The $0.99$ and $0.999$ quantiles for $Z$ are $3.228$ and $4.124$. The three
terms of the inequalities of Theorem 1 for $Z$ and $\nu=4.124$ are: the tail
function $1.00\,10^{-3}$, the enhanced __ Markov bound $1.08\,10^{-3}$ and the
traditional Markov bound $0.242$. The new inequality is thousands times
better, see also Table 1.

Table 1 compares the approximations of the tail function by the enhanced and
traditional Markov bounds for the half-normal and exponential distributions
with expected value $1$. The two functions are evaluated up to $\nu=8$, which
is far from the expected value. The comparison between tail function fits does
not depend on the scale parameter. The Table shows that the enhanced Markov
bound follows the tail function very closely, unlike the traditional Markov
bound. The approximations of the enhanced Markov bound follow the tail
function in all orders of magnitude, up to $10^{-10},$ while the traditional
Markov is thousands of times less accurate in both cases for samples larger
than $10^{3}$.

## References

  * Arnold (2015) B. Arnold.  _Pareto distribution_.  Taylor & Francis, Boca Raton, FL, 2015.
  * Bhat and Kosuru (2022) A. Bhat and S. Kosuru.  Generalizations of some concentration inequalities.  _Statistics & Probability Letters_, 182:109298, 2022.
  * Casella and Berger (2002) G. Casella and R. Berger.  _Statistical inference_.  Thomson Learning Inc., Pacific Grove, CA, 2002.
  * Chebyshev (1867) P. Chebyshev.  Des valeurs moyennes.  _Journal de Mathématiques Pures et Appliquées_ , 12:177–184, 1867.
  * Chernoff (1952) H. Chernoff.  A measure of asymptotic efficiency for tests of a hypothesis based on the sum of observations.  _The Annals of Mathematical Statistics_ , pages 493–507, 1952.
  * Choi and Kim (2023) W. Choi and I. Kim.  Averaging p-values under exchangeability.  _Statistics & Probability Letters_, 194:109748, 2023.
  * Cohen (2015) J. Cohen.  Markov’s inequality and chebyshev’s inequality for tail probabilities: A sharper image.  _The American Statistician_ , 69(1):5–7, 2015\.
  * Huber (2019) M. Huber.  Halving the bounds for the markov, chebyshev, and chernoff inequalities using smoothing.  _The American Mathematical Monthly_ , 126(10):915–927, 2019.
  * Lorenz (1905) M. Lorenz.  Methods of measuring the concentration of wealth.  _Publications of the American statistical association_ , 9(70):209–219, 1905.
  * Markov (1884) A. Markov.  On certain applications of algebraic continued fractions.  _Unpublished Ph. D. thesis, St Petersburg_ , 1884.
  * Ogasawara (2020) H. Ogasawara.  The multivariate markov and multiple chebyshev inequalities.  _Communications in Statistics-Theory and Methods_ , 49(2):441–453, 2020.
  * Ren et al. (2021) Z. Ren, Y. Wei, and E. Candès.  Derandomizing knockoffs.  _Journal of the American Statistical Association_ , pages 1–11, 2021\.
  * Ross (2014) S. Ross.  _Introduction to probability models_.  Academic press, Burlington, MA, 2014.
  * Royston and Parmar (2013) P. Royston and M. Parmar.  Restricted mean survival time: an alternative to the hazard ratio for the design and analysis of randomized trials with a time-to-event outcome.  _BMC medical research methodology_ , 13(1):1–15, 2013.
  * Satyanarayana (2014) A. Satyanarayana.  Intelligent sampling for big data using bootstrap sampling and chebyshev inequality.  In _2014 IEEE 27th Canadian Conference on Electrical and Computer Engineering (CCECE)_ , pages 1–6. IEEE, 2014.
  * Solari and Djordjilović (2022) A. Solari and V. Djordjilović.  Multi split conformal prediction.  _Statistics & Probability Letters_, 184:109395, 2022.


# Topological delocalization transitions and mobility edges in the
nonreciprocal Maryland model

Longwen Zhou<EMAIL_ADDRESS>College of Physics and Optoelectronic
Engineering, Ocean University of China, Qingdao, China 266100  Yongjian Gu
<EMAIL_ADDRESS>College of Physics and Optoelectronic Engineering, Ocean
University of China, Qingdao, China 266100

###### Abstract

Non-Hermitian effects could trigger spectrum, localization and topological
phase transitions in quasiperiodic lattices. We propose a non-Hermitian
extension of the Maryland model, which forms a paradigm in the study of
localization and quantum chaos by introducing asymmetry to its hopping
amplitudes. The resulting nonreciprocal Maryland model is found to possess a
real-to-complex spectrum transition at a finite amount of hopping asymmetry,
through which it changes from a localized phase to a mobility edge phase.
Explicit expressions of the complex energy dispersions, phase boundaries and
mobility edges are found. A topological winding number is further introduced
to characterize the transition between different phases. Our work introduces a
unique type of non-Hermitian quasicrystal, which admits exactly obtainable
phase diagrams, mobility edges, and holding no extended phases at finite
nonreciprocity in the thermodynamic limit.

††preprint: APS/123-QED

##  I Introduction

Non-Hermitian topological matter has attracted great attention in the past
decade (see NHRev1 ; NHRev2 ; NHRev3 ; NHRev4 ; NHRev5 ; NHRev6  for reviews).
Theoretically, these unique phases have been classified according to their
symmetries and exceptional topologies NHClass1 ; NHClass2 ; NHClass3 ;
NHClass4 ; NHClass5 ; NHClass6 . Experimentally, non-Hermitian topological
phases have been engineered in a broad range of physical platforms NHExp1 ;
NHExp2 ; NHExp3 ; NHExp4 ; NHExp5 ; NHExp6 ; NHExp7 ; NHExp8 ; NHExp9 ;
NHExp10 ; NHExp11 ; NHExp12 ; NHExp13 ; NHExp14 , yielding potential
applications such as topological lasers TILZ1 ; TILZ2 ; TILZ3  and high-
performance sensors NHSens1 ; NHSens2 ; NHSens3 ; NHSens4 .

The localization problem in non-Hermitian systems was introduced early on by
Hatano and Nelson HNM1 , and followed up by a series of related studies HNM2 ;
HNM3 ; HNM4 ; HNM5 . In the Hatano-Nelson model, the asymmetry in the hopping
amplitude of the one-dimensional Anderson model was found to be able to induce
a transition from an insulator to a metallic phase. Recently, the development
of non-Hermitian topological matter has re-evoked the interest in non-
Hermitian systems with spatial aperiodicity. Specially, the interplay between
non-Hermitian effects and quasiperiodic modulations has been found to induce
rich patterns of localization-delocalization transitions, topological phase
transitions and mobility edges in one-dimensional (1D) non-Hermitian
quasicrystals (NHQCs) NHQC1 ; NHQC2 ; NHQC3 ; NHQC4 ; NHQC5 ; NHQC6 ; NHQC7 ;
NHQC8 ; NHQC9 ; NHQC10 ; NHQC11 ; NHQC12 ; NHQC13 ; NHQC14 ; NHQC15 ; NHQC16 ;
NHQC17 ; NHQC18 ; NHQC19 ; NHQC20 ; NHQC21 ; NHQC22 ; NHQC23 ; NHQC24 ; NHQC25
. Typical systems considered in the study of 1D NHQCs include various
extensions of the Aubry-André-Harper NHQC1 ; NHQC2 ; NHQC3 ; NHQC4 ; NHQC5 ,
Fibonacci NHQC17 , Su-Schrieffer-Heeger NHQC21 , Kitaev chain NHQC24  and
Maryland NHQC25  models. Localization and topological transitions induced by
time-periodic driving fields have also been investigated in the context of
Floquet NHQCs NHQC26 . Another way to make a disordered system non-Hermitian
is to couple it to the outside world by a lead, as explored earlier in OpenQC
.

In this work, we introduce a non-Hermitian extension of the quasiperiodic
Maryland model MM1 ; MM2 ; MM3 ; MM4 ; MM5 ; MM6  by adding nonreciprocity to
its hopping amplitudes. The resulting system exhibits two distinct NHQC phases
and a phase transition induced by the hopping asymmetry. We introduce our
model in Sec. II and reveal its key features in Sec. III. Besides
characterizing the real-to-complex spectrum transition and the delocalization
transition from an insulator to a mobility edge phase with coexisting extended
and localized states in Secs. III.1 and III.2, we also introduce a topological
winding number to describe different phases and transitions in the
nonreciprocal Maryland model in Sec. III.3. We summarize our results and
discuss potential future work in Sec. IV.

##  II Model

The Maryland model describes particles hopping in a tight-binding lattice and
subject to an unbounded onsite superlattice potential. Its Hamiltonian in
position representation takes the form $H_{0}=\sum_{n}(J|n\rangle\langle
n+1|+{\rm H.c.}+V\tan(\pi\alpha n)|n\rangle\langle n|)$. Here $J$ is the
nearest-neighbor (NN) hopping amplitude, $V$ is the amplitude of onsite
potential, $n\in\mathbb{Z}$ is the lattice site index and $\\{|n\rangle\\}$
forms a complete basis of the lattice. $\alpha$ is chosen as an irrational
number to realize quasiperiodic modulations. The Hermitian Maryland model
$H_{0}$ is first introduced as an integrable model to study localization
problems in quantum chaotic systems MM1 ; MM2 ; MM3 ; MM4 ; MM5 ; MM6 . Later,
it is also utilized to understand the localization in higher dimensions MM7 ;
MM8 ; MM9  and topological features of integer quantum Hall effects MM10 .
Experimentally, the Maryland model might be realized in photonic systems by
engineering the light propagation in polygonal optical waveguide lattices
MMExp1 .

In this work, we focus on the localization problem in a non-Hermitian
extension of the Maryland model, which is obtained by incorporating asymmetry
into the hopping amplitudes of $H_{0}$. The Hamiltonian of the resulting
system, which is dubbed the nonreciprocal Maryland model (NRMM), takes the
following form

$\displaystyle H$ $\displaystyle=J\sum_{n}(e^{-\gamma}|n\rangle\langle
n+1|+e^{\gamma}|n+1\rangle\langle n|)$ $\displaystyle+V\sum_{n}\tan(\pi\alpha
n)|n\rangle\langle n|.$ (1)

Here $J,V,\gamma\in\mathbb{R}$. $\gamma$ measures the degree of asymmetry
between left-to-right and right-to-left NN hopping amplitudes. We take the
periodic boundary condition (PBC) for all calculations below by identifying
$|n\rangle=|n+L\rangle$, where $n=1,2,...,L$ and $L$ is the length of lattice.
When $\alpha=p/q$ ($p,q$ being coprime integers) is chosen to be a rational
number and $L$ is an integer multiple of $q$, the system is in the
commensurate regime and under the PBC it is expected to hold charge density
wave like states. In this work, we instead take $\alpha=\frac{\sqrt{5}-1}{2}$,
i.e., the inverse golden ratio to yield a quasiperiodic potential. Inserting
the expansion of state $|\psi\rangle=\sum_{n}\psi_{n}|n\rangle$ into the
eigenvalue equation $H|\psi\rangle=E|\psi\rangle$, we obtain

$J(e^{-\gamma}\psi_{n+1}+e^{\gamma}\psi_{n-1})+V\tan(\pi\alpha
n)\psi_{n}=E\psi_{n}.$ (2)

Here $E$ is the energy of state $|\psi\rangle$, which is in general complex as
$H\neq H^{\dagger}$. $\psi_{n}\equiv\langle n|\psi\rangle$ represents the
amplitude of right eigenvector $|\psi\rangle$ on the $n$th lattice site. The
solution of Eq. (2) under the PBC then yields all the possible eigenenergies
and eigenstates of the NRMM. Note that in numerical calculations, we take a
rational approximation for $\alpha$ by setting $\alpha\simeq F_{l}/F_{l+1}$,
where $F_{l}$ and $F_{l+1}$ are two adjacent elements of the Fibonacci
sequence.

In the Hermitian limit ($\gamma=0$), due to the unbounded nature of onsite
potential $V_{n}=V\tan(\pi\alpha n)$, all eigenstates of $H$ are localized
with energy-dependent localization lengths for irrational $\alpha$ and $V\neq
0$. Away from the Hermitian limit, however, we find a nonreciprocity induced
transition of the system from a localized phase with real spectrum to a
mobility edge phase with complex spectrum at a finite $\gamma=\gamma_{c}$,
which is thus of non-Hermitian origin. Note that the mobility edge phase means
a phase in which extended and localized states coexist and are separated in
their energies by a mobility edge. We present systematic characterizations of
this transition and the resulting energy-dependent mobility edges in the
following section.

##  III Results

In this section, we investigate the spectrum, delocalization and topological
transitions of the NRMM. In Sec. III.1, we study the energies of NRMM and find
a real-to-complex spectral transition at a finite hopping asymmetry
$\gamma=\gamma_{c}$, whose expression as a function of the hopping amplitude
and onsite potential is obtained. In Sec. III.2, the spectrum transition is
connected to a transition of the system from localized to mobility edge
phases. The mobility edge separating localized and extended states is further
picked up and its expression is found to be energy-dependent. In Sec. III.3, a
topological winding number is introduced to distinguish phases with different
transport nature and characterize the transitions between them in the NRMM,
thus yielding a complete phase diagram. For ease of reference, we summarize
our main results about the NRMM in Table 1.

Phase | Localized | Mobility edge
---|---|---
Condition | $|2J\sinh\gamma|<|V|$ | $|2J\sinh\gamma|>|V|$
Spectrum | Real | Complex (see Eq. (3))
IPR |  $>0$ for all states |  $>0$ and $\simeq 0$ coexist
Mobility edge equation | $\frac{V^{2}}{(2J\sinh\gamma)^{2}}+\frac{({\rm Re}E)^{2}}{(2J\cosh\gamma)^{2}}=1$
Winding number | $w=\int_{0}^{2\pi}\frac{d\theta}{2\pi i}\partial_{\theta}\ln\det[H(\theta)-E_{0}]=\begin{cases}0&{\rm Localized}\\\ \pm 1&{\rm Mobility\,\,edge}\end{cases}$
Table 1: Summary of results for the NRMM. $J$, $\gamma$ and $V$ control the
hopping amplitude, hopping asymmetry and onsite potential. $E$ is the
eigenenergy. $H(\theta)$ is obtained from $H$ in Eq. (1) by setting
$e^{\pm\gamma}\rightarrow e^{\pm\gamma\pm i\theta/L}$, with $L$ being the
length of lattice. The base energy is set as $E_{0}=0$ in the calculation of
$w$.

###  III.1 Real-to-complex spectrum transition

We first study the spectrum of NRMM by solving the eigenvalue Eq. (2). Two
typical examples of the spectrum are presented in Figs. 1(a) and 1(b), where
${\rm Re}E$ and ${\rm Im}E$ refer to the real and imaginary parts of energy
$E$, respectively. We observe that with weak hopping asymmetry $\gamma$, the
eigenvalues of $H$ could retain real. When $\gamma$ goes beyond a critical
value $\gamma_{c}$, a finite amounts of eigenenergies deviate from the real
axis, and the spectrum undergoes a real-to-complex transition. After the
transition, the complex part of eigenenergies develop a loop on the ${\rm
Re}E$-${\rm Im}E$ plane surrounding a base energy $E_{0}=0$. The points along
the loop in Fig. 1(b) satisfy the equation

$E_{\pm}=2J\cos(\beta-i\gamma)\pm iV,\quad\beta\in[-\pi,\pi)$ (3)

under the constraint ${\rm Im}{E_{-}}>{\rm Im}{E_{+}}$.

Figure 1: Spectrum and DOS of NRMM under the PBC. System parameters are $J=1$,
$\alpha=\frac{\sqrt{5}-1}{2}$ for all panels, and the length of lattice is
$L=377$. (a) and (b) show two typical examples of the spectrum on the complex
plane with $V=1$. (c) and (d) show the maximal imaginary parts of
eigenenergies $\max|{\rm Im}E|$ and DOS with complex energies $\rho$ at
different sets of $(V,\gamma)$. The red dashed lines in (c) and (d) highlight
the boundaries between phases with real and complex spectrum, which are
determined by Eq. (4).

To find the critical point $\gamma_{c}$ as a function of system parameters, we
evaluate the maximum of imaginary parts of energy $\max|{\rm Im}E|$ and the
density of states (DOS) with complex eigenvalues $\rho$ at different $V$ and
$\gamma$, with results presented in Figs. 1(c) and 1(d). In numerical
calculations, we count $E$ as a complex eigenvalue if $|{\rm Im}E|>10^{-5}$.
It is clear that once $V\neq 0$, we could obtain spectrum transitions from
real to complex in the NRMM with the increase of $|\gamma|$. By setting
Im$E_{\pm}=0$ in Eq. (3), we find the critical values of hopping asymmetry
$\gamma=\gamma_{c}$ where spectrum transitions happen, i.e.,

$\gamma_{c}=\pm{\rm arcsinh}\left(\frac{V}{2J}\right).$ (4)

When $|\gamma|<|\gamma_{c}|$, all eigenvalues of $H$ are found to be real,
whereas a finite portion of the spectrum becomes complex for
$|\gamma|>|\gamma_{c}|$. We plot these exact phase boundaries by red dashed
lines in Figs. 1(c) and 1(d), and find that they are coincide with numerical
calculations of the spectrum. Meanwhile, Eq. (4) provides us with a guideline
for the study of transport nature of the NRMM, as will be discussed in the
next subsection.

###  III.2 Delocalization transition and mobility edge

In NHQCs, spectrum transitions usually accompany state transitions regarding
their spatial profiles NHQC1 ; NHQC5 . In the NRMM, we also discover a
transition from an insulator phase with no extended eigenstates to a mobility
edge phase, in which extended and localized eigenstates coexist. To see this,
we first inspect the inverse participation ratio (IPR), which is defined for
the $i$th normalized eigenstate $|\psi^{i}\rangle$ of $H$ in the lattice
representation as ${\rm
IPR}_{i}={\textstyle\sum_{n=1}^{L}}|\psi_{n}^{i}|^{4}$. Here the amplitude
$\psi_{n}^{i}=\langle n|\psi^{i}\rangle$ and $i=1,2,...,L$. In the localized
phase, all eigenstates have finite IPRs. Extended states start to appear when
the minimum of IPRs, denoted as $\min({\rm IPR})$, starts to approach zero.

In Fig. 2(a), we show the minimum, maximum and average of IPRs for the NRMM.
It is clear that the $\max({\rm IPR})\simeq 1$ due to the unbounded nature of
the onsite potential $V_{n}$, which indicates that there are localized states
at any hopping asymmetry in the limit $L\rightarrow\infty$. Notably, the
$\min({\rm IPR})$ decreases to zero when $\gamma$ goes beyond a critical value
$\gamma_{c}$, which happens to be coincide with the critical point of spectrum
transition in Eq. (4). Therefore, when the hopping asymmetry is tuned from
below to above the critical point $\gamma_{c}$, the NRMM transforms from a
localized phase with real spectrum to a mobility edge phase with complex
spectrum. The number of extended states in the mobility edge phase further
increases with the increase of $\gamma$, as hinted by the ${\rm ave}({\rm
IPR})$ in Fig. 2(a). In Fig. 2(b), we present the $\min({\rm IPR})$ as a
function of system parameters $(V,\gamma)$, and indeed observe two distinct
phases characterized by $\min({\rm IPR})>0$ and $\min({\rm IPR})\simeq 0$.
Their boundaries are highlighted by the red dashed lines and given exactly by
Eq. (4). To the best of our knowledge, the NRMM contributes the first example
of a 1D NHQC with only localized and mobility edge phases, but no metallic
phases at any amounts of non-Hermiticity.

Figure 2: IPRs of NRMM under the PBC. The length of lattice is $L=987$. System
parameters are $J=1$, $\alpha=\frac{\sqrt{5}-1}{2}$ for all panels. In (a),
the solid, dashed and dash-dotted lines show the minimum, maximum and average
of IPRs versus the hopping asymmetry $\gamma$ at $V=1$. The crossing point of
the dotted line and the horizontal axis corresponds to the critical value of
$\gamma=\gamma_{c}={\rm asinh}(V/2J)\approx 0.4812$, where the system
undergoes a transition from the localized to the mobility edge phase. (b)
shows the minimum of IPRs at different parameters $(V,\gamma)$. The red dashed
lines refer to the boundaries between localized and mobility edge phases,
which are given by Eq. (4). The IPRs of all states versus the real parts of
their energies and $V$ ($\gamma$) are shown in (c) ((d)) with $\gamma=0.5$
($V=1$). The magenta dashed lines obtained following Eq. (5) are mobility
edges separating extended and localized states.

To give a more detailed look at the mobility edge, we show the IPRs of all
eigenstates of the NRMM versus the real parts of their energies and the
potential amplitude $V$ (hopping asymmetry $\gamma$) for two typical examples
in Figs. 2(c) and 2(d). We observe that the states with ${\rm IPR}\simeq 0$
and ${\rm IPR}>0$ are clearly separated in both figures. Moreover, with
thorough numerical analysis, we find an equation that describes the mobility
edge of the NRMM, i.e.,

$\frac{V^{2}}{(2J\sinh\gamma)^{2}}+\frac{({\rm
Re}E)^{2}}{(2J\cosh\gamma)^{2}}=1.$ (5)

Trajectories determined by this equation, presented by the magenta dashed
lines in Figs. 2(c) and 2(d), separate states with vanishing and finite IPRs
in the energy-parameter plane of the system. Moreover, the Eq. (5) is well-
defined at finite $V$ only if $\gamma\neq 0$. Therefore, mobility edges in the
NRMM are solely originated from non-Hermitian effects encoded in the hopping
asymmetry of the lattice.

###  III.3 Topological invariant and phase diagram

In recent studies, a spectral winding number has been introduced to depict the
transitions between different NHQC phases NHQC1 , following a strategy that is
different from the topological characterization of Hermitian quasicrystals
(see QCRev1  for a review). The general idea is to incorporate a periodic
parameter $\theta$ into the Hamiltonian $H$ of the system, usually achieved by
imposing the twist boundary condition (flux insertion), and then track the
spectral flow of the parameter-dependent Hamiltonian $H(\theta)$ with respect
to a certain base energy on the complex plane during the change of $\theta$
over a cycle. If the spectrum of $H(\theta)$ is real, all eigenvalues collapse
onto the real axis and their winding number $w$ must vanish with respect to
any base energies. If $H(\theta)$ possesses complex energies, their flow with
respect to $\theta$ could form loops around certain points on the complex
plane, and the spectral winding number of $H(\theta)$ around these points
could be nonzero and quantized. The value of $w$ jumps whenever the spectral
of the system changes from real to complex, or vice versa. Therefore, $w$ can
be naturally employed as a topological order parameter to distinguish NHQC
phases with real and complex eigenspectrum. Interestingly, it was found in
some typical NHQC models NHQC1 ; NHQC26  that the real-complex spectral
transition could go together with a localization-delocalization transition
(with a possible exception reported in Ref. NHQC23 ). Therefore, we may adopt
the same winding number to signify the emergence of mobility edge phase in our
system, since the extended states therein are those whose energies possess
nonvanishing imaginary parts.

For the NRMM, we construct a spectral winding number as

$w=\frac{1}{2\pi
i}\int_{0}^{2\pi}\partial_{\theta}\ln\det[H(\theta)-E_{0}]d\theta.$ (6)

Here $H(\theta)$ is obtained from $H$ by letting $e^{\pm\gamma}\rightarrow
e^{\pm\gamma\pm i\theta/L}$ in Eq. (1), with $L$ being the length of lattice.
This winding number counts the number of times the spectrum of $H(\theta)$
winds around the base energy $E_{0}$ when $\theta$ changes over a cycle from
zero to $2\pi$. Referring to the spectrum presented in Fig. 1, we choose
$E_{0}=0$ in our calculation of $w$ without loss of generality. It is clear
that in the Hermitian region ($\gamma=0$) we have $w=0$, since in this case
$H(\theta)$ has only real energies. At a finite hopping asymmetry
$\gamma=\gamma_{c}$, we expect a quantized jump of $w$ from zero to $\pm 1$,
which should be accompanied by the spectrum transition of the NRMM from real
to complex.

Figure 3: Winding numbers of the NRMM. System parameters are $J=1$ and
$\alpha=\frac{\sqrt{5}-1}{2}$. The length of lattice is $L=377$ with the twist
boundary condition. Each region with a uniform color corresponds to a phase
with common spectrum and transport features, whose winding number $w$ is
denoted explicitly therein. The red dashed lines separating different regions
denote the phase boundaries obtained from Eq. (4).

In Fig. 3, we present the winding number $w$ of NRMM versus the onsite
potential $V$ and hopping nonreciprocity $\gamma$, which is obtained directly
from Eq. (6). The red dashed lines in Fig. 3 highlight the exact phase
boundaries, which are given by Eq. (4). We observe that the winding number $w$
takes a quantized change when crossing the border between two NHQC phases, and
remain constant elsewhere. More specially, we find $w=0$ when the system is
prepared in the localized phase with real spectrum, and $w=\pm 1$ if the
system moves into the mobility edge phase with complex spectrum. These
observations confirm that the topological winding number $w$ can indeed be
utilized to discriminate the NHQC phases of NRMM with distinct spectrum and
transport nature, and characterize the transitions between them in the
meantime.

For completeness, we have checked other values of the irrational parameter
$\alpha$ (e.g., $\alpha=(\sqrt{5}+1)/2$, $1/\sqrt{2}$) in our calculations of
spectrum, IPRs and winding numbers of the NRMM. The results we obtained are
all consistent with those reported in Figs. 1–3, which implies that the
conclusion we drew from this section is general to other incommensurate values
of the onsite potential in Eq. (1).

##  IV Summary and discussion

In this work, we uncover non-Hermiticity induced spectral, localization and
topological phase transitions in the nonreciprocal Maryland model. A
transformation of the system from a localized phase with real spectrum to a
mobility edge phase with complex spectrum can be obtained only if there are
finite amounts of hopping asymmetry. The equations satisfied by the complex
spectrum, phase boundaries and energy-dependent mobility edges are found
exactly. A topological order parameter is further employed to build the phase
diagram and characterize transitions between different NHQC phases. Notably,
due to the unbounded nature of onsite potential in the NRMM, we find no
extended phases at any finite hopping nonreciprocity, which is distinct from
the situation found in a Maryland model with complex onsite potential NHQC25 .
Our study thus enriches the family of NHQCs by unveiling a particular type of
system with only localized and mobility edge phases, whose properties can be
characterized exactly.

In future work, it would be interesting to consider many-body effects in
Maryland-type NHQCs and investigate their dynamical properties. The interplay
between non-Hermiticity and quantum chaos in the NRMM can further be studied
following the mapping discussed in Appendix A. In this work, our calculations
are performed under the PBC and the non-Hermitian skin effects (NHSEs) NHSE1 ;
NHSE2 ; NHSE3 ; NHSE4 ; NHSE5  are expected to have no impacts on the results.
Under the open boundary condition, the nonreciprocal hoppings in our model
could possibly induce non-Hermitian skin modes. Recently, it was shown that
(de)localization transitions and (pseudo) mobility edges could even emerge in
clean systems due to the NHSEs, and topological characterizations of these
intriguing phenomena have been proposed PMBZhou . Therefore, it is expected
that in the presence of both NHSEs and spatial quasiperiodicity, richer
patterns of spectrum, localization and topological transitions could appear in
generic NHQCs, which deserve more thorough explorations.

###### Acknowledgements.

L.Z. is supported by the National Natural Science Foundation of China (Grant
No. 11905211), the China Postdoctoral Science Foundation (Grant No.
2019M662444), the Fundamental Research Funds for the Central Universities
(Grant No. 841912009), the Young Talents Project at Ocean University of China
(Grant No. 861801013196), and the Applied Research Project of Postdoctoral
Fellows in Qingdao (Grant No. 861905040009).

##  Appendix A NRMM and Floquet system

The Hermitian Maryland model can be mapped to a mathematically equivalent
Floquet system MM1 . A similar mapping can also be constructed for the NRMM,
which may serve as an entrance for the study of the interplay between non-
Hermiticity and quantum chaos.

We start by rewriting the eigenvalue Eq. (2) as

$\frac{E}{V}\psi_{n}-\frac{e^{-\gamma}}{V}\psi_{n+1}-\frac{e^{\gamma}}{V}\psi_{n-1}=\tan(\pi\alpha
n)\psi_{n},$ (A.1)

where we have set $J=1$ as the unit of energy. Multiplying the imaginary unit
$i$ from both sides of Eq. (A.1), we obtain

$\frac{V\psi_{n}-i(E\psi_{n}-e^{-\gamma}\psi_{n+1}-e^{\gamma}\psi_{n-1})}{V\psi_{n}+i(E\psi_{n}-e^{-\gamma}\psi_{n+1}-e^{\gamma}\psi_{n-1})}=e^{-i2\pi\alpha
n}.$ (A.2)

For a 1D quasicrystal, the amplitude $\psi_{n}$ can be expressed as a
superposition of plane waves NHQC25 , i.e.,

$\psi_{n}=\sum_{\ell}\varphi_{\ell}e^{ik_{\ell}n}.$ (A.3)

Here for any $\ell\neq\ell^{\prime}$, the difference between wave numbers
$k_{\ell}$ and $k_{\ell^{\prime}}$ is an integer multiple of $2\pi\alpha$.
Taking $\psi_{n}$ as amplitudes, we can further construct a series

$\displaystyle\Psi(x)$
$\displaystyle=\sum_{n}\psi_{n}e^{inx}=\sum_{\ell}\varphi_{\ell}\sum_{n}e^{i(x+k_{\ell})n}$
$\displaystyle=2\pi\sum_{\ell,n}\varphi_{\ell}\delta(x+k_{\ell}-2\pi n),$
(A.4)

where we used the Poisson summation formula to arrive at the last equality. If
we now multiply $e^{inx}$ to Eq. (A.2) and take the summation over $n$, we
will obtain with the help of Eq. (A.4) that

$\displaystyle\left[1-i\frac{E}{V}+i\frac{2}{V}\cos(x-i\gamma)\right]\Psi(x)$
(A.5) $\displaystyle=$
$\displaystyle\left[1+i\frac{E}{V}-i\frac{2}{V}\cos(x-2\pi\alpha-i\gamma)\right]\Psi(x-2\pi\alpha).$

To make the connection between the NRMM and its Floquet equivalent more
transparent, we introduce the function

$\Phi(x)=\left[1+i\frac{E}{V}-i\frac{2}{V}\cos(x-i\gamma)\right]\Psi(x).$
(A.6)

Using Eq. (A.6), we can express Eq. (A.5) as

$\Phi(x)=\frac{1+i\frac{E}{V}-i\frac{2}{V}\cos(x-i\gamma)}{1-i\frac{E}{V}+i\frac{2}{V}\cos(x-i\gamma)}\Phi(x-2\pi\alpha).$
(A.7)

Performing the Taylor expansion of $\Phi(x-2\pi\alpha)$, we find

$\Phi(x-2\pi\alpha)=\sum_{n}\frac{(-2\pi\alpha)^{n}}{n!}\frac{\partial^{n}\Phi(x)}{\partial
x^{n}}=e^{-2\pi\alpha\partial_{x}}\Phi(x).$ (A.8)

Meanwhile, we can introduce a function ${\cal K}(x)$ that satisfies

$\displaystyle e^{-i{\cal K}(x)}$ $\displaystyle=\cos[{\cal K}(x)]-i\sin[{\cal
K}(x)]$
$\displaystyle=\frac{1+i\frac{E}{V}-i\frac{2}{V}\cos(x-i\gamma)}{1-i\frac{E}{V}+i\frac{2}{V}\cos(x-i\gamma)},$
(A.9)

yielding

${\cal K}(x)=2\arctan\left[\frac{2}{V}\cos(x-i\gamma)-\frac{E}{V}\right].$
(A.10)

Plugging Eqs. (A.8)–(A.10) into Eq. (A.7), we finally obtain

$\Phi(x)=e^{-i{\cal K}(x)}e^{-2\pi\alpha\partial_{x}}\Phi(x),$ (A.11)

which can be interpreted as describing the one-period evolution of a particle
with linear dispersion in its kinetic energy $-2\pi\alpha i\partial_{x}$ and
subject to a delta-kicking potential ${\cal K}(x)$ within the driving period
$T=1$. The corresponding Schrödinger equation takes the form

$i\partial_{t}\Phi=-2\pi\alpha i\partial_{x}\Phi+{\cal
K}(x)\sum_{m\in{\mathbb{Z}}}\delta(t-m)\Phi,$ (A.12)

and the quasienergy can be viewed as $\varepsilon=0$. The above analysis
establishes a relationship between the spectrum nature of NRMM and the
dynamics of a periodically kicked particle with linearized kinetic energy.

## References

  * (1) R. El-Ganainy, K. G. Makris, M. Khajavikhan, Z. H. Musslimani, S. Rotter, and D. N. Christodoulides, Non-Hermitian physics and PT symmetry, Nat. Phys. 14, 11-19 (2018).
  * (2) V. M. Martinez Alvarez, J. E. Barrios Vargas, M. Berdakin, and L. E. F. Foa Torres, Topological states of non-Hermitian systems, Eur. Phys. J. Spec. Top. 227, 1295 (2018).
  * (3) A. Ghatak and T. Das, New topological invariants in non-Hermitian systems, J. Phys.: Condens. Matter 31, 263001 (2019).
  * (4) C. Coulais, R. Fleury, and J. van Wezel, Topology and broken Hermiticity, Nat. Phys. 17, 9-13 (2021).
  * (5) E. J. Bergholtz, J. C. Budich, and F. K. Kunst, Exceptional topology of non-Hermitian systems, Rev. Mod. Phys. 93, 015005 (2021).
  * (6) Y. Ashida, Z. Gong, and M. Ueda, Non-Hermitian physics, Adv. Phys. 69, 249-435 (2021).
  * (7) H. Shen, B. Zhen, and L. Fu, Topological band theory for non-Hermitian Hamiltonians, Phys. Rev. Lett. 120, 146402 (2018).
  * (8) 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).
  * (9) K. Kawabata, K. Shiozaki, M. Ueda and M. Sato, Symmetry and topology in non-Hermitian physics, Phys. Rev. X 9, 041015 (2019).
  * (10) H. Zhou and J. Y. Lee, Periodic table for topological bands with non-Hermitian symmetries, Phys. Rev. B 99, 235112 (2019).
  * (11) C. C. Wojcik, X. Sun, T. Bzdu$\check{\rm s}$ek, and S. Fan, Homotopy characterization of non-Hermitian Hamiltonians, Phys. Rev. B 101, 205417 (2020).
  * (12) K. Shiozaki and S. Ono, Symmetry indicator in non-Hermitian systems, Phys. Rev. B 104, 035424 (2021).
  * (13) Y. Xu, S.-T. Wang, and L.-M. Duan, Weyl Exceptional Rings in a Three-Dimensional Dissipative Cold Atomic Gas, Phys. Rev. Lett. 118, 045701 (2017).
  * (14) J. Li, A. K. Harter, J. Liu, L. d. Melo, Y. N. Joglekar, and L. Luo, Observation of parity-time symmetry breaking transitions in a dissipative Floquet system of ultracold atoms, Nat. Commun. 10, 855 (2019).
  * (15) W. Gou, T. Chen, D. Xie, T. Xiao, T.-S. Deng, B. Gadway, W. Yi, and B. Yan, Tunable Nonreciprocal Quantum Transport through a Dissipative Aharonov-Bohm Ring in Ultracold Atoms, Phys. Rev. Lett. 124, 070402 (2020).
  * (16) J. M. Zeuner, M. C. Rechtsman, Y. Plotnik, Y. Lumer, S. Nolte, M. S. Rudner, M. Segev, and A. Szameit, Observation of a Topological Transition in the Bulk of a Non-Hermitian System. Phys. Rev. Lett. 115, 040402 (2015).
  * (17) S. Weimann, M. Kremer, Y. Plotnik, Y. Lumer, S. Nolte, K. G. Makris, M. Segev, M. C. Rechtsman, and A. Szameit, Topologically protected bound states in photonic parity-time-symmetric crystals, Nat. Mater. 16, 433-438 (2017).
  * (18) K. Wang, X. Qiu, L. Xiao, X. Zhan, Z. Bian, B. C. Sanders, W. Yi, and P. Xue, Observation of emergent momentum-time skyrmions in parity-time-symmetric non-unitary quench dynamics, Nat. Commun. 10, 2293 (2019).
  * (19) L. Xiao, T. Deng, K. Wang, G. Zhu, Z. Wang, W. Yi, and P. Xue, Non-Hermitian bulk-boundary correspondence in quantum dynamics, Nat. Phys. 16, 761-766 (2020).
  * (20) T. Hofmann, T. Helbig, F. Schindler, N. Salgo, M. Brzezińska, M. Greiter, T. Kiessling, D. Wolf, A. Vollhardt, A. Kaba$\check{\rm s}$i, C. H. Lee, A. Bilu$\check{\rm s}$ić, R. Thomale, and T. Neupert, Reciprocal skin effect and its realization in a topolectrical circuit, Phys. Rev. Res. 2, 023265 (2020).
  * (21) T. Helbig, T. Hofmann, S. Imhof, M. Abdelghany, T. Kiessling, L. W. Molenkamp, C. H. Lee, A. Szameit, M. Greiter, and R. Thomale, Generalized bulk-boundary correspondence in non-Hermitian topolectrical circuits, Nat. Phys. 16, 747-750 (2020).
  * (22) S. Liu, S. Ma, C. Yang, L. Zhang, W. Gao, Y. J. Xiang, T. J. Cui, and S. Zhang, Gain- and Loss-Induced Topological Insulating Phase in a Non-Hermitian Electrical Circuit, Phys. Rev. Appl. 13, 014047 (2020).
  * (23) W. Zhu, X. Fang, D. Li, Y. Sun, Y. Li, Y. Jing, and H. Chen, Simultaneous Observation of a Topological Edge State and Exceptional Point in an Open and Non-Hermitian Acoustic System, Phys. Rev. Lett. 121, 124501 (2018).
  * (24) C. Shen, J. Li, X. Peng, and S. A. Cummer, Synthetic exceptional points and unidirectional zero reflection in non-Hermitian acoustic systems, Phys. Rev. Mater. 2, 125203 (2018).
  * (25) H. Gao, H. Xue, Q. Wang, Z. Gu, T. Liu, J. Zhu, and B. Zhang, Observation of topological edge states induced solely by non-Hermiticity in an acoustic crystal, Phys. Rev. B 101, 180303(R) (2020).
  * (26) Y. Wu, W. Liu, J. Geng, X. Song, X. Ye, C.-K. Duan, X. Rong, and J. Du, Observation of parity-time symmetry breaking in a single-spin system, Science 364, 878-880 (2019).
  * (27) G. Harari, M. A. Bandres, Y. Lumer, M. C. Rechtsman, Y. D. Chong, M. Khajavikhan, D. N. Christodoulides, and M. Segev, Topological insulator laser: Theory. Science 359, 4003 (2018).
  * (28) M. A. Bandres, S. Wittek, G. Harari, M. Parto, J. Ren, M. Segev, D. N. Christodoulides, and M. Khajavikhan, Topological insulator laser: Experiments. Science 359, 4005 (2018).
  * (29) Y. V. Kartashov and D. V. Skryabin, Two-Dimensional Topological Polariton Laser, Phys. Rev. Lett. 122, 083902 (2019).
  * (30) J. Wiersig, Enhancing the Sensitivity of Frequency and Energy Splitting Detection by Using Exceptional Points: Application to Microcavity Sensors for Single-Particle Detection, Phys. Rev. Lett. 112, 203901 (2014).
  * (31) H. Hodaei, A. U. Hassan, S. Wittek, H. Garcia-Gracia, R. El-Ganainy, D. N. Christodoulides, and M. Khajavikhan, Enhanced sensitivity at higher-order exceptional points, Nature 548, 187-191 (2017).
  * (32) W. Chen, S. K. Özdemir, G. Zhao, J. Wiersig, and L. Yang, Exceptional points enhance sensing in an optical microcavity, Nature 548, 192-196 (2017).
  * (33) H.-K. Lau and A. A. Clerk, Fundamental limits and non-reciprocal approaches in non-Hermitian quantum sensing, Nat. Commun. 9, 4320 (2018).
  * (34) N. Hatano and D. R. Nelson, Localization Transitions in Non-Hermitian Quantum Mechanics, Phys. Rev. Lett. 77, 570 (1996).
  * (35) N. Hatano and D. R. Nelson, Vortex pinning and non-Hermitian quantum mechanics, Phys. Rev. B 56, 8651 (1997).
  * (36) J. Feinberg and A. Zee, Non-Hermitian localization and delocalization, Phys. Rev. E 59, 6433 (1999).
  * (37) J. Feinberg and A. Zee, Spectral curves of non-hermitian hamiltonians, Nucl. Phys. B 552, 599-623 (1999).
  * (38) N. Hatano and J. Feinberg, Chebyshev-polynomial expansion of the localization length of Hermitian and non-Hermitian random chains, Phys. Rev. E 94, 063305 (2016).
  * (39) A. Jazaeri and I. I. Satija, Localization transition in incommensurate non-Hermitian systems, Phys. Rev. E 63, 036222 (2001).
  * (40) Q. Zeng, S. Chen, and R. Lü, Anderson localization in the non-Hermitian Aubry-André-Harper model with physical gain and loss, Phys. Rev. A 95, 062118 (2017).
  * (41) S. Longhi, Topological Phase Transition in non-Hermitian Quasicrystals, Phys. Rev. Lett. 122, 237601 (2019).
  * (42) H. Jiang, L. Lang, C. Yang, S. Zhu, and S. Chen, Interplay of non-Hermitian skin effects and Anderson localization in nonreciprocal quasiperiodic lattices, Phys. Rev. B 100, 054301 (2019).
  * (43) S. Longhi, Metal-insulator phase transition in a non-Hermitian Aubry-André-Harper model, Phys. Rev. B 100, 125157 (2019).
  * (44) T. Liu, H. Guo, Y. Pu, and S. Longhi, Generalized Aubry-André self-duality and mobility edges in non-Hermitian quasiperiodic lattices, Phys. Rev. B 102, 024205 (2020).
  * (45) Y. Liu, X.-P. Jiang, J. Cao, and S. Chen, Non-Hermitian mobility edges in one-dimensional quasicrystals with parity-time symmetry, Phys. Rev. B 101, 174205 (2020).
  * (46) Q. Zeng, Y. Yang, and R. Lü, Topological phases in one-dimensional nonreciprocal superlattices, Phys. Rev. B 101, 125418 (2020).
  * (47) Q. Zeng, Y. Yang, and Y. Xu, Topological phases in non-Hermitian Aubry-André-Harper models, Phys. Rev. B 101, 020201(R) (2020).
  * (48) Q. Zeng and Y. Xu, Winding numbers and generalized mobility edges in non-Hermitian systems, Phys. Rev. Res. 2, 033052 (2020).
  * (49) L. Zhai, S. Yin, and G. Huang, Many-body localization in a non-Hermitian quasiperiodic system, Phys. Rev. B 102, 064206 (2020).
  * (50) S. Longhi, Phase transitions in a non-Hermitian Aubry-André-Harper model, Phys. Rev. B 103, 054203 (2021).
  * (51) Y. Liu, Y. Wang, Z. Zheng, and S. Chen, Exact non-Hermitian mobility edges in one-dimensional quasicrystal lattice with exponentially decaying hopping and its dual lattice, Phys. Rev. B 103, 134208 (2021).
  * (52) Y. Liu, Y. Wang, X. Liu, Q. Zhou, and S. Chen, Exact mobility edges, PT-symmetry breaking, and skin effect in one-dimensional non-Hermitian quasicrystals, Phys. Rev. B 103, 014203 (2021).
  * (53) Z. Xu and S. Chen, Dynamical evolution in a one-dimensional incommensurate lattice with PT symmetry, Phys. Rev. A 103, 043325 (2021).
  * (54) X. Cai, Boundary-dependent self-dualities, winding numbers, and asymmetrical localization in non-Hermitian aperiodic one-dimensional models, Phys. Rev. B 103, 014201 (2021).
  * (55) L. Tang, G. Zhang, L. Zhang, and D. Zhang, Localization and topological transitions in non-Hermitian quasiperiodic lattices, Phys. Rev. A 103, 033325 (2021).
  * (56) T. Liu, S. Cheng, H. Guo, and X. Gao, Fate of Majorana zero modes, exact location of critical states, and unconventional real-complex transition in non-Hermitian quasiperiodic lattices, Phys. Rev. B 103, 104203 (2021).
  * (57) L. Zhai, G. Huang, and S. Yin, Cascade of the delocalization transition in a non-Hermitian interpolating Aubry-André-Fibonacci chain, Phys. Rev. B 104, 014202 (2021).
  * (58) X. Xia, K. Huang, S. Wang, and X. Li, A new class of exact mobility edges in non-Hermitian quasiperiodic models, arXiv:2105.12640.
  * (59) L. Zhou and W. Han, Non-Hermitian quasicrystal in dimerized lattices, Chin. Phys. B 30, 100308 (2021).
  * (60) Y. Liu, Q. Zhou, and S. Chen, Localization transition, spectrum structure, and winding numbers for one-dimensional non-Hermitian quasicrystals, Phys. Rev. B 104, 024201 (2021).
  * (61) A. P. Acharya, A. Chakrabarty, and D. K. Sahu, Localization, PT-Symmetry Breaking and Topological Transitions in non-Hermitian Quasicrystals, arXiv:2108.03181.
  * (62) X. Cai, Localization and topological phase transitions in non-Hermitian Aubry-André-Harper models with _p_ -wave pairing, Phys. Rev. B 103, 214202 (2021).
  * (63) S. Longhi, Non-Hermitian Maryland model, Phys. Rev. B 103, 224206 (2021).
  * (64) L. Zhou, Floquet engineering of topological localization transitions and mobility edges in one-dimensional non-Hermitian quasicrystals, Phys. Rev. Research 3, 033184 (2021).
  * (65) H. Kunz and B. Shapiro, Statistics of resonances in a semi-infinite disordered chain, Phys. Rev. B 77, 054203 (2008).
  * (66) D. B. Grempel, S. Fishman, and B. E. Prange, Localization in an Incommensurate Potential: An Exactly Solvable Model, Phys. Rev. Lett. 49, 833 (1982).
  * (67) S. Fishman, D. R. Grempel, and R. E. Prange, Chaos, Quantum Recurrences, and Anderson Localization, Phys. Rev. Lett. 49, 509 (1982).
  * (68) D. R. Grempel, R. E. Prange, and S. Fishman, Quantum dynamics of a nonintegrable system, Phys. Rev. A 29, 1639 (1984).
  * (69) M. V. Berry, Incommensurability in an Exactly-Soluble Quantal and Classical Model for a Kicked Rotator, Physica, 10D, 369 (1984).
  * (70) B. Simon, Almost periodic Schrödinger operators IV. The maryland model, Ann. Phys. 159, 157 (1985).
  * (71) F. M. Izrailev, Simple models of quantum chaos: Spectrum and eigenfunctions, Phys. Rep. 196, 299 (1990).
  * (72) J. Bellissard, R. Lima and E.Scoppola, Localization in $\nu$-Dimensional Incommensurate Structures, Comm. Math. Phys. 88, 465 (1983).
  * (73) A. L. Figotin and L. A. Pastur, An Exactly Solvable model of a Multidimensional Incommensurate Structure, Comm. Math. Phys. 95, 401 (1984).
  * (74) S. Fishman, D. R. Grempel and R. E. Prange, Localization in a $d$-Dimensional Incommensurate Structure, Phys. Rev. B 29, 4272 (1984).
  * (75) S. Ganeshan, K. Kechedzhi, and S. Das Sarma, Critical integer quantum Hall topology and the integrable Maryland model as a topological quantum critical point , Phys. Rev. B 90, 041405 (2014).
  * (76) S. Longhi, Maryland model in optical waveguide lattices, Opt. Lett. 46, 637-640 (2021).
  * (77) O. Zilberberg, Topology in quasicrystals, Opt. Mater. Express 11, 1143 (2021).
  * (78) T. E. Lee, Anomalous Edge State in a Non-Hermitian Lattice, Phys. Rev. Lett. 116, 133903 (2016).
  * (79) S. Yao and Z. Wang, Edge States and Topological Invariants of Non-Hermitian Systems, Phys. Rev. Lett. 121, 086803 (2018).
  * (80) 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).
  * (81) 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(R) (2018).
  * (82) C. H. Lee and R. Thomale, Anatomy of skin modes and topology in non-Hermitian systems, Phys. Rev. B 99, 201103(R) (2019).
  * (83) S. Mu, L. Zhou, L. Li, and J. Gong, Non-Hermitian pseudo mobility edge in a coupled chain system, arXiv:2111.11914.