# Topology in shallow-water waves:
A spectral flow perspective

Clément Tauber<EMAIL_ADDRESS>Institut de Recherche
Mathématique Avancée, UMR 7501 Université de Strasbourg et CNRS, 7 rue René-
Descartes, 67000 Strasbourg, France  Guo Chuan Thiang
<EMAIL_ADDRESS>Beijing International Center for Mathematical
Research, Peking University, No. 5 Yiheyuan Road Haidian District, Beijing,
P.R. China 100871

###### Abstract

In the context of topological insulators, the shallow-water model was recently
shown to exhibit an anomalous bulk-edge correspondence. For the model with a
boundary, the parameter space involves both longitudinal momentum and boundary
conditions, and exhibits a peculiar singularity. We resolve the anomaly in
question by defining a new kind of edge index as the spectral flow around this
singularity. Crucially, this edge index samples a whole family of boundary
conditions, and we interpret it as a boundary-driven quantized pumping. Our
edge index is stable due to the topological nature of spectral flow, and we
prove its correspondence with the bulk Chern number index using scattering
theory and a relative version of Levinson’s theorem. The full spectral flow
structure of the model is also investigated.

##  1 Introduction

Bulk-edge correspondence is a central concept of topological insulators. It
states that topological indices defined independently for an infinite sample –
the bulk – and for a half-infinite sample with boundary – the edge – actually
coincide. Originally investigated in Quantum Hall effect [17], bulk-edge
correspondence theorems have then been established for a wide range of models,
both discrete [18, 28, 16, 1, 25, 13] and continuous [2, 3, 8, 5, 12].
Applications go beyond condensed matter physics, such as in optics [27, 10],
acoustics [26], or fluid dynamics [9], actually to any wave phenomenon that
can be described by a self-adjoint operator.

Recently, the bulk-edge correspondence has been shown to be violated for the
first time in the shallow-water model [30, 15], that describes oceanic layers
on Earth. Once regularized by an odd-viscous term, such a model has a well-
defined bulk index: the Chern number. However, the edge picture is anomalous:
there exists a family of self-adjoint boundary conditions such that the number
of edge modes changes with the choice of boundary condition. The origin of
this anomaly is rooted in the unbounded nature of the operator’s spectrum and
was analyzed via scattering theory and a relative version of Levinson’s
theorem, originally developed in [16].

The aim of this paper is to circumvent this anomaly and restore a bulk-edge
correspondence theorem for shallow-water waves, by defining the edge index in
a manner which can sample a whole family of boundary conditions at once. We
focus on the same family of boundary conditions studied in [15], and first
show that this provides a continuous parametrization of boundary operators.
There is a special point in the joint parameter space of momentum and boundary
conditions, at which self-adjointness is lost. We define the edge index as the
spectral flow along a loop encircling this special point, and show that it
defines a topological index which coincides with the bulk Chern number. The
situation is similar to that of Weyl semimetals, which exhibits spectral flow
around singularities in the boundary momentum space where the Fredholm
condition is lost [31].

Spectral flow is a powerful concept that has been used for bulk-edge
correspondences in 2D discrete models of topological insulators in [16, 11,
6], and 3D Weyl semimetals in [31, 14]. In our context, the spectral flow
along a loop counts the signed number of crossings of edge mode branches with
an energy level, as the loop is traversed. Usually, a boundary condition is
fixed when computing the number of edge modes, but this way of counting leads
to an anomalous mismatch with the bulk index [15]. With hindsight, it is
unsurprising that the correct edge index should sample different boundary
conditions, since the bulk index does not pick out any particular boundary
condition. Furthermore, our edge index reveals the intriguing occurrence of
quantized pumping over a cycle of boundary conditions.

The interpretation of spectral flow as a topological quantum pump has a
longstanding tradition, where the system is usually probed via cycles of
external parameters such as fluxes [20, 21, 2]. To our knowledge, our edge
index is the first to involve a cycle of genuine boundary conditions of the
sample. Fictitious cycles of boundary conditions were previously considered by
the second author in [31, 7].

Finally, we show that our new edge index is stable against perturbations, a
crucial feature of topological quantities. This relies on a generalization of
the naive spectral flow definition, based on [22, 4], which deals with the
occurrence of possible pathological edge mode branches. As a byproduct we
investigate the complete spectral flow structure of the model.

##### Outline.

In Section 2, we provide, as a warm-up, an elementary example of the new type
of topological quantum pump discussed above. Our main results on the bulk-
boundary correspondence for the shallow-wave model are presented in Section 3.
The proofs are detailed in Sections 4 and 5. Finally, in Section 6, we discuss
the full spectral flow structure of the shallow-water model on a half-space.

##  2 Charge pumping via a cycle of boundary conditions

On a line, the free Schrödinger particle governed by the Hamiltonian
$\Delta=-\frac{d^{2}}{dx^{2}}$ has spectrum $[0,\infty)$. There appears to be
nothing topological about this operator, i.e. it has no bulk topological
index.

We might therefore expect the same for the particle on the half-line $x\geq
0$. This is not the case. To conserve probability, we need self-adjoint
boundary conditions at $x=0$, and these are given by (see pp. 144 of [24])

$\psi^{\prime}(0)+a\psi(0)=0,\qquad a\in\mathbb{R}\cup\\{\infty\\}.$

Here, $a=0$ labels the Neumann Laplacian on the half-line, while the Dirichlet
Laplacian is labelled by $a=\infty$, which is identified with $a=-\infty$. The
remaining $a\neq 0,\infty$ label Robin boundary conditions. Altogether, the
parameter space of boundary conditions is the compactified line
$\mathbb{R}\cup\\{\infty\\}$ (i.e. a circle), and we write $\Delta_{a}$ for
the half-line Laplacian subject to the $a$-boundary condition.

The essential spectrum of each $\Delta_{a}$ remains $[0,\infty)$. However,
once $a>0$, the operator $\Delta_{a}$ acquires an eigenvalue $-a^{2}$ with
normalizable eigenfunction $e^{-ax}$. Thus, as the parameter $a$ is increased
along the loop $\mathbb{R}\cup\\{\infty\\}$, one eigenvalue emerges out of the
essential spectrum when $a=0$, and it decreases towards $-\infty$ as $a$ is
increased towards $\infty$. This “charge pump” is illustrated in Figure 1
below.

$a$Spectrum$-\infty$$\infty$$0$$\circ$-1 Figure 1: The Laplace operator on the
half-line has boundary-driven spectral flow of $-1$ across any negative energy
level, as the boundary condition parameter $a$ is varied around a loop.

To achieve the above non-trivial spectral flow, it is important that we are
dealing with unbounded operators, so that the negative eigenvalue does not
have to rejoin the negative essential spectrum (which is empty). The family
$\\{\Delta_{a}+1\\}_{a\in\mathbb{R}\cup\\{\infty\\}}$ is an unbounded self-
adjoint Fredholm loop, for which a precise theory of spectral flow and its
topological invariance was developed in [4, 22].

This example indicates that we need to carefully account for the topology of
boundary conditions, when investigating bulk-boundary correspondence in
continuum models which are spectrally unbounded, such as the shallow-water
wave model. There are, however, two main differences with the simple model
above. First, the shallow-water model is two-dimensional with translation
invariance along the boundary, so that the parameter space is actually a
(punctured) cylinder instead of a circle. Second, its essential spectrum is
unbounded from above and below, which leads to more subtle situations for
eigenvalue branches.

##  3 Shallow-water model and main results

###  3.1 Bulk model and Chern number

The linear, rotating and odd-viscous shallow-water model describes a two-
dimensional thin layer of fluid of height $\eta$ and velocity $(u,v)$. Such
variables are ruled by a system of linear partial differential equations that
can be rewritten as a Schrödinger equation [29, 15] :

${\mathrm{i}}\partial_{t}\psi=\mathcal{H}\psi,\qquad\psi=\begin{pmatrix}\eta\\\
u\\\ v\end{pmatrix},\qquad\mathcal{H}=\begin{pmatrix}0&p_{x}&p_{y}\\\
p_{x}&0&-{\mathrm{i}}(f-\nu p^{2})\\\ p_{y}&{\mathrm{i}}(f-\nu
p^{2})&0\end{pmatrix},$ (1)

with $p_{x}=-{\mathrm{i}}\partial_{x}$, $p_{y}=-{\mathrm{i}}\partial_{y}$ and
${p}^{2}=p_{x}^{2}+p_{y}^{2}$. Here, $f,\nu$ are model constants satisfying
$f>0$, $\nu>0$ and $1-4f\nu>0$. The operator $\mathcal{H}$ is self-adjoint on
the domain $H^{1}(\mathbb{R}^{2})\oplus H^{2}(\mathbb{R}^{2})\oplus
H^{2}(\mathbb{R}^{2})$ (denoting the usual Sobolev spaces) in
$L^{2}(\mathbb{R}^{2},\mathbb{C}^{3})$. Due to translation invariance, the
solutions of (1) decompose into Fourier modes
$\psi=\widehat{\psi}{\mathrm{e}}^{{\mathrm{i}}(k_{x}x+k_{y}y-\omega t)}$, with
momentum $(k_{x},k_{y})\in\mathbb{R}^{2}$ and frequency $\omega\in\mathbb{R}$.
This reduces (1) to $H\widehat{\psi}=\omega\widehat{\psi}$ where
$H(k_{x},k_{y})$ is a family of $3\times 3$ Hermitian matrices. Solving the
eigenvalue equation leads to three frequency bands:

$\omega_{\pm}({k_{x},k_{y}})=\pm\sqrt{k^{2}+(f-\nu
k^{2})^{2}},\qquad\omega_{0}({k_{x},k_{y}})=0,$ (2)

with $k^{2}=k_{x}^{2}+k_{y}^{2}$. The bands are separated by two spectral
gaps: $(0,f)$ and $(-f,0)$. It was shown in [29, 15] that each band is
associated to a well defined topological index, the Chern number, with
respective values $C_{\pm}=\pm 2$ and $C_{0}=0$, attesting to non-trivial
topology in the bulk (namely when $\mathcal{H}$ acts in
$L^{2}(\mathbb{R}^{2},\mathbb{C}^{3})$).

###  3.2 Half-space model and its singularity

One anticipates a corresponding topological invariant of edge states in the
half-plane problem. We therefore consider $\mathcal{H}$ of Eq. (1) acting on
the upper half-plane $y\geq 0$, denoting it by $\mathcal{H}^{\sharp}$. As a
general convention, we will use a $\sharp$ superscript to denote half-space
operators. We analyse translation-invariant (in the $x$-direction) boundary
conditions, so that the momentum $k_{x}$ is still conserved.

When we Fourier transform $\mathcal{H}$ of Eq. (1) only in the $x$-direction,
we obtain a family $H(k_{x}),\,k_{x}\in\mathbb{R}$ of ordinary differential
operators acting on a line, explicitly given by the expression on the right
side of Eq. (4) below. The spectrum of $H(k_{x})$ is the union over
$k_{y}\in\mathbb{R}$ of the spectrum of $H(k_{x},k_{y})$ (found in Eq. (2)
above), namely,

$\sigma(H(k_{x}))=\;(-\infty,-\sqrt{k_{x}^{2}+(f-\nu
k_{x}^{2})^{2}}]\,\cup\,\\{0\\}\,\cup\,[\sqrt{k_{x}^{2}+(f-\nu
k_{x}^{2})^{2}},\infty).$ (3)

Note that $0$ is infinitely-degenerate, thus an essential spectral point of
$H(k_{x})$.

Let $H^{\sharp}(k_{x})$ denote the same formal differential operator as
$H(k_{x})$, but acting only on the _half-line_ $y\geq 0$:

$H^{\sharp}(k_{x})=\begin{pmatrix}0&k_{x}&-{\mathrm{i}}\frac{d}{dy}\\\
k_{x}&0&-{\mathrm{i}}(f-\nu(k_{x}^{2}-\frac{d^{2}}{dy^{2}}))\\\
-{\mathrm{i}}\frac{d}{dy}&{\mathrm{i}}(f-\nu(k_{x}^{2}-\frac{d^{2}}{dy^{2}}))&0\end{pmatrix}.$
(4)

Then $H^{\sharp}(k_{x})$ is symmetric (i.e. formally self-adjoint) on the
initial dense domain $C_{c}^{\infty}((0,\infty);\mathbb{C}^{3})\subset
L^{2}((0,\infty);\mathbb{C}^{3})$, and we use the same symbol for its closure.
It may be extended to a self-adjoint operator by specifying appropriate
boundary conditions at $y=0$. As in [15], we shall focus on the following
family of boundary conditions:

$v|_{y=0}=0,\qquad({\mathrm{i}}k_{x}u+a\partial_{y}v)|_{y=0}=0.$ (5)

Here, the real parameters $k_{x},a$ initially parametrize a plane, but
$a=\infty$ is also allowed, and defines the same boundary condition as
$a=-\infty$. So $a\in\mathbb{R}\cup\\{\infty\\}$ parametrizes a compactified
line, i.e. a circle, and $(k_{x},a)$ parametrizes an infinite cylinder
$\mathbb{R}\times(\mathbb{R}\cup\\{\infty\\})$.

Crucially, the $k_{x}=0=a$ case gives a _vacuous_ and hence non self-adjoint
condition, and must be excluded. Thus the correct parameter space is a
_punctured_ cylinder,

$(k_{x},a)\in(\mathbb{R}\times(\mathbb{R}\cup\\{\infty\\}))\setminus\\{(0,0)\\}=:\mathring{C},$

as illustrated in Figure 2. In Section 6, we will see that this singularity of
self-adjointness is the source of the topology of the half-plane model.

$\circ$$\cdots$$\cdots$$k_{x}$$a$$\ell_{0}$$\ell_{-}$$\ell_{+}$ Figure 2: The
parameter space $\mathring{C}$ of longitudinal momentum $k_{x}\in\mathbb{R}$
and boundary condition $a\in\mathbb{R}\cup\\{\infty\\}$. The singularity
$(0,0)$ is removed, so that $\mathring{C}$ is a punctured cylinder. The
directed loops $\ell_{0},\ell_{+},\ell_{-}$ are non-contractible in
$\mathring{C}$.

The following continuity result is proved in Section 4.

######  Theorem 3.1.

For each point $(k_{x},a)\in\mathring{C}$, the boundary condition
$\eqref{eq:boundary_condition}$ provides a self-adjoint extension of
$H^{\sharp}(k_{x})$ that we denote by $H^{\sharp}(k_{x},a)$. Moreover, the
resolvent assignment

$(k_{x},a)\mapsto\left(H^{\sharp}(k_{x},a)\pm{\mathrm{i}}\right)^{-1}$

is norm-continuous on the punctured cylinder $\mathring{C}$.

The continuity of the family $H^{\sharp}(k_{x},a)$ is an essential step to
prove the stability of the edge index defined below. Moreover, a continuity
property with respect to a boundary condition is not straightforward and
requires von Neumann theory of deficiency indices for self-adjoint extensions.
See Section 4 below. Finally, because we deal with unbounded operators, it is
natural to work with the topology of norm-resolvent convergence.

###  3.3 Bulk-edge correspondence

For any fixed $R>0$, consider the following non-contractible loop in the
parameter space $\mathring{C}$:

$\mathcal{C}_{R}=\\{(k_{x},a)=(R\cos\theta,R\sin\theta),\theta\in[-\pi,\pi]\\}\subset\mathring{C}.$
(6)

It is homotopic to $\ell_{0}$ from Figure 2. Each point in $\mathcal{C}_{R}$
defines a self-adjoint operator

$H^{\sharp}_{R,\theta}:=H^{\sharp}(R\cos\theta,R\sin\theta),$

continuously parametrized by $\theta$ due to Theorem 3.1.

The essential spectrum of $H^{\sharp}_{R,\theta}$ is obtained by putting
$k_{x}=R\cos\theta$ in Eq. (3) for the spectrum of the bulk operator
$H(k_{x})$,

$\sigma_{\mathrm{ess}}(H^{\sharp}_{R,\theta})=(-\infty,-\omega_{\theta}]\cup\\{0\\}\cup[\omega_{\theta},+\infty),\qquad\omega_{\theta}=\sqrt{R^{2}\cos(\theta)^{2}+(f-\nu
R^{2}\cos(\theta)^{2})^{2}},$ (7)

while discrete eigenvalues may appear inside the gaps
$(-\omega_{\theta},0)\cup(0,\omega_{\theta})$. As $\theta$ varies, such
eigenvalues trace out continuous edge mode branches that live inside the local
essential spectral gap of the family $\\{H_{R,\theta}^{\sharp}\\}_{\theta}$,
see Figure 3. We shall focus on eigenvalue branches in the upper gap
$(0,\omega_{\theta})$ from now on. Recall that the global upper gap is
$(0,f)$, so that $(0,f)\subset(0,\omega_{\theta})$ for all $\theta$.

######  Definition 3.2.

For $\mu\in(0,f)$, the number of edge modes $n(\mathcal{C}_{R},\mu)$ is
defined as the signed number of crossings of the edge modes branches of
$H^{\sharp}_{R,\theta}$ with the fiducial line $\omega=\mu$, counted
positively (resp. negatively) if the branch has negative slope (resp. positive
slope) at the crossing, as $\theta$ increases.

We illustrate this definition in Figure 3 for various values of $R$.

Figure 3: Spectrum of $H_{R,\theta}^{\sharp}$ for $f=1$ and $\epsilon=0.2$,
along $\mathcal{C}_{R}$ with $R=1,\,2$ and $4$, respectively on the left,
middle and right. Shaded blue regions correspond to extended solutions whereas
yellow branches are edge modes confined near the boundary. The number
$n(\mathcal{C}_{R},\mu)$ of signed intersections of such branches with the red
line $\omega=\mu$ is always $2$, for any choice of $R>0$ and $\mu\in(0,f)$.

######  Theorem 3.3 (Bulk-edge correspondence).

The number of edge modes $n(\mathcal{C}_{R},\mu)$ is independent of $R>0$ and
$\mu\in(0,f)$, and matches with the difference of bulk Chern numbers:

$n(\mathcal{C}_{R},\mu)=C_{+}-C_{0}.$

In [15] one attempted to count the number of edge modes for a fixed boundary
condition $a\neq 0$. This violated a more standard formulation of bulk-edge
correspondence along the lines of [16, 18]. In fact, the number of edge mode
branches crossing $\omega=\mu$ depends on the choice of $\mu\in(0,f)$, see
[15, Fig. 2], so that another definition for edge mode counting has to be used
([15, Definition 4]). Still, the system remains anomalous: the number of edge
modes changes with $a$. The origin of the anomaly is rooted in the infinite
region of the spectrum, as $k_{x}$ and $\omega$ go to infinity. In this paper,
we restrict the parameters to a compact loop $\mathcal{C}_{R}$, and Theorem
3.3 above shows that the bulk-edge correspondence can be restored. The edge
index $n(\mathcal{C}_{R},\mu)$ can be interpreted as a quantized pumping of
edge modes from the upper to the middle band when both $k_{x}$ and $a$ are
driven together along the loop $\mathcal{C}_{R}$, see Remark 3.7.

The proof of Theorem 3.3 is done in Section 5. We extend the approach from
[15] based on scattering amplitudes and Levinson’s theorem, originally
developed in [16] for discrete models.

###  3.4 Spectral flow stability

An essential feature of topological indices is their stability against
perturbations. In Theorem 3.3, the edge mode count via transversal crossings,
$n(\mathcal{C}_{R},\mu)$, is well-defined due to regularity properties of the
edge modes branches for the unperturbed $H_{R,\theta}$, a family of order 2
ordinary differential operators with constant coefficients. For example, at
each $\theta$, there are only finitely many edge modes, so they do not
accumulate into the bulk spectrum. Moreover, as $\theta$ is increased, the
number of merging points with the bulk spectrum, used in Definition 5.1 later,
is also finite. Such properties may not persist if we perturb $H_{R,\theta}$.

It is therefore worth pointing out that $n(\mathcal{C}_{R},\mu)$ is, up to a
minus sign, a special case of _spectral flow_ a la Phillips [22, 4], whose
precise definition and homotopy invariance is recalled in Section 6,
Definition 6.1. Crucially, the spectral flow formulation of edge mode counting
still makes sense in the perturbed setting, where the aforementioned
pathologies in the edge mode branches may occur. The only requirement is
continuity with respect to the path parameter, which is ensured by Theorem
3.1.

######  Corollary 3.4 (Bulk-edge correspondence, perturbed version).

Let $V=\\{V(k_{x})\\}_{k_{x}\in\mathbb{R}}$ be a continuous and uniformly
bounded family of self-adjoint operators on
$L^{2}((0,\infty),\mathbb{C}^{3})$. Assume each $V(k_{x})$ to be relatively
compact with respect to $H^{\sharp}(k_{x})$. Let

$\ell_{0}:[0,2\pi]\rightarrow\mathring{C},\qquad\theta\mapsto\ell_{0}(\theta)\equiv(k_{x}(\theta),a(\theta))$

be any continuous loop winding around $(k_{x},a)=(0,0)$ once in the
anticlockwise sense. Then the operator loop

$\tilde{\ell_{0}}:\theta\mapsto
H^{\sharp}(\ell_{0}(\theta))+V(k_{x}(\theta))-\mu$

has well-defined spectral flow ${\rm Sf}(\tilde{\ell}_{0})$ across any
$\mu\in(0,f)$, which equals

${\rm Sf}(\tilde{\ell}_{0})=-C_{+}.$

###### Proof.

The essential spectrum of the perturbed system
$H^{\sharp}(k_{x},a)+V(k_{x})-\mu$ does not depend on $V$, and it has a well-
defined spectral flow along any loop in $\mathring{C}$ due to Theorem 3.1. By
homotopy invariance of spectral flow, we may continuously turn the
perturbation $V$ off and deform $\ell_{0}$ to the loop $\mathcal{C}_{R}$
without changing the spectral flow. Thus $-{\rm Sf}(\tilde{\ell}_{0})=-{\rm
Sf}(\mathcal{C}_{R})\equiv n(\mathcal{C}_{R},\mu)=C_{+}$ using Theorem 3.3. ∎

######  Example 3.5.

Suppose the presence of the boundary induces an $x$-independent potential
function $g\equiv g(y)$ ($3\times 3$ Hermitian matrix-valued), which is
continuous and decays to 0 as $y\to\infty$. The perturbation $V(k_{x})$ is
simply the multiplication operator $M_{g}$ for all $k_{x}$.

Section 6 is dedicated to an exhaustive study of the spectral flow structure
of the family $\\{H^{\sharp}(k_{x},a)\\}_{(k_{x},a)\in\mathring{C}}$. There,
we extend the definition of spectral flow across any energy curve lying in an
essential spectral gap. In particular, $\mathrm{Sf}^{+}$ denotes the spectral
flow across any fiducial curve in the upper gap. Generally, the spectral flow
taken along any loop in $\mathring{C}$ could serve as a possible edge
topological invariant. We actually show that all possible spectral flows are
already determined by the one along $\mathcal{C}_{R}$. More precisely,
$\pi_{1}(\mathring{C})$ is generated by two loops $\ell_{+}$ and $\ell_{-}$
oriented along increasing $a$ (Figure 2), and the main result is

######  Proposition 3.6.

In the shallow-water wave model, the spectral flow structure is

${\rm Sf}^{+}(\ell_{+})=-{\rm Sf^{+}}(\ell_{-})=-\frac{C_{+}}{2}=-1,$

where the loop $\ell_{\pm}$ can be taken at any fixed $k_{x}\gtrless 0$.

Thus our bulk-edge correspondence, Theorem 3.3, or the perturbed version
Corollary 3.4, gives a _complete_ correspondence of bulk and edge topological
invariants for the shallow-water model, with boundary conditions given in Eq.
(5).

######  Remark 3.7.

When $k_{x}>0$ is fixed and the boundary condition $a$ is varied once around a
complete loop, there is pumping of a positive and fixed momentum edge mode
from the upper band to the flat band; the pumping goes in the reverse
direction for negative momentum modes. This is another instance of charge
pumping via a cycle of boundary conditions, as introduced in Section 2. Loops
at fixed $k_{x}$ such as $\ell_{\pm}$ are probably more easy to implement than
$\ell_{0}$ in practice.

##  4 Continuity of the self-adjoint family in the half-space model

In this Section we prove Theorem 3.1 in two steps. We first provide a
continuous parametrization of all the abstract self-adjoint extensions of
$H^{\sharp}(k_{x})$ via the theory of von Neumann. Then we derive Eq. (16),
showing how the parametrization by boundary conditions (5) is continuously
embedded into the abstract scheme.

###  4.1 Universal family of self-adjoint extensions

We first decompose $H^{\sharp}(k_{x})=H^{\sharp}_{0}+B(k_{x})$ with

$H^{\sharp}_{0}:=\begin{pmatrix}0&0&-{\mathrm{i}}\frac{d}{dy}\\\
0&0&-{\mathrm{i}}\nu\frac{d^{2}}{dy^{2}}\\\
-{\mathrm{i}}\frac{d}{dy}&{\mathrm{i}}\nu\frac{d^{2}}{dy^{2}}&0\end{pmatrix},\qquad
B(k_{x}):=\begin{pmatrix}0&k_{x}&0\\\ k_{x}&0&-{\mathrm{i}}f+{\mathrm{i}}\nu
k_{x}^{2}\\\ 0&{\mathrm{i}}f-{\mathrm{i}}\nu k_{x}^{2}&0\end{pmatrix}.$ (8)

Since $B(k_{x})$ is bounded, it suffices to find the domains of self-
adjointness for $H^{\sharp}_{0}$ to obtain those for $H^{\sharp}(k_{x})$. The
von Neumann theory (see X.1 of [24] for a pedagogical treatment) provides a
systematic way to do so.

As a preliminary simplification, observe that with the scaling substitution
$\tilde{y}=\frac{y}{2\nu}$, we have the formal differential operator

$H^{\sharp}_{0}=\frac{1}{2\nu}\begin{pmatrix}0&0&-{\mathrm{i}}\frac{d}{d\tilde{y}}\\\
0&0&-\frac{{\mathrm{i}}}{2}\frac{d^{2}}{d\tilde{y}^{2}}\\\
-{\mathrm{i}}\frac{d}{d\tilde{y}}&\frac{{\mathrm{i}}}{2}\frac{d^{2}}{d\tilde{y}^{2}}&0\end{pmatrix}.$

This defines a symmetric operator on the initial domain
$C_{c}^{\infty}(\mathbb{R}_{+};\mathbb{C}^{3})\subset
L^{2}(\mathbb{R}_{+};\mathbb{C}^{3})$, where $\mathbb{R}_{+}=(0,\infty)$
denotes the half-line, and we take $H^{\sharp}_{0}$ to be the closure. The
adjoint $(H^{\sharp}_{0})^{*}$ is an extension of $H^{\sharp}_{0}$ to the
domain $H^{1}(\mathbb{R}_{+})\oplus H^{2}(\mathbb{R}_{+})\oplus
H^{2}(\mathbb{R}_{+})$ (the Sobolev spaces on $\mathbb{R}_{+}$) with no
boundary condition imposed whatsoever. This latter domain is too large in the
sense that $(H^{\sharp}_{0})^{*}$ has $L^{2}$-eigenvalues off the real axis.
That is, there are non-trivial _deficiency subspaces_

$\displaystyle\mathcal{V}_{\pm}:={\rm
ker}\left((H^{\sharp}_{0})^{*}\mp\frac{{\mathrm{i}}}{2\nu}\right)={\rm
Ran}\left(H^{\sharp}_{0}\pm\frac{{\mathrm{i}}}{2\nu}\right)^{\perp},$

as captured by the _deficiency indices_ 111The scaling by $2\nu$ is convenient
for our problem and does not change the deficiency indices ([24], Theorem
X.1). $n_{\pm}={\rm dim}\,\mathcal{V}_{\pm}$, which we now compute.

The $+{\mathrm{i}}/2\nu$ eigenfunctions of $(H^{\sharp}_{0})^{*}$ can be found
by the exponential ansatz $e^{-\lambda y}$, with the characteristic roots
$\lambda$ satisfying

$0=4\nu^{4}\lambda^{4}-4\nu^{2}\lambda^{2}+1=(2\nu^{2}\lambda^{2}-1)^{2}.$ (9)

For normalizability, we need the positive root $1/\sqrt{2}\nu$, which is
repeated, and conclude that $n_{+}=2$. That $n_{-}=2$ is deduced in a similar
way.

######  Proposition 4.1.

The deficiency indices of $(H^{\sharp}_{0})^{*}$ are $(n_{+},n_{-})=(2,2)$, so
that the domains of the self-adjoint extensions of $H^{\sharp}_{0}$, thus also
those of $H^{\sharp}(k_{x})$, are in bijection with ${\rm U}(2)$.

In more detail, given a unitary isomorphism
$U:\mathcal{V}_{+}\rightarrow\mathcal{V}_{-}$, we define the operator
$H^{\sharp}_{0}(U)$ acting on the extended domain

$\displaystyle{\rm Dom}(H^{\sharp}_{0}(U))$
$\displaystyle=\\{\phi+\psi_{+}+U\psi_{+}\;|\;\phi\in{\rm
Dom}(H^{\sharp}_{0}),\psi_{+}\in\mathcal{V}_{+}\\},$ $\displaystyle
H^{\sharp}_{0}(U)(\phi+\psi_{+}+U\psi_{+})$
$\displaystyle=H^{\sharp}_{0}(\phi)+\frac{{\mathrm{i}}}{2\nu}\psi_{+}-\frac{{\mathrm{i}}}{2\nu}U\psi_{+}.$
(10)

Then $H^{\sharp}_{0}(U)$ is a self-adjoint extension of $H^{\sharp}_{0}$, and
all self-adjoint extensions of $H^{\sharp}_{0}$ are of this form for some
unitary $U$. We mention that a choice of orthonormal basis for
$\mathcal{V}_{+}$ and for $\mathcal{V}_{-}$ is needed in order to identify $U$
with a matrix in ${\rm U}(2)$.

Since $H^{\sharp}_{0}(U)$ is now self-adjoint, the resolvent
$(H^{\sharp}_{0}(U)\pm{\mathrm{i}})^{-1}$ exists as a bounded operator from
$L^{2}((0,\infty);\mathbb{C}^{3})\rightarrow{\rm Dom}(H^{\sharp}_{0}(U))$.

######  Lemma 4.2.

The map ${\rm U}(2)\ni U\mapsto(H^{\sharp}_{0}(U)\pm{\mathrm{i}})^{-1}$ is
norm continuous.

###### Proof.

An equivalent statement is the continuity of
$U\mapsto(H^{\sharp}_{0}(U)+{\mathrm{i}}/2\nu)^{-1}$ ([23] Theorem VIII.19).
There is an orthogonal splitting of the Hilbert space as ${\rm
Ran}(H^{\sharp}_{0}+{\mathrm{i}}/2\nu)\oplus\mathcal{V}_{+}$. The domain of
$H^{\sharp}_{0}(U)$ is the vector space sum ${\rm Dom}(H^{\sharp}_{0})+I(U)$,
where $I(U):=\\{\psi_{+}+U\psi_{+}\,|\,\psi_{+}\in\mathcal{V}_{+}\\}.$ From
its construction, Eq. (10), we have

$(H^{\sharp}_{0}(U)+{\mathrm{i}}/2\nu)(\phi+\psi_{+}+U\psi_{+})=(H^{\sharp}_{0}+{\mathrm{i}}/2\nu)\phi+({\mathrm{i}}/\nu)\psi_{+}.$

Thus $(H^{\sharp}_{0}(U)+{\mathrm{i}}/2\nu)^{-1}$ maps the first component
${\rm Ran}(H^{\sharp}_{0}+{\mathrm{i}}/2\nu)$ back to ${\rm
Dom}(H^{\sharp}_{0})$, independently of the choice of $U$. On the second
component $\mathcal{V}_{+}$, the resolvent takes $\psi_{+}$ to
$\frac{\nu}{{\mathrm{i}}}(1+U)\psi_{+}\in I(U)$, whence we see that its
dependence on $U$ is (norm-)continuous. ∎

######  Definition 4.3.

For $U\in{\rm U}(2)$, we write $H^{\sharp}_{U}(k_{x})$ for the self-adjoint
extension of $H^{\sharp}(k_{x})$ given by
$H^{\sharp}_{U}(k_{x}):=H^{\sharp}_{0}(U)+B(k_{x})$ (the right side is defined
in Eq. (10) and (8)), acting on the domain

${\rm Dom}(H^{\sharp}_{U}(k_{x}))={\rm
Dom}(H^{\sharp}_{0}(U))=\\{\phi+\psi_{+}+U\psi_{+}\;|\;\phi\in{\rm
Dom}(H^{\sharp}_{0}),\psi_{+}\in\mathcal{V}_{+}\\}.$ (11)

######  Proposition 4.4.

The resolvent $(H^{\sharp}_{U}(k_{x})\pm{\mathrm{i}})^{-1}$ depends (jointly)
continuously on $U\in{\rm U(2)}$ and on $k_{x}\in\mathbb{R}$.

###### Proof.

Suppose $(U^{\prime},k_{x}^{\prime})\rightarrow(U,k_{x})$. Using a standard
identity for the resolvent of a perturbed operator, we get the desired
convergence,

$\displaystyle\left(H^{\sharp}_{U^{\prime}}(k_{x}^{\prime})+{\mathrm{i}}\right)^{-1}$
$\displaystyle\equiv\left(H^{\sharp}_{0}(U^{\prime})+B(k_{x}^{\prime})+{\mathrm{i}}\right)^{-1}$
$\displaystyle=\left(H^{\sharp}_{0}(U^{\prime})+{\mathrm{i}}\right)^{-1}\left(1+B(k_{x}^{\prime})(H^{\sharp}_{0}(U^{\prime})+{\mathrm{i}})^{-1}\right)^{-1}$
$\displaystyle\longrightarrow\left(H^{\sharp}_{0}(U)+{\mathrm{i}}\right)^{-1}\left(1+B(k_{x})(H^{\sharp}_{0}(U)+{\mathrm{i}})^{-1}\right)^{-1}\equiv\left(H^{\sharp}_{U}(k_{x})+{\mathrm{i}}\right)^{-1}.$

Here we used continuity of $k_{x}\mapsto B(k_{x})$, and
$U\mapsto(H^{\sharp}_{0}(U)+{\mathrm{i}})^{-1}$ (Lemma 4.2), as well as joint
continuity of algebraic operations on bounded operators. Similarly,
$(H^{\sharp}_{U^{\prime}}(k_{x}^{\prime})-{\mathrm{i}})^{-1}\rightarrow(H^{\sharp}_{U}(k_{x})-{\mathrm{i}})^{-1}$.
∎

###  4.2 Sub-family parametrized by self-adjoint boundary conditions

We proceed to identify which of the abstract self-adjoint domains in Section
4.1 (labelled by $U$) are realized by the concrete boundary conditions of (5)
(labelled by $(k_{x},a)$). As preparation, we work out the deficiency
subspaces $\mathcal{V}_{\pm}$ explicitly.

We may verify that

$\psi_{1,\pm}(\tilde{y})=\begin{pmatrix}\sqrt{2}\tilde{y}-1\\\
\sqrt{2}-\tilde{y}\\\
\pm\tilde{y}\end{pmatrix}e^{-\sqrt{2}\tilde{y}},\qquad\psi_{2,\pm}(\tilde{y})=\begin{pmatrix}\sqrt{2}\tilde{y}+1\\\
-\tilde{y}\\\ \pm(\tilde{y}+\sqrt{2})\end{pmatrix}e^{-\sqrt{2}\tilde{y}},$

are $\pm{\mathrm{i}}/2\nu$ eigenfunctions of $(H^{\sharp}_{0})^{*}$, so that
$\\{\psi_{1,\pm},\psi_{2,\pm}\\}$ spans $\mathcal{V}_{\pm}$. This pair of
bases may not look optimal, but we will shortly see why it was chosen in this
way.

Notice that
$\langle\psi_{i,+}|\psi_{j,+}\rangle=\langle\psi_{i,-}|\psi_{j,-}\rangle$ for
all $i,j=1,2$. So for each $\beta\in{\rm U}(1)$, the transformation of basis
vectors,

$U_{\beta}\psi_{1,+}:=\beta\psi_{1,-},\qquad U_{\beta}\psi_{2,+}:=\psi_{2,-},$

defines a unitary map $U_{\beta}:\mathcal{V}_{+}\rightarrow\mathcal{V}_{-}$ .
In turn, we obtain (through Definition 4.3) a ${\rm U}(1)$-subfamily of self-
adjoint extensions, $\beta\mapsto H^{\sharp}_{U_{\beta}}(k_{x})$. This
subfamily may be understood through boundary conditions as follows.

The basis vectors $\psi_{2,\pm}\equiv(\eta_{2,\pm},u_{2,\pm},v_{2,\pm})$ were
engineered to ensure that
$\psi_{2,+}+U_{\beta}\psi_{2,+}=\psi_{2,+}+\psi_{2,-}$ has the following
boundary conditions in the last two $u,v$ components,

$(u_{2,+}+u_{2,-})|_{\tilde{y}=0}=(v_{2,+}+v_{2,-})|_{\tilde{y}=0}=(v^{\prime}_{2,+}+v^{\prime}_{2,-})|_{\tilde{y}=0}=0,$

while its first component $(\eta_{2,+}+\eta_{2,-})|_{\tilde{y}=0}$ is
unrestricted. This means that for functions in the vector space

$\displaystyle I(U_{\beta})$
$\displaystyle=\\{\psi_{+}+U_{\beta}\psi_{+}\;:\;\psi_{+}\in\mathcal{V}_{+}\\}$
$\displaystyle=\\{c_{1}(\psi_{1,+}+\beta\psi_{1,-})+c_{2}(\psi_{2,+}+\psi_{2,-})\,:\,c_{1},c_{2}\in\mathbb{C}\\},$

only the $c_{1}(\psi_{1,+}+\beta\psi_{1,-})$ part contributes to the boundary
conditions,

$(\psi_{+}+U_{\beta}\psi_{+})|_{\tilde{y}=0}=c_{1}(\psi_{1,+}+\beta\psi_{1,-})|_{\tilde{y}=0}=c_{1}\begin{pmatrix}(\sqrt{2}\tilde{y}-1)(1+\beta)\\\
(\sqrt{2}-\tilde{y})(1+\beta)\\\
\tilde{y}(1-\beta)\end{pmatrix}e^{-\sqrt{2}\tilde{y}}\Big{|}_{\tilde{y}=0}.$
(12)

Boundary conditions for the functions $(\eta,u,v)^{\rm t}$ belonging to

${\rm Dom}(H^{\sharp}_{U_{\beta}}(k_{x}))={\rm
Dom}(H^{\sharp}_{0}({U_{\beta}}))={\rm Dom}(H^{\sharp}_{0})+I(U_{\beta})$

come from the $I(U_{\beta})$ part, and these conditions are read off from Eq.
(12) as

$v|_{y=0}=0,\qquad(1-\beta)u|_{y=0}=\sqrt{2}(1+\beta)v^{\prime}|_{y=0},\qquad\eta|_{y=0}\;{\rm
unrestricted}.$ (13)

Therefore, any boundary condition expressed in terms of $(k_{x},a)$ in Eq. (5)
has an equivalent expression in terms of some $\beta=\beta(k_{x},a)$ in Eq.
(13), via the relation

$\frac{{\mathrm{i}}a}{\sqrt{2}k_{x}}=\frac{u|_{y=0}}{\sqrt{2}v^{\prime}|_{y=0}}=:\frac{1+\beta(k_{x},a)}{1-\beta(k_{x},a)}\;\;\Leftrightarrow\;\;H^{\sharp}(k_{x},a)=H^{\sharp}_{U_{\beta(k_{x},a)}}(k_{x}).$
(14)

Recall the Cayley transform homeomorphisms

$\displaystyle{\rm Cay}$
$\displaystyle:\mathbb{R}\cup\\{\infty\\}\rightarrow{\rm
U}(1),\qquad\gamma\mapsto\frac{\gamma-{\mathrm{i}}}{\gamma+{\mathrm{i}}},$
$\displaystyle{\rm Cay}^{-1}$ $\displaystyle:{\rm
U}(1)\rightarrow\mathbb{R}\cup\\{\infty\\},\qquad\beta\mapsto{\mathrm{i}}\frac{1+\beta}{1-\beta}.$

The relation in Eq. (14) is uniquely satisfied by

$\beta(k_{x},a)={\rm
Cay}\left(-\frac{a}{\sqrt{2}k_{x}}\right)=\frac{a+{\mathrm{i}}\sqrt{2}k_{x}}{a-{\mathrm{i}}\sqrt{2}k_{x}}.$

Note that the above formula also works in the $k_{x}=0$ case, where every
$a\neq 0$ defines the same boundary condition corresponding to $\beta(0,a)=1$.
(Recall that $(k_{x},a)=(0,0)$ is inadmissible as a boundary condition.)
Altogether, we have constructed the continuous classifying map

$\displaystyle U_{\beta}:\mathring{C}$ $\displaystyle\rightarrow{\rm
U}(1)\subset{\rm U}(2)$ $\displaystyle(k_{x},a)$
$\displaystyle\overset{\beta}{\mapsto}\frac{a+{\mathrm{i}}\sqrt{2}k_{x}}{a-{\mathrm{i}}\sqrt{2}k_{x}}\mapsto
U_{\beta(k_{x},a)}.$ (15)

###### Proof of Theorem 3.1.

By definition, the assignment $(k_{x},a)\mapsto H^{\sharp}(k_{x},a)$ factors
as

$(k_{x},a)\mapsto U_{\beta(k_{x},a)}\mapsto
H^{\sharp}_{U_{\beta(k_{x},a)}}(k_{x})\equiv H^{\sharp}(k_{x},a),$ (16)

so the continuity result follows immediately from Prop. 4.4. ∎

######  Remark 4.5.

The map $\beta$ in Eq. (15) restricts to a winding number $\mp 1$ map when
$k_{x}\gtrless 0$, and degenerates to the constant map 1 at $k_{x}=0$. The
puncture at $(k_{x},a)=(0,0)$ means that at $k_{x}=0$, the map $\beta$ is only
defined along a punctured circle
$(\mathbb{R}\cup\\{\infty\\})\setminus\\{0\\}$, and has ill-defined winding
number. This allows the winding number of $\beta(k_{x},\cdot)$ to
“continuously” switch signs when $k_{x}$ switches sign.

##  5 Bulk-edge correspondence via scattering theory

In this section we prove Theorem 3.3 using scattering theory. Such an approach
was developed in [15] for a fixed value of $a$, so we extend it here to an
$a$-dependent framework. The strategy is the following: We first define the
scattering amplitude $S$ and establish its main properties. Then we relate the
number of edge modes with the winding number of $S$ along some limiting loop
approaching $\mathcal{C}_{R}$. Furthermore, the Chern number $C_{+}$ is also
related to a winding number of $S$ along some other loop, which is independent
of $a$. Finally we show that the two winding numbers coincide.

Before that, we reformulate the number of edge modes in a more suitable way.

######  Definition 5.1.

The number of edge modes $n_{\mathrm{b}}(R)$ below a bulk spectral band is the
signed number of edge mode branches emerging (counted positively) or
disappearing (counted negatively) at the lower band limit, as $\theta$
increases. The number $n_{\mathrm{a}}(R)$ of edge modes above a band is
counted likewise up to a global sign change.

The family $H_{R,\theta}^{\sharp}$ has three bulk bands, see (7), that we
denote by $+,\,-$ and $0$. The definition above is more general than
Definition 3.2 and works beyond compact parameters, see [15]. However, the
parameter $\theta$ is compact here and since the edge mode branches are
continuous, the two definitions coincide in our case. One has

$n_{\mathrm{b}}^{+}(R)=n(\mathcal{C}_{R},\mu)=n_{\mathrm{a}}^{0}(R).$ (17)

One could think of $n_{\mathrm{b}}^{+}(R)$ as a crossing counting with a
fiducial line $\omega=\mu$ that has been continuously deformed to the bottom
of the $+$ band curve $\mu_{\theta}=\omega_{\theta}$ (see Figure 6). The
number $n_{b}$ can then be related to the Chern number via scattering theory.
Moreover, notice that the middle band $\omega_{0}=0$ is completely flat so
that scattering theory cannot be developed there. However, because it is also
topologically trivial ($C_{0}=0$), we can simply ignore it and reduce the
proof of Theorem 3.3 to the relation $n_{\mathrm{b}}^{+}(R)=C_{+}$. Finally,
notice that Section 6 discusses the possibility of extending Definition 5.1
via Phillips spectral flow across an energy curve $\mu_{\theta}$.

###  5.1 Scattering amplitude

This subsection mostly imports from [15] the required formalism such as bulk
sections, scattering state and amplitude. We include it for self-consistency
of the paper, to set notations and to provide explicit expressions that are
used later. We also emphasize the $a$ dependence.

##### Bulk data.

A Fourier decomposition of bulk equation (1) with modes
$\psi=\widehat{\psi}{\mathrm{e}}^{{\mathrm{i}}(k_{x}x+k_{y}y-\omega t)}$ leads
to the eigenvalue equation:

$H\widehat{\psi}=\omega\widehat{\psi},\qquad\widehat{\psi}=\begin{pmatrix}\hat{\eta}\\\
\hat{u}\\\ \hat{v}\end{pmatrix}\qquad
H({k_{x},k_{y}})=\begin{pmatrix}0&k_{x}&k_{y}\\\
k_{x}&0&-{\mathrm{i}}(f-\nu{k}^{2})\\\
k_{y}&{\mathrm{i}}(f-\nu{k}^{2})&0\end{pmatrix},$ (18)

with ${k}^{2}=k_{x}^{2}+k_{y}^{2}$ and $H({k_{x},k_{y}})$ a Hermitian matrix.
It admits three frequency bands $\omega_{\pm}$ and $\omega_{0}$ given in (2).
We shall focus on the upper band $\omega_{+}$ from now on. With $\nu>0$, the
problem can be compactified at $k\to\infty$, and we identify the compactified
$k$-plane with the Riemann sphere $\mathbb{C}\cup\\{\infty\\}\cong S^{2}$ via
$z=k_{x}+{\mathrm{i}}k_{y}\equiv(k_{x},k_{y})$. We call
$\widehat{\psi}(k_{x},k_{y})$ a bulk section. Since $C_{+}=2$, it is
impossible to find a global section that is regular for all $z\in S^{2}$. We
need at least two distinct ones, that are regular locally on two overlapping
patches to cover the sphere. One section is given by

$\widehat{\psi}^{\infty}(k_{x},k_{y})=\dfrac{1}{k_{x}-{\mathrm{i}}k_{y}}\begin{pmatrix}{k}^{2}/\omega_{+}\\\
k_{x}-{\mathrm{i}}k_{y}q\\\ k_{y}+{\mathrm{i}}k_{x}q\end{pmatrix}\,,\quad
q(k_{x},k_{y}):=\tfrac{f-\nu{k}^{2}}{\omega_{+}}\,,\quad\omega_{+}=\omega_{+}(k_{x},k_{y})\,.$
(19)

Notice that $q\rightarrow 1$ (resp. $-1$) as $k\rightarrow 0$ (resp.
$\infty$). Thus (19) defines a section of the eigenbundle of $\omega_{+}$ that
is smooth and non zero for all $z\in\mathbb{C}$, including $z=0$, but not at
$\infty$, where it is singular and winds like $z/\bar{z}$. We can move the
singularity by defining for
$\zeta=\zeta_{x}+{\mathrm{i}}\zeta_{y}\in\mathbb{C}$,

$\widehat{\psi}^{\zeta}=t^{\zeta}_{\infty}\widehat{\psi}^{\infty},\qquad
t^{\zeta}_{\infty}(z)=\frac{\bar{z}-\bar{\zeta}}{z-\zeta}.$ (20)

Notice that $\widehat{\psi}^{\zeta}$ is regular on
$S^{2}\setminus\\{\zeta\\}$. On the other hand, $t^{\zeta}_{\infty}$ is
regular for all $z\in S^{2}\setminus\\{\infty,\zeta\\}$ and singular at the
two omitted points.

##### Scattering amplitude.

Back in the half-space, translation invariance is broken along $y$, so normal
modes $\psi=\widehat{\psi}{\mathrm{e}}^{{\mathrm{i}}(k_{x}x+k_{y}y-\omega t)}$
are not solutions of the boundary problem, and $k_{y}$ is not conserved.
However for $\kappa>0$ we shall consider such modes with $k_{y}=\kappa$ and
$k_{y}=-\kappa$ as outgoing and incoming plane waves with respect to the
boundary. Such modes have the same frequency since
$\omega(k_{x},\kappa)=\omega(k_{x},-\kappa)$. There are actually two other
values of $k_{y}$ with the same frequency, however they are purely imaginary
and read

$\kappa_{\mathrm{\mathrm{ev}/\mathrm{div}}}(k_{x},\kappa)=\pm{\mathrm{i}}\sqrt{\kappa^{2}+2k_{x}^{2}+\dfrac{1-2\nu
f}{\nu^{2}}}\in\pm{\mathrm{i}}\mathbb{R}_{+}\,,$ (21)

When $k_{y}=\kappa_{\mathrm{ev}}$ the evanescent mode is exponentially
decaying as $y\to\infty$, whereas the diverging mode for
$k_{y}=\kappa_{\mathrm{div}}$ diverges.

######  Remark 5.2.

Another perspective is to look at the solutions of

$k_{x}^{2}+k_{y}^{2}+(f-\nu(k_{x}^{2}+k_{y}^{2}))^{2}=\omega^{2}$

for fixed $k_{x}\in\mathbb{R}$ and $\omega^{2}>k_{x}^{2}+(f-\nu
k_{x}^{2})^{2}$. Such an equation has four solutions: two are real and two
purely imaginary. It turns out that they can all be expressed in terms of the
real positive one. If we call the latter $\kappa>0$ we recover the expressions
above.

Let $U_{\mathrm{out}}\subset\mathbb{R}^{2}$ be an open subset, and let
$U_{\mathrm{in}}\subset\mathbb{R}^{2}$ and
$U_{\mathrm{ev}}\subset\mathbb{R}\times{\mathrm{i}}\mathbb{R}$ be the images
under the maps $(k_{x},\kappa)\mapsto(k_{x},-\kappa)$ and
$(k_{x},\kappa)\mapsto(k_{x},\kappa_{\mathrm{ev}})$. Consider bulk section
$\widehat{\psi}_{\mathrm{in/out/ev}}$ with momentum $k_{x}$ and
$k_{y}=-\kappa$, $\kappa$ and $\kappa_{\mathrm{ev}}$ for $k_{x},\kappa\in
U_{\mathrm{out}}$. We assume the sections to be non-vanishing and regular on
their respective domains, and that they are of amplitude one, namely
$\langle\psi_{\mathrm{in}}(k_{x},-\kappa),\psi_{\mathrm{in}}(k_{x},-\kappa)\rangle=1$
and similarly for out and ev.

The scattering state is the linear combination

$\psi_{s}=\big{(}\widehat{\psi}_{\mathrm{in}}{\mathrm{e}}^{-{\mathrm{i}}\kappa
y}+S\widehat{\psi}_{\mathrm{\mathrm{out}}}{\mathrm{e}}^{{\mathrm{i}}\kappa
y}+T\widehat{\psi}_{\mathrm{ev}}{\mathrm{e}}^{{\mathrm{i}}\kappa_{\mathrm{ev}}y}\big{)}{\mathrm{e}}^{{\mathrm{i}}(k_{x}x-\omega
t)}$ (22)

which satisfies the boundary condition (5). It has well defined momentum
$k_{x}$ and frequency $\omega=\omega_{+}(k_{x},\kappa)$. Such a state is
uniquely defined up to multiples, and for any self-adjoint boundary condition,
the quantity $S(k_{x},\kappa)\in U(1)$ is called the scattering amplitude, see
[15]. It depends on the choice of bulk sections and on the boundary condition,
so we shall rather write $S(k_{x},\kappa,a)$.

##### Explicit expression and properties.

For $\kappa>0$ we shall work with $\zeta={\mathrm{i}}$ (any
$\zeta={\mathrm{i}}\zeta_{y}$ with $\zeta_{y}>0$ would fit) and

$\widehat{\psi}_{\mathrm{in}}(k_{x},-\kappa)=\widehat{\psi}^{\zeta}(k_{x},-\kappa),\qquad\widehat{\psi}_{\mathrm{out}}(k_{x},\kappa)=\widehat{\psi}^{\zeta}(k_{x},\kappa),\qquad\widehat{\psi}_{\mathrm{ev}}(k_{x},\kappa_{\mathrm{ev}})=\widehat{\psi}^{\infty}(k_{x},\kappa_{\mathrm{ev}})$
(23)

so that $U_{\mathrm{out}}=S^{2}\setminus\\{\zeta\\}$,
$U_{\mathrm{in}}=S^{2}\setminus\\{-\zeta\\}$ and $U_{\mathrm{ev}}=S^{2}$. In
particular $U_{\mathrm{in}}\cup U_{\mathrm{out}}=S^{2}$ and
$U_{\mathrm{in}}\cap U_{\mathrm{out}}=S^{2}\setminus\\{\pm\zeta\\}$. Also
notice that $\widehat{\psi}_{\mathrm{ev}}$ is regular for all
$(k_{x},\kappa)\in S^{2}$ (yet it is not a global section on $S^{2}$ because
it has momentum $k_{x},\kappa_{\mathrm{ev}}\neq k_{x},\kappa$). Unless stated,
we shall always work with this choice of sections to define $S$.

For $\zeta\in S^{2}$ we write

$\widehat{\psi}^{\zeta}=\begin{pmatrix}\eta^{\zeta}\\\ u^{\zeta}\\\
v^{\zeta}\end{pmatrix},$

so that the boundary condition (5) implies for the scattering state

$\displaystyle
v^{\zeta}(-\kappa)+Sv^{\zeta}(\kappa)+Tv^{\infty}(\kappa_{\mathrm{ev}})=0$
(24) $\displaystyle k_{x}u^{\zeta}(-\kappa)-a\kappa
v^{\zeta}(-\kappa)+S(k_{x}u^{\zeta}(\kappa)+a\kappa
v^{\zeta}(\kappa))+T(k_{x}u^{\infty}(\kappa_{\mathrm{ev}})+a\kappa_{\mathrm{ev}}v^{\infty}(\kappa_{\mathrm{ev}}))=0$
(25)

where we omitted the $k_{x}$ dependence, leading to

$S(k_{x},\kappa,a)=-\dfrac{g(k_{x},-\kappa,a)}{g(k_{x},\kappa,a)},$

where

$g(k_{x},\kappa,a)=\begin{vmatrix}k_{x}u^{\zeta}(k_{x},\kappa)+a\kappa
v^{\zeta}(k_{x},\kappa)&k_{x}u^{\infty}(k_{x},\kappa_{\mathrm{ev}})+a\kappa_{\mathrm{ev}}v^{\infty}(k_{x},\kappa_{\mathrm{ev}})\\\
v^{\zeta}(k_{x},\kappa)&v^{\infty}(k_{x},\kappa_{\mathrm{ev}})\end{vmatrix}.$
(26)

Recall that $\kappa_{\mathrm{ev}}$ depends on $k_{x},\kappa$, see (21). It was
shown in [15] that $S$ is well-defined (in particular $g$ is non-vanishing) as
long as the boundary condition is self-adjoint and the sections
$\widehat{\psi}_{\mathrm{in}}$ and $\widehat{\psi}_{\mathrm{out}}$ are
regular. Thus $S(k_{x},\kappa,a)$ is defined on

$D_{S}=\Big{(}\mathbb{R}\times\mathbb{R}_{+}^{*}\times\mathbb{R}\Big{)}\setminus\Big{(}\\{(0,\kappa,0)\,|\,\kappa>0\\}\cup\\{(0,\zeta_{y},a)\,|\,a\in\mathbb{R}\\}\Big{)}$

illustrated in Figure 5 below. Moreover it is easy to see from its expression
that $S$ is smooth on the domain $D_{S}$.

###  5.2 Number of edge modes and relative Levinson’s theorem

In order to study a relevant scattering amplitude, consider the following
curve for any $\epsilon>0$

$\mathcal{C}_{R}^{\epsilon}=\\{(k_{x},\kappa,a)=(R\cos\theta,\epsilon,R\sin\theta)\quad|\quad\theta\in\mathbb{[}-\pi,\pi]\\}.$

######  Proposition 5.3.

For any $\epsilon$ such that $0<\epsilon<\zeta_{y}$, the scattering amplitude
is well-defined and satisfies

$\lim_{\epsilon\to
0}\dfrac{1}{2\pi{\mathrm{i}}}\int_{\mathcal{C}_{R}^{\epsilon}}S^{-1}{\mathrm{d}}S=n_{\mathrm{b}}^{+}(R).$

This is a consequence of the relative Levinson’s theorem, see [15, Thm 13].
This general statement is true for any continuously parametrized compact
domain, such as the loops $C_{R}^{\epsilon}$ parametrized by $\theta$. As
$\kappa\to 0$ the scattering amplitude is computed near the bottom of the
upper band, and feels the bound states below it. As $\theta$ increases, it
then counts the change in the number of bound states, that is the number of
edge modes from the definition above. We illustrate it for specific values of
$R$ in Figure 4.

Figure 4: Lower panel : spectrum of $H^{\sharp}_{R,\theta}$ with respect to
$\theta$ for $R=1$, $2$ and $4$ respectively. Upper panel : argument of the
scattering amplitude with respect to $\theta$ for $\epsilon=0.05$. The
argument jumps precisely when an edge mode branch disappears or emerges from
the upper bulk band. The total change of argument of $S$ matches with the
number of edge modes, that is always $2$.

###  5.3 Scattering and Chern number

Here we consider a different curve in the parameter space. For $0<\delta<1$:

$\Gamma_{\delta,a_{0}}=\\{(k_{x},\kappa,a)=(\delta\cos\alpha,\delta\sin\alpha+1,a_{0})\,|\,\alpha\in[-\pi,\pi]\\}.$

This is a circle of radius $\delta$ in the $(k_{x},\kappa)$ plane with
$\kappa>0$, centered around $\zeta=(0,1)$. Any closed curve with that property
would fit, in particular $\delta$ is not necessarily small here. Then we have

######  Proposition 5.4.

For any $a_{0}\neq 0$ and $\delta>0$,

$\dfrac{1}{2\pi{\mathrm{i}}}\int_{\Gamma_{\delta,a_{0}}}S^{-1}{\mathrm{d}}S=C_{+}$

where $C_{+}$ is the Chern number of the upper band.

This is called the bulk-scattering correspondence, and is proved in [15, Thm
9]. The proof goes as follows. Pick a section $\widehat{\psi}_{\mathrm{in}}$
on $U_{\mathrm{in}}\cap\\{\kappa<0\\}$. The boundary condition required in the
definition of the scattering state naturally defines a scattering map
$\mathcal{S}:\mathcal{E}_{k_{x},-\kappa}\to\mathcal{E}_{k_{x},\kappa}$ where
$\mathcal{E}_{k_{x},\kappa}$ is the fiber above $(k_{x},\kappa)$. Thus one
naturally has the abstract section denoted by
$\widecheck{\psi}_{\mathrm{out}}$, and given by
$\widecheck{\psi}_{\mathrm{out}}(k_{x},\kappa)=\mathcal{S}\widehat{\psi}_{\mathrm{in}}(k_{x},-\kappa)$.
On the other hand, according to (23) and (22) one uses a common reference
section for in and out on $U_{\mathrm{in}}\cap U_{\mathrm{out}}$, namely
$\widehat{\psi}_{\mathrm{in}}=\widehat{\psi}_{\mathrm{out}}$ so that the
abstract section is actually proportional to $\widehat{\psi}_{\mathrm{in}}$.
Explicitly, we have
$\widecheck{\psi}_{\mathrm{out}}(k_{x},\kappa)=S(k_{x},\kappa,a)\widehat{\psi}_{\mathrm{in}}(k_{x},-\kappa)$.
Thus $S$ is nothing but a transition function between in and out sections.
Consequently, if $U_{\mathrm{in}}$ and $U_{\mathrm{out}}$ cover $S^{2}$ and if
the loop $\Gamma_{\delta,a_{0}}$ splits $S^{2}$ with interior contained in
$U_{\mathrm{in}}$ and exterior contained in $U_{\mathrm{out}}$, then the
winding of $S$ along it is the Chern number. One can check that this is the
case with the choice (23) for $S$ as long as $\zeta$ is inside
$\Gamma_{\delta,a_{0}}$.

###  5.4 The correspondence

The main claim in this section, together with the two propositions above,
implies $n_{b}^{+}(R)=C_{+}$, and ends the proof of Theorem 3.3.

######  Proposition 5.5.

The scattering amplitude $S$ satisfies:

$\dfrac{1}{2\pi{\mathrm{i}}}\int_{\mathcal{C}_{R}^{\epsilon}}S^{-1}{\mathrm{d}}S=\dfrac{1}{2\pi{\mathrm{i}}}\int_{\Gamma_{\delta,a_{0}}}S^{-1}{\mathrm{d}}S$

for any $\epsilon$ sufficiently small for the left hand side to be well
defined.

###### Proof.

The scattering amplitude is defined explicitly through (23), which ensures
that it is well defined and smooth for $\kappa>0$ and away from $k_{x}=a=0$
and $k_{x}=0,\kappa=\zeta_{y}$. We summarize all that in the left panel of
Figure 5.

$\kappa$$a$$k_{x}$$\epsilon$$\Gamma_{\delta,a_{0}}$$\mathcal{C}_{R,\epsilon}$$\zeta_{y}$$\kappa$$a$$k_{x}$$\check{\Gamma}$$\mathcal{C}$$\kappa$$a$$k_{x}$$L$
Figure 5: The winding of $S$ along $\mathcal{C}_{R}^{\epsilon}$ tends to
$n_{b}^{+}(R)$, and the winding along $\Gamma_{\delta,a_{0}}$ gives the Chern
number $C_{+}$. Up to some continuous deformation, the difference of such
winding numbers is encoded in the one around the loop $L$, that turns out to
be zero.

Apart from the region where $S$ is singular (red lines and $\kappa\leq 0$),
the curves can be continuously deformed leaving the winding numbers unchanged.
Thus, without loss of generality, we can set $a_{0}=-1$, $\zeta_{y}=1$,
$\epsilon<1$ and $\delta<1$, and then deform $\mathcal{C}_{R}^{\epsilon}$ to a
square loop of center $(0,\epsilon,0)$ and length $1$ in the plane
$\kappa=\epsilon$, called $\mathcal{C}$. Similarly, we deform
$\Gamma_{\delta,a_{0}}$ to a square loop of center $(0,1,-1)$ and length $1$
in the plane $a=-1$, called $\Gamma$. $\check{\Gamma}$ is the same loop but
with reversed orientation. See the middle panel of Figure 5. Then we have

$\int_{\mathcal{C}}S^{-1}{\mathrm{d}}S-\int_{\Gamma}S^{-1}{\mathrm{d}}S=\int_{L}S^{-1}{\mathrm{d}}S$

where $L$ is a deformation of $\mathcal{C}\cup\check{\Gamma}$ which lies in
the half-plane $\mathcal{P}$ defined by $\kappa+a=1$, $\kappa>0$ (any half-
plane of the form $\lambda\kappa+a=\lambda$, $\kappa>0$ for some $\lambda>0$
would actually fit). See the right panel of Figure 5. Thus the winding numbers
of $S$ along $\mathcal{C}_{R}^{\epsilon}$ and $\Gamma_{\delta,a_{0}}$ coincide
if the winding of $S$ along any loop in $\mathcal{P}$ around $(0,1,0)$
vanishes. Consider for example for $0<\alpha<1$

$\ell_{\alpha}=\big{\\{}(\alpha\cos(\theta),1-\alpha\sin(\theta),\alpha\sin(\theta))\,|\,\theta\in[0,2\pi]\big{\\}},$

which is homotopic to $L$ in $\mathcal{P}$ up to orientation.

Now we expand (26) on $\ell$ near $\alpha=0$. We use the fact that, for
$k_{x}=\alpha\cos(\theta)$ and $k_{y}=1-\alpha\sin(\theta)$,

$\lim_{\alpha\to
0}u^{\zeta}=\dfrac{f-\nu}{\sqrt{1+(f-\nu)^{2}}}{\mathrm{e}}^{2{\mathrm{i}}\theta},\qquad\lim_{\alpha\to
0}v^{\zeta}={\mathrm{i}}{\mathrm{e}}^{2{\mathrm{i}}\theta}.$

Moreover one has
$\kappa_{\mathrm{ev}}\to{\mathrm{i}}\sqrt{1+\frac{1}{\nu^{2}}-\frac{2f}{\nu}}$
and

$\lim_{\alpha\to
0}u^{\infty}(k_{x},\kappa_{\mathrm{ev}})=\dfrac{1-f\nu+\nu^{2}}{\nu\sqrt{1+(f-\nu)^{2}}},\qquad\lim_{\alpha\to
0}v^{\infty}(k_{x},\kappa_{\mathrm{ev}})={\mathrm{i}}.$

Consequently, one has to the leading order in $\alpha$

$g\sim-{\mathrm{i}}\alpha{\mathrm{e}}^{2{\mathrm{i}}\theta}(A\cos\theta-(B+{\mathrm{i}})\sin(\theta))$

where $A,B>0$ are positive constant depending on $f$ and $\nu$. We then
perform a similar computation for $g$ but flip $\kappa\to-\kappa$ so that
$k_{x}=\alpha\cos(\theta)$ and $k_{y}=-1+\alpha\sin(\theta)$. To leading order
near $\alpha=0$ we get

$g\sim{\mathrm{i}}\alpha(A\cos\theta-(B-{\mathrm{i}})\sin(\theta))$

Recalling that
$S(k_{x},\kappa,a)=-\frac{g(k_{x},-\kappa,a)}{g(k_{x},\kappa,a)}$ we deduce

$\lim_{\alpha\to
0}S={\mathrm{e}}^{-2{\mathrm{i}}\theta}\dfrac{A\cos\theta-(B-{\mathrm{i}})\sin(\theta)}{A\cos\theta-(B+{\mathrm{i}})\sin(\theta)}.$

Although this limit depends on $\theta$, it is easy to see that the winding
number of the expression on the right hand side vanishes, and so does the
winding of $S$ along any $\ell_{\alpha}$, which ends the proof.

∎

##  6 Full spectral flow structure

As discussed in Section 3.4, the stability of the edge index requires a
generalization of spectral flow from Definition 3.2 to take into account
pathological edge mode branches that may occur when perturbing the system.

Let $\mathscr{F}^{\rm sa}$ denote the space of self-adjoint (possibly
unbounded) Fredholm operators, and equip it with the topology of norm-
resolvent convergence. In what follows, the characteristic function on an
interval $[a,b]\subset\mathbb{R}$ is denoted $\chi_{[a,b]}$.

######  Proposition/Definition 6.1 ([22, 4]).

Let $F:I=[0,2\pi]\rightarrow\mathscr{F}^{\rm sa}$ be a continuous path. There
exists a partition of $I$, $0=\theta_{0}<\theta_{1}<\ldots<\theta_{N}=2\pi,$
and corresponding $a_{i}>0,i=1,\ldots,N$, such that each subpath of spectral
projections

$[\theta_{i-1},\theta_{i}]\ni\theta\mapsto\chi_{[-a_{i},a_{i}]}(F(\theta))$

is finite-rank and (norm-)continuous. For
$\theta\in[\theta_{i-1},\theta_{i}]$, write $r^{+}_{i}(\theta):={\rm
dim}\,\chi_{[0,a_{i}]}(F(\theta))$. The spectral flow $F$ (across 0 energy) is
defined to be

${\rm Sf}(F)\equiv{\rm Sf}(\\{F(\theta)\\}_{\theta\in
I}):=\sum_{i=1}^{N}r^{+}_{i}(\theta_{i})-r^{+}_{i}(\theta_{i-1}).$ (27)

This integer is independent of the choices of partition and $a_{i}$, and is a
homotopy invariant which is additive under concatenation of paths.

Figure 6: The spectral flow across 0 energy is defined using any jagged
cylinder (black dotted lines) of sub-intervals with no leaking of branches on
the horizontal edges. It extends Definition 3.2 to situations where the edge
mode branches may behave pathologically. The same spectral flow is obtained
across a fiducial curve (red dotted curve) lying slightly below the positive
essential spectrum.

The above definition of spectral flow is illustrated in Figure 6. Along the
$i$-th subpath, continuity of $\chi_{[-a_{i},a_{i}]}(F(\theta))$ ensures
constancy of the rank, so that no “leakage” of eigenvalues occurs across the
local upper/lower energy bounds $\pm a_{i}$. Then the spectral flow across 0
during this subpath is simply the net increase in the number of non-negative
eigenvalues at the left endpoint $\theta_{i-1}$ as compared to that at the
right endpoint $\theta_{i}$. Summing up these local spectral flows gives the
spectral flow along the full path.

Notice that any eigenvalues above/below $\pm a_{i}$ are, by construction,
irrelevant to the local spectral flows, so it does not matter that they might
accumulate into the essential spectrum. Also, the transversality of eigenvalue
curves across 0 is not required.

###  6.1 Spectral flow across an energy curve

In Eq. (17), we wrote that in the unperturbed model, the edge mode crossings
can be taken with respect to a constant energy $\mu$ in the bulk gap, or a
fiducial energy curve $\mu_{\rm fid}$ slightly below/above the upper/flat bulk
band. For the latter, a notion of emergence/disappearance from a bulk band is
used (Definition 5.1). For a perturbed model, these counts may become ill-
defined, but their replacements by spectral flows a la Phillips are always
well-defined.

Usually, for a path $F:[0,2\pi]\rightarrow\mathscr{F}^{\rm sa}$, the integer
${\rm Sf}(F)$ measures the net eigenvalue flow across the 0-energy reference
curve. More generally, we may wish to adopt another continuous fiducial
reference curve $\mu_{\rm fid}:[0,2\pi]\rightarrow\mathbb{R}$ such that

$\mu_{\rm fid}(\theta)\not\in\sigma_{\rm
ess}(F(\theta)),\;\;\forall\theta\in[0,2\pi],$

and consider the eigenvalue flow across $\mu_{\rm fid}$, namely, the quantity
${\rm Sf}(F-\mu_{\rm fid})$.

The original path $F$ may be considered as being homotopic to a concatenation
of three continuous paths: (i) turning on $\mu_{\rm fid}$, (ii) $F-\mu_{\rm
fid}$, (iii) turning off $\mu_{\rm fid}$. Spectral flow is homotopy invariant
and additive under concatenation of paths. This means that we have

$\displaystyle{\rm Sf}(F)$ $\displaystyle={\rm Sf}\left(\\{F(0)-t\mu_{\rm
fid}(0)\\}_{t\in[0,1]}\right)+{\rm Sf}\left(F-\mu_{\rm fid}\right)+{\rm
Sf}\left(\\{F(2\pi)-\mu_{\rm fid}(2\pi)+t\mu_{\rm
fid}(2\pi)\\}_{t\in[0,1]}\right)$ $\displaystyle=-\\#_{[0,\mu_{\rm
fid}(0))}F(0)+{\rm Sf}\left(F-\mu_{\rm fid}\right)+\\#_{[0,\mu_{\rm
fid}(2\pi))}F(2\pi).$ (28)

Here, $\\#_{[0,\lambda)}$ denotes the number of eigenvalues lying in the
energy interval $[0,\lambda)$, counted with multiplicity. From Eq. (28), we
deduce that _for an operator loop_ , $F(0)=F(2\pi)$, _the spectral flow is
independent of the choice of reference energy loop_ (with $\mu_{\rm
fid}(0)=\mu_{\rm fid}(2\pi)$),

${\rm Sf}(F)={\rm Sf}(F-\mu_{\rm fid}),$ (29)

generalizing Eq. (17).

####  6.1.1 Relation to (e)merging events at bulk-band edges

In Definition 6.1 of spectral flow of an operator loop, we use a partition by
$\theta_{i}\in[0,2\pi]$ and local energy bounds $\pm a_{i}$ for the $i$-th
subpath. The union of the rectangles
$[\theta_{i-1},\theta_{i}]\times[-a_{i},a_{i}]$ over $i=1,\ldots N$ is a
jagged cylinder with jagged upper (horizontal) boundary across which no
leakage of eigenvalues occurs, see Figure 6.

Write $\Lambda(\theta)$ for the bottom of the positive essential spectrum of
$F(\theta)$. We pick the fiducial reference energy curve $\mu_{\rm fid}$ to
lie slightly below $\Lambda$ and above the jagged boundary (Figure 6).
Suppose, in addition, that the discrete spectrum of $F(\theta)$ is finite for
all $\theta$ (this is satisfied in our unperturbed model). Since at each
$\theta_{i},i=0,\ldots,N$, there is a maximal eigenvalue below the positive
essential spectrum (no accumulation occurs), we may furthermore arrange for
$\mu_{\rm fid}$ to be large enough so that

$\\#_{[\mu_{\rm fid}(\theta_{i}),\Lambda(\theta_{i}))}=0,\qquad i=0,\ldots,N.$
(30)

Compare, for the $i$-th subpath, the spectral flow across $\mu_{\rm fid}$ with
the spectral flow across $a_{i}$. By construction, the latter vanishes (“no
leakage”). We deduce, in the same way as Eq. (28), that

$\displaystyle{\rm Sf}(\\{F(\theta)-\mu_{\rm
fid}(\theta)\\}_{\theta\in[\theta_{i-1},\theta_{i}]})$
$\displaystyle=\\#_{[a_{i},\mu_{\rm
fid}(\theta_{i-1}))}F(\theta_{i-1})-\\#_{[a_{i},\mu_{\rm
fid}(\theta_{i}))}F(\theta_{i})$ (31)
$\displaystyle=\\#_{[a_{i},\Lambda(\theta_{i-1}))}F(\theta_{i-1})-\\#_{[a_{i},\Lambda(\theta_{i}))}F(\theta_{i}),$

with the last equality due to Eq. (30). Thus the $i$-th local spectral flow is
simply the change in the number of bound states lying between $a_{i}$ and
$\Lambda$. Up to a sign, this is Definition 5.1 applied to the $i$-th subpath,
for which a relative Levinson’s theorem like Proposition 5.3 applies. The full
spectral flow of $F$ is then the sum of such local changes in the bound state
count.

Notice that we cannot generally count the bound states starting from a
_single_ global energy level $a$ for the entire loop — this would entail a
global “no leakage” condition across $a$, and therefore no spectral flow at
all. The point is that bound states are allowed to disappear into or emerge
from a lower band of essential spectrum (a lower-lying bulk band). In
contrast, for more usual applications of original Levinson’s theorem to
bounded-below operators, there does exist some global lower energy bound $a$
which is never breached by the bound states.

######  Remark 6.2.

If accumulation of eigenvalues occurs, we have to count the (change in the)
number of states lying below a fiducial energy slightly smaller than the
essential spectrum, as in Eq. (31).

###  6.2 Spectral flow structure of shallow-water wave model

The scattering theory analysis of [15] and Section 5 show that

$\sigma_{\rm
ess}(H^{\sharp}(k_{x},a))=\sigma(H(k_{x}))=\;(-\infty,-\sqrt{k_{x}^{2}+(f-\nu
k_{x}^{2})^{2}}]\,\cup\,\\{0\\}\,\cup\,[\sqrt{k_{x}^{2}+(f-\nu
k_{x}^{2})^{2}},\infty).$

Thus $H^{\sharp}(k_{x},a)$ has a negative essential spectral gap, and a
positive essential spectral gap.

Theorem 3.1 on the continuity of the parametrization $(k_{x},a)\mapsto
H^{\sharp}(k_{x},a)$ means that the spectral flow along any path
$I\rightarrow\mathring{C}$, across any continuous energy curve $\mu_{\rm fid}$
in the positive essential spectral gap, is well-defined. In the case of a loop
in $\mathring{C}$, the choice of reference energy loop $\mu_{\rm fid}$ is
immaterial (Eq. (29)), and the following definition is meaningful.

######  Definition 6.3.

Let $\ell:I\rightarrow\mathring{C}$ be any continuous loop. We define

${\rm Sf}^{+}(\ell):={\rm Sf}\left(\left\\{(H^{\sharp}-\mu_{\rm
fid})(\ell(\theta))\right\\}_{\theta\in I}\right),$

where $\mu_{\rm fid}\equiv\mu_{\rm fid}(\cdot)$ is any continuous fiducial
energy function valued in the positive essential spectral gap of
$H^{\sharp}(\cdot)$. This assignment of integers ${\rm Sf}^{+}(\ell)$ to each
loop $\ell$ in $\mathring{C}$ is called the _spectral flow structure_ of the
shallow-water wave model on the half-plane.

The spectral flow only depends on the homotopy class of $\ell$, so ${\rm
Sf}^{+}$ descends to a group homomorphism

${\rm Sf}^{+}(\pi_{1}(\mathring{C}))\rightarrow\mathbb{Z},$

and it is enough to work this out for a set of generators of
$\pi_{1}(\mathring{C})$. It is easy to see that $\pi_{1}(\mathring{C})\cong
F_{2}$, the free group on two generators, since $\mathring{C}$ is homotopy
equivalent to a wedge of two circles. Thus the generating loops in
$\mathring{C}$ can be taken to be a loop $\ell_{+}$ at any fixed $k_{x}>0$,
together with a loop $\ell_{-}$ at any fixed $k_{x}^{\prime}<0$, oriented
along increasing $a$ (Figure 2).

There now appears to be two independent edge topological invariants, ${\rm
Sf}^{+}(\ell_{+})$ and ${\rm Sf}^{+}(\ell_{-})$, whereas there is only one
bulk Chern number. Actually, we have the following relationship:

######  Proposition 6.4.

Let $\ell_{\pm}:I\rightarrow\mathring{C}$ be the loop at some constant
$k_{x}\gtrless 0$, parametrized by increasing
$a\in\mathbb{R}\cup\\{\infty\\}$. Then ${\rm Sf}^{+}(\ell_{+})=-{\rm
Sf}^{+}(\ell_{-})$.

###### Proof.

For $0\leq t\leq 1$, consider the self-adjoint operators

$H^{\sharp}(k_{x},a;t):=\begin{pmatrix}0&tk_{x}&-{\mathrm{i}}\frac{d}{dy}\\\
tk_{x}&0&-{\mathrm{i}}(f-\nu(k_{x}^{2}-\frac{d^{2}}{dy^{2}}))\\\
-{\mathrm{i}}\frac{d}{dy}&{\mathrm{i}}(f-\nu(k_{x}^{2}-\frac{d^{2}}{dy^{2}}))&0\end{pmatrix},\qquad{\rm
subject\;to}\;\eqref{eq:boundary_condition},$

which at $t=1$ are just the half-line operators $H^{\sharp}(k_{x},a)$ that we
have been studying. Fix some large $\tilde{k}_{x}>\sqrt{f/\nu}$, then the
positive essential spectral gap of $H^{\sharp}(\pm\tilde{k}_{x},a;t)$ is
$(0,\sqrt{t^{2}\tilde{k}_{x}^{2}+(f-\nu\tilde{k}_{x}^{2})^{2}})$ and remains
open as $t$ is decreased from $1$ to $0$. We shall take $\mu_{\rm fid}$ to be
some small constant positive number $\mu$ (say $\mu=f/2$) which stays in this
gap. We can compute ${\rm Sf}^{+}(\ell_{\pm})$ as the spectral flow along the
loop at $\pm\tilde{k}_{x}$ (due to homotopy invariance),

${\rm Sf}^{+}(\ell_{\pm})={\rm
Sf}\left(a\mapsto(H^{\sharp}(\pm\tilde{k}_{x},a;t=1)-\mu)\right)\overset{t\rightarrow
0}{=}{\rm Sf}\left(a\mapsto(H^{\sharp}(\pm\tilde{k}_{x},a;t=0)-\mu)\right).$
(32)

Now observe that $H^{\sharp}(\tilde{k}_{x},a;0)$ and
$H^{\sharp}(-\tilde{k}_{x},a;0)$ are the same operators, except that the
former is subject to boundary condition $a/\tilde{k}_{x}$ whereas the latter
is subject to $a/(-\tilde{k}_{x})=(-a)/\tilde{k}_{x}$, i.e., the loop
parameter $a$ is swapped for $-a$. Therefore, the right side of Eq. (32)
acquires a sign change when $\ell_{+}$ is changed to $\ell_{-}$, giving the
desired result. ∎

The proof of Proposition 3.6 is now straightforward. The anticlockwise loop
$\ell_{0}$ encircling the singularity $(0,0)$ is (homotopy equivalent to) the
concatenation of $\ell_{+}$ and $\ell_{-}^{\rm op}$, where $(\cdot)^{\rm op}$
denotes orientation reversal. Thus

${\rm Sf}^{+}(\ell_{0})={\rm Sf}^{+}(\ell_{+})+{\rm Sf}^{+}(\ell_{-}^{\rm
op})={\rm Sf}^{+}(\ell_{+})-{\rm Sf}^{+}(\ell_{-})=2\,{\rm Sf}^{+}(\ell_{+}),$

where the last equality follows from Prop. 6.4. In Corollary 3.4, we saw that
the spectral flow around the singularity is minus the Chern number of the
upper bulk band, ${\rm Sf}^{+}(\ell_{0})=-C_{+}=-2$.

######  Remark 6.5.

We can similarly define ${\rm Sf}^{-}$ as the spectral flow structure across
energy curves in the lower essential spectral gap. The lower bulk band has
Chern number $C_{-}=-2$, and the bound state counting above this band via
Levinson’s theorem needs a sign change. Overall, the conclusion is that ${\rm
Sf}^{-}={\rm Sf}^{+}$.

######  Remark 6.6.

In the 3D Weyl semimetal setting from [31], one encounters a singularity in
the 2D boundary momentum space parameter due to a closing of the essential
spectral gap at the projected _Weyl point_ , leading to a loss of the Fredholm
condition there. This singularity does not involve the boundary condition, so
the _Fermi arcs_ as determined by the spectral flows are present whatever the
boundary condition. A peculiar feature of the shallow-water model is that the
parameter space singularity is not due to gap closing, but rather a loss of
self-adjointness.

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