# Existence and uniqueness of solutions to Bogomol’nyi-Prased-Sommerfeld
equations on graphs

Yuanyang Hu  School of Mathematics and Statistics, Henan University, Kaifeng,
Henan 475004, P. R. China<EMAIL_ADDRESS>
(Date: February 6, 2020)

###### Abstract.

Let $G=(V,E)$ be a connected finite graph. We investigate two Bogomol’nyi-
Prased-Sommerfeld equations on $G$. We establish necessary and sufficient
conditions for the existence and uniqueness of solutions to the two BPS
equations.

Mathematics Subject Classification (2010) 35A01, 35R02.

Key words: BPS equations, finite graph, equation on graphs

##  1\. Introduction

Vortices involve momentous roles in multitudinous fields of theoretical
physics comprizing quantum Hall effect, condensed-matter physics, electroweak
theory, superconductivity theory, optics, and cosmology. Taubes established
the multiple vortex static solutions for the Abelian Higgs model for the first
time in [31, 32, 41]. After that, a large number of work related to vortex
equations has been accomplished; see, for example, [4, 15, 33, 35, 39] and the
references therein. Recently, Lin and Yang pursued a systematic research [22,
23] of the multiple vortex equations obtained in [7, 8, 9, 12, 27, 28, 29,
30]. They established a great deal of sharp existence and uniqueness theorems.
The approach they employ include a priori estimates, monotone iterations, and
a degree-theory argument, and constrained minimization. In [40], Yang and Lieb
presented a series of sharp existence and uniqueness theorems for the
solutions of some non-Abelian vortex equations derived in [5,6], which provide
an essential mechanism for linear confinement. Recently, Han [15] established
the existence of multipe vortex solutions for the BPS equations derived in
[26] from the theory of multi-intersection of D-branes. Chen and Yang [6]
proved two sharp existence theorems for the non-Abelian BPS vortex equations
arising in the supersymmetric $U(1)\times SU(N)$ gauge theory. In this paper,
we investigate the existence and uniqueness of the solutions to the two BPS
equations in [6, 15] on a connected finite graph.

During the recent time, the investigation on the equations on graphs has drawn
a following and considerable attention from scholars; see, for example, [3,
10, 11, 13, 16, 17, 18, 19, 20, 21, 34, 37, 36] and the references therein.
Grigor’yan, Lin and Yang [13] study the Kazdan-Warner equation

$\Delta u=c-he^{u}$

on graph. Huang, Lin and Yau [19] proved the existence of solutions to mean
field equations

$\Delta u+e^{u}=\rho\delta_{0}$

and

$\Delta u=\lambda e^{u}\left(e^{u}-1\right)+4\pi\sum_{j=1}^{M}\delta_{p_{j}}$

on graphs. Huang, wang and Yang study the Mean field equation

$\Delta
u+\rho\left(\frac{he^{u}}{\int_{M}he^{u}dx}-\frac{1}{|M|}\right)=4\pi\sum_{j=1}^{N}\alpha_{q_{j}}\left(\delta_{q_{j}}-\frac{1}{|M|}\right)$

and the relativistic Ablian Chern-Simons equations

$\left\\{\begin{array}[]{l}\Delta u_{1}=\lambda
e^{u_{2}}\left(e^{u_{1}}-1\right)+4\pi\sum_{j=1}^{k_{1}}\alpha_{j}\delta_{p_{j}},\\\
\Delta u_{2}=\lambda
e^{u_{1}}\left(e^{u_{2}}-1\right)+4\pi\sum_{j=1}^{k_{2}}\beta_{j}\delta_{q_{j}}\end{array}\right.$

on the finite connected graph. Lin and Wu [24] investigated blow-up
phenomenons for the nonlinear parabolic equation $u_{t}=\Delta u+f(u)$ on a
locally finite graph.

The paper is organized as follows. In Section 2, we introduce preliminaries
and then state our main results. Section 3 is devoted to the proof of Theorem
2.1. In section 4, we give the proof of Theorem 2.2.

##  2\.  Settings and Main Results

Let $G=(V,E)$ be a connected finite graph, where $V$ denotes the vetex set and
$E$ denotes the edge set. Inspired by the work of Huang-Lin-Yau [19], we study
two BPS equations on $G$. One is

$\Delta
u_{j}=e^{u_{j}}+\sum\limits_{i=1}^{l}\mathrm{e}^{u_{i}}-(l+1)+4\pi\sum_{s=1}^{N_{j}}\delta_{p_{j,s}},~{}j=1,2,\cdots,l$
(2.1)

$l$ is an integer, $N_{j}(j=1,\dots,l)$ are positive integers,
$P_{j,s}(j=1,\dots,l,s=1,\dots,N_{j})$ are arbitrarily chosen distinct
vertices on the graph, and $\delta_{p}$ is the Dirac mass at $p$. The other is

$\Delta
u_{i}=\sum_{j=1}^{N}a_{ij}\left(\mathrm{e}^{u_{j}}-v^{2}\right)+4\pi\sum_{s=1}^{n_{i}}\delta_{p_{i,s}}(x),\quad
i=1,\ldots,N,$ (2.2)

where

$a_{ij}=\frac{1}{N}\left(\frac{e^{2}}{2}-\frac{g^{2}}{2N}\right)+\delta_{ij}\frac{g^{2}}{2N},\quad
i,j=1,\ldots,N,$ (2.3)

$n_{i}(i=1,\dots,N)$ are positive integers, $v,e,g$ are constants and $N$ is a
positive integer.

Let $\mu:V\to(0,+\infty)$ be a finite measure, and
$|V|$=$\text{Vol}(V)=\sum\limits_{x\in V}\mu(x)$ be the volume of $V$. For
each edge $xy\in E$, we assume that its weight $w_{xy}>0$ and that
$w_{xy}=w_{yx}$. For any function $u:V\to\mathbb{R}$, the Laplacian of $u$ is
defined by

$\Delta u(x)=\frac{1}{\mu(x)}\sum_{y\sim x}w_{yx}(u(y)-u(x)),$ (2.4)

where $y\sim x$ means $xy\in E$. The gradient $\nabla$ of function $f$ is
defined by a vector

$\nabla
f(x):=\left(\left[f(y)-f(x)\right]\sqrt{\frac{w_{xy}}{2\mu(x)}}\right)_{y\in
V}.$

The associated gradient form reads

$\Gamma(u,v)(x)=\frac{1}{2\mu(x)}\sum_{y\sim x}w_{xy}(u(y)-u(x))(v(y)-v(x)).$
(2.5)

We denote the length of the gradient of $u$ by

$|\nabla u|(x)=\sqrt{\Gamma(u,u)(x)}=\left(\frac{1}{2\mu(x)}\sum_{y\sim
x}w_{xy}(u(y)-u(x))^{2}\right)^{1/2}.$

Denote, for any function $u:V\rightarrow\mathbb{R}$, an integral of $u$ on $V$
by $\int\limits_{V}ud\mu=\sum\limits_{x\in V}\mu(x)u(x)$.

Our main results can be stated as following:

######  Theorem 2.1.

Equations (2.1) admits a unique solution if and only if $\max\limits_{1\leq
j\leq l}\\{N_{j}\\}<\frac{(l+1)}{4\pi}|V|$.

######  Theorem 2.2.

Equations (2.2) admits a unique solution if and only if

$n_{i}<\frac{g^{2}v^{2}}{8\pi
N}|V|+\frac{1}{N}\left(1-\frac{1}{N}\left(\frac{g}{e}\right)^{2}\right)n,\quad
i=1,\ldots,N.$

As in [13], we define a sobolev space and a norm by

$H^{1}(V)=W^{1,2}(V)=\left\\{u:V\rightarrow\mathbb{R}:\int\limits_{V}\left(|\nabla
u|^{2}+u^{2}\right)d\mu<+\infty\right\\},$

and

$\|u\|_{H^{1}(V)}=\|u\|_{W^{1,2}(V)}=\left(\int\limits_{V}\left(|\nabla
u|^{2}+u^{2}\right)d\mu\right)^{1/2}.$

We next give the following Sobolev embedding and Poincare inequality which
will be used later in the paper.

######  Lemma 2.1.

([13, Lemma 5]) Let $G=(V,E)$ be a finite graph. The sobolev space
$W^{1,2}(V)$ is precompact. Namely, if ${u_{j}}$ is bounded in $W^{1,2}(V)$,
then there exists some $u\in W^{1,2}(V)$ such that up to a subsequence,
$u_{j}\to u$ in $W^{1,2}(V)$.

######  Lemma 2.2.

([13, Lemma 6]) Let $G=(V,E)$ be a finite graph. For all functions
$u:V\to\mathbb{R}$ with $\int\limits_{V}ud\mu=0$, there exists some constant
$C$ depending only on $G$ such that $\int\limits_{V}u^{2}d\mu\leq
C\int\limits_{V}|\nabla u|^{2}d\mu$.

##  3\. The proof of Theorem 2.1

Since $\int\limits_{V}4\pi\sum\limits_{s=1}^{N_{j}}\delta_{p_{j,s}}d\mu=4\pi
N_{j}$, $j=1,2,\cdots,l$, there exists $u_{j}^{0}$ such that

$\Delta u_{j}^{0}=4\pi\sum\limits_{s=1}^{N_{j}}\delta_{p_{j,s}}-\frac{4\pi
N_{j}}{|V|},~{}x\in V,~{}j=1,2,\cdots,l.$ (3.1)

It is well-known that $(u_{1}^{0},u_{2}^{0},\dots,u_{l}^{0})$ is the unique
solution of (3.1) by up to a constant vector $\mathbf{C}$. Set
$u_{j}=u_{j}^{0}+v_{j}$, $j=1,2,\cdots,l$. Then we know that

$\Delta
v_{j}=e^{u_{j}^{0}+v_{j}}+\sum\limits_{i=1}^{l}\mathrm{e}^{u^{0}_{i}+v_{i}}-(l+1)+\frac{4\pi
N_{j}}{|V|},~{}j=1,\cdots,l.$ (3.2)

The following lemma gives a necessary condition for $\eqref{11}$ to have a
solution.

######  Lemma 3.1.

If equations $\eqref{11}$ admits a solution, then $\max\limits_{1\leq j\leq
l}\\{N_{j}\\}<\frac{(l+1)}{4\pi}|V|$.

###### Proof.

Integering (3.2) on $V$, we deduce that

$\int\limits_{V}\mathrm{e}^{u_{j}^{0}+v_{j}}\mathrm{~{}d}\mu=|V|-4\pi
N_{j}+\frac{4\pi}{l+1}\sum_{i=1}^{l}N_{i}=:K_{j},\quad j=1,\ldots,l,$ (3.3)

and hence that $\max\limits_{1\leq j\leq l}\\{N_{j}\\}<\frac{(l+1)}{4\pi}|V|$.

We now finish the proof. ∎

Hereafter, we use boldfaced letters to denote column vectors.

The following lemma gives a sufficient condition for (2.1) to have a solution.

######  Lemma 3.2.

If $\max\limits_{1\leq j\leq l}\\{N_{j}\\}<\frac{(l+1)}{4\pi}|V|$, then (2.1)
has a solution.

###### Proof.

Rewrite (3.2), we have

$\Delta\textbf{v}=A\textbf{E}-\bm{\Phi},$ (3.4)

where $\Delta\textbf{v}=(\Delta v_{1},\cdots,\Delta v_{l})^{T}$,
$\textbf{E}=(e^{u_{1}^{0}+v_{1}},e^{u_{2}^{0}+v_{2}},\cdots,e^{u_{l}^{0}+v_{l}})^{T},$
$\bm{\Phi}=(l+1-\frac{4\pi N_{1}}{|V|},\cdots,l+1-\frac{4\pi N_{l}}{|V|})$ and

$A=(a_{ij})_{l\times l}=\left(\begin{array}[]{ccccc}2&1&1&\ldots&1\\\
1&2&1&\ldots&1\\\ 1&1&2&\ldots&1\\\ \vdots&\vdots&\vdots&\ddots&\vdots\\\
1&1&1&\ldots&2\end{array}\right).$ (3.5)

In order to find the energy functional of (3.2), we have the following
argument.

It is easy to see that $A=\textbf{B}\textbf{B}^{T}$, where
$B=(b_{ij})_{l\times l}$ and

$b_{ij}=\begin{cases}\sqrt{\frac{1+i}{i}},~{}i=j,\\\
\sqrt{\frac{1}{j(j+1)}},~{}i\not=j,~{}i>j,\\\ 0,~{}i\not=j,~{}i<j.\end{cases}$

Let

$\textbf{q}=B^{-1}\textbf{v},$ (3.6)

where $\textbf{v}=(v_{1},\cdots,v_{l})^{T}$ and $B^{-1}$ is the inverse of
matrix $B$. Then we know that

$\displaystyle\Delta q_{1}=$
$\displaystyle\sqrt{2}\exp\left(u_{1}^{0}+\sqrt{2}q_{1}\right)$ (3.7)
$\displaystyle+\frac{1}{\sqrt{2}}\sum_{i=2}^{l}\exp\left(u_{i}^{0}+\sum_{k=1}^{i-1}\frac{q_{k}}{\sqrt{k(k+1)}}+\sqrt{\frac{i+1}{i}}q_{i}\right)-g_{1},$
$\displaystyle\Delta q_{j}=$
$\displaystyle\sqrt{\frac{j+1}{j}}\exp\left(u_{j}^{0}+\sum_{k=1}^{j-1}\frac{q_{k}}{\sqrt{(k+1)k}}+\sqrt{\frac{j+1}{j}}q_{j}\right)$
(3.8)
$\displaystyle+\sqrt{\frac{1}{j(j+1)}}\sum_{i=j+1}^{l}\exp\left(u_{i}^{0}+\sum_{k=1}^{i-1}\frac{q_{k}}{\sqrt{k(k+1)}}+\sqrt{\frac{i+1}{i}}q_{i}\right)-g_{j},$
$\displaystyle j=1,\cdots,l,$ $\Delta
q_{l}=\sqrt{\frac{l+1}{l}}\exp\left(u_{l}^{0}+\sum_{k=1}^{l-1}\frac{q_{k}}{\sqrt{(k+1)k}}+\sqrt{\frac{l+1}{l}}q_{l}\right)-g_{l},$
(3.9)

where $\textbf{g}=(g_{1},\cdots,g_{l})^{T}=B^{-1}\bm{\Phi}$. Now we define the
energy functional

$\begin{gathered}I(\mathbf{q})=I\left(q_{1},\ldots,q_{l}\right)=\int\limits_{V}\left\\{\frac{1}{2}\sum_{i=1}^{l}\Gamma({q}_{i},{q}_{i})+\exp\left(u_{1}^{0}+\sqrt{2}q_{1}\right)-g_{1}q_{1}\right.\\\
\left.+\sum_{i=2}^{l}\exp\left(u_{i}^{0}+\sum_{k=1}^{i-1}\frac{q_{k}}{\sqrt{k(k+1)}}+\sqrt{\frac{i+1}{i}}q_{i}\right)-g_{i}q_{i}\right\\}\mathrm{d}\mu.\end{gathered}$
(3.10)

It is easy to check that if $I$ has a critical point, then it is a solution of
equations (3.7)-(3.9).

We next prove $I$ has a critical point.

Define $K:=(k_{1},\dots,k_{l})^{T}$. By (3.3), we have
$K=|V|(B^{T})^{-1}\mathbf{g}.$ For any $q\in H^{1}(V)$, we can write

$q=\bar{q}+\hat{q},$ (3.11)

where $\int\limits_{V}\hat{q}d\mu=0$ and
$\bar{q}=\frac{\int\limits_{V}qd\mu}{|V|}$.

For any $\textbf{q}\in\underbrace{H^{1}(V)\times\dots\times H^{1}(V)}_{l}$,
from (3.6), we conclude that

$\displaystyle I(\mathbf{q})$
$\displaystyle=\int\limits_{V}\frac{1}{2}\sum_{i=1}^{l}\Gamma(\hat{q}_{i},\hat{q}_{i})\mathrm{d}\mu+\int\limits_{V}\sum_{i=1}^{l}\exp\left(u_{0}+\hat{v}_{i}+\bar{v}_{i}\right)\mathrm{d}\mu-K^{T}\bar{\mathbf{v}}$
(3.12)
$\displaystyle=\int\limits_{V}\frac{1}{2}\sum_{i=1}^{l}\Gamma(\hat{q}_{i},\hat{q}_{i})\mathrm{d}\mu+\int\limits_{V}\sum_{i=1}^{l}\exp\left(u_{0}+\hat{v}_{i}+\bar{v}_{i}\right)\mathrm{d}\mu-\sum_{i=1}^{l}K_{i}\bar{v}_{i}.$

By Jensen’s inequality, we obtain

$exp\left(\frac{\int\limits_{V}u_{i}^{0}+\hat{v}_{i}+\bar{v}_{i}d\mu}{|V|}\right)\leq\frac{1}{|V|}\int\limits_{V}exp\left(u_{i}^{0}+\hat{v}_{i}+\bar{v}_{i}\right)d\mu.$
(3.13)

Thus we deduce that

$\int\limits_{V}exp\left(u_{i}^{0}+\hat{v}_{i}+\bar{v}_{i}\right)d\mu\geq|V|exp\left(\frac{1}{|V|}\int\limits_{V}u_{i}^{0}d\mu\right)e^{\bar{v}_{i}}=:c_{i}e^{\bar{v}_{i}}~{}\text{for}~{}i=1,\cdots,l.$
(3.14)

By virtue of $\max\limits_{1\leq j\leq l}\\{N_{j}\\}<\frac{(l+1)}{4\pi}|V|$,
we obtain $K_{i}>0$ for $i=1,\cdots,l$. Thus we deduce that

$\displaystyle I(\mathbf{q})$
$\displaystyle\geq\int_{V}\frac{1}{2}\sum\limits_{i=1}^{l}|\nabla\hat{q}_{i}|^{2}d\mu+\sum\limits_{i=1}^{l}(c_{i}e^{\bar{v}_{i}}-K_{i}\bar{v}_{i})$
(3.15)
$\displaystyle\geq\int_{V}\frac{1}{2}\sum\limits_{i=1}^{l}|\nabla\hat{q}_{i}|^{2}d\mu+\sum\limits_{j=1}^{l}K_{j}log{\frac{c_{j}}{K_{j}}}.$

It follows that $I$ is bounded from below in $H^{1}(V)$. Thus we can choose a
minimizing sequence $\\{q_{1,k},\dots,q_{l,k}\\}_{k=1}^{\infty}$ of the
following minimization problem

$\inf\limits\\{I(\mathbf{q})|\mathbf{q}=(q_{1},\dots,q_{l})\in\underbrace{H^{1}(V)\times\dots\times
H^{1}(V)}_{l}\\}.$

In view of $\lim\limits_{t\to\pm\infty}c_{i}e^{t}-K_{i}t=\pm\infty$ for
$i=1,\dots,l$. we deduce from (3.15) that $\\{\bar{v}_{i,k}\\}_{k=1}^{\infty}$
is a bounded for $i=1,\dots,l$. By (3.6), we know that
$\\{\bar{q}_{i,k}\\}_{k=1}^{\infty}$ is bounded for $i=1,\cdots,l$. From
(3.15), we see that $\\{\nabla\hat{q}_{i,k}\\}_{k=1}^{\infty}$ is bounded in
$L^{2}(V)$ for $i=1,\cdots,l.$ From Lemma 2.2, we conclude that
$\\{\hat{q}_{i,k}\\}_{k=1}^{\infty}$ is bounded in $L^{2}(V)$, $i=1,\cdots,l.$
Thus, $\\{q_{i,k}\\}_{k=1}^{\infty}$ is bounded in $L^{2}(V)$. Therefore,
$\\{q_{i,k}\\}_{k=1}^{\infty}$ is bounded in $H^{1}(V).$ Therefore, by Lemma
2.1, there exists
$\textbf{q}_{\infty}:=\left(q_{1,\infty},\dots,q_{l,\infty}\right)\in\underbrace{H^{1}(V)\times\dots\times
H^{1}(V)}_{l}$ so that, by passing to a subsequent,

$q_{i,k}\to q_{i,\infty}$ (3.16)

uniformly for $x\in V$ as $k\to+\infty$ for $i=1,\cdots,l$. Thus,
$\mathbf{q}_{\infty}$ is a critical point of $I$. Clearly, $I$ is srtictly
convex in $H^{1}(V)$. Thus, we know that the solution of equations (3.7)-(3.9)
is unique.

We now complete the proof. ∎

######  Lemma 3.3.

There exists at most one solution of equations (2.1).

###### Proof.

Suppose that $\mathbf{u}=(u_{1},\dots,u_{l})^{T}$ and
$\mathbf{v}=(v_{1},\dots,v_{l})^{T}$ are two solutions of the equations (2.1).
By mean value theorem, there exists $\bm{\xi}=(\xi_{1},\dots,\xi_{l})^{T}$
such that

$\Delta(\bm{u}-\bm{v})=AE(\bm{u}-\bm{v}),$

where $E=diag(e^{\xi_{1}},\dots,e^{\xi_{l}})$.

We now prove that $u_{1}\equiv v_{1}$ on $V$. Let
$M=\max\limits_{V}=(u_{1}-v_{1})(x_{0})$. We claim that $M\leq 0$. Otherwise,
$M>0$. Thus, we conclude that

$\Delta(u_{1}-v_{1})(x_{0})=[e^{\xi_{1}}(u_{1}-v_{1})](x_{0})>0.$ (3.17)

From (2.4), we obtain

$\Delta(u_{1}-v_{1})(x_{0})\leq 0.$

This is impossible. Therefore, we have $u_{1}\leq v_{1}$ on $V$. Thus, we see
that

$u_{1}\equiv v_{1}~{}\text{on}~{}V.$ (3.18)

By a similar argument, we can show that

$u_{i}\equiv v_{i}~{}\text{on}~{}V,$ (3.19)

for $i=2,\dots,l.$ Therefore, we conclude that $\mathbf{u}=\mathbf{v}$ on $V$.

The proof is finished. ∎

###### Proof of Theorem 2.1.

The desired conclusion follows directly from Lemmas 3.1, 3.2 and 3.3. ∎

##  4\. The proof of Theorem 2.2

Set $u_{i}^{0}$ be a solution of

$\Delta u_{i}^{0}=-\frac{4\pi
n_{i}}{|V|}+4\pi\sum_{s=1}^{n_{i}}\delta_{p_{i,s}}(x),\quad i=1,\ldots,N.$
(4.1)

Set

$u_{i}=u_{i}^{0}+U_{i},\quad i=1,\ldots,N.$ (4.2)

Then (2.2) is transformed into

$\Delta
U_{i}=\sum_{j=1}^{N}a_{ij}\left(\mathrm{e}^{u_{j}^{0}+U_{j}}-v^{2}\right)+\frac{4\pi
n_{i}}{|V|},\quad i=1,\ldots,N.$ (4.3)

We now write (4.3) in the vector form

$\Delta\mathbf{U}=\hat{A}\mathbf{G}+\mathbf{F},$ (4.4)

where

$\mathbf{U}=\left(U_{1},\ldots,U_{N}\right)^{T},\quad\mathbf{G}=\left(\mathrm{e}^{u_{1}^{0}+U_{1}},\ldots,\mathrm{e}^{u_{N}^{0}+U_{N}}\right)^{T},$
(4.5) $\mathbf{F}=\left(\frac{4\pi
n_{1}}{|V|}-v^{2}\sum_{j=1}^{N}a_{1j},\ldots,\frac{4\pi
n_{N}}{|V|}-v^{2}\sum_{j=1}^{N}a_{Nj}\right)^{T}=:\left(f_{1},\ldots,f_{N}\right)^{T},$
(4.6)

and $\hat{A}=\left(a_{ij}\right)_{N\times N}$ is a $N\times N$ matrix.
Applying the Cholesky decomposition theorem, we can write

$\hat{A}=T^{T}T,$ (4.7)

where $T=(t_{ij})_{N\times N}$ is an upper triangular matrix,

$\displaystyle t_{11}=\sqrt{a+b},\quad
t_{12}=t_{13}=\cdots=t_{1N}=\frac{a}{t_{11}}\equiv\alpha_{1}>0,$
$\displaystyle t_{22}=\sqrt{(a+b)-\alpha_{1}^{2}},\quad
t_{23}=t_{24}=\cdots=t_{2N}=\frac{a-\alpha_{1}^{2}}{t_{22}}\equiv\alpha_{2}>0,$
$\displaystyle\ldots$

$t_{(N-1)(N-1)}=\sqrt{(a+b)-\sum_{i=1}^{N-2}\alpha_{i}^{2}},\quad
t_{(N-1)N}=\frac{a-\sum_{i=1}^{N-2}\alpha_{i}^{2}}{t_{(N-1)(N-1)}}\equiv\alpha_{N-1}>0,$
$t_{NN}=\sqrt{(a+b)-\sum_{i=1}^{N-1}\alpha_{i}^{2}},$

$a=(1/N)\left(e^{2}/2-g^{2}/2N\right)$ and $b=g^{2}/2N$. Set
$\mathbf{v}=\left(v_{1},\ldots,v_{N}\right)^{T},L=\left(T^{T}\right)^{-1}=:\left(l_{ij}\right)_{N\times
N}$ and

$\mathbf{v}=L\mathbf{U}.$

By (4.4), we have

$\Delta\mathbf{v}=T\mathbf{G}+L\mathbf{F}.$ (4.8)

We rewrite (4.8) as

$\Delta
v_{i}=t_{ii}\mathrm{e}^{u_{i}^{0}+t_{ii}v_{i}+\sum_{k=1}^{i-1}\alpha_{k}v_{k}}+\alpha_{i}\sum_{k=1}^{N}\mathrm{e}^{u_{j}^{0}+t_{jj}v_{j}+\sum_{k=1}^{j-1}\alpha_{k}v_{k}}+\sum_{j=1}^{i}l_{ij}f_{j},\quad
i=1,\ldots,N.$ (4.9)

Define the energy functional

$J(\mathbf{v})=\int\limits_{V}\left\\{\frac{1}{2}\sum_{i=1}^{N}\Gamma(v_{i},v_{i})+\sum_{i=1}^{N}\mathrm{e}^{u_{i}^{0}+t_{ii}v_{i}+\sum_{k=1}^{i-1}\alpha_{k}v_{k}}+\sum_{i=1}^{N}\left(\sum_{j=1}^{i}l_{ij}f_{j}\right)v_{i}\right\\}\mathrm{d}\mu.$
(4.10)

The following lemma gives a necessary condition for (2.2) to admit a solution.

######  Lemma 4.1.

If (2.2) admits a solution, then

$n_{i}<\frac{g^{2}v^{2}}{8\pi
N}|V|+\frac{1}{N}\left(1-\frac{1}{N}\left[\frac{g}{e}\right]^{2}\right)n,\quad
i=1,\ldots,N.$ (4.11)

###### Proof.

Let

$q_{i}=\int_{V}\mathrm{e}^{u_{i}^{0}+t_{ii}v_{i}+\sum_{k=1}^{i-1}\alpha_{k}v_{k}}\mathrm{~{}d}\mu,\quad
i=1,\ldots,N$ (4.12)

From (4.9), we get

$t_{ii}q_{i}+\alpha_{i}\sum_{j=i+1}^{N}q_{j}=-|V|\sum_{j=1}^{i}l_{ij}f_{j}=:p_{i},\quad
i=1,\ldots,N$ (4.13)

By (4.13), we deduce that

$T\mathbf{q}=-|V|L\mathbf{F}=\mathbf{p},$ (4.14)

where $\mathbf{p}=(p_{1},\dots,p_{N})^{T}$ and
$\mathbf{q}=(q_{1},\dots,q_{N})^{T}$. It is easy to check that

$A^{-1}=\frac{1}{b(Na+b)}\left(\begin{array}[]{cccc}(N-1)a+b&-a&\cdots&-a\\\
-a&(N-1)a+b&\cdots&-a\\\ \cdots&\cdots&\cdots&\cdots\\\
-a&-a&\cdots&(N-1)a+b\end{array}\right).$ (4.15)

Therefore, we deduce that

$q_{i}=v^{2}|V|+\frac{4\pi
a}{b(Na+b)}\sum_{j=1}^{N}n_{i}-\frac{4\pi}{b}n_{i},\quad i=1,\ldots,N.$ (4.16)

Thus, we have

$q_{i}=v^{2}|V|+8\pi\left(\frac{1}{g^{2}}-\frac{1}{Ne^{2}}\right)n-\frac{8\pi
N}{g^{2}}n_{i}>0,\quad i=1,\ldots,N.$ (4.17)

We now complete the proof. ∎

We give a sufficient condition for equations (4.3) to have a solution in the
following lemma.

######  Lemma 4.2.

If

$n_{i}<\frac{g^{2}v^{2}}{8\pi
N}|V|+\frac{1}{N}\left(1-\frac{1}{N}\left(\frac{g}{e}\right)^{2}\right)n,\quad
i=1,\ldots,N,$ (4.18)

then (4.3) admits a solution.

###### Proof.

For any
$\mathbf{v}=(v_{1},\dots,v_{N})\in\underbrace{H^{1}(V)\times\dots\times
H^{1}(V)}_{N}$. By the notation (3.11) and Jensen’s inequality, we have

$\displaystyle\int_{V}\mathrm{e}^{u_{i}^{0}+t_{ii}\left(\bar{v}_{i}+\hat{v}_{i}\right)+\sum_{k=1}^{i-1}\alpha_{k}\left(\bar{v}_{k}+\hat{v}_{k}\right)}\mathrm{d}\mu$
$\displaystyle\geq|V|\exp\left(\frac{1}{|V|}\int_{V}u_{i}^{0}\mathrm{~{}d}\mu\right)\exp\left(t_{ii}\bar{{v}}_{i}+\sum_{k=1}^{i-1}\alpha_{k}\bar{{v}}_{k}\right)$
(4.19)
$\displaystyle=:\sigma_{i}\mathrm{e}^{t_{ii}\bar{v}_{i}+\sum_{k=1}^{i-1}\alpha_{k}\bar{v}_{k}},\quad
i=1,\ldots,N.$

Set

$\bar{w}_{i}=t_{ii}\bar{v}_{i}+\sum_{k=1}^{i-1}\alpha_{k}\bar{v}_{k},\quad
i=1,\ldots,N.$ (4.20)

From (4.14), we conclude that

$\displaystyle
J(\mathbf{v})-\frac{1}{2}\int\limits_{V}\sum_{i=1}^{N}\left|\nabla\hat{v}_{i}\right|^{2}\mathrm{~{}d}\mu$
$\displaystyle\geq\sum_{i=1}^{N}\left(\sigma_{i}\mathrm{e}^{\bar{w}_{i}}-q_{i}\bar{w}_{i}\right)$
(4.21)
$\displaystyle\geq\sum_{i=1}^{N}q_{i}\left(1+\ln\left[\frac{\sigma_{i}}{q_{i}}\right]\right).$

Considering the following minization problem

$\eta\equiv\inf\left\\{J(\mathbf{v})\mid\mathbf{v}\in\underbrace{H^{1}(V)\times\dots\times
H^{1}(V)}_{N}\right\\}.$ (4.22)

Let $\\{(v_{1,k},\dots,v_{N,k})\\}_{k=1}^{\infty}$ be a minimizing sequence of
(4.22), by a similar argument as the proof of Lemma 3.3, we can show that
there exists $\mathbf{v_{\infty}}:=(v_{1,\infty},\dots,v_{N,\infty})^{T}$ such
that, by passing to a subsequence,

$v_{i,k}\to v_{i,\infty}$ (4.23)

uniformly for $x\in~{}V$ as $k\to+\infty$ for $i=1,\dots,N.$ Thus,
$\mathbf{v}_{\infty}$ is a critical point of $J$. It’s easy to check that $J$
is strictly convex in $H^{1}(V)$. Thus, we know that the solution of equations
(4.9) is unique.

The proof is finished. ∎

The following lemma gives the uniqueness of solutions to equations (2.2)

######  Lemma 4.3.

There exists at most one solution of equations (2.1).

###### Proof.

Assume that $\mathbf{u}=(u_{1},\dots,u_{N})^{T}$ and
$\mathbf{v}=(v_{1},\dots,v_{N})^{T}$ are two solutions of the equations (2.2).
By mean value theorem, there exists $\bm{\xi}=(\xi_{1},\dots,\xi_{N})^{T}$
such that

$\Delta(\bm{u}-\bm{v})=\hat{A}E(\bm{u}-\bm{v}),$

where $E=diag(e^{\xi_{1}},\dots,e^{\xi_{N}})$.

We now prove that $u_{1}\equiv v_{1}$ on $V$. Let
$M=\max\limits_{V}=(u_{1}-v_{1})(x_{0})$. We claim that $M\leq 0$. Otherwise,
$M>0$. Thus, we conclude that

$\Delta(u_{1}-v_{1})(x_{0})=[e^{\xi_{1}}(u_{1}-v_{1})](x_{0})>0.$ (4.24)

From (2.4), we obtain

$\Delta(u_{1}-v_{1})(x_{0})\leq 0.$

This is impossible. Therefore, we have $u_{1}\leq v_{1}$ on $V$. Thus, we see
that

$u_{1}\equiv v_{1}~{}\text{on}~{}V.$ (4.25)

By a similar argument as above, we can show that

$u_{i}\equiv v_{i}~{}\text{on}~{}V,$ (4.26)

for $i=2,\dots,N$, which implies that $\mathbf{u}\equiv\mathbf{v}$ on $V$.

We now complete the proof. ∎

###### Proof of Theorem 2.2.

The desired conclusion follows directly from Lemmas 4.1, 4.2 and 4.3. ∎

Next, we give a constrained minization approach to the problem.

Denote

$I_{i}(\mathbf{v})=\int\limits_{V}\mathrm{e}^{u_{i}^{0}+t_{ii}v_{i}+\sum_{k=1}^{i-1}\alpha_{k}v_{k}}\mathrm{~{}d}\mu=q_{i},\quad
i=1,\ldots,N,$ (4.27)

We consider the following constrained minization problem

$\gamma=\inf\left\\{J(\mathbf{v})\mid\mathbf{v}\in\underbrace{H^{1}(V)\times\dots\times
H^{1}(V)}_{N},I_{1}(\mathbf{v})=q_{1},\ldots,I_{N}(\mathbf{v})=q_{N}\right\\}.$
(4.28)

We now investigate whether the constraints in (4.28) give rise to the so-
called ”constraints” problem due to the issue of the Lagrange multipliers. For
this purpose, let $\mathbf{v}=(v_{1},\dots,v_{N})^{T}$ be a critical point of
$J$ subject to the constraints

$I_{i}(\mathbf{v})=q_{N},~{}i=1,\dots,N.$ (4.29)

Then we can find real numbers $\sigma_{1},\dots,\sigma_{\textsc{N}}$ such that

$d_{i}J=\sum\limits_{j=1}^{N}\sigma_{j}d_{i}I_{j},~{}i=1,\dots,N,$ (4.30)

where $d_{i}(i=1,\dots,N)$ denote the Fr$\acute{e}$chet differention with
respect to the i-th arguments, respectively. It follows that, for any
$z_{1},\dots,z_{N}\in H^{1}(V),$

$\displaystyle\int\limits_{V}$
$\displaystyle\left\\{\Gamma(v_{i},z_{i})+\left(t_{ii}\mathrm{e}^{u_{i}^{0}+t_{ii}v_{i}+\sum\limits_{k=1}^{i-1}\alpha_{k}v_{k}}+\alpha_{i}\sum_{j=i+1}^{N}\mathrm{e}^{u_{j}^{0}+t_{jj}v_{j}+\sum\limits_{k=1}^{j-1}\alpha_{k}v_{k}}+\sum_{j=1}^{i}l_{ij}f_{j}\right)z_{i}\right\\}\mathrm{d}\mu$
(4.31)
$\displaystyle=\sigma_{i}t_{ii}\int\limits_{V}\mathrm{e}^{u_{i}^{0}+t_{ii}v_{i}+\sum\limits_{k=1}^{i-1}\alpha_{k}v_{k}}z_{i}\mathrm{~{}d}\mu+\alpha_{i}\sum\limits_{j=i+1}^{N}\sigma_{j}\int\limits_{V}\mathrm{e}^{u_{j}^{0}+t_{jj}v_{j}+\sum\limits_{k=1}^{j-1}\alpha_{k}v_{k}}z_{i}\mathrm{~{}d}\mu.$

Taking $z_{1},\dots,z_{N}=1$ in (4.31), we deduce that

$\sigma_{i}t_{ii}q_{i}+\alpha_{i}\sum_{j=i+1}^{N}\sigma_{j}q_{j}=0,\quad
i=1,\ldots,N,$ (4.32)

and hence that

$\xi_{N}=\xi_{N-1}=\dots=\xi_{1}=0,$ (4.33)

which reveals that all terms in (4.30) arising from the Lagrange multiplies
are automatically absent. Thus, a solution of (4.28) satisfies (4.9). Applying
the notation (3.11), we rewrite (4.27) as

$\mathrm{e}^{t_{ii}\bar{v}_{i}+\sum\limits_{k=1}^{i-1}\alpha_{k}\bar{v}_{k}}\int\limits_{V}\mathrm{e}^{u_{i}^{0}+t_{ii}\hat{v}_{i}+\sum\limits_{k=1}^{i-1}\alpha_{k}\hat{v}_{k}}\mathrm{~{}d}\mu=q_{i},\quad
i=1,\ldots,N,$ (4.34)

from which we deduce that

$\bar{v}_{i}=\sum_{j=1}^{i}l_{ij}\left(\log q_{j}-\log
J_{j}(\hat{\mathbf{v}})\right),\quad i=1,\ldots,N,$ (4.35)

where

$\hat{\mathbf{v}}=\left(\hat{v}_{1},\ldots,\hat{v}_{N}\right)^{T}.$

Thus, we can rewrite (4.10) as

$\displaystyle
J(\mathbf{v})-\sum_{i=1}^{N}\int\limits_{V}\frac{1}{2}\Gamma(\hat{v}_{i},\hat{v}_{i})\mathrm{~{}d}\mu$
$\displaystyle=\sum_{i=1}^{N}q_{i}-\sum_{i=1}^{N}p_{i}\bar{v}_{i}$ (4.36)
$\displaystyle=\sum_{i=1}^{N}\sum_{j=1}^{i}p_{i}l_{ij}\ln
I_{j}(\hat{\mathbf{v}})+\sum_{i=1}^{N}\left(q_{i}-p_{i}\sum_{j=1}^{i}l_{ij}\ln
q_{j}\right).$

From (4.14), there exists constant $C=C(L,\mathbf{p},\mathbf{q})$ such that

$J(\mathbf{v})=\frac{1}{2}\sum_{i=1}^{N}\int\limits_{V}\Gamma(\hat{v}_{i},\hat{v}_{i})\mathrm{~{}d}\mu+\sum_{i=1}^{N}q_{i}\ln
I_{i}(\hat{\mathrm{v}})-C.$ (4.37)

By the Jensen inequality, we deduce that

$I_{i}(\hat{\mathbf{v}})\geq|V|\exp\left(\int_{V}u_{i}^{0}\mathrm{~{}d}\mu\right)=:\mu_{i},\quad
i=1,\ldots,N.$ (4.38)

Thus we obtain

$J(\mathbf{v})-\frac{1}{2}\sum_{i=1}^{N}\int\limits_{V}\left|\nabla\hat{v}_{i}\right|^{2}\mathrm{~{}d}\mu\geq\sum_{i=1}^{N}q_{i}\ln\mu_{i}-C.$
(4.39)

Set $\\{(v_{1,k},\dots,v_{N,k})\\}_{k=1}^{\infty}$ be a minimizing sequence of
(4.28). By (4.39), $\\{(\hat{v}_{1,k},\dots,\hat{v}_{N,k})\\}_{k=1}^{\infty}$
is bounded in $L^{2}(V)$. By Lemma 2.2,
$\\{(\nabla\hat{v}_{1,k},\dots,\nabla\hat{v}_{N,k})\\}_{k=1}^{\infty}$ is
bounded in $L^{2}(V)$. Hence, we see that
$\\{(v_{1,k},\dots,v_{N,k})\\}_{k=1}^{\infty}$ is bounded in $H^{1}(V)$. By
Lemma 2.1, we deduce that, by passing to a subsequence,

$v_{i,k}\to w_{i,\infty}$ (4.40)

uniformly for $x\in~{}V$ as $k\to+\infty$ for $i=1,\dots,N.$ Thus, by the
notation (3.11), we know that $\bar{w}_{i,\infty}$ satisfies (4.35) for
$i=1,\dots,N$, and that

$\gamma=J(\mathbf{w}_{\infty}),$

where $\mathbf{w}_{\infty}:=(w_{1,\infty},\dots,w_{N,\infty})$. Therefore, we
know that $\mathbf{w}_{\infty}$ is a solution to the problem (4.28). It
follows that $\mathbf{w}_{\infty}$ is a solution to the problem (4.9).

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# Enhancement of Rural Connectivity by Recycling TV Towers with Massive MIMO
Techniques

Ammar El Falou, and Mohamed-Slim Alouini

###### Abstract

Nowadays, the digital divide is one of the major issues facing the global
community. Around $3$ billion people worldwide are still not-connected or
under-connected. In this article, we investigate the use of TV towers with
multi user (MU) massive multiple input multiple output (mMIMO) techniques to
offer connectivity in rural areas. Specifically, the coverage range is
assessed for a MU mMIMO base station (BS) mounted on a high tower as a TV
tower, and compared with a legacy mMIMO BS. The obtained results show that one
high tower BS can cover an area at least $25$ times larger than the area
covered by a legacy BS. This is of high interest as recycling TV towers can
enhance the rural connectivity with low expenditures. We apply the proposed
solution to a realistic case study in an Ethiopian rural area, based on
population densities and locations of current BS and TV towers. Our study
shows that a high number of people can be covered by existing TV towers.
Additional possible solutions to enhance rural connectivity are discussed in
the last section.

##  I Introduction

###  I-A Digital Divide

People in developed countries are benefiting from the experience offered by 4G
and 5G communication systems in their work, education, health system, social
life, and entertainment. On the other side, around $3$ billion people in
undeveloped countries and rural areas in some developed countries are still
not-connected or under-connected, a problem globally known as the “digital
divide” [1, 2, 3]. These regions generally suffer from bad infrastructures
with regular power outages and lack of roads and transportation, or have a low
density of population. Such reasons have made the cost of deploying cellular
sites in rural regions not economically viable. Operators need to account for
powering and backhauling costs in their business models and compare them with
possible revenues to take deployment decisions.

###  I-B Wireless Connectivity and Super Cells

Nowadays, and especially after the spread of the COVID-19 pandemic, providing
ubiquitous connectivity is becoming a necessity and is gaining more interest
[1, 2, 4, 5, 6, 3, 7, 8, 9, 10, 11, 12]. Maintaining a seamless connectivity
is mandatory for efficient online education, remote working, e-governance, and
online business. To this end, recent efforts in 6G communications are focused
on decreasing the digital divide gap to meet future sustainable hyper-
connectivity demands [1].

In many non-connected or under-connected areas, wireless connectivity can be
seen as the only option for access and backhaul communications [1, 13].
Indeed, wired communications require high expenses that cannot be justified by
the limited economical revenues in rural regions. Several technologies and
approaches are proposed for wireless connectivity in rural areas as: high-
altitude platform systems (HAPS), satellite communications, unmanned aerial
vehicle base station (UAV-BS) and super cells (SCs) (see [1, 2, 4, 5, 6, 3, 7,
8, 9, 10, 11, 12, 13, 14] and references therein). Among them, the technology
of SCs, i.e., cells with a high tower and large coverage area can be
considered as the most cost-effective [1, 13, 15]. The advantage of this
solution is further amplified when using available TV towers to provide
coverage to SCs.

###  I-C Current Situation of TV Transmitters

Television or TV service appeared at the beginning of the $20$th century, and
gained popularity after $1950$s. Terrestrial TV service is typically
transmitted via a high tower in order to cover a large area of tens of
kilometers. TV towers are connected to the electrical grid and backhauled to
broadcast served TV channels. The use of TV towers for extending rural
connectivity is mainly motivated by:

  1. (i)

The availability of electricity and backhauling in current TV towers thus
operational expenditures (OPEX) are lower than other solutions.

  2. (ii)

The high height which helps in providing large coverage areas. Indeed, high
towers provide a propagation advantage as a line of sight (LoS) connection is
highly probable.

  3. (iii)

Large cells are more suitable for rural areas, as the cell coverage is the
main limitation in this case, rather than the capacity as in urban areas.

  4. (iv)

The prior re-allocation of parts of the TV spectrum, mainly the sub-700 MHz
band, for communication purposes [16, 17].

For all these reasons, taking full advantage of TV towers for connectivity can
be seen as an evident choice to mitigate the digital divide.

###  I-D MIMO and 6G Systems

Multiple-input multiple-output (MIMO) techniques are key technologies to
increase the system capacity and coverage, thus improving the user experience
[18]. They are therefore introduced in wireless communication standards as 5G
and are essential in future standards as 6G [8]. Indeed, MIMO precoding
techniques aim to:

  1. (i)

Focus the power transmitted by the serving BS on the targeted devices through
beamforming, thus increasing the coverage range.

  2. (ii)

Decrease multi-user interference thanks to spatial selectivity and increase
the rate of users.

###  I-E Related Works

In this work, we explore the potential of recycling TV towers with massive
MIMO (mMIMO) precoding for rural connectivity. Similar works can be listed as
follows.

  1. 1.

Meta Engineering Super Cell: The term SC was proposed in [13] by Meta
Connectivity as a large-area coverage cell that leverages towers up to $250$ m
height, equipped with high-gain, narrow-sectored antennas in order to increase
the coverage range and the capacity. Drive tests were used to assess the
advantage of this solution. A 36-sector SC with Luneberg Lens is deployed and
it is shown that one SC can replace 15 to 25 traditional macrocells. Single
outdoor user is considered in the drive tests.

  2. 2.

Curvalux: Curvalux [19] proposed a phased array multi-beam antenna to be
installed on legacy cellular towers. This array forms 16 high-gain beams and
cover a sector having an angle equal to 60∘. Many proof-of-concept (POC)
trials have been done in several countries as Philippines, Mongolia,
Indonesia, Kuwait. These POC validate the feasibility of the proposed multi-
beam antenna but technical details and measurement are not published.

  3. 3.

TV Towers: The potential of mMIMO on TV towers for cellular coverage extension
has been investigated in [15]. The Ericsson 9999 rural path-loss model and
independent Rayleigh fading are considered. Different carrier frequencies have
been considered mainly 700 MHz, 1800 MHz and 3500 MHz and a bandwidth of 20
MHz is used with all frequencies. A single-cell BS, operating in a time
division duplex (TDD) mode, with a big size mMIMO uniform cylindrical array
(UCyA) of $2$ m radius and up to $10$ m height, is considered. Under these
assumptions, authors in [15] have shown that an area of radius $70$ km can be
covered around a TV tower using the mMIMO technology.

###  I-F Proposed Work and Contributions

In this article, we investigate the use of multi-user (MU) mMIMO linear
precoding at the top of high towers, as TV towers, to improve the downlink
connectivity of rural areas. As a benchmark, we consider a MU mMIMO legacy BS.
The user is considered as connected in the downlink when its rate exceeds $10$
Mbps, which guarantees a good quality of service to several applications
including online education, remote working, and online business [1]. By
evaluating the coverage range, we show that recycled high towers implementing
MU mMIMO technology can cover an area at least $25$ times larger than the area
covered by a legacy BS. We apply this solution to a realistic case study in a
low-income country suffering from a high telecommunication service imbalance
as in Ethiopia [20, 21]. We take into consideration the population density
maps and the location of present cellular sites and TV towers. We show that a
large number of persons can be covered by recycling TV towers for
connectivity. Finally, we discuss some new technologies that can further
enhance the connectivity in rural areas as: hybrid mMIMO with non orthogonal
multiple access (NOMA), reconfigurable intelligent surface (RIS), and low-
earth orbiting (LEO) satellites.

The main differences between the proposed work and previous works in the
literature are:

  1. 1.

In [13, 19] fixed beam high-gain antennas are used regardless of the location
of users. This solution might not be energy-efficient. In our work and in
[15], mMIMO beamforming is employed, adapting based on the channel state
information (CSI) of served users. However, channel estimation is necessary in
this case.

  2. 2.

In [15], a zero-forcing (ZF) linear precoder is considered which might not be
efficient when users are close to each other [22, 23, 24]. We consider a
regularized ZF (RZF) precoder which is more suitable than ZF in practice, with
limited added complexity and cost [18].

  3. 3.

In our work, we consider the 3GPP channel models [25] based on realistic
measurements rather than the Ericsson 9999 model. These models are generated
using the open source geometry-based quasi deterministic radio channel
generator (QUADRIGA) stochastic channel simulator v.2.6 [26]. Simulations are
done with different types of 3GPP 38.901 models mainly line of sight (LoS) for
high tower BS and non LOS (NLoS) for legacy BS. On the other hand, authors in
[15] use Ericsson 9999 model with independent Rayleigh fading channel. This
assumption is not realistic especially in the case of high towers where LoS is
paramount.

  4. 4.

Authors in [15] consider a mMIMO BS with a large UCyA that can accommodate up
to $293\times 464=135,952$ antennas. ZF precoding requires as much radio-
frequency (RF) chains as antennas. The implementation of this number of RF
chains is not possible in practice. Moreover, the implementation of a UCyA
with a radius of $2$ m and a height of $10$ m might be challenging to resist
against wind in practice.

  5. 5.

Different from [15], we complement our work by taking into consideration a
real case study in Ethiopian rural area, with population density and location
of legacy BSs and TV towers. We differentiate between three cases: low,
medium, and high number of served users per cell. This study is important to
motivate network operators to recycle TV towers for connectivity.

TABLE I: System Parameters Parameter | Legacy BS | High Tower BS (TV Tower)
---|---|---
Tx Height [m] | $25$ | $150$
Rx Height [m] | $8$ | $8$
Channel Model | 3GPP 38.901 RMa NLOS | 3GPP 38.901 RMa LOS
Nb. of Active Users $K$ | $20,50,100$ | $20,50,100$
Target DL Rate [Mbps] | $10$ | $10$
Number of Antennas |  $256$ single/$512$ dual polarized |  $256$ single/$512$ dual polarized
$M_{h}\times M_{v}$ |  $32\times 8$ (single)/$32\times 8\times 2$ (dual) |  $32\times 8$ (single)/$32\times 8\times 2$ (dual)
Bandwidth $B$ [MHz] |  $10,20,100$ MHz |  $10,20,100$ MHz
Transmit Power [dBm] | $46$ | $50$
Cyclic Prefix Overhead | $5\%$ | $5\%$
Carrier Frequency $f_{c}$ [MHz] |  | $700$ | $1800$ | $3500$
---|---|---
| $700$ | $1800$ | $3500$
---|---|---
Height $h_{a}$ [m] |  | $1.50$ | $0.59$ | $0.30$
---|---|---
| $1.50$ | $0.59$ | $0.30$
---|---|---
Radius $r_{a}$ [m] |  | $1.09$ | $0.42$ | $0.21$
---|---|---
| $1.09$ | $0.42$ | $0.21$
---|---|---

##  II System Model

###  II-A DL MU Massive MIMO Base Stations

We consider a downlink (DL) MU mMIMO single-cell BS with $256$ single-
polarized antennas (or $512$ dual-polarized antennas) forming a UCyA as
depicted in Fig. 1. We restrict the number of antennas in order to have a
system with a reasonable size and complexity. The number of horizontal
antennas $M_{h}=32$ and vertical antennas $M_{v}=8$. The system parameters are
listed in Table I. We note that in practical scenarios, antennas in the
azimuth can help in the separation of users in the horizontal direction and
antennas in the elevation can help in the separation of users in the vertical
direction. For rural areas it is preferable to have antennas in the azimuth
more than in the elevation as users are generally distributed in the
horizontal direction.

Figure 1: System model: A mMIMO base station.

According to [1], we consider a target rate per user of 10 Mbps. To ensure
fairness between users, a max-min power control is done maintaining similar
signal to interference plus noise ratio (SINR) values for all users [18]. We
assume that the receiver is placed at a height of $8$ m111An omnidirectional
antenna is considered at the receiver side. An antenna with certain gain at
the receiver will help in increasing: either the BS coverage range or the data
rate of its respective user.. This can represent the case when the receiver
antenna is placed at the roof of the house unit in rural areas. The number of
active users per cell $K\in\\{20,50,100\\}$ reflects a low, medium, and high
number of active users distributed uniformly around the BS.

The cyclic prefix overhead for orthogonal frequency division multiplexing
(OFDM) is considered at $5\%$. Three carrier frequencies in the sub-$6$ GHz
are considered: $700$, $1800$, and $3500$ MHz. The corresponding bandwidths
are: $10$ MHz, $20$ MHz, and $100$ MHz as in the 5G NR frequency bands [17].
Note that systems using the former frequencies, $700$ and $1800$ MHz, operate
in frequency division duplex (FDD) mode where the full bandwidth is used in
DL, while those using the later frequency, $3500$ MHz, operate in a TDD mode
where $3/4$ proportion of the bandwidth is for DL222For mMIMO systems, TDD
mode is preferred as the channel estimation scales with the number of users
and not with the number of antennas. However, for sub-$6$ GHz frequencies, the
number of antennas is relatively low and both FDD and TDD are used [17].. The
RZF linear precoder is used as an efficient and low-complex precoder for MU
MIMO systems. Other precoders such as: ZF, maximum ratio (MR) or minimum mean-
squared error (MMSE) are not recommended. Indeed, ZF is not efficient when
users are close to each other. MR is low-complex but has a limited data rate
while MMSE is optimal but very complex for practical implementation [18].
Two type of single-cell BSs are considered:

  * •

High tower BS (TV tower): A BS with a transmitter height of $150$ m and a
transmit power of $100$ W. The value of the tower height is inspired from
digital terrestrial television broadcasting (DTTB) where TV tower effective
height is equal to $150$ m in rural scenarios [16].

  * •

Legacy BS: A BS with a transmitter height of $25$ m and a transmit power of
$40$ W. These parameters are considered as a benchmark.

###  II-B Antenna Array and Channel Models

TABLE II: Coverage Distance for a Legacy BS and a High Tower BS (TV Tower) Type | $K$ |  $f_{c}$ [MHz] | Duplex Mode |  $B$ [MHz] |  $d_{\text{cov}}$ single [Km] |  $d_{\text{cov}}$ dual [Km]
---|---|---|---|---|---|---
Legacy BS | 20 |  |  |  | 4.1 | 4.9
50 | 700 | FDD | 10 | 2.7 | 3.4
100 |  |  |  | 1.7 | 2.1
20 |  |  |  | 2.9 | 3.5
50 | 1800 | FDD | 20 | 1.9 | 2.5
100 |  |  |  | 1.2 | 1.6
20 |  |  |  | 2.2 | 2.7
50 | 3500 | TDD (DL: 3/4) | 100 | 1.5 | 1.9
100 |  |  |  | 1 | 1.4
High Tower BS (TV Tower) | 20 |  |  |  | 28 | 37
50 | 700 | FDD | 10 | 15 | 21
100 |  |  |  | 9.5 | 12.5
20 |  |  |  | 14.5 | 16.5
50 | 1800 | FDD | 20 | 10.5 | 13
100 |  |  |  | 7.5 | 9.5
20 |  |  |  | 13.5 | 15
50 | 3500 | TDD (DL: 3/4) | 100 | 10 | 13
100 |  |  |  | 8 | 10

We use in this work two 3GPP channel models for rural areas [25], mainly, the
3GPP rural macro NLoS (3GPP $38.901$ RMa NLoS) for the legacy BS and the 3GPP
rural macro LoS (3GPP $38.901$ RMa LoS) for the high tower BS. Note that these
models are based on real measurements. In our simulations, we generate the
channel coefficients using the QUADRIGA channel simulator v.2.6 [26]. The BS
is placed in the center of the cell and the UCyA is placed at the top of the
tower. The separation between antennas is equal to $\lambda/2$ where $\lambda$
is the wavelength. The use of 3GPP rural channel models and QUADRIGA will make
the presented simulations more realistic333We note that 3GPP models in [25]
are proposed for distances $\leq 10$ km. However, we did not see any anomaly
for higher distances., as they account for several geometric aspects of the
system as users locations, correlation between antennas, and heights.

###  II-C Coverage Range Results

We conducted extensive Monte-Carlo simulations in order to compute the
coverage range of the two types of BSs. The coverage range is defined as the
range for which more than $95\%$ of users have a rate higher than $10$ Mpbs.
The results are presented in Table II where $d_{\text{cov}}$ stands for the
coverage distance, or the maximum value of the coverage range. It is clear in
Table II that the coverage distance is multiplied by more than $5$ when using
a high tower BS instead of a legacy BS with the same mMIMO configuration. This
reflects that one high tower BS can cover a $25$ times bigger area than a
legacy BS. This result is consistent with the measurements done in [13] when
studying super cells. Note that for a low number of active users, e.g.,
$K=20$, and a low carrier frequency $f_{c}=700$ MHz, this ratio can go up to
$57$ times with dual-polarized antennas.

##  III Mapping to a real case

In this section, we map the obtained results to the case of a rural area in
Ethiopia. We consider the area depicted in Fig. 2(a). This rural area is at
the borders of Ethiopia, specifically between latitude:
$10^{\circ}30^{\prime}$N- $12^{\circ}$N, and longitude
$35^{\circ}$E-$36^{\circ}$E. Fig. 2(a) shows the population density per km2,
denoted by $p$, and the location of legacy cellular BSs and TV towers inside
this area. The population density is obtained from Facebook/Meta density
population data [27]. The total number of people in this area is about
$200,000$ people. The locations and types of cells (3G/4G) are obtained from
[28] and the locations of TV towers are obtained from [29].

From Fig. 2(a), it is clear that:

  1. 1.

A large part of the considered area is not covered by any 3G BS. Legacy BSs
are located in some places with high population density.

  2. 2.

4G BSs are not available in the considered area.

  3. 3.

Some TV towers are available.

(a) Current TV towers

(b) Relocated towers

Figure 2: Rural area in Ethiopia: population density per km2 $p$ and coverage
areas.

In order to assess the benefit of recycling TV towers for connectivity, we
also plot in Fig. 2(a) the coverage circles around legacy BSs and TV tower BSs
when both use MU mMIMO. The favored frequency for propagation, i.e.,
$f_{c}=700$ MHz is considered. The percentage of active users from the
population is between $1\%$ and $2\%$, identified as a loaded scenario for
rural areas [7]. In practice, this value depends on the adoption rate of
Internet service and the percentage of active users.

From Fig. 2(a), we can conclude that a substantial increase in the covered
area is seen by recycling TV towers. Indeed, the actual cellular network can
cover only $9,000$ persons out of the total population of $200,000$ even when
using MU mMIMO. This represents $4.5\%$ of the total population. By recycling
TV towers for communication, $19,500$ more persons can be covered in the
considered scenario, and the percentage of covered persons reaches $14.25\%$.

Furthermore, it is clear from Fig. 2(a) that two TV towers out of the three
available ones are covering areas with a low population density, i.e., $p<10$
persons per km2. If a relocation of the TV towers is possible, the number of
covered persons by high towers can be increased to $30,000$ as shown in Fig.
2(b) using coverage circles with solid lines. The percentage of covered
persons then becomes $19.5\%$.

For the early adoption of Internet in rural areas, the number of covered
persons can largely increase and the percentage of covered persons can reach
$63\%$. The corresponding coverage areas are shown in Fig. 2(b) with dashed
lines. This early adoption corresponds to the case where $2\%$ of persons
subscribe for Internet service (adoption rate) while $1/20$ out of them are
active at a certain time, as assumed in [7] in their study on the engineering-
economics aspects of satellite broadband constellations for connectivity
purposes. In practice, the optimization of the location of high towers and
their parameters as: transmit power, number of antennas, and height, can
further increase the coverage percentage.

##  IV Possible enhancements

Figure 3: Possible enhancements to the system.

In this article, we have shown that recycling TV towers for enhancing rural
connectivity is a promising solution. However, this solution is not sufficient
to cover the entire rural areas. Possible enhancements to improve the
performance of the proposed solution include NOMA and hybrid mMIMO,
reconfigurable intelligent surfaces, and LEO satellites. Fig. 3 shows the
proposed enhancements.

###  IV-A NOMA and Hybrid MIMO

In results presented in Section II-C, we have considered that active users are
uniformly distributed around the cell tower. In reality, the distribution of
persons in rural areas is not uniform as shown in Fig. 2. Therefore, active
users might not be uniformly distributed causing multi-user interference for
users close to each other. Multiple access techniques should be used in order
to mitigate this interference. NOMA techniques [30] are preferred over
orthogonal multiple access (OMA) techniques as time-division multiple access
(TDMA) or frequency-division multiple access (FDMA), as they offer higher data
rates. Indeed, NOMA allows the service of two or more users using the same
beam as depicted in Fig. 3. Users are then separated thanks to interference
cancellation techniques.

Furthermore, in order to increase the beamforming gain and the coverage range,
the number of antennas should be increased. This requires as many RF chains as
antennas for a digital precoding system. These additional RF chains will
increase the cost and complexity of the system. Hybrid MIMO is a possible way
to increase the number of antennas while maintaining the cost and complexity
order of the system. Hybrid MIMO and NOMA can be combined together to benefit
from high beamforming gains while maintaining an acceptable number of RF
chains at the transmitter [31].

###  IV-B Reconfigurable Intelligent Surfaces

RISs have been proposed to reconfigure the wireless propagation environment
and enhance transmission performance. An RIS is made of passive reconfigurable
elements and can vary the phase and amplitude of the incident signals in order
to reflect them in the desired directions [32]. Unlike other technologies as
active relays, RISs do not require power-hungry RF chains and power
amplifiers, making them low-cost and energy efficient solutions for coverage
extensions. RIS-aided massive MIMO communications [33, 34] can therefore be
used to extend the coverage range in rural areas, as shown in Fig. 3. The
location of RISs should be carefully chosen to direct the beams in the
direction of users, taking into account the existence of blockages in the
terrain.

###  IV-C LEO Satellites

LEO satellites can be used to cover rural areas that are not reachable by the
deployed towers with mMIMO support. The advantage of LEO constellations is
that they can provide low-latency high-rate communication links for rural
users [2]. However, their high deployment cost requires expensive subscription
fees to make their business model lucrative. This makes LEO constellations
more suitable for rural areas in rich developed countries than undeveloped
poor countries. Again, mMIMO can be used with LEO satellites to increase the
beamforming gains [35], and decrease the OPEX in order to reduce further the
subscription fees.

##  V Conclusion

In this article, we have shown that recycling TV towers with mMIMO precoding
provides a low-cost enabler of connectivity in rural areas. Indeed, one high
tower BS can cover an area at least $25$ times larger than a Legacy BS.
Furthermore, we highlighted the benefits of the proposed solution in a
realistic use case (Ethiopian rural area) where we showed that a large number
of people can be covered by deploying a BS at the top of available TV towers.
Furthermore, we have discussed some future directions to enhance the
performance of the proposed solution. An important perspective of this work is
to consider the terrain information (buildings, valleys, trees, mountains,
hills) in addition to the population density maps and towers locations in the
coverage study. This will permit drawing a coverage map similar to our
proposed one where network operators can obtain directly the number of
possible subscribers in a region and include this number in their business
model.

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Ammar El Falou [M]<EMAIL_ADDRESS>received in 2013 his Ph.D.
degree in communication and information science from IMT Atlantique, Brest,
France. In 2014, he worked as a postdoctoral at Orange Labs, Sophia Antipolis,
France. From 2015 to 2020, he was an Assistant Professor at Lebanese
International University (LIU). From 2020 to 2021, he was an Assistant
Professor at ESIEE Paris, France. He is currently a Research Engineer at King
Abdullah University of Science and Technology (KAUST), Saudi Arabia. His
current research interests include massive MIMO systems, rural connectivity
and power-efficient wireless communication systems.
---
Mohamed-Slim Alouini [F]<EMAIL_ADDRESS>was born in Tunis,
Tunisia. He received the Ph.D. degree in Electrical Engineering from the
California Institute of Technology (Caltech), Pasadena, CA, USA, in 1998. He
served as a faculty member in the University of Minnesota, Minneapolis, MN,
USA, then in the Texas A&M University at Qatar, Education City, Doha, Qatar
before joining King Abdullah University of Science and Technology (KAUST),
Thuwal, Makkah Province, Saudi Arabia as a Professor of Electrical Engineering
in 2009. His current research interests include the modeling, design, and
performance analysis of wireless communication systems.
---


# The Fractional Orbital Chern Hall Effect

Christopher Sims  Elmore Family school of Electrical and Computer Engineering,
Purdue University<EMAIL_ADDRESS>
(September 3, 2024)

###### Abstract

The fractional quantum hall effect (FQHE) is a milestone of modern day
physics, its disovery paved the way for the study of fractional charges which
do not obey abelian physics. However, all FQHE require an external magnetic
field in order for there to be fractionally charged electrons. This work shows
a theoretical study of strongly interaction electrons in the Kagome lattice
with magnetism. Under optimal conditions, a band gap opens on the surface of
the material which hosts fractional Fermions. These fractional Fermions form
composite quasi-electrons without the need for external magnetic field. These
states are predicted to host the FQHE without the need for an external
magnetic field.

##  I Introduction

The quantum hall effect was discovered in 1980, this discovery was provided an
avenue to study the effect of magnetism on electrons in a system[1, 2].
Shortly after the quantum hall effect was observed, the fractional quantum
hall effect was unveiled[3]. This system differs from the QHE because
electrons are now confined to 2D and the electrons form Laughlin states[4]
under applied magnetic fields[5, 6]. While these two discoveries are essential
to the understanding of electrons in a magnetic field[7], the study of
electrons without a magnetic field was not discovered until 2007[8] (predicted
in 2004[9, 10]). With this discovery, it is found that intrinsic magnetism
from spin interactions in a lattice can lead to a quantum hall effect,
commonly referred to as the quantum spin hall effect (QSHE)[11]. An extension
of this effect is the quantum anomalous hall effect with originates from the
berry curvature (Chern insulators), in which there is quantized edge mode
without magnetic field[12, 13].

With all of these discoveries there still exists a “missing” piece to the
puzzle. A fractional hall effect without an applied magnetic field has not
been experimentally discovered to date. The main issue is there is no
experimental evidence that fractional Fermions can be formed without the need
for an applied magnetic field[14]. It has been predicted that from a strongly
interaction Haldane model with spin orbit coupling[15] that a gapped Chern
insulator can form. Inside the gap, these newly predicted states host
fractional Dirac fermions which do not need a magnetic field in order to
exist[16, 17]. The origin of these fractional Dirac states come from the
fractionally charged hybridized orbitals which have a fractional Chern number
associated with them[17, 18, 19].

It has been predicted that fractional Chern insulators are states beyond the
magnetically supported Laughlin states which make up the FQHE[20]. Therefore,
throughout this work, fractional Chern Fermions will refer to fractional non-
Laughlin states such as the fractional Dirac Fermion. Fractional Chern
Fermions have been observed in van-der-walls heterostructures, however, they
need a magnetic field in order to exhibit the fractional hall effect which
still exists under the genus of the FQHE[21, 22]. Recently, it has been
predicted that strongly gapped Kagome lattices can host fractional Dirac
fermions without the need for an external magnetic field[23, 24].

This work predicts the fractional orbital Chern hall effect, a hall effect
with fractional filling which can be observed without the need for an external
applied magnetic field. This prediction is support by Hamiltonian simulation
and theoretical modeling. In addition, materials and experimental methods are
put forward to observe this effect.

##  II Materials and Methods

In order to model the edge states of the Chern gapped insulator, a spin full
tight binding Haldane[25, 26] model for the Kagome lattice[27, 28, 29] is
constructed:

$\displaystyle H=$ $\displaystyle{}-t\sum_{\langle i,j\rangle\alpha}e^{2\pi
i\Phi_{ij}}c^{\dagger}_{i\alpha}c_{j\alpha}$ (1)
$\displaystyle-\mu\sum_{i\alpha}c^{\dagger}_{i\alpha}c_{j\alpha}$
$\displaystyle+i\frac{\lambda_{SO}}{3\sqrt{3}}\sum_{\ll
i,j\gg\alpha}s^{\alpha}_{z}e^{2\pi
i\Phi_{ij}}v_{ij}c^{\dagger}_{i\alpha}c_{j\alpha}$
$\displaystyle+\lambda_{V}\sum_{i\alpha}t^{i}_{z}c^{\dagger}_{i\alpha}c_{j\alpha}$
$\displaystyle+\lambda_{X}\sum_{i\alpha}s^{\alpha}_{z}c^{\dagger}_{i\alpha}c_{j\alpha}$
$\displaystyle+\lambda_{SX}\sum_{i\alpha}t^{i}_{z}s^{\alpha}_{z}c^{\dagger}_{i\alpha}c_{j\alpha}$
$\displaystyle+\lambda_{SSO}\sum_{\ll
i,j\gg\alpha}t^{i}_{z}s^{\alpha}_{z}e^{2\pi
i\Phi_{ij}}v_{ij}c^{\dagger}_{i\alpha}c_{j\alpha}$

The notations are as follows: $s_{z}=\uparrow\downarrow$, $t_{z}=A,B$,
$s^{\alpha}_{z}=\pm 1$. Where the terms $t$ is the transfer energy. The second
term $\mu$ is the chemical potential. The third term $\lambda_{SO}$ is the
spin orbit coupling. The fourth therm $\lambda_{V}$ is the staggered
sublattice which is effected by external electric field. The fifth term
$\lambda_{X}$ is the exchange term. With the sixth and seventh terms being the
staggered exchange $\lambda_{S}X$ and the staggered spin terms
$\lambda_{S}SO$.

##  III Results & Discussion

A Kagome Lattice is constructed in the tight binding regime in order to
confirm previous DFT results which predict gapped Chern Fermions in the
lattice with intrinsic magnetism. In the full tight binding model there exists
terms which include correlation and spin correlation effects, these terms
($\lambda_{SX}$,$\lambda_{SSO}$,$\lambda_{X}$,$\lambda_{V}$) are set to zero.
The model parameters are set to $t=2.7$, $\lambda_{SO}=0.1t$, and $\mu=0.2$.
From this tight binding model the bulk band structure for the Kagome is
constructed in the $\Gamma$-$K$-$M$-$K$-$\Gamma$ direction. The band structure
reveals a gapped state at the $K$ point with a secondary band gap along the
$K$-$\Gamma$ line (Fig. 3). In order to see the surface states, a stacked
model is constructed to model the bulk and surface states. In the bulk [Fig.
1(A)], there exists no band gap in the Kagome lattice. However, on the surface
[Fig 1(B)], a gap forms with surface states. there exists multiple crossings
due to the strongly interacting nature of the Kagome bands. A Wannier charge
center calculation shows high Chern numbers (n¿1) (Fig 4) which is critical
for the formation of fractionally charged Fermions[30]. The Wannier charge
center analysis does not show fractional charge since these states are the
bulks states projected onto the edge, as opposed to the gapped states that
form on the surface.

Figure 1: Surface edge spectrum of the Kagome Hamiltonian with
$t=2.7$,$\mu=0.2$,$\lambda_{SO}=0.1t$. Blue indicates bulk states and red
indicates surface states. (a) Bulk (b) surface

###  III.1 Hall Resistance

In order to model the fractional hall effect in the material candidates,
composite fermion theory is utilized in order to construct the fractional
filling factors with their associated Chern bands. In this work the
parameterization of the field $p$ is characterized by the intrinsic magnetic
field in the material. Fractional Fermion theory shows that this parameter
scales with the Chern number due to stronger interactions with the intrinsic
fields. In high Chern number materials such as Kagome magnets, the base Chern
number bands in the ungapped bulk can reach Chern numbers of $C=\geq 20$. Here
For the landau levels, six are selected for the base bands with
fractionalization occurring on the surface.

Figure 2: $R_{XY}$ as a function of $\mu$ the chemical potential, blue
indicates $C_{\uparrow}$, red indicates $C_{\downarrow}$,

In the case of Kagome magnets RMn6Ge6 (R = ) the states inside the gap form
fractionally charged Dirac Fermions on the surface which are spin polarized.
Although the fractional states are formed from orbital Chern effects the
surface (FOCHE), these state are spin polarized with opposite chirality, this
particular hall effect is most accurately described by the term Fractional
Quantum spin Hall effect (FQSHE). Due to the fact that both the positive
($C_{+}$) and negative ($C_{-}$) Chern bands exist at the same energy level
there needs to be a penalty function in the chemical potential of each band
added in order to observe the fractional filling in the hall effect, otherwise
there will be no experimental signature [Fig. 2(A)]. In the spinfull kagome
lattice, the spin Chern numbers
$C_{spin}=\frac{1}{2}(C_{\uparrow}-C_{\downarrow}$ are slightly offset due to
spliting caused by the intrinsic magnetic field. By apply asymmetric gates to
the top ($V_{top}$) and the bottom($V_{bottom}$) of a mesoscopic sample.
$\mu(V_{g},B)$ shows a parameterization of the spin polarized $C_{\uparrow}$
and $C_{\downarrow}$ hall conductivities with respect to the net gate voltage
on each state from asymmetric gating. $\mu_{offset}$ is increased only for
$C_{\uparrow}$ with of 5,10, and 15 showing an increase in hall plateaus with
gating [Fig. 2(C-D)].

##  IV Conclusions

In conclusion, it is predicted that Chern gapped insulators exhibit the FQHE
without the need for external magnetic field. This new effect is termed the
Fractional Quantum Spin Hall Effect (FQSHE) for gapped, spin polarized Dirac
Fermions. This effect is predicted to exist in Kagome materials RMn6Ge6 under
dual gate conditions. This work provides motivation for the study of
fractional topology without the need for an external magnetic field in kagome
magnets. In future work, the Kagome lattice model can be extended to include
more exchange parameters and be extended to see the Hofstadter’s response in
the presence of a magnetic field.

* The bulk band structure shows good agreement with previous experimental and theoretical investigations into Kagome materials. The flat band is missing due to the fact that only 3 orbitals are used for this calculation.

Figure 3: Bulk band dispersion of the Kagome lattice with 3 bands with $t=2.7$
and $\lambda_{SO}=0.1t$

The Wilson loop analysis shows good agreement with previous studies which
found high Chern numbers in the Bulk tight binding Hamiltonians. Fractional
Chern numbers are not seen due to the fact that there is no surface component
in this edge model

Figure 4: (A) Wilson loop calculation and (B) edge bands of the bulk states of
the tight binding model

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