# Energy corrections due to the Non-commutative Phase-Space of the Charged
Harmonic Oscillator in a constant magnetic field in 3D

Muhittin Cenk Eser<EMAIL_ADDRESS>Department of Physics, Eastern
Mediterranean University, 99628 Famagusta, North Cyprus  Mustafa Riza
<EMAIL_ADDRESS>Department of Physics, Eastern Mediterranean
University, 99628 Famagusta, North Cyprus

###### Abstract

In this paper, we study the effects of non-commutative quantum mechanics in
three dimensions on the energy levels of a charged harmonic oscillator in the
presence of a constant magnetic field in the $z$-direction. The extension of
this problem to three dimensions proves to be non-trivial. We obtain the
first-order corrections to the energy-levels in closed form in the low energy
limit of weak non-commutativity.

Non-commutative Quantum Mechanics, Non-commutative Phase space, 3D Harmonic
Oscillator, constant magnetic field, Landau Levels

###### pacs:

11.10 Nx, 03.65-w

00footnotetext: $\scriptstyle\copyright$2013 Chinese Physical Society and the
Institute of High Energy Physics of the Chinese Academy of Sciences and the
Institute of Modern Physics of the Chinese Academy of Sciences and IOP
Publishing Ltd

##  I Introduction

With Heisenberg’s introduction of the uncertainty principle [1], the classical
paradigm that position and momentum commutate at no cost crushed.
Additionally, the discussion of a charged particle in an electromagnetic field
in the framework of quantum mechanics leads us to the introduction of the
kinetic momentum operator, which in contrast to the canonical momentum
operator, also does not commute. These two facts, emerging from the nature of
quantum mechanics, evidently brings up the question, is the assumption of the
commutation of the position and momentum operators among themselves an
accurate assumption? Or, under which conditions is the assumption of the
vanishing commutators of $[x_{i},x_{j}]=0$ and $[p_{i},p_{j}]=0$ correct? In
his pioneering work Hartland Snyder [2] noticed that Lorenz invariance does
not necessarily require non-commutativity of the position and momentum
operators in field theory, manifesting the foundation of the detailed
discussion of the Quantum Field Theory in Non-commutative spaces by R. Szabo
[3, 4]. Non-commutativity is generally associated with the effect of the
geometry of the space [5, 6]. The magnetic field’s impact on non-commutativity
has been discussed in various works, especially in the framework of the Landau
problem [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18]. There are several
discussions on the non-commutative Quantum Hall effect [19, 20, 21, 22].

Let us recall the basic properties of Non-commutative Quantum Mechanics [23].
Let us denote $\hat{x}_{i}$ as the non-commutative position operator and
$\hat{p}_{i}$ as the non-commutative momentum operator; then the commutation
relationships are given as

$\left[\hat{x}_{i},\hat{x}_{j}\right]=i\theta_{ij},\quad\left[\hat{p}_{i},\hat{p}_{j}\right]=i\eta_{ij}$
(1)

where $\theta_{ij}$ and $\eta_{ij}$ are both antisymmetric tensors.

Consequently, as one can verify easily, the relationship between non-
commutative operators $\hat{x}_{i}$ and $\hat{p}_{i}$ with their commutative
counterparts can be written as

$\displaystyle\hat{x}_{i}$ $\displaystyle=$ $\displaystyle\alpha
x_{i}-\frac{1}{2\alpha\hbar}\theta_{ij}p_{j}$ (2) $\displaystyle\hat{p}_{i}$
$\displaystyle=$ $\displaystyle\alpha
p_{i}+\frac{1}{2\alpha\hbar}\eta_{ij}x_{j},$ (3)

where $\alpha\in(0,1)$ is the scaling constant related to the non-
commutativity of the phase-space and and $\eta_{ij}$ and $\theta_{ij}$ are
antisymmetric tensors with the properties

$\eta_{ij}=\begin{cases}\eta&\text{if }ij=12,23,31\\\ -\eta&\text{if
}ij=21,32,13\\\ 0&\text{else }\end{cases}$ (4)

and

$\theta_{ij}=\begin{cases}\theta&\text{if }ij=12,23,31\\\ -\theta&\text{if
}ij=21,32,13\\\ 0&\text{else }\end{cases}\qquad,$ (5)

respectivly.

Mathematically, the non-commutativity of the base manifold can be realized by
application the Weyl-Moyal star product [24]

$\left(f\star
g\right)(x,p)=e^{i\frac{1}{2\alpha^{2}}\theta_{ij}\partial_{i}^{x}\partial_{j}^{x}+i\frac{1}{2\alpha^{2}}\eta_{ij}\partial_{i}^{p}\partial_{j}^{p}}f(x)g(y)=\\\
=f(x,p)g(x,p)+\frac{i\theta_{ij}}{2\alpha^{2}}\partial_{i}^{x}f\partial_{j}^{x}g\Biggr{|}_{x_{i}=x_{j}}+\frac{i\eta_{ij}}{2\alpha^{2}}\partial_{i}^{p}f\partial_{j}^{p}g\Biggr{|}_{p_{i}=p_{j}}+\\\
+{\cal{O}}\left(\theta_{ij}^{2}\right)+{\cal{O}}\left(\eta_{ij}^{2}\right)+{\cal{O}}\left(\theta_{ij}\eta_{ij}\right)$
(6)

So, the shift from ordinary Quantum Mechanics to non-commutative Quantum
Mechanics is performed by employing the Weyl-Moyal product (6) instead of the
ordinary product. So, the non-commutative time-independent Schrödinger
equation becomes

$H(x,p)\star\psi(x)=E\psi(x).$ (7)

By employing the Bopp’s shift [25], we can turn the Weyl-Moyal product again
to the ordinary product by substituting $x$ and $p$ in the non-commutative
equation by $\hat{x}$ and $\hat{p}$, namely

$H(x,p)\star\psi(x)=H(\hat{x},\hat{p})\psi(x).$ (8)

Based on Harko [26], we can see that the non-commutativity parameters $\eta$
and $\theta$ can be considered as energy-dependent and that both become
sufficiently small in the low energy limit. We will use this fact in order to
extend the discussions of a charged particle in a harmonic oscillator with the
presence of a magnetic field. The isotropic charged harmonic oscillator in a
magnetic field could be solved exactly in the framework of 2D non-
commutativity [27]. Let us now include the $z$-direction into the non-
commutative framework. In the low energy limit, we can consider the parameters
$\eta$ and $\eta$ small, and we can calculate the effect of the non-
commutativity in 3D using first-order perturbation in $\eta$ and $\theta$.

In light of this, we will discuss the non-commutative charged harmonic
oscillator in the presence of a constant magnetic field employing non-
commutativity to all three spacial parameters. In section, we are going to
discuss the non-commutative Hamiltonian of the charged particle in a parabolic
potential in the presence of a constant magnetic field. Then we will expand
the Hamiltonian in terms of $\theta$ and $\eta$, and calculate the energy
correction due to non-commutativity in the domain of weak non-commutativity,
i.e., in the low energy limit. Finally, we will compare the energy corrections
with respect to the magnetic field’s magnitude, noting that the sign of the
correction is a function of the magnitude of the employed magnetic field.Then
we will close with some concluding remarks.

##  II 3D non-commutative charged harmonic oscillator in a constant magnetic
field

Our starting point is the commutative Hamiltonian for the charged harmonic
oscillator presence of a constant magnetic field.

$H_{0}(x,p)=\frac{1}{2m}\left(\vec{p}-\frac{q}{c}\vec{A}\right)^{2}+\frac{1}{2}m\omega^{2}\left(x^{2}+y^{2}+z^{2}\right)$
(9)

Without loss of generality we will choose the direction of the constant
magnetic field in $z$-direction, i.e., $\vec{B}=B\hat{k}$ and
$\vec{A}(\vec{x},t)=\frac{1}{2}\left(-yB\hat{i}+xB\hat{j}\right)$ in Coulomb
gauge. So, our Hamiltonian $H_{0}$ modifies to

$H_{0}(x,p)=\frac{1}{2m}\left(\left(p_{x}+\frac{qB}{2c}y\right)^{2}+\left(p_{y}-\frac{qB}{2c}x\right)^{2}+p_{z}^{2}\right)+\\\
+\frac{1}{2}m\omega^{2}\left(x^{2}+y^{2}+z^{2}\right).$ (10)

Expanding the Hamiltonian (10) and regrouping the terms we get

$H_{0}(x,p)=\frac{1}{2m}\left(p_{x}^{2}+p_{y}^{2}+p_{z}^{2}\right)-\frac{1}{2}\omega_{c}L_{z}+\\\
+\frac{1}{2}m\tilde{\omega}^{2}\left(x^{2}+y^{2}\right)+\frac{1}{2}m\omega^{2}z^{2},$
(11)

where $L_{z}=xp_{y}-yp_{x}$ is the $z$-component of the angular momentum
operator, $\omega_{c}=\frac{qB}{mc}$ the cyclotron frequency, and
$\tilde{\omega}^{2}=\omega^{2}+\frac{\omega_{c}^{2}}{4}$ is the modified
frequency of the harmonic oscillator in the $xy$-plane. From (8) we know that
the Weyl-Moyal product can be turned into a normal product by substituting
commutative $x$ and $p$ by the non-commutative operators $\hat{x}$ and
$\hat{p}$, so let us first consider the Hamiltonian $H_{0}(\hat{x},\hat{p})$.

$H_{0}(\hat{x},\hat{p})=\frac{1}{2m}\left(\hat{p}_{x}^{2}+\hat{p}_{y}^{2}+\hat{p}_{z}^{2}\right)-\frac{1}{2}\omega_{c}\hat{L}_{z}+\\\
+\frac{1}{2}m\tilde{\omega}^{2}\left(\hat{x}^{2}+\hat{y}^{2}\right)+\frac{1}{2}m\omega^{2}\hat{z}^{2}$
(12)

Let us state all non-commutative operators explicitly in the 3D version using
(2) and (3)together with (4) and (5), respectively.

$\displaystyle\hat{x}$ $\displaystyle=$ $\displaystyle\alpha
x-\frac{\theta}{2\alpha\hbar}p_{y}+\frac{\theta}{2\alpha\hbar}p_{z}$ (13)
$\displaystyle\hat{y}$ $\displaystyle=$ $\displaystyle\alpha
y-\frac{\theta}{2\alpha\hbar}p_{z}+\frac{\theta}{2\alpha\hbar}p_{x}$ (14)
$\displaystyle\hat{z}$ $\displaystyle=$ $\displaystyle\alpha
z-\frac{\theta}{2\alpha\hbar}p_{x}+\frac{\theta}{2\alpha\hbar}p_{y}$ (15)
$\displaystyle\hat{p}_{x}$ $\displaystyle=$
$\displaystyle\alpha{p}_{x}+\frac{\eta}{2\alpha\hbar}y-\frac{\eta}{2\alpha\hbar}z$
(16) $\displaystyle\hat{p}_{y}$ $\displaystyle=$
$\displaystyle\alpha{p}_{y}+\frac{\eta}{2\alpha\hbar}z-\frac{\eta}{2\alpha\hbar}x$
(17) $\displaystyle\hat{p}_{z}$ $\displaystyle=$
$\displaystyle\alpha{p}_{z}\frac{\eta}{2\alpha\hbar}x-\frac{\eta}{2\alpha\hbar}y$
(18)

Based on the position and momentum operators defined in the equations
(13)-(18), we can construct all other operators needed in this calculation.

As a consequence the non-commutative angular momentum operator $\hat{L}_{z}$
can be stated explicitly as following

$\hat{L}_{z}=\hat{x}\hat{p}_{y}-\hat{y}\hat{p}_{x}=\alpha^{2}L_{z}+\frac{\theta}{2\hbar}\left(-p_{x}^{2}-p_{y}^{2}+p_{x}p_{z}+p_{y}p_{z}\right)+\\\
+\frac{\eta}{2\hbar}\left(-x^{2}-y^{2}+xz+yz\right)+\frac{\theta\eta}{4\alpha^{2}\hbar^{2}}\left(L_{x}+L_{y}+L_{z}\right).$
(19)

Furthermore, the sum of the squares of the components of the non-commutative
momentum operator $\hat{p}_{x}^{2}+\hat{p}_{y}^{2}+\hat{p}_{z}^{2}$

$\hat{p}_{x}^{2}+\hat{p}_{y}^{2}+\hat{p}_{z}^{2}=\alpha^{2}\left(p_{x}^{2}+p_{y}^{2}+p_{z}^{2}\right)-\frac{\eta}{\hbar}\left(L_{x}+L_{y}+L_{z}\right)+\\\
+\frac{\eta^{2}}{2\alpha^{2}\hbar^{2}}\left(x^{2}-xy+y^{2}-xz-
yz+z^{2}\right),$ (20)

and the sum of the squares of the $x$ and $y$ components of the non-
commutative squared position operator $\hat{x}^{2}+\hat{y}^{2}$

$\hat{x}^{2}+\hat{y}^{2}=\alpha^{2}\left(x^{2}+y^{2}\right)+\frac{\theta}{\hbar}\left(-L_{z}+(x-y)p_{z}\right)+\\\
+\frac{\theta^{2}}{4\alpha^{2}\hbar^{2}}\left(p_{x}^{2}+p_{y}^{2}+2p_{z}^{2}-2p_{x}p_{z}-2p_{y}p_{z}\right),$
(21)

and finally square of the $z$ component of the non-commutative position
operator $\hat{z}^{2}$

$\hat{z}^{2}=\alpha^{2}z^{2}+\frac{\theta}{\hbar}z\left(p_{y}-p_{x}\right)+\frac{\theta^{2}}{4\alpha^{2}\hbar^{2}}\left(p_{x}-p_{y}\right)^{2}.$
(22)

Substituting (19)-(22) into (12) gives the non-commutative Hamiltonian in the
commutative algebra. After regrouping and summarizing all terms we get the
expanded non-commutative Hamiltonian in the commutative space.

$H_{0}(\hat{x},\hat{p})=\alpha^{2}H_{0}(x,p)+\frac{\eta}{\hbar}H_{\eta}(x,p)+\frac{\theta}{\hbar}H_{\theta}(x,p)+\\\
+\frac{\eta\theta}{\hbar^{2}}H_{\eta\theta}(x,p)+\frac{\eta^{2}}{\hbar^{2}}H_{\eta^{2}}(x,p)+\frac{\theta^{2}}{\hbar^{2}}H_{\theta^{2}}(x,p)$
(23)

with

$\displaystyle H_{\eta}$ $\displaystyle=$
$\displaystyle-\frac{1}{2m}\left(L_{x}+L_{y}+L_{z}\right)-$ (24)
$\displaystyle-\frac{1}{4}\omega_{c}\left(-x^{2}-y^{2}+xz+yz\right)$
$\displaystyle H_{\theta}$ $\displaystyle=$
$\displaystyle-\frac{1}{4}\omega_{c}\left(-p_{x}^{2}-p_{y}^{2}+p_{x}p_{z}+p_{y}p_{z}\right)+$
(25)
$\displaystyle+\frac{1}{2}m\tilde{\omega}^{2}\bigl{(}-L_{z}+(x-y)p_{z}\bigr{)}+$
$\displaystyle+\frac{1}{2}m\omega^{2}z\left(p_{y}-p_{x}\right)$ $\displaystyle
H_{\eta\theta}$ $\displaystyle=$
$\displaystyle\frac{\omega_{c}}{8\alpha^{2}}\left(L_{x}+L_{y}+L_{z}\right)$
(26) $\displaystyle H_{\eta^{2}}$ $\displaystyle=$
$\displaystyle\frac{1}{4m\alpha^{2}}\left(x^{2}-xy+y^{2}-xz-yz+z^{2}\right)$
(27) $\displaystyle H_{\theta^{2}}$ $\displaystyle=$
$\displaystyle\frac{1}{4\alpha^{2}}\Biggl{[}\frac{1}{2}m\tilde{\omega}^{2}\left(p_{x}^{2}+p_{y}^{2}+2p_{z}^{2}-2p_{x}p_{z}-2p_{y}p_{z}\right)$
(28)
$\displaystyle+\frac{1}{2}m\omega^{2}\left(p_{x}-p_{y}\right)^{2}\Biggr{]}$

Obviously we return to the well known commutative case if
$\alpha=1,\theta=\eta=0$.

##  III Perturbative approach

According to Harko et al. [26], the contribution of the second-order terms
$\eta^{2}$, $\theta^{2}$, and $\eta\theta$ are small compared to the terms in
$\eta$ and $\theta$ in the low energy limit. Consequently, we can determine
the effect of the non-commutativity on the binding energy by employing first-
order perturbation theory.

To determine the impact of non-commutativity on the energy levels of a charged
harmonic oscillator in 3D in the presence of a constant magnetic field, we
first have to revisit the well-known commutative case. The Hamiltonian in the
commutative case in cylindrical coordinates is then given as

$H_{0}(x,p)=-\frac{\hbar^{2}}{2m}\left(\frac{1}{\rho}\frac{\partial}{\partial\rho}\left(\rho\frac{\partial}{\partial\rho}\right)+\frac{1}{\rho^{2}}\frac{\partial^{2}}{\partial\varphi^{2}}\right)-\frac{1}{2}\omega_{c}\frac{\hbar}{i}\frac{\partial}{\partial\varphi}+\\\
+\frac{1}{2}m\left(\omega^{2}+\frac{\omega_{c}^{2}}{4}\right)\rho^{2}-\frac{\hbar^{2}}{2m}\frac{\partial^{2}}{\partial
z^{2}}+\frac{1}{2}m\omega^{2}z^{2},$ (29)

where $x=\rho\cos\varphi$, $y=\rho\sin\varphi$ consequently
$\rho^{2}=x^{2}+y^{2}$,
$L_{z}=\frac{\hbar}{i}\frac{\partial}{\partial\varphi}$, and
$p_{\rho}^{2}=-\hbar^{2}\left(\frac{1}{\rho}\frac{\partial}{\partial\rho}\left(\rho\frac{\partial}{\partial\rho}\right)\right)$.
With $\tilde{\omega}^{2}=\omega^{2}+\frac{\omega_{c}^{2}}{4}$ we get

$H_{0}(x,p)=-\frac{\hbar^{2}}{2m}\left(\frac{1}{\rho}\frac{\partial}{\partial\rho}\left(\rho\frac{\partial}{\partial\rho}\right)+\frac{1}{\rho^{2}}\frac{\partial^{2}}{\partial\varphi^{2}}\right)-\frac{1}{2}\omega_{c}\frac{\hbar}{i}\frac{\partial}{\partial\varphi}+\\\
+\frac{1}{2}m\tilde{\omega}^{2}\rho^{2}-\frac{\hbar^{2}}{2m}\frac{\partial^{2}}{\partial
z^{2}}+\frac{1}{2}m\omega^{2}z^{2}.$ (30)

In cylindrical coordinates, the time-independent Schrödinger equation for a
particle in an isotropic harmonic oscillator in the presence of a constant
magnetic field can be solved by separation of variables as

$\psi_{n_{\rho},\mu,n_{z}}(x)=\chi(\rho)e^{i\mu\varphi}\zeta(z).$ (31)

After substitution into the time independent Schrödinger equation we get the
eigenfunction as:

$\displaystyle\zeta(z)$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2^{n}n!}}\left(\frac{m\omega}{\pi\hbar}\right)^{1/4}e^{-\frac{m\omega
z}{2\hbar}}H_{n_{z}}\left(\sqrt{\frac{m\omega}{\hbar}}z\right)$ (32)
$\displaystyle\chi(\rho)$ $\displaystyle=$ $\displaystyle
A_{1}\left(\sqrt{2}\rho\right)^{|\mu|}e^{-\frac{m\tilde{\omega}\rho^{2}}{2\hbar}}U\left(-n_{\rho},1+|\mu|,\frac{m\tilde{\omega}\rho^{2}}{2\hbar}\right)+$
(33)
$\displaystyle+A_{2}\left(\sqrt{2}\rho\right)^{|\mu|}e^{-\frac{m\tilde{\omega}\rho^{2}}{2\hbar}}L_{n_{\rho}}^{|\mu|}\left(-n_{\rho},\frac{m\tilde{\omega}\rho^{2}}{2\hbar}\right)$

and the eigenvalue

$E_{n_{\rho},\mu,n_{z}}=\hbar\tilde{\omega}(2n_{\rho}+|\mu|+1)+\frac{1}{2}\hbar\omega_{c}\mu+\hbar\omega\left(n_{z}+\frac{1}{2}\right).$
(34)

The corrections to the binding energy for weak non-commutativity in first-
order perturbation theory are then according to (23) given as

$\Delta
E^{(1)}=\frac{\eta}{\hbar}\Bigl{\langle}n_{\rho},\mu,n_{z}\Bigr{|}H_{\eta}\Bigr{|}n_{\rho},\mu,n_{z}\Bigr{\rangle}+\\\
+\frac{\theta}{\hbar}\Bigl{\langle}n_{\rho},\mu,n_{z}\Bigr{|}H_{\theta}\Bigr{|}n_{\rho},\mu,n_{z}\Bigr{\rangle}$
(35)

Due to the symmetry of the problem all following matrix elements vanish:

$\Bigl{\langle}n_{\rho},\mu,n_{z}\Bigr{|}L_{x}\Bigr{|}n_{\rho},\mu,n_{z}\Bigr{\rangle}=\Bigl{\langle}n_{\rho},\mu,n_{z}\Bigr{|}L_{y}\Bigr{|}n_{\rho},\mu,n_{z}\Bigr{\rangle}=\\\
=\Bigl{\langle}n_{\rho},\mu,n_{z}\Bigr{|}xz\Bigr{|}n_{\rho},\mu,n_{z}\Bigr{\rangle}=\Bigl{\langle}n_{\rho},\mu,n_{z}\Bigr{|}yz\Bigr{|}n_{\rho},\mu,n_{z}\Bigr{\rangle}=\\\
=\Bigl{\langle}n_{\rho},\mu,n_{z}\Bigr{|}p_{x}p_{z}\Bigr{|}n_{\rho},\mu,n_{z}\Bigr{\rangle}=\Bigl{\langle}n_{\rho},\mu,n_{z}\Bigr{|}p_{y}p_{z}\Bigr{|}n_{\rho},\mu,n_{z}\Bigr{\rangle}=\\\
=\Bigl{\langle}n_{\rho},\mu,n_{z}\Bigr{|}yp_{z}\Bigr{|}n_{\rho},\mu,n_{z}\Bigr{\rangle}=\Bigl{\langle}n_{\rho},\mu,n_{z}\Bigr{|}xp_{z}\Bigr{|}n_{\rho},\mu,n_{z}\Bigr{\rangle}=\\\
=\Bigl{\langle}n_{\rho},\mu,n_{z}\Bigr{|}p_{x}\Bigr{|}n_{\rho},\mu,n_{z}\Bigr{\rangle}=\Bigl{\langle}n_{\rho},\mu,n_{z}\Bigr{|}p_{y}\Bigr{|}n_{\rho},\mu,n_{z}\Bigr{\rangle}=0$

So, the only matrix elements that are non-vanishing are

$\Delta
E^{(1)}=\frac{\eta}{\hbar}\Bigl{\langle}n_{\rho},\mu,n_{z}\Bigr{|}-\frac{1}{2m}L_{z}-\frac{\omega_{c}}{4}\rho^{2}\Bigr{|}n_{\rho},\mu,n_{z}\Bigr{\rangle}+\\\
+\frac{\theta}{\hbar}\Bigl{\langle}n_{\rho},\mu,n_{z}\Bigr{|}-\frac{\omega_{c}}{4}p_{\rho}^{2}-\frac{1}{2}m\tilde{\omega}^{2}L_{z}\Bigr{|}n_{\rho},\mu,n_{z}\Bigr{\rangle}.$
(36)

With the help of [28, 29] the lengthy integrals can be solved in closed form
and we get for the first-order corrections in $\eta$

$\Delta
E_{\eta}^{(1)}=-\frac{\eta|\mu|}{2m}-\frac{\eta\omega_{c}}{4m\tilde{\omega}}\left(2n_{\rho}+|\mu|+1\right)$
(37)

and $\theta$

$\Delta E_{\theta}^{(1)}=-\frac{1}{2}\theta
m\tilde{\omega}\left(\tilde{\omega}-\frac{1}{2}\omega_{c}f(n_{\rho},|\mu|)\right)$
(38)

with

$f(n_{\rho},\mu)=2\begin{pmatrix}n_{\rho}+\mu\\\
\mu\end{pmatrix}-4\mu\begin{pmatrix}\mu+n_{\rho}+2\\\
n_{\rho}-1\end{pmatrix}-\\\
-\mu(1+\mu)\Biggl{[}2\begin{pmatrix}\mu+n_{\rho}\\\
n_{\rho}\end{pmatrix}+4\begin{pmatrix}\mu+n_{\rho}-2\\\
n_{\rho}\end{pmatrix}+\\\ +\begin{pmatrix}\mu+n_{\rho}+1\\\
n_{\rho}\end{pmatrix}-\begin{pmatrix}\mu+n_{\rho}+2\\\
n_{\rho}-1\end{pmatrix}\Biggr{]}.$ (39)

A short dimensional analysis shows that $\eta$ has the dimension of
$mass^{2}\frac{Length^{2}}{Time^{2}}$, and $\theta$ has the dimension of
$Length^{2}$. So, the calculated corrections have the correct dimension of
energy.

So, finally, we can summarize the results of our calculation in first-order
perturbation theory. Recalling the non-commutative Hamiltonian (23) we see
that the unperturbed energy is

$E_{n_{\rho},\mu,n_{z}}^{(0)}=\Bigl{\langle}n_{\rho},|\mu|,n_{z}\Bigr{|}\alpha^{2}H_{0}(x,p)\Bigr{|}n_{\rho},|\mu|,n_{z}\Bigr{\rangle}=\\\
=\alpha^{2}\left[\hbar\tilde{\omega}(2n_{\rho}+|\mu|+1)+\frac{1}{2}\hbar\omega_{c}\mu+\hbar\omega\left(n_{z}+\frac{1}{2}\right)\right].$
(40)

The first order energy corrections are

$\Delta
E^{(1)}=-\frac{\eta|\mu|}{2m}-\frac{\eta\omega_{c}}{4m\tilde{\omega}}\left(2n_{\rho}+|\mu|+1\right)-\\\
-\frac{1}{2}\theta
m\tilde{\omega}\left(\tilde{\omega}-\frac{1}{2}\omega_{c}f(n_{\rho},|\mu|)\right)$
(41)

with $f(\rho,|\mu|)$ given in (39). These results hold for the situations,
where $\eta\ll\hbar m\omega_{c}$ and $\theta\ll\frac{\hbar}{m\tilde{\omega}}$.

##  IV Conclusion

We studied the charged harmonic oscillator within a constant magnetic field in
the context of the non-commutative quantum mechanics. In this line, we assumed
the commutators to be up to linear order of $\theta$ and $\eta$ in eq.(1)
based on the definition given in eq.(4) and (5). Following the standard
Schrödingerequation in non-commutative quantum mechanics, eq.(6), we
constructed the non-commutative Hamiltonian in the equation (23). Finally, we
have solved the Schrödinger equation, using the time-independent perturbation
method. We obtained the corrections in the energy levels, up to the first
order of the non-commutative parameters. Our final energy correction in
eq.(41) turns zero if the non-commutative parameters vanish.

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# Design monolayer iodinenes based on halogen bond and tiling theory

Kejun Yu  Centre for Quantum Physics, Key Laboratory of Advanced
Optoelectronic Quantum Architecture and Measurement (MOE), School of Physics,
Beijing Institute of Technology, Beijing, 100081, China  Beijing Key Lab of
Nanophotonics & Ultrafine Optoelectronic Systems, School of Physics, Beijing
Institute of Technology, Beijing, 100081, China  Botao Fu  College of Physics
and Electronic Engineering, Center for Computational Sciences, Sichuan Normal
University, Chengdu, 610068, China  Runwu Zhang  Centre for Quantum Physics,
Key Laboratory of Advanced Optoelectronic Quantum Architecture and Measurement
(MOE), School of Physics, Beijing Institute of Technology, Beijing, 100081,
China  Beijing Key Lab of Nanophotonics & Ultrafine Optoelectronic Systems,
School of Physics, Beijing Institute of Technology, Beijing, 100081, China
Da-shuai Ma  Institute for Structure and Function & Department of Physics &
Chongqing Key Laboratory for Strongly Coupled Physics, Chongqing University,
Chongqing 400044, P. R. China  Center of Quantum materials and devices,
Chongqing University, Chongqing 400044, P. R. China  Xiao-ping Li  School of
Physical Science and Technology, Inner Mongolia University, Hohhot 010021,
China  Zhi-Ming Yu<EMAIL_ADDRESS>Centre for Quantum Physics, Key
Laboratory of Advanced Optoelectronic Quantum Architecture and Measurement
(MOE), School of Physics, Beijing Institute of Technology, Beijing, 100081,
China  Beijing Key Lab of Nanophotonics & Ultrafine Optoelectronic Systems,
School of Physics, Beijing Institute of Technology, Beijing, 100081, China
Cheng-Cheng Liu<EMAIL_ADDRESS>Centre for Quantum Physics, Key Laboratory
of Advanced Optoelectronic Quantum Architecture and Measurement (MOE), School
of Physics, Beijing Institute of Technology, Beijing, 100081, China  Beijing
Key Lab of Nanophotonics & Ultrafine Optoelectronic Systems, School of
Physics, Beijing Institute of Technology, Beijing, 100081, China  Yugui Yao
Centre for Quantum Physics, Key Laboratory of Advanced Optoelectronic Quantum
Architecture and Measurement (MOE), School of Physics, Beijing Institute of
Technology, Beijing, 100081, China  Beijing Key Lab of Nanophotonics &
Ultrafine Optoelectronic Systems, School of Physics, Beijing Institute of
Technology, Beijing, 100081, China

###### Abstract

Xenes, two-dimensional (2D) monolayers composed of a single element, with
graphene as a typical representative, have attracted widespread attention.
Most of the previous Xenes, X from group-IIIA to group-VIA elements have
bonding characteristics of covalent bonds. In this work, we for the first time
unveil the pivotal role of a halogen bond, which is a distinctive type of
bonding with interaction strength between that of a covalent bond and a van
der Waals interaction, in 2D group-VIIA monolayers. Combing the ingenious non-
edge-to-edge tiling theory and state-of-art ab initio method with refined
local density functional M06-L, we provide a precise and effective bottom-up
construction of 2D iodine monolayer sheets, iodinenes, primarily governed by
halogen bonds, and successfully design a category of stable iodinenes,
encompassing herringbone, Pythagorean, gyrated truncated hexagonal, i.e.
diatomic-kagome, and gyrated hexagonal tiling pattern. These iodinene
structures exhibit a wealth of properties, such as flat bands, nontrivial
topology, and fascinating optical characteristics, offering valuable insights
and guidance for future experimental investigations. Our work not only unveils
the unexplored halogen bonding mechanism in 2D materials but also opens a new
avenue for designing other non-covalent bonding 2D materials.

Introduction. — Two-dimensional (2D) materials have been a hot topic since the
discovery of graphene. Up to now, as one of the most important members of 2D
materials, increasing Xenes made from group-III to VI elements have been
predicted and synthesized, e.g., boropheneMannix  _et al._ (2015); Wu  _et
al._ (2012), siliceneAufray  _et al._ (2010), phosphoreneLiu  _et al._ (2014)
and tellureneZhu  _et al._ (2017). Yet, as one of the most intriguing members
of the Xene family, group-VII Xenes have only begun to demonstrate research
advancements in recent yearsQian  _et al._ (2020); Kuang  _et al._ (2021); You
_et al._ (2023); Ye  _et al._ (2023). Generally, disparate valence electron
configurations among diverse main group elements engender markedly distinct
structural arrangements, bonding behaviors, and material characteristics
within the various Xene family. Undoubtedly, exploring the theoretical
foundations of group-VII Xenes will enhance our understanding of 2D materials
and generate a more extensive range of practical applications.

Solid halogen crystals exhibit a layered structure in the Cmca space groupvan
Bolhuis  _et al._ (1967), with stacked planar monolayers. Among the halogens,
only iodine maintains its crystalline phase under ambient conditions. Van der
Waals force between the layers enables the feasibility of exfoliating few-
layer iodinenes from bulk iodine. The halogen bond (XB, X stands for halogen),
akin to hydrogen bonds as a non-covalent interactionBrunet  _et al._ (1997);
Lin  _et al._ (2019), plays a crucial role in connecting iodine molecules to
form a monolayer structure, despite often being overlooked. Recently few-layer
iodinene nanosheets have been experimentally obtained from bulk iodine by
physical exfoliation, but strangely, the monolayers of these nanosheets
maintain the outline in the original bulk phase but seem to destroy the
halogen bonds in the layerQian  _et al._ (2020); Kuang  _et al._ (2021); You
_et al._ (2023); Ye  _et al._ (2023). It is noteworthy that XBs has been
utilized for fabricating halogen-bonded organic frameworks (XOFs)Gong  _et
al._ (2021); Chongboriboon  _et al._ (2020) successfully with reliable
stability in both solid and solution phases. Then a question arises naturally:
could 2D iodinenes be constructed by utilizing halogen bonding interactions?

In this work, based on the XBs, we predict a series of monolayer iodinenes as
a new category of 2D materials utilizing the tiling theory and first-principle
calculations. First of all, through our meticulous DFT calculations, including
the calculations of lattice parameters and phonon spectra, we find that
M06-LZhao and Truhlar (2006) is an appropriate exchange-correlation functional
for evaluating the XB and realize that the XB plays a key role in the
formation of stable structures of halogen crystals. Herringbone iodinene, as
monolayer iodinene exfoliated from bulk iodine, shows dynamic stability
resulting from XB networks. Afterwards, according to the tiling theory and
taking into account the diversity of XB networks, we systematically design a
series of structures of iodinenes based on halogen bonded iodine molecules.
Finally, we calculate the electronic structures of our designed iodinenes. All
of those halogen bonded iodinenes are semiconductors, including topological
nontrivial Pythagorean, gyrated truncated hexagonal (GTH), i.e. diatomic-
kagome and gyrated hexagonal (GH) iodinenes, and herringbone iodinene
revealing potential on photoelectronic application with the absorption
coefficient exceeds $10^{5}cm^{-1}$ in the visible region.

Figure 1: (a) 3D crystal structure (the upper figure) and monolayer (the lower
figure) of bulk iodine. Black bold and red dashed lines indicate
intramolecular covalent bond and intermolecular halogen bond, respectively.
(b) Electrostatic potential (ESP) map on 0.015 a.u. charge density isosurface
of monolayer herringbone iodinene. $\sigma$-hole occurs at the blue tip on the
isosurface with positive electrostatic potential. (c, d) Phonon spectra and
band structures of monolayer herringbone iodinene. Grey arrows indicate the
band gap. Blue dashed lines and black lines are the band structures without
and with spin-orbit coupling (SOC) respectively.

The halogen bond and selection of DFT functional. — The figure presented in
Fig. 1(a) depicts the layered structure of bulk iodine. Specifically, each
planar layer within the bulk iodine is formed through the connection of iodine
molecules via XBs. The electrostatic interaction model offers an intuitive
image of XBs, as depicted in Fig. 1(b), which displays the electrostatic
potential (ESP) map of a monolayer iodine sheet. The unique ESP map is derived
from the inherent anisotropic charge density distribution (see Fig. S1(b)) of
iodine molecules themselves, where electron density is accumulated around the
equator and depleted on the elongation of the covalent bond. The depleted
region is the so called $\sigma$-holeClark  _et al._ (2007), corresponding to
the blue tip in Fig. 1(b) with positive ESP. The XB is defined as the
attraction between a nucleophilic region and the positive electrostatic region
of a $\sigma$-hole, analogous to a hydrogen bond Desiraju  _et al._ (2013).
Additionally, the orientation of layers along the $\mathbf{c}$ axis in the
bulk iodine crystal is attributed to the weak vdW forces. This arrangement
reveals the potential for exfoliation to yield a freestanding monolayer of
iodinene, given the significantly greater strength of the XB compared to vdW
interactions.

A comparative analysis of various GGA and meta-GGA functionals was conducted
to evaluate their computational accuracy in modeling the crystal structure of
iodine. The results indicate that M06-L accurately captures the attractive XB
interaction in bulk iodine, whereas SCAN does not exhibit this capability (see
Fig. S2). This finding is consistent with the study by George et al.George
_et al._ (2015). It is pertinent to highlight that the utilization of PBE and
PBEsol functionals for assessing the interaction associated with halogen
bonding (as delineated in Fig. S2 and Table S1) is deemed inappropriate,
despite their application in recent researchZhu  _et al._ (2023); Huran  _et
al._ (2021). To account for long-range dispersive interactions, the DFT-D3
methodGrimme  _et al._ (2010) was employed, resulting in a more accurate
geometry configuration for bulk iodine (as demonstrated in Table. S1).
Therefore, M06-L+D3 was chosen for the DFT calculations. Further computational
details can be found in the supplementary materials.

Figure 2: (a) Schematic diagram of halogen bond (XB) between two iodine
molecules. The iodine atoms, $p_{y}$ orbitals (lone pairs), $\sigma_{p_{x}}$
bond and $\sigma^{*}_{p_{x}}$ antibond orbital ($\sigma$-hole) are depicted as
purple balls, green spindles, yellow ellipsoid and blue ellipsoid,
respectively. $p_{z}$ and $s$ orbitals are not shown here because $p_{z}$ sits
out of the plane and $s$ is the core orbital. The red dotted line signifies
the presence of the XB, involving the transfer of electrons from a lone pair
to a $\sigma$-hole through a donor-acceptor interaction. (b) Schematic diagram
of XB angle. Figure 3:  (a1-a5) Five tilings that meet the requirements of
designing the 2D iodinenes and (b1-b5) corresponding relaxed crystal
structures. The nomenclature of those tilings is adopted in a visualized way.
R-herringbone: rectangle herringbone tiling; GTH: gyrated truncated hexagonal
tiling, which is topologically equivalent to diatomic-kagome structure; GH:
gyrated hexagonal tiling; TDT: truncated distorted trihexagonal tiling. In
(b1-b5), the covalent bond and halogen bond are depicted by black and red
lines respectively. Grey dot lines indicate the unit cell.

Design iodinenes with the tiling theory. — A monolayer iodinene is exfoliated
from bulk iodine, and we call it as herringbone iodinene since the profile
resembles herringbone. After relaxation, lattice constant $a=10.66$ Å is
stretched by 6.6% and $b=4.38$ Å is compressed by 6.8% compared to the bulk
phase. The intramolecular bond length and XB length are 2.75 and 3.60 Å
respectively with only minor deviations from the case of bulk iodine.
Additionally, the electrostatic potential (ESP) map of the herringbone
iodinene is almost indistinguishable from that of the monolayer iodine sheet
observed in bulk (refer to Fig. 1(b)). There is no imaginary frequency in
phonon spectra (Fig. 1(c)) of freestanding monolayer iodinene, revealing its
dynamic stability. The halogen bonding network plays a crucial role in
enabling individual iodine molecules to come together to comprise a planar
structure. The band structures are depicted at Fig. 1(d), where a quasi-direct
band gap (1.165 eV) emerges between the conduction band minima (CBM) at
$\Gamma$ and valence band maxima (VBM) near $\Gamma$. Notablely, the highest
occupied band from $\Gamma$ to VBM is almost flat with an energy shift about
only 8 meV (31 meV for HSE06), so it is a direct gap semiconductor
approximately. The in-plane absorption coefficient curve is presented in Fig.
S12(a), with two distinct peaks (around 2.04 and 2.74 eV) exceeding $3\times
10^{5}$ $cm^{-1}$ in the visible region. This value is considerably higher
than that observed in many other 2D materials and organic perovskite materials
($10^{4}\sim 10^{5}cm^{-1}$) Huang  _et al._ (2022). As a result, herringbone
iodinene is promising for photoelectric conversion of solar energy.

Considering the crucial role of XB in forming stable bulk iodine and
exfoliated monolayer, we now provide a general and effective bottom-up
construction of monolayer iodinenes based on XB, with the usage of ingenious
non-edge-to-edge tiling theory. Geometrically, it is evident that herringbone
iodinene can be achieved by placing iodine atoms at each vertex of a hexagonal
herringbone tiling, with the long edges representing the XBs. Therefore, we
aim to construct iodinenes by bonding diatomic iodine molecules with XBs and
utilizing the tiling theoryLu  _et al._ (2019); Guan  _et al._ (2014); Jin
_et al._ (2017).

The molecular orbital schematic diagram of the halogen bonding between two
iodine molecules provides a valuable visualization of the underlying molecular
interactions essential to construct halogen bonding network under
investigation. As shown in Fig. 2(a), each iodine molecule has one
$\sigma$${}_{p_{x}}$ bond, two $p_{y}$ orbitals (lone pairs) and one
$\sigma^{*}_{p_{x}}$ antibond orbital. $s$ orbital forms a core pair and the
$p_{z}$ orbital lies out of the plane, so they are ignored. Electrons on
$p_{y}$ lone pair (electron donor) could be transferred from the left diiodine
to the $\sigma^{*}_{p_{x}}$ antibond orbital (serve as electron acceptor
Cavallo  _et al._ (2016)) on the right diiodine, which is the orbital
interaction picture of halogen bonding. For the case of herringbone iodinene,
every molecule acts as both a donor and an acceptor, constituting a XB net
(red dashed line in Fig. 1(a)) stabilizing those molecules inside. From the XB
interaction between two iodine molecules, several essential points can be
derived to guide the construction of iodinene. (1) Each diiodine molecule
exhibits two $\sigma$-holes and two lone pairs within the same plane, enabling
each atom to form two connections with other diiodines. Constructing a 2D
iodinene based on the XB network is equivalent to a one-to-one correspondence
between the $\sigma$-hole and the lone pair. (2) Iodinene is expected to be
planar, devoid of buckling, as all $\sigma$-holes and lone pairs involved in
the XB are confined to the same plane. (3) Every iodine atom should be
considered equivalent to one another in principle. There is no justification
for discrimination among them, as the XBs occur uniformly between diiodine
molecules.

The tiling theory can help us construct iodinenes systematically. If any two
polygons are either disjoint or share one vertex or one entire edge in common,
tiling by convex polygons is called edge-to-edgeGrunbaum and Shephard (1977);
Chavey . In contrast, if adjacent tiles share only part of their edge, this is
so-called non-edge-to-edge tiling. Some works have predicted structures of 2D
covalent materials utilizing the tiling theoryLu  _et al._ (2019); Guan  _et
al._ (2014); Jin  _et al._ (2017), more specifically, within the edge-to-edge
classification. Regarding iodinenes, there are refined concepts that involve
non-edge-to-edge tilings. Firstly, each atom is treated as a vertex, and
intramolecular bonds and XBs are represented as edges in the tiling. Secondly,
each vertex is connected to three edges, two of which have equal length, which
corresponds to the case that each iodine atom connects a covalent bond and two
equidistant XBs. Additionally, adjacent edges with equal lengths cannot be
collinear, because $\sigma$-hole and $p_{y}$ lone pair on the same atom cannot
be in alignment. Lastly, all vertices should be equivalent, meaning they are
related by the symmetry of the tiling, known as vertex-transitive or uniform
tiling Grunbaum and Shephard (1977); Chavey . Based on this analysis, we can
identify the required tilings from the existing non-edge-to-edge
patternsGrünbaum and Shephard (1978). The results are presented in Fig. 3.

Five non-edge-to-edge tilings (see Fig. 3(a1)$\sim$(a5)) are selected from the
existing 91 types of isogonal tilingsGrünbaum and Shephard (1978) to map the
tiling pattern to iodinenes by placing an atom on every vertex. Hence, the
nomenclatures of those iodinenes could be labeled by their tiling patterns.
The initial length of the short and long edges are set as 2.75 and 3.60 Å
respectively, according to the intramolecular and XB length in the case of
monolayer herringbone iodinene, and all XBs’angle (see definition of XB angle
at Fig. 2(b)) are set as 180$\degree$. After structural relaxation, all the
initial structures are slightly distorted without breaking the lattice
symmetry, with those XB’ angles deviating from the initial 180$\degree$ to
different extents. Bonding features could be seen simply from the charge
density and electrostatic potential (ESP) map (Fig. S6 & S7). The rectangle
herringbone pattern evolves to a hexagonal herringbone pattern (Fig. 3(b1)),
which is totally equivalent to the foregoing monolayer herringbone iodinene.
The Pythagorean tiling (Fig. 3(a2)) is tessellated by two different size
squares, whose name originates from the Pythagoras theorem. Slight rotation of
the XB occurs after structural relaxation, i.e., from 180$\degree$ to
179.1$\degree$, making the edge on the bigger square slightly distorted.
Interestingly, in all squares, internal angles remain exactly 90$\degree$, so
we call this structure Pythagorean iodinene (Fig. 3(b2)). Gyrated truncated
hexagonal tiling (GTH, see Fig. 3(a3)) is composed of regular hexagons and
regular trigons, resembling the diatomic-kagome lattice topologically and
possessing a novel electronic structure that will be illustrated later. As for
the gyrated hexagonal tiling (GH, see (Fig. 3(a4))), XB forms the edge of the
hexagon and the partial edge of the trigon, which is reversed from GTH. The
fifth pattern is truncated distorted trihexagonal tiling (TDT, see (Fig.
3(a5))), which is composed of small regular trigons and big distorted
truncated trigons. TDT bears the maximal distortion of XB angle from
180$\degree$ to 163.9$\degree$ during relaxation. More details such as space
group and Wyckoff position of those relaxed structures are shown in Table. S2.
The phonon spectra of the five iodinene structures were calculated (Fig. S10),
revealing that herringbone, Pythagorean, and GTH/diatomic-kagome exhibited no
imaginary frequencies. GH displayed a small imaginary frequency near the
$\Gamma$ point, but it could be eliminated with a mere 4$\%$ tensile strain.
While these configurations were all selected based on the same principles, TDT
was the only one with a considerable imaginary frequency, warranting a
separate investigation. Formation energy calculations (Table. S2) indicated
that herringbone iodinene possessed the lowest-energy configuration, while
Pythagorean, TDT, GH, and GTH exhibited progressively higher energies in
order.

To comprehensively explore the potential structures of monolayer iodinenes, we
employed additional conventional methods to generate more 2D configurations.
(a) We assume some simple lattices for iodinenes such as square, rectangle,
triangle, honeycomb, lieb, and kagome structures (see Fig. S14). Iodinenes
mapped on those simple lattices behave as conventional covalent 2D crystals
where iodine serves as a polyvalent element. And those structures are highly
energetic and dynamically unstable (Fig. S16), part of which is even more
energetic than gas diiodine that are not likely to form a stable freestanding
crystal against dissociation to gas diiodine (see Table. S2). (b) Furthermore,
we manually connected diiodine-based building blocks (see Fig. S8) solely
through XBs, without employing the knowledge of the tiling theory. Through
this approach, we have obtained a total of nine configurations (see Fig. S9),
with two of them (“T4” and “T4+T4m”, as detailed in the supplementary
materials) differing from the previously five non-edge-to-edge tiling
patterns. However, both “T4” and “T4+T4m” exhibit dynamic instability (Fig.
S10), despite having lower energy than GTH. This phenomenon may be attributed
to the violation of the requirement for all iodine atoms to be equivalent. (c)
Additionally, a structure search using Particle Swarm Optimization (PSO) Wang
_et al._ (2012) is conducted for comparative analysis with the designed
structures (see Fig. S14). The PSO search identifies the emergence of
herringbone and Pythagorean iodinenes in our design. However, a few low-energy
configurations, such as GTH and GH iodinenes, are not detected. This contrast
highlights the distinct effectiveness of our design.

Formation energy of all configurations derived is listed in Table. S2, with
the energy of gas diiodine being the reference zero, where both M06-L+D3 and
PBE functionals are used. Certainly, PBE leads to inaccurate results, but it
tells us the significance of using M06-L+D3 exactly in the comparison. Under
the PBE scheme, the formation energy of Pythagorean iodinene reaches 412 meV,
which exceeds the bulk iodine and herringbone iodinene, where the correct
inclusion of XB energy is missed. Under the M06-L+D3 scheme, herringbone
iodinene is the most low-energetic configuration apart from bulk iodine,
followed by Pythagorean iodinene. The seven configurations with the lowest
energy (herringbone, Pythagorean, “T4+T4m”, TDT, GH, “T4”, GTH) consist of
iodine molecules linked by halogen bonds. Conversely, alternative energetic
configurations comprise iodine molecules without XB nets, like ITSS (isosceles
triangle snub square) and OR (octagon rhombus), and exhibit typical behavior
of a covalent crystal lattice (refer to Fig. S15). These outcomes collectively
demonstrate the genuine inclination of monolayer iodinenes toward arrangements
formed through halogen-bonded iodine molecules, thereby substantiating the
efficiency of our devised approach.

Electronic structures of our designed 2D iodinenes

Figure 4: Band structures of freestanding (a) Pythagorean, (b) gyrated
truncated hexagonal and (c) gyrated hexagonal iodinene. Blue dash and black
lines are the band structures without and with SOC respectively. Topological
Z2 invariants are labeled above the occupied states defined.

Pythagorean iodinene is the second lowest energetic structure among all
monolayer iodinenes ($\Delta$E = -457 meV/molecule). We acquire Pythagorean
iodinene by mapping _Pythagorean_ tiling to monolayer iodinene. After
optimization, intramolecular bond and XB length are 2.77 and 3.52 Å
respectively, and XBs form a orthogonal network creating a non-edge-to-edge
_Pythagorean_ tiling. If the noncovalent interactions of XB is not included,
the imaginary frequency will arise like the case of bulk iodine when PBE was
used (see Fig. S5). The calculated band structures reveal that it is an
insulator with a direct band gap being 1.69 eV with VBM and CBM at $\Gamma$
point, as shown in Fig. 4 (a). And its HSE06 gap is 2.17 eV (Fig. S11(b)),
indicating its potential application in optoelectronics. Moreover, band
crossing between the highest valence band and its lower band is opened by
spin-orbit coupling (SOC), which induces a band inversion. And similar case
occurs between the conduction band and its upper band. Spatial inversion and
time reversal symmetry are preserved, so energy bands are twofold degenerate.
For spatial inversion invariant systemsFu and Kane (2007), we calculate the
topological Z2 invariant by counting the parity of the occupied states (see
Fig. 4(a)) and find that Pythagorean iodinene is 2D topological insulators
with the Fermi energy shifted above the lowest unoccupied band (1.6 eV) or
below highest occupied band (-0.6 eV). In Fig. S13, gapless Dirac edge states
connect the bulk states, corresponding to the Z2 values labeled in Fig. 4(a),
which originates from the bulk-boundary correspondence. Considering 2D
materials are fabricated generally on substrates, the Fermi level could be
shifted by choosing an appropriate substrate or dopping. Hence Pythagorean
iodinene could serve as a potential topological insulator.

Besides Pythagorean iodinene, freestanding GTH/diatomic-kagome and GH iodinene
have topological nontrivial band structures as well. As can be seen from Figs.
4(b) and (c), there are direct band gaps with 1.42 eV and 1.67 eV
respectively. Their topological properties are similar to Pythagorean iodinene
that nontrivial band topology occurs above the lowest unoccupied band or below
the highest occupied band. Moreover, GTH possesses a diatomic-kagome lattice
leading to a flat bandSethi  _et al._ (2021); Liu  _et al._ (2021) around 1.9
eV (see Fig. 4(b)). With an appropriate substrate or heterojunction, the flat
band is promising for a strong correlation platform. Besides, the computed in-
plane optical absorption coefficients for Pythagoren, GTH/diatomic-kagome and
GH iodinenes all exceed $10^{5}cm^{-1}$ within the visible range (with peaks
between $2\sim 3$ eV, see Fig. S12), indicating their potential applications
in photoelectronic devices.

Conclusions and prospects. — An innovative approach to the design of monolayer
iodinenes has been introduced, involving the mapping of unique non-edge-to-
edge tilings onto iodinene structures through the halogen bond between
diiodine molecules. The effectiveness of this method has been validated
through first-principles calculations, including the identification of four
dynamically stable structures: herringbone, Pythagorean, GTH/diatomic-kagome,
and GH. These iodinenes exhibit direct or quasi-direct band gaps, with
herringbone displaying exceptional anisotropic light absorption coefficients
within the visible light range, holding potential for optoelectronic
application. Pythagorean, GTH/diatomic-kagome, and GH iodinenes showcase
nontrivial band topologies, and a flat band emerges in GTH/diatomic-kagome
iodinene, enhancing the potential for diverse applications. Notably,
traditional approaches, including PSO searching, yield structures devoid of
halogen bonds. These structures exhibit higher energies than those formed by
iodine molecules interconnected through halogen bonds, and they display
dynamic instability. This outcome underscores the preference of iodinenes for
construction from iodine molecules united by halogen bonds rather than
conventional 2D covalent crystals, further substantiating the rationality of
our design approach. Furthermore, the M06-L exchange-correlation functional is
noteworthy for its balanced combination of computational accuracy and
efficiency in the DFT simulation of halogen bond interactions within periodic
systems.

Different from other Xenes as covalent crystals, iodinene belongs to a new
category of the 2D crystals where XBs dominate. As XBs have extensive
applications in crystal engineeringTeyssandier  _et al._ (2020); Metrangolo
_et al._ (2007), catalysisSutar and Huber (2019), supramolecular
chemistryPriimagi  _et al._ (2013); Metrangolo and Resnati (2001),
biomolecular systemsAuffinger  _et al._ (2004), self-assemblyCorradi  _et al._
(2000); Zha  _et al._ (2016) and drug designLu  _et al._ (2009), etc. already,
monolayer iodinenes including herringbone, Pythagorean, GTH/diatomic-kagome
and GH patterns would provide a new platform to explore more innovations.

Acknowledgements. — The work is supported by the National Key R&D Program of
China (Grant No. 2020YFA0308800) and National Natural Science Foundation of
China (Grant No. 12204330). We would like to thank Jin Cao, Chaoxi Cui,
Xiaodong Zhou, Xiuxian Yang and Liping Liu for helpful discussions.

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