# A Two-Scale Effective Model for Defect-Induced Localization Transitions in
Non-Hermitian Systems

B. Davies1,2, S. Barandun3, E. O. Hiltunen4, R. V. Craster1,2,5,6 and H.
Ammari3 1Department of Mathematics, Imperial College London, London SW7 2AZ,
United Kingdom
2Centre for Plasmonics and Metamaterials, Imperial College London, London SW7
2AZ, United Kingdom
3Department of Mathematics, ETH Zurich, Zurich 8092, Switzerland
4Department of Mathematics, Yale University, New Haven, CT 06511, USA
5Department of Mechanical Engineering, Imperial College London, London SW7
2AZ, United Kingdom
6UMI 2004 Abraham de Moivre-CNRS, Imperial College London, London SW7 2AZ,
United Kingdom

###### Abstract

We illuminate the fundamental mechanism responsible for the transition between
the non-Hermitian skin effect and defect-induced localization in the bulk. We
study a Hamiltonian with non-reciprocal couplings that exhibits the skin
effect (the localization of all eigenvectors at one edge) and add an on-site
defect in the center. Using a two-scale asymptotic method, we characterize the
long-scale growth and decay of the eigenvectors and derive a simple and
intuitive effective model for the transition that occurs when the defect is
sufficiently large that one of the modes is localized at the defect site,
rather than at the edge of the system.

††preprint: APS/123-QED

Introduction.—Understanding and harnessing systems that are non-Hermitian but
have real spectra represents one of the most exciting and active frontiers in
the physical sciences [1, 2, 3, 4, 5]. For example, lattices with non-
reciprocal couplings have been studied at length for their ability to support
spectra composed exclusively of eigenmodes that are exponentially localized at
one edge of the system [6, 7, 8, 9, 10, 11, 12, 13, 6, 14, 15, 16, 17, 18].
This phenomenon, known as the _non-Hermitian skin effect_ , has deep
implications for localization and transport properties and has been realised
in settings including quantum systems [10], acoustic metamaterials [11, 12],
elastic media [13], dimerized systems [6, 14] and multi-dimensional models
[15, 16, 17, 18].

The non-Hermitian skin effect is a very strong localization phenomenon and
there is significant interest in understanding how it behaves in the presence
of imperfections and disorder [19, 20, 21, 22, 23, 24, 25, 26, 27]. The effect
is known to persist in the presence of small disorder; however, when
sufficiently large defects are added, Anderson-type effects can cause
eigenvectors to be localized within the bulk of the system (away from the
edge). The most common setting in which to study these effects is the non-
Hermitian Anderson model, which is typically a nearest-neighbour Hamiltonian
with a random potential (independent on each site) and non-reciprocal coupling
strengths [19, 20]. Many studies have considered this model’s spectral
properties in the presence of periodic boundary conditions, to understand its
topological features and how its complex spectrum behaves in the presence of
disorder [19, 20, 21, 22, 23, 24, 25]. However, one of the most exciting
features of non-Hermitian systems is that altering the boundary conditions can
drastically affect their bulk properties [25]. Indeed, in a finite-sized model
with open or fixed boundary conditions at the edges, the skin effect can occur
[1, 6, 7, 8, 9, 10, 11, 12, 13, 6, 14, 15, 16, 17, 18]. In this case, when
imperfections are introduced there is the additional challenging question of
predicting where eigenvectors will be localized since the site where a given
eigenvector is localized is the result of competition between defect-induced
localization in the bulk and localization due to the skin effect at the edge.
This Letter develops an effective model to illuminate this competition and the
resulting transition between the two regimes.

The defect-induced localization transition in non-Hermitian systems is
fundamentally different from classical Hermitian analogues [28, 29]. In
Hermitian settings, if localization within the bulk occurs, it will typically
do so for any arbitrarily small defect size. In our non-Hermitian setting,
however, the localizing strength of the skin effect means that small defects
do not induce localization within the bulk. Instead, there is some non-zero
critical defect size at which the defect is sufficiently large to induce
localization. This transition can be predicted from the topological origins of
the non-Hermitian skin effect [24, 25, 30]. Due to the theory of Toeplitz
operators, eigenvectors corresponding to eigenvalues falling within a
specified region of the complex plane (specified by the winding number of the
associated symbol) must be localized at the edge [31, 32]. The boundary of
this region is where the localization transition occurs as eigenmodes escaping
it (due to the influence of defects) can be localized within the bulk. This
can be used to quantify the robustness of the skin effect [26, 27] and to
characterize the critical defect size at which an eigenvalue crosses the
boundary and a localization transition occurs.

In this Letter, the localization transition due to defects in a system
exhibiting the non-Hermitian skin effect is revealed. We develop a two-scale
effective model for the localization transition that occurs when sufficiently
large defects are added to a lattice with non-reciprocal couplings. This
characterizes the competition between localization at the edge due to the non-
Hermitian skin effect and defect-induced localization in the bulk. Our
approach to characterizing the effective properties of the system near to this
localization transition is a variant of the two-scale asymptotic methods known
variously as $\mathbf{k}\cdot\mathbf{p}$ theory [33, 34] or high-frequency
homogenization (HFH) [35]. These two-scale methods have a long history in
mathematics [36, 37] and have been applied to a variety of settings including
optics [38, 39], phononic crystals [40], semiconductors [34], elastic
composites [41], frame and lattice structures [42, 43], materials with
imperfect interfaces [44, 45] and forced problems [46]. Most relevant to our
work are methods for discrete systems [47] and for describing localization due
to defects [28, 29].

The two-scale effective model proposed in this work characterizes eigenvectors
near to the localization transition as the product of a short-scale modified
standing mode and a long-scale amplitude function. The short-scale modified
standing mode is the eigenvector that exists when the system is exactly at the
localization transition point. This is the non-Hermitian analogue of the
reference Bloch mode used in $\mathbf{k}\cdot\mathbf{p}$ theory and HFH [33,
34, 35, 29]. We derive an equation for the long-scale amplitude function which
characterizes the effective properties of the system in an asymptotic
neighbourhood of the transition point. Solutions of this effective model will
be either exponentially decaying or growing away from the defect, revealing
whether an eigenvector is either localized at the defect or the edge,
respectively. The switch between these two regimes, captured by the effective
model, is the localization transition in question.

To demonstrate our two-scale method, we consider a model that is a variant of
the non-Hermitian Hamiltonians introduced by Hatano and Nelson [19, 20], which
has non-reciprocal coupling terms and a modulated on-site potential function.
In order to illuminate the fundamental physics of the localization transition,
we consider the simplest model that captures its key features. This is a
homogeneous potential with a single defect in the center of the array. We
derive the critical defect size at which the localization transition occurs
and introduce a two-scale ansatz to derive an effective model that holds close
to this point. This gives quantitative predictions for eigenvectors on either
side of the transition, which are localized either at the edge or at the
defect, as captured by the effective model.

Figure 1: Hamiltonians with non-reciprocal couplings exhibit the non-Hermitian
skin effect whereby eigenvalues falling within a specified region $\mathbb{W}$
are all localized at one edge of the system, as shown in (c), (d) and (e). All
the eigenvalues for this 29-site system are shown in the complex plane in (a),
along with the region $\mathbb{W}$. When a defect $d$ is added to the
potential at one of the sites, this can cause an eigenvalue to leave
$\mathbb{W}$ and be localized at the defect site, as shown in (b). We propose
a two-scale effective model for this transition.

The unperturbed skin effect.—We are interested in the eigenvalues of the non-
Hermitian Hamiltonian

$H\psi_{n}=V_{n}\psi_{n}+e^{-\gamma}\psi_{n-1}+e^{\gamma}\psi_{n+1},$ (1)

where $V_{n}:\mathbb{Z}\to\mathbb{R}$ is some potential function and
$\gamma>0$ is a real constant that characterizes the non-reciprocity of the
system.

In the unperturbed case, when $V_{n}=v\in\mathbb{R}$ for all $n$, (1) is a
truncated Toeplitz matrix with symbol given by

$f_{H}(s)=v-e^{\gamma}s-e^{-\gamma}s^{-1},$ (2)

for $s\in S^{1}\subset\mathbb{C}$, the unit circle in the complex plane [31].
Let $\mathbb{W}$ be the set of all $z\in\mathbb{C}$ such that the winding
number of $f_{H}$ around $z$ is negative. Then, it is a general property of
truncated Toeplitz matrices that any eigenvalue falling within $\mathbb{W}$
will correspond to an eigenvector that is exponentially localized at one edge
[31, 32]. This phenomenon is the non-Hermitian skin effect. Some examples of
eigenvectors that are localized at an edge due to the skin effect are shown in
Fig. 1(c)-(e). The corresponding eigenvalues all fall within the shaded grey
region $\mathbb{W}$ in the complex plane in Fig. 1(a). This spectrum is that
of a system of 29 sites with a defect at the center. This defect is
sufficiently large that one of the eigenvalues has escaped $\mathbb{W}$ and
the corresponding eigenvector is localized at the defect site, as shown in
Fig. 1(b). Since the eigenvalues are always real valued, they can only leave
the region $\mathbb{W}$ at the points $E=v\pm 2\cosh\gamma$. How the
eigenvectors change as the eigenvalues cross these points is the transition we
will capture in this work.

Figure 2: The maximal or minimal eigenvectors of a 29-site system with a
defect in the center. (a)-(e) show the eigenvector with the largest eigenvalue
in a system with a positive defect $d=2\sinh\gamma+\alpha\epsilon$. The
effective amplitude predicted by our two-scale model (6) is shown in a dotted
line. Inset shows the eigenvalues in the complex plane with $\mathbb{W}$
shaded and the eigenvalue of the plotted eigenvector circled in red. (f)-(j)
show the analogous eigenpairs for a negative defect
$d=-(2\sinh\gamma+\alpha\epsilon)$. $\gamma=0.4$ and $v=1$ are used throughout
and the values of $\alpha\in\\{-1,1\\}$ and $\epsilon>0$ are shown above each
plot.

Defect-induced localization.—To examine the fundamental mechanism of the
transition from localization due to the skin effect to defect-induced
localization in the bulk, we study the spectrum of the Hamiltonian (1) with a
single defect in the on-site potential. That is, we suppose that
$V_{n}=v+d\delta_{n,0}$ where $d\in\mathbb{R}$ characterizes the magnitude of
the defect and $\delta_{n,m}$ is the Kronecker delta. Adding a small defect
will not be sufficient to cause localization in the bulk as the eigenvalues
will still be within $\mathbb{W}$. The localization transition occurs at a
non-zero defect size given by $d=\pm 2\sinh\gamma$. This critical defect size
causes an eigenvalue to take the value $E=v\pm 2\cosh\gamma$ which falls at
the transition points on the boundary of $\mathbb{W}$. The critical defect
size does not depend on the background potential $v$ since changing $v$
translates both the spectrum and the region $\mathbb{W}$. The corresponding
eigenvectors have the form

$U^{\pm}(n)=\begin{cases}(\pm 1)^{n}&n\leq 0,\\\ (\pm
e^{-2\gamma})^{n}&n>0.\end{cases}$ (3)

They are modified standing modes in the sense that they are standing modes
(which are either periodic or anti-periodic) on the left of the defect and
decay quickly to the right. Variants of $U^{\pm}$ for a finite-sized system
are shown in Figs. 2(c) and 2(h).

To develop an effective model for the transition we consider a system that is
an $\epsilon$-perturbation of the critical system for some $0<\epsilon\ll 1$.
We consider the potential

$V_{n}=v\pm(2\sinh\gamma+\alpha\epsilon)\delta_{n,0},$ (4)

where $\alpha\in\\{-1,1\\}$ captures the sign of the perturbation away from
the critical defect size.

The eigenvectors plotted in Figs. 1 and 2 display typical two-scale behaviour:
oscillations on a short length scale with a slowly varying amplitude that is
modulated on a longer scale. To capture this, we introduce the continuous
long-scale variable $\eta=\epsilon n$ and seek a two-scale solution as an
asymptotic series
$\psi_{n}=\Psi^{(0)}(\eta,n)+\epsilon\Psi^{(1)}(\eta,n)+\dots$ with eigenvalue
$E=E^{(0)}+\epsilon E^{(1)}+\epsilon^{2}E^{(2)}+\dots$. Details of this
asymptotic analysis can be found in the Supplemental Material, which has some
crucial differences from the traditional application to Hermitian systems. The
leading-order form of the eigenvector is

$\Psi^{(0)}(\eta,n)=f(\eta)U^{\pm}(n),$ (5)

where $U^{\pm}(n)$ are the modified standing modes given in (3) and $f(\eta)$
is a long-scale amplitude modulation that solves the Schrödinger eigenvalue
problem

$\cosh\gamma f_{\eta\eta}(\eta)+\alpha\delta(\eta)f(\eta)=\pm E^{(2)}f(\eta).$
(6)

This effective model characterizes the transition between localization at the
edge and at the defect site. It has a solution which decays away from the
defect (so that $f\to 0$ as $\eta\to\pm\infty$) if and only if $\alpha>0$.
Conversely, when $\alpha<0$ any solution of (6) must be exponentially growing
away from the defect, meaning the eigenvector will be localized at the edge.
Thus, the model shows that there can be localization at the defect site if and
only if the defect is larger than the critical size. The first eigensolution
of (6) is

$f(\eta)=\exp\left(-\frac{\alpha}{2\cosh\gamma}|\eta|\right),$ (7)

with eigenvalue $E^{(2)}=\pm(4\cosh\gamma)^{-1}$. Collecting the other terms
in the asymptotic expansion for the eigenvalue (see the Supplemental Material
for details) gives

$E=v\pm
2\cosh\gamma\pm\epsilon\alpha\tanh\gamma\pm\epsilon^{2}\frac{1}{4\cosh\gamma}+O(\epsilon^{3}).$
(8)

The leading-order approximation of the eigenvector (18) is compared to
numerical solutions in Fig. 2. The numerical solutions are obtained for a
finite-sized system with 29 sites and Dirichlet boundary conditions at either
ends. As a result, the eigenvectors experience some edge effects (visible at
the left edge) which are not accounted for in the effective model. However, we
see very good agreement away from the edges including, crucially, near to the
defect site. First, we observe in 2(c) and 2(h) that when $\epsilon=0$ the
eigenvector is the modified standing mode given by $U^{\pm}$ in (3). When
$\alpha<0$ the eigenvectors are localized at the left edge of the system due
to the skin effect. This is shown in Figs. 2(a), 2(b), 2(f) and 2(g), where we
see that the effective amplitude (with $f(\eta)$ being exponentially growing
away from the defect) is accurately predicting the profile of the eigenvector
near to the defect. Finally, when $\alpha>0$ the modes are localized at the
defect site ($n=0$). Again, the effective amplitude gives a good prediction of
the profile (with the localization being due to the fact that $f(\eta)$ is
exponentially decaying in this case). The good agreement shown in Fig. 2 is in
spite of the fact that the system is relatively small (and the eigenvectors
experience edge effects at the ends of the array) and holds even for
relatively large values of the asymptotic parameter $\epsilon$.

The asymptotic formula (8) for the minimal or maximal eigenvalue (which may
escape $\mathbb{W}$ for sufficiently large defects) is compared to numerical
values in Fig. 3. The same system of 29 sites from Fig. 2 is used. Again, we
see excellent agreement, even for large $\epsilon$ and in spite of the edge
effects that occur in the finite-sized system.

Figure 3: The maximal or minimal eigenvalues for a system with either (a) a
positive defect or (b) a negative defect, respectively. Numerical values for a
29-site system with a single defect $d=\pm(2\sinh\gamma+\alpha\epsilon)$ at
the center are compared to the asymptotic formula (8) for $\gamma=0.4$ and
$v=1$.

Conclusions.— Our results show that, despite the exotic features of non-
Hermitian systems and their sometimes counter-intuitive localization
properties, defect-induced localization transitions can be described concisely
and intuitively. Our effective model means that the competition between
localization at the edge and at a defect site within the bulk can be predicted
without intensive computations. As non-Hermitian systems are increasingly used
in applications such as sensing [48, 49], tunneling [50] and more [3, 7], it
is imperative to have at hand a broad array of analytical and numerical tools.
Our results add a valuable new approach to the toolbox for studying
localization phenomena in non-Hermitian systems.

The two-scale effective model developed here gives insight into the
fundamental mechanism that causes eigenvectors to transition between
localization at an edge and at a defect site in the bulk due. This effective
model reduces localization at either the edge or the defect site to the growth
or decay of a long-scale modulation function. Meanwhile, the short-scale
oscillations of eigenvectors are captured by a modified standing wave. As well
as providing new intuition, this model gives accurate predictions of the
profiles of eigenvectors on either side of the localization transition, even
for large values of the asymptotic parameter and in small arrays (with edge
effects due to fixed boundary conditions).

An exciting future direction for investigation is extending these methods to
non-reciprocal lattices with potentials given by independent random variables
on multiple sites (known variously as the non-Hermitian Anderson model [19,
20, 21, 22]). Our work considered the simplest possible model that displayed
the essential features of the transition between skin and defect localization.
This has illuminated the fundamental mechanism responsible for the competition
between these two localization effects.

###### Acknowledgements.

B.D. is supported by EPSRC under grant number EP/X027422/1. R.V.C.’s work was
funded by UK Research and Innovation (UKRI) under the UK government’s Horizon
Europe funding guarantee [grant number 10033143].

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Supplementary Material

##  I Positive defect

We are interested in solutions of the difference equation

$v\psi_{n}+(2\sinh\gamma+\alpha\epsilon)\delta_{n,0}\psi_{n}+e^{-\gamma}\psi_{n-1}+e^{\gamma}\psi_{n+1}=E\psi_{n},$
(9)

where the defect $d=2\sinh\gamma+\alpha\epsilon$ has been chosen to be an
$O(\epsilon)$ perturbation away from the critical defect size $d=2\sinh\gamma$
(which leads to the largest eigenvalue being exactly on the boundary of
$\mathbb{W}$). We introduce the continuous long-scale variable $\eta=\epsilon
n$ and will seek a two-scale solution in terms of $n$ and $\eta$. For
convenience, we elect to switch the $O(\epsilon)$ part of the defect onto the
long-scale variable $\eta$. To do so, we make the association
$\delta_{n,0}=\epsilon\delta(\eta)$. This is a standard equivalence (used in
_e.g._ [28]) and is taken to give the appropriate normalization between the
Kronecker and Dirac deltas when re-scaling. This can be seen formally by
comparing the expressions

$1=\int_{-\infty}^{\infty}\delta(\eta)\
\mathrm{d}\eta=\int_{-\infty}^{\infty}\delta(\epsilon n)\epsilon\
\mathrm{d}n,$ (10)

and

$1=\sum_{n=-\infty}^{\infty}\delta_{n,0}=\int_{-\infty}^{\infty}\delta_{n,0}\
\mathrm{d}n,$ (11)

where integration with respect to the discrete variable $n$ should use the
counting measure. This leads to the difference equation

$v\psi_{n}+2\sinh\gamma\delta_{n,0}\psi_{n}+\alpha\epsilon^{2}\delta(\eta)\psi_{n}+e^{-\gamma}\psi_{n-1}+e^{\gamma}\psi_{n+1}=E\psi_{n}.$
(12)

We seek a two-scale solution $\psi_{n}=\Psi(\eta,n)$. We can Taylor expand in
the long-scale variable to see that

$\displaystyle\psi_{n-1}$
$\displaystyle=\Psi(\eta-\epsilon,n-1)=\Psi(\eta,n-1)-\epsilon\Psi_{\eta}(\eta,n-1)+\frac{1}{2}\epsilon^{2}\Psi_{\eta\eta}(\eta,n-1)+\dots,$
(13) $\displaystyle\psi_{n+1}$
$\displaystyle=\Psi(\eta+\epsilon,n+1)=\Psi(\eta,n+1)+\epsilon\Psi_{\eta}(\eta,n+1)+\frac{1}{2}\epsilon^{2}\Psi_{\eta\eta}(\eta,n+1)+\dots,$
(14)

where the subscripts are used to denote differentiation with respect to that
variable. In addition to this Taylor expansion, we will seek solutions of (12)
in terms of asymptotic expansions:

$\displaystyle\Psi$
$\displaystyle=\Psi^{(0)}+\epsilon\Psi^{(1)}+\epsilon^{2}\Psi^{(2)}+\dots,$
(15) $\displaystyle E$ $\displaystyle=E^{(0)}+\epsilon
E^{(1)}+\epsilon^{2}E^{(2)}+\dots.$ (16)

Substituting (13)–(16) into (12) yields a hierarchy of equations in
$\epsilon$. At leading order, we have

$v\Psi^{(0)}(\eta,n)+2\sinh\gamma\delta_{n,0}\Psi^{(0)}(\eta,n)+e^{-\gamma}\Psi^{(0)}(\eta,n-1)+e^{\gamma}\Psi^{(0)}(\eta,n+1)=E^{(0)}\Psi^{(0)}(\eta,n).$
(17)

This has solution

$\Psi^{(0)}(\eta,n)=f(\eta)U^{+}(n),$ (18)

with eigenvalue $E^{(0)}=v+2\cosh\gamma$, where

$U^{+}(n)=\begin{cases}1&n\leq 0,\\\ (e^{-2\gamma})^{n}&n>0.\end{cases}$ (19)

and $f(\eta)$ is some long-scale function that is not yet determined. This
form of the leading-order solution (18) is consistent with the traditional
applications of high-frequency homogenization to Hermitian systems [35]. It is
given by $U^{+}(n)$, which is the mode that exists when $\epsilon=0$ (and is
reminiscent of a standing wave, at least on the left of the defect site),
modulated by some slowly-varying amplitude function $f(\eta)$. Determining
$f(\eta)$, which describes the long-scale variation of the eigenmode at
leading order and thus captures whether the eigenmode is localized at the
defect or not, is the primary goals of this analysis.

Collecting the terms at order $\epsilon$, we have the equation

$\begin{split}v\Psi^{(1)}(\eta,n)+2\sinh\gamma\delta_{n,0}\Psi^{(1)}(\eta,n)+e^{-\gamma}\Psi^{(1)}(\eta,n-1)-e^{-\gamma}\Psi^{(0)}_{\eta}(\eta,n-1)&+e^{\gamma}\Psi^{(1)}(\eta,n+1)+e^{\gamma}\Psi^{(0)}_{\eta}(\eta,n+1)\\\
&=E^{0}\Psi^{(1)}(\eta,n)+E^{(1)}\Psi^{(0)}(\eta,n).\end{split}$ (20)

A solution for $\Psi^{(1)}$ with the same form as (18) exists provided that

$-e^{-\gamma}\Psi^{(0)}_{\eta}(\eta,n-1)+e^{\gamma}\Psi^{(0)}_{\eta}(\eta,n+1)=E^{(1)}\Psi^{(0)}(\eta,n).$
(21)

Inserting the expression (18) for $\Psi^{(0)}$ gives different behaviour
depending on the sign of $n$ (and, consequently, of $\eta$). If $n<0$ we have
that

$2\sinh\gamma f_{\eta}(\eta)=E^{(1)}f(\eta),$ (22)

whereas for $n>0$

$-2\sinh\gamma f_{\eta}(\eta)=E^{(1)}f(\eta).$ (23)

We elect to capture this sign change using the continuous variable $\eta$
(since $\mathrm{sgn}(\eta)=\mathrm{sgn}(n)$, where $\mathrm{sgn}(x)=|x|/x$ is
the standard sign function), yielding the first-order ODE

$2\sinh\gamma f_{\eta}(\eta)=-E^{(1)}\mathrm{sgn}(\eta)f(\eta).$ (24)

This equation represents a crucial difference from the application of high-
frequency homogenization to the corresponding Hermitian system, as in [28].
When $\gamma=0$ and the system is Hermitian, the left-hand side vanishes
meaning $E^{(1)}=0$ and the defect eigenvalue is an $O(\epsilon^{2})$
perturbation of $E^{(0)}$. In this case, if $\gamma\neq 0$ then $E^{(1)}\neq
0$. However, (24) is not sufficient to determine either the function $f(\eta)$
or the constant $E^{(1)}$. We must proceed to higher orders of $\epsilon$ and
return to (24) later.

We can also collect the terms at order $\epsilon^{2}$, which yields the
equation

$\begin{split}&v\Psi^{(2)}(\eta,n)+2\sinh\gamma\delta_{n,0}\Psi^{(2)}(\eta,n)+\alpha\delta(\eta)\Psi^{(0)}(\eta,n)+e^{-\gamma}\Psi^{(2)}(\eta,n-1)-e^{-\gamma}\Psi^{(1)}_{\eta}(\eta,n-1)+\frac{1}{2}e^{-\gamma}\Psi^{(0)}_{\eta\eta}(\eta,n-1)\\\
&\quad+e^{\gamma}\Psi^{(2)}(\eta,n+1)+e^{\gamma}\Psi^{(1)}_{\eta}(\eta,n+1)+\frac{1}{2}e^{\gamma}\Psi^{(0)}_{\eta\eta}(\eta,n+1)=E^{(0)}\Psi^{(2)}(\eta,n)+E^{(1)}\Psi^{(1)}(\eta,n)+E^{(2)}\Psi^{(0)}(\eta,n).\end{split}$
(25)

Once again, a solution for $\Psi^{(2)}$ with short-scale dependence given by
(19) can be constructed provided that the other terms cancel. Assuming the
$\eta$-dependent part of $\Psi^{(1)}$ also satisfies (24), we are left with

$\alpha\delta(\eta)\Psi^{(0)}(\eta,n)+\frac{1}{2}e^{-\gamma}\Psi^{(0)}_{\eta\eta}(\eta,n-1)+\frac{1}{2}e^{\gamma}\Psi^{(0)}_{\eta\eta}(\eta,n+1)=E^{(2)}\Psi^{(0)}(\eta,n).$
(26)

Substituting $\Psi^{(0)}(\eta,n)=f(\eta)U^{+}(n)$ from (18) gives

$\alpha\delta(\eta)f(\eta)U^{+}(n)+\frac{1}{2}e^{-\gamma}f_{\eta\eta}(\eta)U^{+}(n)+\frac{1}{2}e^{\gamma}f_{\eta\eta}(\eta)U^{+}(n)=E^{(2)}f(\eta)U^{+}(n),$
(27)

for $n<0$ and

$\alpha\delta(\eta)f(\eta)U^{+}(n)+\frac{1}{2}e^{-\gamma}e^{2\gamma}f_{\eta\eta}(\eta)U^{+}(n)+\frac{1}{2}e^{\gamma}e^{-2\gamma}f_{\eta\eta}(\eta)U^{+}(n)=E^{(2)}f(\eta)U^{+}(n),$
(28)

for $n>0$. In either case, this can be rearranged to give a Schrödinger
eigenvalue problem $f(\eta)$ and $E^{(2)}$:

$\cosh\gamma f_{\eta\eta}(\eta)+\alpha\delta(\eta)f(\eta)=E^{(2)}f(\eta).$
(29)

The Schrödinger eigenvalue problem (29) is our effective model for the
localization at the defect site. We observe, firstly, that it has a decaying
solution $f(\eta)\to 0$ as $\eta\to\pm\infty$ for positive $E^{(2)}$ if and
only if $\alpha>0$. To see this, we multiply (29) by $f$ and integrate to
obtain

$\left[\cosh\gamma
f_{\eta}f\right]_{\infty}^{\infty}-\cosh\gamma\int_{-\infty}^{\infty}(f_{\eta})^{2}\
\mathrm{d}\eta+\alpha f(0)^{2}=E^{(2)}\int_{-\infty}^{\infty}f^{2}\
\mathrm{d}\eta,$ (30)

from which it is clear that $\lim_{\eta\to\pm\infty}f_{\eta}f=0$ only if
$\alpha>0$ (since we expect $E^{(2)}>0$ so that the eigenvalue will fall
outside of $\mathbb{W}$ and not be localized at the left edge of the system
due to the skin effect). When $\alpha>0$, the solution can be found using
Fourier transforms. We let $\alpha=1$ and then in Fourier space (29) becomes

$-\cosh\gamma\xi^{2}\hat{f}(\xi)+f(0)=E^{(2)}\hat{f}(\xi),$ (31)

from which we find that

$\hat{f}(\xi)=\frac{f(0)}{E^{(2)}+\cosh\gamma\xi^{2}}.$ (32)

Applying the inverse Fourier transform yields the solution

$f(\eta)=\frac{f(0)}{2\sqrt{E^{(2)}\cosh\gamma}}\exp\left(-\sqrt{\frac{E^{(2)}}{\cosh\gamma}}|\eta|\right).$
(33)

Inspecting the value at $\eta=0$ shows that $2\sqrt{E^{(2)}\cosh\gamma}=1$,
from which we have that

$E^{(2)}=\frac{1}{4\cosh\gamma}.$ (34)

Fixing the normalization of $f(\eta)$ to be such that
$\sup_{\eta}|f(\eta)|=f(0)=1$, gives that

$f(\eta)=\exp\left(-\frac{1}{2\cosh\gamma}|\eta|\right).$ (35)

Now that we have identified $f(\eta)$, we can use (24) to find the value of
$E^{(1)}$. Differentiating (35) yields

$f_{\eta}(\eta)=-\frac{1}{2\cosh\gamma}\mathrm{sgn}(\eta)f(\eta),$ (36)

which can be compared with (24) to see that

$E^{(1)}=\tanh\gamma.$ (37)

Bringing this all together, we have found that, if and only if $\alpha=1>0$,
there exists an eigenmode localized at the defect site $n=0$ which has
eigenvalue

$E=v+2\cosh\gamma+\epsilon\tanh\gamma+\epsilon^{2}\frac{1}{4\cosh\gamma}+O(\epsilon^{3}),$
(38)

and leading-order profile

$\psi_{n}=U^{+}(n)\exp\left(-\frac{1}{2\cosh\gamma}|\epsilon
n|\right)+O(\epsilon),$ (39)

where $U^{+}(n)$ was specified in (19).

When $\alpha<0$, the effective model (29) is still solvable but its solution
will not decay as $\eta\to\pm\infty$. Indeed, we expect the general solution
of (29) to be of the form
$f(\eta)=A\exp(-\sqrt{E^{(2)}/\cosh\gamma}\eta)+B\exp(\sqrt{E^{(2)}/\cosh\gamma}\eta)$.
The constants $A$ and $B$ will be different for $\eta<0$ and $\eta>0$ to
account for the discontinuity in the derivative $f_{\eta}(\eta)$ at $\eta=0$
that is introduced by the $\delta(\eta)$ term. Matching the solutions such
that $f(\eta)$ is continuous and $f_{\eta}(\eta)$ has a jump of magnitude
$(\cosh\gamma)^{-1}$ at $\eta=0$ leads to the normalized solution

$f(\eta)=\exp\left(\frac{1}{2\cosh\gamma}|\eta|\right),$ (40)

where we have let $\alpha=-1$. Noting that

$f_{\eta}(\eta)=\mathrm{sgn}(\eta)\frac{1}{2\cosh\gamma}f(\eta),$ (41)

and

$f_{\eta\eta}(\eta)=\frac{1}{4(\cosh\gamma)^{2}}f(\eta)+\frac{1}{\cosh\gamma}\delta(\eta)f(\eta),$
(42)

we can see that substitution into (29) yields the corresponding eigenvalue
$E^{(2)}=(4\cosh\gamma)^{-1}$ (which is unchanged from $\alpha<0$). However,
the switch from decay to growth on either side of the defect has led to a
change of sign in (41), such that substitution into (24) leads to
$E^{(1)}=-\tanh\gamma$.

Combining the analysis for the $\alpha>0$ and $\alpha<0$ cases, we find the
leading-order profile of the eigenvector to be

$\psi_{n}=U^{+}(n)\exp\left(-\frac{\alpha}{2\cosh\gamma}|\epsilon
n|\right)+O(\epsilon),$ (43)

with associated eigenvalue

$E=v+2\cosh\gamma+\epsilon\alpha\tanh\gamma+\epsilon^{2}\frac{1}{4\cosh\gamma}+O(\epsilon^{3}).$
(44)

##  II Negative defect

The above analysis can be repeated for the case of a negative defect. Here, we
study the difference equation

$v\psi_{n}-2\sinh\gamma\delta_{n,0}\psi_{n}-\alpha\epsilon^{2}\delta(\eta)\psi_{n}+e^{-\gamma}\psi_{n-1}+e^{\gamma}\psi_{n+1}=E\psi_{n},$
(45)

and proceed to seek solutions in terms of similar expansions similar to (15)
and (16). The leading-order equation is

$v\Psi^{(0)}(\eta,n)-2\sinh\gamma\delta_{n,0}\Psi^{(0)}(\eta,n)+e^{-\gamma}\Psi^{(0)}(\eta,n-1)+e^{\gamma}\Psi^{(0)}(\eta,n+1)=E^{(0)}\Psi^{(0)}(\eta,n).$
(46)

This admits the solution

$\Psi^{(0)}(\eta,n)=f(\eta)U^{-}(n),$ (47)

with eigenvalue $E^{0}=v-2\cosh\gamma$, where

$U^{-}(n)=\begin{cases}(-1)^{n}&n\leq 0,\\\
(-e^{-2\gamma})^{n}&n>0.\end{cases}$ (48)

Once again, the goal is to find the long-scale function $f(\eta)$.

Collecting the terms at order $\epsilon$ gives the equation

$\begin{split}v\Psi^{(1)}(\eta,n)-2\sinh\gamma\delta_{n,0}\Psi^{(1)}(\eta,n)&+e^{-\gamma}\Psi^{(1)}(\eta,n-1)-e^{-\gamma}\Psi^{(0)}_{\eta}(\eta,n-1)+e^{\gamma}\Psi^{(1)}(\eta,n+1)\\\
&\quad+e^{\gamma}\Psi^{(0)}_{\eta}(\eta,n+1)=E^{(0)}\Psi^{(1)}(\eta,n)+E^{(1)}\Psi^{(0)}(\eta,n).\end{split}$
(49)

As above, we want that

$-e^{-\gamma}\Psi^{(0)}_{\eta}(\eta,n-1)+e^{\gamma}\Psi^{(0)}_{\eta}(\eta,n+1)=E^{(1)}\Psi^{(0)}(\eta,n),$
(50)

which leads to the first-order ODE

$f_{\eta}(\eta)=\frac{E^{(1)}}{2\sinh\gamma}\mathrm{sgn}(\eta)f(\eta),$ (51)

which is the same as (24) up to a change of sign of the constant on the right-
hand side.

Proceeding to the $\epsilon^{2}$ terms, we find the equation

$\begin{split}v\Psi^{(2)}(\eta,n)&-2\sinh\gamma\delta_{n,0}\Psi^{(2)}(\eta,n)-\alpha\delta(\eta)\Psi^{0}(\eta,n)+e^{-\gamma}\Psi^{2}(\eta,n-1)-e^{-\gamma}\Psi^{1}_{\eta}(\eta,n-1)+\frac{1}{2}e^{-\gamma}\Psi^{0}_{\eta\eta}(\eta,n-1)\\\
&\quad+e^{\gamma}\Psi^{2}(\eta,n+1)+e^{\gamma}\Psi^{1}_{\eta}(\eta,n+1)+\frac{1}{2}e^{\gamma}\Psi^{0}_{\eta\eta}(\eta,n+1)=E^{0}\Psi^{2}(\eta,n)+E^{1}\Psi^{1}(\eta,n)+E^{2}\Psi^{0}(\eta,n).\end{split}$
(52)

Using the same logic as above, we arrive at the equation

$-\alpha\delta(\eta)f(\eta)U^{-}(n)-\frac{1}{2}e^{-\gamma}f_{\eta\eta}(\eta)U^{-}(n-1)-\frac{1}{2}e^{\gamma}f_{\eta\eta}(\eta)U^{-}(n+1)=E^{(2)}f(\eta)U^{-}(n),$
(53)

which leads to a Schrödinger eigenvalue problem for $f(\eta)$ and $E^{(2)}$,
which is analogous to (29):

$\cosh\gamma f_{\eta\eta}(\eta)+\alpha\delta(\eta)f(\eta)=-E^{(2)}f(\eta).$
(54)

Repeating the analysis of (29) but with the sign of the eigenvalue changed, we
see that (54) admits a decaying solution for $f(\eta)$ if and only if
$\alpha>0$ and $E^{(2)}<0$. Thus, letting $\alpha=1$ leads to the solution

$f(\eta)=\exp\left(-\frac{1}{2\cosh\gamma}|\eta|\right),$ (55)

along with

$E^{(2)}=-\frac{1}{4\cosh\gamma}.$ (56)

Using (51), this leads to

$E^{(1)}=-\tanh\gamma.$ (57)

Finally, we can conclude that, if and only if $\alpha=1>0$, there exists an
eigenmode localized at the defect site $n=0$ which has eigenvalue

$E=v-2\cosh\gamma-\epsilon\tanh\gamma-\epsilon^{2}\frac{1}{4\cosh\gamma}+O(\epsilon^{3}),$
(58)

and leading-order profile

$\psi_{n}=U^{-}(n)\exp\left(-\frac{1}{2\cosh\gamma}|\epsilon
n|\right)+O(\epsilon),$ (59)

where $U^{-}(n)$ was specified in (48).

The analysis for the $\alpha<0$ case is, again, similar to the previous
section. We find that the exponentially decaying solution (55) switches to an
exponentially growing mode. Thus, we obtain the general expressions for the
eigenvector

$\psi_{n}=U^{-}(n)\exp\left(-\frac{\alpha}{2\cosh\gamma}|\epsilon
n|\right)+O(\epsilon),$ (60)

and its associated eigenvalue

$E=v-2\cosh\gamma-\epsilon\alpha\tanh\gamma-\epsilon^{2}\frac{1}{4\cosh\gamma}+O(\epsilon^{3}),$
(61)

which hold for $\alpha\in\\{-1,1\\}$.