# Geometry-induced Monopole Magnetic Field and Quantum Spin Hall Effect

Yong-Long Wang1,2 Email<EMAIL_ADDRESS>Hao Zhao1,2 Hua Jiang1 Hui
Liu2<EMAIL_ADDRESS>Yan-Feng Chen3 Email<EMAIL_ADDRESS>1 School of
Physics and Electronic Engineering, Linyi University, Linyi, 276000, China  2
Department of Physics, Nanjing University, Nanjing 210093, China  3 National
Laboratory of Solid State Microstructures, Department of Materials Science and
Engineering, Nanjing University, Nanjing, 210093, China

###### Abstract

The geometric effects of two-dimensional curved systems have been an
interesting topic for a long time. A Möbius surface is specifically
considered. For a relativistic particle confined to the nontrivial surface, we
give the effective Dirac equation in the thin-layer quantization formalism,
and we find a geometric gauge potential that results from the rotation
transformation of the local frame moving on Möbius strip, and an effective
mass that is from the rescaling transformation. Intriguingly, the geometric
gauge potential can play a role of monopole magnetic field for the particles
with spin, and which can produce quantum spin Hall effects. As potential
applications, effective monopole magnetic fields and spin Hall phenomena can
be generated and manipulated by designing the geometries and topologies of
two-dimensional nanodevices.

PACS Numbers: 73.50.-h, 73.20.-r, 03.65.-w, 02.40.-k

##  I Introduction

With the rapid development of artificial microstructure technology,
theoretical and experimental physicists are always trying to discover novel
phenomena of Hall physics in the thin films with complex geometries. Those
investigations boost the interests in the effective quantum dynamics of curved
surface. A much suitable scheme, the thin-layer quantization approach was
primitively employed to study the geometric quantum effects of curved surface
by introducing a confining potential Jensen and Koppe (1971); da Costa (1981),
and then the approach was generalised to low-dimensional manifolds embedded in
high-dimensional manifolds Schuster and Jaffe (2003). In order to eliminate
the ambiguity of calculation order, the quantization approach was clearly
regularized in a fundamental formalism Wang and Zong (2016), in which the
geometric effects mainly manifest as a scalar geometric potential da Costa
(1981), a geometric momentum Liu  _et al._ (2011); Wang  _et al._ (2017), a
geometric orbital angular momentum Wang  _et al._ (2017), and a geometric
gauge potential Schuster and Jaffe (2003); Wang  _et al._ (2018). The scalar
geometric potential has been proved that can construct a topological band
structure for periodically minimal surfaces Aoki  _et al._ (2001), can
generate bound states for spirally rolled-up nanotubes Ortix and van den Brink
(2010), can eliminate the reflection for bent waveguides Campo  _et al._
(2014), can provide the transmission gaps for periodically corrugated thin
layers Wang  _et al._ (2016); Cao  _et al._ (2019) and so on. The geometric
momentum and the geometric angular momentum can additionally contribute Lai
_et al._ (2019) and modify the spin-orbit coupling Ortix  _et al._ (2014);
Wang  _et al._ (2017); Armitage  _et al._ (2018). As empirical evidences, the
scalar geometric potential has been realized by an optical analogue in a
topological crystal Szameit  _et al._ (2010), and the geometric momentum was
observed to affect the propagation of surface plasmon polaritons on metallic
wires Spittel  _et al._ (2015). In other words, the thin-layer quantization
formalism is valid for a curved surface system.

As far as we know, the thin-layer quantization formalism has been successfully
employed to deduce the effective Schrödinger equation Jensen and Koppe (1971);
da Costa (1981, 1982); Encinosa and Etemadi (1998); Encinosa and Mott (2003);
Gravesen and Willatzen (2005); Ferrari and Cuoghi (2008); Jensen and Dandoloff
(2009); Ortix and van den Brink (2011); de Oliveira (2014) and the effective
Pauli equation Kosugi (2011); Kenneth and Avron (2014); Wang  _et al._ (2014).
Most of the known geometric effects are induced by curvature that is
determined by the relationship between the three-dimensional metric tensor and
the two-dimensional metric tensor Jensen and Koppe (1971); da Costa (1981),
which can be also determined by a diffeomorphism transformation. As a crucial
ingredient, the geometric potential in the non-relativistic curved system does
not appear in the relativistic case, the effective Dirac equation Matsutani
(1993); Burgess and Jensen (1993); Brandt and Sánchez-Monroy (2016); Yang  _et
al._ (2020). Therefore, for the relativistic particle confined to a curved
surface the geometric effects need a further investigation. Due to the spinor
being not a representation of a dffeomorphism group, we have to consider not
only the diffeomorphism transformations, but also the rotation transformation
of dreibein fields that is a local Lorentz rotation Parrikar  _et al._ (2014).
In the thin-layer quantization formalism, the effective quantum dynamics is
obtained by reducing the normal degree of freedom, and the geometric effects
can be induced not only by curvature, but also by torsion Wang  _et al._
(2018). The torsion-induced effects have been demonstrated in a twisted
quantum ring Taira and Shima (2010a), on a Möbius ladder Zhao  _et al._
(2009); de Souza  _et al._ (2017) and a space curve Brandt and Sánchez-Monroy
(2015) as a torsion-induced magnetic moment, a torsion-induced Zeeman-like
coupling and an anomalous phase shift Taira and Shima (2010b), respectively.
Therefore, in the torsion-induced gauge structure, the nontrivial properties
of geometric magnetic field and the topological properties of torsion Landau
levels need further investigations.

The Dirac magnetic monopole is an Abelian monopole that results from the
singularities of electromagnetic field Dirac (1931). Decades later a non-
Abelian magnetic field was generalised for the non-Abelian Yang-Mills gauge
field Wu and Yang (1975); Nishino  _et al._ (2018); Deguchi and Fujikawa
(2020). Although the magnetic monopole is not an active research topic, the
related researches are still published from time to time. Theoretically, the
magnetic monopole was constructed in the pure Yang-Mills theory Wu and Yang
(1975), in the Georgi-Glashow model ’tHooft (1974) and in the ”complementary”
gauge-scalar model Kondo (2016). Experimentally, the magnetic monopole has
been proposed to be realised in bilayer graphene San-Jose  _et al._ (2012);
Uri  _et al._ (2020). Particularly, the discussions of magnetic monopole were
triggered by the appearance of SU(2) gauge theory in the classification of
four-manifolds Donaldson and Kronheimer (1990). In a most general case, the
effective quantum dynamics on a two-dimensional manifold or a one-dimensional
manifold can be endowed SU(2) gauge structures by reducing two degrees of
freedom with a confining potential of SO(3) symmetries Brandt and Sánchez-
Monroy (2015); Liang  _et al._ (2020). For a specific case, it was proved that
the effective dynamics confined to a certain curved surface is endowed U(1)
gauge structure by introducing a confining potential with SO(2) symmetries
Wang  _et al._ (2020). Mathematically, those Abelian and non-Abelian gauge
fields can be induced by the geometries of curved systems that is like the
strain-induced pseudomagnetic field Zhang  _et al._ (2014) and the tensor
monopole Tan  _et al._ (2021). As a crucial result in the present paper, an
effective SU(2) gauge potential can be induced by the geometries of Möbius
strip for spin, and behaves as a monopole effective magnetic field.

A new topological state of quantum matter, the quantum spin Hall effect is
different from the traditional quantum Hall effect without externally applying
magnetic fields. It is strikingly that the quantum spin Hall effect was
primitively and independently predicted as the effects of spin-orbit
interactions Kane and Mele (2005a, b) and as the result of the presence of a
strain gradients Bernevig and Zhang (2006). Subsequently, the new topological
phenomenon was experimentally realised in HgTe quantum wells Bernevig  _et
al._ (2006). For the quantum Hall effect, the quantized Hall conductances are
determined by the quantum Landau levels that are created by the externally
applied magnetic fields, while the quantum spin Hall effect is determined by
the degenerate quantum Landau levels that is created by the spin-orbit
coupling in conventional semiconductors Hasan and Kane (2010); Qi and Zhang
(2011). Therefore, it is relative hard to realise experimentally. In view of
that the geometries of curved systems can play the role of gauge potential for
particles with orbital spin Olpak (2012); Wang  _et al._ (2018), therefore
there appears an interesting topic to study the relationship between the
topologies of quantum Hall state and those of curved systems. As another
important result, the quantum spin Hall effects can be induced by the geometry
of Möbius strip.

As previously mentioned, we will deduce the effective Dirac equation for a
relativistic particle that is confined to Möbius strip, and will specifically
study the quantum effects that are induced by the nontrivial properties of
Möbius strip. The present paper is organized as follows. In Sec. the
geometrical properties of a Möbius strip are reviewed. In Sec. III, the thin-
layer quantization formalism is briefly discussed, and then it is employed to
deduce the effective Dirac equation for the relativistic particle confined to
Möbius strip. In Sec. IV, the geometry-induced monopole magnetic field is
discussed. In Sec. V, the quantum spin Hall effects induced by geometry are
investigated. In Sec. V, the conclusions and discussions are given.

##  II The geometrical properties of Möbius strip

For a curved surface $\mathbb{S}^{2}$, one can adapt a suitable curvilinear
coordinate system to describe it as $\textbf{r}(q_{1},q_{2})$, a function of
$q_{1}$ and $q_{2}$, where $q_{1}$ and $q_{2}$ are two tangent coordinate
variables of $\mathbb{S}^{2}$. For a point near $\mathbb{S}^{2}$, one has to
introduce a coordinate variable $q_{3}$ normal to $\mathbb{S}^{2}$ to
parameterize it as
$\textbf{R}(q_{1},q_{2},q_{3})=\textbf{r}(q_{1},q_{2})+q_{3}\textbf{n}$, where
n denotes the basis vector normal to $\mathbb{S}^{2}$. The presence of
$\mathbb{S}^{2}$ will deform its near space that is denoted as
$\Xi\mathbb{S}^{2}$. The deformation can be described by a diffeomorphism
transformation which belongs to $GL(3,\mathbb{R})$. As usual, a relativistic
particle can be described by a spinor. Due to without a spinor representation,
$GL(3,\mathbb{R})$ can not be employed to describe the action of
diffeomorphism transformation on the spinor that is confined to
$\mathbb{S}^{2}$. However, the confined spinor will obey the rotation
transformation connecting the local frames of the different points on
$\mathbb{S}^{2}$ Bertlmann (1996), which is a generator of $SO(3)$. In what
follows, a Möbius strip will be considered.

Figure 1: Schematic of a Möbius strip. $\vec{e}_{r}$ and $\vec{e}_{s}$ are two
tangent unit basis vectors, and $\vec{e}_{n}$ is the normal unit basis vector
of Möbius strip. The local frames $F_{1}$, $F_{2}$ and $F_{3}$ are localized
at $\theta=0,\frac{4\pi}{3},\frac{8\pi}{3}$, respectively.

A Möbius surface of half-width $w$ with midcircle of radius $R$ and at height
$z=0$, which can be parametrized by $\textbf{r}(r,\theta)=(r_{x},r_{y},r_{z})$
with

$\begin{split}&r_{x}(r,\theta)=[R+r\cos(\frac{\theta}{2})]\cos\theta,\\\
&r_{y}(r,\theta)=[R+r\cos(\frac{\theta}{2})]\sin\theta,\\\
&r_{z}(r,\theta)=r\sin(\frac{\theta}{2}),\end{split}$ (1)

where $r$ and $\theta$ are two variables with $r\in[-w,w]$ and
$\theta\in[0,4\pi]$, $R$ can be taken as a constant. The Möbius strip is
denoted by $\mathbb{M}^{2}$ and sketched in Fig. 1. For convenience, we can
adapt an orthogonal frame spanned by two tangent vectors $\textbf{e}_{r}$ and
$\textbf{e}_{s}$ and a normal vector $\textbf{e}_{n}$. According to Eq. (1),
we can obtain the three basis vectors as

$\begin{split}&\textbf{e}_{r}=(\cos\frac{\theta}{2}\cos\theta,\cos\frac{\theta}{2}\sin\theta,\sin\frac{\theta}{2}),\\\
&\textbf{e}_{s}=\frac{2}{N}(-(R\sin\theta+\frac{3}{2}r\cos\theta\sin\frac{\theta}{2}+r\sin\frac{\theta}{2}),\\\
&\quad\quad\quad
R\cos\theta+\frac{1}{4}r\cos\frac{\theta}{2}+\frac{3}{4}r\cos\frac{3\theta}{2},\frac{1}{2}r\cos\frac{\theta}{2}),\\\
&\textbf{e}_{n}=\frac{2}{N}(R\sin\frac{\theta}{2}\cos\theta-r\sin^{2}\frac{\theta}{2}\sin\theta,\\\
&\quad\quad\quad
R\sin\frac{\theta}{2}\sin\theta+\frac{1}{2}r(\sin^{2}\theta+\cos\theta),\\\
&\quad\quad\quad-R\cos\frac{\theta}{2}-r\cos^{2}\frac{\theta}{2}),\end{split}$
(2)

respectively, where $N$ is a normalized constant as

$N=(4R^{2}+8Rr\cos\frac{\theta}{2}+2r^{2}\cos\theta+3r^{2})^{\frac{1}{2}}.$
(3)

It is straightforward that $(\textbf{e}_{r},\textbf{e}_{s},\textbf{e}_{n})$
can be obtained from the usual basis vectors
$(\textbf{e}_{x},\textbf{e}_{y},\textbf{e}_{z})$ by a rotation transformation
${\rm{U}}_{\mathcal{R}}$ in the following form

$\left[\begin{array}[]{ccc}\textbf{e}_{r}\\\ \textbf{e}_{s}\\\
\textbf{e}_{n}\end{array}\right]={\rm{U}}_{\mathcal{R}}(r,\theta)\left[\begin{array}[]{ccc}\textbf{e}_{x}\\\
\textbf{e}_{y}\\\ \textbf{e}_{z}\end{array}\right],$ (4)

where ${\rm{U}}_{\mathcal{R}}$ can be specifically expressed as

${\rm{U}}_{\mathcal{R}}(r,\theta)=\left[\begin{array}[]{ccc}{e_{r}}^{x}&{e_{r}}^{y}&{e_{r}}^{z}\\\
{e_{s}}^{x}&{e_{s}}^{y}&{e_{s}}^{z}\\\
{e_{n}}^{x}&{e_{n}}^{y}&{e_{n}}^{z}\end{array}\right],$ (5)

through the dreibeins ${e_{\alpha}}^{i}$, $(\alpha=r,s,n)$ and $(i=x,y,z)$,
which can be written as

$\begin{split}&{e_{r}}^{x}=\cos\frac{\theta}{2}\cos\theta,\\\
&{e_{r}}^{y}=\cos\frac{\theta}{2}\sin\theta,\\\
&{e_{r}}^{z}=\sin\frac{\theta}{2},\\\
&{e_{s}}^{x}=-\frac{1}{N}(4R\cos\frac{\theta}{2}+3r\cos\theta+2r)\sin\frac{\theta}{2},\\\
&{e_{s}}^{y}=\frac{1}{N}(2R\cos\theta+\frac{1}{2}r\cos\frac{\theta}{2}+\frac{3}{2}r\cos\frac{3\theta}{2}),\\\
&{e_{s}}^{z}=\frac{1}{N}r\cos\frac{\theta}{2},\\\
&{e_{n}}^{x}=\frac{2}{N}(R\sin\frac{\theta}{2}\cos\theta-r\sin^{2}\frac{\theta}{2}\sin\theta),\\\
&{e_{n}}^{y}=\frac{1}{N}[2R\sin\frac{\theta}{2}\sin\theta+r(\sin^{2}\theta+\cos\theta)],\\\
&{e_{n}}^{z}=-\frac{2}{N}(R\cos\frac{\theta}{2}+r\cos^{2}\frac{\theta}{2}).\end{split}$
(6)

It is easy to check that ${\rm{U}}_{\mathcal{R}}(r,\theta)$ is a generator of
$SU(2)$ group, the universal cover of $SO(3)$, for $\theta\in[0,4\pi]$.

As another result, the geometry of Möbius strip will deform its near space
denoted as $\Xi\mathbb{M}^{2}$. The deformation leads to the influence of the
normal space on the tangent space, which can be described by a rescaling
factor that relates the three-dimensional metric tensor $G_{\alpha\beta}$ and
the two-dimensional metric tensor $g_{ab}$. With the definition,
$g_{ab}=\partial_{a}\textbf{r}\cdot\partial_{b}\textbf{r}$ $(a,b=1,2)$, the
metric tensor defined on $\mathbb{M}^{2}$ can be obtained as

$g_{ab}=\left[\begin{array}[]{ccc}1&0\\\ 0&\frac{N^{2}}{4}\end{array}\right],$
(7)

and the corresponding determinant and inverse matrix can be then obtained as

$g=\frac{N^{2}}{4},$ (8)

and

$g^{ab}=\left[\begin{array}[]{ccc}1&0\\\ 0&\frac{4}{N^{2}}\end{array}\right],$
(9)

respectively. With the definition,
$h_{ab}=\textbf{e}_{n}\cdot\partial^{2}\textbf{r}/\partial q^{a}\partial
q^{b}$ $(a,b=1,2)$, the second fundamental form $h_{ab}$ can be written as

$h_{ab}=\left[\begin{array}[]{ccc}0&R/N\\\
R/N&\frac{N^{2}+r^{2}}{2N}\sin\frac{\theta}{2}\end{array}\right].$ (10)

Further, the Weingarten curvature matrix can be calculated as

${\alpha_{a}}^{b}=\left[\begin{array}[]{ccc}0&-4R/N^{3}\\\
-R/N&-\frac{2(N^{2}+r^{2})}{N^{3}}\sin\frac{\theta}{2}\end{array}\right]$ (11)

with ${\alpha_{a}}^{b}=-h_{ac}g^{cb}$.

In terms of $\mathbb{M}^{2}$, the position vector of a point in
$\Xi\mathbb{M}^{2}$ can be parameterized by

$\textbf{R}(r,\theta,q_{3})=\textbf{r}(r,\theta)+q_{3}\textbf{e}_{n},$ (12)

where $q_{3}$ is a coordinate variable normal to $\mathbb{M}^{2}$. With the
definition,
$G_{\alpha\beta}=\partial_{\alpha}\textbf{R}\cdot\partial_{\beta}\textbf{R}$
$(\alpha,\beta=1,2,3)$, the covariant elements $G_{ab}$ can be described by

$G_{ab}=g_{ab}+(\alpha g+g^{T}\alpha^{T})_{ab}q_{3}+(\alpha
g\alpha^{T})_{ab}(q_{3})^{2}$ (13)

through $g_{ab}$ and ${\alpha_{a}}^{b}$, and $G_{33}=1$ and the rest elements
vanishing. It is easy to prove that $g$, the determinant of $g_{ab}$, and $G$,
the determinant of $G_{\alpha\beta}$, satisfy the following simple
relationship

$G=f^{2}g,$ (14)

where $f$ is the rescaling factor with the following form

$f=1+{\rm{Tr}}(\alpha)q_{3}+{\rm{det}}(\alpha)(q_{3})^{2}.$ (15)

In relativistic case, the Dirac equation contains only one-order derivative
operator, therefore $f^{\frac{1}{2}}$ and $f^{-\frac{1}{2}}$ can be further
approximated as

$f^{\mp\frac{1}{2}}\approx 1\mp\frac{1}{2}{\rm{Tr}(\alpha)}q_{3}.$ (16)

Obviously, the determinant of ${\alpha_{a}}^{b}$ disappears from Eq. (16). It
means that the geometric effects of rescaling factor depend only on the mean
curvature ${\rm{M}}$ of $\mathbb{M}^{2}$, not on the Gaussian curvature
${\rm{K}}$. ${\rm{M}}$ and ${\rm{K}}$ can be expressed in the following form

$\begin{split}&{\rm{M}}=\frac{1}{2}{\rm{Tr}}(\alpha)=\frac{(N^{2}+r^{2})\sin\frac{\theta}{2}}{N^{3}},\\\
&{\rm{K}}={\rm{det}}(\alpha)=-\frac{4R^{2}}{N^{4}},\end{split}$ (17)

respectively. However, in the non-relativistic limit the effective Hamiltonian
will contain two-order derivative operators, the factor $f^{\frac{1}{2}}$ and
$f^{-\frac{1}{2}}$ have to contain the terms of $(q_{3})^{2}$, and ${\rm{K}}$
will then contribute additional geometric effects. As a result, in the case of
a curved surface with nonvanishing ${\rm{K}}$, the thin-layer scheme does not
commutate with the non-relativistic limit.

##  III The effective Dirac equation on Möbius strip

In this section, for the relativistic particle confined to $\mathbb{M}^{2}$,
the effective Dirac equation will be deduced in the thin-layer quantization
formalism. In the semiclassical formalism, a confining potential is first
introduced to reduce the degree of freedom normal to $\mathbb{M}^{2}$. Without
loss of generality, the strength of confining potential can not be strong
enough to create particle and anti-particle pairs Burgess and Jensen (1993).
In the other words, the number of particles is conserved in the quantization
procedure.

###  III.1 A Rescaling Transformation

In quantum mechanics, the particle number conservation can be described by

$\begin{split}\int|\Psi^{{\dagger}}\Psi|d\tau&=\int|\Psi|^{2}\sqrt{G}d^{3}q\\\
&=\int|\sqrt{f}\Psi|^{2}(\sqrt{g}drds)dq_{3},\end{split}$ (18)

where $\Psi$ is a wave function that describes a relativistic particle in
three-dimensional space, $q$ stands for the three curvilinear coordinate
variables of an adapted frame, $G$ is the determinant of the metric tensor
$G_{\alpha\beta}$ defined in the subspace $\Xi\mathbb{M}^{2}$ and $g$ is the
determinant of the metric tensor $g_{ab}$ defined on the curved surface
$\mathbb{M}^{2}$. The final aim of the thin-layer quantization scheme is to
give the effective Dirac equation that is separated from the normal quantum
motion analytically. It is worthwhile to notice that the rescaling factor $f$
is generally a function of $q_{1}$, $q_{2}$ and $q_{3}$. The $q$-dependence is
determined by the diffeomorphism transformation induced by $\mathbb{M}^{2}$.
In order to divide $\sqrt{G}d^{3}q$ into a $q_{3}$-independent part and a
$q_{1,2}$-independent one analytically, a new wave function $\chi$,
$\chi=\sqrt{f}\Psi$, has to be introduced. The replacement can eliminate $f$
from $\sqrt{G}d^{3}q$.

Under the diffeomorphism transformation, the wave function $\Psi$ and an
ordinary physical operator $\hat{{\rm{O}}}$ satisfy the following
transformations

$\begin{split}&\chi=f^{\frac{1}{2}}\Psi,\\\
&\hat{{\rm{O}}}^{\prime}=f^{\frac{1}{2}}\hat{{\rm{O}}}f^{-\frac{1}{2}}.\end{split}$
(19)

In view of the original intention of the introduction of $\chi$, the above
transformation can describe the redistribution of the spatial probability of
particle near $\mathbb{M}^{2}$. However, this transformation can not well
describe the influences on spinor. As a result, for the spinor on
$\mathbb{M}^{2}$, one has to consider the rotation transformation defined by
the background dreibein field of $\mathbb{M}^{2}$.

###  III.2 A Frame Rotation Transformation

In comparison with the particle described by Schödinger equation, the particle
described by Dirac equation has an additional intrinsic degree of freedom,
spin. Since the spinor is not a representation of the group
$GL(3,\mathbb{R})$, the diffeomorphism transformation can not describe the
dynamics of spinor on $\mathbb{M}^{2}$. As usual, in $\Xi\mathbb{M}^{2}$ the
spinor is taken as the eigenstate of $\sigma_{3}$, where $\sigma_{3}$ is a
Pauli matrix. It is straightforward to check that the basis vectors
$(\textbf{e}_{r},\textbf{e}_{s},\textbf{e}_{n})$ can be obtained by
$(\textbf{e}_{x},\textbf{e}_{y},\textbf{e}_{z})$ through a rotation
transformation ${\rm{U}}_{\mathcal{R}}$, which connects the local frames of
the different points of $\mathbb{M}^{2}$. Specifically, $\sigma_{3}$ can be
also obtained from $\sigma_{z}$ by performing ${\rm{U}}_{\mathcal{R}}$. In the
usual Pauli-Dirac representation, for the spinor on $\mathbb{M}^{2}$ the new
wave function $\chi$ and the rescaled physical operator $\hat{O}^{\prime}$
satisfy the following transformations

$\begin{split}&\chi^{\prime}={\rm{U}}_{\mathcal{R}}\chi,\\\
&\hat{O}^{\prime\prime}={\rm{U}}_{\mathcal{R}}\hat{O}^{\prime}{\rm{U}}_{\mathcal{R}}^{-1},\end{split}$
(20)

where ${\rm{U}}_{\mathcal{R}}$ denotes the rotation connecting
$(\textbf{e}_{r},\textbf{e}_{s},\textbf{e}_{n})$ and
$(\textbf{e}_{x},\textbf{e}_{y},\textbf{e}_{z})$, and
${\rm{U}}_{\mathcal{R}}^{-1}$ is the inverse of ${\rm{U}}_{\mathcal{R}}$.

As a conclusion, for a spinor on $\mathbb{S}^{2}$, the wave function $\Psi$
and an ordinary physical operator $\hat{{\rm{O}}}$ in general need to satisfy
the following transformations,

$\begin{split}&\Psi^{\prime}={\rm{U}}_{\mathcal{R}}f^{\frac{1}{2}}\Psi,\\\
&\hat{{\rm{O}}}^{\prime}={\rm{U}}_{\mathcal{R}}f^{\frac{1}{2}}\hat{{\rm{O}}}f^{-\frac{1}{2}}{\rm{U}}_{\mathcal{R}}^{-1}.\end{split}$
(21)

Furthermore, the effective physical operator describing the spinor on
$\mathbb{S}^{2}$ can be determined by

$\hat{{\rm{O}}}_{{\rm{eff}}}=\lim_{q_{3}\to
0}({\rm{U}}_{\mathcal{R}}f^{\frac{1}{2}}\hat{{\rm{O}}}f^{-\frac{1}{2}}{\rm{U}}_{\mathcal{R}}^{-1})-\hat{{\rm{O}}}_{\bot},$
(22)

where $\hat{{\rm{O}}}_{\bot}$ denotes the normal component of $\hat{\rm{O}}$.
This equation is key in the present paper, which condenses the initial spirit
of the thin-layer quantization formalism Wang and Zong (2016). Interestingly,
it is can be extended to vector fields, such as electromagnetic field Lai  _et
al._ (2019).

###  III.3 Effective Dirac Equation

In the spirit of the thin-layer quantization scheme, a relativistic particle
is initially considered that is described by a Dirac equation,

$(i\gamma^{\mu}\partial_{\mu}-m)\Psi=0,$ (23)

where $\partial_{\mu}$ is a derivative operator in (3+1)-dimensional
spacetime, $m$ is the static mass of particle, and $\gamma^{\mu}$
$(\mu=0,1,2,3)$ are $4\times 4$ Dirac matrices that satisfy the following
anticommutation relationship

$[\gamma^{\mu},\gamma^{\nu}]_{+}=\gamma^{\mu}\gamma^{\nu}+\gamma^{\nu}\gamma^{\mu}=2\eta^{\mu\nu},$
(24)

where $\eta^{\mu\nu}={\rm diag}(1,-1,-1,-1)$. In Eq. (23), $\Psi$ is a four-
component spinor, which can be separated into a scalar component $\psi$ and a
vector component $\hat{s}$ under the transformations Eqs. (19) and (20). In
the Pauli-Dirac representation, the Dirac matrices $\gamma^{\mu}$ can be
directly expressed as

$\gamma^{0}=\left(\begin{array}[]{ccc}I&0\\\
0&I\end{array}\right),\quad\gamma^{i}=\left(\begin{array}[]{ccc}0&\sigma^{i}\\\
-\sigma^{i}&0\end{array}\right),$ (25)

where $I$ is a unit $2\times 2$ matrix, and $\sigma^{i}$ $(i=x,y,z)$ are Pauli
matrices

$\sigma^{x}=\left(\begin{array}[]{ccc}0&1\\\
1&0\end{array}\right),\sigma^{y}=\left(\begin{array}[]{ccc}0&-i\\\
i&0\end{array}\right),\sigma^{z}=\left(\begin{array}[]{ccc}1&0\\\
0&-1\end{array}\right),$ (26)

respectively.

In the thin-layer quantization formalism, for a relativistic particle in
$\Xi\mathbb{M}^{2}$ the four-component wave function $\Psi$ and the Dirac
Hamiltonian $\rm{H}$ in Eq. (23) should satisfy the transformations Eq. (21)
as

$\begin{split}&\chi={\rm{U}}_{\mathcal{R}}f^{\frac{1}{2}}\Psi,\\\
&{\rm{H}}^{\prime}={\rm{U}}_{\mathcal{R}}f^{\frac{1}{2}}{\rm{H}}f^{-\frac{1}{2}}{\rm{U}}_{\mathcal{R}}^{-1}.\end{split}$
(27)

Subsequently, a confining potential is introduced to reduce the normal degree
of freedom. In the presence of introduced potential the relativistic particle
is going to longtime stay at the ground state of the dimension normal to
$\mathbb{M}^{2}$. With the normal ground state and Eq. (22), the effective
Dirac Hamiltonian can be specifically determined by

$\begin{split}{\rm{H}}_{{\rm{eff}}}&=\lim_{\varepsilon\to
0}\langle\chi_{{\bot}_{0}}|({\rm{U}}_{\mathcal{R}}f^{\frac{1}{2}}{\rm{H}}f^{-\frac{1}{2}}{\rm{U}}_{\mathcal{R}}^{-1})-{\rm{H}}_{\bot}|\chi_{{\bot}_{0}}\rangle\\\
&=\langle\chi_{{\bot}_{0}}|({\rm{U}}_{\mathcal{R}}f^{\frac{1}{2}}{\rm{H}}f^{-\frac{1}{2}}{\rm{U}}_{\mathcal{R}}^{-1})-{\rm{H}}_{\bot}|\chi_{{\bot}_{0}}\rangle_{0},\end{split}$
(28)

where $\varepsilon$ describes the scaling size of the normal degree of
freedom, and $\chi_{\bot_{0}}$ is the normal ground state. In order to obtain
the solution of ground state, the transformed wave function $\chi$ is
necessarily divided into a tangent part $\chi_{||}(r,s)$ and a normal part
$\chi_{\perp}(q_{3})$ analytically,

$\chi(q)=\chi_{\bot}(q_{3})\chi_{\|}(r,s).$ (29)

In the process, the time dimension is conveniently taken as a constant
variable, the introduced confining potential can not strong enough to create
particle and antiparticle pairs Burgess and Jensen (1993) to conserve the
particle number invariance. For the sake of simplicity, the confining
potential $V_{c}$ can be chosen Encinosa  _et al._ (2005) in the following
form

$V_{c}=\lim_{\varepsilon\to 0}\begin{cases}0,\quad-\frac{\varepsilon}{2}\leq
q_{3}\leq\frac{\varepsilon}{2},\\\ \infty,\quad
q_{3}<-\frac{\varepsilon}{2},\quad q_{3}>\frac{\varepsilon}{2},\end{cases}$
(30)

where $\varepsilon$ describes the width of the potential well. And the Dirac
equation (23) is then rewritten as

$(i\gamma^{\alpha}\partial_{\alpha}-m+V_{c})\Psi=E\Psi,$ (31)

where $\alpha=1,2,3$ describing the tree coordinate variables in an adapted
curvilinear coordinate system, and $E$ is the eigenenergy of Dirac particle.

For the extreme limit of $V_{c}$, the normal component of Dirac equation can
be directly written into

$(i\gamma^{3}\partial_{3}+V_{c})\chi_{\bot}=E_{\bot}\chi_{\bot},$ (32)

where $E_{\bot}$ is the normal component of energy that satisfies
$E_{\|}+E_{\bot}=E$, wherein $E_{\|}$ is the tangent component. In the
presence of $V_{c}$, the Dirac particle is strictly confined in the interval
$[-\varepsilon/2,\varepsilon/2]$, and thus Eq. (32) can be simplified as

$i\gamma^{3}\partial_{3}\chi_{\bot}=E_{\bot}\chi_{\bot}$ (33)

in the interval. Without considering the spin degree of freedom, the above
equation can be further rewritten as

$-\partial_{3}^{2}\chi_{\bot}=E_{\bot}^{2}\chi_{\bot}.$ (34)

With the boundary continuous conditions, the solution of $\chi_{\bot}$ can be
given as

$\chi_{\bot}(q_{3})=\sqrt{\frac{2}{\varepsilon}}\cos(k_{n}q_{3}),$ (35)

where $k_{n}$ is the normal component of momentum that is quantized as
$k_{n}=(2n+1)\pi/\varepsilon$. With the limit $\varepsilon\to 0$, the gap
between the ground state and the first excited state becomes enough large, the
Dirac particle will permanently lie in the ground state $|\chi_{\bot
0}\rangle$.

In the Pauli-Dirac representation, the spinor on $\mathbb{M}^{2}$ can be taken
as the eigenstates of $\sigma_{3}$, where $\sigma_{3}$ is a Pauli matrix in a
curvilinear coordinate system. Based on the previously discussions, the new
Dirac matrices $\gamma^{\alpha}$ and the new derivative operator $D_{\alpha}$
can be described by the Dirac matrices $\gamma^{i}$ and the derivative
operator $\partial_{i}$ in flat space with a rotation transformation
$\rm{U}_{\mathcal{R}}$,

$\begin{split}&\gamma^{\alpha}=\rm{U}_{\mathcal{R}}\gamma^{i}={e^{\alpha}}_{i}\gamma^{i},\\\
&D_{\alpha}=\rm{U}_{\mathcal{R}}^{-1}\partial_{i}+\partial_{i}(\rm{U}_{\mathcal{R}}^{-1})={e_{\alpha}}^{i}\partial_{i}+(\partial_{i}{e^{i}}_{\alpha}).\end{split}$
(36)

Practically, the rotation transformation $\rm{U}_{\mathcal{R}}$ can be
accomplished by performing three rotation transformations ${\rm{U}}_{z}$,
${\rm{U}}_{y}$ and ${\rm{U}}_{x}$ in order, where ${\rm{U}}_{z}$ is a rotation
around $\textbf{e}_{z}$, ${\rm{U}}_{y}$ is a rotation around
$\textbf{e}_{y}^{\prime}$ which stands for the $y$-axis rotated by
${\rm{U}}_{z}$, and ${\rm{U}}_{x}$ stands for a rotation around
$\textbf{e}_{x}^{\prime\prime}$ which stands for the $x$-axis rotated by
performing ${\rm{U}}_{z}$ and then ${\rm{U}}_{y}$. In Cartesian coordinate
system, ${\rm{U}}_{z}$ can be deduced as

${\rm{U}}_{z}(\theta)=\left[\begin{array}[]{ccc}\cos\theta&\sin\theta&0\\\
-\sin\theta&\cos\theta&0\\\ 0&0&1\end{array}\right],$ (37)

${\rm{U}}_{y}$ can be

${\rm{U}}_{y}(\theta)=\left[\begin{array}[]{ccc}\cos\frac{\theta}{2}&0&-\sin\frac{\theta}{2}\\\
0&1&0\\\ \sin\frac{\theta}{2}&0&\cos\frac{\theta}{2}\end{array}\right],$ (38)

and ${\rm{U}}_{x}$ can be

${\rm{U}}_{x}(r,\theta)=\left[\begin{array}[]{ccc}1&0&0\\\
0&\cos\theta_{x}&\sin\theta_{x}\\\
0&-\sin\theta_{x}&\cos\theta_{x}\end{array}\right],$ (39)

respectively, here $\cos\theta_{x}=2(R+r\cos\frac{\theta}{2})/N$ and
$\sin\theta_{x}=r/N$. It is easy to check that
${\rm{U}}_{\mathcal{R}}={\rm{U}}_{x}{\rm{U}}_{y}{\rm{U}}_{z}$. In the Pauli-
Dirac presentation of $\sigma_{3}$, ${\rm{U}}_{\mathcal{R}}$ can be
reexpressed as

${\rm{U}}_{\mathcal{R}}(r,\theta)=e^{-i{\bm{\theta}}\cdot{\bm{\sigma}}},$ (40)

where ${\bm{\theta}}=(\theta_{x},\theta_{y},\theta_{z})$ and
${\bm{\sigma}}=\bm{e}_{n}\sigma_{3}$, with $\theta_{x}={\rm{arcsin}}(r/N)$,
$\theta_{y}=\theta/2$, $\theta_{z}=\theta$.

Substitute Eqs. (16), (35) and (40) into Eq. (28), we can obtain the effective
Hamiltonian $\rm{H}_{eff}$ that describes the relativistic particle on
$\mathbb{M}^{2}$ as

${\rm{H}_{eff}}=i\gamma^{a}(\partial_{a}-i\sigma_{3}\mathcal{A}_{a})-(m+m_{\rm
eff}),$ (41)

where $\mathcal{A}$ is a geometric gauge potential with
$\mathcal{A}_{a}=\partial_{a}({\bm{\theta}}\cdot{\bm{\sigma}})$ $(a=r,s)$ and
$m_{\rm eff}$ is an effective mass induced by geometry, $m_{\rm
eff}=\frac{1}{2}{\rm{Tr}}(\alpha)$, wherein $\alpha$ denotes the Weingartein
curvature matrix. In terms of Eq. (41), the effective Dirac equation can be
written into the following form

$[i\gamma^{a}(\partial_{a}-i\sigma_{3}\mathcal{A}_{a})-(m+m_{\rm
eff})]|\chi_{\|}\rangle=E_{\|}|\chi_{\|}\rangle,$ (42)

where $E_{\|}$ is the tangent component of energy eigenvalue, and
$|\chi_{\|}\rangle$ stands for the tangent part of wave function. Strikingly,
the geometry of $\mathbb{M}^{2}$ plays a role of gauge potential and the mean
curvature of $\mathbb{M}^{2}$ contributes an effective mass.

Obviously, the geometric gauge potential $\mathcal{A}_{a}$ is given by
$\partial_{a}$ acting on ${\rm{U}_{\mathcal{R}}}^{-1}$, and the effective mass
$m_{\rm{eff}}$ results from the action of $\partial_{3}$ on
$f^{-\frac{1}{2}}$. It is worthwhile to notice that all the high order terms
of $q_{3}$ in $f^{-\frac{1}{2}}$ are vanished by performing the integral
$\langle\chi_{\bot_{0}}|\chi_{\bot_{0}}\rangle_{0}$. Meaningly, the high power
terms are missing the actions of $\partial_{3}$ before performing the non-
relativistic limit. In other words, the thin-layer quantization scheme does
not commute with the non-relativistic limit process. Importantly, the
effective Pauli equation on $\mathbb{M}^{2}$ should be performed in a certain
order Wang and Zong (2016), specifically, the non-relativistic limit is prior
to the thin-layer quantization scheme.

##  IV Geometry-induced Monopole Magnetic Field

(a) (b)
(c) (d)

Figure 2: (a) In the case of a half-integer linking number, there is an
effective monopole magnetic field $\mathcal{\bm{B}}$ that orientates inward.
(b) The monopole field is projected onto a sphere. (c) In the case of a
integer linking number, there is an effective common magnetic field. (d) The
common field is projected onto a sphere.

The rotation ${\rm{U}}_{\mathcal{R}}$ describes the connection of the local
frames of different points on $\mathbb{M}^{2}$, which can be described by two
tangent coordinate variables $r$ and $s$ of $\mathbb{M}^{2}$ without the
normal coordinate variable $q_{3}$. It is easy to obtain that the three
components of the geometric gauge potential are $\mathcal{A}_{r}=2R/N^{2}$,

$\mathcal{A}_{s}=\frac{1}{\sqrt{N^{2}-r^{2}}}\sin^{2}\frac{\theta}{2}\sin^{2}\theta_{x}(\cos\theta\cos^{2}\theta_{x}+\cos\frac{\theta}{2}\sin
2\theta_{x})+\frac{1}{N}[(\sin\theta\sin\frac{\theta}{2}+2\cos\frac{\theta}{2})\cos\theta_{x}-\cos\theta\sin\theta_{x}],$
(43)

and $\mathcal{A}_{n}=0$, respectively. And the geometric magnetic field can be
then calculated as

$\mathcal{B}_{n}=\partial_{r}\mathcal{A}_{s}-\partial_{s}\mathcal{A}_{r},$
(44)

and $\mathcal{B}_{r}=\mathcal{B}_{s}=0$, because $\mathcal{A}_{r}$ and
$\mathcal{A}_{s}$ both do not depend on $q_{3}$, and $\mathcal{A}_{3}=0$. It
is apparent that $\mathcal{B}_{n}$ is along the normal direction of
$\mathbb{M}^{2}$, and orientates inward that is sketched in Fig. 2 (a) as an
effective monopole magnetic field. The result is described in Fig. 2 (b). The
particular phenomenon just displays in the case of a half-integer linking
number Korte and van der Heijden (2009) $\frac{1}{2\pi}\int{\tau
d\theta}=n+\frac{1}{2}$ $(n\in Z)$, $Z$ is an integer. For an integer linking
number $\frac{1}{2\pi}\int{\tau d\theta}=n+1$, the effective magnetic field
becomes common that is sketched in Fig. 2(c), which is also projected onto a
sphere described in Fig. 2 (d). It is apparently that the monopole magnetic
field is completely determined by the topological structure of
$\mathbb{M}^{2}$, a single side. And the geometric magnetic field
$\mathcal{B}_{n}$ is specifically described in the plane spanned by $r$ and
$\theta$ as sketched in Fig. 3.

Figure 3: The contours of geometric magnetic field in the plant spanned by $r$
and $\theta$ with $R=4$, $r\in[-1,1]$ and $\theta\in[0,4\pi]$.

Noticeably, the geometric magnetic field is distinctly different from the
common magnetic field. The geometric magnetic field acts on a particle by
coupling with spin, while the common magnetic field acts on a particle by
coupling with electric charge. Specifically, the effective magnetic field is
determined by the geometry of $\mathbb{M}^{2}$, the nontrivial monopole
properties are defined by the nontrivial topological properties of
$\mathbb{M}^{2}$. Once the nontrivial single side vanishes, the nontrivial
monopole would disappear and the common effective magnetic field would appear.
Furthermore, the geometric magnetic field is in general a non-Abelian gauge
field, which determines the gauge structure of the effective Hamiltonian
confined on $\mathbb{M}^{2}$. In other words, the gauge structure of effective
dynamics can be constructed by the geometry of a particular curved surface. As
potential applications, the effective gauge field can be generated by
designing the geometries two-dimensional nanodevices.

##  V Geometry-induced Quantum Spin Hall Effect

Figure 4: Schematic of a spin Hall effect on a part of Möbius strip. The blue
arrow denotes the geometric magnetic field, the green balls with pink upward
arrows are the electrons of spin up, the green balls with cyan downward arrows
are the electrons of spin down, and the the red arrows stand for the moving
direction of electrons.

In the presence of $\mathcal{\bm{A}}$, the Dirac particle moving on
$\mathbb{M}^{2}$ will then feel a pseudo-Lorentz force, which is induced by
the geometry of $\mathbb{M}^{2}$. Because that the spin of particle is
initially coupled with the geometry of $\mathbb{M}^{2}$ as the term
$\sigma_{3}\mathcal{A}$ in Eq. (42) Wang  _et al._ (2020), which plays the
role of the spin-orbit coupling Kane and Mele (2005a) in conventional
semiconductor and that in the presence of a strain gradient Bernevig and Zhang
(2006), and which can be rewritten as $u_{3}\mathcal{B}_{n}$ in the non-
relativistic limit, where $u_{3}$ denotes the normal component of spin
magnetic moment. In light of the left hand rule, for a certain section of
$\mathbb{M}^{2}$ the spin-outward particles gather to one side, the spin-
inward particles aggregate toward the other side, which are sketched in Fig.
4. The spin-outward means that the spin orientates outward along the normal
direction of $\mathbb{M}^{2}$, the spin-inward means that the spin orientates
inward $\mathbb{M}^{2}$. As a result, on $\mathbb{M}^{2}$ the spin-outward
particles and the spin-inward particles are completely separated into two
groups, they are gathering on the two sides for a certain section,
respectively, and they are collecting on two different faces for a certain
edge, respectively. Most strikingly, the spin-inward particles and the spin-
outward particles are moving in a same direction with same spin polarization
as a full spin current for a certain edge that sketched in Fig. 5 (a).
Interestingly, the spin polarization is entirely determined by the nontrivial
topological properties of $\mathbb{M}^{2}$, which determines the degeneracy of
pseudo-Landau levels in momentum space. The two degenerate pseudo-Landau
levels are separated in configuration space with a gap that is about the width
of $\mathbb{M}^{2}$. Mathematically, the Möbius strip is a two-dimensional
compact manifold with a single boundary and a one-sided surface. In other
words, the topological structure of $\mathbb{M}^{2}$ can spontaneously flip
the ”spin-down” into the ”spin-up” to provide a pure spin current. For a half-
integer number $\frac{1}{2\pi}\oint\tau ds=n+\frac{1}{2}$ $(n\in\mathbb{Z})$,
the pure spin current commonly attributes to the spin-outward particles and
the spin-inward particles. In the case of $\frac{1}{2}$, the big cyan arrows
and the big pink ones contribute equally to the spin current along the same
edge. In the case of an integer number $\frac{1}{2\pi}\oint\tau ds=n$, the
pure spin current differently attributes one kind of particle contribution,
either the spin-outward particles or the spin-inward ones. The result is
sketched in Fig. 5(b). Distinguishingly, the spin-outward particles and the
spin-inward ones do not have the same contribution to the spin current, and
they are impossibly along the same edge.

Figure 5: (a) Spin Hall effect on the Möbius strip of a half-integer linking
number $\frac{1}{2}$. (b) Spin Hall effect on the Möbius strip of an integer
linking number $1$. The blue arrows stand for the geometric magnetic field,
the green balls with pink outward arrows denote the spin-outward particles,
the yellow balls with cyan inward arrows are the spin-inward particles. The
big pink arrows stand for the current consisting of spin-outward particles,
and the big cyan arrows stand for the current consisting of the spin-inward
ones.

As a classical analogy, this can be thought of in terms of the magnus effect,
a spinning soccer ball will ”stray” from its normal straight path in a
direction dependent on it’s sense of rotation. Therefore, the spin-outward
particles will initially gather toward one side, the spin-inward ones
aggregate toward the other side for a certain section of $\mathbb{M}^{2}$. In
the case of half-integer linking number, the ”so-called” two edges are really
the same one, the emergence of quantum spin Hall effect is eventually
determined by the geometry of $\mathbb{M}^{2}$. In the case of integer linking
number, the two edges are completely different, the spin current vanishes for
the whole of $\mathbb{M}^{2}$, the spin-outward current emerges along one
edge, while the spin-inward current does along the other.

##  VI Conclusions and discussions

For the quantum particles confined to a two-dimensional curved system, the
geometric effects are a standing-long interesting topic. The required
effective dynamics can be given by using the thin-layer quantization scheme,
which is suitable and valid. Because that two important geometric effects, the
geometric potential and the geometric momentum, have been proved by
experiments. The two effects both result from the rescaling factor that depend
on the normal coordinate variable. The dependence of normal space directly
originates from the metric tensor that is defined in the three-dimensional
subspace spanned by two tangent coordinate variables and a normal one of
$\mathbb{S}^{2}$. So far, the fundamental framework of thin-layer quantization
scheme is suitable and sufficient for the geometric potential and geometric
momentum da Costa (1981); Liu  _et al._ (2011); Wang  _et al._ (2017).
Unfortunately, the fundamental formalism can not give the geometric gauge
potential that is related to the symmetries of the introduced confining
potential Wang  _et al._ (2018) for the curved surface embedded in the usual
three-dimensional Euclidean space Wang  _et al._ (2020). In order to remove
the difficulty, the formula of geometric effects are rediscussed, which is
clearly evidenced by that the geometric effects are not only from the
rescaling transformation, but also from the rotation transformation intimately
connected to the local frames. These results will enable the thin-layer
quantization formalism to play a more important and effective role in the
effective quantum dynamics for the particles confined to low-dimensional
curved systems, especially with the development of low-dimensional
nanodevices.

For the particles confined to a Möbius strip, the effective Dirac equation is
given by using the developed thin-layer quantization formalism. There are two
important results for the geometric effects. One is the effective mass that
results from the rescaling factor. The other is the effective gauge potential
that results from the rotation transformation connected to the local frames.
The presence of the rescaling factor determines that the thin-layer
quantization formalism does not commutate with the non-relativistic limit.
Interestingly, the effective magnetic field is monopole that is determined by
the single face of Möbius strip. This result provides a feasible way to
generate a non-Abelian monopole magnetic field, the gauge structure can be
constructed by designing the geometry of two-dimensional curved system. In the
presence of monopole effective magnetic field, the spin-outward particles and
the spin-inward ones are completely separated as full spin polarization. The
pure spin current is also determined by the geometry of Möbius strip due to
the coupling of geometry and spin. As a conclusion, the nontrivial topological
properties of Möbius strip entirely determine the emergences of the monopole
magnetic field and the quantum spin Hall effect. In other words, the complex
geometries and topologies of two-dimensional systems can provide a new
perspective to investigate new phenomena of Hall physics implied in a high-
dimensional space.

With the rapid development of flexible electronics, flexible spintronics and
metastucture physics, the geometric quantum effects play a more and more
important role in the effective quantum dynamics for two-dimensional curved
systems. The geometries of nanodevices can be employed to improve the
development of nanodevices and topological quantum computation and topological
quantum commutation. Altogether, our results demonstrate a viable manner to
control the electronic levels and transpose properties of two-dimensional
curved system, shedding new light on the design of novel electronics devices
by geometry engineering.

## Acknowledgments

This work is jointly supported by the Natural Science Foundation of Shandong
Province of China (Grant No. ZR2020MA091), the National Key R&D Program of
China (Grant No. 2017YFA0303702), the National Nature Science Foundation of
China (Grants, No. 11625418).

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