# Universal properties of boundary and interface charges in continuum models
of one-dimensional insulators

Sebastian Miles  Institut für Theorie der Statistischen Physik, RWTH Aachen,
52056 Aachen, Germany and JARA - Fundamentals of Future Information Technology
Dante M. Kennes  Institut für Theorie der Statistischen Physik, RWTH Aachen,
52056 Aachen, Germany and JARA - Fundamentals of Future Information Technology
Max Planck Institute for the Structure and Dynamics of Matter, Center for Free
Electron Laser Science, 22761 Hamburg, Germany  Herbert Schoeller  Institut
für Theorie der Statistischen Physik, RWTH Aachen, 52056 Aachen, Germany and
JARA - Fundamentals of Future Information Technology  Mikhail Pletyukhov
<EMAIL_ADDRESS>Institut für Theorie der Statistischen Physik,
RWTH Aachen, 52056 Aachen, Germany and JARA - Fundamentals of Future
Information Technology

###### Abstract

We study single-channel continuum models of one-dimensional insulators induced
by periodic potential modulations which are either terminated by a hard wall
(the boundary model) or feature a single region of dislocations and/or
impurity potentials breaking translational invariance (the interface model).
We investigate the universal properties of excess charges accumulated near the
boundary and the interface, respectively. We find a rigorous analytic proof
for the earlier observed linear dependence of the boundary charge on the phase
of the periodic potential modulation as well as extend these results to the
interface model. The linear dependence on the phase shows a universal value
for the slope, and is intersected by discontinuous jumps by plus or minus one
electron charge at the phase points where localized states enter or leave a
band of extended states. Both contributions add up such that the periodicity
of the excess charge in the phase over a $2\pi$-cycle is maintained. While in
the boundary model this property is usually associated with the bulk-boundary
correspondence, in the interface model a correspondence of scattering state
and localized state contributions to the total interface charge is unveiled on
the basis of the so-called nearsightedness principle.

##  I Introduction

The notion of topology has provided a powerful, novel viewpoint on a wide
variety of phenomena in physics [JackiwRebbi, -hatsugai_1993, ], successfully
solidifying and expanding our understanding of condensed matter systems
[hasan_kane_2010, -asboth_etal_16, ]. Two fields that, e.g., have greatly
profited from the paradigm of topology are the Quantum Hall effect (QHE)
[LaughlinQHE, ] and the microscopic theory of polarization, now known as
modern theory of polarization (MTP) [king_smith_vanderbilt_93, ].

The field of the QHE significantly profited from the insight that the integer
$n$ characterizing the Quantum Hall conductance $\sigma_{xy}=ne^{2}/h$ is
linked to the topological invariant characterizing bands, now known as the
TKNN invariant [thouless_etal_prl_82, ] and mathematically identified
[thouless_1983, ] with the first Chern index $C_{1}$. The importance of phase-
singular points of the Bloch states $|\psi_{k\alpha}\rangle$ (with the
quasimomentum $k$ in the occupied bands $\alpha$) for both the conductance
quantization and for the above mentioned identification of $n=C_{1}$ was
quickly recognized [kohmoto_1985, ]. This understanding has subsequently led
Y. Hatsugai to the formulation [hatsugai_1993, ] of the seminal bulk-boundary
correspondence, which links the first Chern index (and thereby the topological
characterization of the Bloch states in the Brillouin zone in terms of the
Bloch wavefunction’s phase vorticities) to the number of edge modes contained
within the band gaps of the insulating system.

On the other hand, the MTP relies heavily on concepts of topology since it
establishes a connection between localized charges $Q_{B}$ at boundaries and
the Zak-Berry phase
$\frac{\gamma_{\alpha}}{2\pi}=i\int_{BZ}\frac{dk}{2\pi}\langle\psi_{k\alpha}|\frac{d}{dk}\psi_{k\alpha}\rangle
dk$ [zak_89, ] of the bulk Bloch bands even beyond any symmetry constraints
[king_smith_vanderbilt_93, ]. This is the so-called _surface charge theorem_
[king_smith_vanderbilt_surf_93, ,marzari_etal_12, ]. However, this connection
is in general not unique but only determined up to an unknown integer
[vanderbilt_18, ], since the MTP does not give a prescription which specific
gauge of $|\psi_{k\alpha}\rangle$ should be used: Any nontrivial gauge
transformation with a nonzero winding number changes this relation by an
integer.

Recently, systems have been proposed that capture both, the QHE and aspects of
the MTP, simultaneously establishing an analogy between the two [FracInDots,
,thakurati_etal_18, ]. Following the key idea of the mapping between a two-
dimensional QHE model, which is translationally invariant in one dimension,
and a one-dimensional (1D) model of an insulator with a modulated periodic
potential, establishes the correspondence between the Hall conductance and the
derivative of the boundary charge $Q_{B}$ with respect to the phase of the
modulations of the 1D periodic potential. Studying a lattice model of a 1D
insulator with quite a large number ($Z=10$) of sites per unit cell, reveals a
linear dependence of $Q_{B}$ on the phase, which is accompanied by
discontinuous jumps by plus/minus one electron charge at the phase points
where an edge state enters/leaves the band. Thereby, $Q_{B}$ was elevated to
be a relevant physical observable quantifying the spectral flow of the
boundary eigenvalue problem. The slope of the linear dependence is universally
given by the number of the occupied bands (in units of $2\pi/e$), suggesting
that it should be identified not only with $n$, but with $C_{1}$ as well.

A wider class of generalized Aubry-André-Harper models [aa_model,
,harper_model, ] has been later studied analytically [pletyukhov_etal_prbr_20,
,pletyukhov_etal_prb_20, ] in the semi-infinite geometry with a hard-wall
boundary condition at the origin. The main conclusions of [thakurati_etal_18,
] have been confirmed, also proving that the slope of the linear dependence is
indeed determined by the Chern index. In addition, it was shown that on top of
the linear dependence for every finite $Z$ there is an additional non-
universal $\frac{2\pi}{Z}$-periodic function, contributing to $Q_{B}$, which
is expected to vanish at larger $Z$. Moreover, a distinguished gauge has been
identified, in which the relation between the Zak-Berry phase and $Q_{B}$
holds exactly. This gauge is fixed by choosing the last component of the Bloch
vector to be real. As a consequence, the winding and the phase-singular points
of the first Bloch vector component were shown to determine the universal
values of the change $\Delta Q_{B}$ under a lattice shift by one site and for
the jumps in $Q_{B}$, respectively.

In addition, in Ref. [weber_etal_prl_20, ] quantum fluctuation of the boundary
charge have been studied, and the so-called surface fluctuation theorem has
been formulated, relating the boundary charge fluctuations generically to the
fluctuations of the bulk polarization. Furthermore, for one-dimensional
systems, a universal $1/E_{g}$ low-energy scaling was established, with
$E_{g}$ denoting the size of the gap in which the chemical potential lies.
Other universal low-energy properties of the Wannier functions in these models
have been recently addressed in Ref. [wannier_paper, ].

An interface model combining the two semi-infinite lattice models with a
mutual phase mismatch has been studied in Ref. [RatBCPlet, ]. It has been
shown that the interface charge $Q_{I}$ is given — up to an integer — by the
dipole moment mismatch of the two subsystems. Furthermore, it was shown how to
determine the integer for a specific type of the interface (realized by a
hopping attenuation between the two subsystems at no phase mismatch)
expressing it as a winding number of some complex function defined in the
Brillouin zone. A principle stating that a local perturbation to a many-
electron system may only lead to an integer-valued change in accumulated
charges is called the nearsightedness principle [NearSightednessPrin, ]. On
its basis we expect an ubiquitous occurrence of winding number expressions for
this kind of changes: A charge robustness with respect to numerous system
parameters and its sharp quantization can be only guaranteed by a topological
expression.

In this paper, we study single-channel continuum models of one-dimensional
insulators. Mathematically they correspond to the limit $Z\to\infty$ (number
of sites per unit cell), $a\to 0$ (lattice spacing) at the fixed $L=Za$ (unit
cell length) of the previously studied lattice models. We investigate
properties of both boundary and interface localized charges and reveal their
universal features. Most of them are expected on the basis of our previous
studies [pletyukhov_etal_prbr_20, -RatBCPlet, ]: A linear phase dependence of
$Q_{B}$ and $Q_{I}$, discontinuous jumps at the points where a localized state
enters/leaves a band of extended states, both the surface charge and surface
fluctuation theorems and the bulk-boundary correspondence. However, there are
additional features which are specific only to continuum models. In
particular, the absence of the umklapp processes leads to the rigid values of
the Chern indices ($=1$ for each band), as well as to additional constraints
on numbers of edge states leaving and entering a specific band and on Bloch
wavefunction’s vorticity values. We find that these constraints stem from the
properties of the Hill’s equation which were thoroughly studied in the
mathematical literature (see, e.g., Refs. [arscott, -MagnusWinkler, ]). As an
additional motivation to study continuum models we literally quote the
citation from Ref. [wang_troyer, ] which we share without reserve: ”There are
several advantages to working with a continuum model instead of a tight-
binding lattice model… Continuum models apply to a broader range of
experimental situations…”.

As for the continuum interface models, we establish that the quantized part of
$Q_{I}$ is intimately related to analytic properties of the reflection
coefficient. This observation accords with the recent results of Ref. [nakata,
]. In particular, extracting a function $\tilde{d}_{k\alpha}$, which is
defined in the Brillouin zone for every band $\alpha$, from a scattering
matrix denominator, we demonstrate that $\tilde{d}_{k\alpha}$ plays a role
similar to that of the Bloch state phase in the boundary problem.

In sum, the present paper consolidates the findings of our previous papers
[pletyukhov_etal_prbr_20, -RatBCPlet, ] on the subject and firmly establishes
the universal properties of both the boundary and the interface charges
drawing a unified analogy between the two quantities.

The paper is organised as follows. In Sec. II a general construction of the
Bloch states based on the approach of [kohn_59, ] for a generic but periodic
1D potential is discussed. In Sec. III we first determine eigenstates, both
extended and localized, of the boundary problem with a hard-wall boundary
introduced into the origin of the translationally invariant system. Next, we
define the boundary charge which contains contributions from all eigenstates
below the chemical potential put into the gap $\nu$. We rigorously prove the
surface charge theorem, emphasizing the role of the gauge choosing the last
component of the Bloch state to be real. Then we establish the phase
dependence of edge state dispersions, recover the bulk-boundary
correspondence, and state the universal form for both the total boundary
charge and the band’s individual contributions to it. In Sec. IV we formulate
the interface eigenvalue problem and construct both its scattering and
localized eigenstates. Next, we evaluate the total interface charge discussing
individual contributions to it from both of these types of eigenstates. This
section is concluded with a novel expression for the phase dependence of the
interface charge. All revealed properties are listed in the Summary Section V.
The appendices provide detailed mathematical proofs of all intermediate
statements and identities used in our derivations.

Figure 1: Potential landscape
$V(x)=a_{0}\cos\frac{2\pi\omega_{0}}{L}x+a_{1}\sin\frac{2\pi\omega_{1}}{L}x+a_{2}\sin\frac{2\pi\omega_{2}}{L}x$
used in exemplary calculations. The parameters are
$(a_{0},a_{1},a_{2})=(0.1,0.4,0.2)$ and
$(\omega_{0},\omega_{1},\omega_{2})=(1,3,4)$. Figure 2: a) The Lyapunov
function $D(E)$ [Eq. (14)] for the potential choice depicted in Fig. 1. In the
calculation we set the mass $m=1$ and the unit cell length $L=10$. The bands
given by the condition $|D(E)|\leq 1$ are indicated as the colored regions.
Typicallymatveev  there are infinitely many bands (and band gaps), only the
first four of them being shown in the figure. b) The band structure for the
same model in dependence of $k$.

##  II Bloch states construction for a periodic 1D potential

In this section we describe a construction of the Bloch states for a periodic
one-dimensional potential following the ideas of W. Kohn [kohn_59, ]. In
particular, we review properties of these eigenstates and thereby fix
notations for subsequent discussions. Throughout the paper we use the units
$e=\hbar=1$.

The 1D Schrödinger equation

$\displaystyle-\frac{1}{2m}\psi^{\prime\prime}(x)+V(x)\,\psi(x)=E\,\psi(x)$
(1)

with a periodic potential $V(x)=V(x+L)$ is a particular form of the Hill’s
equation whose properties are comprehensively described in the mathematical
literature (see, e.g., [arscott, -MagnusWinkler, ]). For the following
illustrations we arbitrarily choose a specific potential landscape shown in
Fig. 1, which does not have any internal symmetries. It will be used in all
exemplary calculations of this paper.

At fixed $E$, a general solution of (1) can be expressed as combination of two
linearly independent solutions, $\psi_{1}(x,E)\equiv\psi_{1}(x)$ and
$\psi_{2}(x,E)\equiv\psi_{2}(x)$, that is

$\displaystyle\psi(x)=c_{1}\,\psi_{1}(x)+c_{2}\,\psi_{2}(x).$ (2)

Choosing $\psi_{1}(x)$ and $\psi_{2}(x)$ to satisfy the initial conditions
$\psi_{1}(0)=1$, $\psi^{\prime}_{1}(0)=0$ and $\psi_{2}(0)=0$,
$\psi^{\prime}_{2}(0)=1$, respectively, we fix the fundamental matrix of the
equation

$\displaystyle\left(\begin{array}[]{cc}\psi_{1}(x)&\psi_{2}(x)\\\
\psi^{\prime}_{1}(x)&\psi^{\prime}_{2}(x)\end{array}\right).$ (5)

Its determinant, the Wronskian,

$\displaystyle
W(x)=\psi_{1}(x)\,\psi^{\prime}_{2}(x)-\psi_{2}(x)\,\psi^{\prime}_{1}(x)\equiv
1\neq 0$ (6)

is a nonzero constant: It follows that $W^{\prime}(x)=0$, and the specific
value 1 results from the chosen initial conditions. This property guarantees
linear independence of the vectors $\begin{pmatrix}\psi_{1}(x)\\\
\psi^{\prime}_{1}(x)\end{pmatrix}$ and $\begin{pmatrix}\psi_{2}(x)\\\
\psi^{\prime}_{2}(x)\end{pmatrix}$ at all $x$. Note that all elements of (5)
are real.

According to the Bloch theorem, a smooth and bounded solution of (1) can be
represented in the form

$\displaystyle\psi_{k}(x)=u_{k}(x)\,e^{ikx},$ (7)

where $u_{k}(x)$ is a smooth and periodic function, $u_{k}(x)=u_{k}(x+L)$, and
$k\in[-\frac{\pi}{L},\frac{\pi}{L})$. Thus, we obtain the boundary conditions
for $\psi_{k}(x)$

$\displaystyle\psi_{k}(0)$ $\displaystyle=\psi_{k}(L)\,e^{-ikL},$ (8)
$\displaystyle\psi^{\prime}_{k}(0)$
$\displaystyle=\psi^{\prime}_{k}(L)\,e^{-ikL}.$ (9)

On the other hand, the representation (2) is still valid, implying a $k$
dependence of the coefficients,

$\displaystyle\psi_{k}(x)=c_{1,k}\,\psi_{1}(x)+c_{2,k}\,\psi_{2}(x).$ (10)

From the boundary conditions (8), (9) we find

$\displaystyle 0$
$\displaystyle=c_{1,k}\,[\psi_{1}(L)-e^{ikL}]+c_{2,k}\,\psi_{2}(L),$ (11)
$\displaystyle 0$
$\displaystyle=c_{1,k}\,\psi^{\prime}_{1}(L)+c_{2,k}\,[\psi^{\prime}_{2}(L)-e^{ikL}].$
(12)

To find nonzero $c_{1,k}$ and $c_{2,k}$, we claim

$\displaystyle\,[\psi_{1}(L)-e^{ikL}]\,[\psi^{\prime}_{2}(L)-e^{ikL}]-\psi_{2}(L)\,\psi^{\prime}_{1}(L)=0.$
(13)

Taking into account (6), we establish the dispersion relation

$\displaystyle\cos kL=\frac{1}{2}\,[\psi_{1}(L)+\psi^{\prime}_{2}(L)]\equiv
D(E),$ (14)

where $D(E)$ is the so called Lyapunov function. Solving (14), one
traditionally obtains the bands $\epsilon_{k\alpha}$ labelled by the band
index $\alpha$ and possessing the property
$\epsilon_{k\alpha}=\epsilon_{-k,\alpha}$ (see Fig. 2). As $E\to-\infty$, we
can neglect $V(x)$ in (1) and thereby establish that asymptotically
$D(E)\approx\cosh(L\sqrt{2m|E|})\to+\infty$. Since $D(E)$ is a continuous
function, this means that the bottom energy of the lowest band ($\alpha=1$)
satisfies the equation $D(E)=+1$. In addition, within the band $\alpha$, the
function $D(E)$ is monotonous [which follows, e.g., from the Schwarz
inequality (208)], and

$\displaystyle\text{sign}\left[\frac{\partial D(E)}{\partial
E}\right]=(-1)^{\alpha}.$ (15)

Then, from (11) we find

$\displaystyle\frac{c_{2,k}}{c_{1,k}}=-\frac{\psi_{1}(L)-e^{ikL}}{\psi_{2}(L)},$
(16)

and inserting this into (10) we obtain

$\displaystyle\psi_{k}(x)=c_{1,k}\left[\psi_{1}(x)-\frac{\psi_{1}(L)-e^{ikL}}{\psi_{2}(L)}\,\psi_{2}(x)\right]$
(17)
$\displaystyle=\frac{1}{\sqrt{N_{k}}}\left[\psi_{1}(x)\,\psi_{2}(L)-\psi_{1}(L)\,\psi_{2}(x)+\psi_{2}(x)\,e^{ikL}\right].$
(18)

Hereby we introduced the normalization $N_{k}$ by virtue of the relation

$\displaystyle\int_{0}^{L}dx\,\,|\psi_{k}(x)|^{2}=1.$ (19)

Noticing that (see Appendix A for details)

$\displaystyle-\psi_{2}(x-L)=\psi_{1}(x)\,\psi_{2}(L)-\psi_{1}(L)\,\psi_{2}(x),$
(20)

and explicitly writing the band index $\alpha$, we get the Bloch states in the
form

$\displaystyle\psi_{k\alpha}(x)$
$\displaystyle=\frac{1}{\sqrt{N_{k\alpha}}}\left[-\psi_{2}(x-L)+\psi_{2}(x)\,e^{ikL}\right],$
(21)

along with

$\displaystyle u_{k\alpha}(x)$
$\displaystyle=\frac{e^{-ikx}}{\sqrt{N_{k\alpha}}}\left[-\psi_{2}(x-L)+\psi_{2}(x)\,e^{ikL}\right].$
(22)

Here $\psi_{2}(x)=\psi_{2}(x,\epsilon_{k\alpha})$ depends on $k$ and $\alpha$
via $\epsilon_{k\alpha}$, but in the following we omit for brevity an explicit
indication of this dependence.

Note that $\psi_{k\alpha}(x)=\psi_{k+\frac{2\pi}{L},\alpha}(x)$ and
$u_{k\alpha}(x)=u_{k\alpha}(x+L)$, while the reciprocal relations are not
valid, i.e. $u_{k\alpha}(x)\neq u_{k+\frac{2\pi}{L},\alpha}(x)$,
$\psi_{k\alpha}(x)\neq\psi_{k\alpha}(x+L)$.

It is remarkable that

$\displaystyle\psi_{k\alpha}(0)=u_{k\alpha}(0)=u_{k\alpha}(L)=\frac{\psi_{2}(L)}{\sqrt{N_{k\alpha}}}$
(23)

is real. This property fixes the gauge of the Bloch states which will prove to
be advantageous for the calculation of the boundary charge in Sec. III.

In general, we also have the relations

$\displaystyle\psi^{*}_{k\alpha}(x)$ $\displaystyle=\psi_{-k,\alpha}(x),$ (24)
$\displaystyle u^{*}_{k\alpha}(x)$ $\displaystyle=u_{-k,\alpha}(x),$ (25)

which can be realised in arbitrary gauge.

Finally, we introduce the Bloch states

$\displaystyle\Psi_{k\alpha}(x)=\sqrt{\frac{L}{2\pi}}\,\psi_{k\alpha}(x)=\sqrt{\frac{L}{2\pi}}\,u_{k\alpha}(x)\,e^{ikx},$
(26)

which satisfy the following normalization and completeness relations

$\displaystyle\int
dx\,\,\Psi_{k\alpha}^{*}(x)\,\Psi_{k^{\prime}\alpha^{\prime}}(x)=\delta(k-k^{\prime})\,\delta_{\alpha\alpha^{\prime}},$
(27)
$\displaystyle\sum_{\alpha}\int_{-\pi/L}^{\pi/L}dk\,\,\Psi_{k\alpha}(x)\,\Psi_{k\alpha}^{*}(x^{\prime})=\delta(x-x^{\prime}).$
(28)

Note that the prefactor $\sqrt{L/(2\pi)}$ in (26) is needed in order to
reconcile (27) with the normalization
$\int_{0}^{L}dx\,|u_{k\alpha}(x)|^{2}=1$.

Later on the states (26) are used for constructing physical observables. Some
properties of $N_{k\alpha}$, $\epsilon_{k\alpha}$, and
$\psi_{2}(L,\epsilon_{k\alpha})$, which are also useful for this purpose, are
quoted in Appendix B.

Figure 3: Envelope function in the right half-infinite system: $f(x)=1$ for
all unit cells lying to the left of $X_{1}$, while decaying smoothly to zero
far from the boundary on a length scale $X_{2}$ much larger than the unit cell
length $L$.

##  III Boundary charge

In this section we discuss solutions of the Schrödinder equation (1) in the
half-space $x>0$. This eigenvalue problem is equipped with the open boundary
condition $\psi(x=0)=0$. Analogously to the case of 1D single-channel lattice
models [pletyukhov_etal_prb_20, ], extended eigenstates of the present
boundary problem can be constructed in terms of the Bloch states introduced in
the previous section. In addition, we construct edge states which are
exponentially localized near the boundary.

The central object of our study in this section is the boundary charge $Q_{B}$
accumulated near the boundary. For a system’s filling determined by the
chemical potential $\mu_{\nu}$ lying in the $\nu$th band gap, $Q_{B}$ receives
contributions not only from an edge state eventually residing in this gap, but
also from all occupied bands and the lower-gaps’ edge states. We extend our
previous lattice considerations [pletyukhov_etal_prb_20, ] to the case of
continuum models and establish specific expressions for all contributions to
$Q_{B}$. In particular, we provide a new rigorous proof of the surface charge
theorem also fixing an expression for the integer part of $Q_{B}$ which is
usually left unknown (cf., e.g., Ref. [vanderbilt_18, ]).

Studying the dependence of $Q_{B}$ on the modulation phase of the potential
(i.e., on the distance by which $V(x)$ is being shifted towards the boundary),
we observe a manifestation of the bulk-boundary correspondence in the spirit
of Ref. [hatsugai_1993, ]. We also firmly establish the universal linear phase
dependence of $Q_{B}$, with the slope being given by the Chern index of a band
(bands). Thereby we provide a rigorous proof to the earlier numerical findings
of Ref. [thakurati_etal_18, ]. Moreover, we reveal that in continuum models
the Chern index for each band is strictly equal to $+1$, and that the number
of bounces of an edge state energy in a certain gap between two adjacent band
edges during one pump cycle is given by the gap index.

Finally, we investigate the boundary charge fluctuations and conclude that in
the narrow-gap limit they possess universal properties satisfying the surface
fluctuation theorem which was stated in the previous works [weber_etal_prl_20,
,wannier_paper, ].

###  III.1 Eigenstates of the semi-infinite model

In this subsection we construct the extended eigenstates of (1) in the right
half-space $x>0$ and use them to evaluate each band’s contribution to the
boundary charge.

The boundary problem eigenstates can be expressed in terms of the states (26):

$\displaystyle\Psi_{\text{b},k\alpha}(x)=\Psi_{k\alpha}(x)-\Psi_{-k,\alpha}(x),\quad
k\in\left[0,\frac{\pi}{L}\right],$ (29)

They have the eigenenergies $\epsilon_{k\alpha}$ (i.e. form the same bands as
in the translationally invariant model) and satisfy the boundary condition

$\displaystyle\Psi_{\text{b},k\alpha}(0)=0.$ (30)

Their normalization reads

$\displaystyle\int_{0}^{\infty}dx\,\,\Psi^{*}_{\text{b},k\alpha}(x)\,\Psi_{\text{b},k^{\prime}\alpha^{\prime}}(x)=\delta(k-k^{\prime})\,\delta_{\alpha\alpha^{\prime}}.$
(31)

Note that the set of all $\Psi_{\text{b},k\alpha}(x)$ is not complete, since
localized edge states are also permitted. These edge states will be discussed
in Sec. III.4.

The simplicity of the form of (29) is due to the special gauge choice which is
fixed by the condition (23). Using (29) we define the boundary charge
[thakurati_etal_18, ,pletyukhov_etal_prb_20, ] associated with the band
$\alpha$

$\displaystyle Q_{B,\alpha}$
$\displaystyle=\int_{0}^{\infty}dx\,\,[\rho_{\alpha}(x)-\bar{\rho}_{\alpha}]\,f(x)$
(32)

in terms of the charge density

$\displaystyle\rho_{\alpha}(x)$
$\displaystyle=\int_{0}^{\pi/L}dk\,\,|\Psi_{\text{b},k\alpha}(x)|^{2},$ (33)

the average charge density in the bulk

$\displaystyle\bar{\rho}_{\alpha}$
$\displaystyle=\frac{1}{L}\int_{0}^{L}dx\int_{-\pi/L}^{\pi/L}dk\,\,|\Psi_{k\alpha}(x)|^{2}=\frac{1}{L},$
(34)

and the envelope function $f(x)$. The latter possesses the following
properties: 1) $f(x)\approx 1$ in the range $|x|<X_{1}$; 2) in the range
$X_{1}<|x|<X_{1}+X_{2}$ it smoothly crosses over to zero; 3) $f(x)\approx 0$
for $|x|>X_{1}+X_{2}$ (see Fig. 3). Here $X_{1,2}\gg\xi\gg L$ are scales large
in comparison with the localization length $\xi=\kappa^{-1}$ (see Sec. III.4
for the latter definition).

Figure 4: a) Band structure including right/left (solid/dashed) edge state
dispersions depending on $x_{\varphi}$ for $V(x)$ from Fig. 1. The number of
touching points with adjacent bands equals the gap index implying a net
difference of one touching point between leaving and entering edge states per
band. By virtue of the bulk-boundary correspondence [see Sections III.3 and
III.7] this fact reflects the constant band Chern index
$C_{1,\alpha}=M_{\alpha}^{(-)}-M_{\alpha}^{(+)}=1$. b) Close up of
$\epsilon_{k,2}$. Edge states in the right system ($\kappa>0$) and left system
($\kappa<0$) alternate upon traversing the touching points [see Eq. (114) and
(115)]. c) Boundary charge $\Delta
Q_{B,2}(x_{\varphi},0)=Q_{B,2}(x_{\varphi})-Q_{B,2}(0)$ [Eq. (67)] and
$\mathrm{Im}[k_{\text{e}}(x_{\varphi})]=\kappa(x_{\varphi})$ for the edge
states adjacent to $\epsilon_{k,2}$. The universal linear growth is disrupted
by discontinuous jumps at the positions of the touching points (= zeros of
$\kappa(x_{\varphi})$). Since $C_{1,\alpha}=1$ for each band we regain
periodicity under a shift $x_{\varphi}\rightarrow x_{\varphi}+L$.

It can be shown [pletyukhov_etal_prb_20, ] that the boundary charge (32)
consists of the Friedel charge

$\displaystyle Q_{F,\alpha}=$
$\displaystyle-\text{Re}\,\int_{0}^{\infty}dx\int_{-\pi/L}^{\pi/L}dk\,\,\Psi_{k\alpha}^{2}(x)$
(35) $\displaystyle=$
$\displaystyle-L\,\,\text{Re}\,\int_{0}^{L}dx\int_{-\pi/L}^{\pi/L}\frac{dk}{2\pi}\,\,\psi_{k\alpha}^{2}(x)$
$\displaystyle\times\sum_{n=1}^{\infty}e^{2i(kL+i0^{+})(n-1)L},$ (36)

which results from the charge density modulation near the boundary, and the
polarization charge

$\displaystyle Q_{P,\alpha}$
$\displaystyle=-\frac{1}{L}\int_{0}^{L}dx\,\,x\left(L\int_{-\pi/L}^{\pi/L}\frac{dk}{2\pi}\,\,|\psi_{k\alpha}(x)|^{2}-\frac{1}{L}\right)$
(37)
$\displaystyle=\frac{1}{2}-\int_{0}^{L}dx\,\,x\int_{-\pi/L}^{\pi/L}\frac{dk}{2\pi}\,\,|u_{k\alpha}(x)|^{2},$
(38)

which originates from the crossover region $X_{1}\lesssim x\lesssim
X_{1}+X_{2}$ of the envelope function $f(x)$, and it is expressed via the
dipole moment of the unit cell (see below).

The Friedel charge (36) is independent of the envelope function $f(x)$, since
the corresponding Friedel charge density decays exponentially fast on the
scale $\xi\ll X_{1}$. In turn, the polarization charge (38) relies on the
properties of $f(x)$, and it is obtained from

$\displaystyle Q_{P,\alpha}$
$\displaystyle=\int_{0}^{\infty}dx\,\,[\bar{\rho}_{\alpha}(x)-\bar{\rho}_{\alpha}]\,f(x)$
(39)
$\displaystyle=\sum_{n=0}^{\infty}\int_{0}^{L}d\bar{x}\,\,[\bar{\rho}_{\alpha}(\bar{x})-\bar{\rho}_{\alpha}]\,f(\bar{x}+nL),$
(40) $\displaystyle\bar{\rho}_{\alpha}(x)$
$\displaystyle=\int_{-\pi/L}^{\pi/L}dk\,\,|\Psi_{k\alpha}(x)|^{2}=\bar{\rho}_{\alpha}(x+L),$
(41)

in the following way. Observing that the main contribution to (40) comes from
the range $X_{1}<nL<X_{1}+X_{2}$, we expand the envelope function
$f(\bar{x}+nL)\approx f(nL)+\bar{x}f^{\prime}(nL)$. The leading contribution
identically vanishes after averaging over $\bar{x}$ (which is defined as
$x\\!\mod L$), and we get

$\displaystyle
Q_{P,\alpha}\approx\sum_{n=0}^{\infty}f^{\prime}(nL)\int_{0}^{L}d\bar{x}\,\,\bar{x}\,[\bar{\rho}_{\alpha}(\bar{x})-\bar{\rho}_{\alpha}].$
(42)

Using the properties of the envelope function, we approximately evaluate the
sum

$\displaystyle\sum_{n=0}^{\infty}f^{\prime}(nL)\approx$
$\displaystyle\frac{1}{L}\int_{0}^{\infty}dy\,\,f^{\prime}(y)$ (43)
$\displaystyle=$ $\displaystyle\frac{1}{L}[f(\infty)-f(0)]=-\frac{1}{L},$ (44)

and thus reproduce (37).

Evaluating the sum in (36) we express the Friedel charge in the form

$\displaystyle Q_{F,\alpha}$
$\displaystyle=-L\int_{0}^{L}dx\int_{-\pi/L}^{\pi/L}\frac{dk}{2\pi}\,\,\,\text{Re}\,\left\\{\frac{\psi_{k\alpha}^{2}(x)\,e^{-ikL}}{-2i\,\sin(kL+i0^{+})}\right\\}.$
(45)

Note that a convergence factor $+i0^{+}$ was added to the sum over $n$: We
exchanged this summation with the integral over $k$, and this action had to be
complemented by proper regularization.

Using a variant of the Sokhotski–Plemelj theorem

$\displaystyle\frac{1}{\sin(kL+i0^{+})}=\frac{1}{\sin
kL}-i\pi\,\delta(kL)+i\pi\,\delta(kL-\pi),$ (46)

we obtain

$\displaystyle Q_{F,\alpha}^{\text{I}}=$
$\displaystyle-L\int_{0}^{L}dx\int_{-\pi/L}^{\pi/L}\frac{dk}{2\pi}\,\,\text{Re}\,\left\\{\frac{\psi_{k\alpha}^{2}(x)\,e^{-ikL}}{-2i\,\sin
kL}\right\\},$ (47) $\displaystyle Q_{F,\alpha}^{\text{II}}=$
$\displaystyle-\frac{1}{4}\int_{0}^{L}dx\int_{-\pi/L}^{\pi/L}d(kL)$
$\displaystyle\times\text{Re}\left\\{\psi_{k\alpha}^{2}(x)\,e^{-ikL}\,\,[\delta(kL)-\delta(kL-\pi)]\right\\}$
$\displaystyle=$
$\displaystyle-\frac{1}{4}\int_{0}^{L}dx\,\,\text{Re}\left\\{\psi_{0,\alpha}^{2}(x)+\psi_{\frac{\pi}{L},\alpha}^{2}(x)\right\\}.$
(48)

Since $\psi_{0,\alpha}(x)$ and $\psi_{\frac{\pi}{L},\alpha}(x)$ are both real
and normalized by (19), we get $Q_{F,\alpha}^{\text{II}}=-\frac{1}{2}$.

In turn,

$\displaystyle Q_{F,\alpha}^{\text{I}}=$
$\displaystyle-L\int_{0}^{L}dx\int_{-\pi/L}^{\pi/L}\frac{dk}{2\pi}$
$\displaystyle\times\text{Re}\,\left\\{[\psi_{k\alpha}(x)\,e^{-ikL}-\psi_{-k,\alpha}(x)\,e^{ikL}]\right.$
$\displaystyle\qquad\times\left.\psi_{k\alpha}(x)\,e^{-ikL}\,\frac{e^{ikL}}{-2i\,\sin
kL}\right\\}$
$\displaystyle+\frac{L}{2}\int_{0}^{L}dx\int_{-\pi/L}^{\pi/L}\frac{dk}{2\pi}\,\,|\psi_{k\alpha}(x)|^{2}$
(49) $\displaystyle=$
$\displaystyle-L\int_{0}^{L}dx\int_{-\pi/L}^{\pi/L}\frac{dk}{2\pi}\,\,\frac{\psi_{2}(x-L)}{N_{k\alpha}}$
$\displaystyle\times\left[\psi_{2}(x-L)-\psi_{2}(x)\,\cos
kL\right]+\frac{1}{2}.$ (50)

So finally we get

$\displaystyle Q_{F,\alpha}=$
$\displaystyle-L\int_{0}^{L}dx\int_{-\pi/L}^{\pi/L}\frac{dk}{2\pi}\,\,\frac{\psi_{2}(x-L)}{N_{k\alpha}}$
$\displaystyle\times\left[\psi_{2}(x-L)-\psi_{2}(x)\,\cos kL\right].$ (51)

###  III.2 Berry phase expression for polarization

In this subsection we extend the observation previously made for the lattice
models in Ref. [pletyukhov_etal_prb_20, ] that the boundary charge is given by
the Zak-Berry phase [zak_89, ]. This statement is generally known as the
surface charge theorem [vanderbilt_18, ]; however the role of the gauge of the
Bloch states used for a construction of the Zak-Berry connection is usually
not elucidated. Therefore this theorem is often stated up to an unknown
integer which can be changed by a winding number of the phase of the gauge
transformation.

The importance of the gauge (22) (i.e., with the real last component of the
Bloch vector $u_{k\alpha}$) was first recognized in Refs.
[pletyukhov_etal_prbr_20, ,pletyukhov_etal_prb_20, ]. In particular, it was
shown therein that while working in this specific gauge no additional integer
contribution to $Q_{B}$ arises in lattice models. Below we show that this also
holds true for continuum models.

Let us consider the expression

$\displaystyle\text{Im}\,\left[u_{k\alpha}^{*}(x)\,e^{ik(L-x)}\,\frac{d}{dk}u_{k\alpha}(x)\,e^{-ik(L-x)}\right]$
(52) $\displaystyle=$
$\displaystyle\,\,\text{Im}\,\left[\psi_{k\alpha}^{*}(x)\,e^{ikL}\,\frac{\partial}{\partial
k}\psi_{k\alpha}(x)\,e^{-ikL}\right]$
$\displaystyle+\frac{d\epsilon_{k\alpha}}{dk}\,\,\text{Im}\,\left[\psi_{k\alpha}^{*}(x)\,\frac{\partial}{\partial
E}\psi_{k\alpha}(x)\right]$ (53) $\displaystyle=$
$\displaystyle-\frac{L}{N_{k\alpha}}\left[\psi_{2}(x-L)-\psi_{2}(x)\,\cos
kL\right]\,\psi_{2}(x-L)$
$\displaystyle+\frac{1}{N_{k\alpha}}\frac{d\epsilon_{k\alpha}}{dk}\,\sin
kL\,\,F(x),$ (54)

where

$\displaystyle F(x)=\psi_{2}(x)\,\frac{\partial\psi_{2}(x-L)}{\partial
E}-\psi_{2}(x-L)\,\frac{\partial\psi_{2}(x)}{\partial E}.$ (55)

Due to the property

$\displaystyle\int_{0}^{L}dx\,F(x)=0,$ (56)

whose proof is given in Appendix A, we can write

$\displaystyle Q_{F,\alpha}=$
$\displaystyle\int_{0}^{L}dx\int_{-\pi/L}^{\pi/L}\frac{dk}{2\pi}\,\,\text{Im}\,\left[u_{k\alpha}^{*}(x)\,\frac{d}{dk}u_{k\alpha}(x)\right]$
$\displaystyle-\int_{0}^{L}dx\,\,(L-x)\int_{-\pi/L}^{\pi/L}\frac{dk}{2\pi}\,\,|u_{k\alpha}(x)|^{2}.$
(57)

Combining this expression with (38), we obtain

$\displaystyle Q_{B,\alpha}$
$\displaystyle=-\frac{1}{2}+\int_{-\pi/L}^{\pi/L}\frac{dk}{2\pi}\,\,A_{k\alpha},$
(58) $\displaystyle A_{k\alpha}$
$\displaystyle=-i\int_{0}^{L}dx\,\,u_{k\alpha}^{*}(x)\,\frac{d}{dk}u_{k\alpha}(x),$
(59)

where the last line expresses the Zak-Berry connection.

###  III.3 Universal properties of the boundary charge

The interesting behaviour of $Q_{B}$ under the shift of a potential towards
the boundary was initially observed in numerical studies of Ref.
[thakurati_etal_18, ]. It was revealed that $Q_{B}$ linearly depends on the
distance of the shift (i.e. on the potential modulation phase), the slope
acquiring quantized universal values. In this subsection we provide an
analytic proof of the linear dependence and discuss other universal properties
of the boundary charge.

Suppose we shift our system towards the wall by $x_{\varphi}$. It means that
we have to consider the boundary problem with a new potential

$\displaystyle V^{(\varphi)}(x)=V(x+x_{\varphi}).$ (60)

The new solution $\psi_{k\alpha}^{(\varphi)}(x)$ is related to the old one
$\psi_{k\alpha}(x)$ by

$\displaystyle\psi_{k\alpha}^{(\varphi)}(x)$
$\displaystyle=\psi_{k\alpha}(x+x_{\varphi})\,\,e^{-i\Phi_{k\alpha}(x_{\varphi})},$
(61) $\displaystyle u_{k\alpha}^{(\varphi)}(x)$
$\displaystyle=u_{k\alpha}(x+x_{\varphi})\,\,e^{ikx_{\varphi}}\,\,e^{-i\Phi_{k\alpha}(x_{\varphi})}.$
(62)

To enable the usage of (58), (59) in the shifted system, we must claim
similarly to (23) that $\psi_{k\alpha}^{(\varphi)}(0)$ is real. This condition
fixes (up to the irrelevant sign) the phase factor

$\displaystyle
e^{i\Phi_{k\alpha}(x_{\varphi})}\sim\psi_{k\alpha}(x_{\varphi})=u_{k\alpha}(x_{\varphi})\,\,e^{ikx_{\varphi}}.$
(63)

Note that it is periodic in $k$ due to the analogous property of
$\psi_{k\alpha}(x)$. In addition,

$\displaystyle
e^{i\Phi_{k\alpha}(x_{\varphi}+L)}=e^{i\Phi_{k\alpha}(x_{\varphi})}\,\,e^{ikL}.$
(64)

The boundary charge in the shifted system can be written with respect to the
reference system value $Q_{B,\alpha}(0)\equiv Q_{B,\alpha}$ as

$\displaystyle\Delta Q_{B,\alpha}(x_{\varphi},0)$
$\displaystyle=Q_{B,\alpha}(x_{\varphi})-Q_{B,\alpha}(0)$ (65)
$\displaystyle=\int_{-\pi/L}^{\pi/L}\frac{dk}{2\pi}\,\,[A_{k\alpha}^{(\varphi)}-A_{k\alpha}]$
(66)
$\displaystyle=\frac{x_{\varphi}}{L}-\text{wn}\,[e^{i\Phi_{k\alpha}(x_{\varphi})}],$
(67)

where

$\displaystyle\text{wn}\,[e^{i\Phi_{k\alpha}(x_{\varphi})}]=-\text{wn}\,[e^{-i\Phi_{k\alpha}(x_{\varphi})}]$
(68)
$\displaystyle=-\frac{i}{2\pi}\int_{-\pi/L}^{\pi/L}dk\,\,e^{-i\Phi_{k\alpha}(x_{\varphi})}\,\frac{d}{dk}e^{i\Phi_{k\alpha}(x_{\varphi})}$
(69)

is the winding number of the corresponding phase factor. In this derivation we
used the periodicity of $u_{k\alpha}(x)$ in $x$. Due to the property (64), the
boundary charge $Q_{B,\alpha}(x_{\varphi})$ is manifestly periodic under a
shift $x_{\varphi}\to x_{\varphi}+L$, as it should be.

Piecewise, i.e. outside the points where the winding number jumps, the slope

$\displaystyle
L\,\frac{dQ_{B,\alpha}(x_{\varphi})}{dx_{\varphi}}=C_{1,\alpha}=1$ (70)

is universal and coincides with the first Chern index of the band $\alpha$
defined on the two-dimensional torus $(k,x_{\varphi})$:

$\displaystyle C_{1,\alpha}=$
$\displaystyle-\int_{-\pi/L}^{\pi/L}\frac{dk}{2\pi}\int_{0}^{L}dx_{\varphi}$
$\displaystyle\times
2\,\text{Im}\int_{0}^{L}dx\,\,\frac{d\psi_{k\alpha}^{(\varphi)\,*}(x)}{dk}\,\,\frac{d\psi_{k\alpha}^{(\varphi)}(x)}{dx_{\varphi}}.$
(71)

This index is a sign-weighted count of the phase singular points across the
band $\alpha$ [kohmoto_1985, ,pletyukhov_etal_prb_20, ], thus giving (up to
the sign) the net change of the winding number over one period in
$x_{\varphi}$. The value of $C_{1,\alpha}$ must be synced — as done in the
first equality of (70) — with the slope of the linear part of
$Q_{B,\alpha}(x_{\varphi})$ in order to maintain the periodicity
$Q_{B,\alpha}(x_{\varphi})=Q_{B,\alpha}(x_{\varphi}+L)$. In Appendix C we show
by explicit evaluation of (71) that $C_{1,\alpha}=1$ for each band, i.e. prove
the fulfillment of the second equality in (70).

The result (70) differs to some extent from the findings of Ref.
[pletyukhov_etal_prb_20, ] for a lattice model with $Z$ sites per unit cell.
There, the slope (with respect to $\frac{\varphi}{2\pi}$) could take integer
values $C_{1,\alpha}=1+s_{\alpha}Z$ with some integer $s_{\alpha}$. The limit
$Z\to\infty$ should bring us to the case of the presently studied continuum
models. In order to get finite Chern indices, it is necessary to ensure that
upon taking this limit the potential as a function of the phase variable does
not make any kinks between the lattice sites. This procedure ensures
$s_{\alpha}=0$ and yields the result $C_{1,\alpha}=1$ which is consistent with
that of our present consideration.

The bulk-boundary correspondence [hatsugai_1993, ] implies that the found
value of $C_{1,\alpha}$ should be supported by the same value for the
difference between numbers of edge states entering and leaving the band
$\alpha$ on the interval $0\leq x_{\varphi}<L$. We inspect this property in
the next subsection.

To complete the discussion of the boundary charge we also outline expressions
for the boundary charge in the left subsystem

$\displaystyle Q_{B,\alpha}^{L}$
$\displaystyle=\int_{-\infty}^{0}dx\,\,[\rho_{\alpha}(x)-\bar{\rho}_{\alpha}]\,f(x),$
(72) $\displaystyle=Q_{F,\alpha}^{L}+Q_{P,\alpha}^{L}=-1-Q_{B,\alpha},$ (73)

where

$\displaystyle Q_{F,\alpha}^{L}=$
$\displaystyle-\int_{-\infty}^{0}dx\int_{-\pi/L}^{\pi/L}dk\,\,\text{Re}\,\Psi_{k\alpha}^{2}(x)$
(74) $\displaystyle=$
$\displaystyle-L\,\,\text{Re}\,\int_{0}^{L}dx\int_{-\pi/L}^{\pi/L}\frac{dk}{2\pi}\,\,\psi_{k\alpha}^{2}(x)$
$\displaystyle\times\sum_{n=-\infty}^{0}e^{2i(kL-i0^{+})(n-1)}$ (75)
$\displaystyle=$ $\displaystyle-1-Q_{F,\alpha},$ (76) $\displaystyle
Q_{P,\alpha}^{L}=$ $\displaystyle-Q_{P,\alpha}.$ (77)

For simplicity we choose $f(x)=f(-x)$ in (72). In general, the values of
$X_{1,2}$ may however differ for each half-space, but this does not affect the
results (76) and (77).

###  III.4 Edge states

Besides the eigenstates (29) the semi-infinite model with the open boundary
condition possesses edge localized states. We notice that the function (18)
vanishes at $x=0$, if the condition

$\displaystyle\psi_{2}(L)=0$ (78)

is fulfilled. This can not happen within the energy range of any band, since
according to the property (221) the function $\psi_{2}(L)$ is sign-definite
there. Therefore, the condition (78) can be only fulfilled in band gaps. For
this to happen we should allow for complex valued $k$.

The other condition (14) relating $k$ and $E$ to each other contains the
function $D(E)$, which is real at every $E$. This imposes the constraint that
complex-valued $k$ can only be of the form either $k=\pm\pi/L+i\kappa$ or
$k=i\kappa$, both with real $\kappa$.

The condition (6) is also fulfilled at every $E$. Combining it with (78), we
obtain

$\displaystyle\psi_{1}(L)=\frac{1}{\psi^{\prime}_{2}(L)}.$ (79)

Furthermore, from (14) it follows

$\displaystyle(-1)^{s}\,\cosh\kappa
L=\frac{1}{2}\left(\psi_{1}(L)+\frac{1}{\psi_{1}(L)}\right).$ (80)

Here $s=0$ corresponds to the choice $\text{Re}\,k=0$, and $s=1$ corresponds
to the choice $\text{Re}\,k=\pm\pi/L$.

Equation (80) has two solutions: $\psi_{1}(L)=(-1)^{s}\,e^{\kappa L}$ and
$\psi_{1}(L)=(-1)^{s}\,e^{-\kappa L}$. They give the following eigenfunctions
of edge states

$\displaystyle\psi_{\text{e}}(x)$
$\displaystyle\sim[-\psi_{1}(L)+e^{ikL}]\,\,\psi_{2}(x)$
$\displaystyle=-2\,(-1)^{s}\,\sinh\kappa L\,\,\psi_{2}(x),$ (81)
$\displaystyle\psi_{\text{e}}(x)$
$\displaystyle\sim[-\psi_{1}(L)+e^{ikL}]\,\,\psi_{2}(x)\equiv 0,$ (82)

respectively. Thus, a nontrivial solution has

$\displaystyle e^{\kappa L}=(-1)^{s}\,\psi_{1}(L)>0.$ (83)

Labelling each band gap by the index $\alpha$ of the band which lies beneath
it, we search for the relation between $s$ and $\alpha$. Comparing (206) and
(207), we find that at the edge state energy $E=\epsilon_{\text{e},\alpha}$ it
holds

$\displaystyle\text{sign}\,\,\psi_{1}(L,\epsilon_{\text{e},\alpha})=\text{sign}\,\frac{\partial\psi_{2}(L,\epsilon_{\text{e}})}{\partial
E}=\text{sign}\,\,D(\epsilon_{\text{e},\alpha}).$ (84)

But $\text{sign}\,D(\epsilon_{\text{e},\alpha})=(-1)^{\alpha}$ (see Fig. 2),
and therefore

$\displaystyle(-1)^{s}=(-1)^{\alpha},$ (85)

i.e. edge states residing in odd gaps have $\text{Re}\,k=\pm\pi/L$, while edge
states residing in even gaps have $\text{Re}\,k=0$.

Being primarily interested in the right subsystem ($x>0$), we impose the
condition $\kappa>0$ in order to have a normalizable (decaying) solution. From
(83) it follows that a nontrivial edge state solution in the right subsystem
exists, if

$\displaystyle|\psi_{1}(L)|>1\quad\Rightarrow\quad\kappa=\frac{1}{L}\,\ln|\psi_{1}(L)|>0.$
(86)

In turn, a nontrivial edge state solution in the left subsystem exists, if

$\displaystyle|\psi_{1}(L)|<1\quad\Rightarrow\quad\kappa=\frac{1}{L}\,\ln|\psi_{1}(L)|<0.$
(87)

In Appendix D we prove that in each band gap there is precisely one edge
state, corresponding either to the right subsystem ($x>0$, $\kappa>0$) or to
the left subsystem ($x<0$, $\kappa<0$). Its normalized wavefunction reads

$\displaystyle\Psi_{\text{e},\alpha}(x)=\frac{\Theta(x\,\,\text{sign}\,\kappa)}{\sqrt{N_{\text{e},\alpha}}}\,\psi_{2}(x),$
(88)

with the corresponding normalizations

$\displaystyle N_{\text{e},\alpha}$
$\displaystyle=\int_{0}^{\infty}dx\,\,\psi_{2}^{2}(x)=\sum_{n=0}^{\infty}\int_{0}^{L}dx\,\,\psi_{2}^{2}(x+nL)$
$\displaystyle=\frac{(-1)^{\alpha}}{4m}\,\frac{\partial\psi_{2}(L)}{\partial
E}\,\frac{1}{\sinh\kappa L},\quad\kappa>0,$ (89) $\displaystyle
N_{\text{e},\alpha}$
$\displaystyle=\int_{-\infty}^{0}dx\,\,\psi_{2}^{2}(x)=\sum_{n=0}^{\infty}\int_{-L}^{0}dx\,\,\psi_{2}^{2}(x-nL)$
$\displaystyle=-\frac{(-1)^{\alpha}}{4m}\,\frac{\partial\psi_{2}(L)}{\partial
E}\,\frac{1}{\sinh\kappa L},\quad\kappa<0.$ (90)

These expressions are derived with help of (191), (205) and (20), (206). We
remark that at the edge state energy $E=\epsilon_{\text{e},\alpha}$ it always
holds $(-1)^{\alpha}\,\frac{\partial\psi_{2}(L)}{\partial E}>0$ (see the
discussion in Appendix D).

###  III.5 Phase dependence of edge states

If we shift the potential $V(x)$ towards the hard wall by $x_{\varphi}$, the
edge state energy $\epsilon_{\text{e},\alpha}$ will change its position within
the band gap $\alpha$. At some values of $x_{\varphi}$ the function
$\epsilon_{\text{e},\alpha}(x_{\varphi})$ can touch either lower or upper
bands adjacent to the corresponding band gap. In the following we study the
edge state energy dependence on $x_{\varphi}$.

We start with expressing the fundamental system
$\\{\psi_{1}^{(\varphi)}(x),\psi_{2}^{\varphi}(x)\\}$ of the shifted model
with $V^{(\varphi)}(x)=V(x+x_{\varphi})$ in terms of the original reference
system $\\{\psi_{1}(x),\psi_{2}(x)\\}$:

$\displaystyle\psi_{1}^{(\varphi)}(x)$
$\displaystyle=c_{11}\,\psi_{1}(x+x_{\varphi})+c_{12}\,\psi_{2}(x+x_{\varphi}),$
(91) $\displaystyle\psi_{2}^{(\varphi)}(x)$
$\displaystyle=c_{21}\,\psi_{1}(x+x_{\varphi})+c_{22}\,\psi_{2}(x+x_{\varphi}).$
(92)

Applying the defining relations

$\displaystyle\psi_{1}^{(\varphi)}(0)$
$\displaystyle=1,\quad\psi_{1}^{(\varphi)\,\prime}(0)=0,$ (93)
$\displaystyle\psi_{2}^{(\varphi)}(0)$
$\displaystyle=0,\quad\psi_{2}^{(\varphi)\,\prime}(0)=1,$ (94)

we obtain the equations

$\displaystyle 1$
$\displaystyle=c_{11}\,\psi_{1}(x_{\varphi})+c_{12}\,\psi_{2}(x_{\varphi}),$
(95) $\displaystyle 0$
$\displaystyle=c_{21}\,\psi^{\prime}_{1}(x_{\varphi})+c_{22}\,\psi^{\prime}_{2}(x_{\varphi}),$
(96)

and

$\displaystyle 0$
$\displaystyle=c_{21}\,\psi_{1}(x_{\varphi})+c_{22}\,\psi_{2}(x_{\varphi}),$
(97) $\displaystyle 1$
$\displaystyle=c_{21}\,\psi^{\prime}_{1}(x_{\varphi})+c_{22}\,\psi^{\prime}_{2}(x_{\varphi}).$
(98)

Solving them we get

$\displaystyle\left(\begin{array}[]{cc}c_{11}&c_{12}\\\
c_{21}&c_{22}\end{array}\right)=\left(\begin{array}[]{cc}\psi^{\prime}_{2}(x_{\varphi})&-\psi^{\prime}_{1}(x_{\varphi})\\\
-\psi_{2}(x_{\varphi})&\psi_{1}(x_{\varphi})\end{array}\right).$ (103)

An immediate consequence of this solution is that the function $D(E)$ is
independent of $x_{\varphi}$: Considering

$\displaystyle D^{(\varphi)}(E)=$
$\displaystyle\frac{1}{2}\left\\{\psi_{1}^{(\varphi)}(L)+\psi_{2}^{(\varphi)\,\prime}(L)\right\\}$
(104) $\displaystyle=$
$\displaystyle\frac{1}{2}\left\\{\psi^{\prime}_{2}(x_{\varphi})\,\psi_{1}(L+x_{\varphi})-\psi^{\prime}_{1}(x_{\varphi})\,\psi_{2}(L+x_{\varphi})\right.$
$\displaystyle\left.-\psi_{2}(x_{\varphi})\,\psi^{\prime}_{1}(L+x_{\varphi})+\psi_{1}(x_{\varphi})\,\psi^{\prime}_{2}(L+x_{\varphi})\right\\},$

and employing (190), (191), and (6) we obtain

$\displaystyle
D^{(\varphi)}(E)=\frac{1}{2}\left\\{\psi_{1}(L)+\psi^{\prime}_{2}(L)\right\\}=D(E).$
(105)

Due to this property all band edges are flat with respect to $x_{\varphi}$ in
Fig. 4. We also note that the relation (61) implies the equality

$\displaystyle\frac{\psi_{k\alpha}^{(\varphi)\,\prime}(0)}{\psi_{k\alpha}^{(\varphi)}(0)}=\frac{\psi^{\prime}_{k\alpha}(x_{\varphi})}{\psi_{k\alpha}(x_{\varphi})}.$
(106)

With help of (91), (92), (103) and (192)-(195) we verify that it actually
holds for all energies.

The edge state conditions in the shifted system

$\displaystyle\psi_{2}^{(\varphi)}(L)$ $\displaystyle=0,$ (107)
$\displaystyle\psi_{1}^{(\varphi)}(L)$
$\displaystyle=(-1)^{\alpha}\,e^{\kappa(x_{\varphi})L}$ (108)

are expressed with help of the above relations as

$\displaystyle\frac{\psi_{1}(L+x_{\varphi})}{\psi_{1}(x_{\varphi})}=\frac{\psi_{2}(L+x_{\varphi})}{\psi_{2}(x_{\varphi})}=(-1)^{\alpha}\,e^{\kappa(x_{\varphi})L}.$
(109)

Solving them we find the edge state dispersions
$\epsilon_{\text{e},\alpha}(x_{\varphi})$ and showcase them in Fig. 4(a,b).

###  III.6 Band-touching points and vorticities

At the band-touching point $x^{*}_{\varphi}$ the condition

$\displaystyle\kappa(x_{\varphi}^{*})=0$ (110)

is fulfilled. Following Ref. [hatsugai_1993, ], we assign to this point the
vorticity

$\displaystyle\text{sign}\,\frac{d\kappa(x_{\varphi}^{*})}{dx_{\varphi}}.$
(111)

Using in the following the right system ($x>0$) as the reference one, we
observe that the vorticity $-1$ counts an edge state entering the band (change
of $\kappa$ from positive to negative), while the vorticity $+1$ counts an
edge state leaving the band (change of $\kappa$ from negative to positive).

Evaluating (111) on the basis of (109) and (190), (191) we obtain

$\displaystyle\text{sign}\,\frac{d\kappa(x_{\varphi}^{*})}{dx_{\varphi}}$
$\displaystyle=(-1)^{\alpha}\,\text{sign}\,\left[\psi^{\prime}_{1}(L)\,\frac{d}{dx_{\varphi}}\frac{\psi_{2}(x_{\varphi}^{*})}{\psi_{1}(x_{\varphi}^{*})}\right]$
$\displaystyle=(-1)^{\alpha}\,\text{sign}\,\left[\psi^{\prime}_{1}(L)\right]$
(112)

and

$\displaystyle\text{sign}\,\frac{d\kappa(x_{\varphi}^{*})}{dx_{\varphi}}$
$\displaystyle=(-1)^{\alpha}\,\text{sign}\,\left[\psi_{2}(L)\,\frac{d}{dx_{\varphi}}\frac{\psi_{1}(x_{\varphi}^{*})}{\psi_{2}(x_{\varphi}^{*})}\right]$
$\displaystyle=-(-1)^{\alpha}\,\text{sign}\,\left[\psi_{2}(L)\right].$ (113)

From these expressions (which are equivalent, see below) we make an important
conclusion: The vorticity value is a property of the band edge, i.e. it has
the same value for all touching points at the same band edge. Distinguishing
between the top edge of the band $\alpha$ and the bottom edge of the band
$(\alpha+1)$, both facing the same band gap $\alpha$, and using (221) we
establish

$\displaystyle\text{sign}\,\frac{d\kappa(x_{\varphi}^{*})}{dx_{\varphi}}\bigg{|}_{\alpha+1,\,\text{bottom}}$
$\displaystyle=-1,$ (114)
$\displaystyle\text{sign}\,\frac{d\kappa(x_{\varphi}^{*})}{dx_{\varphi}}\bigg{|}_{\alpha,\,\text{top}}$
$\displaystyle=+1.$ (115)

Due to the continuity of $\epsilon_{\text{e},\alpha}(x_{\varphi})$ in
$x_{\varphi}$ the edge state trajectory for the right system originates in the
lower band (at the top of the band $\alpha$) and terminates in the upper band
(at the bottom of the band $\alpha+1$).

To demonstrate the equivalence of (112) and (113), we relate the functions
$\psi^{\prime}_{1}(L)$ and $\psi_{2}(L)$ at the band edge, where
$D(E)=(-1)^{\alpha}$, via

$\displaystyle\psi^{\prime}_{1}(L)\,\psi_{2}(L)$
$\displaystyle=-1+\psi^{\prime}_{2}(L)\,\psi_{1}(L)$
$\displaystyle=-1+[2(-1)^{\alpha}-\psi_{1}(L)]\,\,\psi_{1}(L)$
$\displaystyle=-[(-1)^{\alpha}-\psi_{1}(L)]^{2}\,\leq\,0.$ (116)

This means that
$\text{sign}\,[\psi^{\prime}_{1}(L)]=-\text{sign}\,[\psi_{2}(L)]$ (unless we
have a touching point at $x_{\varphi}^{*}=0$, which can be always excluded by
choosing a different reference system).

The observation about the vorticity values made above allows us to conclude
that

$\displaystyle M_{\alpha}^{(-)}=M_{\alpha+1}^{(+)},$ (117)

where $M_{\alpha}^{(\pm)}$ are the numbers of edge states which enter/leave
the band $\alpha$ (labels correspond to vorticities $\mp 1$). These properties
can be recognized in Fig. 4, where, within a gap, the edge dispersions touch
the adjacent bands the same number of times thus demonstrating the property
(117).

###  III.7 Bulk-boundary correspondence

To prove the bulk-boundary correspondence, we have to show that

$\displaystyle M_{\alpha}^{(-)}-M_{\alpha}^{(+)}=C_{1,\alpha}=+1.$ (118)

Let us prove that

$\displaystyle M_{\alpha}^{(-)}$ $\displaystyle=\alpha,$ (119) $\displaystyle
M_{\alpha}^{(+)}$ $\displaystyle=\alpha-1.$ (120)

These relations automatically imply (118).

First, we summarize the conditions satisfied at the touching points. Combining
(109) and (110) as well as using (190) and (191) we derive

$\displaystyle\psi_{2}(L)\,\psi_{1}(x^{*}_{\varphi})+\psi_{2}(x^{*}_{\varphi})\,\psi^{\prime}_{2}(L)$
$\displaystyle=(-1)^{\alpha}\,\psi_{2}(x^{*}_{\varphi}).$ (121)

Eliminating $\psi^{\prime}_{2}(L)=2(-1)^{\alpha}-\psi_{1}(L)$, we express this
condition in the form

$\displaystyle\psi_{1}(x_{\varphi}^{*})\,\psi_{2}(L)-[\psi_{1}(L)-(-1)^{\alpha}]\,\psi_{2}(x_{\varphi}^{*})=0.$
(122)

Hereby one can recognize an equation for the roots of the periodic (even
$\alpha$) and antiperiodic (odd $\alpha$) solutions (18) of Eq. (1).

A problem of counting zeros of periodic and antiperiodic solutions of a linear
differential equation with periodic coefficients has a long tradition in the
mathematical literature [arscott, ; ince, ; mckean, ]. Adapting the existing
statements and their proofs initially formulated in Ref. [ince, ] to our
present notations (see Appendix E for details), we eventually conclude that

$\displaystyle M^{(-)}_{\alpha}=M_{\alpha+1}^{(+)}=\alpha.$ (123)

This relation is equivalent to (119) and (120), and the bulk-boundary
correspondence (118) is confirmed.

The discussed property is demonstrated in Fig. 4(c): In order to ensure the
periodicity of $Q_{B,\alpha}(x_{\varphi})$ which grows linearly in
$x_{\varphi}$ with the slope $\frac{1}{L}$ on the interval $L$, there must be
an overall discontinuous jump by $-1$. The jumps occur only when an edge state
either leaves or enters the band. Thus, the bulk-boundary correspondence (118)
expresses the necessary charge balance during a pump cycle.

###  III.8 Total boundary charge

Setting the chemical potential $\mu_{\nu}=\epsilon_{\frac{\nu\pi}{L},\nu}$ at
the top of the uppermost valence band, we sum up all contributions to the
total boundary charge, coming from both the bands and the edge states. Thus we
get

$\displaystyle
Q_{B}^{(\nu)}=\sum_{\alpha=1}^{\nu}Q_{B,\alpha}+\sum_{\alpha=1}^{\nu-1}Q_{\text{e},\alpha}.$
(124)

The edge state contribution from the gap $\alpha$ equals
$Q_{\text{e},\alpha}=1$, when $\kappa_{\alpha}>0$, and zero otherwise. Based
on the fact that the boundary charge for a single band is periodic in
$x_{\varphi}$ (see Sec. III.3) we recognize that this also holds true for the
total boundary charge. Furthermore, considering the individual contributions
to Eq. (124), it can be seen that jumps in $Q_{B,\alpha}$ occurring at the
touching points are compensated by edge states themselves, besides those jumps
which occur at the topmost band edge below $\mu_{\nu}$. Therefore,

$\displaystyle
Q_{B}^{(\nu)}(x_{\varphi})=Q_{B}^{(\nu)}(0)+\frac{\nu}{L}x_{\varphi}-\sum_{j=1}^{\nu}\Theta(x_{\varphi,j}^{*}),$
(125)

where $\\{x_{\varphi,j}^{*}\\}$ is a set of touching points at which an edge
state leaves the top edge of the uppermost valence band $\nu$. According to
(123) there are precisely $\nu$ such points, and this is the number needed to
ensure the periodicity
$Q_{B}^{(\nu)}(x_{\varphi})=Q_{B}^{(\nu)}(x_{\varphi}+L)$. Fig. 5 illustrates
this behaviour which was already observed by the authors of
[thakurati_etal_18, ] in a lattice model with quite large $Z=10$ rendering
their results to be tractable in terms of the present continuum theory.

Figure 5: Total boundary charge for $\mu=\epsilon_{\frac{\pi}{L},3}$.
According to Eq. (123), $\nu=3$ jumps downward are observed at the positions
of the touching points with the adjacent band edge hence restoring periodicity
over a shift $x_{\varphi}\rightarrow x_{\varphi}+L$.

Following the previous studies [weber_etal_prl_20, ,wannier_paper, ], we quote
the expression for the fluctuations of (125)

$\displaystyle X_{2}\,(\Delta Q_{B}^{(\nu)})^{2}$
$\displaystyle=\sum_{\alpha,\beta=1}^{\nu}\int_{-\pi/L}^{\pi/L}\frac{dk}{2\pi}\,\,(\mathcal{Q}_{k})_{\alpha\beta}.$
(126)

Here $(\mathcal{Q}_{k})_{\alpha\beta}$ is the geometric tensor [see (237)],
and the relation

$\displaystyle\frac{1}{X_{2}}=\int_{-\infty}^{\infty}dx\,\,[f^{\prime}(x)]^{2}$
(127)

quantifies the crossover range of the envelope function $f(x)$. In the narrow-
gap limit we re-derive (see Appendix F) the previously established universal
low-energy scaling of (126) – the so called surface fluctuation theorem
[weber_etal_prl_20, ,wannier_paper, ]

$\displaystyle X_{2}\,(\Delta
Q_{B}^{(\nu)})^{2}\approx\frac{v_{F,\nu}}{8E_{g,\nu}},$ (128)

expressed in terms of the Fermi velocity $v_{F,\nu}$ and the gap value
$E_{g,\nu}$.

##  IV Interface charge

This section discusses the properties of interface charges $Q_{I}$ that
accumulate at interfaces with nontrivial conjunctions of two half-infinite
subsystems. Similarly to our proceedings in Sec. III and particularly in Ref.
[RatBCPlet, ] (see Appendix C therein) we determine the scattering eigenstates
by modelling the interface microscopically. This construction, in turn,
provides the scattering matrix that has already been shown to be determining
for the properties at the interface [fulga_2011, ,fulga_2012, ]. In addition,
we construct interface localized states — exponentially localized states with
bilateral support. Energetically these states reside in the gaps between bands
formed by the scattering eigenstates.

At the interface we allow for a mutual phase mismatch $x_{\varphi}\neq
x_{\varphi\prime}$ of the potentials in the two subsystems on the right and
left in addition to a local impurity potential $\lambda\,\delta(x)$. Fixing
the value of $x_{\varphi^{\prime}}$ and cyclically pumping $x_{\varphi}$, we
establish a number of similarities between $Q_{I}(x_{\varphi})$ and the
boundary charge discussed in the previous section. In particular, we derive
for $Q_{I}(x_{\varphi})$ analogs of the universal expressions (67) and (125).
In doing so we identify a function $\tilde{d}_{k\alpha}(x_{\varphi})$ whose
windings and phase singular points play a tremendous determining role in
describing a spectral flow of the interface problem, not to speak of the
equation $\tilde{d}_{k\alpha}(x_{\varphi})=0$ determining the interface
localized states.

It is then recognized that the linear change of $Q_{I}(x_{\varphi})$ is
associated with an incomplete compensation of the dipole-moment contributions
(38) from the right and left subsystems, which accords with the main motif of
Ref. [RatBCPlet, ]. Other changes to $Q_{I}(x_{\varphi})$ are found to be
discontinuously integer-valued, and this serves as an additional justification
for the nearsightedness principle, which was heuristically postulated in Ref.
[NearSightednessPrin, ].

###  IV.1 Scattering eigenstates

We consider the periodic potential $V^{(\varphi)}(x)=V(x+x_{\varphi})$ in the
right half-space, and the periodic potential
$V^{(\varphi^{\prime})}(x)=V(x+x_{\varphi^{\prime}})$ in the left half-space.
In addition, we add an extra impurity potential at the interface
$V_{\text{imp}}(x)=\lambda\,\delta(x)$. The overall interface potential used
in numerical calculations is depicted in Fig. 6.

In this subsection we construct scattering eigenstates of this model using the
two bases, $\Psi_{k\alpha}^{(\varphi)}(x)$ and
$\Psi_{k\alpha}^{(\varphi^{\prime})}(x)$, which are the eigenstates of the
bulk eigenvalue problems with the corresponding periodic potentials. Note that
these bulk eigenvalue problems are isospectral, since they are mutually
related by the unitary transformation — the translation by
$x_{\varphi}-x_{\varphi^{\prime}}$. Therefore, the bands $\epsilon_{k\alpha}$
of the scattering states in the interface problem also coincide with those of
the bulk models.

We make the following ansatz for the scattering eigenstates

$\displaystyle\Psi_{k\alpha}^{(r)}(x)$
$\displaystyle=\Theta(-x)\,\left[\Psi_{k\alpha}^{(\varphi^{\prime})}(x)+r_{k\alpha}\,\Psi_{-k,\alpha}^{(\varphi^{\prime})}(x)\right]$
$\displaystyle+\Theta(x)\,\,t_{k\alpha}\,\Psi_{k\alpha}^{(\varphi)}(x),$ (129)
$\displaystyle\Psi_{k\alpha}^{(l)}(x)$
$\displaystyle=\Theta(-x)\,\,t^{\prime}_{k\alpha}\,\Psi_{-k,\alpha}^{(\varphi^{\prime})}(x)$
$\displaystyle+\Theta(x)\left[\Psi_{-k,\alpha}^{(\varphi)}(x)+r^{\prime}_{k\alpha}\,\Psi_{k\alpha}^{(\varphi)}(x)\right],$
(130)

labeled by $k\in[0,\frac{\pi}{L}]$ and the band index $\alpha$.

Figure 6: Potential
$\Theta(x)\,V(x+x_{\varphi})+\Theta(-x)\,V(x+x_{\varphi^{\prime}})+\lambda\,\delta(x)$
of the interface problem (black) and exemplary interface localized state
(green) belonging to the gap $\alpha=1$ and evaluated for the parameters
$x_{\varphi}=0.1L$, $x_{\varphi^{\prime}}=0$, and $\lambda=1$.

To establish the transmission ($t_{k\alpha},t^{\prime}_{k\alpha}$) and
reflection ($r_{k\alpha},r^{\prime}_{k\alpha}$) amplitudes, we employ the
wavefunction’s matching conditions at the interface

$\displaystyle\Psi_{k\alpha}^{(\eta)}(0^{-})$
$\displaystyle=\Psi_{k\alpha}^{(\eta)}(0^{+}),$ (131)
$\displaystyle\frac{d\Psi_{k\alpha}^{(\eta)}}{dx}(0^{+})-\frac{d\Psi_{k\alpha}^{(\eta)}}{dx}(0^{-})$
$\displaystyle=2m\lambda\,\Psi_{k\alpha}^{(\eta)}(0),$ (132)

where $\eta=r,l$. Thereby we get

$\displaystyle
t_{k\alpha}\,\psi_{k\alpha}^{(\varphi)}(0)-r_{k\alpha}\,\psi_{-k,\alpha}^{(\varphi^{\prime})}(0)=\psi_{k\alpha}^{(\varphi^{\prime})}(0),$
(133) $\displaystyle
t_{k\alpha}\,[\psi_{k\alpha}^{(\varphi)\,\prime}(0)-2m\lambda\,\psi_{k\alpha}^{(\varphi)}(0)]-r_{k\alpha}\,\psi_{-k,\alpha}^{(\varphi^{\prime})\,\prime}(0)=\psi_{k\alpha}^{(\varphi^{\prime})\,\prime}(0),$
(134)

and

$\displaystyle
t^{\prime}_{k\alpha}\,\psi_{-k,\alpha}^{(\varphi^{\prime})}(0)-r^{\prime}_{k\alpha}\,\psi_{k\alpha}^{(\varphi)}(0)=\psi_{-k,\alpha}^{(\varphi)}(0),$
(135) $\displaystyle
t^{\prime}_{k\alpha}\,[\psi_{-k,\alpha}^{(\varphi^{\prime})\,\prime}(0)+2m\lambda\,\psi_{-k,\alpha}^{(\varphi^{\prime})}(0)]-r^{\prime}_{k\alpha}\,\psi_{k\alpha}^{(\varphi)\,\prime}(0)=\psi_{-k,\alpha}^{(\varphi)\,\prime}(0),$
(136)

where

$\displaystyle\psi_{k\alpha}^{(\varphi)}(0)$
$\displaystyle=\psi_{-k,\alpha}^{(\varphi)}(0)=\frac{\psi_{2}^{(\varphi)}(L)}{\sqrt{N_{k\alpha}^{(\varphi)}}},$
(137) $\displaystyle\psi_{k\alpha}^{(\varphi)\,\prime}(0)$
$\displaystyle=\frac{-\psi_{1}^{(\varphi)}(L)+e^{ikL}}{\sqrt{N_{k\alpha}^{(\varphi)}}}.$
(138)

The solutions of these linear equations read

$\displaystyle t_{k\alpha}$ $\displaystyle=\frac{2i\,\sin
kL\,\,\psi_{2}^{(\varphi^{\prime})}(L)}{d_{k\alpha}}\,\sqrt{\frac{N_{k\alpha}^{(\varphi)}}{N_{k\alpha}^{(\varphi^{\prime})}}},$
(139) $\displaystyle-r_{k\alpha}$ $\displaystyle=1-\frac{2i\,\sin
kL\,\,\psi_{2}^{(\varphi)}(L)}{d_{k\alpha}},$ (140)

and

$\displaystyle t^{\prime}_{k\alpha}$ $\displaystyle=\frac{2i\,\sin
kL\,\,\psi_{2}^{(\varphi)}(L)}{d_{k\alpha}}\,\sqrt{\frac{N_{k\alpha}^{(\varphi^{\prime})}}{N_{k\alpha}^{(\varphi)}}},$
(141) $\displaystyle-r^{\prime}_{k\alpha}$ $\displaystyle=1-\frac{2i\,\sin
kL\,\,\psi_{2}^{(\varphi^{\prime})}(L)}{d_{k\alpha}},$ (142)

where

$\displaystyle d_{k\alpha}=$
$\displaystyle\sqrt{N_{k\alpha}^{(\varphi)}N_{k\alpha}^{(\varphi^{\prime})}}\left\\{-\psi_{k\alpha}^{(\varphi)}(0)\,\psi_{-k,\alpha}^{(\varphi^{\prime})\,\prime}(0)\right.$
$\displaystyle\left.+[\psi_{k\alpha}^{(\varphi)\,\prime}(0)-2m\lambda\,\psi_{k\alpha}^{(\varphi)}(0)]\,\,\psi_{-k,\alpha}^{(\varphi^{\prime})}(0)\right\\}$
(143) $\displaystyle=$
$\displaystyle\psi_{2}^{(\varphi)}(L)\,[\psi_{1}^{(\varphi^{\prime})}(L)-e^{-ikL}]$
$\displaystyle-[\psi_{1}^{(\varphi)}(L)-e^{ikL}+2m\lambda\,\psi_{2}^{(\varphi)}(L)]\,\,\psi_{2}^{(\varphi^{\prime})}(L).$
(144)

This function is expressed in term of two different fundamental systems
referring to potentials shifted by $x_{\varphi}$ and $x_{\varphi^{\prime}}$.
Using the transformation laws (91), (92), (103) of a fundamental system under
the potential’s shift, we express $d_{k\alpha}$ in terms of the unshifted
fundamental system. This gives

$\displaystyle d_{k\alpha}=$
$\displaystyle\frac{\sqrt{N_{k\alpha}^{(\varphi)}N_{k\alpha}^{(\varphi^{\prime})}}}{N_{k\alpha}}\,\tilde{d}_{k\alpha}\,e^{-i\Phi_{k\alpha}(x_{\varphi})\,+\,i\Phi_{k\alpha}(x_{\varphi^{\prime}})},$
(145) $\displaystyle\frac{\tilde{d}_{k\alpha}}{N_{k\alpha}}=$
$\displaystyle-\psi_{k\alpha}(x_{\varphi})\,\psi^{\prime}_{-k,\alpha}(x_{\varphi^{\prime}})+\psi^{\prime}_{k\alpha}(x_{\varphi})\,\psi_{-k,\alpha}(x_{\varphi^{\prime}})$
$\displaystyle-2m\lambda\,\psi_{k\alpha}(x_{\varphi})\,\psi_{-k,\alpha}(x_{\varphi^{\prime}}).$
(146)

The introduced function $\tilde{d}_{k\alpha}$ plays an important role in the
forthcoming analysis of the interface properties, which will be elucidated
later.

The transmission and reflection coefficients form the scattering matrix

$\displaystyle
S_{k\alpha}=\left(\begin{array}[]{cc}t_{k\alpha}&r^{\prime}_{k\alpha}\\\
r_{k\alpha}&t^{\prime}_{k\alpha}\end{array}\right),$ (149)

which is unitary (see Appendix G for verification). This property implies the
relations

$\displaystyle|t_{k\alpha}|^{2}+|r_{k\alpha}|^{2}=1,$ (150)
$\displaystyle|t^{\prime}_{k\alpha}|^{2}+|r^{\prime}_{k\alpha}|^{2}=1,$ (151)
$\displaystyle
t_{k\alpha}^{*}\,r^{\prime}_{k\alpha}+r_{k\alpha}^{*}\,t^{\prime}_{k\alpha}=0,$
(152)

which guarantee the normalization and orthogonality of the states (129) and
(130).

###  IV.2 Interface localized states

To find eigenstates which are exponentially localized near the interface, we
make the ansatz

$\displaystyle\psi_{\text{i},\alpha}(x)=$
$\displaystyle\Theta(-x)\,a_{\alpha}\,[-\psi_{2}^{(\varphi^{\prime})}(x-L)+\psi_{2}^{(\varphi^{\prime})}(x)\,(-1)^{\alpha}\,e^{\kappa
L}]$ $\displaystyle+$
$\displaystyle\Theta(x)\,b_{\alpha}\,[-\psi_{2}^{(\varphi)}(x-L)+\psi_{2}^{(\varphi)}(x)\,(-1)^{\alpha}\,e^{-\kappa
L}],$ (153)

which uses the (unnormalized) Bloch states (21)

$\displaystyle-\psi_{2}^{(\varphi)}(x-L)+\psi_{2}^{(\varphi)}(x)\,(-1)^{\alpha}\,e^{\mp\kappa
L}\propto\psi_{\frac{\alpha\pi}{L}\pm i\kappa}^{(\varphi)}(x)$ (154)

with complex $k=\frac{\alpha\pi}{L}\pm i\kappa$ and $\kappa>0$. Applying the
matching conditions (131), (132) we obtain the equations

$\displaystyle
b_{\alpha}\,\psi_{2}^{(\varphi)}(L)=a_{\alpha}\,\psi_{2}^{(\varphi^{\prime})}(L),$
(155) $\displaystyle
b_{\alpha}\left[-\psi_{1}^{(\varphi)}(L)+(-1)^{\alpha}\,e^{-\kappa
L}-2m\lambda\,\psi_{2}^{(\varphi)}(L)\right]$
$\displaystyle=a_{\alpha}\,[-\psi_{1}^{(\varphi^{\prime})}(L)+(-1)^{\alpha}\,e^{\kappa
L}].$ (156)

Figure 7: a) Energy spectrum of the interface eigenvalue problem for the
potential shown in Fig. 6 with $\lambda=1$. The solid lines in the bandgaps
represent energies of the interface localized states. b) Total interface
charge for a system with the chemical potential
$\mu=\epsilon_{\frac{\pi}{L},3}$, i.e. lying on the top of the third band.
Downward jumps occur whenever the localized states leave the topmost occupied
band edge. c) Close-up of the first gap: The continuous curve connecting both
physical ($\kappa>0$) and unphysical ($\kappa<0$) solutions of (160) exhibits
the double period $2L$. d) Close-up of the second band. e) Second band’s
contribution to the interface charge. Jumps occur in both downward and upward
directions at the touching points (which are the roots of the functions
$\tilde{d}_{\frac{\pi}{L},2}(x_{\varphi})=0$ and
$\tilde{d}_{0,2}(x_{\varphi})=0$) depending on the direction of the localized
state’s spectral flow.

The condition for the existence of a nontrivial solution yields the equation

$\displaystyle\psi_{2}^{(\varphi^{\prime})}(L)\left[-\psi_{1}^{(\varphi)}(L)+(-1)^{\alpha}\,e^{-\kappa
L}-2m\lambda\,\psi_{2}^{(\varphi)}(L)\right]$
$\displaystyle=\psi_{2}^{(\varphi)}(L)\,[-\psi_{1}^{(\varphi^{\prime})}(L)+(-1)^{\alpha}\,e^{\kappa
L}].$ (157)

Noticing that by virtue of (137), (138), and (106) it holds

$\displaystyle\frac{-\psi_{1}^{(\varphi)}(L)+(-1)^{\alpha}\,e^{\mp\kappa
L}}{\psi_{2}^{(\varphi)}(L)}=\frac{\psi^{\prime}_{k\alpha}(x_{\varphi})}{\psi_{k\alpha}(x_{\varphi})}\bigg{|}_{k=\frac{\alpha\pi}{L}\pm
i\kappa},$ (158)

we cast (157) to the form

$\displaystyle\left[\frac{\psi^{\prime}_{k\alpha}(x_{\varphi})}{\psi_{k\alpha}(x_{\varphi})}-\frac{\psi^{\prime}_{-k,\alpha}(x_{\varphi^{\prime}})}{\psi_{-k,\alpha}(x_{\varphi^{\prime}})}\right]_{k=\frac{\alpha\pi}{L}+i\kappa}-2m\lambda=0,$
(159)

which is equivalent to

$\displaystyle\tilde{d}_{\frac{\alpha\pi}{L}+i\kappa,\alpha}=0.$ (160)

In addition, we have the relations between $\kappa$ and the localized state’s
energy $\epsilon_{\text{i},\alpha}$

$\displaystyle(-1)^{\alpha}\,\cosh\kappa
L=D^{(\varphi)}(\epsilon_{\text{i},\alpha})=D^{(\varphi^{\prime})}(\epsilon_{\text{i},\alpha})=D(\epsilon_{\text{i},\alpha}),$
(161)

where $D^{(\varphi)}(E)$ is defined in (104), and in (105) it is shown to be
independent of $x_{\varphi}$. Note that to solve (159) for $\kappa>0$ we have
to replace $e^{\pm\kappa L}$ with $|D|\pm\sqrt{D^{2}-1}$.

Figure 8: Same as in Fig. 7 for $\lambda=-1$.

From the normalization condition

$\displaystyle\int_{-\infty}^{\infty}dx\,\,|\psi_{\text{i},\alpha}(x)|^{2}=1$
(162)

we derive the equation for $|a_{\alpha}|^{2}$:

$\displaystyle\frac{1}{|a_{\alpha}|^{2}}$
$\displaystyle=\sum_{n=1}^{\infty}e^{-2n\kappa L}$ (163)
$\displaystyle\times\int_{0}^{L}dx\left[-\psi_{2}^{(\varphi^{\prime})}(x-L)+\psi_{2}^{(\varphi^{\prime})}(x)\,(-1)^{\alpha}\,e^{\kappa
L}\right]^{2}$
$\displaystyle+\left(\frac{\psi_{2}^{(\varphi^{\prime})}(L)}{\psi_{2}^{(\varphi)}(L)}\right)^{2}\sum_{n=0}^{\infty}e^{-2n\kappa
L}$
$\displaystyle\times\int_{0}^{L}dx\left[-\psi_{2}^{(\varphi)}(x-L)+\psi_{2}^{(\varphi)}(x)\,(-1)^{\alpha}\,e^{-\kappa
L}\right]^{2}.$

Using the identities (202), (205), and (206) we then find

$\displaystyle\frac{1}{|a_{\alpha}\,\psi_{2}^{(\varphi^{\prime})}(L)|^{2}}=\frac{1}{|b_{\alpha}\,\psi_{2}^{(\varphi)}(L)|^{2}}$
$\displaystyle=$
$\displaystyle\frac{1}{2m}\,\frac{d}{dE}\left[\frac{-\psi_{1}^{(\varphi)}(L)+(-1)^{\alpha}\,e^{-\kappa
L}}{\psi_{2}^{(\varphi)}(L)}\right.$
$\displaystyle\left.\qquad\qquad-\frac{-\psi_{1}^{(\varphi^{\prime})}(L)+(-1)^{\alpha}\,e^{\kappa
L}}{\psi_{2}^{(\varphi^{\prime})}(L)}\right].$ (164)

Note that the function under the derivative in the right-hand side is the same
— up to the constant $-2m\lambda$ — as the one in Eq. (159), whose roots
determine interface localized states. This implies that a second-order root is
impossible (since otherwise the denominators in the left-hand side of (164)
becomes infinite, and the corresponding state is unnormalizable). Hence, any
two physical roots of (160) may not coalesce, and on this basis we conclude
that interface localized states are always nondegenerate.

The wavefunction of an exemplary interface localized state lying in the first
gap is shown in Fig. 6. An important question is for which parameter values
such a state is present, and how many of them can be accommodated in each gap?
To systematically study this, we fix $x_{\varphi^{\prime}}=0$ and vary the
parameter $x_{\varphi}$. There are two characteristic cases: $\lambda>0$ (Fig.
7(a,c,d)) and $\lambda<0$ (Fig. 8(a,c,d)). The bands of the scattering
eigenstates energetically coincide with those of the bulk eigenvalue problem.
The localized states dispersions in $x_{\varphi}$ are depicted by solid lines
lying in the bandgaps. They are continuously prolonged by grey dashed lines
which represent unphysical solutions (that is with $\kappa<0$) of the equation
(160). We remark that the whole continuous curve has a double period $2L$ (see
particularly close-ups of the first gap in Figs. 7(c) and 8(c) for a
confirmation that this property holds in each gap). This is qualitatively
different from the behavior of an edge state dispersion in the boundary
problem [Fig. 4(a,b)], which has a period $L$. The only exception from the
double-period rule is the localized state residing beneath the lowest band: It
has a period $L$, and there is no counterpart for this state in the boundary
problem. Note that this state may occur not only for negative $\lambda$, but
also for positive $\lambda$ (see additionally Figs. 12(a,c) and 13(a,c) in
Appendix H for other values of $\lambda$). It is also remarkable that, unlike
in the boundary problem, there are physical dispersions which both enter and
leave a band at the same band edge (as shown e.g. in the first gap in Fig.
12(c)). Overall, we may have either zero, one, or two localized states in each
gap for a fixed value of $x_{\varphi}$, as well as either zero or one state
beneath the lowest band.

The case $\lambda=0$ is realized by the limits $\lambda\to 0^{\pm}$, as shown
in Fig. 9. Note that these limits are especially nontrivial close to the
translationally invariant point $x_{\varphi}=x_{\varphi^{\prime}}=0$. In
particular, for $\lambda>0$ the localized state detaches from the top of the
lower band, while for $\lambda<0$ it detaches from the bottom of the upper
band.

###  IV.3 Interface charge: Band’s contribution

The $\alpha$th band’s contribution to the interface charge is defined by

$\displaystyle Q_{I,\alpha}=$ $\displaystyle\int_{-\infty}^{\infty}dx\,\,f(x)$
$\displaystyle\times\left[\int_{0}^{\pi/L}dk\left(|\Psi_{k\alpha}^{(r)}(x)|^{2}+|\Psi_{k\alpha}^{(l)}(x)|^{2}\right)-\bar{\rho}_{\alpha}\right],$
(165)

with a symmetric envelope function $f(-x)=f(x)$. Due to the unitarity of the
scattering matrix and the property (256), the average bulk charge density
$\bar{\rho}_{\alpha}$ is cancelled out, and (165) can be represented as a sum
$Q_{I,\alpha}=Q_{I,\alpha}^{R}+Q_{I,\alpha}^{L}$ of the two charge
contributions integrated over the right and left half-space, respectively.
They amount to

$\displaystyle Q_{I,\alpha}^{R}=$ $\displaystyle
L\,\,\text{Re}\,\int_{-\pi/L}^{\pi/L}\frac{dk}{2\pi}\int_{0}^{L}dx\,\,r^{\prime}_{k\alpha}\,\psi_{k\alpha}^{(\varphi)\,2}(x)$
$\displaystyle\times\sum_{n=1}^{\infty}e^{2i(kL+i0^{+})(n-1)}+Q_{P,\alpha}(x_{\varphi})$
(166)

and

$\displaystyle Q_{I,\alpha}^{L}=$ $\displaystyle
L\,\,\text{Re}\,\int_{-\pi/L}^{\pi/L}\frac{dk}{2\pi}\int_{0}^{L}dx\,\,r^{*}_{k\alpha}\,\psi_{k\alpha}^{(\varphi^{\prime})\,2}(x)$
$\displaystyle\times\sum_{n=-\infty}^{0}e^{2i(kL-i0^{+})(n-1)}-Q_{P,\alpha}(x_{\varphi^{\prime}}),$
(167)

where the polarization charge

$\displaystyle Q_{P,\alpha}(x_{\varphi})=$
$\displaystyle-\frac{1}{L}\int_{0}^{L}dx\,\,x\left(L\int_{-\pi/L}^{\pi/L}\frac{dk}{2\pi}\,|u^{(\varphi)}_{k\alpha}(x)|^{2}-\frac{1}{L}\right)$
$\displaystyle=$ $\displaystyle\frac{x_{\varphi}}{L}+Q_{P,\alpha}$ (168)

is an analog of (38). Note that the last equality follows from (62) and the
periodicity of $u^{(\varphi)}_{k\alpha}(x)$ in $x$.

Figure 9: Localized states dispersions for $\lambda=-0.1,0,0.1$: Illustration
of the nontrivial limits $\lambda\to 0^{\pm}$ close to the translationally
invariant point $x_{\varphi}=x_{\varphi^{\prime}}=0$ (shown is the third gap).
In particular, for $\lambda\to 0^{+}$ there is a state in the gap at
$\epsilon_{\frac{\pi}{L},3}+0^{+}$, while for $\lambda\to 0^{-}$ there is a
state in the gap at $\epsilon_{\frac{\pi}{L},4}-0^{+}$. At $\lambda=0$ these
states touch the bands.

Taking into account the special structure of the reflection coefficients (140)
and (142), the relation (73), and the boundary charge formula (67) we deduce

$\displaystyle Q_{I,\alpha}=$
$\displaystyle-1+\frac{x_{\varphi}-x_{\varphi^{\prime}}}{L}-\text{wn}\,[e^{i\Phi_{k\alpha}(x_{\varphi})}]+\text{wn}\,[e^{i\Phi_{k\alpha}(x_{\varphi^{\prime}})}]$
$\displaystyle+\bar{Q}_{I,\alpha},$ (169)

where

$\displaystyle\bar{Q}_{I,\alpha}=$
$\displaystyle-L\,\,\text{Re}\,\int_{-\pi/L}^{\pi/L}\frac{dk}{2\pi}\int_{0}^{L}dx$
$\displaystyle\times$
$\displaystyle\frac{\psi_{2}^{(\varphi^{\prime})}(L)\,\psi_{k\alpha}^{(\varphi)\,2}(x)\,e^{-ikL}+\psi_{2}^{(\varphi)}(L)\,\psi_{-k,\alpha}^{(\varphi^{\prime})\,2}(x)\,e^{ikL}}{d_{k\alpha}}.$
(170)

It can be shown that $\bar{Q}_{I,\alpha}$ is an integer contribution given in
terms of the winding number of $d_{k\alpha}$:

$\displaystyle\bar{Q}_{I,\alpha}=i\int_{-\pi/L}^{\pi/L}\frac{dk}{2\pi}\,\frac{1}{d_{k\alpha}}\,\frac{d}{dk}d_{k\alpha}=-\text{wn}\,[d_{k\alpha}],$
(171)

see Appendix I for details of this derivation. By virtue of (145), (146) we
similarly obtain Eq. (169) in terms of $\tilde{d}_{k\alpha}$ as

$\displaystyle Q_{I,\alpha}=$
$\displaystyle-1+\frac{x_{\varphi}-x_{\varphi^{\prime}}}{L}-\text{wn}\,[\tilde{d}_{k\alpha}].$
(172)

This is the first main result of this section. It establishes the universal
form of a band’s contribution to the interface charge. This form is similar to
the one obtained for the boundary charge (67). The phase of the function
$\tilde{d}_{k\alpha}$ plays here a role similar to that of the phase
$e^{i\Phi_{k\alpha}}$ in the boundary problem.

Without loss of generality we set $x_{\varphi^{\prime}}=0$. Then, taking into
account (21) as well as (137), (138), we conclude

$\displaystyle\tilde{d}_{k\alpha}(x_{\varphi})=$
$\displaystyle-[-\psi_{2}(x_{\varphi}-L)+\psi_{2}(x_{\varphi})e^{ikL}]$
$\displaystyle\times[-\psi_{1}(L)+e^{-ikL}+2m\lambda\,\psi_{2}(L)]$
$\displaystyle+[-\psi^{\prime}_{2}(x_{\varphi}-L)+\psi^{\prime}_{2}(x_{\varphi})e^{ikL}]\,\psi_{2}(L).$
(173)

Considering $Q_{I,\alpha}$ as a function of $x_{\varphi}$, we notice its
periodicity $Q_{I,\alpha}(x_{\varphi})=Q_{I,\alpha}(x_{\varphi}+L)$, which is
due to the properties
$\tilde{d}_{k\alpha}(x_{\varphi}+L)=\tilde{d}_{k\alpha}(x_{\varphi})e^{ikL}$
and
$\text{wn}\,[\tilde{d}_{k\alpha}(x_{\varphi}+L)]=\text{wn}\,[\tilde{d}_{k\alpha}(x_{\varphi})]+1$.
Note that the latter follow from the definitions (173) and (69) of
$\tilde{d}_{k\alpha}(x_{\varphi})$ and of the winding number, respectively.

The behavior $Q_{I,\alpha=2}(x_{\varphi})$ is illustrated in Figs. 7(e) (for
$\lambda=1$) and 8(e) (for $\lambda=-1$) as well as in the analogous panels in
Appendix H (for other values of $\lambda$). For a detection of the touching
points we plot the real-valued functions
$\tilde{d}_{\frac{\pi}{L},2}(x_{\varphi})$ (for the bottom band edge) and
$\tilde{d}_{0,2}(x_{\varphi})$ (for the top band edge) and determine their
roots. Their roots give the band’s touching points at the corresponding edge.
Whenever the localized state enters/leaves the band, the value of
$Q_{I,\alpha=2}(x_{\varphi})$ jumps by $\pm 1$. A relation between the winding
number change and the spectral flow of the localized states into/out of the
bands is discussed in greater detail in the next subsection.

###  IV.4 Total interface charge

The total interface charge in the system with the chemical potential
$\mu_{\nu}$ at the top of the $\nu$th band is given by a sum of contributions
from all states below $\mu_{\nu}$,

$\displaystyle
Q_{I}^{(\nu)}(x_{\varphi})=\sum_{\alpha=1}^{\nu}Q_{I,\alpha}(x_{\varphi})+\sum_{\alpha=0}^{\nu-1}Q_{\text{i},\alpha}(x_{\varphi}).$
(174)

Here $Q_{\text{i},\alpha}(x_{\varphi})$ is a number of the interface localized
states in the gap $\alpha$. Note that in contrast to (124) the summation in
the second term of (174) begins from $\alpha=0$ (”zeroth gap”, i.e. the energy
range beneath the lowest band), since an interface localized can occur there
as well.

To establish the $x_{\varphi}$ dependence of $Q_{I}^{(\nu)}$, we first derive
$Q_{I}^{(\nu)}(0)$ at finite $\lambda$, that is the zero phase mismatch
result. Then, exploiting the continuity of the spectral flow with
$x_{\varphi}$, we address the sought-after dependence.

In the zero mismatch limit $x_{\varphi}=0$, the bands’ contributions are
evaluated in terms of

$\displaystyle\tilde{d}_{k\alpha}(0)=2\,\psi_{2}(L)\,[i\sin
kL-m\lambda\,\psi_{2}(L)].$ (175)

This expression immediately follows from (173) after applying the defining
relations of the functions $\psi_{1}$ and $\psi_{2}$, stated after Eq. (2),
and the properties (192), (193). Since the function
$\psi_{2}(L)=\psi_{2}(L,\epsilon_{k\alpha})$ does not change its sign within
the band $\alpha$ [see (221)], the complex function $\tilde{d}_{k\alpha}(0)$
does not wind with $k$ around the origin, leaving
$\text{wn}\,[\tilde{d}_{k\alpha}(0)]=0$. Then from (172) it follows that at
$x_{\varphi}=x_{\varphi^{\prime}}=0$ each band provides the $-1$ contribution
to the interface charge, that is $Q_{I,\alpha}(0)=-1$.

To compute $Q_{I}^{(\nu)}(0)$ by means of (174), we also need to establish the
number of localized states. At $x_{\varphi}=x_{\varphi^{\prime}}=0$ they obey
the equation

$\displaystyle-(-1)^{\alpha}m\lambda\,\psi_{2}(L)=\sinh\kappa L,$ (176)

which follows from $\tilde{d}_{k\alpha}(0)=0$ [note that $\psi_{2}(L)=0$ does
not give a nontrivial solution inside a gap away from bands’ edges]. It is
convenient to rewrite (176) as

$\displaystyle-m\lambda\,\frac{\psi_{2}(L)}{D(E)}=\frac{\sqrt{D^{2}(E)-1}}{|D(E)|}.$
(177)

According to (221) and (207), the function in the left-hand side is
monotonically decreasing across the gap from a positive to a negative value
for $\lambda>0$, and it is monotonically increasing from a negative to a
positive value for $\lambda<0$. In turn, the function in the right-hand side
is positive within the gap and vanishes at its ends. Since both sides of (177)
are continuous in $E$, they have a single intersection point for any
$\lambda\neq 0$, which gives the energy of the interface localized state.
Thus, each gap hosts one localized state.

It is necessary to additionally inspect the ”zeroth gap”, i.e. the energy
range below the first band. The right-hand side of (177) is monotonically
decreasing from $+1$ (at $E=-\infty$) down to $0^{+}$. Remarking that

$\displaystyle\psi_{2}(L)\stackrel{{\scriptstyle
E\to-\infty}}{{\approx}}\frac{\sinh(L\sqrt{2m|E|})}{\sqrt{2m|E|}},$ (178)

we conclude that the left-hand side of (177) for $\lambda>0$ is monotonically
decreasing from $0^{-}$ (at $E=-\infty$) to a negative value, and for
$\lambda<0$ it is monotonically increasing from $0^{+}$ (at $E=-\infty$) to a
positive value. Thus, only in the latter case we obtain an additional
localized solution. It is a remnant of the bound state in the attractive
($\lambda<0$) delta-potential and the flat background potential, which has the
energy $E=-\frac{m\lambda^{2}}{2}$ [note that this result is easily recovered
from (177) with help of (178)].

Summarizing the above results in the zero mismatch limit $x_{\varphi}=0$, we
obtain

$\displaystyle Q_{I}^{(\nu)}(0)=\begin{cases}-1,&\lambda>0,\\\
0,&\lambda<0.\end{cases}$ (179)

Next, we investigate how these universal values change with $x_{\varphi}$.

Figure 10: A sketch of the proof of (182). a) Fixing notations for the present
and absent localized state before and after the touching point, respectively.
b) The winding number changes upon traversing the touching point: The contour
of $\tilde{d}_{k\alpha}(x_{\varphi})$ is swept across the origin. The precise
value of the change ($+1$ or $-1$) depends on a mutual orientation of the
normal and the tangent vectors to the contour in the vicinity of the origin.

Moving along the localized state dispersion in $x_{\varphi}$ [Fig. 10(a)], we
establish that at the touching point $x_{\varphi}^{*}$ [obeying the equation
$\tilde{d}_{k\alpha}(x_{\varphi}^{*})=0$ with $\text{Re}\,k=0,\pi/L$ and
$\kappa=0$] it holds

$\displaystyle\frac{dE(x_{\varphi}^{*})}{dx_{\varphi}}=(-1)^{\alpha}L\,\sinh\kappa
L\,\frac{d\kappa(x_{\varphi}^{*})}{dx_{\varphi}}=0,$ (180)

as well as

$\displaystyle\frac{d\kappa(x_{\varphi}^{*})}{dx_{\varphi}}=-\frac{\frac{\partial\tilde{d}_{k\alpha}}{\partial
x_{\varphi}}}{\frac{\partial\tilde{d}_{k\alpha}}{\partial\kappa}}$
$\displaystyle=\frac{(-1)^{\alpha}\frac{\partial\tilde{d}_{k\alpha}}{\partial
x_{\varphi}}}{L\psi_{2}(L)[\psi_{1}(x_{\varphi}^{*})+\psi^{\prime}_{2}(x_{\varphi}^{*})-2m\lambda\,\psi_{2}(x_{\varphi}^{*})]}.$
(181)

On the other hand, looking at the touching point from the band’s perspective,
we notice that the winding number of $\tilde{d}_{k\alpha}$ associated with the
band $\alpha$ changes upon going across the touching point by

$\displaystyle\Delta\,\text{wn}\,[\tilde{d}_{k\alpha}]$
$\displaystyle=\text{wn}\,[\tilde{d}_{k\alpha}(x_{\varphi}^{*}+\varepsilon)]-\text{wn}\,[\tilde{d}_{k\alpha}(x_{\varphi}^{*}-\varepsilon)]$
$\displaystyle=\text{sign}\,\left[\frac{\partial\tilde{d}_{k\alpha}}{\partial
x_{\varphi}}\,\frac{\partial\,\text{Im}\,\tilde{d}_{k\alpha}}{\partial
k}\right],$ (182)

where

$\displaystyle\frac{\partial\,\text{Im}\,\tilde{d}_{k\alpha}}{\partial k}=$
$\displaystyle(-1)^{\alpha}L\psi_{2}(L)$
$\displaystyle\times[\psi_{1}(x_{\varphi}^{*})+\psi^{\prime}_{2}(x_{\varphi}^{*})-2m\lambda\,\psi_{2}(x_{\varphi}^{*})].$
(183)

The derivation of (182) is sketched in Fig. 10(b). For $k=0,\pi/L$ the
function $\tilde{d}_{k\alpha}$ becomes real. These values correspond to points
on the real axis, where the parametric plot of $\tilde{d}_{k\alpha}$ (or the
loop) intersects the horizontal axis. Moving one of these points (say, the
left one, as shown on the sketch) across the origin (where the touching point
condition is satisfied) by means of tuning $x_{\varphi}$ leads to a change in
the winding number. This change (by either $+1$ or $-1$) depends on the
orientation of the tangent vector to the loop in the vicinity of the origin. A
careful inspection of all possible scenarios leads us to the expression (182).

By analogy with (111) we introduce the interface vorticity as a sign of (181).
It quantifies whether the localized interface state leaves the band (the value
$+1$ indicates the change of $\kappa$ from negative to positive) or enters the
band (the value $-1$ indicates the change of $\kappa$ from positive to
negative). Comparing (181) with (182), (183) we show that this vorticity
equals

$\displaystyle\text{sign}\,\left[\frac{d\kappa_{\alpha}(x_{\varphi}^{*})}{dx_{\varphi}}\bigg{|}_{\alpha}\right]=\Delta\,\text{wn}\,[\tilde{d}_{k\alpha}].$
(184)

Here in the left-hand side we additionally mark that the localized state in
the gap $\alpha$ (encoded in $\kappa_{\alpha}$) touches the band $\alpha$
(encoded in the lowest subscript $\alpha$). Analogously we show that upon
touching the upper band it holds

$\displaystyle\text{sign}\,\left[\frac{d\kappa_{\alpha}(x_{\varphi}^{*})}{dx_{\varphi}}\bigg{|}_{\alpha+1}\right]=\Delta\,\text{wn}\,[\tilde{d}_{k,\alpha+1}].$
(185)

The expressions (184) and (185) have a transparent physical meaning: When the
localized state enters/leaves the band, the band’s contribution to the total
interface charge changes accordingly, so that there is no net contribution to
$Q_{I}^{(\nu)}$. The only uncompensated changes occur at the topmost occupied
band edge (since we put there the chemical potential and therefore do not
count the localized states touching that band edge). Hence we obtain the
following expression for the total interface charge

$\displaystyle Q_{I}^{(\nu)}(x_{\varphi})=$ $\displaystyle
Q_{I}^{(\nu)}(0)+\frac{\nu}{L}x_{\varphi}$
$\displaystyle-\sum_{j=1}^{\nu+l}\Theta(x_{\varphi,j}^{-})+\sum_{j=1}^{l}\Theta(x_{\varphi,j}^{+}).$
(186)

This is the second main result of this section: It states the universal
dependence of the total interface charge on the pumping parameter
$x_{\varphi}$. It is similar to (125), the main difference being that the
localized state energy may now both enter and leave a band at the same band
edge. Here $\\{x_{\varphi,j}^{\pm}\\}$ is a (model-specific) set of points,
where the interface localized state enters/leaves the topmost occupied band
edge. The difference between the numbers of $x_{\varphi,j}^{-}$ and
$x_{\varphi,j}^{+}$ must be exactly equal to $\nu$: This is needed to ensure
the periodicity $Q_{I}^{(\nu)}(x_{\varphi}+L)=Q_{I}^{(\nu)}(x_{\varphi})$.

The behavior $Q_{I}^{(3)}(x_{\varphi})$ for the three occupied bands is
illustrated in Figs. 7(b) (for $\lambda=1$) and 8(b) (for $\lambda=-1$) as
well as in the analogous panels in Appendix H for the values $\lambda=\pm
0.1$, which additionally give an idea about the behavior of
$Q_{I}^{(3)}(x_{\varphi})$ close to $\lambda=0$.

##  V Summary

In this work, we have studied universal properties of the boundary change
$Q_{B}(x_{\varphi})$ in one-dimensional single-channel continuum insulator
models. We have found a rigorous proof of the linear dependence of $Q_{B}$ on
the phase $x_{\varphi}$ of the periodic potential modulation. In continuum
models, this result appears in an undisguised form: All nonuniversal
$\frac{2\pi}{Za}$-periodic contributions previously found in the lattice model
considerations [pletyukhov_etal_prb_20, ] are suppressed in the continuum
limit $Z\to\infty$ (where $Z$ is a number of sites in a unit cell, and $a$ is
a lattice spacing).

The total boundary charge consists of both band contributions and potentially
edge states residing in the band gaps. We have extended the earlier findings
[thakurati_etal_18, ,pletyukhov_etal_prb_20, ] for both the individual band
contributions $Q_{B,\alpha}$ and the total boundary charge to the class of
continuum models studied here. In particular, in the first case the slope of
the linear dependence is given by the first Chern index $C_{1,\alpha}$ of band
$\alpha$, which acquires the same value $C_{1,\alpha}=1$ for each band. This
value is in tune with the difference between the numbers $M_{\alpha}^{(\mp)}$
of the touching points, at which the edge state dispersions (evolved in
$x_{\varphi}$) leave and enter the band. We have shown that over one pumping
cycle $x_{\varphi}\to x_{\varphi}+L$ (i.e. pushing the periodic potential
towards the boundary over one period), in continuum models, there are exactly
$\alpha$ edge states which leave the band $\alpha$ from its top side, and
$\alpha-1$ edge states which enter the band from its bottom side. At the
corresponding touching points, $Q_{B,\alpha}$ acquires discontinuous jump
contributions $\pm 1$ in such a way that the periodicity
$Q_{B,\alpha}(x_{\varphi}+L)=Q_{B,\alpha}(x_{\varphi})$ is maintained. Thus,
the equality $C_{1,\alpha}=M_{\alpha}^{(-)}-M_{\alpha}^{(+)}$ is not only a
manifestation of the bulk-boundary correspondence, but also an expression for
charge conservation.

An analogous result has been obtained for the total boundary charge
$Q_{B}^{(\nu)}$ in a system with $\nu$ fully occupied bands: The slope $\nu$
(with respect to $\frac{x_{\varphi}}{L}$) of the linear dependence is
compensated by the occurrence of exactly $\nu$ edges states removing a single-
electron charge during one pumping cycle.

Studying the boundary charge fluctuations, we have confirmed the earlier
stated surface fluctuation theorem [weber_etal_prl_20, ,wannier_paper, ],
which expresses the universal low-energy $1/E_{g}$ scaling of these
fluctuations with the gap size $E_{g}$.

The other interesting facet of the excess charges studied in the present work
is the interface charge $Q_{I}$ and its universal properties. Bringing two
semi-infinite systems with different phases $x_{\varphi}\neq
x_{\varphi^{\prime}}$ of the periodic potential modulation into contact and
introducing an additional delta-potential barrier between them, we have
revealed similar universal dependencies of $Q_{I}$ on $x_{\varphi}$ at fixed
$x_{\varphi^{\prime}}$, i.e., a slope 1 for the single band contribution and a
slope $\nu$ for the total interface charge (in a system with $\nu$ occupied
bands). To restore the periodicity $Q_{I}(x_{\varphi})=Q_{I}(x_{\varphi}+L)$,
the linear growth is accompanied by discontinuous jumps $\pm 1$ at the points
where the interface localized state enters/leaves the band. Thereby, the
required charge conservation is respected. However, in interface models there
is no analog of the bulk-boundary correspondence, since the spectral flow of
the interface localized states is in accord with changes in winding numbers of
a novel function called $\tilde{d}_{k\alpha}$, and not with those of the
phases of the Bloch states. The function $\tilde{d}_{k\alpha}$ which we
introduced not only represents a specific combination of the Bloch states in
the bulk of both right and left subsystems, but also contains information
about their interface. It connects to the denominator of the scattering
matrix, and thus inherits the analytic features of the reflection coefficient.
We have elucidated the tremendous role of the function $\tilde{d}_{k\alpha}$
in describing the universal properties of the interface charge; in particular,
the contribution from its winding number to $Q_{I}$ and the equation
$\tilde{d}_{k\alpha}=0$ to determine interface localized states. The integer-
valued changes of $Q_{I}$ analysed in our study, which are associated with the
winding number changes and an emergence/cease of localized states, serve as a
manifestation of the nearsightedness principle [NearSightednessPrin, ].

A perspective extension of the present work consists in studying a
multichannel generalization of the continuum one-dimensional insulator models
and revealing similar dependencies for both boundary and interface charges
[mueller_etal_2021, ].

##  VI Acknowledgments

We appreciate a lasting fruitful exchange of ideas on the boundary charge
subject with J. Klinovaja and D. Loss. The work was supported by the Deutsche
Forschungsgemeinschaft via RTG 1995.

##  Appendix A Properties of the Hill equation

To prove the validity of (20), we notice that both left- and right-hand sides
this relation satisfy (1) and have the same initial conditions at $x=L$

$\displaystyle\psi_{1}(L)\,\psi_{2}(L)-\psi_{1}(L)\,\psi_{2}(L)$
$\displaystyle=0=-\psi_{2}(0),$ (187)
$\displaystyle\psi^{\prime}_{1}(L)\,\psi_{2}(L)-\psi_{1}(L)\,\psi^{\prime}_{2}(L)$
$\displaystyle=-1=-\psi^{\prime}_{2}(0).$ (188)

Here the second identity holds on the basis of (6). Since for given initial
conditions a solution of a differential equation is unique, the functions on
the both sides of (20) appear to be identical.

Analogously we prove the relations

$\displaystyle\psi_{1}(x-L)$
$\displaystyle=\psi_{2}(x)\,\psi^{\prime}_{1}(-L)+\psi_{1}(x)\,\psi_{1}(-L),$
(189) $\displaystyle\psi_{1}(x+L)$
$\displaystyle=\psi_{2}(x)\,\psi^{\prime}_{1}(L)+\psi_{1}(x)\,\psi_{1}(L),$
(190) $\displaystyle\psi_{2}(x+L)$
$\displaystyle=\psi_{2}(x)\,\psi^{\prime}_{2}(L)+\psi_{1}(x)\,\psi_{2}(L),$
(191)

and remark that together with (20) they imply the following consequences

$\displaystyle\psi_{2}(-L)$ $\displaystyle=-\psi_{2}(L),$ (192)
$\displaystyle\psi^{\prime}_{2}(-L)$ $\displaystyle=\psi_{1}(L),$ (193)
$\displaystyle\psi_{1}(-L)$ $\displaystyle=\psi^{\prime}_{2}(L),$ (194)
$\displaystyle\psi^{\prime}_{1}(-L)$ $\displaystyle=-\psi^{\prime}_{1}(L).$
(195)

To prove (56), we consider the following equations

$\displaystyle-\frac{1}{2m}\frac{\partial\psi^{\prime\prime}_{2}(x)}{\partial
E}+[V(x)-E]\frac{\partial\psi_{2}(x)}{\partial E}$
$\displaystyle=\psi_{2}(x),$ (196)
$\displaystyle-\frac{1}{2m}\frac{\partial\psi^{\prime\prime}_{2}(x-L)}{\partial
E}+[V(x)-E]\frac{\partial\psi_{2}(x-L)}{\partial E}$
$\displaystyle=\psi_{2}(x-L).$ (197)

Multiplying the first equation with $\frac{\partial\psi_{2}(x-L)}{\partial
E}$, the second equation with $\frac{\partial\psi_{2}(x)}{\partial E}$, and
subtracting the obtained results from each other, we find

$\displaystyle-2m\,F(x)$ $\displaystyle=\frac{\partial\psi_{2}(x-L)}{\partial
E}\,\frac{\partial\psi^{\prime\prime}_{2}(x)}{\partial
E}-\frac{\partial\psi_{2}(x)}{\partial
E}\,\frac{\partial\psi^{\prime\prime}_{2}(x-L)}{\partial E}.$ (198)

Integrating both sides over $x$ from $0$ to $L$ and applying the integration
by parts, we obtain

$\displaystyle-2m\int_{0}^{L}dx\,\,F(x)$
$\displaystyle=\left[\frac{\partial\psi_{2}(x-L)}{\partial
E}\,\frac{\partial\psi^{\prime}_{2}(x)}{\partial
E}-\frac{\partial\psi_{2}(x)}{\partial
E}\,\frac{\partial\psi^{\prime}_{2}(x-L)}{\partial E}\right]_{0}^{L}.$ (199)

But the right-hand side equals zero, since

$\displaystyle\frac{\partial\psi_{2}(0)}{\partial
E}=\frac{\partial\psi^{\prime}_{2}(0)}{\partial E}=0$ (200)

(we recall that $\psi_{2}(0)=0$ and $\psi^{\prime}_{2}(0)=1$ are constants).
This proves the identity (56).

Below we list further properties of the Hill’s equation.

Multiplying (196) with $\psi_{2}(x-L)$ and integrating over $x$ from $0$ to
$L$ (also using the integration by parts), we obtain

$\displaystyle\frac{1}{2m}\,\psi^{\prime}_{2}(x-L)\,\frac{\partial\psi_{2}(x)}{\partial
E}\,\bigg{|}_{0}^{L}-\frac{1}{2m}\,\psi_{2}(x-L)\,\frac{\partial\psi^{\prime}_{2}(x)}{\partial
E}\,\bigg{|}_{0}^{L}$
$\displaystyle=\int_{0}^{L}dx\,\,\psi_{2}(x-L)\,\psi_{2}(x).$ (201)

It follows

$\displaystyle\frac{1}{2m}\,\frac{\partial\psi_{2}(L)}{\partial
E}=\int_{0}^{L}dx\,\,\psi_{2}(x-L)\,\psi_{2}(x).$ (202)

In the analogous manner we derive

$\displaystyle\frac{1}{2m}\left[\psi^{\prime}_{2}(x)\,\frac{\partial\psi_{2}(x)}{\partial
E}-\psi_{2}(x)\,\frac{\partial\psi^{\prime}_{2}(x)}{\partial
E}\right]_{0}^{L}=\int_{0}^{L}dx\,\,\psi_{2}^{2}(x)$ (203)

and

$\displaystyle\frac{1}{2m}\left[\psi^{\prime}_{2}(x-L)\,\frac{\partial\psi_{2}(x-L)}{\partial
E}-\psi_{2}(x-L)\,\frac{\partial\psi^{\prime}_{2}(x-L)}{\partial
E}\right]_{0}^{L}$ $\displaystyle=\int_{0}^{L}dx\,\,\psi_{2}^{2}(x-L),$ (204)

giving

$\displaystyle\frac{1}{2m}\left[\psi^{\prime}_{2}(L)\frac{\partial\psi_{2}(L)}{\partial
E}-\psi_{2}(L)\frac{\partial\psi^{\prime}_{2}(L)}{\partial E}\right]$
$\displaystyle=\int_{0}^{L}dx\,\,\psi_{2}^{2}(x),$ (205)

and

$\displaystyle\frac{1}{2m}\left[\psi_{1}(L)\frac{\partial\psi_{2}(L)}{\partial
E}-\psi_{2}(L)\frac{\partial\psi_{1}(L)}{\partial
E}\right]=\int_{0}^{L}dx\,\psi_{2}^{2}(x-L),$ (206)

respectively.

Summing up (205) and (206), we obtain

$\displaystyle\frac{1}{m}\left[D(E)\,\frac{\partial\psi_{2}(L)}{\partial
E}-\psi_{2}(L)\,\frac{\partial D(E)}{\partial E}\right]$
$\displaystyle=\int_{0}^{L}dx\,\,\psi_{2}^{2}(x)+\int_{0}^{L}dx\,\,\psi_{2}^{2}(x-L).$
(207)

On the basis of (202) and (207) we state the Schwarz inequality

$\displaystyle D(E)\,\frac{\partial\psi_{2}(L)}{\partial
E}-\psi_{2}(L)\,\frac{\partial D(E)}{\partial
E}\geq\bigg{|}\frac{\partial\psi_{2}(L)}{\partial E}\bigg{|}.$ (208)

Note that from this inequality it follows that strictly inside a band, i.e.
for $-1<D(E)<1$, the function $D(E)$ may not have extrema, that is
$\frac{\partial D(E)}{\partial E}\neq 0$, since otherwise we get the
contradiction

$\displaystyle\bigg{|}\frac{\partial\psi_{2}(L)}{\partial
E}\bigg{|}>D(E)\,\frac{\partial\psi_{2}(L)}{\partial
E}\geq\bigg{|}\frac{\partial\psi_{2}(L)}{\partial E}\bigg{|}.$ (209)

Similar relations can be derived for the function $\psi_{1}$:

$\displaystyle-\frac{1}{2m}\,\frac{\partial\psi^{\prime}_{1}(L)}{\partial
E}=\int_{0}^{L}dx\,\,\psi_{1}(x-L)\,\psi_{1}(x),$ (210)
$\displaystyle\frac{1}{2m}\left[\psi^{\prime}_{1}(L)\frac{\partial\psi_{1}(L)}{\partial
E}-\psi_{1}(L)\frac{\partial\psi^{\prime}_{1}(L)}{\partial
E}\right]=\int_{0}^{L}dx\,\psi_{1}^{2}(x),$ (211)
$\displaystyle\frac{1}{2m}\left[\psi^{\prime}_{1}(L)\frac{\partial\psi^{\prime}_{2}(L)}{\partial
E}-\psi^{\prime}_{2}(L)\frac{\partial\psi^{\prime}_{1}(L)}{\partial
E}\right]=\int_{0}^{L}dx\,\psi_{1}^{2}(x-L).$ (212)

Summing up (211) and (212) we obtain

$\displaystyle\frac{1}{m}\left[\psi^{\prime}_{1}(L)\,\frac{\partial
D(E)}{\partial E}-D(E)\,\frac{\partial\psi^{\prime}_{1}(L)}{\partial
E}\right]$
$\displaystyle=\int_{0}^{L}dx\,\,\psi_{1}^{2}(x)+\int_{0}^{L}dx\,\,\psi_{1}^{2}(x-L).$
(213)

Analogously to (208) we state the other Schwarz inequality

$\displaystyle\psi^{\prime}_{1}(L)\,\frac{\partial D(E)}{\partial
E}-D(E)\,\frac{\partial\psi^{\prime}_{1}(L)}{\partial
E}\geq\bigg{|}\frac{\partial\psi^{\prime}_{1}(L)}{\partial E}\bigg{|}.$ (214)

##  Appendix B Some properties of the Bloch states

Differentiating (1) for $\psi_{k\alpha}(x)$ with respect to $k$, then
multiplying it with $\psi^{*}_{k\alpha}(x)$, and integrating over $x$ from $0$
to $L$ (also using the integration by parts), we get

$\displaystyle 2m\,\frac{d\epsilon_{k\alpha}}{dk}=$
$\displaystyle-\psi_{k\alpha}^{*}(L)\,\frac{d\psi^{\prime}_{k\alpha}(L)}{dk}+\psi_{k\alpha}^{*}(0)\,\frac{d\psi^{\prime}_{k\alpha}(0)}{dk}$
$\displaystyle+\psi^{\prime\,*}_{k\alpha}(L)\,\frac{d\psi_{k\alpha}(L)}{dk}-\psi^{\prime\,*}_{k\alpha}(0)\,\frac{d\psi_{k\alpha}(0)}{dk}$
$\displaystyle=$
$\displaystyle\,2L\,\text{Im}\,[\psi_{k\alpha}^{*}(0)\,\psi^{\prime}_{k\alpha}(0)],$
(215)

where the last equality follows from (8) and (9).

Expressing from (21)

$\displaystyle\psi_{k\alpha}(0)$
$\displaystyle=\frac{\psi_{2}(L)}{\sqrt{N_{k\alpha}}},$ (216)
$\displaystyle\psi^{\prime}_{k\alpha}(0)$
$\displaystyle=\frac{1}{\sqrt{N_{k\alpha}}}\,[e^{ikL}-\psi_{1}(L)],$ (217)

we establish

$\displaystyle\frac{d\epsilon_{k\alpha}}{dk}=\frac{L}{m}\,\frac{\psi_{2}(L)}{N_{k\alpha}}\,\sin
kL.$ (218)

Differentiating (14) with respect to $k$

$\displaystyle-L\,\sin kL=\frac{\partial D(E)}{\partial
E}\,\frac{d\epsilon_{k\alpha}}{dk},$ (219)

we also get an alternative form of (218)

$\displaystyle N_{k\alpha}=-\frac{\psi_{2}(L)}{m}\frac{\partial D(E)}{\partial
E}=-\frac{\psi_{2}(L)}{2m}\left(\frac{\partial\psi_{1}(L)}{\partial
E}+\frac{\partial\psi^{\prime}_{2}(L)}{\partial E}\right).$ (220)

Comparing it with (15) we conclude that within the band $\alpha$ the following
sign

$\displaystyle\text{sign}\,[\psi_{2}(L,\epsilon_{k\alpha})]=-(-1)^{\alpha}$
(221)

is constant and solely determined by the band index $\alpha$.

##  Appendix C Evaluation of the Chern index

To evaluate (71), we represent

$\displaystyle\int_{0}^{L}dx\,\,\text{Im}\,\frac{d\psi_{k\alpha}^{(\varphi)\,*}(x)}{dk}\,\frac{d\psi_{k\alpha}^{(\varphi)}(x)}{dx_{\varphi}}$
$\displaystyle=$
$\displaystyle\int_{0}^{L}dx\,\,\text{Im}\,\frac{d\psi_{k\alpha}^{*}(x+x_{\varphi})}{dk}\,\psi^{\prime}_{k\alpha}(x+x_{\varphi})$
(222)
$\displaystyle-\frac{d\Phi_{k\alpha}(x_{\varphi})}{dx_{\varphi}}\,\frac{1}{2}\,\frac{d}{dk}\int_{0}^{L}dx\,\,|\psi_{k\alpha}(x+x_{\varphi})|^{2}$
(223)
$\displaystyle+\frac{d\Phi_{k\alpha}(x_{\varphi})}{dk}\,\frac{1}{2}\,\int_{0}^{L}dx\,\,\frac{d}{dx}|\psi_{k\alpha}(x+x_{\varphi})|^{2},$
(224)

where we used (61) and the periodicity of
$|\psi_{k\alpha}(x)|^{2}=|u_{k\alpha}(x)|^{2}$ in $x$. Apparently, the
contributions (223) and (224) identically vanish.

Transforming the contribution (222)

$\displaystyle\int_{0}^{L}dx\,\,\text{Im}\,\frac{d\psi_{k\alpha}^{*}(x+x_{\varphi})}{dk}\,\psi^{\prime}_{k\alpha}(x+x_{\varphi})$
$\displaystyle=$
$\displaystyle\frac{d}{dk}\int_{0}^{L}dx\,\,\text{Im}\,\psi_{k\alpha}^{*}(x+x_{\varphi})\,\psi^{\prime}_{k\alpha}(x+x_{\varphi})$
$\displaystyle-\int_{0}^{L}dx\,\,\text{Im}\,\psi_{k\alpha}^{*}(x+x_{\varphi})\,\frac{d\psi^{\prime}_{k\alpha}(x+x_{\varphi})}{dk}$
(225) $\displaystyle=$
$\displaystyle\frac{d}{dk}\int_{0}^{L}dx\,\,\text{Im}\,\psi_{k\alpha}^{*}(x+x_{\varphi})\,\psi^{\prime}_{k\alpha}(x+x_{\varphi})$
$\displaystyle-\left[\text{Im}\,\psi_{k\alpha}^{*}(x+x_{\varphi})\,\frac{d\psi_{k\alpha}(x+x_{\varphi})}{dk}\right]_{0}^{L}$
$\displaystyle+\int_{0}^{L}dx\,\,\text{Im}\,\psi_{k\alpha}^{\prime\,*}(x+x_{\varphi})\,\frac{d\psi_{k\alpha}(x+x_{\varphi})}{dk},$
(226)

we establish

$\displaystyle\int_{0}^{L}dx\,\,\text{Im}\,\frac{d\psi_{k\alpha}^{*}(x+x_{\varphi})}{dk}\,\psi^{\prime}_{k\alpha}(x+x_{\varphi})$
$\displaystyle=$
$\displaystyle\frac{1}{2}\,\frac{d}{dk}\int_{0}^{L}dx\,\,\text{Im}\,\psi_{k\alpha}^{*}(x+x_{\varphi})\,\psi^{\prime}_{k\alpha}(x+x_{\varphi})$
(227) $\displaystyle-\frac{L}{2}\,|u_{k\alpha}(x_{\varphi})|^{2}.$ (228)

Since $\psi_{k\alpha}(x)$ is periodic in $k$, the term (227) drops out under
the $k$-integration, and we get

$\displaystyle C_{1,\alpha}=$ $\displaystyle
L\,\int_{0}^{L}dx_{\varphi}\int_{-\pi/L}^{\pi/L}\frac{dk}{2\pi}\,\,|u_{k\alpha}(x_{\varphi})|^{2}=1.$
(229)

##  Appendix D One edge state per gap

Consider odd gap $\alpha$, that is
$E\in[\epsilon_{\frac{\pi}{L},\alpha},\epsilon_{\frac{\pi}{L},\alpha+1}]$,
characterized by $D(E)\leq-1$. According to (221), the function
$\psi_{2}(L,E)$ must change its sign across this gap from positive to negative
value, and this guarantees the existence of at least one edge state per gap.

Let us show that it is impossible to have more than one edge state per gap.
Suppose that we have multiple roots of $\psi_{2}(L,E)$ on the indicated above
interval, see Fig. 11. Because of the sign-changing property, their number
must be odd, degenerate roots (that is, with
$\psi_{2}(L,E)=\frac{\partial}{\partial E}\psi_{2}(L,E)=0$) being excluded on
the basis of (207). This entails that at every even root we should have
$\frac{\partial}{\partial E}\psi_{2}(L,E)>0$. But this contradicts to (207),
since its left-hand side appears to be negative at even roots. Therefore, even
roots are impossible, and thus we have only a single edge state per gap.

Analogously, we prove the same property for even gaps.

Figure 11: Illustration of the impossibility to have multiple roots of
$\psi_{2}(L,E)$ within gaps. Assume an odd gap index $\alpha$ ($D(E)\leq-1$)
which hosts an odd number of roots of $\psi_{2}(L,E)$. If the latter number is
greater than one, there must necessarily be one root that satisfies both
$\psi_{2}(L,E)=0$ and $\frac{\partial}{\partial E}\psi_{2}(L,E)>0$. Since this
implies $\text{sign}[D(E)\frac{\partial}{\partial E}\psi_{2}(L,E)]<0$ we
obtain a contradiction to (207). A similar argumentation holds for even gap
index $\alpha$.

##  Appendix E Zeros of (anti)periodic solutions of (1)

Consider a family of Hamiltonians

$\displaystyle H_{\xi}=-\frac{1}{2m}\frac{d^{2}}{dx^{2}}+\zeta\,V(x),\quad
V(x)=V(x+L),$ (230)

where $\zeta$ is continuously varied on the interval $(0,1]$. Equipping the
eigenvalue problem (230) with the periodic boundary conditions, we obtain
periodic eigenfunctions $\psi_{l;\zeta}^{(p)}(x)=\psi_{l;\zeta}^{(p)}(x+L)$
labelled by $l=0,1,2,\ldots$, which are continuous in $\zeta$. The
corresponding eigenenergies are denoted by $\epsilon_{l;\zeta}^{(p)}$. In
addition, we consider the antiperiodic boundary conditions leading to the
solutions $\psi_{l;\zeta}^{(a)}(x)=-\psi_{l;\zeta}^{(a)}(x+L)$ of (230) with
eigenenergies $\epsilon_{l;\zeta}^{(a)}$, labelled by $l=1,2,\ldots$.

The energies $\epsilon_{l;\zeta}^{(p)}$ and $\epsilon_{l;\zeta}^{(a)}$ are
identified with the bands’ edges of the eigenvalue problem (1):

$\displaystyle\epsilon_{k=0,\alpha;\zeta}$
$\displaystyle=\epsilon_{\alpha-1;\zeta}^{(p)},$ (231)
$\displaystyle\epsilon_{k=\frac{\pi}{L},\alpha;\zeta}$
$\displaystyle=\epsilon_{\alpha;\xi}^{(a)},$ (232)

where the band index $\alpha\geq 1$. Analogously we identify the corresponding
eigenfunctions

$\displaystyle\psi_{k=0,\alpha;\zeta}(x)$
$\displaystyle=\psi_{\alpha-1;\zeta}^{(p)}(x),$ (233)
$\displaystyle\psi_{k=\frac{\pi}{L},\alpha;\zeta}(x)$
$\displaystyle=\psi_{\alpha;\zeta}^{(a)}(x).$ (234)

The main observation is that a number of zeros of either
$\psi_{l;\zeta}^{(p)}(x)$ or $\psi_{l;\zeta}^{(a)}(x)$ on the interval
$x\in[0,L)$ does not depend on the value of $\zeta$. Indeed, to change a
number of zeros of a continuous eigenfunction we would need to make its two
nearby roots coalesce at some value $\zeta^{*}$ and point $x^{*}$. Thereby we
get a second-order root with
$\psi_{l;\zeta^{*}}^{(p,a)}(x^{*})=\psi^{(p,a)\,\prime}_{l;\zeta^{*}}(x^{*})=0$.
But this implies that $\psi_{l;\zeta^{*}}^{(p,a)}(x)\equiv 0$. This is
impossible, since there must be a nontrivial solution corresponding to the
energy $\epsilon_{l;\zeta^{*}}^{(p,a)}$.

Let us find numbers of zeros of the periodic functions
$\psi_{\alpha;\zeta}^{(p)}(x)$ and $\psi_{\alpha;\zeta}^{(a)}(x)$ for
infinitesimal $\zeta\ll 1$.

In this limit, the eigenstates $\psi_{\alpha-1;\zeta}^{(p)}(x)$ and
$\psi_{\alpha;\zeta}^{(p)}(x)$ with even $\alpha$ become nearly degenerate
with the energy $\epsilon_{\alpha}^{(0)}=\frac{\alpha^{2}\pi^{2}}{2mL^{2}}$.
Using $\frac{1}{\sqrt{L}}e^{\pm i\frac{\pi\alpha}{L}x}$ as a basis in this
two-dimensional degenerate subspace, we find in the first-order degenerate
perturbation theory
$\epsilon_{\alpha}^{(p)}=\epsilon_{\alpha}^{(0)}+\zeta|\tilde{V}_{\alpha}|$
and
$\epsilon_{\alpha-1}^{(p)}=\epsilon_{\alpha}^{(0)}-\zeta|\tilde{V}_{\alpha}|$
with the corresponding eigenstates

$\displaystyle\psi_{\alpha;\zeta\ll 1}^{(p)}(x)$
$\displaystyle\approx\sqrt{\frac{2}{L}}\cos\left(\frac{\pi\alpha}{L}x+\frac{\varphi_{\alpha}}{2}\right),$
(235) $\displaystyle\psi_{\alpha-1;\zeta\ll 1}^{(p)}(x)$
$\displaystyle\approx\sqrt{\frac{2}{L}}\sin\left(\frac{\pi\alpha}{L}x+\frac{\varphi_{\alpha}}{2}\right),$
(236)

where
$\tilde{V}_{\alpha}=|\tilde{V}_{\alpha}|e^{i\varphi_{\alpha}}=\frac{1}{L}\int_{0}^{L}dx\,V(x)\,e^{-i\frac{2\pi\alpha}{L}x}$
is the $\alpha$th Fourier component of the periodic potential. Both function
(235) and (236) have $\alpha$ zeros on the interval $x\in[0,L)$. An exception
in this consideration is $\alpha=0$: the eigenfunction
$\psi_{\alpha=0;\zeta\ll 1}^{(p)}(x)\approx\frac{1}{\sqrt{L}}$ remains
nondegenerate. It has no zeros.

Analogously we consider $\psi_{\alpha+1;\zeta}^{(a)}(x)$ and
$\psi_{\alpha;\zeta}^{(a)}(x)$ with odd $\alpha$, and find them to be of the
form (235) and (236), respectively. This implies that they also have $\alpha$
zeros on the interval $x\in[0,L)$.

Thus, by virtue of (233) and (234), the functions $\psi_{k=0,\alpha+1}(x)$ and
$\psi_{k=0,\alpha}(x)$ with even $\alpha$ have $\alpha$ zeros, as well as the
functions $\psi_{k=\frac{\pi}{L},\alpha+1}(x)$ and
$\psi_{k=\frac{\pi}{L},\alpha}(x)$ with odd $\alpha$, have $\alpha$ zeros.
These properties ensure the relation (123).

##  Appendix F Universal scaling of the boundary charge fluctuations

The geometric tensor appearing in (126) is defined by

$\displaystyle(\mathcal{Q}_{k})_{\alpha\beta}$
$\displaystyle=\langle\frac{du_{k\alpha}}{dk}|\frac{du_{k\beta}}{dk}\rangle\,\delta_{\alpha\beta}-|\langle
u_{k\alpha}|\frac{du_{k\beta}}{dk}\rangle|^{2}.$ (237)

Here we employ the shorthand notation $\langle
F|G\rangle=\int_{0}^{L}dx\,F^{*}(x)G(x)$. Using the completeness relation
$\sum_{\beta}|u_{k\beta}\rangle\langle u_{k\beta}|=\hat{1}$, we alternatively
express

$\displaystyle\sum_{\alpha,\beta=1}^{\nu}(\mathcal{Q}_{k})_{\alpha\beta}=\sum_{\alpha=1}^{\nu}\sum_{\beta=\nu+1}^{\infty}|\langle
u_{k\alpha}|\frac{du_{k\beta}}{dk}\rangle|^{2}.$ (238)

In the obtained double sum the index $\alpha$ runs through the valence bands,
while the index $\beta$ runs through the conduction bands.

For $\alpha\neq\beta$ we establish

$\displaystyle-i\,\langle\psi_{k\alpha}|\frac{d\psi_{k\beta}}{dk}\rangle=-i\,\langle
u_{k\alpha}|\frac{du_{k\beta}}{dk}\rangle+\langle
u_{k\alpha}|\,x\,|u_{k\beta}\rangle$ (239) $\displaystyle=$
$\displaystyle\frac{L}{2m(\epsilon_{k\alpha}-\epsilon_{k\beta})}\left[\psi_{k\alpha}^{*}(0)\,\psi^{\prime}_{k\beta}(0)-\psi_{k\alpha}^{*\,\prime}(0)\,\psi_{k\beta}(0)\right]$
(240) $\displaystyle=$
$\displaystyle\frac{1}{2m(\epsilon_{k\alpha}-\epsilon_{k\beta})}\int_{0}^{L}dx\,\,[\psi_{k\alpha}^{*}(x)\,\psi^{\prime}_{k\beta}(x)-\psi_{k\alpha}^{*\,\prime}(x)\,\psi_{k\beta}(x)]$
$\displaystyle+\langle u_{k\alpha}|\,x\,|u_{k\beta}\rangle.$ (241)

To obtain (240) from (239), we consider the equation

$\displaystyle-\frac{1}{2m}\frac{d\psi^{\prime\prime}_{k\beta}(x)}{dk}+[V(x)-\epsilon_{k\beta}]\,\frac{d\psi_{k\beta}(x)}{dk}=\frac{d\epsilon_{k\beta}}{dk}\,\psi_{k\beta}(x),$
(242)

and project it onto the state $\langle\psi_{k\alpha}|$ with $\alpha\neq\beta$,
such that $\langle\psi_{k\alpha}|\psi_{k\beta}\rangle=0$. It follows

$\displaystyle\frac{1}{2m}\int_{0}^{L}dx\left[\psi^{*}_{k\alpha}(x)\frac{d\psi^{\prime\prime}_{k\beta}(x)}{dk}-\psi^{*\,\prime\prime}_{k\alpha}(x)\frac{d\psi_{k\beta}(x)}{dk}\right]$
(243) $\displaystyle=$
$\displaystyle(\epsilon_{k\alpha}-\epsilon_{k\beta})\,\langle\psi_{k\alpha}|\frac{d\psi_{k\beta}}{dk}\rangle.$
(244)

Integrating (243) by parts, we obtain (240).

To derive (241) from (240), we multiply the Schrödinger equation for
$|\psi_{k\beta}\rangle$ with $x$ and then project the result onto the state
$\langle\psi_{k\alpha}|$. This gives

$\displaystyle\frac{1}{2m}\int_{0}^{L}dx\,\left[x\,\psi_{k\alpha}^{*}(x)\,\psi^{\prime\prime}_{k\beta}(x)-\,x\,\psi_{k\alpha}^{*\,\prime\prime}(x)\psi_{k\beta}(x)\right]$
(245)
$\displaystyle=(\epsilon_{k\alpha}-\epsilon_{k\beta})\,\langle\psi_{k\alpha}|\,x\,|\psi_{k\beta}\rangle.$
(246)

Integrating (245) by parts and using
$\langle\psi_{k\alpha}|x|\psi_{k\beta}\rangle=\langle
u_{k\alpha}|x|u_{k\beta}\rangle$ as well as
$\psi_{k\alpha}(L)=e^{ikL}\psi_{k\alpha}(0)$,
$\psi^{\prime}_{k\alpha}(L)=e^{ikL}\psi^{\prime}_{k\alpha}(0)$, we obtain
(241).

To further simplify (241) we notice that

$\displaystyle\int_{0}^{L}dx\,\,[\psi_{k\alpha}^{*}(x)\,\psi^{\prime}_{k\beta}(x)-\psi_{k\alpha}^{*\,\prime}(x)\,\psi_{k\beta}(x)]$
$\displaystyle=$
$\displaystyle\frac{2}{\epsilon_{k\alpha}-\epsilon_{k\beta}}\int_{0}^{L}dx\,\,V(x)\,\frac{d}{dx}\,[\psi_{k\alpha}^{*}(x)\,\psi_{k\beta}(x)].$
(247)

Thus, we get

$\displaystyle-i\,\langle u_{k\alpha}|\frac{du_{k\beta}}{dk}\rangle$
$\displaystyle=$
$\displaystyle\frac{1}{m(\epsilon_{k\alpha}-\epsilon_{k\beta})^{2}}\int_{0}^{L}dx\,\,V(x)\,\frac{d}{dx}\,[\psi_{k\alpha}^{*}(x)\,\psi_{k\beta}(x)].$
(248)

The validity of (247) follows from the two equations

$\displaystyle-\frac{1}{2m}\int_{0}^{L}dx\,\,\psi_{k\alpha}^{*\,\prime}(x)\,\psi^{\prime\prime}_{k\beta}(x)$
$\displaystyle+\int_{0}^{L}dx\,\,\psi_{k\alpha}^{*\,\prime}(x)\,[V(x)-\epsilon_{k\beta}]\,\psi_{k\beta}(x)=0,$
(249)
$\displaystyle-\frac{1}{2m}\int_{0}^{L}dx\,\,\psi_{k\alpha}^{*\,\prime\prime}(x)\,\psi^{\prime}_{k\beta}(x)$
$\displaystyle+\int_{0}^{L}dx\,\,\psi_{k\alpha}^{*\,}(x)\,[V(x)-\epsilon_{k\alpha}]\,\psi^{\prime}_{k\beta}(x)=0.$
(250)

Adding them up, we establish

$\displaystyle\int_{0}^{L}dx\,\,V(x)\,\frac{d}{dx}[\psi_{k\alpha}^{*}(x)\,\psi_{k\beta}(x)]$
$\displaystyle=$
$\displaystyle\epsilon_{k\alpha}\int_{0}^{L}dx\,\,\psi_{k\alpha}^{*\,}(x)\,\psi^{\prime}_{k\beta}(x)+\epsilon_{k\beta}\int_{0}^{L}dx\,\,\psi_{k\alpha}^{*\,\prime}(x)\,\psi_{k\beta}(x)$
$\displaystyle=$
$\displaystyle\frac{\epsilon_{k\alpha}-\epsilon_{k\beta}}{2}\int_{0}^{L}dx\,\,[\psi_{k\alpha}^{*\,}(x)\,\psi^{\prime}_{k\beta}(x)-\psi_{k\alpha}^{*\,\prime}(x)\,\psi_{k\beta}(x)]$
$\displaystyle+\frac{\epsilon_{k\alpha}+\epsilon_{k\beta}}{2}\int_{0}^{L}dx\,\,\frac{d}{dx}[\psi_{k\alpha}^{*\,}(x)\,\psi_{k\beta}(x)].$
(251)

Due to the periodicity of $[\psi_{k\alpha}^{*\,}(x)\,\psi_{k\beta}(x)]$ the
last term vanishes, and we obtain (247).

In the narrow-gap limit we find the low-energy approximation for (126) (cf.
[wannier_paper, ] and Appendix E): The leading contribution is received from
the valence band
$\epsilon_{k\nu}\approx\epsilon_{\nu}^{(0)}-\sqrt{(v_{F,\nu}k)^{2}+|\tilde{V}_{\nu}|^{2}}$
and the conduction band
$\epsilon_{k,\nu+1}\approx\epsilon_{\nu}^{(0)}+\sqrt{(v_{F,\nu}k)^{2}+|\tilde{V}_{\nu}|^{2}}$,
which are adjacent to the chemical potential
$\mu_{\nu}=\epsilon_{\nu}^{(0)}=\frac{\nu^{2}\pi^{2}}{2mL^{2}}$ (assuming even
$\nu$ and performing the expansion near $k=0$; for odd $\nu$ one first has to
shift $k\to k-\pi/L$). Here
$v_{F,\nu}=\frac{\sqrt{2m\mu_{\nu}}}{m}=\frac{\nu\pi}{mL}$ is the Fermi
velocity, and the $\nu$th Fourier component
$\tilde{V}_{\nu}=\frac{1}{L}\int_{0}^{L}dx\,V(x)\,e^{-i\frac{2\pi\nu}{L}x}$ of
the potential $V(x)$ determines the $\nu$th gap’s size
$E_{g,\nu}=2|\tilde{V}_{\nu}|$. To approximate (248) we use (233), (234) as
well as (235), (236), and obtain

$\displaystyle\int_{0}^{L}dx\,\,V(x)\,\frac{d}{dx}\,[\psi_{k\alpha}^{*}(x)\,\psi_{k\beta}(x)]$
$\displaystyle\approx\int_{0}^{L}dx\,\,V(x)\,\frac{d}{dx}\,\frac{1}{L}\sin\left(\frac{2\pi\nu}{L}x+\varphi_{\nu}\right)$
(252)
$\displaystyle=\frac{2\pi\nu}{L}|\tilde{V}_{\nu}|=2mv_{F,\nu}|\tilde{V}_{\nu}|.$
(253)

Combining all approximations made together, we establish

$\displaystyle X_{2}\,(\Delta
Q_{B}^{(\nu)})^{2}\approx\frac{1}{2\pi}\int_{-\infty}^{\infty}dk\,\,|\langle
u_{k\nu}|\frac{du_{k,\nu+1}}{dk}\rangle|^{2}$
$\displaystyle\approx\frac{1}{8\pi}\int_{-\infty}^{\infty}dk\,\,\frac{v_{F,\nu}^{2}|\tilde{V}_{\nu}|^{2}}{[(v_{F,\nu}k)^{2}+|\tilde{V}_{\nu}|^{2}]^{2}}=\frac{v_{F,\nu}}{16|\tilde{V}_{\nu}|},$
(254)

which is equivalent to (128).

##  Appendix G Unitarity of the scattering matrix (149)

To check the unitarity of (149), we first observe that by virtue of the
identity

$\displaystyle\frac{\psi_{2}^{(\varphi)}(L)}{N_{k\alpha}^{(\varphi)}}=\frac{\psi_{2}^{(\varphi^{\prime})}(L)}{N_{k\alpha}^{(\varphi^{\prime})}}=-\frac{m}{\frac{\partial
D(E)}{\partial E}},$ (255)

following from (220), it holds

$\displaystyle t_{k\alpha}=t^{\prime}_{k\alpha}=-\frac{m}{\frac{\partial
D(E)}{\partial E}}\,\frac{2i\,\sin
kL}{d_{k\alpha}}\,\sqrt{N_{k\alpha}^{(\varphi)}N_{k\alpha}^{(\varphi^{\prime})}}.$
(256)

Next, we establish that

$\displaystyle|r_{k\alpha}|^{2}=$ $\displaystyle 1-\frac{4\sin
kL\,\,\psi_{2}^{(\varphi)}(L)}{|d_{k\alpha}|^{2}}\,\text{Im}\,d_{k\alpha}$
$\displaystyle+\frac{4\sin^{2}kL\,\,[\psi_{2}^{(\varphi)}(L)]^{2}}{|d_{k\alpha}|^{2}},$
(257) $\displaystyle|r^{\prime}_{k\alpha}|^{2}=$ $\displaystyle 1-\frac{4\sin
kL\,\,\psi_{2}^{(\varphi^{\prime})}(L)}{|d_{k\alpha}|^{2}}\,\text{Im}\,d_{k\alpha}$
$\displaystyle+\frac{4\sin^{2}kL\,\,[\psi_{2}^{(\varphi^{\prime})}(L)]^{2}}{|d_{k\alpha}|^{2}}.$
(258)

Since

$\displaystyle\text{Im}\,d_{k\alpha}=[\psi_{2}^{(\varphi)}(L)+\psi_{2}^{(\varphi^{\prime})}(L)]\,\sin
kL,$ (259)

we get

$\displaystyle|r_{k\alpha}|^{2}=|r^{\prime}_{k\alpha}|^{2}=1-\frac{4\sin^{2}kL\,\,\psi_{2}^{(\varphi)}(L)\,\psi_{2}^{(\varphi^{\prime})}(L)}{|d_{k\alpha}|^{2}}.$
(260)

To demonstrate the validity of (150) and (151) it suffices to notice that

$\displaystyle\left[\frac{m}{\frac{\partial D(E)}{\partial
E}}\right]^{2}N_{k\alpha}^{(\varphi)}N_{k\alpha}^{(\varphi^{\prime})}=\psi_{2}^{(\varphi)}(L)\,\psi_{2}^{(\varphi^{\prime})}(L),$
(261)

which holds true by virtue of (255).

The relation (152) requires

$\displaystyle\frac{1}{d_{k\alpha}^{*}}\left[1-\frac{2i\sin
kL\,\,\psi_{2}^{(\varphi^{\prime})}(L)}{d_{k\alpha}}\right]$ $\displaystyle-$
$\displaystyle\frac{1}{d_{k\alpha}}\left[1+\frac{2i\sin
kL\,\,\psi_{2}^{(\varphi)}(L)}{d_{k\alpha}^{*}}\right]=0,$ (262)

which indeed follows from (259).

Thus, the unitarity of (149) is verified.

Figure 12: Same as in Fig. 7 (besides a separate first gap close-up) for
$\lambda=0.1$. Figure 13: Same as in Fig. 7 (besides a separate first gap
close-up) for $\lambda=-0.1$.

##  Appendix H Additional illustrations

This section contains additional Figures 12 and 13 to highlight the previously
discussed properties of energy spectrum of the interface eigenvalue problem.
The chosen parameters $\lambda=\pm 0.1$ serve to illustrate the limit of the
absent impurity potential $\lambda\to 0^{\pm}$, which is most nontrivially
achieved close to the translationally invariant point
$x_{\varphi}=x_{\varphi^{\prime}}=0$.

##  Appendix I Derivation of (171)

Let us consider

$\displaystyle\frac{d}{dk}d_{k\alpha}=\frac{d\psi_{2}^{(\varphi)}(L)}{dk}\,[\psi_{1}^{(\varphi^{\prime})}(L)-e^{-ikL}]$
$\displaystyle-[\psi_{1}^{(\varphi)}(L)-e^{ikL}+2m\lambda\,\psi_{2}^{(\varphi)}(L)]\,\frac{d\psi_{2}^{(\varphi^{\prime})}(L)}{dk}$
$\displaystyle+\psi_{2}^{(\varphi)}(L)\,\frac{d\psi_{1}^{(\varphi^{\prime})}(L)}{dk}$
$\displaystyle-\left[\frac{d\psi_{1}^{(\varphi)}(L)}{dk}+2m\lambda\,\frac{d\psi_{2}^{(\varphi)}(L)}{dk}\right]\psi_{2}^{(\varphi^{\prime})}(L)$
$\displaystyle+iL\left\\{\psi_{2}^{(\varphi)}(L)\,e^{-ikL}+\psi_{2}^{(\varphi^{\prime})}(L)\,e^{ikL}\right\\},$
(263)

and compare it with

$\displaystyle P_{k\alpha}=$ $\displaystyle
L\int_{0}^{L}dx\,\,\psi_{2}^{(\varphi^{\prime})}(L)\,\psi_{k\alpha}^{(\varphi)\,2}(x)\,e^{-ikL}$
$\displaystyle+$ $\displaystyle
L\int_{0}^{L}dx\,\,\psi_{2}^{(\varphi)}(L)\,\psi_{-k,\alpha}^{(\varphi^{\prime})\,2}(x)\,e^{ikL},$
(264)

which appears in (170), that is

$\displaystyle\bar{Q}_{I,\alpha}=-\text{Re}\,\int_{-\pi/L}^{\pi/L}\frac{dk}{2\pi}\,\frac{P_{k\alpha}}{d_{k\alpha}}.$
(265)

With help of (202), (205) and (218) we evaluate

$\displaystyle\int_{0}^{L}dx\,\,\psi_{k\alpha}^{(\varphi)\,2}(x)$
$\displaystyle=1+\int_{0}^{L}dx\,\,\psi_{k\alpha}^{(\varphi)}(x)\left[\psi_{k\alpha}^{(\varphi)}(x)-\psi_{-k,\alpha}^{(\varphi)}(x)\right]$
$\displaystyle=1-\frac{i}{L}\,\frac{d\ln\psi_{2}^{(\varphi)}(L)}{dk}$
$\displaystyle+\frac{i}{L}\,\left[\psi^{(\varphi)\,\prime}_{2}(L)\,\frac{d\ln\psi_{2}^{(\varphi)}(L)}{dk}-\frac{d\psi^{(\varphi)\,\prime}_{2}(L)}{dk}\right]\,e^{ikL}.$
(266)

Thus

$\displaystyle P_{k\alpha}$
$\displaystyle=L\,\left\\{\psi_{2}^{(\varphi)}(L)\,e^{ikL}+\psi_{2}^{(\varphi^{\prime})}(L)\,e^{-ikL}\right\\}$
$\displaystyle-i\,e^{-ikL}\,\psi_{2}^{(\varphi^{\prime})}(L)\,\frac{d\ln\psi_{2}^{(\varphi)}(L)}{dk}$
$\displaystyle+i\,\psi_{2}^{(\varphi^{\prime})}(L)\,\left[\psi^{(\varphi)\,\prime}_{2}(L)\,\frac{d\ln\psi_{2}^{(\varphi)}(L)}{dk}-\frac{d\psi^{(\varphi)\,\prime}_{2}(L)}{dk}\right]$
$\displaystyle+i\,\psi_{2}^{(\varphi)}(L)\,\frac{d\ln\psi_{2}^{(\varphi^{\prime})}(L)}{dk}\,e^{ikL}$
$\displaystyle-i\,\psi_{2}^{(\varphi)}(L)\,\left[\psi^{(\varphi^{\prime})\,\prime}_{2}(L)\,\frac{d\ln\psi_{2}^{(\varphi^{\prime})}(L)}{dk}-\frac{d\psi^{(\varphi^{\prime})\,\prime}_{2}(L)}{dk}\right]$
$\displaystyle=L\,\left\\{\psi_{2}^{(\varphi)}(L)\,e^{-ikL}+\psi_{2}^{(\varphi^{\prime})}(L)\,e^{ikL}\right\\}$
$\displaystyle-i\,e^{-ikL}\,\psi_{2}^{(\varphi^{\prime})}(L)\,\frac{d\ln\psi_{2}^{(\varphi)}(L)}{dk}$
$\displaystyle+i\,\psi_{2}^{(\varphi^{\prime})}(L)\,\left[\psi^{(\varphi)\,\prime}_{2}(L)\,\frac{d\ln\psi_{2}^{(\varphi)}(L)}{dk}+\frac{d\psi^{(\varphi)}_{1}(L)}{dk}\right]$
$\displaystyle+i\,\psi_{2}^{(\varphi)}(L)\,\frac{d\ln\psi_{2}^{(\varphi^{\prime})}(L)}{dk}\,e^{ikL}$
$\displaystyle-i\,\psi_{2}^{(\varphi)}(L)\,\left[\psi^{(\varphi^{\prime})\,\prime}_{2}(L)\,\frac{d\ln\psi_{2}^{(\varphi^{\prime})}(L)}{dk}+\frac{d\psi^{(\varphi^{\prime})}_{1}(L)}{dk}\right].$
(267)

Replacing in (263) $[\psi_{1}^{(\varphi^{\prime})}(L)-e^{-ikL}]$ by

$\displaystyle\frac{d_{k\alpha}+[\psi_{1}^{(\varphi)}(L)-e^{ikL}+2m\lambda\,\psi_{2}^{(\varphi)}(L)]\,\psi_{2}^{(\varphi^{\prime})}(L)}{\psi_{2}^{(\varphi)}(L)},$
(268)

and $[\psi_{1}^{(\varphi)}(L)-e^{ikL}+2m\lambda\,\psi_{2}^{(\varphi)}(L)]$ by

$\displaystyle\frac{[\psi_{1}^{(\varphi^{\prime})}(L)-e^{-ikL}]\,\psi_{2}^{(\varphi)}(L)-d_{k\alpha}}{\psi_{2}^{(\varphi^{\prime})}(L)},$
(269)

we evaluate

$\displaystyle\frac{d}{dk}d_{k\alpha}-i\,P_{k\alpha}=d_{k\alpha}\,\frac{d}{dk}\ln[\psi_{2}^{(\varphi)}(L)\,\psi_{2}^{(\varphi^{\prime})}(L)],$
(270)

and

$\displaystyle\text{Im}\,\int_{-\pi/L}^{\pi/L}\frac{dk}{2\pi}\,\,\frac{1}{d_{k\alpha}}\frac{d}{dk}d_{k\alpha}-\text{Re}\,\int_{-\pi/L}^{\pi/L}\frac{dk}{2\pi}\,\,\frac{P_{k\alpha}}{d_{k\alpha}}=0.$
(271)

Hence,

$\displaystyle\bar{Q}_{I,\alpha}=-\text{Im}\,\int_{-\pi/L}^{\pi/L}\frac{dk}{2\pi}\,\,\frac{1}{d_{k\alpha}}\,\frac{d}{dk}d_{k\alpha},$
(272)

which is equivalent to (171).

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