Geometry-induced Monopole Magnetic Field and Quantum Spin Hall Effect

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.

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.

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.

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.

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.

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.

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.

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.

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.

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).