# Topological edge spectrum along curved interfaces

Alexis Drouot  University of Washington, Seattle, USA<EMAIL_ADDRESS>and
Xiaowen Zhu  University of Washington, Seattle, USA<EMAIL_ADDRESS>
###### Abstract.

We prove that that if the boundary of a topological insulator divides the
plane in two regions containing arbitrarily large balls, then it acts as a
conductor. Conversely, we show that topological insulators that fit within
strips do not need to admit conducting boundary modes.

##  1\. Introduction and main results

###  1.1. Introduction

Topological insulators are novel materials with striking properties. They are
phases of matter insulating in their bulk (the Hamiltonian has a spectral
gap), but that turn into conductors when truncated to half-spaces (the
spectral gap fills). The resulting edge conductance is equal to the difference
of the bulk topological invariants across the cut, a principle known as the
bulk-edge correspondence [KS02, EG02, EGS05, ASV13, GP13, GT18, ST19, B19,
D21, LT22]. Here, we consider truncations of topological insulators in regions
more sophisticated than half-spaces (for example, sectors or filled
parabolas). We investigate how the shape of the resulting material affects the
spectrum.

Figure 1. The first three truncations systematically give rise to edge
spectrum. In the last case, we construct Haldane-type topological insulators
$H_{-},H_{+}$ such that $H_{e}$ is also an insulator.

The main operator in our analysis is approximately equal to

$H_{e}:=\begin{cases}H_{+}\quad\text{ on }&\Omega,\\\ H_{-}\quad\text{ on
}&\Omega^{c}={\mathbb{Z}}^{2}\setminus\Omega,\end{cases}$ (1.1)

for two Hamiltonians $H_{\pm}$ on
$\ell^{2}({\mathbb{Z}}^{2},{\mathbb{C}}^{d})$ with distinct bulk invariants
within a joint spectral gap ${\mathcal{G}}$ (potentially one of them
representing the vacuum); and $\Omega$ a subset of ${\mathbb{Z}}^{2}$. We
refer to §1.2 for precise definitions and assumptions. We ask for geometric
conditions on $\Omega$ that guarantee that $H_{e}$ has spectrum filling
${\mathcal{G}}$.

Our main result, Theorem 1, asserts that if both $\Omega$ and $\Omega^{c}$
contain arbitrarily large balls then $H_{e}$ has spectrum filling
${\mathcal{G}}$ (referred to as edge spectrum). Hence $H_{e}$ behaves like a
conductor near $\partial\Omega$. Examples of such domains $\Omega$ include
half-planes, sectors, regions enclosed by hyperbolas, and so on; they exclude
strips or half-strips – see Figure 1. In this last case, we actually show that
there exist examples of distinct topological insulators $H_{\pm}$ such that
$H_{e}$ remains an insulator – Theorem 3. Therefore, topological materials
fitting in strips can violate the bulk-edge correspondence: boundaries or
interfaces are not systematically conductors. This violation was suggested in
a question of G.M. Graf in an online talk by G.C. Thiang [GT20].

Our experience of the world, is by nature, an approximation of the reality.
Experiment samples – here, $\Omega$ – are always finite and spectral
measurements are valid only up to an uncertainty $\delta>0$. Hence, in
practice $\Omega$ never contains arbitrarily large balls – but neither can
experiments assess with full certainty that spectral gaps completely fill:
they can only measure emergence of a $\delta$-dense set of spectrum. Theorem 2
is a quantitative formulation of Theorem 1 that relates to these observations.
It predicts that there exist constants $\alpha,R_{0}$ such that for all $R\geq
R_{0}$, the following holds. Assume that both $\Omega,\Omega^{c}$ contain a
ball of radius $R$. Then the spectrum of $H_{e}$ within ${\mathcal{G}}$ is
$C\ln(R)/R$-dense:

$\forall\lambda_{*}\in{\mathcal{G}},\ \ \exists\lambda\in\Sigma(H_{e}),\ \
|\lambda-\lambda_{*}|\leq\dfrac{\alpha\ln(R)}{R}.$ (1.2)

This justifies why topological insulators truncated to sufficiently large
balls appear in experiments conducting along their boundaries.

###  1.2. Topological insulators and interface operators

We briefly review standard facts from condensed matter physics. Electronic
propagation through a given material is described via a selfadjoint operator
$H$ on a Hilbert space – here $\ell^{2}({\mathbb{Z}}^{2},{\mathbb{C}}^{d})$.
The spectrum $\Sigma(H)$ of $H$ characterizes the electronic nature of the
material: $H$ is a conductor at energy $\lambda$ if and only if
$\lambda\in\Sigma(H)$ and an insulator otherwise.

In the rest of this paper, $\nu\in(0,1]$ is a fixed parameter.

We work here with short-range Hamiltonians: operators on
$\ell^{2}({\mathbb{Z}}^{2},{\mathbb{C}}^{d})$ whose kernels satisfy

$\forall{\bm{x}},{\bm{y}}\in{\mathbb{Z}}^{2},\qquad\big{|}H({\bm{x}},{\bm{y}})\big{|}\leq\nu^{-1}e^{-2\nu|{\bm{x}}-{\bm{y}}|},\qquad|{\bm{x}}-{\bm{y}}|:=|x_{1}-y_{1}|+|x_{2}-y_{2}|.$
(1.3)

Under (1.3), one can define the bulk conductance of $H$ at an insulating
energy. For $\lambda\notin\Sigma(H)$, let
$P_{\lambda}(H)={\mathbb{1}}_{(-\infty,\lambda)}(H)$ be the spectral projector
below energy $\lambda$, and $\Lambda_{1}$ (respectively $\Lambda_{2}$) the
indicator function of ${\mathbb{N}}\times{\mathbb{Z}}$ (respectively
${\mathbb{Z}}\times{\mathbb{N}}$). Then the operator
$P_{\lambda}(H)\big{[}[P_{\lambda}(H),\Lambda_{1}],[P_{\lambda}(H),\Lambda_{2}]\big{]}$
is trace-class (see [EGS05] and Remark 1 below) and

$\sigma(H,\lambda):=-2\pi
i\operatorname{Tr}\left(P_{\lambda}(H)\big{[}[P_{\lambda}(H),\Lambda_{1}],[P_{\lambda}(H),\Lambda_{2}]\big{]}\right)$
(1.4)

is well-defined. We comment that if ${\mathcal{G}}$ is a subinterval of
$\Sigma(H)^{c}$ (referred to below as a spectral gap), then

$\lambda,\lambda^{\prime}\in{\mathcal{G}}\ \Rightarrow\
\sigma(H,\lambda)=\sigma(H,\lambda^{\prime}).$ (1.5)

Therefore, there is no ambiguity in using the notation
$\sigma(H,{\mathcal{G}})$ for $\sigma(H,\lambda)$, $\lambda\in{\mathcal{G}}$.
It represents the bulk conductance for energies in ${\mathcal{G}}$ [ASS94].
Under the gap condition $\lambda\notin\Sigma(H)$, $\sigma(H,\lambda)$ is an
integer that measures topological aspects of the Hamiltonian $H$.

In this work, we ask under which conditions interfaces between two
topologically distinct insulators – the bulk materials – carry currents. We
make the following assumption on the bulk components:

######  Assumption 1.

$H_{\pm}$ are two selfadjoint, short-range Hamiltonians on
$\ell^{2}({\mathbb{Z}}^{2},{\mathbb{C}}^{d})$, with a common spectral gap
${\mathcal{G}}$ (an interval contained in
$\Sigma(H_{+})^{c}\cap\Sigma(H_{-})^{c}$) and distinct bulk conductances
within ${\mathcal{G}}$:

$\sigma(H_{+},{\mathcal{G}})\neq\sigma(H_{-},{\mathcal{G}}).$ (1.6)

Given a domain $\Omega\in{\mathbb{Z}}^{2}$, we define its boundary by

$\partial\Omega:=\\{{\bm{x}}\in\Omega,\
B_{1}({\bm{x}})\not\subset\Omega\\}\cup\\{{\bm{x}}\in\Omega^{c},\
B_{1}({\bm{x}})\not\subset\Omega^{c}\\},$ (1.7)

where $B_{r}({\bm{x}})$ denotes the $\ell^{1}$-ball of radius $r$ centered at
${\bm{x}}$ in ${\mathbb{Z}}^{2}$. We will denote the distance from ${\bm{x}}$
to $\partial\Omega$ by $d({\bm{x}},\partial\Omega)$. We make the following
assumptions on the interface operator:

######  Assumption 2.

$H_{e}$ is a selfadjoint, short-range Hamiltonian on
$\ell^{2}({\mathbb{Z}}^{2},{\mathbb{C}}^{d})$ satisfying the kernel condition:

$\forall{\bm{x}},{\bm{y}}\in{\mathbb{Z}}^{2},\qquad\big{|}E({\bm{x}},{\bm{y}})\big{|}\leq\nu^{-1}e^{-2\nu
d({\bm{x}},\partial\Omega)},\qquad
E:=H_{e}-{\mathbb{1}}_{\Omega}H_{+}{\mathbb{1}}_{\Omega}-{\mathbb{1}}_{\Omega^{c}}H_{-}{\mathbb{1}}_{\Omega^{c}}.$
(1.8)

The condition (1.8) means that $H_{e}$ is equal to $H_{+}$ on $\Omega$ and
$H_{-}$ on $\Omega^{c}$, up to corrections decaying exponentially away from
$\partial\Omega$.

###  1.3. Main results

To formulate our main results, we will need the notion of filling radius for a
subset $\Omega$ of ${\mathbb{Z}}^{2}$:

${\operatorname{Fr}}(\Omega)=\sup\\{r:\ \exists\ {\bm{x}}\in{\mathbb{Z}}^{2},\
B_{r}({\bm{x}})\cap{\mathbb{Z}}^{2}\subset\Omega\\}.$ (1.9)

It measures the size of the largest ball contained in $\Omega$:
${\operatorname{Fr}}(\Omega)\geq r$ if and only if $\Omega$ contains a ball of
radius $r$.

######  Theorem 1.

Let $H_{\pm},H_{e}$ satisfying Assumptions 1 and 2. If
${\operatorname{Fr}}(\Omega)={\operatorname{Fr}}(\Omega^{c})=\infty$, then
${\mathcal{G}}\subset\Sigma(H_{e})$.

This means that if the boundary of a topological insulators divides the plane
in two regions of infinite filling radius, then it is a conductor. Theorem 1
will actually follow from a more quantitative statement. Given $\delta>0$, we
say that a set $S\subset{\mathcal{G}}$ is $\delta$-dense within
${\mathcal{G}}$ if either $|{\mathcal{G}}|<\delta$ or if

$\forall\lambda_{*}\in{\mathcal{G}},\ \exists\lambda\in S\text{ s.t.
}|\lambda_{*}-\lambda|\leq\delta.$ (1.10)

######  Theorem 2.

There exist constants $\alpha_{\nu}>0$ and $R_{\nu}>0$, depending on $\nu$
only, such that the following holds. For $H_{\pm},H_{e}$ satisfying
Assumptions 1 and 2 and any $R\geq R_{\nu}$:

${\operatorname{Fr}}(\Omega),\ {\operatorname{Fr}}(\Omega^{c})\geq
R\quad\Rightarrow\quad\Sigma(H_{e})\cap{\mathcal{G}}\ \ $ is
$\alpha_{\nu}\frac{\ln R}{R}$–dense within ${\mathcal{G}}$.

Theorem 2 has the following physical interpretation. Assume that $H_{e}$
represents the truncation of a topological insulator $H_{+}$ in the ball
$B_{R}(0)$, and that we have a measurement procedure that can infer if an
energy is within $\delta$ of $\Sigma(H_{e})$. Theorem 2 asserts that if
$\ln(R)/R\ll\delta$, then experiments measure that the spectral gap of $H_{+}$
closes when truncating it to $B_{R}(0)$. This imperfect conclusion ($H_{e}$
actually has discrete spectrum when truncated to $B_{R}(0)$) is due to the
limitation of the measuring procedure.

Theorem 2 implies Theorem 1: if
${\operatorname{Fr}}(\Omega)={\operatorname{Fr}}(\Omega^{c})=\infty$, then
${\operatorname{Fr}}(\Omega)$ and ${\operatorname{Fr}}(\Omega^{c})$ are larger
than $R$ for any $R$, so $\Sigma(H_{e})\cap{\mathcal{G}}$ is $\delta$-dense
within ${\mathcal{G}}$ for any $\delta>0$. This means that
$\Sigma(H_{e})\cap{\mathcal{G}}$ is actually dense within ${\mathcal{G}}$;
since it is a closed subset of ${\mathcal{G}}$ we conclude that
$\Sigma(H_{e})\cap{\mathcal{G}}={\mathcal{G}}$, equivalently
${\mathcal{G}}\subset\Sigma(H_{e})$.

A natural question is whether the conclusion of Theorem 1 fails for unbounded
sets $\Omega$ with finite filling radius.

######  Theorem 3.

Fix $L>0$ and $\Omega\subset[-L,L]\times{\mathbb{Z}}$. There exists operators
$H_{\pm}$ satisfying Assumption 1 for a gap ${\mathcal{G}}$ containing $0$;
and $H_{e}$ satisfying Assumption 2 such that $0\notin\Sigma(H_{e})$.

In other words, a topological insulator fitting in a strip does not have to be
a conductor. This constitutes a violation of the bulk-edge correspondence: an
interface between two distinct topological phases does not have to support
edge states, for instance if the interface is the boundary of a half-strip.

Figure 2. A region $\Omega$ that does not satisfy the assumption of Theorem 1:
the largest ball that fits in $\Omega^{c}$ has radius $1$, so
${\operatorname{Fr}}(\Omega^{c})=1<\infty$. There actually exist operators
$H_{\pm},H_{e}$ satisfying Assumptions 1, 2 such that
${\mathcal{G}}\not\subset\Sigma(H_{e})$.

###  1.4. Sketch of proof.

We explain here the main ideas leading to Theorem 1. Strictly speaking, the
paper will focus on quantitative forms of these ideas to obtain Theorem 2,
which (as explained above) implies Theorem 1.

Let $H$ be a selfadjoint, short-range Hamiltonian on
$\ell^{2}({\mathbb{Z}}^{2},{\mathbb{C}}^{d})$ and $\lambda\notin\Sigma(H)$.
Our main argument is the observation that $\sigma(H,\lambda)$ can be computed
from the sole knowledge of $H$ within a ball centered at any point n of
${\mathbb{Z}}^{2}$, as long as its radius $R$ is sufficiently large (depending
on $\nu$, but independent of n). While this may seem paradoxical, the global
condition $\lambda\notin\Sigma(H)$ actually ensures that the result of the
computation is independent of n.

We proceed now by contradiction. Let $H_{\pm},H_{e}$ satisfy the assumptions
of Theorem 1, and assume that $\lambda\in{\mathcal{G}}\setminus\Sigma(H_{e})$.
We can then define $\sigma(H_{e},\lambda)$, and, thanks to the above
observations, compute it from sole knowledge of $H_{e}$ on balls of the form
$B_{R}({\textbf{n}})$.

Pick now n so that $B_{R}({\textbf{n}})$ lies deep in $\Omega$, i.e. in
$\Omega$ and far from $\partial\Omega$; this is possible because under
${\operatorname{Fr}}(\Omega)=\infty$, $\Omega$ contains arbitrarily large
balls. From Assumption 2, $H_{e}$ is roughly $H_{+}$ there and we deduce
$\sigma(H_{e},\lambda)=\sigma(H_{+},\lambda)$. Likewise, pick n so that
$B_{R}({\textbf{n}})$ lies deep in $\Omega^{c}$ and deduce
$\sigma(H_{e},\lambda)=\sigma(H_{-},\lambda)$. This contradicts the assumption
$\sigma(H_{+},\lambda)\neq\sigma(H_{-},\lambda)$. Therefore,
$\lambda\in\Sigma(H_{e})$.

We comment that this proof also applies to materials made off three or more
topological insulators, with at least two of them with different bulk
invariant filling regions with infinite filling radius.

###  1.5. Relation to existing results

The question of how the shape of the truncation affects the edge spectrum has
considered before. The bulk-edge correspondence predicts the emergence of edge
spectrum for half-space truncations: it gives the resulting interface
conductance as a difference of Chern numbers [KS02, EG02, EGS05, ASV13, GP13,
B19, D21, LT22].

In [FGW00], the authors focus on truncated quantum Hall Hamiltonians and
derive a global analytic condition on $\Omega$ for the emergence of edge
spectrum. They verify that this condition holds for regions with
asymptotically flat boundary. This includes local perturbation of sectors.

More recently the techniques have drifted to coarse geometry and K-theory. In
[T20, O22] the authors prove that magnetic Hamiltonians truncated to corners
or sectors, and their local perturbations, have edge spectrum. The furthest-
reaching work is due to Ludewig–Thiang [LT22]. It predicts that truncations of
general topological insulators to domains coarsely equivalent to half-planes
admit edge spectrum.

Our work derives a simple visual criterion for emergence of edge spectrum:
both $\Omega$ and $\Omega^{c}$ contain arbitrary large balls (equivalently,
${\operatorname{Fr}}(\Omega)={\operatorname{Fr}}(\Omega^{c})=\infty$). It is
not evident how our condition relates to those of [FGW00, LT22]. Our proof
returns to an analytic approach, which has the benefit to come up with a
quantitative form of the result. This version explains in what sense
experimentalists observe edge spectrum in bounded samples.

The shape of edge states matters in technological applications: they are the
vectors of conduction along the edge. When the boundary is weakly curved –
which correspond to the adiabatic or semiclassical regime – several works
constructed edge states as wavepackets [bal2021edge, bal2022semiclassical,
D22, PPY22]. The assumptions in the present work are significantly weaker – we
only assume existence of a spectral gap – but the result is also significantly
weaker: we only prove existence of edge spectrum.

###  1.6. Open problems

An open problem is whether the bulk-edge correspondence generalizes to
truncations to domains $\Omega$ satisfying
${\operatorname{Fr}}(\Omega)={\operatorname{Fr}}(\Omega^{c})=\infty$. We
believe that this condition will need to be strengthened to something more
quantitative for the bulk-edge correspondence to hold. There are already
results that use the K-theoretic and coarse geometry framework [LT22]; it
would be nice to provide analytic proofs.

It has been shown that the bulk-edge correspondence holds when the gap
condition (${\mathcal{G}}\cap\Sigma(H_{\pm})$ is empty) is replaced by a
mobility gap condition ($H_{\pm}$ exhibits dynamical localization within
${\mathcal{G}}$); see [EGS05]. At this point we do not know if relaxing
Assumption 1 to a mobility gap gives rise to edge spectrum.

###  1.7. Notations

We will use the following notations:

  * •

${\bm{x}}=(x_{1},x_{2})$ denotes an element of ${\mathbb{Z}}^{2}$.

  * •

$|{\bm{x}}|=|x_{1}|+|x_{2}|$ denotes the $\ell^{1}$-norm on
${\mathbb{Z}}^{2}$.

  * •

$B_{r}({\bm{x}}):=\\{{\bm{y}}\in{\mathbb{Z}}^{2}:|{\bm{y}}-{\bm{x}}|\leq r\\}$
is the ball of radius $r\in{\mathbb{R}}^{+}$ centered at
${\bm{x}}\in{\mathbb{Z}}^{2}$.

  * •

If $A\subset{\mathbb{R}}^{2}$ and $x\in{\mathbb{R}}^{2}$, $d(x,A)$ denotes the
distance from $x$ to $A$.

  * •

Given an operator $H:\ell^{2}({\mathbb{Z}}^{2})\to\ell^{2}({\mathbb{Z}}^{2})$,
we let $H({\bm{x}},{\bm{y}})=\langle H\delta_{\bm{x}},\delta_{\bm{y}}\rangle$
be the kernel of $H$. We let
$P_{\lambda}(H):=\mathbb{1}_{(-\infty,\lambda)}(A)$ be the spectral projection
below energy $\lambda$.

  * •

In the whole paper, $C_{\nu}$ denotes a constant that can vary from line to
line but depends only on the parameter $\nu$ from §1.2.

###  1.8. Acknowledgements

This problem was motivated in part by a question from G.M. Graf at a lecture
by G.C. Thiang [GT20] during the online workshop “Mathematics of topological
insulators” in 2020 at the American Institute of Mathematics. We are very
grateful to the staff at AIM and the organizers of the workshop, D. Freed,
G.M. Graf, R. Mazzeo and M.I. Weinstein.

We gratefully acknowledge support from National Science Foundation DMS 2054589
and the Pacific Institute for the Mathematical Sciences. The contents of this
work are solely the responsibility of the authors and do not necessarily
represent the official views of PIMS.

##  2\. The main proposition

We proved Theorem 1 using Theorem 2 in §1.3. Theorem 2 will essentially follow
from Proposition 1 below. It essentially asserts that two insulators that
coincide on a large enough ball (with radius depending on $\nu$ but not on the
center of the ball) must have the same bulk conductance.

######  Assumption 3.

$H$ is a selfadjoint, short-range operator on
$\ell^{2}({\mathbb{Z}}^{2};{\mathbb{C}}^{d})$, such that for some
$\lambda\in{\mathbb{R}}$ and $\delta\in(0,1)$,

$(\lambda-\delta,\lambda+\delta)\cap\Sigma(H)=\emptyset.$ (2.1)

######  Proposition 1.

There exists a constant $C_{\nu}\geq 1$ such that the following holds. Let
$\epsilon>0$, $r>0$, ${\textbf{n}}\in{\mathbb{Z}}^{2}$, and $H_{1}$, $H_{2}$
satisfy Assumption 3 such that

$|(H_{1}-H_{2})({\bm{x}},{\bm{y}})|\leq\epsilon,\quad{\bm{x}},{\bm{y}}\in
B_{4r}({\textbf{n}}).$ (2.2)

Then we have:

$|\sigma(H_{1},\lambda)-\sigma(H_{2},\lambda)|\leq\frac{C_{\nu}}{\delta^{12}}\left(e^{-\frac{\delta
r}{2C_{\nu}}}+\epsilon^{1/2}\right).$ (2.3)

###### Proof of Theorem 2 assuming Proposition 1.

1\. We recall that $\nu$ is a fixed parameter. In this first step, we set the
values the constants $\alpha_{\nu}$ and $R_{\nu}$. Let $C_{\nu}$ be given by
Proposition 1; we set $\alpha_{\nu}=300C_{\nu}$. We now define $R_{\nu}$. The
quantity $R^{12}e^{-\nu R/2}$ goesto $0$ as $R$ goes to infinity. Therefore,
there exists $R_{\nu}\geq 4$ such that for all $R\geq R_{\nu}$,

$R^{12}e^{-\nu R/2}<\dfrac{1}{2}.$ (2.4)

Fix now $R\geq R_{\nu}$ (in particular, (2.4) holds); and define

$\delta=\alpha_{\nu}\frac{\ln R}{3R}=100C_{\nu}\dfrac{\ln R}{R}.$ (2.5)

We will prove that ${\mathcal{G}}\cap\Sigma(H_{e})$ is $3\delta$-dense within
${\mathcal{G}}$, that is, $\alpha_{\nu}\frac{\ln R}{R}$-dense within
$\Sigma(H_{e})$.

2\. Let us assume for now that the following statements hold,

$\displaystyle(\lambda_{*}-\delta,\lambda_{*}+\delta)\cap\Sigma(H_{e})=\emptyset,$
(2.6)
$\displaystyle(\lambda_{*}-\delta,\lambda_{*}+\delta)\cap\Sigma(H_{+})=\emptyset,$
(2.7)
$\displaystyle(\lambda_{*}-\delta,\lambda_{*}+\delta)\cap\Sigma(H_{-})=\emptyset,$
(2.8)

and let’s aim for a contradiction. Note that these statements imply that
$H_{e},H_{\pm}$ satisfy Assumption 3.

Since $\Omega$ has filling radius at least $R$, there exists
${\textbf{n}}\in{\mathbb{Z}}^{2}$ such that
$B_{8r}({\textbf{n}})\subset\Omega$, $r=R/8$. We now look at
$(H_{e}-H_{+})({\bm{x}},{\bm{y}})$ for ${\bm{x}},{\bm{y}}$ in
$B_{4r}({\textbf{n}})$. Because ${\bm{x}},{\bm{y}}\in\Omega$, we have:

$(H_{e}-H_{+})({\bm{x}},{\bm{y}})=(H_{e}-{\mathbb{1}}_{\Omega}H_{+}{\mathbb{1}}_{\Omega}-{\mathbb{1}}_{\Omega^{c}}H_{-}{\mathbb{1}}_{\Omega^{c}})({\bm{x}},{\bm{y}})=E({\bm{x}},{\bm{y}}),$
(2.9)

where $E$ is the operator defined in Assumption 2. Moreover,

$d({\bm{x}},\partial\Omega)\geq
d({\textbf{n}},\partial\Omega)-|{\bm{x}}-{\textbf{n}}|\geq 8r-4r=4r,$ (2.10)

because $B_{8r}({\textbf{n}})=B_{R}({\textbf{n}})\subset\Omega$. It follows
from (1.8) that

$\big{|}(H_{e}-H_{+})({\bm{x}},{\bm{y}})\big{|}\leq\nu^{-1}e^{-8\nu
r},\qquad{\bm{x}},{\bm{y}}\in B_{4r}({\textbf{n}}).$ (2.11)

Proposition 1 then yields

$\big{|}\sigma(H_{e},\lambda_{*})-\sigma(H_{+},\lambda_{*})\big{|}\leq\dfrac{C_{\nu}}{\delta^{12}}\exp\left(-\dfrac{\delta
r}{2C_{\nu}}\right)+\dfrac{C_{\nu}\nu^{-1/2}}{\delta^{12}}e^{-4\nu r}.$ (2.12)

We recall that $\delta$ has the value (2.5). Therefore, since $C_{\nu}\geq 1$
and $R\geq R_{\nu}\geq 4$,

$\displaystyle\dfrac{C_{\nu}}{\delta^{12}}\exp\left(-\dfrac{\delta
r}{2C_{\nu}}\right)$ $\displaystyle=\dfrac{C_{\nu}R^{12}}{(100C_{\nu}\ln
R)^{12}}\exp\left(-\dfrac{100C_{\nu}\ln R}{R}\cdot\dfrac{R}{16C_{\nu}}\right)$
(2.13) $\displaystyle\leq R^{12}\exp\left(-\dfrac{25}{4}R\right)\leq
R^{-1/2}\leq\dfrac{1}{2}.$ (2.14)

The last two inequalities are true for $R\geq 4$. Likewise, because $R$
satisfies (2.4),

$\dfrac{C_{\nu}\nu^{-1/2}}{\delta^{12}}e^{-4\nu
r}=\dfrac{C_{\nu}R^{12}}{(100C_{\nu}\ln R)^{12}}\exp\left(-\dfrac{\nu
R}{2}\right)\leq R^{12}\exp\left(-\dfrac{\nu R}{2}\right)<\dfrac{1}{2}.$
(2.15)

Going back to (2.7), we conclude that

$\big{|}\sigma(H_{e},\lambda_{*})-\sigma(H_{+},\lambda_{*})\big{|}<1.$ (2.16)

Since bulk conductances are integers (see [EGS05]*Proposition 3 and Remark 1
below), we conclude that
$\sigma(H_{e},\lambda_{*})=\sigma(H_{+},\lambda_{*})$.

Similarly, we conclude that
$\sigma(H_{e},\lambda_{*})=\sigma(H_{-},\lambda_{*})$. This cannot be true,
since $\sigma(H_{+},\lambda_{*})\neq\sigma(H_{-},\lambda_{*})$. We conclude
that for each $\lambda_{*}\in{\mathcal{G}}$, one of the statements among (2.7)
must fail. In other words, for all $\lambda_{*}\in{\mathcal{G}}$, there exists
some $\lambda\in\Sigma(H_{e})\cup\Sigma(H_{+})\cup\Sigma(H_{-})$ such that
$|\lambda-\lambda_{*}|\leq\delta$.

It remains to show that $\Sigma(H_{e})\cap{\mathcal{G}}$ is $3\delta$-dense
within ${\mathcal{G}}$. Write ${\mathcal{G}}=(a,b)$ with $b-a>3\delta$
(otherwise any subset is $3\delta$-dense by definition). Let
$\lambda_{*}\in(a+\delta,b-\delta)$ and
$\lambda\in\Sigma(H_{e})\cup\Sigma(H_{+})\cup\Sigma(H_{-})$ such that
$|\lambda-\lambda_{*}|\leq\delta$. In particular,
$\lambda\in(a,b)={\mathcal{G}}$, which is a spectral gap of $H_{\pm}$, so
$\lambda\in\Sigma(H_{e})$. Let now $\lambda_{*}\in(a,\mu_{*})$,
$\mu_{*}=a+2\delta$; since $b-a>3\delta$, $\mu_{*}\in(a+\delta,b-\delta)$ and
by the previous step there exists $\mu\in\Sigma(H_{e})$ such that
$|\mu-\mu_{*}|\leq\delta$. In particular, $|\mu-\lambda_{*}|\leq 3\delta$. A
similar argument works for $\lambda_{*}\in(b-2\delta,b)$. We conclude that
${\mathcal{G}}\cap\Sigma(H_{e})$ is $3\delta$-dense within ${\mathcal{G}}$. ∎

##  3\. Proof of Proposition 1

###  3.1. On short-range Hamiltonians

Throughout the proofs below, we will use the following estimates, proved in
Appendix A: For $a\in(0,1]$, $R>0$, we have

$\displaystyle\sum_{s\in{\mathbb{Z}}}e^{-2a|s|}$
$\displaystyle\leq\dfrac{2}{a},\quad\sum\limits_{{\bm{x}}\in{\mathbb{Z}}^{2}}e^{-2a|{\bm{x}}|}\leq\dfrac{4}{a^{2}},\quad\text{and}\quad\sum_{|{\bm{x}}|\geq
R,{\bm{x}}\in{\mathbb{Z}}^{2}}e^{-2a|{\bm{x}}|}\leq\dfrac{8}{a^{2}}e^{-aR}.$
(3.1)

We make here a few observations on the selfadjoint, short-range Hamiltonians
$H$ on $\ell^{2}({\mathbb{Z}}^{2},{\mathbb{C}}^{d})$. First, they are bounded
in terms of the (fixed) parameter $\nu\in(0,1]$ quantifying the short-range
condition (1.3). Specifically, an application of Schur’s test gives:

$\|H\|\leq\dfrac{4}{\nu^{3}}.$ (3.2)

We refer to Appendix A for the proof.

As in [EGS05, AW15], we introduce

$S_{\alpha}:=\sup_{{\bm{x}}\in{\mathbb{Z}}^{2}}\sum_{{\bm{y}}\in{\mathbb{Z}}^{2}}|H({\bm{x}},{\bm{y}})|\left(e^{\alpha|{\bm{x}}-{\bm{y}}|}-1\right).$
(3.3)

We note that if $H$ is short range under the definition (1.3), then for any
$\alpha\in(0,2\nu)$, $S_{\alpha}<+\infty$. Also, for later use, if
$\alpha\in(0,\nu]$:

$\dfrac{S_{\alpha}}{\alpha}\leq\dfrac{S_{\nu}}{\nu}\leq\dfrac{16}{\nu^{4}},$
(3.4)

Again, see Appendix A for the proof.

We recall the Combes–Thomas inequality [CT73]:

######  Proposition 2.

[AW15]*Theorem 10.5 Let $H$ be a selfadjoint, short-range operator on
$\ell^{2}({\mathbb{Z}}^{2};{\mathbb{C}}^{d})$. If $\alpha\in(0,2\nu)$ and
$z\in{\mathbb{C}}$ are such that $\Delta:=d(z,\Sigma(H))>S_{\alpha}$, then we
have

$|(H-z)^{-1}({\bm{x}},{\bm{y}})|\leq\frac{1}{\Delta-
S_{\alpha}}\exp(-\alpha|{\bm{x}}-{\bm{y}}|).$ (3.5)

###  3.2. Spectral projections

An application of the Combes–Thomas inequality is control of the kernels of
spectral projections:

######  Lemma 3.1.

There exists a constant $C_{\nu}$, such that for any $H,\lambda$ satisfying
Assumption 3:

$\big{|}P_{\lambda}(H)({\bm{x}},{\bm{y}})\big{|}\leq\frac{C_{\nu}}{\delta}\exp\left(-\dfrac{\delta}{C_{\nu}}|{\bm{x}}-{\bm{y}}|\right).$
(3.6)

###### Proof.

Set $\alpha=2^{-5}\nu^{4}\delta$, so that $\alpha\leq\nu$ and
$S_{\alpha}\leq\frac{16\alpha}{\nu^{4}}\leq\frac{\delta}{2}$, see (3.4). Let
$\gamma$ be a contour enclosing $\Sigma(H)\cap(-\infty,\lambda)$, at least
$\delta$-distant from $\Sigma(H)$. For $z\in\gamma$, we have:

$(H-z)^{-1}({\bm{x}},{\bm{y}})\leq\dfrac{e^{-\alpha|{\bm{x}}-{\bm{y}}|}}{\Delta-
S_{\alpha}}\leq\frac{e^{-\alpha|{\bm{x}}-{\bm{y}}|}}{\delta-\frac{\delta}{2}}\leq\frac{2}{\delta}\exp\left(-\alpha|{\bm{x}}-{\bm{y}}|\right).$
(3.7)

Integrating this over $\gamma$, we have:

$\big{|}P_{\lambda}(H)({\bm{x}},{\bm{y}})\big{|}=\left|\dfrac{1}{2\pi
i}\oint_{\gamma}(H-z)^{-1}({\bm{x}},{\bm{y}})dz\right|\leq\frac{|\gamma|}{\pi\delta}\exp\left(-\alpha|{\bm{x}}-{\bm{y}}|\right).$
(3.8)

Without loss of generalities, we may assume that $|\gamma|\leq 2\pi+4\|H\|$ –
see Figure 3 From the bound (3.2) on $\|H\|$, we deduce that

$\dfrac{|\gamma|}{\pi}\leq 2+\dfrac{16}{\pi\nu^{3}}\leq\dfrac{8}{\nu^{3}}.$
(3.9)

We conclude that

$\big{|}P_{\lambda}(H)({\bm{x}},{\bm{y}})\big{|}\leq\frac{8}{\nu^{3}\delta}\exp\left(-\dfrac{\nu^{4}\delta}{32}|{\bm{x}}-{\bm{y}}|\right).$
(3.10)

This yields (3.6) (with for instance $C_{\nu}=32\nu^{-4}$). ∎

Figure 3. The contour $\gamma$. We note that the spectrum of $H$ is contained
in $[-\|H\|,\|H\|]$, so it has diameter less than $2\|H\|$. Since $\delta\leq
1$, the two half-circles have perimeter less than $2\pi$ and the length of
$\gamma$ is less than $4\|H\|+2\pi$. Note that $\gamma$ does not need to
enclose $\lambda$.

######  Remark 1.

As a result, if $H$, $\lambda$ satisfy Assumption 3, then the open bounded
interval $(\lambda-\delta,\lambda+\delta)$ satisfies condition [EGS05]*(1.2);
and it follows that $\sigma(H,\lambda)$ is well-defined and is an integer
[EGS05]*Proposition 3.

######  Lemma 3.2.

There exists a constant $C_{\nu}$ such that the following holds. Let
$r,\epsilon>0$, and two triplets $H_{1},\lambda,\delta$ and
$H_{2},\lambda,\delta$ satisfying Assumption 3, such that for
${\bm{x}},{\bm{y}}\in B_{4r}(0)$,

$|H_{1}({\bm{x}},{\bm{y}})-H_{2}({\bm{x}},{\bm{y}})|\leq\epsilon.$ (3.11)

Then for $({\bm{x}},{\bm{y}})$ in $B_{2r}(0)$:

$|P_{\lambda}(H_{1})({\bm{x}},{\bm{y}})-P_{\lambda}(H_{2})({\bm{x}},{\bm{y}})|\leq\dfrac{C_{\nu}}{\delta^{6}}\left(\exp\left(-\dfrac{\delta
r}{C_{\nu}}\right)+\epsilon\right).$ (3.12)

###### Proof.

Let $\alpha,\gamma$ be as in the proof of Lemma 3.1. We have:

$\big{(}P_{\lambda}(H_{1})-P_{\lambda}(H_{2})\big{)}({\bm{x}},{\bm{y}})=\sum_{{\bm{x}}^{\prime},{\bm{y}}^{\prime}}\dfrac{1}{2\pi
i}\oint_{\gamma}(H_{2}-z)^{-1}({\bm{x}},{\bm{x}}^{\prime})\big{(}H_{2}({\bm{x}}^{\prime},{\bm{y}}^{\prime})-H_{1}({\bm{x}}^{\prime},{\bm{y}}^{\prime})\big{)}(H_{1}-z)^{-1}({\bm{y}}^{\prime},{\bm{y}})dz.$
(3.13)

From (3.7) then (3.9), and using the bound (3.9) for $|\gamma|$:

$\displaystyle\big{|}\big{(}P_{\lambda}(H_{1})-P_{\lambda}(H_{2})\big{)}({\bm{x}},{\bm{y}})\big{|}$
$\displaystyle\leq\dfrac{2|\gamma|}{\pi\delta^{2}}\sum_{{\bm{x}}^{\prime},{\bm{y}}^{\prime}}\exp\left(-\alpha|{\bm{x}}-{\bm{x}}^{\prime}|-\alpha|{\bm{y}}-{\bm{y}}^{\prime}|\right)\big{|}H_{2}({\bm{x}}^{\prime},{\bm{y}}^{\prime})-H_{1}({\bm{x}}^{\prime},{\bm{y}}^{\prime})\big{|}$
(3.14)
$\displaystyle\leq\dfrac{2^{4}}{\nu^{3}\delta^{2}}\sum_{{\bm{x}}^{\prime},{\bm{y}}^{\prime}}\exp\left(-\alpha|{\bm{x}}-{\bm{x}}^{\prime}|-\alpha|{\bm{y}}-{\bm{y}}^{\prime}|\right)\big{|}H_{2}({\bm{x}}^{\prime},{\bm{y}}^{\prime})-H_{1}({\bm{x}}^{\prime},{\bm{y}}^{\prime})\big{|}.$
(3.15)

We split the RHS sum in two parts, depending whether
${\bm{x}}^{\prime},{\bm{y}}^{\prime}\in B_{4r}(0)$ or not. When they do, we
can use the bound (3.11) on the kernel of $H_{1}-H_{2}$. It yields

$\displaystyle\sum_{x^{\prime},y^{\prime}\in
B_{4r}(0)}\exp\left(-\alpha|{\bm{x}}-{\bm{x}}^{\prime}|-\alpha|{\bm{y}}-{\bm{y}}^{\prime}|\right)\big{|}H_{2}(x^{\prime},y^{\prime})-H_{1}(x^{\prime},y^{\prime})\big{|}$
(3.16) $\displaystyle\leq$
$\displaystyle\epsilon\sum_{x^{\prime},y^{\prime}\in{\mathbb{Z}}^{2}}\exp\left(-\alpha|{\bm{x}}-{\bm{x}}^{\prime}|-\alpha|{\bm{y}}-{\bm{y}}^{\prime}|\right)$
(3.17) $\displaystyle\leq$
$\displaystyle\epsilon\sum_{x^{\prime},y^{\prime}\in{\mathbb{Z}}^{2}}\exp\left(-\alpha|{\bm{x}}^{\prime}|-\alpha|{\bm{y}}^{\prime}|\right)\leq\epsilon\left(\dfrac{16}{\alpha^{2}}\right)^{2},$
(3.18)

where in the last line we used (3.1).

When now restricting to ${\bm{x}}^{\prime}$ or ${\bm{y}}^{\prime}\in
B_{4r}(0)^{c}$, we note that for ${\bm{x}},{\bm{y}}\in B_{2r}(0)$ either
$|{\bm{x}}-{\bm{x}}^{\prime}|\geq 2r$ or $|{\bm{y}}-{\bm{y}}^{\prime}|\geq
2r$. Thus, we have

$\displaystyle\sum_{{\bm{x}}^{\prime}\text{ or }{\bm{y}}^{\prime}\in
B_{2r}(0)^{c}}\exp\left(-\alpha|{\bm{x}}-{\bm{x}}^{\prime}|-\alpha|{\bm{y}}-{\bm{y}}^{\prime}|\right)\big{|}H_{2}({\bm{x}}^{\prime},{\bm{y}}^{\prime})-H_{1}({\bm{x}}^{\prime},{\bm{y}}^{\prime})\big{|}$
(3.19) $\displaystyle\leq$
$\displaystyle\dfrac{2}{\nu}\sum_{{\bm{x}}^{\prime}\text{ or
}{\bm{y}}^{\prime}\in
B_{4r}(0)^{c}}\exp\left(-\alpha|{\bm{x}}-{\bm{x}}^{\prime}|-\alpha|{\bm{y}}-{\bm{y}}^{\prime}|\right)\leq\dfrac{4}{\nu}\sum_{{\bm{x}}^{\prime}\in
B_{2r}(0)^{c},{\bm{y}}^{\prime}\in{\mathbb{Z}}^{2}}\exp\left(-\alpha|{\bm{x}}^{\prime}|-\alpha|{\bm{y}}^{\prime}|\right)$
(3.20) $\displaystyle\leq\dfrac{4}{\nu}\left(\frac{32}{\alpha^{2}}e^{-\alpha
r}\right)\left(\frac{16}{\alpha^{2}}\right)=\frac{8}{\nu}\left(\frac{2^{8}}{\alpha^{4}}\right)e^{-\alpha
r}.$ (3.21)

In the last line, we applied (3.1). So, heading back to (3.15) and using the
value $\alpha=2^{-5}\nu^{4}\delta$ from the proof of Lemma 3.1, we obtain

$\big{|}\big{(}P_{\lambda}(H_{1})-P_{\lambda}(H_{2})\big{)}(x,y)\big{|}\leq\dfrac{2^{7}}{\nu^{4}\delta^{2}}\left(\dfrac{2^{8}}{\alpha^{4}}\right)(e^{-\alpha
r}+\epsilon)=\dfrac{2^{35}}{\nu^{20}\delta^{6}}\left(\exp\left(-\dfrac{\nu^{4}\delta}{32}r\right)+\epsilon\right).$
(3.22)

This yields (3.12) (with $C_{\nu}=2^{35}\nu^{-20}$ – we made no attempts to
minimize this constant). ∎

###  3.3. Technical result.

The key technical step in the proof of Proposition 1 is:

######  Proposition 3.

Fix $\varepsilon>0,\ C>0,\ r>0,\ \beta\in(0,1]$. Let $A_{0},A_{1},A_{2}$ be
three operators on $\ell^{2}({\mathbb{Z}}^{2},{\mathbb{C}}^{d})$ with the
following properties:

  * (i)

For $j\in\\{0,1,2\\}$, $|A_{j}({\bm{x}},{\bm{y}})|\leq Ce^{-2\beta|x-y|}$;

  * (ii)

There exists $k\in\\{0,1,2\\}$ such that if ${\bm{x}},{\bm{y}}\in
B_{2r}(0)$,then $\big{|}A_{k}({\bm{x}},{\bm{y}})\big{|}\leq C\varepsilon$.

Then $B:=A_{0}[A_{1},\Lambda_{1}][A_{2},\Lambda_{2}]$ is trace-class and

$\big{|}\operatorname{Tr}(B)\big{|}\leq
C^{3}\frac{2^{16}}{\beta^{6}}\left(e^{-\beta r}+\varepsilon^{1/2}\right).$
(3.23)

Let us start with a simple result:

######  Lemma 3.3.

Let $A$ be a bounded operator on $\ell^{2}({\mathbb{Z}}^{2},{\mathbb{C}}^{d})$
with $|A({\bm{x}},{\bm{y}})|\leq e^{-2\beta|{\bm{x}}-{\bm{y}}|}$. Then

$\big{|}[A,\Lambda_{1}]({\bm{x}},{\bm{y}})\big{|}\leq
e^{-2\beta|x_{1}|-2\beta|y_{1}|-2\beta|x_{2}-y_{2}|}.$ (3.24)

If moreover $|A({\bm{x}},{\bm{y}})|\leq\varepsilon$ for
$({\bm{x}},{\bm{y}})\in B_{2r}(0)$, then

$\big{|}[A,\Lambda_{1}]({\bm{x}},{\bm{y}})\big{|}\leq\varepsilon^{1/2}e^{-\beta|x_{1}|-\beta|y_{1}|-\beta|x_{2}-y_{2}|},\qquad{\bm{x}},{\bm{y}}\in
B_{2r}(0).$ (3.25)

###### Proof.

The kernel of $[A,\Lambda_{1}]$ is

$[A,\Lambda_{1}]({\bm{x}},{\bm{y}})=A({\bm{x}},{\bm{y}})(\Lambda_{1}({\bm{y}})-\Lambda_{1}({\bm{x}})).$
(3.26)

We note that $|\Lambda_{1}({\bm{y}})-\Lambda_{1}({\bm{x}})|=0$ if
$x_{1},y_{1}$ are both positive or both negative; and it is at most $1$
otherwise, that is if $x_{1}y_{1}\leq 0$. Therefore, we have the bound

$\big{|}[A,\Lambda_{1}]({\bm{x}},{\bm{y}})\big{|}\leq
Ce^{-2\beta|{\bm{x}}-{\bm{y}}|}\mathbb{1}_{x_{1}y_{1}\leq 0}.$ (3.27)

Whenever $x_{1}y_{1}\leq 0$, we have

$|{\bm{x}}-{\bm{y}}|=|x_{1}-y_{1}|+|x_{2}-y_{2}|=|x_{1}|+|y_{1}|+|x_{2}-y_{2}|.$
(3.28)

It follows that

$\big{|}[A,\Lambda_{1}]({\bm{x}},{\bm{y}})\big{|}\leq
e^{-2\beta|x_{2}-y_{2}|-2\beta|x_{1}|-2\beta|y_{1}|}.$ (3.29)

This completes the proof of (3.24). To prove (3.25), we recall that
$|\Lambda_{1}({\bm{y}})-\Lambda_{1}({\bm{x}})|\leq 1$; hence
$\big{|}[A,\Lambda_{1}]({\bm{x}},{\bm{y}})\big{|}\leq\varepsilon$. It suffices
then to interpolate this bound with (3.24). ∎

For the proof of Proposition 3, we will use the following inequality: for
$\beta\in(0,1]$, ${\bm{x}},{\bm{w}}\in{\mathbb{Z}}^{2}$,

$\sum_{{\bm{y}},{\bm{z}}\in{\mathbb{Z}}^{2}}e^{-2\beta|{\bm{x}}-{\bm{y}}|-2\beta|y_{2}-z_{2}|-2\beta|y_{1}|-2\beta|z_{1}|-2\beta|z_{1}-w_{1}|-2\beta|z_{2}|-2\beta|w_{2}|}\leq\left(\dfrac{4}{\beta}\right)^{4}e^{-\beta|{\bm{x}}|-\beta|{\bm{w}}|}.$
(3.30)

We refer to Appendix A for a proof.

###### Proof of Proposition 3.

1\. By a scaling argument, we can assume that $C=1$. We first control the
kernel of $B$:

$|B({\bm{x}},{\bm{w}})|=\left|\sum_{y,z\in{\mathbb{Z}}^{2}}A_{0}({\bm{x}},{\bm{y}})B_{1}({\bm{y}},{\bm{z}})B_{2}({\bm{z}},{\bm{x}})\right|,$
(3.31)

where $B_{j}=[A_{j},\Lambda_{j}]$. We control the kernels of $A_{0}$,
$B_{1},B_{2}$ using assumption $(i)$ and (3.24). It yields:

$\displaystyle|B({\bm{x}},{\bm{w}})|$
$\displaystyle=\left|\sum_{{\bm{y}},{\bm{z}}\in{\mathbb{Z}}^{2}}A_{0}({\bm{x}},{\bm{y}})B_{1}({\bm{y}},{\bm{z}})B_{2}({\bm{z}},{\bm{w}})\right|$
(3.32)
$\displaystyle\leq\sum_{{\bm{y}},{\bm{z}}\in{\mathbb{Z}}^{2}}e^{-2\beta|{\bm{x}}-{\bm{y}}|-2\beta|y_{2}-z_{2}|-2\beta|y_{1}|-2\beta|z_{1}|-2\beta|z_{1}-w_{1}|-2\beta|z_{2}|-2\beta|w_{2}|}$
(3.33)
$\displaystyle\leq\left(\dfrac{4}{\beta}\right)^{4}e^{-\beta|{\bm{x}}|-\beta|{\bm{w}}|}.$
(3.34)

Thus $|B({\bm{x}},{\bm{w}})|$ decays exponentially, hence $B$ is trace-class;
moreover

$\displaystyle\operatorname{Tr}(B)\leq\sum_{{\bm{x}}\in{\mathbb{Z}}^{2}}|B({\bm{x}},{\bm{x}})|\leq$
$\displaystyle\sum_{{\bm{x}},{\bm{y}},{\bm{z}}\in{\mathbb{Z}}^{2}}e^{-2\beta|{\bm{x}}-{\bm{y}}|-2\beta|y_{2}-z_{2}|-2\beta|y_{1}|-2\beta|z_{1}|-2\beta|z_{1}-x_{1}|-2\beta|z_{2}|-2\beta|x_{2}|}$
(3.35) $\displaystyle:=$
$\displaystyle\sum_{{\bm{x}},{\bm{y}},{\bm{z}}\in{\mathbb{Z}}^{2}}f(2\beta,{\bm{x}},{\bm{y}},{\bm{z}}).$
(3.36)

2\. We now split the sum in (3.36) in two pieces: $|{\bm{x}}|\geq r$ and
$|{\bm{x}}|\leq r$. Thanks to (3.30) and (3.1), we have

$\sum_{|{\bm{x}}|\geq
r}f(2\beta,{\bm{x}},{\bm{y}},{\bm{z}})\leq\left(\dfrac{4}{\beta}\right)^{4}\sum_{|{\bm{x}}|\geq
r}e^{-2\beta|{\bm{x}}|}\leq\left(\dfrac{4}{\beta}\right)^{4}\dfrac{8}{\beta^{2}}e^{-\beta
r}=\frac{2^{11}}{\beta^{6}}e^{-\beta r}.$ (3.37)

We focus below on $|{\bm{x}}|\leq r$.

3\. If $k=0$ in (ii), then we split the sum in (3.36) according to
$|{\bm{y}}|\geq 2r$ and $|{\bm{y}}|\leq 2r$. In the former case,
$|{\bm{x}}-{\bm{y}}|\geq r$. Therefore, when $|{\bm{x}}|\leq r,\
|{\bm{y}}|\geq 2r$, we deduce that

$\displaystyle\big{|}A_{0}({\bm{x}},{\bm{y}})\big{|}$ $\displaystyle\leq
e^{-2\beta|{\bm{x}}-{\bm{y}}|}\leq e^{-\beta r-\beta|{\bm{x}}-{\bm{y}}|},$
(3.38)
$\displaystyle\big{|}A_{0}({\bm{x}},{\bm{y}})B_{1}({\bm{y}},{\bm{z}})B_{2}({\bm{z}},{\bm{x}})\big{|}$
$\displaystyle\leq e^{-\beta r}f(\beta,{\bm{x}},{\bm{y}},{\bm{z}}).$ (3.39)

If now $|{\bm{y}}|\leq 2r$ (and $|{\bm{x}}|\leq r\leq 2r$), then we can use
(ii). Interpolating with (i) gives, for $|{\bm{x}}|\leq r,\ |{\bm{y}}|\leq
2r$:

$\displaystyle\big{|}A_{0}({\bm{x}},{\bm{y}})\big{|}$
$\displaystyle\leq\varepsilon^{1/2}e^{-\beta|{\bm{x}}-{\bm{y}}|},$ (3.40)
$\displaystyle\big{|}A_{0}({\bm{x}},{\bm{y}})B_{1}({\bm{y}},{\bm{z}})B_{2}({\bm{z}},{\bm{x}})\big{|}$
$\displaystyle\leq\varepsilon^{1/2}f(\beta,{\bm{x}},{\bm{y}},{\bm{z}}).$
(3.41)

Summing the bounds (3.39) and (3.41) produces:

$\displaystyle\sum_{\begin{subarray}{c}|{\bm{x}}|\leq r,\\\
{\bm{y}},{\bm{z}}\in{\mathbb{Z}}^{2}\end{subarray}}\big{|}A_{0}({\bm{x}},{\bm{y}})B_{1}({\bm{y}},{\bm{z}})B_{2}({\bm{z}},{\bm{x}})\big{|}$
$\displaystyle\leq e^{-\beta r}\sum_{\begin{subarray}{c}|{\bm{x}}|\leq r,\\\
|{\bm{y}}|\geq
2r,{\bm{z}}\in{\mathbb{Z}}^{2}\end{subarray}}f(\beta,{\bm{x}},{\bm{y}},{\bm{z}})+\varepsilon^{1/2}\sum_{\begin{subarray}{c}|{\bm{x}}|\leq
r,\\\ |{\bm{y}}|\leq
2r,{\bm{z}}\in{\mathbb{Z}}^{2}\end{subarray}}f(\beta,{\bm{x}},{\bm{y}},{\bm{z}})$
(3.42) $\displaystyle\leq\left(e^{-\beta
r}+\varepsilon^{1/2}\right)\sum_{{\bm{x}},{\bm{y}},{\bm{z}}\in{\mathbb{Z}}^{2}}f(\beta,{\bm{x}},{\bm{y}},{\bm{z}})$
(3.43) $\displaystyle\leq\frac{2^{16}}{\beta^{6}}\left(e^{-\beta
r}+\varepsilon^{1/2}\right).$ (3.44)

where we used (3.30) and (3.1) to get

$\sum_{{\bm{x}},{\bm{y}},{\bm{z}}\in{\mathbb{Z}}^{2}}f(\beta,{\bm{x}},{\bm{y}},{\bm{z}})\leq\left(\frac{8}{\beta}\right)^{4}\sum_{{\bm{x}}\in{\mathbb{Z}}^{2}}e^{-\beta|{\bm{x}}|}\leq\left(\frac{8}{\beta}\right)^{4}\frac{16}{\beta^{2}}=\frac{2^{16}}{\beta^{6}}.$
(3.45)

4\. We now work on $k=1$. We split the sum in ${\bm{y}},{\bm{z}}\in B_{2r}(0)$
and ${\bm{y}}$ or ${\bm{z}}$ outside $B_{2r}(0)$. In the latter case, either
$|{\bm{x}}-{\bm{y}}|\geq r$ or $|{\bm{z}}-{\bm{x}}|\geq r$. So either

$\big{|}A_{0}({\bm{x}},{\bm{y}})\big{|}\leq e^{-\beta
r}e^{-\beta|{\bm{x}}-{\bm{y}}|}\text{ or
}\big{|}B_{2}({\bm{z}},{\bm{x}})\big{|}\leq e^{-\beta
r}e^{-\beta|{\bm{z}}-{\bm{x}}|}.$ (3.46)

In either case, we recover the bound (3.39). In the former case, we can use
(3.25), and recover (3.41). Since (3.39) and (3.41) lead to (3.44), we obtain
this bound here as well.

5\. The case $k=2$ follows the same path as $k=0$ . This completes the proof.
∎

###  3.4. Comparison of bulk conductances

We will use the following result [EGS05]*Lemma 7(ii), which essentially states
that the bulk conductance is independent of $\Lambda_{1},\Lambda_{2}$:

######  Proposition 4.

Let $H$ be a short-range operator on
$\ell^{2}({\mathbb{Z}}^{2},{\mathbb{C}}^{d})$ and $\lambda\notin\Sigma(H)$.
For any n,

$\sigma(H,\lambda)=-2\pi
i\operatorname{Tr}\left(P_{\lambda}(H)\big{[}[P_{\lambda}(H),\Lambda_{1}(\cdot-
n_{1})],[P_{\lambda}(H),\Lambda_{2}(\cdot-n_{2})]\big{]}\right).$ (3.47)

We are now ready to prove Proposition 1.

###### Proof of Proposition 1.

1\. For simplicity, use the notation $P_{j}=P_{\lambda}(H_{j})$. Let $T$ be
the translation by n: $Tu(\cdot)=u(\cdot-{\textbf{n}})$. We have
$\Lambda_{j}(\cdot-n_{j})=T\Lambda_{j}(\cdot)T^{*}$ and
$T^{*}P_{\lambda}(H)T=P_{\lambda}(T^{*}HT)$. Using these and Proposition 4, as
well as the cyclicity of the trace, we obtain

$\displaystyle\sigma(H_{j},\lambda)$ $\displaystyle=-2\pi
i\operatorname{Tr}\big{(}P_{j}\big{[}[P_{j},T\Lambda_{1}T^{*}],[P_{j},T\Lambda_{2}T^{*}]\big{]}\big{)}$
(3.48) $\displaystyle=-2\pi
i\operatorname{Tr}\big{(}P_{j}\big{[}T[T^{*}P_{j}T,\Lambda_{1}]T^{*},T[T^{*}P_{j}T,\Lambda_{2}]T^{*}\big{]}\big{)}$
(3.49) $\displaystyle=-2\pi
i\operatorname{Tr}\big{(}P_{j}T\big{[}[T^{*}P_{j}T,\Lambda_{1}],[T^{*}P_{j}T,\Lambda_{2}]\big{]}T^{*}\big{)}$
(3.50) $\displaystyle=-2\pi
i\operatorname{Tr}\big{(}T^{*}P_{j}T\big{[}[T^{*}P_{j}T,\Lambda_{1}],[T^{*}P_{j}T,\Lambda_{2}]\big{]}\big{)}=\sigma(T^{*}H_{j}T,\lambda).$
(3.51)

Therefore, by replacing $H_{j}$ by $T^{*}H_{j}T$, we can simply assume that
${\textbf{n}}=0$.

2\. Now we write

$\sigma(H_{1},\lambda)-\sigma(H_{2},\lambda)=T(P_{1}-P_{2},P_{1},P_{1})+T(P_{2},P_{1}-P_{2},P_{1})+T(P_{2},P_{2},P_{1}-P_{2}),$
(3.52)

where $T(A_{0},A_{1},A_{2})$ is the trilinear form

$T(A_{0},A_{1},A_{2})=-2\pi
i\operatorname{Tr}\left(A_{0}[A_{1},\Lambda_{1}][A_{2},\Lambda_{2}]\right)+2\pi
i\operatorname{Tr}\left(A_{0}[A_{2},\Lambda_{2}][A_{1},\Lambda_{1}]\right).$
(3.53)

From Lemmas 3.1 and 3.2, we have (for a constant $C_{\nu}$ depending on $\nu$
only):

$\displaystyle\big{|}P_{j}({\bm{x}},{\bm{y}})\big{|}$
$\displaystyle\leq\dfrac{C_{\nu}}{\delta}e^{-\delta|{\bm{x}}-{\bm{y}}|/C_{\nu}},\qquad
j=1,2;$ (3.54)
$\displaystyle\big{|}P_{1}({\bm{x}},{\bm{y}})-P_{2}({\bm{x}},{\bm{y}})\big{|}$
$\displaystyle\leq\dfrac{C_{\nu}}{\delta^{6}}\left(e^{-\delta|{\bm{x}}-{\bm{y}}|/C_{\nu}}+\epsilon\right),\qquad{\bm{x}},{\bm{y}}\in
B_{2r}({\textbf{n}}).$ (3.55)

Therefore, the triplets $(A_{0},A_{1},A_{2})$ involved in (3.52) satisfy the
assumptions of Proposition 3, with constants

$C=\dfrac{C_{\nu}}{\delta},\quad\beta=\dfrac{\delta}{2C_{\nu}},\quad\varepsilon=\dfrac{1}{\delta^{5}}\left(e^{-\delta
r/C_{\nu}}+\epsilon\right).$ (3.56)

So, we deduce that

$\big{|}\sigma(H_{1},\lambda))-\sigma(H_{2},\lambda)\big{|}\leq\dfrac{C_{\nu}}{\delta^{3+6}}\left(e^{\frac{\delta
r}{2C_{\nu}}}+\dfrac{1}{\delta^{5/2}}(e^{-\frac{\delta
r}{C_{\nu}}}+\epsilon)^{1/2}\right)\leq\dfrac{C_{\nu}}{\delta^{12}}\left(e^{-\frac{\delta
r}{2C_{\nu}}}+\epsilon^{1/2}\right).$ (3.57)

This completes the proof of Proposition 1. ∎

##  4\. Violation of the bulk-edge correspondence in strips

In this section, we show that topological insulators lying within strips do
not necessarily support edge states along their boundary; this means that
geometrically, $\Omega$ needs to be unbounded in all directions for the bulk
spectral gaps to systematically fill.

Specifically, for any $L>0$, we construct an edge operator $H_{e}$ satisfying
Assumption 2 with:

  * •

The bulk operators $H_{\pm}$ are insulating at energy $0$, with bulk
conductance $\mp 1$;

  * •

$\Omega\subset{\mathbb{Z}}\times[-L,L]$, in particular
${\operatorname{Fr}}(\Omega)\leq L$;

  * •

$0\notin\Sigma(H_{e})$: the bulk gap did not fully close.

Hence, although the bulk operators $H_{\pm}$ represent topologically distinct
topological phases, the interface $\partial\Omega$ does not support conducting
states for $H_{e}$. In particular, a material made of topologically distinct
insulators across $\partial\Omega$, $\Omega={\mathbb{N}}\times[-L,L]$,
violates the bulk-edge correspondence. This was suspected by G.M. Graf, but
the problem was left open in an online talk by G.C. Thiang [GT20].

###  4.1. Haldane model

Our bulk operators are based on Haldane’s model [H88], which we review
briefly. For each site of the honeycomb lattice, the Haldane Hamiltonian
models tunneling to the three nearest neighbors and complex coupling (for
instance induced by a periodic magnetic field) to the six second-nearest
neighbors, see Figure 4. We will use a version based on the
${\mathbb{Z}}^{2}$-lattice (which only differ from the standard honeycomb
version by a linear change of variable):

$H_{\pm}=H_{0}\pm S$

where $H_{0},S$ are selfadjoint, short-range Hamiltonians on
$\ell^{2}({\mathbb{Z}}^{2},{\mathbb{C}}^{d})$ given by:

$\begin{split}&(H_{0}\psi)_{n}=\begin{bmatrix}\psi_{n}^{B}+\psi_{n-e_{1}}^{B}+\psi_{n-e_{2}}^{B}\\\
\psi_{n}^{A}+\psi_{n+e_{1}}^{A}+\psi_{n+e_{2}}^{A}\end{bmatrix},\\\
&(S\psi)_{n}=is\begin{bmatrix}\psi_{n+e_{1}}^{A}-\psi_{n-e_{1}}^{A}+\psi_{n-e_{2}}^{A}-\psi_{n+e_{2}}^{A}+\psi_{n+e_{2}-e_{1}}^{A}-\psi_{n+e_{1}-e_{2}}^{A}\\\
-\psi_{n+e_{1}}^{B}+\psi_{n-e_{1}}^{B}-\psi_{n-e_{2}}^{B}+\psi_{n+e_{2}}^{B}-\psi_{n+e_{2}-e_{1}}^{B}+\psi_{n+e_{1}-e_{2}}^{B}\end{bmatrix}.\end{split}$

The parameter $s$ above quantifies the ratio between first and second-nearest
neighbor coupling. We restrict it to $(0,1]$ here.

Figure 4. The black arrows represent tunneling to the three nearest neighbors;
the dashed blue arrows represent complex coupling to six second-nearest
neighbors in the Haldane model [H88].

As a result, the discrete Fourier transform w.r.t. ${\mathbb{Z}}^{2}$ is

$\begin{split}\hat{H}_{\pm}(\xi)&=\begin{bmatrix}\pm
2s\eta(\xi)&\overline{\omega(\xi)}\\\ \omega(\xi)&\mp
2s\eta(\xi)\end{bmatrix},\qquad\xi\in[-\pi,\pi]^{2},\\\
\omega(\xi)&:=1+e^{i\xi_{1}}+e^{i\xi_{2}},\\\
\eta(\xi)&:=\sin(\xi_{1})-\sin(\xi_{2})+\sin(\xi_{2}-\xi_{1}).\end{split}$

The eigenvalues of $\hat{H}_{\pm}(\xi)$ are $\pm\lambda(\xi)$ with

$\lambda(\xi)=\sqrt{(2s\eta(\xi))^{2}+|\omega(\xi)|^{2}}.$ (4.1)

The functions $\eta$ and $\omega$ do not vanish simultaneously; therefore
$\lambda$ never vanishes and the Hamiltonians $H_{\pm}$ are insulating at
energy $0$: $0\notin\Sigma(H_{\pm})$, equivalently $H_{\pm}$ are invertible.
Because of translational invariance, it is known their bulk conductance equals
the Chern number of their low energy eigenbundle (see e.g. [S15]*Equations
(13) and Corollary 8.4.4) and we have $\sigma(H_{\pm};0)=\mp 1$, see
[BH13]*§8. In particular, the insulators described by $H_{\pm}$ are
topologically distinct and $H_{\pm}$ satisfy Assumption 1.

###  4.2. Edge operator

Fix $L>0$ and $\Omega\subset[-L,L]\times{\mathbb{N}}$, we define the edge
operator by

$H_{e}:=H_{+}-2{\mathbb{1}}_{\Omega^{c}}S{\mathbb{1}}_{\Omega^{c}}.$

Then formally, $H_{e}$ is $H_{+}$ in $\Omega$ and $H_{-}$ in $\Omega^{c}$. We
prove Theorem 3, formulated here as:

######  Proposition 5.

$H_{e}$ satisfies Assumption 2 at energy $0$; however there exists a numerical
constant $\rho_{0}>0$ such that if $0<s\leq\rho_{0}L^{-1}$ then
$0\notin\Sigma(H_{e})$.

This implies that an interface lying in a strip, between two topologically
distinct insulating phases does not necessarily fill the bulk spectral gap.

######  Remark 2.

A general argument implies that the edge conductance of $H_{e}$ across the
$x_{2}$-axis is $0$. Indeed, this conductance is stable under perturbations
within strips orthogonal to the $x_{2}$-axis, such as
$2{\mathbb{1}}_{\Omega^{c}}S{\mathbb{1}}_{\Omega^{c}}$; so it is equal to that
of $H_{e}+2{\mathbb{1}}_{\Omega^{c}}S{\mathbb{1}}_{\Omega^{c}}=H_{+}$, which
is $0$.

To the best of our knowledge, there is no general argument that implies that
$H_{e}$ has edge conductance across the $x_{1}$-axis equal to $0$. For
$0<s\leq\rho_{0}L^{-1}$, it is a consequence of Proposition 5: $H_{e}$ has no
state with energy near $0$. For $\Omega={\mathbb{N}}\times[-L,L]$, this
implies that no quantum particle may travel from one end of $\partial\Omega$
to the other with high probability.

Proposition 5 is a consequence of the uncertainty principle: a function
localized in frequency may not be localized in position. For $s$ small, the
Fourier transforms of $H_{\pm}$ have eigenvalues of order $1$, unless $\xi$ is
near the zeroes $\xi_{\pm}^{*}=\pm 2\pi/3(1,-1)$ of $\omega(\xi)$ – in which
case they are of order $s$. Therefore, a $O(s)$-perturbation (such as $S$) may
not close the gap unless it generates states for $H_{e}$ that are concentrated
in frequency near $\xi_{\pm}^{*}$. By (a taylored version of) the uncertainty
principle, such states may not be localized within a strip (such as $\Omega$).

###  4.3. Proof

We prove Proposition 5 here. We will need the following lemmas:

######  Lemma 4.1.

  1. (1)

There exists $\lambda_{0}>0$, such that

$\lambda(\xi)\geq\lambda_{0}\cdot
s,\quad\text{~{}for~{}all~{}}\quad\xi\in[-\pi,\pi]^{2},\quad s\in(0,1].$

  2. (2)

There exists $\mu_{0}>0$ such that

$|\omega(\xi)|\geq\mu_{0}\cdot
d(\xi,\\{\xi_{\pm}^{*}\\}),\quad\text{~{}for~{}all~{}}\quad\xi\in[-\pi,\pi]^{2}.$

######  Remark 3.

At the physical level these are well-known bounds; we taylor them here to our
needs. Part (1) means that $H_{\pm}$ have a spectral gap at energy $0$; part
(2) means that the Wallace Hamiltonian $H_{0}$ have a Dirac cone.

###### Proof of Lemma 4.1.

(1) From (4.1) and $s\in(0,1]$, we have

$\lambda(\xi)=\sqrt{(2s\eta(\xi))^{2}+|\omega(\xi)|^{2}}\geq
s\sqrt{(2\eta(\xi))^{2}+|\omega(\xi)|^{2}}.$ (4.2)

Moreover,

$\begin{split}\omega(\xi)=0\quad&\Leftrightarrow\quad\xi=\xi^{*}_{\pm}=\pm\dfrac{2\pi}{3}(1,-1)\quad\Rightarrow\quad\eta(\xi^{*}_{\pm})=\pm\frac{3\sqrt{3}}{2}.\end{split}$

Thus $\omega(\xi)$ and $\eta(\xi)$ cannot vanish simultaneously and
$\sqrt{(2\eta(\xi))^{2}+|\omega(\xi)|^{2}}$ never vanishes. By continuity,
$\sqrt{(2\eta(\xi))^{2}+|\omega(\xi)|^{2}}\geq\lambda_{0}$ for some
$\lambda_{0}>0$. This proves (1) by going back to (4.2).

(2) We first write down $\omega$ as a function valued in ${\mathbb{R}}^{2}$
instead of ${\mathbb{C}}$:

$\omega(\xi)=(1+\cos(\xi_{1})+\cos(\xi_{2}),\sin(\xi_{1})+\sin(\xi_{2})).$
(4.3)

With this notation,

$\nabla(\omega(\xi))=\begin{bmatrix}-\sin(\xi_{1})&-\sin(\xi_{2})\\\
\cos(\xi_{1})&\cos(\xi_{2})\end{bmatrix},\qquad\nabla(\omega(\xi^{*}_{\pm}))=\dfrac{1}{2}\begin{bmatrix}\mp\sqrt{3}&\pm\sqrt{3}\\\
-1&-1\end{bmatrix}.$

As a result, for any $u=(u_{1},u_{2})^{T}$,

$\begin{split}|\nabla\omega(\xi^{*}_{\pm})u|^{2}&=\frac{1}{4}[3(u_{1}-u_{2})^{2}+(u_{1}+u_{2})^{2}]=\frac{1}{4}(4u_{1}^{2}+4u_{2}^{2}-4u_{1}u_{2})\geq\frac{1}{2}|u|^{2}.\end{split}$

Assume for any $n$, there is $\xi_{n}\neq\xi_{\pm}^{*}\in[-\pi,\pi]^{2}$ such
that

$|\omega(\xi_{n})|\leq\frac{d(\xi_{n},\\{\xi_{\pm}^{*}\\})}{n}.$ (4.4)

By compactness of $[-\pi,\pi]^{2}$, there exists a subsequence of $\xi_{n}$
that converges to some $\xi_{\infty}$. From (4.4) and
$d(\xi_{n},\\{\xi_{\pm}^{*}\\})\leq 4\pi$, we deduce
$|\omega(\xi_{\infty})|=0$ hence $\xi_{\infty}$ is either $\xi^{*}_{+}$ or
$\xi_{-}^{*}$. As a result, as $n\rightarrow\infty$,

$\begin{split}\frac{1}{n}\geq\frac{|\omega(\xi_{n})-\omega(\xi_{\infty})|}{|\xi_{n}-\xi_{\infty}|}&=\frac{|\nabla\omega(\xi_{\infty})(\xi_{n}-\xi_{\infty})|+O(|\xi_{n}-\xi_{\infty}|^{2})}{|\xi_{n}-\xi_{\infty}|}\\\
&\geq\frac{1}{\sqrt{2}}+O(|\xi_{n}-\xi_{\infty}|)\to\frac{1}{\sqrt{2}}.\end{split}$

We get a contradiction. Thus there is some $\mu_{0}>0$ such that
$|\omega(\xi)|\geq\mu_{0}|\xi-\xi_{\pm}^{*}|$.∎

######  Lemma 4.2.

There exists $C_{0}>0$ such that for all $s\in(0,1]$, $L>0$ and
$u\in\ell^{2}({\mathbb{Z}}^{2},{\mathbb{C}}^{2})$:

$\operatorname{Supp}u\subset{\mathbb{Z}}\times[-L,L]\qquad\Rightarrow\qquad\|H_{+}^{-1}u\|_{\ell^{2}}\leq
C_{0}L^{1/3}s^{-2/3}\|u\|_{\ell^{2}}.$

###### Proof of Lemma 4.2.

Recall that $\omega^{-1}(0)=\\{\xi^{*}_{+},\xi^{*}_{-}\\}$. Since eigenvalues
of $\hat{H}_{+}^{-1}(\xi)$ are $\pm\frac{1}{\lambda(\xi)}$, for any
$\delta>0$, we have

$\begin{split}\|H_{+}^{-1}u\|^{2}_{\ell^{2}({\mathbb{Z}}^{2})}&=\|\hat{H}_{+}^{-1}(\xi)\hat{u}(\xi)\|_{L^{2}([-\pi,\pi]^{2})}^{2}\\\
&\leq\int_{[-\pi,\pi]^{2}}\frac{1}{(\lambda(\xi))^{2}}|\hat{u}(\xi)|^{2}d\xi\\\
&\leq\int_{B_{\delta}(\xi^{*}_{\pm})}\frac{1}{\lambda_{0}^{2}s^{2}}|\hat{u}(\xi)|^{2}d\xi+\int_{\left(B_{\delta}(\xi^{*}_{\pm})\right)^{c}}\frac{1}{\mu_{0}^{2}\delta^{2}}|\hat{u}(\xi)|^{2}d\xi\\\
&\leq\int_{B_{\delta}(\xi^{*}_{\pm})}\frac{1}{\lambda_{0}^{2}s^{2}}|\hat{u}(\xi)|^{2}d\xi+\frac{(2\pi)^{2}}{\mu_{0}^{2}\delta^{2}}\|u\|_{\ell^{2}}^{2}\end{split}$
(4.5)

thanks to Plancherel’s formula
$\|\hat{u}\|_{L^{2}([-\pi,\pi]^{2})}=2\pi\|u\|_{\ell^{2}({\mathbb{Z}}^{2})}$.
If $\operatorname{Supp}(u)\subset{\mathbb{Z}}\times[-L,L]$, then

$\hat{u}(\xi_{1},n_{2})=\sum\limits_{n_{1}}e^{-n_{1}\xi_{1}}u(n_{1},n_{2})=0,\text{~{}if~{}}n_{2}\notin[-L,L].$

Thus we have

$\begin{split}\int_{B_{\delta}(\xi^{*}_{\pm})}\frac{1}{\lambda_{0}^{2}s^{2}}|\hat{u}(\xi)|^{2}d\xi&\leq\frac{1}{\lambda_{0}^{2}}\int_{B_{\delta}(\xi^{*}_{\pm})}\left|\sum\limits_{n_{2}\in[-L,L]}e^{-in_{2}\xi_{2}}\hat{u}(\xi_{1},n_{2})\right|^{2}d\xi\\\
&\leq\frac{1}{\lambda_{0}^{2}s^{2}}\int_{B_{\delta}(\xi^{*}_{\pm})}2L\sum\limits_{n_{2}\in[-L,L]}|\hat{u}(\xi_{1},n_{2})|^{2}d\xi\\\
&\leq\frac{2L}{\lambda_{0}^{2}s^{2}}\int_{|\xi_{2}-(\xi^{*}_{\pm})_{2}|\leq\delta}\int_{-\pi}^{\pi}\sum\limits_{n_{2}}|\hat{u}(\xi_{1},n_{2})|^{2}d\xi_{1}d\xi_{2}\\\
&=\frac{2L\cdot
4\delta}{\lambda_{0}^{2}s^{2}}\int_{-\pi}^{\pi}\sum\limits_{n_{2}}|\hat{u}(\xi_{1},n_{2})|^{2}d\xi_{1}=\frac{8\delta
L\cdot 2\pi}{\lambda_{0}^{2}s^{2}}\|u\|_{\ell^{2}}\end{split}$

where we use the Plancherel’s formula on $n_{1}$-coordinates only for the last
line. Combining with the earlier estimates, we get

$\|H_{+}^{-1}u\|_{\ell^{2}({\mathbb{Z}}^{2})}^{2}\leq\left(\frac{16\pi\delta
L}{\lambda_{0}^{2}s^{2}}+\frac{4\pi^{2}}{\mu_{0}^{2}\delta^{2}}\right)\|u\|_{\ell^{2}({\mathbb{Z}}^{2})}^{2}.$

In particular, taking
$\delta=\left(\frac{\pi\lambda_{0}^{2}s^{2}}{4\mu_{0}L}\right)^{\frac{1}{3}}$,
we get

$\|H_{+}^{-1}u\|_{\ell^{2}({\mathbb{Z}}^{2})}^{2}\leq
C_{0}^{2}\left(\frac{L}{s^{2}}\right)^{\frac{2}{3}}\|u\|_{\ell^{2}}^{2},\qquad
C_{0}=2^{\frac{13}{6}}\pi^{\frac{2}{3}}\mu_{0}^{-\frac{2}{3}}\lambda_{0}^{-\frac{2}{3}}.$

This completes the proof. ∎

###### Proof of Proposition 5.

(1) We have

$\begin{split}H_{e}&={\mathbb{1}}_{\Omega}H_{+}{\mathbb{1}}_{\Omega}+{\mathbb{1}}_{\Omega^{c}}(H_{+}-2S){\mathbb{1}}_{\Omega^{c}}+{\mathbb{1}}_{\Omega}H_{+}{\mathbb{1}}_{\Omega^{c}}+{\mathbb{1}}_{\Omega^{c}}H_{+}{\mathbb{1}}_{\Omega}\\\
&={\mathbb{1}}_{\Omega}H_{+}{\mathbb{1}}_{\Omega}+{\mathbb{1}}_{\Omega^{c}}H_{-}{\mathbb{1}}_{\Omega^{c}}+{\mathbb{1}}_{\Omega}H_{+}{\mathbb{1}}_{\Omega^{c}}+{\mathbb{1}}_{\Omega^{c}}H_{+}{\mathbb{1}}_{\Omega}.\end{split}$
(4.6)

By (4.6),

$E=H_{e}-{\mathbb{1}}_{\Omega}H_{+}{\mathbb{1}}_{\Omega}+{\mathbb{1}}_{\Omega^{c}}H_{-}{\mathbb{1}}_{\Omega^{c}}={\mathbb{1}}_{\Omega}H_{+}{\mathbb{1}}_{\Omega^{c}}+{\mathbb{1}}_{\Omega^{c}}H_{+}{\mathbb{1}}_{\Omega}.$

Since
${\mathbb{1}}_{\Omega}H_{+}{\mathbb{1}}_{\Omega^{c}}(x,y)={\mathbb{1}}_{\Omega}(x)H_{+}(x,y){\mathbb{1}}_{\Omega^{c}}(y)=0$
if $|x-y|>2$, we have $E$ satisfies (1.8); thus, $H_{e}$ satisfies Assumption
2.

(2) Recall that since $\lambda(\xi)$ never vanish, $H_{+}$ is invertible. Thus

$H_{e}=H_{+}-2{\mathbb{1}}_{\Omega^{c}}S{\mathbb{1}}_{\Omega^{c}}\quad\Leftrightarrow\quad
H_{+}^{-1}H_{e}=\operatorname{Id}-2H_{+}^{-1}{\mathbb{1}}_{\Omega^{c}}S{\mathbb{1}}_{\Omega^{c}}.$

To show $H_{e}$ is invertible, it is enough to show
$\|2H_{+}^{-1}{\mathbb{1}}_{\Omega^{c}}S{\mathbb{1}}_{{\Omega^{c}}}\|<1$.
Since $\|S\|\leq 6s$,

$\begin{split}\|2H_{+}^{-1}{\mathbb{1}}_{{\Omega^{c}}}S{\mathbb{1}}_{\Omega^{c}}u\|_{\ell^{2}}&\leq
C_{0}L^{1/3}s^{-2/3}\|{\mathbb{1}}_{{\Omega^{c}}}S{\mathbb{1}}_{\Omega^{c}}u\|_{\ell^{2}}\\\
&\leq 6C_{0}L^{\frac{1}{3}}s^{\frac{1}{3}}\|u\|_{\ell^{2}}.\end{split}$ (4.7)

Thus when $s<\rho_{0}L^{-1}$, $\rho_{0}=6^{-3}C_{0}^{-3}$, we have
$\|2H_{+}^{-1}{\mathbb{1}}_{{\Omega^{c}}}S{\mathbb{1}}_{\Omega^{c}}\|<1$. Thus
$H_{e}$ is invertible.∎

######  Remark 4.

Numerics actually yield the values

$\lambda_{0}=1,\qquad\mu_{0}=\dfrac{3}{\pi\sqrt{26}}\simeq 0.18,\qquad
C_{0}\simeq 31,\qquad s<1.5\cdot 10^{-7}L^{-1}.$ (4.8)

That is, if the second-nearest neighbor hopping is much smaller than the
first-nearest neighbor hopping (depending on $L$), then a topological
insulator fitting in a strip of width $L$ may not have edge spectrum.

##  Appendix A proof of some estimates

###### Proof of (3.1).

Fix $a\in(0,1]$. Then:

$\sum_{s\in{\mathbb{Z}}}e^{-2a|s|}=\dfrac{2}{1-e^{-2a}}-1=\dfrac{1+e^{-2a}}{1-e^{-2a}}=\dfrac{1}{\tanh(a)}\leq\frac{2}{a}$
(A.1)

where the last inequality follows from the fact that $\tanh(x)$ is concave
when $x>0$; thus $\tanh(a)\geq\tanh(1)a\geq a/2$ for $a\in(0,1]$. This yields
the first inequality in (3.1). The second inequality follows immediately since
$e^{-2a|{\bm{x}}|}=e^{-2a|x_{1}|}e^{-2a|x_{2}|}$.

Now fix $r\geq 0$. If $|{\bm{x}}|\geq r$, then either $|x_{1}|\geq r/2$ or
$|x_{2}|\geq r/2$. This induces a splitting in two mutually summetric sums:

$\sum_{|{\bm{x}}|\geq r}e^{-2a|{\bm{x}}|}\leq 2\sum_{|x_{1}|\geq r/2,\
x_{2}\in{\mathbb{Z}}}e^{-2a|{\bm{x}}|}\leq
2\left(\dfrac{2}{a}\right)^{2}e^{-ar}.$ (A.2)

This completes the proof. ∎

###### Proof of (3.2).

We apply Schur’s test. For a selfadjoint operator, it reads:

$\displaystyle\|H\|\leq\sup_{{\bm{x}}\in{\mathbb{Z}}^{2}}\sum_{{\bm{y}}\in{\mathbb{Z}}^{2}}\big{|}H({\bm{x}},{\bm{y}})\big{|}$
$\displaystyle\leq\dfrac{1}{\nu}\sup_{{\bm{x}}\in{\mathbb{Z}}^{2}}\sum_{{\bm{y}}\in{\mathbb{Z}}^{2}}e^{-2\nu|{\bm{x}}-{\bm{y}}|}$
(A.3)
$\displaystyle=\dfrac{1}{\nu}\sum_{{\bm{y}}\in{\mathbb{Z}}^{2}}e^{-2\nu|{\bm{y}}|}=\dfrac{1}{\nu}\left(\sum_{y_{1}\in{\mathbb{Z}}}e^{-2\nu|y_{1}|}\right)^{2}\leq\dfrac{4}{\nu^{3}}.$
(A.4)

In the last inequality, we used the first inequality in (3.1), which is valid
since $\nu\in(0,1]$. ∎

###### Proof of (3.4).

By the slope inequality for convex functions $f(x)=e^{xs}$ with any $s\geq 0$,
we have for $0<\alpha\leq\nu$:

$\dfrac{e^{\alpha s}-1}{\alpha}\leq\dfrac{e^{\nu s}-1}{\nu}.$ (A.5)

By applying this inequality to $s=|{\bm{x}}-{\bm{y}}|$, we deduce that
$\frac{S_{\alpha}}{\alpha}\leq\frac{S_{\nu}}{\nu}$. We now estimate $S_{\nu}$.
We have

$\displaystyle
S_{\nu}=\sup_{{\bm{x}}\in{\mathbb{Z}}^{2}}\sum_{{\bm{y}}\in{\mathbb{Z}}^{2}}|H({\bm{x}},{\bm{y}})|\left(e^{\nu|{\bm{x}}-{\bm{y}}|}-1\right)$
$\displaystyle\leq\dfrac{1}{\nu}\sup_{{\bm{x}}\in{\mathbb{Z}}^{2}}\sum_{{\bm{y}}\in{\mathbb{Z}}^{2}}e^{-2\nu|{\bm{x}}-{\bm{y}}|}\left(e^{\nu|{\bm{x}}-{\bm{y}}|}-1\right)$
(A.6)
$\displaystyle\leq\dfrac{1}{\nu}\sum_{{\bm{y}}\in{\mathbb{Z}}^{2}}e^{-\nu|{\bm{y}}|}\leq\dfrac{16}{\nu^{3}},$
(A.7)

where we used (3.1) again. ∎

###### Proof of (3.30).

1\. We first note that we have, by $|t-s|+|s|\geq|t|$ and (3.1):

$\sum_{s\in{\mathbb{Z}}}e^{-2\beta|t-s|-2\beta|s|}\leq
e^{-\beta|t|}\sum_{s\in{\mathbb{Z}}}e^{-\beta|t-s|-\beta|s|}\leq
e^{-\beta|t|}\sum_{s\in{\mathbb{Z}}}e^{-\beta|s|}\leq\frac{4e^{-\beta|t|}}{\beta}.$
(A.8)

2\. We now control $S$, the sum in the LHS of (3.30). To this end, we apply
(A.8) four times: first to $(t,s)=(x_{1},y_{1})$, then $(w_{1},z_{1})$, then
$(y_{2},z_{2})$ and finally $(x_{2},y_{2})$. This gives:

$\displaystyle S$
$\displaystyle\leq\dfrac{4}{\beta}e^{-\beta|x_{1}|-2\beta|w_{2}|}\sum_{y_{2},{\bm{z}}}e^{-2\beta|x_{2}-y_{2}|-2\beta|y_{2}-z_{2}|-2\beta|z_{1}|-2\beta|z_{1}-w_{1}|-2\beta|z_{2}|}$
(A.9)
$\displaystyle\leq\left(\dfrac{4}{\beta}\right)^{2}e^{-\beta|x_{1}|-\beta|w_{1}|-\beta|w_{2}|}\sum_{y_{2},z_{2}}e^{-2\beta|x_{2}-y_{2}|-2\beta|y_{2}-z_{2}|-2\beta|z_{2}|}$
(A.10)
$\displaystyle\leq\left(\dfrac{4}{\beta}\right)^{3}e^{-\beta|x_{1}|-\beta|{\bm{w}}|}\sum_{y_{2}}e^{-2\beta|x_{2}-y_{2}|-2\beta|y_{2}|}\leq\left(\dfrac{4}{\beta}\right)^{4}e^{-\beta|{\bm{x}}|-\beta|{\bm{w}}|}.$
(A.11)

This is (3.30). ∎

## References


∎

11institutetext: F. Author 22institutetext: Post and Telecommunication
Institute of Technology
22email<EMAIL_ADDRESS>
# Photonic Network Coding and Partial Protection for Optical Core Networks:
Two for a Tango

Dao Thanh Hai

(Received: date / Accepted: date)

###### Abstract

The age of acceleration is taking place, driven by the revolutionary digital
transformation creating basically a digital version of our physical world and
the currency in that digital space is data. Massive amount of data has been
generated ranging from wearable devices monitoring our physical health every
single millisecond to autonomous vehicles generating roughly 5Tb hourly to
even astronomical activities producing an order of Exabytes on daily basis and
then ultra-broadband Internet comes into play, moving such data to the cloud.
Internet traffic therefore has been experiencing explosive growth and in this
context, optical transport networks forming the backbone of the Internet are
pushed for transformation in system capacity. While the intuitive solution of
deploying multiple fibers can address the pressing demand for increased
capacity, doing so does not bring improvement in economic of scales in terms
of cost, power consumption and spectral efficiency. This necessitates for a
different approach so that the fiber capacity could be utilized in a more
efficient manner. In this paper, we focus on innovative techniques, that is,
photonic network coding and partial protection, to reduce the effective
traffic load in order to achieve greater capacity efficiency for optical
transport networks. Specifically, the application of network coding is
examined by upgrading the functionalities of intermediate nodes with all-
optical processing (i.e., encoding and decoding) capabilities. Besides,
partial protection relying on the premise of providing just enough bandwidth
in case of failure events is investigated for saving the redundant protection
capacity. That it takes two to tango, combining photonic network coding and
partial protection therefore bring to light new opportunities and challenges.
In mining such new avenue, we present insights on how to derive compounding
gains to maximize spectral efficiency via a case study.

###### Keywords:

Optical transport networks Elastic optical networks Partial protection Network
Coding fiber optics communication

##  1 Introduction

Data is here, data is there and data is all around you. Gone are the days when
data was on the scale of Megabytes and Gigabytes and today it is the era of
Terabytes and Petabytes. Indeed, thanks to the proliferation of super-cheap
computer chips and the ubiquitous wireless networks, a revolutionary process
is taking place, turning anything, from something as tiny as a pill to
something as sizable as an aeroplane, into a part of the Internet of Things
new3 . Such massive amount of data generated and circulated in Internet have
given the rise to explosive growth of Internet traffic. According to a recent
report from Cisco, Global Internet traffic will grow 3.2-fold from 2016 to
2021 with a compound annual growth rate of $26\%$ and the situation becomes
more severe for the busy hour Internet traffic whose growth will be 4.6-fold
from 2016 to 2021 Cisco20 . In this context, optical transport networks
forming the backbone of the Internet have been pushed for radical
transformation in system capacity to accommodate insatiable traffic demands in
a sustainable way ir4 .

For many years, the capacity of an optical fiber has been viewed as being
almost infinite and indeed, the single channel capacity has undergone leap-
and-bound growth with a spectacular rise from 2.5 Gb/s in around 1990 to
beyond 1 Tb/s in 2020 20years ; new1 . Such factor of 400-fold increase in a
span of roughly 30 years is thanks to convergence of remarkable advances in
electronic, photonic and digital signal processing technologies hai_iet ;
hai_wiley ; hai_nics ; EON . In accompany with system capacity expansion, it
has to be noted that there has been more than 3 orders of magnitude decrease
in cost per $Gb/sec\times km$, from roughly $1,000$ $\textdollar/(Gb/sec\times
km)$ down to less than $0.50$ $\textdollar/(Gb/sec\times km)$ 20years ; new4 .
However fiber throughputs are now coming close to the hard limitation
determined by the well-established nonlinear Shannon limit, so network
operators are exploring innovative ways to enlarge the number of parallel
optical paths by means of spatial-division multiplexing (SDM) new2 . There
have been some technological options for implementing SDM including lighting
up more fiber-pairs per link instead of single fiber-pair, or alternatively
making use of multi-core and/or few-mode fibers. While the first option is
intuitive, it does not result in any improvement on the cost and energy per
bit/sec. The second option is expected to boost the system capacity more than
an order of magnitude and yet a lot of challenges in implementation and
tremendous capital costs remain to be addressed. As an economical consequence,
such option is unlikely to be deployed in the near-term. Different from
technological approaches aiming at expanding the system capacity with major
investments, architectural strategies do not rely on the system capacity
increase to handle more traffic but rather, reduce the effective traffic load
so that more traffic can be carried sla1 ; qos1 .

Network Coding (NC) and Partial Protection (PP) have been emerging as
efficient architectural strategies to address the explosive traffic growth in
the near-term scale by means of better utilizing the fiber capacity
hai_comletter ; hai_oft ; hai_access ; hai_comcom ; hai_comcom2 ; hai_systems
; nc_others1 ; nc_others2 ; nc_others3 . Network Coding, originally proposed
in NC , has soon become a radical technique in networking to achieve higher
throughput, security and capacity. The main idea is to reduce the traffic load
in network by encoding signals at favorable conditions and by transmitting
such encoded signal rather sending individual ones, less capacity is required
hai_rtuwo ; hai_springer ; hai_springer2 . Partial protection, on the other
hand, has the potential of saving spectrum resources by offering just enough
protection capacity for demands instead of over-provisioning as in the
conventional approach sla2 ; qos2 . In this paper, we present, for the first
time, original perspectives on how Network Coding and Partial Protection could
generate remarkable spectral savings in optical networks realm. More
interesting is brought in the case when Network Coding meets Partial
Protection and how to optimize the such additive impact. Our analysis on this
unique situation could serve as a guideline to network operators to realize
extra benefits from combining Network Coding and Partial Protection.

The structure of the paper is organized as followed. In Sect. 2, related works
are briefly presented on the emerging of network coding and QoS-aware
protection for optical networks. Next, in Sect. 3 and Sect. 4, we showcase the
niche for leveraging photonic network coding and partial protection to achieve
greater capacity efficiency, respectively. Section 5 is dedicated to an
innovate case study where photonic network coding meets partial protection and
we provide a detailed analysis on how to derive compounding gain in such
unique cases.

##  2 Related Works

Network coding (NC) has been a paradigm shift in communication networks by
bringing the capability of in-network computing. The central idea is that
intermediate nodes, instead of simply storing and forwarding data as adopted
in traditional networking paradigm, is empowered to manipulate transiting data
and then forward such (non-) linearly combined data to its output. The
successes of network coding has been remarkable and therefore has been a de
facto in future wireless networks s7 . In fact, early attempts of exploiting
NC for optical networks have been proposed in s1 ; s2  for the scenario of
protection where multiple signals could be favorably encoded. Other than
protection realm, NC have been particularly well-suited for the multi-cast
transmission and this was the focus of the work in s5 ; s6 . It has to be
noted that the majority of existing works on NC for optical networks have been
focused on performing NC in electrical domain in combination with optical-
electrical-optical network architecture and this was due to the immature of
photonic signal processing in the past. In recent years, though, the rapid
advances in all-optical signal processing have renewed the interest of
applying NC in optical networks for achieving greater capacity efficiency with
a special focus on performing NC in photonic domain. The work in s3  has taken
advantage of NC in optical transport networks for provisioning cloud radio
access network services with greater capacity efficiency while maintaining
fast responses. The use of photonic NC has also been extended to security
services at physical layer with pioneering works in s4 . Applications of NC
have also been found in mm-wave radio-over-fiber networks nc_others1 , in
visible light communications nc_others2 ; nc_others3 , and in passive optical
networks (PON) nc_others4 . Data-center networking has been an active area for
applying NC to reduce traffic nc_others5 ; nc_others6 ; nc_others8 .

Partial protection has been driven by the observation that roughly $30\%$
traffic traveling in optical fiber belongs to premium class while $70\%$
remaining are best-effort type and hence can be temporarily discarded on
occasion of failures sla1 ; sla2 ; sla3 . This means that instead of providing
full protection, just some part of the traffic could be delivered to receivers
upon failure events and by reducing the amount of protection requirement, less
spare resources might be needed, permitting saving of both operational and
capital expenditures. Partial protection had been raised in the context of
traditional wavelength-division multiplexing (WDM) networks with pioneering
works from pp1 ; pp2 ; pp3 . However, due to the fixed nature of WDM networks,
there has been little room for further improvement with partial protection and
consequently the concept of partial protection had been somehow faded. The
development of elastic optical networks (EONs) with adaptive resource
allocation according to specific traffic requirement and/or transmission
qualities opens up therefore new opportunities for a re-consideration of
partial protection In fact, a closely related strategy is called, quality-of-
service provisioning, in EONs have been studied in pp5 ; pp6 ; pp7 ; pp8 .
Recent works in qos1 ; qos2  have been pushing forward the applicability of
partial protection in EONs by presenting an optimal mathematical framework to
maximize partial protection benefits.

It is important to note that both network coding and partial protection
techniques belong to the architectural approaches to address the continued
explosive traffic in optical core networks. It means that rather than
expanding system capacity, such architectural approaches aim at reducing
effective traffic load in the network so as to save spectral capacity for
future traffic. To the best of our survey, existing works in the literature
have so far addressed the use of each technique separately and this
necessitates for a question of whether partial protection could be jointly
applied with network coding to achieve additive impact. We aim to address this
question in this paper.

Our original contribution is indeed a proposal for combining photonic network
coding and partial protection in elastic optical core networks based on
flexible channel spacing and how such combination can result in greater
capacity efficiency than making use of individual technique. Furthermore, we
pinpoint an interesting issue on how to maximize the combination impact, that
is, the selection of protection level for each demand so that on one hand, the
overall partial protection specification is met and on the other hand, the
coding opportunity is optimized.

##  3 Network Coding for Greater Capacity Efficiency

In this part, we showcase a niche of applying NC in optical transport networks
that could bring about major spectrum savings by reducing the effective
traffic load in the network. Let’s consider an elastic optical network shown
in Fig. 1 and assume that there are two traffic demands with different bit-
rates, from node $A$ to node $Z$ and from node $B$ to node $Z$ with 100 Gbps
and 50 Gbps respectively. We also assume the modulation format to be used in
entire network is 16-QAM with polarization multiplexing and the standard
spectrum slice width of 6.25 GHz. Having such assumptions, provisioning those
demands with dedicated protection then involves finding a pair of link-
disjointed route and allocate spectrum slices along such routes for each
demand as illustrated in Fig. 1. It can be seen that due to the non-overlap
spectrum constraint, three spectrum slices are needed along the link $XE$ and
$EZ$. This is the traditional approach without network coding operations.

Figure 1: Traditional Approach without Network Coding Functions

Now let’s turn to the situation when node $X$ is armed with encoding
capability. In particular, supposing that at node $X$, two protection traffic
$A_{p}$ and $B_{p}$ could be XOR-encoded and the encoded signal $A_{p}\oplus
B_{p}$ operates at 100 Gbps, occupies two spectrum slots and goes all the way
to the receiving node $Z$. It is noticed that by doing so one spectrum slice
is saved along the route $XE$ and $EZ$, making up roughly $30\%$ spectral
savings. At the destination node $Z$, three signals are received including
$A$, $B$ and $A_{p}\oplus B_{p}$ and if one signal is lost due to a failure,
it can be recovered (almost) immediately from the other signals that are
received. The encoding and decoding process is clearly shown in Fig. 2. The
underlying idea behind this trick is that by combining signals at specific
nodes, rather than duplicating the signals end-to-end, network resources may
be used more efficiently.

Figure 2: Network Coding-enabled Approach

##  4 Partial Protection for Greater Capacity Efficiency

Partial protection is inspired by the observation that the premium traffic
accounts for roughly $30\%$ of the whole traffic traveling in optical fiber
links today whereas the remaining is best-effort. While the premium type
demands rapid recovery upon failures, the best-effort one could be dropped and
restored after a few tens of seconds without incurring any penalties.
Nevertheless, conventional approach treats both traffic types with the same
level of 1+1 expensive fast protection and this opens up new opportunities to
offer protection bandwidth in a more granular manner via a concept called
partial protection. Besides, partial protection is also backed by the case
that a service can accept some degradation (i.e., decreased rate) upon a
failure in exchange for a reduced cost and network operators, for the sake of
optimizing network resources and reduce cost, may wish to support degraded
services.

A case for illustration of applying partial protection is examined in this
part. In the traditional full protection shown in Fig. 3, four spectrum slices
are needed along the protection route of that 100 Gbps demand from node A to
node B. However, if we take into account the fact that such 100 Gbps traffic
consists of two types, namely, premium and best-effort one, new opportunities
for spectrum saving arise. For example, in Fig. 4, assuming that there is only
half traffic of premium type and thus, needs to guarantee rapid recovery upon
failures while the rest can be discarded temporarily. Doing so therefore saves
the protection bandwidth of two spectrum slices along the protection route,
offering a remarkable $50\%$ spectral savings. Clearly the amount of saving is
dependent to the amount of protected traffic and the less protected traffic
is, the more saving it could be achieved.

Figure 3: Traditional Full Protection Figure 4: Partial Protection Approach

##  5 When Network Coding meets Partial Protection: An Interesting Game Arises

As highlighted in above sections about the capability of network coding and
partial protection in reducing the effective traffic load in the network and
therefore utilizing better fiber capacity, the next step pinpoints to an
interesting situation if both network coding and partial protection is
applied. In such cases, we are interested in a guideline so that the optimal
spectral gain could be achieved and it will be tackled in this part.

Consider the scenario as shown in Fig. 5 for an elastic optical network
adopting the modulation format QPSK (polarization multiplexing) for the whole
network and a standard spectrum slice width of 6.25 GHz. Assuming that there
are two traffic demands from node $A$ and node $B$ to node $Z$ with 100 Gbps
and 150 Gbps respectively. With such assumption, four spectrum slices are
occupied by the first demand while the second demand requires six slices
(i.e., each spectrum slice carries 25 Gbps). For this scenario, in addition to
the opportunities of reducing protection traffic by means of partial
protection, the fact that $A_{p}$ and $B_{p}$ shares the same link $XZ$ paves
the way also for exploiting network coding. Table I draws a performance
comparison measured by number of spectrum slices on link $XZ$ if network
coding and partial protection is separately applied. It can be observed that
applying network coding alone brings about $40\%$ spectral savings while for
applying partial protection, the saving is dependent on the amount of
protected traffic. For a total of 250 Gbps traffic occupying 10 spectrum
slices, each $10\%$ reduction in protection traffic is translated to one
spectrum slice saving and it is detailed in Table 1.

Figure 5: An Exemplary Case to Combine Network Coding and Partial Protection Table 1: Performance Comparison when Applying Network Coding and Partial Protection Separately Scenario |  $N^{o}$ spectrum slices on link $XZ$
---|---
Full protection w/o NC | 10
Full protection with NC | 6
Partial Protection ($90\%$) w/o NC | 9
Partial Protection ($80\%$) w/o NC | 8
…
Partial Protection ($10\%$) w/o NC | 1

Let us now focus on the joint application of network coding and partial
protection. Of the total 250 Gbps traffic by two demands, if partial
protection is applied and the choice then arises on how much reduction is
applied on the first demand (100 Gbps) and how much the reduction is applied
to the second demand (150 Gbps) so as to optimally profit from network coding
operation. For convenience, we assume the reduction in traffic occurs at a
step of 25 Gbps (i.e., corresponds to one spectrum slice and $10\%$ total
traffic).

Basically, the process determining the optimal protection traffic for each
demand consists of two steps. The first step is to list all possible
configurations of providing protection level for each demand so that the
overall protection traffic is met. Next, for each configuration, the second
step is to perform network coding for protected traffic of each demand and
determine the amount of needed spectrum slices. After performing encoding for
all possible configurations, an optimal one is identified in terms of
minimizing number of used spectrum slices.

For illustration, consider the $10\%$ degradation case ($90\%$ protected), we
have two options, that is, either, that $10\%$ is reduced on first demand
($A_{p}$) or on the second demand ($B_{p}$). Since the second demand carries
higher bit-rate and consequently occupies larger amount of slices than the
first demand, the outcome signal of encoding second demand and first demand
will occupy the same amount of slices as with the second demand. This
observation suggests that if $10\%$ is reduced by the second demand, the
number of spectrum slices needed on link XZ after encoding will be five.
Otherwise, if that $10\%$ is reduced by the first demand, the number of
spectrum slices needed on link XZ after encoding will be still six. By same
way of reasoning, Table 2 provides the optimal configuration with respect to
relative amount of protection traffic and the corresponding number of slices
on link XZ after performing network coding.

Table 2: Performance Comparison when Network Coding is Optimally Combined with Partial Protection Scenario | Configuration |  $N^{o}$ slices
---|---|---
$90\%$ | 100 + 125 | 5
$80\%$ | 100 + 100 | 4
$70\%$ | 100 + 75 (75 + 100) | 4
$60\%$ | 75 + 75 | 3
$50\%$ | 75 + 50 (50 + 75) | 3
$40\%$ | 50 + 50 | 2
$30\%$ | 50 + 25 (25 + 50) | 2
$20\%$ | 25 + 25 | 1
$10\%$ | 25 + 0 (0 + 25) | 1

Such optimal configuration provided in Table 2 allows network operators to
decide the best amount of reduction for each demand so as to achieve the
maximal benefit of network coding operations.

##  6 Summary

Inspired by the observation that optical transport networks have been under
critical pressures due to the explosive traffic growth in “data, data, data”
era, we have proposed architectural strategies including Network Coding and
Partial Protection to better utilize the fiber capacity and hence, achieve
greater capacity efficiency. This is possible thanks to the unique opportunity
enabled by Network Coding and Partial Protection to reduce the effective
traffic load in the network. We have pinpointed and analyzed the original
prospect when Network Coding meets Partial Protection and how to optimize such
additive impact. That analysis might play a guideline role for network
operators in operating infrastructure in a more spectrally efficient manner.

It has, nevertheless, be noted that our proposed solution does not aim to
replace long-term technological innovations of expanding system capacity, but
rather serve as consolatory and complementary solutions to mitigate the
approaching of capacity crunch. Our proposal should therefore be paid
attention to before jumping to next curve of technological paradigm such as
spatial-division multiplexing framework.

## Conflict of interest

The authors declare that they have no conflict of interest.

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