$z_{0}\in\mathbb{C}\setminus\mathbb{R}$ (in which case, its true for every
$z_{0}\in\mathbb{C}$).

  * (b)

If $\det J=0$ and $z_{0}\in\mathbb{C}\setminus\mathbb{R}$ then the local
index-1 hypotheses (see Def. 4.14) are true for $H-z_{0}W,W$ with respect to
$J$ on the interval $\mathbb{R}$ if and only if the following conditions are
satisfied:

(5.48) $\displaystyle W_{11}\in M_{n_{1}}(L^{1}_{loc}(\mathbb{R})),$ (5.49)
$\displaystyle H_{11}-H_{12}(H_{22}-z_{0}W_{22})^{-1}H_{21}\in
M_{n_{1}}(L^{1}_{loc}(\mathbb{R})).$

Moreover, the statements (a) and (b) are still true if we replace
$L^{1}_{loc}(\mathbb{R})$ by $L^{1}([0,d])$.

###### Proof.

Suppose (5.2) is satisfied. We begin by proving (a). First, suppose $H,W\in
M_{n}(L^{1}_{loc}(\mathbb{R}))$. Let $z_{0}\in\mathbb{C}$. Then we have
$H-z_{0}W\in M_{n}(L^{1}_{loc}(\mathbb{R}))$. Hence, if $\det J\not=0$ then it
follows immediately from Def. 4.14 that the local index-1 hypotheses (see Def.
4.14) are true for $H-z_{0}W,W$ with respect to $J$ on the interval
$\mathbb{R}$. Conversely, suppose $\det J\not=0$ and the local index-1
hypotheses (see Def. 4.14) are true for $H-z_{0}W,W$ with respect to $J$ on
the interval $\mathbb{R}$, for some $z_{0}\in\mathbb{C}\setminus\mathbb{R}$.
Then $\operatorname{Im}z_{0}\not=0$ and
$H-z_{0}W,H-\overline{z_{0}}W=(H-z_{0}W)^{*}\in
M_{n}(L^{1}_{loc}(\mathbb{R}))$. Hence,

$\displaystyle
W=\frac{1}{\operatorname{Im}z_{0}}\frac{1}{2i}[(H-\overline{z_{0}}W)-(H-z_{0}W)]\in
M_{n}(L^{1}_{loc}(\mathbb{R}))$

which implies $H=(H-z_{0}W)+z_{0}W\in M_{n}(L^{1}_{loc}(\mathbb{R})).$ This
proves statement (a).

We now prove statement (b). Assume $\det J=0$ and let
$z_{0}\in\mathbb{C}\setminus\mathbb{R}$. Then the local index-1 hypotheses
(see Def. 4.14 for $H-z_{0}W,W$ with respect to $J$ on the interval
$\mathbb{R}$ by Lemma 4.15 and Lemma 5.11 are equivalent to the following:

(5.50) $\displaystyle(H-z_{0}W)/(H-z_{0}W)_{22},W_{11}\in
M_{n_{1}}(L^{1}_{loc}(\mathbb{R})),$ (5.51)
$\displaystyle(H-z_{0}W)_{12}(H-z_{0}W)_{22}^{-1}W_{22}^{1/2}\in
M_{n_{1}\times n_{2}}(L^{2}_{loc}(\mathbb{R})),$ (5.52) $\displaystyle
W_{22}^{1/2}(H-z_{0}W)_{22}^{-1}(H-z_{0}W)_{21}\in M_{n_{2}\times
n_{1}}(L^{2}_{loc}(\mathbb{R})).$

Considering this even further, using Holder’s inequality and the facts that
$(H-z_{0}W)_{12}=H_{12}-z_{0}W_{12},(H-z_{0}W)_{21}=H_{21}-z_{0}W_{21},$ and
$W_{22}^{1/2}(H-z_{0}W)_{22}^{-1}W_{22}^{1/2}\in
M_{n_{2}}(L^{\infty}_{loc}(\mathbb{R}))$ (by Lemma 5.11), and
$W_{22}^{-1/2}W_{21}\in M_{n_{2}\times n_{1}}(L^{2}_{loc}(I))$ (see the proof
of Lemma 4.15), we see that those conditions are equivalent to the following:

(5.53) $\displaystyle(H-z_{0}W)/(H-z_{0}W)_{22},W_{11}\in
M_{n_{1}}(L^{1}_{loc}(\mathbb{R})),$ (5.54) $\displaystyle
H_{12}(H-z_{0}W)_{22}^{-1}W_{22}^{1/2}\in M_{n_{1}\times
n_{2}}(L^{2}_{loc}(\mathbb{R})),$ (5.55) $\displaystyle
W_{22}^{1/2}(H-z_{0}W)_{22}^{-1}H_{21}\in M_{n_{2}\times
n_{1}}(L^{2}_{loc}(\mathbb{R})).$

Therefore, by expanding

$\displaystyle(H-z_{0}W)/(H-z_{0}W)_{22}=(H-z_{0}W)_{11}-(H-z_{0}W)_{12}(H-z_{0}W)_{22}^{-1}(H-z_{0}W)_{21}$
$\displaystyle=H_{11}-z_{0}W_{11}-H_{12}(H-z_{0}W)_{22}^{-1}H_{21}$
$\displaystyle+z_{0}H_{12}(H-z_{0}W)_{22}^{-1}W^{1/2}_{22}W^{-1/2}_{22}W_{21}+z_{0}W_{12}W^{-1/2}_{22}W^{1/2}_{22}(H-z_{0}W)_{22}^{-1}H_{21}$
$\displaystyle-
z_{0}^{2}W_{12}W^{-1/2}_{22}W^{1/2}_{22}(H-z_{0}W)_{22}^{-1}W^{1/2}_{22}W^{-1/2}_{22}W_{21},$

we see that those conditions are equivalent to the following conditions:

(5.56) $\displaystyle W_{11}\in M_{n_{1}}(L^{1}_{loc}(\mathbb{R})),$ (5.57)
$\displaystyle H_{12}(H_{22}-z_{0}W_{22})^{-1}W_{22}^{1/2}\in M_{n_{1}\times
n_{2}}(L^{2}_{loc}(\mathbb{R})),$ (5.58) $\displaystyle
W_{22}^{1/2}(H_{22}-z_{0}W_{22})^{-1}H_{21}\in M_{n_{2}\times
n_{1}}(L^{2}_{loc}(\mathbb{R})),$ (5.59) $\displaystyle
H_{11}-H_{12}(H_{22}-z_{0}W_{22})^{-1}H_{21}\in
M_{n_{1}}(L^{1}_{loc}(\mathbb{R})).$

Now considering the imaginary part of the last matrix function [recall, for a
matrix $A\in M_{m}(\mathbb{C}),$ the imaginary part of $A$ is defined by
$\operatorname{Im}A=\frac{1}{2i}(A-A^{*})$ and if $A$ is invertible then
$\operatorname{Im}(A^{-1})=\operatorname{Im}(A^{-1}A^{*}(A^{-1})^{*})=-A^{-1}(\operatorname{Im}A)(A^{-1})^{*}$
and
$\operatorname{Im}(A^{-1})=\operatorname{Im}((A^{-1})^{*}A^{*}A^{-1})=-(A^{-1})^{*}(\operatorname{Im}A)A^{-1}$]
we find that

(5.60)
$\displaystyle\operatorname{Im}[H_{11}-H_{12}(H_{22}-z_{0}W_{22})^{-1}H_{21}]=-(\operatorname{Im}z_{0})H_{12}(H_{22}-z_{0}W_{22})^{-1}W_{22}[(H_{22}-z_{0}W_{22})^{-1}]^{*}H_{21}$
(5.61)
$\displaystyle=-(\operatorname{Im}z_{0})H_{12}(H_{22}-z_{0}W_{22})^{-1}W_{22}^{1/2}[H_{12}(H_{22}-z_{0}W_{22})^{-1}W_{22}^{1/2}]^{*}$

and

(5.62)
$\displaystyle\operatorname{Im}[H_{11}-H_{12}(H_{22}-z_{0}W_{22})^{-1}H_{21}]=-(\operatorname{Im}z_{0})H_{12}[(H_{22}-z_{0}W_{22})^{-1}]^{*}W_{22}(H_{22}-z_{0}W_{22})^{-1}H_{21}$
(5.63)
$\displaystyle=-(\operatorname{Im}z_{0})[W_{22}^{1/2}(H_{22}-z_{0}W_{22})^{-1}H_{21}]^{*}W_{22}^{1/2}(H_{22}-z_{0}W_{22})^{-1}H_{21},$

and we see from this that those conditions are equivalent to the conditions
(5.48)and (5.49). This completes the proof of statement (b).

Finally, the statements (a) and (b) are still true if we replace
$L^{1}_{loc}(\mathbb{R})$ by $L^{1}([0,d])$. The reason is that since $H,W$
are $d$-periodic functions, then $H,W\in M_{n}(L^{1}_{loc}(\mathbb{R}))$ if
and only if $H,W\in M_{n}(L^{1}([0,d])$. Similarly, due to the
$d$-periodicity, (5.48) and (5.49) are true if and only if
$W_{11},H_{11}-H_{12}(H_{22}-z_{0}W_{22})^{-1}H_{21}\in
M_{n_{1}}(L^{1}([0,d]))$. This completes the proof of the proposition. ∎

The next theorem gives very simple conditions which are sufficient for the
hypotheses in Theorem 5.9 to be true.

######  Theorem 5.13.

Suppose (5.2) is satisfied and $\det J=0$. If

(5.64) $\displaystyle H_{11},W_{11},H_{12}W_{22}^{-1}H_{21}\in
M_{n_{1}}(L^{1}_{loc}(\mathbb{R}))$

then the local index-1 hypotheses are true for $H-z_{0}W,W$ with respect to
$J$ on the interval $\mathbb{R}$, for every
$z_{0}\in\mathbb{C}\setminus\mathbb{R}.$ Moreover, the theorem is still true
if we replace $L^{1}_{loc}(\mathbb{R})$ by $L^{1}([0,d])$ in (5.64).

###### Proof.

Assume the hypotheses. Let $z_{0}\in\mathbb{C}\setminus\mathbb{R}$. To prove
the theorem, it suffices by Proposition 5.12.(b), to show that the conditions
(5.48) and (5.49) are satisfied. First, condition (5.48) is true by the
hypotheses. Second, by our hypotheses

(5.65)
$\displaystyle(W_{22}^{-1/2}H_{21})^{*}W_{22}^{-1/2}H_{21}=H_{12}W_{22}^{-1/2}(H_{12}W_{22}^{-1})^{*}=H_{12}W_{22}^{-1}H_{21}\in
M_{n_{1}}(L^{1}_{loc}(\mathbb{R}))$

implying $H_{12}W_{22}^{-1/2}\in M_{n_{1}\times
n_{2}}(L^{2}_{loc}(\mathbb{R})),W_{22}^{-1/2}H_{21}\in M_{n_{2}\times
n_{1}}(L^{2}_{loc}(\mathbb{R}))$. Third, it follows from this, Holder’s
inequality, and Lemma 5.11 that

(5.66) $\displaystyle H_{11}-H_{12}(H_{22}-z_{0}W_{22})^{-1}H_{21}$
$\displaystyle=H_{11}-H_{12}W_{22}^{-1/2}[W_{22}^{1/2}(H_{22}-z_{0}W_{22})^{-1}W_{22}^{1/2}]W_{22}^{-1/2}H_{21}\in
M_{n_{1}}(L^{1}_{loc}(\mathbb{R})).$

This shows that the conditions (5.48) and (5.49) are satisfied which proves
the theorem except for the “Moreover,…” part of this theorem. But this follows
from the fact that since $H_{11},W_{11},H_{12},W_{22},H_{21}$ are $d$-periodic
functions, then (5.64) is true if and only if
$H_{11},W_{11},H_{12}W_{22}^{-1}H_{21}\in M_{n_{1}}(L^{1}([0,d]))$. This
completes the proof of the theorem. ∎

######  Remark 5.14.

As we have just proven, the hypotheses (5.64) in Theorem 5.13 imply those of
(5.48) and (5.49) in Theorem 5.12 for every
$z_{0}\in\mathbb{C}\setminus\mathbb{R}$, provided (5.2) is satisfied and $\det
J=0$. We would like an answer to the following question we leave open: Is the
converse is true or not?

Next, we give an example that shows how useful Theorem 5.13 can be.

######  Example 5.15.

Let $n\in\mathbb{N}$ and $J\in M_{n}(\mathbb{C})$ with $J^{*}=-J\not=0$.
Consider the $d$-periodic DAEs (for any $d>0$) in canonical form:

$J\frac{d}{dt}f+Hf=\lambda Wf,$

where

$\displaystyle H=0,\;W=I_{n},$

that is,

$J\frac{d}{dt}f=\lambda f.$

The DA operator $\mathcal{L}:D(\mathcal{L})\to[\mathcal{M}(\mathbb{R})]^{n}$
associated with $\mathbb{R},J,H,W$ is

$\displaystyle
D(\mathcal{L})=\left\\{f\in[\mathcal{M}(\mathbb{R})]^{n}:Jf\in[W^{1,1}_{loc}(\mathbb{R})]^{n}\right\\}$
$\displaystyle\mathcal{L}f=W^{-1}\left(\frac{d}{dt}Jf+Hf\right)=\frac{d}{dt}Jf.$

Next, the Hilbert space $L^{2}(\mathbb{R};W)$ is just

$\displaystyle L^{2}(\mathbb{R};W)=[L^{2}(\mathbb{R})]^{n},$

and the maximal and minimal operators, $L:D(L)\rightarrow L^{2}(\mathbb{R};W)$
and $L_{0}^{\prime}:D(L_{0}^{\prime})\rightarrow L^{2}(\mathbb{R};W)$,
generated by $\mathcal{L}$ are

$\displaystyle Lf=\mathcal{L}f=\frac{d}{dt}Jf,\;\;\text{for }f\in D(L),$
$\displaystyle\;L_{0}^{\prime}f=\mathcal{L}f=\frac{d}{dt}Jf,\;\;\text{for
}f\in D(L_{0}^{\prime}),$

where

$\displaystyle D(L)=\\{f\in L^{2}(\mathbb{R};W):f\in
D(\mathcal{L}),\mathcal{L}f\in L^{2}(\mathbb{R};W)\\}$
$\displaystyle=\left\\{f\in[L^{2}(\mathbb{R})]^{n}:Jf\in[W^{1,1}_{loc}(\mathbb{R})]^{n},\frac{d}{dt}Jf\in[L^{2}(\mathbb{R})]^{n}\right\\},$
$\displaystyle D(L_{0}^{\prime})=\\{f\in D(L):Jf\text{ has compact support
contained in the interior of }\mathbb{R}\\}$ $\displaystyle=\\{f\in
D(L):(Jf)(t)=0\text{ for all $t$ sufficiently large}\\}.$

Let us now compare and contrast the spectral theory for $L$ in the two
different possible cases: (i) $\det J\not=0$; (ii) $\det J=0$. First, in
either case (i) or (ii), it follows immediately from Theorem 5.13 and
Proposition 5.12 that the local index-1 hypotheses (see Def. 4.14) are true
for $H-z_{0}W,W$ with respect to $J$ on the interval $\mathbb{R}$ for every
$z_{0}\in\mathbb{C}\setminus\mathbb{R}$ and hence by Theorem 5.9 it follows
that $L$ is a self-adjoint operator on $[L^{2}(\mathbb{R})]^{n}$,
$L_{0}^{\prime}$ is essentially self-adjoint with closure $L$, and $L$ has no
eigenvalues of finite multiplicity.

We will now prove explicitly that in the case (i) $\det J\not=0$, the operator
$L$ has no eigenvalues, whereas in the case (ii) $\det J=0$, the operator $L$
has only one eigenvalue, namely, $\lambda=0$ and then we will calculate its
eigenspace $E_{0}$ to show $\dim E_{0}=\infty$ thereby proving $\lambda=0$ is
an eigenvalue of $L$ of infinite multiplicity.

Suppose (i) $\det J\not=0$. Then

$\displaystyle
D(L)=\left\\{f\in[L^{2}(\mathbb{R})]^{n}:Jf\in[W^{1,1}_{loc}(\mathbb{R})]^{n},\frac{d}{dt}Jf\in[L^{2}(\mathbb{R})]^{n}\right\\}$
$\displaystyle=\left\\{f\in[L^{2}(\mathbb{R})]^{n}:f\in[W^{1,1}_{loc}(\mathbb{R})]^{n},\frac{d}{dt}f\in[L^{2}(\mathbb{R})]^{n}\right\\}.$

Hence if $f$ is an eigenvector of $L$ with corresponding eigenvalue $\lambda$
then $f\in D(L)$ and $Lf=\lambda f$ implying $\frac{df}{dt}=\lambda Jf$ and
hence $f(t)=e^{t\lambda J}f(0),\;\forall t\in\mathbb{R}$. As $(iJ)^{*}=iJ$,
this implies that $iJ$ has a orthonormal basis of eigenvectors
$v_{1},\ldots,v_{n}\in\mathbb{C}^{n}$ with corresponding eigenvalues
$\omega_{1},\ldots,\omega_{n}\in\mathbb{R}$ and there exists
$a_{1},\ldots,a_{n}\in\mathbb{C}$ such that $f(0)=\sum_{j=1}^{n}a_{j}v_{j}$
implying $f(t)=\sum_{j=1}^{n}a_{j}e^{-it\lambda\omega_{j}}v_{j},\;\forall
t\in\mathbb{R}$. As $f\in D(L)$ and is an eigenvector of $L$ then
$0<(f,f)<\infty$, but

$\displaystyle(f,f)=\int_{\mathbb{R}}f(t)^{*}f(t)dt=\sum_{j=1}^{n}\sum_{k=1}^{n}\int_{\mathbb{R}}[a_{j}e^{-it\lambda\omega_{j}}v_{j}]^{*}[a_{k}e^{-it\lambda\omega_{k}}v_{k}]dt$
$\displaystyle=\sum_{a_{j}\not=0}|a_{j}|^{2}\int_{\mathbb{R}}\overline{(e^{-it\lambda\omega_{j}})}e^{-it\lambda\omega_{j}}dt$
$\displaystyle=\sum_{a_{j}\not=0}|a_{j}|^{2}\int_{\mathbb{R}}e^{it\omega_{j}(\overline{\lambda}-\lambda)}dt$
$\displaystyle=\sum_{a_{j}\not=0}|a_{j}|^{2}\int_{\mathbb{R}}e^{-i2t\omega_{j}\operatorname{Im}\lambda}dt$

and since the integral

$\displaystyle\int_{\mathbb{R}}e^{-i2t\omega\operatorname{Im}\lambda}dt=\lim_{r\rightarrow-\infty,R\rightarrow+\infty}\int_{r}^{R}e^{-i2t\omega\operatorname{Im}\lambda}dt$

doesn’t exist as a finite limit for any $\omega\in\mathbb{R}$, this yields a
contradiction that $L$ had an eigenvector. This proves that $L$ has no
eigenvalues in the case (i) $\det J\not=0$.

Suppose (ii) $\det J=0$. Then since $(iJ)^{*}=iJ\not=0$, there is a unitary
matrix

$V\in M_{n}(\mathbb{C}),$

such that $V^{-1}JV=[J_{ij}]_{i,j=1,2}$ is in a $2\times 2$ block partitioned
matrix form as in (4.28) with

$\displaystyle J_{ij}\in M_{n_{i}\times
n_{j}}(\mathbb{C}),i,j=1,2,\;J_{ij}=0,\;\;(i,j)\not=(1,1),\;\;\det(J_{11})\neq
0,$

where $n_{1},n_{2}\in\mathbb{N}$ are defined by

$\displaystyle n_{1}:=\operatorname{rank}J\geq 1,\;\;n_{2}:=\dim\ker
J=\operatorname{nullity}(J)=n-n_{1}\geq 1.$

Furthermore, both of the matrices
$H=V^{-1}HV=[H_{ij}]_{i,j=1,2}=0,W=V^{-1}WV=[W_{ij}]_{i,j=1,2}=I_{n}\in
M_{n}(\mathcal{M}(\mathbb{R}))$ are block partitioned already conformal to the
block structure of $V^{-1}JV$ in (4.28), where

$\displaystyle H_{ij}=0,W_{ij}\in M_{n_{i}\times
n_{j}}(\mathcal{M}(\mathbb{R})),\;i,j=1,2;\;W_{11}=I_{n_{1}},\;W_{22}=I_{n_{2}},\;W_{12}=W_{21}=0.$

It now follows that

$\displaystyle
D(L)=\left\\{f\in[L^{2}(\mathbb{R})]^{n}:Jf\in[W^{1,1}_{loc}(\mathbb{R})]^{n},\frac{d}{dt}Jf\in[L^{2}(\mathbb{R})]^{n}\right\\}$
$\displaystyle=\left\\{V\begin{bmatrix}f_{1}\\\
f_{2}\end{bmatrix}:f_{1}\in[W^{1,1}_{loc}(\mathbb{R})]^{n_{1}},f_{1},\frac{df_{1}}{dt}\in[L^{2}(\mathbb{R})]^{n_{1}},f_{2}\in[L^{2}(\mathbb{R})]^{n_{2}}\right\\}$

and

$\displaystyle Lf=\frac{d}{dt}Jf=\frac{d}{dt}JV\begin{bmatrix}f_{1}\\\
f_{2}\end{bmatrix}=V\begin{bmatrix}J_{11}\frac{df_{1}}{dt}\\\
0\end{bmatrix},\;\text{ for all }f=V\begin{bmatrix}f_{1}\\\
f_{2}\end{bmatrix}\in D(L).$

Suppose that $f$ is an eigenvector of $L$ with corresponding eigenvalue
$\lambda$. That is, $0\not=f=V\begin{bmatrix}f_{1}\\\ f_{2}\end{bmatrix}\in
D(L)$ and

$\displaystyle V\begin{bmatrix}J_{11}\frac{df_{1}}{dt}\\\
0\end{bmatrix}=Lf=\lambda f=\lambda V\begin{bmatrix}f_{1}\\\
f_{2}\end{bmatrix},$

that is,

$\displaystyle J_{11}\frac{df_{1}}{dt}=\lambda
f_{1},\;\;f_{1}\in[W^{1,1}_{loc}(\mathbb{R})]^{n_{1}},f_{1},\frac{df_{1}}{dt}\in[L^{2}(\mathbb{R})]^{n_{1}},$
$\displaystyle\lambda f_{2}=0,\;f_{2}\in[L^{2}(\mathbb{R})]^{n_{2}}.$

As $J_{11}^{*}=-J_{11}\in M_{n_{1}}(\mathbb{C})$ and $\det J_{11}\not=0$, it
follows from the case (i) we just considered that we must have $\lambda=0$ and
it is the only eigenvalue of $L$ with eigenspace $E_{0}$ given by

$\displaystyle E_{0}=\left\\{V\begin{bmatrix}0\\\
f_{2}\end{bmatrix}:f_{2}\in[L^{2}(\mathbb{R})]^{n_{2}}\right\\}.$

Moreover, it obvious that $\dim E_{0}=\infty$. This proves that $L$ has only
one eigenvalue, namely, $\lambda=0$ and it is an eigenvalue of $L$ of infinite
multiplicity.

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# In-plane magnetic field induced density wave states
near quantum spin Hall phase transitions

Yongxin Zeng  Department of Physics, The University of Texas at Austin,
Austin, TX 78712  Fei Xue  Department of Physics, University of Alabama at
Birmingham, Birmingham, AL 35294  Allan H. MacDonald  Department of Physics,
The University of Texas at Austin, Austin, TX 78712

###### Abstract

We study the influence of an in-plane magnetic field and Coulomb interactions
on the physics of quantum spin Hall insulators, like those in InAs/GaSb and
HgTe/CdTe quantum wells. Using a Hartree-Fock mean-field theory approximation,
we calculate phase diagrams as functions of the band gap, band hybridization,
and magnetic field strength. We show that when the band hybridization is weak,
the system is unstable against the formation of density wave states. As the
strength of the in-plane magnetic field increases, the density-wave region of
the phase diagram expands and distinct density-wave states appear. We discuss
possible experimental implications of our results.

##  I Introduction

The quantum spin Hall (QSH) insulator is a topologically nontrivial state of
matter characterized by gapless helical edge states protected by time-reversal
symmetry [1, 2]. It was first realized in HgTe/CdTe quantum wells [3, 4], and
later also in other systems like InAs/GaSb quantum wells [5, 6, 7].
Theoretically the physics of QSH insulators is captured by the Bernevig-
Hughes-Zhang (BHZ) model [3], which is a single-particle theory that ignores
interactions and works well in the limit of strong band hybridization.
Interactions become important when the BHZ model band hybridization parameter
$A$ is small, as can be appreciated by considering the limit $A\to 0$, where
coherence between conduction and valence bands, or exciton condensation,
occurs spontaneously when the band gap is smaller than the exciton binding
energy [8, 9, 10, 11, 12, 13, 14, 15]. Recent experiments [16, 17, 18] have
shown excitonic behavior in InAs/GaSb quantum wells. The interplay between
interactions and topology can lead to interesting new phases near the QSH
phase transition [19, 20, 21, 22], which have so far been only lightly
explored.

In this paper we study how an in-plane magnetic field modifies the phase
diagram studied in Ref. 22, which contains time-reversal symmetry-breaking
electron nematic phases. Due to the spatial separation between electron and
hole layers, an in-plane magnetic field shifts the conduction and valence
bands in opposite directions in momentum space. Intuitively this opposite
shift effectively increases the band gap and reduces the hybridization between
the electron and hole bands. When interactions are neglected, an in-plane
magnetic field drives the system into a semi-metallic state. Using a Hartree-
Fock mean-field theory, we show that the nematic states instead break
translational symmetry and become density-wave states. At stronger tunneling
an in-plane magnetic field can drive the system through a variety of different
phases, including quantum anomalous Hall states with and without density-wave
order.

This paper is organized as follows: In Section II we formulate the mean-field
theory we use to describe interaction effects, and explain how we allow the
possibility of translational symmetry breaking. In Section III we summarize
our results by presenting phase diagrams that depend on three parameters: band
gap, hybridization, and the strength of in-plane magnetic field. Finally in
Section IV we discuss the relationship between our work and potential future
experiments, and its relationship to excitonic density-wave states that have
been identified, often controversially, in bulk three-dimensional crystals.

##  II Mean-field theory

We use a four-band BHZ model [3, 5] to describe the InAs/GaSb quantum wells.
The field operators are four-component spinors
$\psi_{\bm{k}}=(a_{c\uparrow\bm{k}},a_{v\uparrow\bm{k}},a_{c\downarrow\bm{k}},a_{v\downarrow\bm{k}})^{T}$,
where $c$ and $v$ denote the conduction and valence bands, and $\uparrow$ and
$\downarrow$ denote two opposite spins. The single-particle physics of the
system under an in-plane magnetic field is described by the modified BHZ
Hamiltonian [23, 21]

$H_{\rm{BHZ}}=\sum_{\bm{k}}\psi_{\bm{k}}^{\dagger}\left(\begin{array}[]{cc}h_{\uparrow}(\bm{k})&0\\\
0&h_{\downarrow}(\bm{k})\end{array}\right)\psi_{\bm{k}},$ (1)

where the two $2\times 2$ matrices $h_{\uparrow}$ and $h_{\downarrow}$ can be
explicitly expressed as

$\begin{split}h_{\uparrow}(\bm{k})&=\left(\begin{array}[]{cc}\frac{\hbar^{2}}{2m_{e}}(\bm{k}-\frac{\bm{Q}}{2})^{2}+\frac{E_{g}}{2}&A(k_{x}+ik_{y})\\\
A(k_{x}-ik_{y})&-\frac{\hbar^{2}}{2m_{h}}(\bm{k}+\frac{\bm{Q}}{2})^{2}-\frac{E_{g}}{2}\end{array}\right),\\\
h_{\downarrow}(\bm{k})&=\left(\begin{array}[]{cc}\frac{\hbar^{2}}{2m_{e}}(\bm{k}-\frac{\bm{Q}}{2})^{2}+\frac{E_{g}}{2}&-A(k_{x}-ik_{y})\\\
-A(k_{x}+ik_{y})&-\frac{\hbar^{2}}{2m_{h}}(\bm{k}+\frac{\bm{Q}}{2})^{2}-\frac{E_{g}}{2}\end{array}\right).\end{split}$
(2)

$m_{e}$ and $m_{h}$ are the effective masses of electrons and holes, $E_{g}$
is the band gap, $A$ is the strength of hybridization between the conduction
and valence bands, and $\bm{Q}$ is the momentum shift due to the in-plane
magnetic field. Without the magnetic field, $\bm{Q}=0$ and
$h_{\uparrow},h_{\downarrow}$ are time-reversal partners:

$h_{\uparrow}(\bm{k})=h_{\downarrow}^{*}(-\bm{k}).$ (3)

When an in-plane magnetic field $\bm{B}=B\hat{y}$ is applied to electron and
hole layers separated by an interlayer distance $d$, the conduction and
valence bands are shifted in momentum by $\mp\bm{Q}=\mp(eBd/\hbar)\hat{x}$ by
Peierls substitution. The in-plane magnetic field breaks the time-reversal
symmetry of the system and induces orbital moments. The electrons and holes
interact via the Coulomb interaction

$H_{I}=\frac{1}{2S}\sum_{bb^{\prime}ss^{\prime}}\sum_{\bm{kk^{\prime}q}}V_{bb^{\prime}}(q)a_{bs\bm{k}+\bm{q}}^{\dagger}a_{b^{\prime}s^{\prime}\bm{k}^{\prime}-\bm{q}}^{\dagger}a_{b^{\prime}s^{\prime}\bm{k}^{\prime}}a_{bs\bm{k}},$
(4)

where $S$ is the area of the two-dimensional system, $b,b^{\prime}$ and
$s,s^{\prime}$ are band and spin indices respectively,
$V_{cc}(q)=V_{vv}(q)=V(q)=2\pi e^{2}/\epsilon q$ is the intralayer Coulomb
interaction, $V_{cv}(q)=V_{vc}(q)=U(q)=V(q)\exp(-qd)$ is the interlayer
Coulomb interaction at interlayer distance $d$, and $\epsilon$ is the
dielectric constant of the surrounding three-dimensional material.

Anticipating the possibility of translational symmetry breaking [21] along the
direction of $\bm{Q}$, we divide momentum space into slabs separated by
$\bm{Q}$. Then apart from the band and spin indices $(b,s)$, the basis states
are labeled by an integer $n$ and a quasi-momentum $\bm{k}$
($|\bm{k}|<|\bm{Q}|/2$) that lies within the first quasi-one-dimensional
Brillouin zone. Together $n$ and $\bm{k}$ refer to the plane-wave state with
momentum $n\bm{Q}+\bm{k}$.

We use a Hartree-Fock mean-field theory to describe the Coulomb interaction.
The Hartree term is

$\begin{split}\Sigma_{H}=\frac{1}{S}\sum_{bb^{\prime}\atop
ss^{\prime}}\sum_{nn^{\prime}n^{\prime\prime}\atop\bm{k}\bm{k}^{\prime}}&V_{bb^{\prime}}((n^{\prime}-n)\bm{Q})\\\
\times&\rho_{b^{\prime}s^{\prime}\,n^{\prime\prime}}^{b^{\prime}s^{\prime}\,n^{\prime\prime}+n^{\prime}-n}(\bm{k}^{\prime})\;a_{bsn^{\prime}\bm{k}}^{\dagger}a_{bsn\bm{k}},\end{split}$
(5)

where the density matrix $\rho$ is defined relative to the density matrix with
valence bands filled and conduction bands empty:

$\rho_{b^{\prime}s^{\prime}n^{\prime}}^{bsn}(\bm{k})=\langle
a_{b^{\prime}s^{\prime}n^{\prime}\bm{k}}^{\dagger}a_{bsn\bm{k}}\rangle-\delta_{bb^{\prime}}\delta_{bv}\delta_{ss^{\prime}}\delta_{nn^{\prime}}.$
(6)

For $n^{\prime}=n$, the Hartree term accounts for the electrostatic potential
energy difference $4\pi e^{2}n_{x}d/\epsilon$ between the electron and hole
layers, where

$n_{x}=\frac{1}{S}\sum_{sn\bm{k}}\rho_{csn}^{csn}(\bm{k})=-\frac{1}{S}\sum_{sn\bm{k}}\rho_{vsn}^{vsn}(\bm{k})$
(7)

is the exciton density. The Fock term

$\begin{split}\Sigma_{F}=-\frac{1}{S}\sum_{bb^{\prime}\atop
ss^{\prime}}\sum_{nn^{\prime}n^{\prime\prime}\atop\bm{k}\bm{k}^{\prime}}&V_{bb^{\prime}}((n^{\prime\prime}-n)\bm{Q}+\bm{k}^{\prime}-\bm{k})\\\
\times&\rho_{bs\,n^{\prime\prime}}^{b^{\prime}s^{\prime}\,n^{\prime\prime}+n^{\prime}-n}(\bm{k}^{\prime})a_{b^{\prime}s^{\prime}n^{\prime}\bm{k}}^{\dagger}a_{bsn\bm{k}}.\end{split}$
(8)

accounts for the exchange interaction. Together, the system is described by
the mean-field Hamiltonian

$H_{\rm{MF}}=H_{\rm{BHZ}}+\Sigma_{H}+\Sigma_{F}.$ (9)

Below we express lengths and energies in terms of characteristic scales
$a_{B}^{*}=\epsilon\hbar^{2}/me^{2}$ and $Ry^{*}=e^{2}/2\epsilon a_{B}^{*}$,
where $m=m_{e}m_{h}/(m_{e}+m_{h})$ is the reduced effective mass. This model
approximates InAs/GaSb quantum wells if we choose
$m_{e}=0.023m_{e},m_{h}=0.4m_{0}$, and $\epsilon=15$ [24], which implies that
$a_{B}^{*}=$36.5\text{\,}\mathrm{n}\mathrm{m}$$ and
$Ry^{*}=$1.3\text{\,}\mathrm{m}\mathrm{e}\mathrm{V}$$. We assume the
interlayer distance
$d=0.3~{}a_{B}^{*}\approx$10\text{\,}\mathrm{n}\mathrm{m}$$. For an in-plane
magnetic field of strength $B=$1\text{\,}\mathrm{T}$$, the momentum shift
$Q=0.606~{}(a_{B}^{*})^{-1}$. For simplicity in Eq. (2) we neglect the
particle-hole asymmetry which does not affect the ground state physics, and
perform numerical calculations with $m_{e}=m_{h}=2m$.

##  III Phase diagrams

(a)  (b)

(c)  (d)

Figure 1: Phase diagrams in $(A,E_{g})$ plane at several fixed magnetic
fields: (a) $Q=0$; (b) $Qa_{B}^{*}=0.6$; (c) $Qa_{B}^{*}=1.2$; (d)
$Qa_{B}^{*}=1.8$. Solid and dotted lines distinguish first-order and
continuous phase transitions. The gray dots in (d) stand for the parameters
used for later illustrative calculations in Figs. 2 and 3. For adjacent InAs
and GaSb layers with thickness $d_{\rm InAs}=d_{\rm
GaSb}=$10\text{\,}\mathrm{n}\mathrm{m}$$, the hybridization strength [25]
$A=$0.37\text{\,}\mathrm{e}\mathrm{V}\,\AA$=0.78\,a_{B}^{*}Ry^{*}$. The
properties of the different phases identified here are described in the main
text.

Given $E_{g}$, $A$ and $Q$, the ground state of the system can be obtained by
solving the Hartree-Fock equations self-consistently and finding the lowest-
energy solution. The $Q=0$ case has been studied in detail in Ref. 22, and the
phase diagram is reproduced in Fig. 1(a). At $A=0$ the number of particles in
the conduction and valence bands are conserved separately. When the bare
energy gap $E_{g}$ is reduced below the $1s$ exciton binding energy, the
excitons condense and coherence is established spontaneously between the
conduction and valence bands. This order survives at finite $A$, where it
breaks rotational symmetry by establishing coherence between $s$-conduction
and $p$-valence electrons, and also breaks time-reversal symmetry by doing so
in a spin-dependent manner. In Fig. 1 we refer to this state as the nematic
insulator state. At large $A$, single-particle physics dominates and the
system undergoes a topological phase transition between the QSH and normal
insulators as $E_{g}$ varies. At moderate values of $A$, the transition
between QSH and normal insulators occurs via an intermediate quantum anomalous
Hall state (QAH) state with broken time-reversal symmetry and a nonzero Chern
number 111In this summary, we have ignored some minor phases that occur in a
small region of the phase diagram near the $A=0$ line..

When an in-plane magnetic field is applied, the QSH state is no longer
protected by time-reversal symmetry. In the simplified BHZ Hamiltonian (1) we
use here, the two spins are decoupled and a spin Chern number can be defined
to distinguish QSH and normal insulators [27]. In the more general case where
spin is not a good quantum number, it has been shown that [28, 29, 30] the
spin Chern number can remain well-defined as a robust topological invariant.
For this reason the QSH-normal insulator transition still exists. In order to
distinguish these two cases we refer to the finite-$B$ QSH state as a time-
reversal symmetry-breaking (TRSB) QSH state [29].

At small hybridization $A$ the momentum-shifted conduction and valence bands
tend to establish coherence by breaking translational symmetry to achieve
better Fermi surface nesting, forming density wave (DW) states with wavevector
$\bm{Q}$. We find that at small but finite $A$ the energetically preferred
state is one in which pairing is between opposite spins, so the order
parameter is $\rho_{v\bar{s}0}^{cs1}(0)$ where $\bar{s}$ denotes the spin
opposite to $s$. At $A=0$ the density matrix element
$\rho_{b^{\prime}\bar{s}0}^{bs1}$ is nonzero only for $b=c,b^{\prime}=v$. When
the band-hybridization parameter $A$ is non-zero, on the the other hand, it is
nonzero for any $b$ and $b^{\prime}$, although the exciton condensate order
parameter ($b=c,b^{\prime}=v$) is always much larger than the other three
($b=b^{\prime}=v$, $b=b^{\prime}=c$, and $b=v,b^{\prime}=c$) density-matrix
elements. When the magnetic field is weak, the DW state exists only near $A=0$
and undergoes a first-order phase transition to the nematic insulator state,
which does not have finite-$Q$ pairing, as $A$ increases (see Fig. 1(b)). The
nematic insulator phase is characterized by the order parameter
$\rho_{v\bar{s}0}^{cs0}(0)$; the coherence that is established does not
accomodate the momentum-space shifts of the conduction and valence bands. We
retain the term nematic insulator used at $Q=0$ even though rotational
symmetry has already been explicitly broken by the in-plane magnetic field.

The boundary between the magnetic-field-stabilized DW state and the nematic
insulator state moves rapidly to the right as the magnetic field strength
increases, eventually squeezing the nematic insulator state out of the phase
diagram as shown in Fig. 1(c). Near the DW phase boundary neighboring the
TRSB-QSH and QAH phase regions, two new phases appear 222We have ignored some
minor phases that appear near the phase boundaries. We find that the QSH/DW
and QAH/DW phases are more stable as the magnetic field strength increases.
that also break translational symmetry along the $\hat{x}$-direction. These
two states are connected to the TRSB-QSH and QAH states via continuous phase
transitions, and we label them as QSH/DW and QAH/DW states respectively, for
reasons that will become clear later. Like the DW state, both the QSH/DW and
QAH/DW states have order parameters of the form
$\rho_{b^{\prime}\bar{s}0}^{bs1}(0)$. Unlike the DW state, however, the
largest pairing terms in the QSH/DW state are between conduction and
conduction, and valence and valence bands, yielding order parameters
$\rho_{v\bar{s}0}^{vs1}(0)(=\rho_{c\bar{s}\bar{1}}^{cs0}(0))$. The QAH/DW
state has different up-to-down and down-to-up spin pairings:
$\rho_{b^{\prime}\uparrow 0}^{b\downarrow 1}\neq\rho_{b^{\prime}\downarrow
0}^{b\uparrow 1}$, with one, say $\rho_{b^{\prime}\uparrow 0}^{b\downarrow
1}$, resembling the DW state with the largest element appearing at
$b=c,b^{\prime}=v$, and the other ($\rho_{b^{\prime}\downarrow 0}^{b\uparrow
1}$) resembling the QSH/DW state with the largest element at $b=b^{\prime}=v$.
The transitions between the DW, QAH/DW and QSH/DW states are all first-order
transitions.

As $Q$ continues to increase, the three density-wave regions keep expanding
and the QAH region shrinks as shown in Fig. 1(d). While the transitions
between the TRSB-QSH, QAH and normal insulator states stay largely unchanged,
the phase boundaries slowly move towards larger $A$ and smaller $E_{g}$ as $Q$
increases, consistent with the intuition that the momentum shift between
conduction and valence bands effectively increases the band gap and weakens
the band hybridization.

Figure 2: Spatial distribution of the charge density (in units of $-e$) in the
electron layer $n_{e}$ at $Qa_{B}^{*}=1.8$ and (i)
$E_{g}=0.1\,Ry^{*},A=0.3\,a_{B}^{*}Ry^{*}$ (DW, yellow); (ii)
$E_{g}=0.1\,Ry^{*},A=0.5\,a_{B}^{*}Ry^{*}$ (QAH/DW, green); (iii)
$E_{g}=-0.4\,Ry^{*},A=0.5\,a_{B}^{*}Ry^{*}$ (QSH/DW, blue).

For $A=0$, the ground states are true exciton condensates and ordering does
not lead to charge-density variations in either layer even if it occurs at
$Q\neq 0$. For $A\neq 0$, the situation changes. The charge densities in the
electron and hole layers (in units of $-e$) are related to the density
matrices $\rho$ by

$\begin{split}n_{e}(\bm{r})&=\frac{1}{S}\sum_{snn^{\prime}\bm{k}}\rho_{csn^{\prime}}^{csn}(\bm{k})e^{-i(n^{\prime}-n)\bm{Q}\cdot\bm{r}},\\\
n_{h}(\bm{r})&=\frac{1}{S}\sum_{snn^{\prime}\bm{k}}\rho_{vsn^{\prime}}^{vsn}(\bm{k})e^{-i(n^{\prime}-n)\bm{Q}\cdot\bm{r}}.\end{split}$
(10)

At $A=0$, the only nonzero $n^{\prime}\neq n$ density-matrix elements in Eq.
(10) are $\rho_{c\bar{s}\,n+1}^{vs\,n}$, so the charge density is uniform in
each layer. When $A\neq 0$, $\rho^{bs\,n}_{bs\,n\pm 1}$ still vanishes because
only opposite spins in neighboring Brillouin zones are coupled. However,
$\rho^{bs\,n}_{bs\,n\pm 2}$ can be nonzero when $A\neq 0$, so the charge
density in each layer oscillates periodically in space, with periodicity
$\pi/Q$. Fig. 2 shows the charge density distribution in the electron layer of
the three density-wave states. We see that the QSH/DW state has larger spatial
charge density fluctuations than the other density-wave states, as expected
because of its conduction-to-conduction and valence-to-valence band couplings.
In our calculation with $m_{e}=m_{h}$, the charge density in the hole layer is
exactly the opposite of that in the electron layer due to the particle-hole
symmetry present for this parameter choice. In the general $m_{e}\neq m_{h}$
case there is partial cancellation between the charge densities in the two
layers, resulting in a weak total charge density oscillation in space.

Figure 3: Wilson loop calculations at (a)
$E_{g}=1\,Ry^{*},A=0.9\,a_{B}^{*}Ry^{*}$ (normal insulator); (b)
$E_{g}=0.1\,Ry^{*},A=0.8\,a_{B}^{*}Ry^{*}$ (QAH); (c)
$E_{g}=-0.4\,Ry^{*},A=0.9\,a_{B}^{*}Ry^{*}$ (TRSB-QSH); (d)
$E_{g}=0.1\,Ry^{*},A=0.3\,a_{B}^{*}Ry^{*}$ (DW); (e)
$E_{g}=0.1\,Ry^{*},A=0.6\,a_{B}^{*}Ry^{*}$ (QAH/DW); (f)
$E_{g}=-0.4\,Ry^{*},A=0.6\,a_{B}^{*}Ry^{*}$ (QSH/DW).

Figure 4: Phase diagram (a) in $(A,Q)$ plane at fixed $E_{g}=0.5\,Ry^{*}$; (b)
in $(Q,E_{g})$ plane at fixed $A=0.6\,a_{B}^{*}Ry^{*}$.

The topological properties of these phases can be studied by performing Wilson
loop calculations [32, 33]. Our results are shown in Fig. 3. For each $k_{x}$,
we calculate the product $D(k_{x})$ of the Berry connection matrices
$F_{i,i+1}^{m,n}=\langle u_{i}^{m}|u_{i+1}^{n}\rangle$ along $k_{y}$, where
$i$ labels steps along $k_{y}$ and $m,n$ label occupied states. We then
calculate the phase angles $\theta(k_{x})$ of the eigenvalues of the matrix
$D(k_{x})$. The topological properties of the system can be read from the
winding behavior of the phase angles. For example, the Chern number is equal
to the net number of times (upwards minus downwards) the evolution curves of
$\theta$ cross a constant reference line parallel to the $k_{x}$-axis. The
results show that the phase angles of the normal insulator (Fig. 3(a)) and DW
(Fig. 3(d)) states always stay near zero, so these two states are both
topologically trivial. In the QAH state (Fig. 3(b)) one of the spins undergoes
band inversion, and the phase angle $\theta$ jumps by $2\pi$ as $k_{x}$ sweeps
from $-Q/2$ to $Q/2$. Interestingly, the Wilson loop of the QAH/DW state (Fig.
3(e)) shows very similar winding behavior as the QAH state. We conclude that
both the QAH and QAH/DW states are topologically nontrivial, characterized by
Chern number $|\mathcal{C}|=1$. For the TRSB-QSH state (Fig. 3(c)) the two
decoupled spin bands are both inverted and the Wilson loops exhibit nontrivial
but opposite winding along $k_{x}$, resulting in zero total Chern number. In
the QSH/DW state (Fig. 3(f)), however, the two spins are coupled via density-
wave order, and the degeneracy at $k_{x}=\pm Q/2$ is lifted. Despite certain
similarity to that of the QSH state, the Wilson loop of the QSH/DW state
suggests that the system is topologically trivial 333In principle we may apply
the definition of spin Chern number in Ref. 28. However, here we follow a more
conservative convention and do not calculate the spin Chern number for the
QSH/DW state, because the definition involves a rather artificial requirement
on adiabatic path [30] and lacks physical significance.. In fact, the QSH/DW
state we find is very similar to the topological charge density wave state
discovered in Ref. 21, except that in Ref. 21 the density-wave order parameter
is between the same spins, whereas we find the system has lower energy when
the coupling is between opposite spins. It is the coupling between different
spins that lifts the degeneracy at $k_{x}=\pm Q/2$ and gives rise to a trivial
Wilson loop as shown in Fig. 3(f).

The evolution of phase diagrams with $Q$ opens up the possibility of tuning
between different phases by applying an in-plane magnetic field. Fig. 4 shows
two phase diagrams in which the in-plane field parameter $Q$ is along one
axis. Fig. 4(a) shows the phase diagram in the $(A,Q)$ plane at fixed
$E_{g}=0.5\,Ry^{*}$. At small $A$, increasing the magnetic field turns the
nematic insulator phase into the DW phase. At large $A$, the magnetic field
drives the QSH state into the normal insulator state. The QAH state shows up
at intermediate $A$. Fig. 4(b) shows the phase diagram in $(Q,E_{g})$ plane at
fixed $A=0.6\,a_{B}^{*}Ry^{*}$. At large positive $E_{g}$, the system stays in
the normal insulator state. At small or negative $E_{g}$, the system starts
from the QAH or QSH state and ends up in one of the three density-wave states
as the magnetic field gets stronger.

##  IV Discussion

This paper describes a study of the influence of in-plane magnetic fields on
the many-electron ground states of two-dimensional electron gas systems with a
conduction band in one layer hybridized with a valence band in a nearby layer.
The interesting regime is one in which the spatially indirect gap is smaller
than the corresponding exciton binding energy, so that the ground state has
electron-hole coherence even when hybridization is neglected. The role of an
in-plane field is to associate a momentum boost with inter-layer tunneling
processes. The phase diagrams we construct as a function of the energy gap
$E_{g}$ and the hybridization strength $A$ can in principle be tested
experimentally by fabricating devices containing interfaces between InAs and
GaSb, or other materials combinations with appropriate band lineups, and using
dual gates to tune $E_{g}$ at fixed electron density. In the InAs/GaSb case
the hybridization strength $A$ can be reduced by inserting an AlSb barrier
layers [17, 18] between the InAs and GaSb layers. Our study builds on earlier
work [22] which studied the influence of interactions on the phase transition
between ordinary and quantum spin Hall insulators in the absence of a magnetic
field, and on work [21] which studied InAs/GaSb interfaces in the presence of
a perpendicular field but did not identify all competing ordered phases. Some
related experimental progress has already been reported in recent transport
experiments by Du and collaborators [16, 17, 18]. Our rich theoretical phase
diagrams suggest that there is much more to discover.

In a non-interacting electron theory, an in-plane magnetic field closes the
hybridization gaps that appear for $E_{g}<0$ and converts the neutral system
from insulators into semimetals. The magnetic field strength needed to close
the gap increases with the strength of the hybridization parameter $A$. When
interactions are included, the ground state remains insulating at all magnetic
fields, by breaking translational symmetry to establish coherence between
electron and hole states that have been boosted to different momenta. The
observation of a gap under a strong in-plane magnetic field in experiment [16,
18] is likely to be of many-body origin, and can be attributed to the density-
wave states studied here. In the parameter range studied in this work, the
semimetal state is never stable against the formation of density waves. More
direct evidence for density-wave states could come from transport measurements
that show nonlinear current-voltage characteristics [35, 36, 37, 38].

Charge density wave states are often observed experimentally in bulk three-
dimensional narrow gap semiconductors or semimetals in which conduction band
minima and valence band maxima occur at different wavevectors. Recent examples
include TiSe2 and Ta2NiSe5 [39, 40, 41, 42]. There is typically some debate
about the origin of charge density-wave states in this type of system. We take
the view that they can almost all be regarded as exciton-insulators 444The use
of the term condensate is usually taken to mean that there is an analogy to
Bose-Einstein condensation, but it should be kept in mind that this analogy is
always imprecise since the number of electrons and holes are not conserved
separately in bulk three dimensional crystals., in the same sense as the
charge density wave states studied in this paper can be regarded as exciton-
insulators. The use of this terminology to classify the type of charge density
wave state is meant to suggest that if only the band gap of the system could
be varied, the ordered state would appear when the minimum energy of the
excitonic collective modes, always present below the interband particle-hole
continuum, vanishes. The bilayer hybridized electron-hole systems studied in
this paper have the advantage that the key microscopic parameters of excitonic
charge-density-wave systems, the energy gap and the ordering wavevector, can
indeed be varied. Their experimental study therefore has the potential to draw
a clear line connecting this type of charge density wave to ideal exciton
condensates.

###### Acknowledgements.

This work was supported by the National Science Foundation through the Center
for Dynamics and Control of Materials: an NSF MRSEC under Cooperative
Agreement No. DMR-1720595.

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# Energy-Efficient Precoding for Multi-User Visible Light Communication with
Confidential Messages

Son T. Duong1, Thanh V. Pham2, Chuyen T. Nguyen1, and Anh T. Pham2 Emails:
<EMAIL_ADDRESS><EMAIL_ADDRESS>2{tvpham,
<EMAIL_ADDRESS>1Hanoi University of Science and Technology, Vietnam.
2Computer Communications Lab., The University of Aizu, Japan.

###### Abstract

In this paper, an energy-efficient precoding scheme is designed for multi-user
visible light communication (VLC) systems in the context of physical layer
security, where users’ messages are kept mutually confidential. The design
problem is shown to be non-convex fractional programming, therefore Dinkelbach
algorithm and convex-concave procedure (CCCP) based on the first-order Taylor
approximation are utilized to tackle the problem. Numerical results are
performed to show the convergence behaviors and the performance of the
proposed solution for different parameter settings.

###### Index Terms:

Multi-user VLC, energy efficiency, physical layer security, precoding.

##  I Introduction

Visible light communication (VLC) has been becoming an attractive wireless
solution to complement existing technologies due to its high-capacity data
transmission with license-free spectrum. Aside from this, the technology also
takes advantage of the widespread deployment of light-emitting diodes (LEDs).
This naturally enables VLC to fit into the future ubiquitous networks.

However, multiple challenges still remain to make VLC more viable, among which
security in terms of information privacy and confidentiality (especially in
public areas) are of the most important issues [1]. In this respect, physical
layer security (PLS) has emerged as novel paradigm to enhance secure
communication by exploiting the randomness of the wireless channels, noise,
and interference. The most promising implication of PLS is that a perfect
secured communication can be achieved from information theoretic point of
view. The secrecy of PLS is quantized by the secrecy rate that defines the
maximum transmission rate at which unauthorized users are unable to extract
any information from the received signals regardless of their computational
capability. While PLS has been a well investigated topic in radio frequency
(RF) communications, it has only been receiving considerable attention in the
past few years in the case of VLC. In practical VLC systems, multiple LED
luminaries should usually be deployed to provide a sufficient illumination. As
such, multiple-input single-output (MISO) channels are more prevalent. In such
scenarios, the degrees of freedom introduced by multiple LED transmitters
enables the use of precoding as a means of secrecy enhancement [2, 3, 4, 5, 6,
7, 8, 9].

Aside from the security, energy efficiency is another essential criterion in
designing a communications system. This increasing attention to energy
consumption comes from the current global effort to reduce the carbon
footprint. In regard to PLS in VLC, there has been a few studies dealing with
precoding design from the energy consumption perspective. In the case of one
legitimate user and multiple eavesdroppers, the authors in [4] and [10]
considered the problem of minimizing the power of the information-bearing
signal while ensuring a minimum achievable secrecy rate. With the same
configuration, [11, 12] focused on designing artificial noise-aided precoding.
The total power consumption of the information-bearing signal and the
artificial noise is then minimized taking into account predefined thresholds
on the signal-to-interference-plus-noise ratios (SINRs) of the legitimate user
and eavesdroppers.

In general, the above mentioned works concerned with the issue of minimizing
the consumed power given that a certain secrecy requirement is fulfilled. This
design approach, however, is not necessarily optimal from the perspective of
energy efficiency, which is defined as the number of bits that can be
transmitted per Joule. In other words, it is the ratio of the achievable
(secrecy) rate to the total power consumption. Considering that an energy-
efficient precoding design for PLS in multi-user (MU) VLC systems has not been
studied, the objective of this paper is to fill this gap. Particularly, we
examine an MU-MISO VLC system where each user treats others as eavesdroppers,
hence its intended message must be kept confidential. The energy efficiency in
the context of PLS of such system is then formulated using a lower bound on
the achievable secrecy sum-rate. Regarding the power consumption, in addition
to the power consumed for the information-bearing signal and circuitry
operations (e.g., as in radio frequency (RF) communications), the lighting
function of VLC requires an additional power for illumination, which also
impacts the achievable secrecy rate. The optimal precoding design problem to
maximize the energy efficiency is shown to be non-convex fractional
programming, which renders finding an exact solution difficult. Hence, our
approach is to find a sub-optimal yet computational efficient solution by
employing the Dinkelbach algorithm and convex-concave procedure (CCCP).
Simulations under different system parameters are conducted to verify the
efficiency of the proposed solution.

The rest of the paper is structured as follows. The system model together with
a formulation of the energy efficiency are described in Section II. Section
III focuses on solving the precoding design to maximize the energy efficiency.
Numerical results and related discussions are given in Section IV. Finally,
Section V concludes the paper.

Notation: $\mathbb{R}^{m\times n}$ denotes the $m\times n$ real-valued
matrices. The uppercase bold symbols, e.g. $\mathbf{M}$, denote matrices,
while the lowercase ones such as $\mathbf{v}$ represent column vectors. The
transpose of $\mathbf{M}$ is written as $\mathbf{M}^{T}$. The $k$-th column
vector of $\mathbf{M}$ is denoted as $\mathbf{m}_{k}$, while its ($i,j$)-th
element is written as $m_{i,j}$. Additionally, $|\cdot|_{1}$,
$\lVert\cdot\rVert$ and $|\cdot|$ are ${L}_{1}$ norm, the Euclidean norm and
absolute value operator, respectively.

##  II System Model

Figure 1: A simple example of the considered MU-MISO VLC system with $N_{T}=4$
LED arrays and $K=3$ users.

Our considered MU-MISO VLC system as illustrated in Fig. 1 consists of $N_{T}$
LED luminaries and $K$ decentralized users, where each user is equipped with a
photodiode (PD). It is reminded that the transmission is considered to be
confidential if each user is not able to decode any information intended to
the others.

###  II-A Signal Model

Let
$\mathbf{d}=\begin{bmatrix}d_{1}&d_{2}&\cdots&d_{K}\end{bmatrix}^{T}\in\mathbb{R}^{K\times
1}$ be the vector of data symbols for all users. Assume that the symbols are
drawn from an $M$-ary pulse amplitude modulation ($M$-PAM) and are modeled as
a random variable (RV) $d$ following a certain distribution over $[-1,1]$ with
zero-mean and variance $\sigma^{2}_{d}$. An information-bearing signal $s_{n}$
for the $n$-th LED transmitter is generated from a linear combination of the
data vector and a precoder
$\mathbf{v}_{n}=\begin{bmatrix}w_{n,1}&w_{n,2}&\cdots&w_{n,K}\end{bmatrix}\in\mathbb{R}^{1\times
K}$ as

$s_{n}=\mathbf{v}_{n}\ \mathbf{d}.$ (1)

For illumination, an DC bias $I_{n}^{\text{DC}}$ should be added to $s_{n}$ to
create a non-negative drive current $x_{n}$ for the LED. The drive current, in
addition, needs to be constrained to a maximum threshold, i.e.,
$I_{\text{max}}$, to ensure that LEDs operate normally. Therefore,

$0\leq x_{n}=s_{n}+I_{n}^{\text{DC}}\leq I_{\text{max}}.$ (2)

The emitted optical power of each LED luminary is given by

${P}_{n}^{s}=\eta\left(s_{n}+I_{n}^{\text{DC}}\right),$ (3)

where $\eta$ is the LED conversion factor. Denote $\mathbf{h}_{k}=[h_{1,k}\
h_{2,k}\ \cdots\ h_{N_{T},k}]^{T}$ is the $k$-th user’s channel matrix, where
$h_{n,k}$ is the line-of-sight (LoS) channel coefficient between the $n$-th
LED array and the user. Details on the VLC channel model can be found in [1]
and references therein. The electrical signal at the PD output is then given
by

$\displaystyle y_{k}$
$\displaystyle=\gamma\mathbf{h}^{T}_{k}\begin{bmatrix}P_{1}^{s}&P_{2}^{s}&\cdots&P^{s}_{N_{T},k}\end{bmatrix}+n_{k}$
$\displaystyle=\gamma\eta\left(\mathbf{h}_{k}^{T}\mathbf{w}_{k}d_{k}+\mathbf{h}_{k}\sum_{i=1,\
i\neq
k}^{K}\mathbf{w}_{i}d_{i}+\mathbf{h}_{k}\mathbf{I}^{\text{DC}}\right)+n_{k},$
(4)

where $\mathbf{w_{k}}=\begin{bmatrix}w_{1,k}\ w_{2,k}\ \cdots\
w_{N_{T},k}\end{bmatrix}^{T}$ is the $k-$th user’s precoder and
$\mathbf{I}^{\text{DC}}=\begin{bmatrix}I_{1}^{\text{DC}}\ I_{2}^{\text{DC}}\
\cdots\ I_{N_{T}}^{\text{DC}}\end{bmatrix}^{T}$. It is noted that since
$|d_{i}|\leq 1$, we have

$\displaystyle-\lVert\mathbf{v}_{k}\rVert_{1}\leq
s_{n}\leq\lVert\mathbf{v}_{k}\rVert_{1}.$ (5)

To ensure both (2) and (5), the following constraint should be satisfied

$\sum_{k=1}^{K}|w_{n,k}|\leq\min\left(I_{n}^{\text{DC}},I_{\text{max}}-I_{n}^{\text{DC}}\right).$
(6)

The receiver noise $n_{k}$ in (II-A) can be modeled as a real-valued zero-mean
Gaussian RV whose variance is given by

$\sigma_{k}^{2}=2\gamma e\overline{P}_{k}^{r}B+4\pi
eA_{r}\gamma\chi_{\text{amb}}(1-\cos(\Psi))B+i_{\text{amp}}^{2}B,$ (7)

where $\overline{P}_{k}^{r}=\eta\mathbf{h}_{k}^{T}\mathbf{I}^{\text{DC}}$ is
the average of received power at the $k$-th user, $A_{r}$ is the area of the
PD, $e$ is the elementary charge, $\Psi$ is the optical field of view (FoV)
ofthe PD, $B$ is the modulation bandwidth, $\chi_{\text{amb}}$ is the ambient
light photo-current, and $i_{\text{amp}}$ is the pre-amplifier noise current
density.

###  II-B Power Consumption

We now analyze the total consumed power at LED transmitters, which can be
expressed by

$P_{\text{total}}=P_{\text{DC}}+P_{\text{AC}},$ (8)

where $P_{\text{DC}}$ and $P_{\text{AC}}$ are the powers of the DC and AC
currents, respectively. The DC current power includes the power used for
illumination by LEDs denoted as $P_{\text{DC, LEDs}}$ and that used by other
circuit components denoted as $P_{\text{DC, circuitry}}$. While $P_{\text{DC,
circuitry}}$ can be considered to be fixed, the power consumption of LEDs can
be adjusted depending on the required dimming level. However, under a specific
usage when the illumination level is not changed, it is reasonable to assume
that $P_{\text{DC, LEDs}}$ is fixed as well. The DC power consumption is then
given as

$P_{\text{DC}}=P_{\text{DC, LEDs}}+P_{\text{DC,circuitry}},$ (9)

with $P_{\text{DC, LEDs}}$ being calculated as

$P_{\text{DC, LEDs}}=\sum_{n=1}^{N_{T}}\ U_{\text{LED}}\ I_{n}^{\text{DC}},$
(10)

where $U_{\text{LED}}$ being the forward voltage of the LEDs.

The AC currents comes from the output current (or voltage) from the precoders
of LED drivers, therefore can be calculated as

$P_{\text{AC}}=r\sum_{k=1}^{K}\sigma_{d}^{2}\lVert\mathbf{w}_{k}\rVert^{2},$
(11)

where $r$ is the equivalent resistance of the AC circuit. Without loss of
generality, we denote $\xi=r\sigma_{d}^{2}$ as equivalent resistance, then
rewrite the total power consumption as

$P_{\text{total}}=P_{\text{DC}}+\xi\sum_{k=1}^{K}\lVert\mathbf{w}_{k}\rVert^{2}.$
(12)

###  II-C Energy Efficiency

For demodulation, the DC component in the received signal in (II-A) is
removed, yielding

$\displaystyle\overline{y}_{k}=\mathbf{h}_{k}^{T}\mathbf{w}_{k}d_{k}+\mathbf{h}_{k}\sum_{i=1,\
i\neq k}^{K}\mathbf{w}_{i}d_{i}+\overline{n}_{k},$ (13)

where $\overline{n}_{k}=\frac{n_{k}}{\gamma\eta}$. According to [7], a lower
bound of achievable confidential secrecy rate of the $k-$th user is given as

$\displaystyle\begin{split}R_{s,k}(\mathbf{W})=&\frac{1}{2}\log_{2}\left(\frac{1+\sum_{i=1}^{K}a_{k}\left(\mathbf{h}_{k}^{T}\mathbf{w}_{i}\right)^{2}}{1+\sum_{i=1,i\neq
k}^{K}b_{k}\left(\mathbf{h}_{k}^{T}\mathbf{w}_{i}\right)^{2}}\right)\\\
&-\frac{1}{2}\log_{2}\left(1+\sum_{i=1,i\neq
k}^{K}b_{i}\left(\mathbf{h}_{i}^{T}\mathbf{w}_{k}\right)^{2}\right),\end{split}$

where $\mathbf{W}=\begin{bmatrix}\mathbf{w}_{1}\ \mathbf{w}_{2}\ \cdots\
\mathbf{w}_{K}\end{bmatrix}$, $a_{k}=\frac{\exp(2h_{d})}{2\pi
e\overline{\sigma}_{k}^{2}}$, and
$b_{k}=\frac{\sigma_{d}^{2}}{\overline{\sigma}_{k}^{2}}$ with
$\overline{\sigma}_{k}^{2}=\frac{\sigma^{2}_{k}}{(\gamma\eta)^{2}}$ and
$h_{d}$ being the differential entropy of $d$. The energy efficiency with
respect to the achievable secrecy sum-rate of the considered system is
therefore given by

$\Phi(\mathbf{W})=\frac{\sum_{k=1}^{K}R_{s,k}(\mathbf{W})}{P_{\text{DC}}+\xi\text{Tr}\left(\mathbf{W}\mathbf{W}^{T}\right)}.$
(14)

##  III Sub-optimal Precoding Design

Our design objective is to maximize the energy efficiency while a minimum
achievable secrecy rate for each user is guaranteed. Hence, the design problem
can be formulated as follows

$\displaystyle\max_{\mathbf{W}}\ \ $ $\displaystyle\ \Phi(\mathbf{W}),$ (15a)
s.t. $\displaystyle\ R_{s,k}(\mathbf{W})\geq\lambda_{k},$ (15b)
$\displaystyle\sum_{k=1}^{K}|w_{n,k}|\leq\min\left(I_{n}^{\text{DC}},I_{\text{max}}-I_{n}^{\text{DC}}\right),$
(15c)

where $\lambda_{k}$ is the threshold for the secrecy rate of the $k$-th user.
It is seen from (14) that the objective function is a non-concave fractional
function with $\mathbf{W}$. This requires the use of Dinkelbach algorithm
[13], which is efficient in solving fractional programming. Let
$N(\mathbf{W})=\sum_{k=1}^{K}R_{s,k}(\mathbf{W})$,
$D(\mathbf{W})=P_{\text{DC}}+\xi\text{Tr}\left(\mathbf{W}\mathbf{W}^{T}\right)$.
Then for some $\mu\geq 0$, the Dinkelbach algorithm involves solving the
following

$\displaystyle\max_{\mathbf{W}}\ \ $ $\displaystyle
N\left(\mathbf{W}\right)-\mu D\left(\mathbf{W}\right),$ (16a) s.t.
$\displaystyle\ R_{s,k}(\mathbf{W})\geq\lambda_{k},$ (16b)
$\displaystyle\sum_{k=1}^{K}|w_{n,k}|\leq\min\left(I_{n}^{\text{DC}},I_{\text{max}}-I_{n}^{\text{DC}}\right).$
(16c)

to obtain a precoder $\mathbf{W}^{\prime}$. The optimal value
$\mu^{*}=\frac{N(\mathbf{W^{*}})}{D(\mathbf{W^{*}})}$ is achieved when
$N(\mathbf{W^{\prime}})-\mu D(\mathbf{W^{\prime}})=0$. This results in the
Dinkelbach algorithm described as follows

Choose the maximum number of iterations ${{L}}_{\text{max},1}$ and the error
tolerance $\epsilon_{1}$.

Initialize $\mu>0$, $l=0$.

while $\text{convergence}==\mathbb{False}\ \text{and}\
l\leq{L}_{\text{max},1}$ do

For a given $\mu$, solve (16) to get $\mathbf{W}^{(l)}$.

if $N\left(\mathbf{W}^{(l)}\right)-\mu
D\left(\mathbf{W}^{(l)}\right)\leq\epsilon_{1}$ then

$\text{convergence}==\mathbf{True}$;

${\mathbf{W}^{*}}={\mathbf{W}^{(l)}}$;

$\mu^{*}=\frac{N\left(\mathbf{W}^{(l)}\right)}{D\left(\mathbf{W}^{(l)}\right)}$;

else

$\text{convergence}==\mathbf{False}$

$l=l+1$;

Update
$\mu=\frac{N\left(\mathbf{W}^{(l)}\right)}{D\left(\mathbf{W}^{(l)}\right)}$;

end if

end while

Return the optimal ${\mathbf{W}^{*}}$ and ${\mu^{*}}$.

Algorithm 1 Dinkelbach-type algorithm for solving (16).

Nevertheless, it should be noted that (16) is not a convex optimization
problem due to the non-concave objective function and the non-convex
constraint in (16b). To overcome this problem, we make use of the CCCP
approach based on the first-order Taylor approximation to approximate the
original problem to a convex one. Specifically, we first introduce the
following slack variables

$\displaystyle r_{1,k}\ \overset{\Delta}{=}\ $
$\displaystyle\frac{1}{2}\log_{2}\left(1+\sum_{i=1}^{K}a_{k}\left(\mathbf{h}_{k}^{T}\mathbf{w}_{i}\right)^{2}\right),$
(17a) $\displaystyle p_{1,k}\ \overset{\Delta}{=}\ $
$\displaystyle\sum_{i=1}^{K}a_{k}\left(\mathbf{h}_{k}^{T}\mathbf{w}_{i}\right)^{2},$
(17b) $\displaystyle r_{2,k}\ \overset{\Delta}{=}\ $
$\displaystyle\frac{1}{2}\log_{2}\left(1+\sum_{i=1,i\neq
k}^{K}b_{k}\left(\mathbf{h}_{k}^{T}\mathbf{w}_{i}\right)^{2}\right),$ (17c)
$\displaystyle p_{2,k}\ \overset{\Delta}{=}\ $ $\displaystyle\sum_{i=1,i\neq
k}^{K}b_{k}\left(\mathbf{h}_{k}^{T}\mathbf{w}_{i}\right)^{2},$ (17d)
$\displaystyle r_{3,k}\ \overset{\Delta}{=}\ $
$\displaystyle\frac{1}{2}\log_{2}\left(1+\sum_{i=1,i\neq
k}^{K}b_{i}\left(\mathbf{h}_{i}^{T}\mathbf{w}_{k}\right)^{2}\right),$ (17e)
$\displaystyle p_{3,k}\ \overset{\Delta}{=}\ $ $\displaystyle\sum_{i=1,i\neq
k}^{K}b_{i}\left(\mathbf{h}_{i}^{T}\mathbf{w}_{k}\right)^{2},$ (17f)

Then, the objective function
$\sum_{k=1}^{K}\left(r_{1,k}-r_{2,k}-r_{3,k}\right)-\mu\left(P_{\text{DC}}+\xi\text{Tr}\left(\mathbf{W}\mathbf{W}^{T}\right)\right)$
is a concave function with respect to $\mathbf{W}$, $r_{1,k}$, $r_{2,k}$, and
$r_{3,k}$. Also, (16) can be rewritten as

$\displaystyle\max_{\begin{subarray}{c}\mathbf{W},r_{1,k},r_{2,k},r_{3,k}\\\
p_{1,k},p_{2,k},p_{3,k}\end{subarray}}\ $
$\displaystyle\sum_{k=1}^{K}\left(r_{1,k}-r_{2,k}-r_{3,k}\right)-\mu\left(P_{\text{DC}}+\xi\text{Tr}\left(\mathbf{W}\mathbf{W}^{T}\right)\right),$
(18a) s.t. $\displaystyle
r_{1,k}\leq\frac{1}{2}\log_{2}\left(1+p_{1,k}\right),$ (18b) $\displaystyle
p_{1,k}\leq\sum_{i=1}^{K}a_{k}\left(\mathbf{h}_{k}^{T}\mathbf{w}_{i}\right)^{2},$
(18c) $\displaystyle r_{2,k}\geq\frac{1}{2}\log_{2}\left(1+p_{2,k}\right),$
(18d) $\displaystyle p_{2,k}\geq\sum_{i=1,i\neq
k}^{K}b_{k}\left(\mathbf{h}_{k}^{T}\mathbf{w}_{i}\right)^{2},$ (18e)
$\displaystyle r_{3,k}\geq\frac{1}{2}\log_{2}\left(1+p_{3,k}\right),$ (18f)
$\displaystyle p_{3,k}\geq\sum_{i=1,i\neq
k}^{K}b_{i}\left(\mathbf{h}_{i}^{T}\mathbf{w}_{k}\right)^{2},$ (18g)
$\displaystyle r_{1,k}-r_{2,k}-r_{3,k}\geq\lambda_{k},$ (18h)
$\displaystyle\sum_{k=1}^{K}|w_{n,k}|\leq\min\left(I_{n}^{\text{DC}},I_{\text{max}}-I_{n}^{\text{DC}}\right).$
(18i)

It is seen that constraints (18c), (18d), and (18f) are still not convex. To
cope with this, the first-order Taylor approximation is employed to
approximate these non-convex constraints. The CCCP is then used to iteratively
solve a sequence of approximating convex problems until a predefined
convergence criterion is met [14]. Specifically, at the $m$-th iteration, the
following problem is solved

$\displaystyle\max_{\begin{subarray}{c}\mathbf{W},r_{1,k},r_{2,k},r_{3,k}\\\
p_{1,k},p_{2,k},p_{3,k}\end{subarray}}\
\sum_{k=1}^{K}\left(r_{1,k}-r_{2,k}-r_{3,k}\right)-\mu\left(P_{\text{DC}}+\xi\text{Tr}\left(\mathbf{W}\mathbf{W}^{T}\right)\right),$
(19a) s.t. $\displaystyle
p_{1,k}\leq\sum_{i=1}^{K}a_{k}\left(\left(\mathbf{h}_{k}^{T}{\mathbf{w}}_{i}^{(m-1)}\right)^{2}\right.$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}\left.+2\left({\mathbf{w}}_{i}^{(m-1)}\right)^{T}\mathbf{h}_{k}\mathbf{h}_{k}^{T}\left(\mathbf{w}_{i}^{(m)}-{\mathbf{w}}_{i}^{(m-1)}\right)\right),$
(19b) $\displaystyle
r_{2,k}\geq\frac{1}{2}\log_{2}\left(1+{p}_{2,k}^{(m-1)}\right)+\frac{\left(p_{2,k}^{(m)}-{p}_{2,k}^{(m-1)}\right)}{2\ln
2\left(1+{p}_{2,k}^{(m-1)}\right)},$ (19c) $\displaystyle
r_{3,k}\geq\frac{1}{2}\log_{2}\left(1+{p}_{3,k}^{(m-1)}\right)+\frac{\left(p_{3,k}^{(m)}-{p}_{3,k}^{(m-1)}\right)}{2\ln
2\left(1+{p}_{3,k}^{(m-1)}\right)},$ (19d)
$\displaystyle\eqref{eqn:r_1k_constraints},\eqref{eqn:p_2k_constraints},\eqref{eqn:p_3k_constraints}-\eqref{eqn:linear_constraints}.$

where ${\mathbf{w}}_{i}^{(m-1)}$, ${p}_{2,k}^{(m-1)}$ and ${p}_{3,k}^{(m-1)}$
are the solutions obtained from the previous iteration. Problem (19) is
convex, thus can be solved efficiently by using standard optimization
softwares , such as CVX [15]. Finally, (18) can be solved by the proposed
Algorithm 2 using CCCP.

Choose the maximum number of iteration ${L}_{\text{max},2}$ and the error
tolerance $\epsilon_{2}>0$.

Choose feasible initial points $\mathbf{W}^{(0)}$, $p_{2,k}^{(0)}$,
$p_{3,k}^{(0)}$ to (19).

Set $m=0$.

while $\text{convergence}==\mathbf{False}$ and $m\leq{L}_{\text{max},2}$ do

$m\mathrel{\mathop{\ordinarycolon}}=m+1$

Solve (19) for $\mathbf{W}^{(m)}$, $p_{2,k}^{(m)}$, $p_{3,k}^{(m)}$ using
$\mathbf{W}^{(m-1)}$, $p_{2,k}^{(m-1)}$, $p_{3,k}^{(m-1)}$ obtained from the
previous iteration.

if
$\frac{\mathinner{\\!\left\lVert\mathbf{W}^{(m)}-\mathbf{W}^{(m-1)}\right\rVert}}{\mathinner{\\!\left\lVert\mathbf{W}^{(m)}\right\rVert}}\leq\epsilon_{2}$
then

$\text{convergence}=\mathbf{True}$

$\mathbf{W}^{*}=\mathbf{W}^{(m)}$

else

$\text{convergence}=\mathbf{False}$.

end if

end while

Return the optimal value $\mathbf{W}^{*}$.

Algorithm 2 CCCP-type algorithm for solving (19)

##  IV Simulation Results and Discussions

In this section, the performance convergence behaviors of the proposed
solution are evaluated. Numerical results are obtained through averaging
$10,000$ different users’ channel realizations. Unless otherwise noted, the
following parameters are used for simulations. LED bandwidth $B=20$ MHz, beam
angle $\phi=120^{\circ}$, LED conversion factor $\eta=2$ W/A, active area of
the PD $A_{r}=1\text{cm}^{2}$, responsivity $\gamma=0.54$ A/W, FoV
$\Psi=60^{\circ}$, optical filter gain $T_{s}(\psi)=1$, refractive index of
the concentrator $\kappa=1.5$, ambient light photocurrent
$\chi_{\text{amp}}=10.93\text{A}/(\text{m}^{2}\cdot\text{Sr}$), and
preamplifier noise current density,
$i_{\text{amp}}=5\text{pA}/\text{Hz}^{-1/2}$. Moreover, the data symbols are
assumed to be uniformly distributed over $[-1,1]$.

((a)) $2\times 2$ layout.

((b)) $2\times 3$ layout.

((c)) $3\times 3$ layout.

Figure 2: Different layouts of LED transmitters.

Figure 3 illustrates the convergence of the proposed solution. For this
purpose, three different scenarios of the numbers of LED transmitters and
users, namely $(N_{T},K)=(4,3)$, $(6,4)$ and $(9,6)$, are taken into
consideration. The configurations of LED transmitter are illustrated in Fig.
2. Additionally, the average emitted optical power of each LED luminary
$\overline{P}_{n}^{s}=\eta I_{n}^{\text{DC}}=30~{}\text{dBm}$,
$U_{\text{LED}}=3.3$V, $P_{\text{DC,circuitry}}=8\text{W}$, $\xi=3\Omega$, and
$\lambda_{k}$’s $=0.5$ bits/s/Hz are set. Firstly, we examine the case that
the initial precoder $\mathbf{W}^{(0)}$ in Algorithm 2 is chosen randomly. It
is shown that considerably large numbers of iterations are required for the
normalized energy efficiency to converge (i.e., $60$, $75$, and $90$
iterations for $(N_{T},K)=(4,3)$, $(6,4)$ and $(9,6)$, respectively). As we
observe that the optimal solution is usually a near zero-forcing (ZF)
precoder, one can choose the initial point as the ZF precoder to speed up the
convergence. Indeed, simulation results revealed significant improvements in
terms of the required number of iterations. Specifically, using ZF precoder as
the initial point, roughly 8, 9, and 20 iterations are needed when
$(N_{T},K)=(4,3)$, $(6,4)$ and $(9,6)$, respectively.

Figure 3: Convergence behaviors of the proposed solution.

In Fig. 4, we assess the energy efficiency with respect to each luminary’s
average emitted optical power $\overline{P}^{s}_{n}$ for different values of
circuitry power consumption where $(N_{T},K)=(4,3)$. It is observed that the
energy efficiency first increases with an increase in $\overline{P}^{s}_{n}$
until its maximum value. It then starts decreasing as $\overline{P}^{s}_{n}$
continues to increase. This phenomenon can be explained as follows. At its low
value region, the increase in the achievable secrecy sum-rate due to
increasing $\overline{P}^{s}_{n}$ is dominant in improving the energy
efficiency. When $\overline{P}^{s}_{n}$ further increases, it becomes the
dominant factor that reduces the energy efficiency as the achievable secrecy
sum-rate only logarithmically increases with $\overline{P}^{s}_{n}$. We also
notice that the optimal point of $\overline{P}^{s}_{n}$ increases in
accordance with $P_{\text{DC, circuitry}}$.

Figure 4: Energy efficiency versus $\overline{P}^{s}_{n}$ with different
$P_{\text{DC, circuitry}}$.

##  V Conclusions

In this paper, we have studied an energy-efficient precoding scheme for PLS in
MU-MISO VLC systems. Due to the non-convex nature of the design problem,
Dinkelbach and CCCP algorithms were employed to find a sub-optimal solution
with lower complexity. Numerical results shown that by choosing ZF precoders
as the initial points for the CCCP algorithm, the rate of convergence of the
proposed solution could be significantly improved. It was also demonstrated
that there exist an optimal value of the average emitted optical power where
the energy efficiency achieves its maximum value.

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## Acknowledgement

This work is supported by the Telecommunications Advancement Foundation (TAF)
under Grant C-2020-2.