# Spin-deformation coupling in two-dimensional polar materials

J. A. Sánchez-Monroy  Departamento de Física, Universidad Nacional de
Colombia, Bogotá, 111321, Colombia<EMAIL_ADDRESS>Carlos Mera Acosta
<EMAIL_ADDRESS>Faculty of Engineering, University of La Sabana,
Chia 250001, Colombia

###### Abstract

The control of the spin degree of freedom is at the heart of spintronics,
which can potentially be achieved by spin-orbit coupling or band topological
effects. In this paper, we explore another potential controlled mechanism
under debate: the spin-deformation coupling (SDC) - the coupling between
intrinsic or extrinsic geometrical deformations and the spin degree of
freedom. We focus on polar-deformed thin films or two-dimensional compounds,
where the Rashba spin-orbit coupling (SOC) is considered as an $SU(2)$ non-
Abelian gauge field. We demonstrate that the dynamics between surface and
normal electronic degrees of freedom can be properly decoupled using the thin-
layer approach by performing a suitable gauge transformation, as introduced in
the context of many-body correlated systems. Our work leads to three
significant results: (i) gauge invariance implies that the spin is uncoupled
from the surface’s extrinsic geometry, challenging the common consensus; (ii)
the Rashba SOC on a curved surface can be included as an $SU(2)$ non-Abelian
gauge field in curvilinear coordinates; and (iii) we identify a previously
unnoticed scalar geometrical potential dependent on the Rashba SOC strength.
This scalar potential, independent of spin, represents the residual effect
remaining after decoupling the normal component of the non-Abelian gauge
field. The outcomes of our work open novel pathways for exploring the
manipulation of spin degrees of freedom through the use of the SDC.

##  I Introduction

Device applications require not only an understanding of the physical
mechanisms behind material properties, but also the possibility of controlling
or switching them. In the particular case of spintronics, which focuses on
manipulating spin-related properties in materials, numerous mechanisms have
been proposed to achieve spin polarization (SP) control Acosta and Fazzio
(2019); Garcia  _et al._ (2010); Žutić  _et al._ (2004); Hoffmann and Bader
(2015), including band topological effects Zhu  _et al._ (2014); Costa  _et
al._ (2021), magnetoelectric effects Kimura  _et al._ (2003); Fusil  _et al._
(2014); Béa  _et al._ (2008), and spin-orbit coupling (SOC)Mera Acosta  _et
al._ (2016, 2019); Acosta  _et al._ (2020); Long  _et al._ (2018); He  _et
al._ (2022); Šmejkal  _et al._ (2018). One of the most studied SOC-driven
phenomena is the SP induced by the breaking of the inversion symmetry, e.g.,
the Rashba effect in polar compounds Mera Acosta  _et al._ (2020); Manchon
_et al._ (2015); Nascimento  _et al._ (2022). In general, the SP directions
are determined by the local symmetry of the chemical environment for the
constituent atoms and the crystallographic point group symmetries Mera Acosta
_et al._ (2021). Consequently, geometric deformations hold the potential to
modulate SP and hence, it offers a route to manipulate spin related
properties. The underlying assumption of such a SP control mechanism in
nanodevices is the existence of the coupling between the spin degree of
freedom and local geometrical deformations Chang  _et al._ (2013); Wang  _et
al._ (2017); Kosugi (2011); Siu  _et al._ (2018); Zhang  _et al._ (2007); Liu
_et al._ (2011); Ortix (2015); Gentile  _et al._ (2015); Ying  _et al._
(2016); Pandey  _et al._ (2018); Salamone  _et al._ (2022), i.e., spin-
deformation coupling (SDC). The theoretically modelling of the SDC requires
having a Hamiltonian that describes the electronic structure of electrons
confined to a deformed surface, denoted as $\mathcal{M}$, where surface modes
are decoupled from normal degrees of freedom. Two dimensional (2D) materials
and surfaces are thus a potential platform to induce deformations and
potentially realize the SDC.

The SDC can be described by the coupling of spin with two distinct
manifestations of the deformation, namely: i) the intrinsic SDC \- geometrical
attributes that exclusively involve the intrinsic geometry of the surface,
devoid of any influence from the embedding space, e.g., the induced metric
tensor ($g_{\mu\nu}$) and the Gaussian curvature (K), and ii) the extrinsic
SDC \- the mathematical constructs depending on the extrinsic geometry,
involving the curvature of a surface within its 3D spatial embedding, such as
the mean curvature (H). Diverse methodologies have been proposed to model the
SDC on curved surfaces Chang  _et al._ (2013); Wang  _et al._ (2017); Kosugi
(2011); Siu  _et al._ (2018) and in quantum wires Zhang  _et al._ (2007); Liu
_et al._ (2011); Ortix (2015); Gentile  _et al._ (2015); Ying  _et al._
(2016); Pandey  _et al._ (2018); Salamone  _et al._ (2022). For instance,
concerning curved two-dimensional Rashba compounds, the prevailing consensus
establishes a SDC involving the coupling of the spin with the extrinsic
curvature, which is typically associated with a term proportional to the
product involving H and a spin component Chang  _et al._ (2013); Wang  _et
al._ (2017); Kosugi (2011); Siu  _et al._ (2018), i.e.,
$\xi\text{H}\tilde{\sigma}^{i}$, where $\tilde{\sigma}^{i}$ denotes the
spatially dependent Pauli matrices, and $\xi$ represents the strength of the
coupling. For the intrinsic SDC in Rashba compounds, the electronic
Hamiltonian incorporates terms coupling the metric on the surface and spin
components Chang  _et al._ (2013); Wang  _et al._ (2017); Kosugi (2011); Siu
_et al._ (2018).

The explorations of the SDC share a common foundation: the thin-layer method.
This method has been widely utilized to describe non-relativistic spinless
charged particles Ferrari and Cuoghi (2008); Brandt and Sánchez-Monroy (2017),
relativistic spinless particles Brandt and Sánchez-Monroy (2017), and
$\frac{1}{2}$-spin particles Sánchez-Monroy (2016) moving on curved surfaces
in the presence of an external electromagnetic field; in all these situations,
the coupling between electromagnetic fields and the extrinsic curvature is
forbidden. Interestingly, when the procedure is not executed accurately, such
as through improper decoupling of degrees of freedom Jensen and Dandoloff
(2009), a fictitious field-deformation coupling resembling the term
$\xi\text{H}\tilde{\sigma}^{i}$ can emerge. This coupling has been identified
as an unphysical gauge-dependent contribution, as demonstrated in Ferrari and
Cuoghi (2008); Ortix and van den Brink (2011). The close analogy between the
SOC and electromagnetic fields suggests exploring the SDC by employing the
well-established interpretation of the Rashba SOC as an effective non-Abelian
field Fröhlich and Studer (1993); Jin  _et al._ (2006); Tokatly (2008), which
is the purpose of this paper.

In this paper, we describe the SDC in the context of two descriptions of the
spin polarization in 2D non-centrosymmetric materials, namely, i) the
phenomenological linear Rashba effect, i.e.,
$\mathcal{H}_{R}=\sigma_{x}k_{y}-\sigma_{y}k_{x}$; and ii) a non-abelian gauge
field capturing all spin effects (including the linear Rashba SOC). In order
to describe the SDC, we first discuss the metric tensor of deformed surfaces
in the context of the thin-layer procedure. Subsequently, we deduce the SDC
based on the two aforementioned types of SOC descriptions. Writing the SOC as
a $SU(2)$ non-Abelian gauge field $\bm{\mathcal{W}}$ and using the thin-layer
method da Costa (1981); Maraner (1995); Schuster and Jaffe (2003), we
demonstrate that the extrinsic SDC proportional to
$\text{H}\tilde{\sigma}^{i}$ is physically forbidden due to gauge invariance.
Indeed, only intrinsic SDC can be used as a SP control mechanism. We identify
a previously unnoticed scalar geometrical spin-dependent potential that is
proportional to the Rashba SOC strength, which represents a shift induced by
the non-abelian gauge field’s normal component.

In order to illustrate the proposed description of the SDC, in the next
section we describe the formalism to describe a curved surface in the context
of the thin-layer method applied to the kinet energy and condinement
potential. In sections III and IV, the SDC is studied describing the Rashba
SOC as a linear phenomenological model and a non-Abelian Gauge field,
respectively. In sections V and VI, we describe the controlability of the SDC
and give a specific example of the proposed formalism. Finally, conclusion are
given in section VII.

##  II Spinless electrons confined in a deformed two-dimensional surface

The general procedure for describing the SDC consists of identifying the
coupling between spin and both the intrinsic and extrinsic curvatures in the
effective Hamiltonian describing a deformed low-dimensional materialda Costa
(1981); Schuster and Jaffe (2003). Consequently, the problem is translated to
include the geometrical deformation in the Hamiltonian, constrained by
confinement within the 2D space. To achieve this goal, we define the
deformation of interest in terms of a metric tensor, and then apply the thin-
layer method to the electronic Hamiltonian $\mathcal{H}_{e}$. For illustrative
purposes, we first focus on the spinless electronic Hamiltonian.

###  II.1 Geometrical deformation parametrization

A local geometric deformation is introduced by considering the electronic
positions on a curved surface $\mathcal{M}$, which is parameterized by
$\bm{r}_{||}=\bm{r}(q_{1},q_{2})$. In a sufficiently small neighborhood of
$\mathcal{M}$, the position vector can be described using the normal
coordinate $q_{3}$, as
$\bm{R}(q_{1},q_{2},q_{3})=\bm{r}_{||}+q_{3}\hat{\bm{n}}$, where
$\hat{\bm{n}}(q_{1},q_{2})$ is an vector field normal to the surface
$\mathcal{M}$. The transformed positions from the Cartesian coordinate system
to the adapted curvilinear coordinates, $(q_{1},q_{2},q_{3})$, defines the
metric tensor

$G_{ij}=\frac{\partial\bm{R}}{\partial
q^{i}}\cdot\frac{\partial\bm{R}}{\partial q^{j}}.$ (1)

Given two tangent vectors to $\mathcal{M}$, e.g.,
$\bm{t}_{\mu}=d\bm{r}_{||}/dq^{\mu}$, we can write the metric on the surface
$g_{\mu\nu}\equiv\bm{t}_{\mu}\cdot\bm{t}_{\nu}$ and the so-called Weingarten
curvature matrix
$\alpha_{\mu\nu}\equiv-\bm{t}_{\mu}\cdot\partial_{\nu}\hat{\bm{n}}$ to
completely characterized the curved surface $\mathcal{M}$, where Greek indexes
(running over 1 and 2) are used for the adapted coordinates on $\mathcal{M}$.
In principle, with exception of geometric or extrinsic torsion $\beta_{\mu
i}^{\ \ k}\eta_{kj}\equiv\hat{\bm{n}}_{i}\cdot\partial_{\mu}\hat{\bm{n}}_{j}$,
all information of the deformation is intrinsically included in $g_{\mu\nu}$
and $\alpha_{\mu\nu}$, which are also known as the first fundamental form
(intrinsic metric) and the second fundamental form (extrinsic curvature),
respectively. One can then differentiate between extrinsic SDC and intrinsic
SDC according to the dependence with respect to the metric on the surface and
the Weingarten curvature matrix. However, the electronic Hamiltonian is not
directly written in terms of $g_{\mu\nu}$ and $\alpha_{\mu\nu}$, but in terms
of the metric tensor $G_{ij}$ and its determinant. Specifically, in the
adapted coordinates, the metric tensor and its determinant, i.e.,

$\displaystyle G_{ij}=\left(\begin{array}[]{ll}\gamma_{\mu\nu}&0\\\
0&1\end{array}\right)$ (4)

and

$|G|=|\gamma|=|g|(1+q^{3}\Tr\alpha_{\mu}^{\
\nu}+(q^{3})^{2}\det\alpha_{\mu}^{\ \nu}),$ (5)

respectively, explicitly involve the intrinsic and extrinsic curvatures
through
$\gamma_{\mu\nu}=g_{\mu\nu}-2q_{3}\alpha_{\mu\nu}+q_{3}^{2}\alpha_{\mu\rho}g^{\rho\sigma}\alpha_{\sigma\nu}$
da Costa (1981). Here, we use the convention convention
$\alpha_{\mu\nu}g^{\nu\beta}=-\alpha_{\mu}^{\ \beta}$ to compute the
determinant of $\gamma_{\mu\nu}$, $|\gamma|$.

###  II.2 The thin-layer method

In the simplest description of a deformed 2D material, the electronic
Hamiltonian $H_{e}$ includes the kinetic energy
$H_{0}=-\hbar^{2}\nabla^{2}/2m_{e}$ and a confining potential $V_{c}$ that
depends only on the normal coordinate to the surface, which is denote by
$q_{3}$, i.e.,

$\displaystyle H_{e}=H_{0}+V_{c}(q_{3}).$ (6)

In the thin-layer method, in opposite to assume that $V_{c}$ becomes infinite
outside the surface da Costa (1981), which is unrealistic because it violates
the Heisenberg uncertainty principle, a perturbative expansion should be
considered Maraner (1995); Schuster and Jaffe (2003); Brandt and Sánchez-
Monroy (2015). Specifically, the normal coordinate is rescaled, i.e.,
$q_{3}\rightarrow\epsilon q_{3}$, so the Hamiltonian $H_{e}$ can be expanded
in the dimensionless parameter $\epsilon$, which provides an effective
Hamiltonian ($\mathcal{H}_{e}$) on the surface based on the perturbative
expansion

$\displaystyle\epsilon^{2}\mathcal{H}_{e}=\mathcal{H}_{e}^{(0)}+\epsilon\mathcal{H}_{e}^{(1)}+\epsilon^{2}\mathcal{H}_{e}^{(2)}.$
(7)

Here, $\mathcal{H}_{e}^{(n)}$ is the $n$-th expansion order of the
Hamiltonian. In the ideal scenario, the zero-order (second-order) Hamiltonian
$\mathcal{H}_{e}^{(0)}$ ($\mathcal{H}_{e}^{(2)}$) is exclusively dependent on
the normal (surface) coordinates. This scenario unravels the electronic
structure of deformed two-dimensional materials, freezing the dynamics of the
normal direction and decoupling it from the tangential degrees of freedom.
Consequently, the wavefunction for the electronic Hamiltonian can be written
as $\ket{\Psi}=\ket{\varphi}\otimes\ket{\psi}$, where the surface (normal)
modes, e.g., $\ket{\psi(q_{1},q_{2})}$ ($\ket{\varphi(q_{3})}$), depends only
on the surface (normal) coordinates $q_{1}$ and $q_{2}$ ($q_{3}$).

The thin-layer method can be summarized in the following steps:

  * a.

Write the differential operator $H_{e}$ in the adapted coordinates and rescale
it as
$\mathcal{H}_{e}\equiv\left(\frac{|G|^{1/4}}{|g|^{1/4}}\right)H_{e}\left(\frac{|g|^{1/4}}{|G|^{1/4}}\right)$.
In this way, one define a new wavefunction $\chi$ as
$\chi\equiv\frac{|G|^{1/4}}{|g|^{1/4}}\Psi$.

  * b.

Insert a dimensionless parameter $\epsilon$ to rescale the normal coordinate,
$q^{3}$, as $q^{3}\rightarrow\epsilon q^{3}$ and expand the rescaled
differential operator in a perturbative expansion on $\epsilon$.

  * c.

Use a product solution and the normalization condition to decoupling the
tangent and the normal degrees of freedom to obtain a effective Hamiltonian in
the surface.

The dimensional reduction procedure consider here has a structure similar to
the Born-Oppenheimer approximation. The first two steps are usually performed
separately for each term in the Hamiltonian.

###  II.3 Thin-layer method for the kinetic energy and confinement potential

To better understand the thin-layer method and its posterior implementation
for the SDC study, we illustrate it for the kinetic energy and the confinement
potential for spinless electrons as an example. We thus now turn our attention
to the use of the metric tensor $G_{ij}$ and its determinant $|G|$ to redefine
the electronic Hamiltonian $H_{e}$ in $\mathcal{M}$ (step a). Our starting
point is the electronic Hamiltonian $H_{e}$ including the kinetic energy
$H_{0}=-\hbar^{2}\nabla^{2}/2m_{e}$ and a confining potential $V_{c}$.

####  II.3.1 Confinement potential

For the step a, one can note that the potential $V_{c}$ has been assumed to
depends only on the normal coordinate $q_{3}$ and hence, there is no change in
it mathematical form. For step b, the procedure is simplified considering that
due to the deep minima, the confinement potential can be approximated, before
rescale, as $V_{c}=\bar{V}_{c}+O(q_{3}^{3})$ Maraner (1995). Where the
potential $\bar{V}_{c}$ is symmetric with respect to the surface, e.g., the
harmonic confinement potential
$\bar{V}_{c}=\frac{1}{2\epsilon^{4}}m\omega^{2}(q_{3})^{2}$. After rescaling,
we can expand $\mathcal{V}_{c}$ in an arbitrary series expansion in $\epsilon$
(step b),

$\displaystyle\epsilon^{2}\mathcal{V}_{c}$ $\displaystyle=$
$\displaystyle\mathcal{V}_{c}^{(0)}+\epsilon\mathcal{V}_{c}^{(1)}+\epsilon^{2}\mathcal{V}_{c}^{(2)}+O(\epsilon^{3})$
(8) $\displaystyle=$
$\displaystyle\frac{1}{2}m\omega^{2}(q_{3})^{2}+O(\epsilon^{3}),$

one can note that the order zero $\mathcal{V}_{c}^{(0)}$ is actually the
quadratic term, meanwhile $\mathcal{V}^{(1)}_{c}$ and
$\mathcal{V}^{(2)}_{c}=0$. We would like to emphasize that by imposing that
$O(q_{3}^{3})$ in $V_{c}$ is independent of $\epsilon$, the Hamiltonian
describing the surface, $\mathcal{H}^{(1)}_{e}$, becomes entirely independent
of the explicit form of $V_{c}$ Maraner (1995). Indeed, the validity of the
thin-layer method relies on the fact that energy scale of $V^{(0)}$ should be
much larger than that of $\epsilon\mathcal{H}^{(1)}_{e}$ and subsequent terms.

####  II.3.2 Kinetic energy

In the curved surface $\mathcal{M}$, the kinetic energy
$H_{0}=-\hbar^{2}\nabla^{2}/2m_{e}$ is rewritten as (step a)

$H_{0}=-\frac{\hbar^{2}}{2m_{e}|G|^{1/2}}\partial_{i}|G|^{1/2}G^{ij}\partial_{j}.$
(9)

Here, $|G|=|g|(1+q_{3}\Tr\alpha_{\mu}^{\ \nu}+q_{3}^{2}\det\alpha_{\mu}^{\
\nu})$ is the determinant of the metric tensor in the adapted curved
coordinates and $m_{e}$ is the electron mass. Recale and expanding the kinetic
energy powers of $\epsilon$ (step a and b)

$\displaystyle\epsilon^{2}\mathcal{H}_{0}=\mathcal{H}_{e}^{(0)}+\epsilon\mathcal{H}_{0}^{(1)}+\epsilon^{2}\mathcal{H}_{0}^{(2)},$
(10)

we find that

$\mathcal{H}_{0}^{(0)}=-\frac{\hbar^{2}}{2m}\partial_{q_{3}}^{2}+\mathcal{V}_{c},$
(11) $\mathcal{H}_{0}^{(1)}=0,$ (12)

and

$\mathcal{H}_{0}^{(2)}=\left[-\frac{\hbar^{2}}{2m|g|^{1/2}}\partial_{\mu}g^{\mu\nu}|g|^{1/2}\partial_{\nu}+\mathcal{V}_{g}\right].$
(13)

The geometrical deformation effects emerge only on the second order of
expansion as a quantum geometric potential, i.e.,
$\mathcal{V}_{g}=-\frac{\hbar^{2}}{2m}(\text{H}^{2}-\text{K})$ and a metric
tensor field $g^{\mu\nu}$ in the curved surface Jensen and Koppe (1971); da
Costa (1981); Maraner (1995); Schuster and Jaffe (2003). As previously
mentioned, two contributions can be identified: the mean curvature
$\text{H}\equiv-\text{tr}\alpha_{\mu}^{\nu}/2$ and the Gaussian curvature
$\text{K}\equiv\det\alpha_{\mu}^{\nu}$. The Gaussian curvature, half of the
scalar curvature, measures the intrinsic curvature of a surface, independent
of its embedding space, while the mean curvature depends on its extrinsic
geometry. Notably empirical evidence of the geometric potential within quantum
systems has been compellingly demonstrated in experiments using the thin-layer
method, such as in metamaterials that exhibit a photonic topological crystal
Szameit  _et al._ (2010) and in the Riemannian geometric effects observed
within a one-dimensional metallic $C_{60}$ polymer Onoe  _et al._ (2012).

On the other hand, note that $\mathcal{H}_{0}^{(0)}$ and
$\mathcal{H}_{0}^{(2)}$ depends only on the normal and surface coordinates,
respectively. Up to the second-order in $\epsilon$, there is no coupling
between $q^{3}$ and $q^{\mu}$, which justifies the separation of variables
$\ket{\Psi}=\ket{\varphi(q^{3})}\otimes\ket{\psi(q^{1},q^{2})}$, where
$\ket{\varphi(q^{3})}$ is the ground state of $\mathcal{H}^{(0)}$. Indeed, in
the kinetic energy and confinement potential, the first order terms in the
expansion are null. The presence of the term $\mathcal{H}_{e}^{(1)}$ could
result in the coupling of normal and surface modes.

In this illustrative demonstration of the formalism, as very well established
for spinless electrons Wachsmuth and Teufel (2010), we verify that there is no
coupling between the surface and normal modes, i.e., the surface modes
$\ket{\psi(q_{1},q_{2})}$ and normal modes $\ket{\varphi(q_{3})}$ are only
described by the Hamiltonian at second and zero order in $\epsilon$,
respectively. The effective Hamiltonian for the surface
$\mathcal{H}_{e}^{(2)}$ is nothing than the term $\mathcal{H}_{0}^{(2)}$. Note
that when $\epsilon$ is small (deep confining potential), the confining energy
($\mathcal{H}_{e}^{(0)}$) is significantly larger than the kinetic energy
($\epsilon^{2}\mathcal{H}_{0}^{(2)}$); thus, the transitions between the
normal and tangential modes are suppressed Wachsmuth and Teufel (2010). The
established steps and analysis of the physical meaning of the terms in the
expansion are the starting point to include the SOC in the surface Hamiltonian
and the posterior study of the SDC.

##  III Spin-deformation coupling based on the phenomenological linear Rashba
effect

In this section, we use the previous parametrization of the surface
deformation and the thin layer method for a electronic Hamiltonian including
the SOC in polar compounds, which is introduced throughout the Rashba effect,
i.e.,

$H_{e}=H_{0}+V_{c}+H_{R}.$ (14)

As demonstrate below, when this procedure is performed, in contrast to the
kinetic energy, the SOC introduces first order in $\epsilon$ terms, i.e.,
$\mathcal{H}_{R}^{(1)}\neq 0$, coupling normal and surface modes.
Consequently, deriving the effective dynamics in 2D translates into the
challenge of formulating a strategy to eliminate the coupling mediated by
$\mathcal{H}_{R}^{(1)}$. Despite the anticipated coupling between normal and
surface modes, we follow the thin-layer method for the Rashba effect,
analogous to Refs. Chang  _et al._ (2013); Wang  _et al._ (2017); Kosugi
(2011); Siu  _et al._ (2018), in order to provide an understanding of the
relation between both the Rashba effect and surface deformation with the SDC,
prior to introducing an $SU(2)$ non-Abelian gauge field for the Rashba spin-
orbit coupling.

The SOC is a relativistic correction that results from the interaction between
the magnetic moment associated with the spin of electrons and the magnetic
field generated by the electron’s motion through an potential gradient
$\bm{\nabla}\phi$, i.e.,
$H_{\text{SOC}}=-\frac{e\hbar}{4m^{2}c^{2}}\bm{\sigma}\cdot\left(\bm{\nabla}\phi\times\bm{p}\right)$.
Emmanuel Rashba noticed that this term for an electron with mass $m$, momentum
$\bm{p}$ and spin $\bm{s}=\frac{\hbar}{2}\bm{\sigma}$ can also be induced by
an external electric field Bychkov and Rashba (1984). Consequently, when the
potential gradient is induced by an uniform electric field perpendicular to
the surface, the SOC can be rewritten as
$\mathcal{H}_{R}=\lambda_{R}(\sigma_{x}k_{y}-\sigma_{y}k_{x})\hat{z}$, which
is typically referred to as the phenomenological linear Rashba effect. In the
last decade, it was realized that the electric dipole in polar compounds
induces a potential gradient leading to an intrinsic Rashba effect Ishizaka
_et al._ (2011), referred to as bulk Rashba effect Mera Acosta  _et al._
(2020). A spin splitting in the wavevector $\bm{k}$ space as well as a helical
spin polarization are the main characteristic of the Rashba effect Mera Acosta
_et al._ (2016), which have been at the heart of the potential application in
spintronics Manchon  _et al._ (2015). This interaction is particularly
significant in atoms where the relativistic effects become appreciable, such
as those with high atomic numbers. The magnitude of $\lambda_{R}$ varies
across materials and is influenced by the atomic number and the electronic
environment. Even if the electric dipole is arbitrarily fixed along the normal
direction, a spin-dependent geometrical potential could emerge.

In the curved surface $\mathcal{M}$, the phenomenological Rashba effect
$H_{R}=S_{ij}\tilde{\sigma}^{i}p^{j}$ is rewritten as

$H_{R}=-i\hbar\mathcal{S}_{j}G^{jk}\partial_{k},$ (15)

where we define $\mathcal{S}_{j}=S_{ij}\tilde{\sigma}^{i}$. Here,
$\tilde{\sigma}^{i}$ are the space-dependent Pauli matrices, and $S_{ij}=0$ if
$i=j$. The space-dependent $\sigma$-matrices and the Rashba tensor $S_{ij}$,
can be defined by

$\displaystyle\tilde{\sigma}^{i}$ $\displaystyle=$
$\displaystyle\frac{\partial q^{i}}{\partial x^{s}}\sigma^{s},$ (16)
$\displaystyle S_{ij}$ $\displaystyle=$ $\displaystyle\frac{\partial
x^{s}}{\partial q^{i}}\frac{\partial x^{t}}{\partial q^{j}}S_{st},$ (17)

where $\sigma^{s}$ and $S_{st}$ are the associated Pauli matrices and the
Rashba tensor in the Cartesian coordinate system, respectively Chang  _et al._
(2013); Wang  _et al._ (2017). Following the confining potential approach we
first rescaled the Rashba Hamiltonian as

$\displaystyle\mathcal{H}_{R}$ $\displaystyle\equiv$
$\displaystyle\frac{|G|^{1/4}}{|g|^{1/4}}H_{R}\frac{|g|^{1/4}}{|G|^{1/4}}.$
(18)

For a consistent perturbative expansion is crucial to take into account that
all functions of $q_{3}$ and the derivatives with respect to $q_{3}$ have to
be rescaled by $q_{3}\rightarrow\epsilon q_{3}$, thus

$\displaystyle\frac{\partial}{\partial
q_{3}}\rightarrow\frac{1}{\epsilon}\frac{\partial}{\partial q_{3}},$ (19)

and $\mathcal{S}_{j}(q^{\mu},\epsilon q_{3})$ is now dependent on $\epsilon$,
threfore it can be Taylor expanded as follows

$\displaystyle\mathcal{S}_{j}(q^{\mu},\epsilon
q_{3})=\mathcal{S}_{j}(q^{\mu},0)+\epsilon
q_{3}\left.\partial_{3}\mathcal{S}_{j}(q^{\mu},q_{3})\right|_{q_{3}=0}+\ldots.\
\ \ $ (20)

Considering now the Rashba effect by expanding the Hamiltonian
$\mathcal{H}_{R}$ up to second order in $\epsilon$, i.e.,
$\epsilon^{2}\mathcal{H}_{R}=\mathcal{H}_{R}^{(0)}+\epsilon\mathcal{H}_{R}^{(1)}+\epsilon^{2}\mathcal{H}_{R}^{(2)}+\ldots$,
we obtain

$\mathcal{H}_{R}^{(0)}=0$ (21)
$\mathcal{H}_{R}^{(1)}=-i\hbar\mathcal{S}_{3}\partial^{3},$ (22)

and

$\mathcal{H}_{R}^{(2)}=-i\hbar\left(\bar{\mathcal{S}}_{\mu}g^{\mu\nu}\partial_{\nu}+\bar{\mathcal{S}}_{3}\text{H}+\frac{\partial\bar{\mathcal{S}}_{3}}{\partial
q^{3}}q^{3}\partial^{3}\right),$ (23)

where we use the following notation
$\bar{\mathcal{S}}_{j}=\mathcal{S}_{j}(q^{\mu},0)$ and
$\partial^{3}\bar{\mathcal{S}}_{3}=\left.\partial^{3}\mathcal{S}_{3}(q^{\mu},q^{3})\right|_{q^{3}=0}$.

We find that there is no contribution of the surface Rashba effect to the
zero-order perturbative expansion (i.e., $\mathcal{H}_{R}^{(0)}=0$). In
$\mathcal{H}_{R}^{(2)}$, the first term resembles that expected from the
Rashba effect on a curved surface, the intrinsic SDC. The last two terms
present in $\mathcal{H}_{R}^{(2)}$ can be interpreted as the extrinsic SDC,
i.e., the coupling of the mean curvature with the spin and other term
associates with $q^{3}\partial^{3}$, which should be adequately frozen in the
ground state of the normal modes. Moreover, we note that
$\mathcal{H}_{R}^{(1)}$ couples the surface and normal modes, indicating the
Rashba effect forbids to write the wavefunction as
$\ket{\Psi}=\ket{\varphi}\otimes\ket{\psi}$.

As implied by the perturbative expansion, effects intrinsic to the surface
electronic dynamics should not modify the normal modes. Therefore, the
transitions between the normal modes promoted by the SOC term
$\mathcal{H}_{R}^{(1)}$ should be carefully study, because the term
$\bar{\mathcal{S}}_{3}\text{H}$ could be a fictitious collateral consequence
of a methodologically inconsistent approach. This is commonly achieved by
imposing $\bra{...}\partial_{q_{3}}f(q_{3})\ket{...}=0$ Chang  _et al._
(2013); Wang  _et al._ (2017), which could result in the inclusion of
fictitious terms in the SDC. Indeed, one could think that the control of the
spin can be achieved throughout geometrical effects, which is erroneous, as we
demonstrate below using a non-abelian field to describe the Rashba SOC.

##  IV Spin-deformation coupling based on a non-Abelian gauge field

In this section, inspired by a common approach in strongly correlated many-
body physics that utilizes a unitary gauge transformation to disentangle
certain degrees of freedom Ashida  _et al._ (2021), we search for an
appropriate unitary transformation to disentangle surface and normal quantum
dynamics. This approach yields an expansion (step b in the thin-layer method)
with a null first-order term, i.e., $\mathcal{H}_{R}^{(1)}=0$. The general
idea is that a highly entangled quantum state in the original frame can be
expressed as a factorable state after the transformation, which has been used
for analyzing quantum impurity systems Lee  _et al._ (1953); Frohlich (1952);
Silbey and Harris (1984, 1984); Ashida  _et al._ (2018), constructing low-
energy effective models Schrieffer and Wolff (1966); Wegner (1994); Głazek and
Wilson (1993); Bravyi  _et al._ (2011), and solving many-body localization or
electron-phonon problems Imbrie (2016); Shi  _et al._ (2020). The starting
point in searching for an convenient gauge transformation is to interpret the
Rashba SOC as an effective non-Abelian vector potential - a vector field
associated with electron spin Fröhlich and Studer (1993); Jin  _et al._
(2006); Tokatly (2008), where different directions of spin can result in
different gauge fields associated with Rashba SOC.

####  IV.0.1 The Rashba effect as a non-Abelian gauge field

In order to maintain generality, we include external semiclassical
electromagnetic fields, encompassing a broad class of systems, among which the
Zeeman coupling is covered. In this scenario, the electronic Hamiltonian is

$H_{e}=H_{0}+H_{R}+H_{Z}+H_{P},$ (24)

which includes the kinetic energy

$H_{0}=\frac{\left(\bm{p}-e\bm{\mathcal{A}}\right)^{2}}{2m},$ (25)

the Rashba effect

$H_{R}=\frac{e\hbar}{4m^{2}c^{2}}\left(\bm{p}-e\bm{\mathcal{A}}\right)\cdot(\bm{\sigma}\times\nabla
V),$ (26)

the Zeeman effect

$H_{Z}=\frac{e\hbar}{2m}\bm{\sigma}\cdot\bm{B},$ (27)

and the electric potential $H_{P}=e\mathcal{A}_{0}$. Here, $\bm{\sigma}$
represents Pauli spin matrices, $\bm{p}$ is the momentum operator,
$\bm{\mathcal{A}}$ and $A_{0}$ are the vector potential and the scalar
potential of the electromagnetic field, respectively, and $V$ is the crystal
potential. As it is well established Fröhlich and Studer (1993) a $SU(2)$ non-
Abelian gauge field $\bm{\mathcal{W}}$ captures the spin-orbit interaction and
the Zeeman coupling when the following identifications are made:

$\mathcal{W}_{i}=\frac{e\hbar}{mc^{2}}\varepsilon_{ija}E_{j}\tau^{a},\ \ \ \ \
\mathcal{W}_{0}=-\frac{e\hbar}{m}B_{a}\tau^{a},$ (28)

with $\tau^{a}=\sigma^{a}/2$. For instance, if the electric dipole is
perpendicular to the two-dimensional material, i.e., $E_{x}=E_{y}=0$, the non-
abelian Gauge field is

$\mathcal{W}_{x}=-\frac{2m\alpha}{\hbar}\tau^{y},\ \ \ \
\mathcal{W}_{y}=\frac{2m\alpha}{\hbar}\tau^{x},\ \ \ \
\mathcal{W}_{0}=-\frac{e\hbar}{m}B_{a}\tau^{a}.$ (29)

This $SU(2)$ non-Abelian gauge field satisfied the Coulomb gauge restriction,
$\nabla\cdot\bm{\mathcal{W}}$ Shikakhwa  _et al._ (2012), and enables the
rewriting of the electronic Hamiltonian as

$H_{e}=\frac{(\bm{p}-e\bm{\mathcal{A}}-\bm{\mathcal{W}})^{2}}{2m}-\frac{1}{2m}\bm{\mathcal{W}}\cdot\bm{\mathcal{W}}+\mathcal{W}_{0}+e\mathcal{A}_{0}.$
(30)

When $\mathcal{W}_{0}=0$ and $\mathcal{A}=0$, the electronic Hamiltonian is
reduced to the Rashba effect, as a matter of discussion we preserved all terms
in the Hamiltonian.

Before decoupling the surface and normal modes, we first introduce a gauge
covariant derivative
$\mathcal{D}_{i}=\partial_{i}-\frac{ie}{\hbar}\mathcal{A}_{i}-\frac{i}{\hbar}\mathcal{W}_{i}$
and the generalized gauge field
$\bm{\mathcal{Z}}=e\bm{\mathcal{A}}-\bm{\mathcal{W}}$. In the adapted
curvilinear coordinates, Eq. (30) becomes

$\displaystyle H_{e}$ $\displaystyle=$
$\displaystyle-\frac{\hbar^{2}}{2m|G|^{1/2}}\mathcal{D}_{i}|G|^{1/2}G^{ij}\mathcal{D}_{j}-\frac{1}{2m}G^{ij}\mathcal{W}_{i}\mathcal{W}_{j}$
(31) $\displaystyle+\mathcal{W}_{0}+e\mathcal{A}_{0}+V_{c}.$

The above Hamiltonian is gauge-invariant, except for the term
$G^{ij}\mathcal{W}_{i}\mathcal{W}_{j}/2m$ which breaks the SU(2) gauge
invariance. This term is commonly absorbed into a scalar potential $\phi(x)$.
If the expansion (step b of the thin-layer method) is directly applied to this
electronic Hamiltonian (Eq. 29), we obtain a result analogous to the previous
section. Specifically, the coupling between normal and surface modes arises
from the first-order term of the expansion. However, in the non-Abelian
formulation, $\mathcal{H}_{e}^{(1)}$ is proportional to the product between
the field $\mathcal{W}_{i}$ and the derivative $\partial^{i}$. As demonstrated
below, a suitable unitary transformation, $\mathbf{U}$, is introduced to
address this term.

####  IV.0.2 Unitary gauge transforamtion

In this subsection, we aim to decouple the fields in a more general context
than previously discussed. We consider a particle confined to an
$m$-dimensional submanifold ($\mathcal{M}$) within an $n$-dimensional manifold
($\mathcal{N}$). While we have thus far focused on the case of a curved
surface with $m=2$ and $n=3$, this generalization allows us to extrapolate our
results to other scenarios, such as a curved wire with $m=1$ and $n=3$ (see
the appendix A for a detailed explanation of the notation).

Analogously to the Abelian case Ferrari and Cuoghi (2008); Ortix and van den
Brink (2011); Brandt and Sánchez-Monroy (2015, 2017), we can decouple the
dynamics by performing a suitable gauge transformation,

$\mathbf{U}=e^{-iT(x^{\mu},\epsilon q^{i})/\hbar}$ (32)

with

$T(x^{\mu},\epsilon
q^{i})=\epsilon\bar{\mathcal{W}}_{i}q^{i}+\frac{\epsilon^{2}}{2}\frac{\partial\bar{\mathcal{W}}_{i}}{\partial
q^{j}}q^{j}q^{i}+\mathcal{O}(\epsilon^{2}).$ (33)

Here, the notations $\bar{\bm{\mathcal{W}}}=\bm{\mathcal{W}}(q^{\mu},0)$, has
been introduced. The transformation $\mathbf{U}$ has the general form used for
disentangle strong correlation in many-body systems
($\mathbf{U}\equiv\exp\left(-i\epsilon T/\hbar\right)$ with a proper choice of
$T$). Note that this transformation can always be performed, even if the
Hamiltonian (31) is not gauge-invariant.

The finite gauge transformation for the component $\mathcal{W}_{i}$ of no-
Abelian field is

$\mathcal{W}^{\prime}_{j}=\mathbf{U}\mathcal{W}_{j}\mathbf{U}^{-1}-\frac{i\hbar}{\epsilon}(\partial_{j}\mathbf{U})\mathbf{U}^{-1}.$
(34)

For the second term in this gauge transformation, we obtain

$\frac{i\hbar}{\epsilon}(\partial_{j}\mathbf{U})\mathbf{U}^{-1}=\bar{\mathcal{W}}_{j}+\frac{\epsilon}{2}\frac{\partial\bar{\mathcal{W}}_{j}}{\partial
q^{k}}q^{k}+\frac{\epsilon}{2}\frac{\partial\bar{\mathcal{W}}_{k}}{\partial
q^{j}}q^{k}+\mathcal{O}(\epsilon^{2}).$ (35)

To calculate the first term in this non-abelian gauge transformation it is
necessary to expand $\mathbf{U}$ on $\epsilon$, i.e,

$\mathbf{U}=\sum_{k=0}^{\infty}\frac{1}{k!}\left(i\epsilon T(x^{\mu},\epsilon
q^{i})\right)^{k},$ (36)

which results in

$\mathbf{U}\mathcal{W}_{j}\mathbf{U}^{-1}=\bar{\mathcal{W}}_{j}+\epsilon\frac{\partial\bar{\mathcal{W}}_{j}}{\partial
q^{i}}q^{i}+i\epsilon
q^{i}[\bar{\mathcal{W}}_{i},\bar{\mathcal{W}}_{j}]+\mathcal{O}(\epsilon^{2})$
(37)

where, we have used that
$\mathbf{U}\mathcal{W}_{j}\mathbf{U}^{-1}=\mathcal{W}_{j}\mathbf{U}\mathbf{U}^{-1}+[\mathbf{U},\mathcal{W}_{j}]\mathbf{U}^{-1}$.
Therefore, by applying a gauge transformation to the normal coordinates, we
obtain

$\mathcal{W}^{\prime}_{i}=-\epsilon\frac{q^{j}}{2}\bar{\mathcal{G}}_{ij}+\mathcal{O}(\epsilon^{2}),$
(38)

where $\bar{\mathcal{G}}_{ij}=\left.\mathcal{G}_{ij}\right|_{q=0}$ is the non-
abelian $SU(2)$ field tensor of the normal components to the surface Liang
_et al._ (2020). There is only one normal component on a surface, so the field
tensor is identically zero since it is antisymmetric. Consequently, through
the application of a gauge transformation, it becomes feasible to eliminate,
to second-order in $\epsilon$, the field component $\mathcal{W}_{q^{3}}$ from
all gauge-invariant terms. When the co-dimension ($n-m$) is greater than 1,
contributions related to the field tensor will appear, see the appendix B for
more details.

Applying a gauge transformation to the coordinates on the surface, one can
show that
$\mathcal{W}^{\prime}_{\mu}=\bar{\mathcal{W}}_{\mu}+\mathcal{O}(\epsilon)$.
Here, only contributions of order $\epsilon$ are considered, as they are the
only ones that contribute to the effective Hamiltonian
$\mathcal{H}_{e}^{(2)}$, see Eq. (LABEL:effectivebeforegt).

In the other hand, for the non-gauge invariant term in the Hamiltonian (Eq.
31), we note however that this term can be written as

$\displaystyle\mathcal{W}_{i}\mathcal{W}^{i}+\mathcal{W}_{\mu}g^{\mu\nu}\mathcal{W}_{\nu}$
$\displaystyle=$
$\displaystyle\mathcal{W}_{ia}\tau^{a}\mathcal{W}^{i}_{b}\tau^{b}+\mathcal{W}_{\mu
a}\tau^{a}\mathcal{W}^{\mu}_{b}\tau^{b}$ (39) $\displaystyle=$
$\displaystyle(\mathcal{W}_{ia}\mathcal{W}^{ia}+\mathcal{W}_{\mu
a}\mathcal{W}^{\mu a})\mathbf{I}_{2},$

where we have used the properties of matrices $\tau^{a}$ and $\mathbf{I}_{2}$
is the $2\times 2$ identity matrix, thus

$\mathbf{U}\left[\mathcal{W}_{i}\mathcal{W}^{i}+\mathcal{W}_{\mu}g^{\mu\nu}\mathcal{W}_{\nu}\right]\mathbf{U}^{-1}=\mathcal{W}_{i}\mathcal{W}^{i}+\mathcal{W}_{\mu}g^{\mu\nu}\mathcal{W}_{\nu}.$
(40)

Hence, this term remains unaffected by the this unitary transformation. 40

Note that in Eq. (40), the fields are those before performing the gauge
transformation. For the surface components, there is no difference because
$\bar{\mathcal{W}}^{\prime}_{\mu}=\bar{\mathcal{W}}_{\mu}$. However, for
normal components, we use the notation $\mathcal{W}_{i}=\mathcal{W}^{B}_{i}$
where the superscript indicates that the field is included before the gauge
transformation.

Once this transformation is performed, as established in the thin-layer
method, the normal coordinates is rescaled, i.e., $q\rightarrow\epsilon q$,
and the perturbative expansion for $\mathcal{H}_{e}$ in powers of $\epsilon$
is considered.

####  IV.0.3 Surface Hamiltonian for the non-Abelian gauge field

When the normal coordinates is rescaled $q\rightarrow\epsilon q$ and the
electronic Hamiltonian is expanded in powers of $\epsilon$, i.e,
$\epsilon^{2}\mathcal{H}_{e}=\mathcal{H}_{e}^{(0)}+\epsilon\mathcal{H}_{e}^{(1)}+\epsilon^{2}\mathcal{H}_{e}^{(2)}$,
we have to consider that $\bm{\mathcal{Z}}(q^{\mu},\epsilon q^{i})$ and
$\bm{\mathcal{W}}(q^{\mu},\epsilon q^{i})$ are also dependent on $\epsilon$.
Specifically, introducing the Taylor expansions

$\bm{\mathcal{Z}}(q^{\mu},\epsilon q^{i})=\bm{\mathcal{Z}}(q^{\mu},0)+\epsilon
q^{j}\left.\frac{\partial\bm{\mathcal{Z}}(q^{\mu},q^{i})}{\partial
q^{j}}\right|_{q=0}+\ldots$ (41)

and

$\bm{\mathcal{W}}(q^{\mu},\epsilon q^{i})=\bm{\mathcal{W}}(q^{\mu},0)+\epsilon
q^{j}\left.\frac{\partial\bm{\mathcal{W}}(q^{\mu},q^{i})}{\partial
q^{j}}\right|_{q=0}+\ldots,$ (42)

we can obtain the electronic Hamiltonian up to the second order in $\epsilon$
(see the appendix for more details). After introducing the Abelian (71) and
non-Abelian (32) gauge transformations, we can decouple the normal and surface
modes, as evidenced by a null first-order term, and zero (second) order
depending only on the normal (surface) coordinate, i.e.,

$\mathcal{H}^{(0)}_{e}=-\frac{\hbar^{2}}{2m}\partial_{3}\partial^{3}+\mathcal{V}_{c},$
(43) $\mathcal{H}^{(1)}_{e}=0,$ (44)

and

$\mathcal{H}^{(2)}_{e}=\frac{-\hbar^{2}}{2m|g|^{1/2}}\mathcal{D}_{\mu}g^{\mu\nu}|g|^{1/2}\mathcal{D}_{\nu}-\frac{\bar{\mathcal{W}}_{\mu}g^{\mu\nu}\bar{\mathcal{W}}_{\nu}}{2m}+\mathcal{Z}_{0}+{\mathcal{R}}.$
(45)

Note that we have introduced the effective geometrical potential in

$\mathcal{R}=\mathcal{V}_{g}-\frac{1}{2m}\bar{\mathcal{W}}^{B}_{3}\bar{\mathcal{W}}^{B3},$
(46)

where the covariant derivative is now expressed as
$\mathcal{D}_{\mu}=\partial_{\mu}-\frac{ie}{\hbar}\bar{\mathcal{A}}_{\mu}-\frac{i}{\hbar}\bar{\mathcal{W}}_{\mu}$.
The effective surface electronic Hamiltonian $\mathcal{H}^{(2)}_{0}$ is
decoupled from normal states. From the point of view of the SDC, the physical
meaning of Eq. 45 is completely different from the surface Hamiltonian
obtained from the phenomenological Rashba effect (Eq. 23), specifically: i)
for the intrinsic SDC, the Rashba effect on a curved surface is incorporated
through the surface non-abelian gauge field $\bar{\mathcal{W}}_{\nu}$, which
couples with the metric and consequently with the intrinsic curvature and ii)
the extrinsic SDC is absent in the Hamiltonian including the term
$\bar{\mathcal{S}}_{3}\text{H}$, as we anticipate by analogy with the
interaction between electromagnetic fields and geometrical deformations. This
can be physically interpreted considering that the gauge invariance of the
Hamiltonian
$\tilde{\mathcal{H}}_{e}=\mathcal{H}_{e}+\frac{1}{2m}\bm{\mathcal{W}}\cdot\bm{\mathcal{W}}$
forbids the extrinsic SDC term $\bar{\mathcal{S}}_{3}\text{H}$. Note that
although the second term in the electronic Hamiltonian Eq.(30) is not gauge
invariant, it does not play a role in the interaction that mediates the
coupling between the surface and normal modes.

Our results show that in opposite to the prevailing consensus, the geometric
deformations of the extrinsic curvature have no effect on the spin dynamics
and cannot be used to control the spin degree of freedom. The utilization of
non-abelian gauge fields for addressing SOC in the context of curved quantum
wires is of particular significance Liang  _et al._ (2020). Curiously, the
coupling term $\bar{\mathcal{W}}_{3}\text{H}$ does not appear in the effective
Hamiltonian in Liang  _et al._ (2020). The absence of extrinsic SDC
contradicting several previous works Zhang  _et al._ (2007); Liu  _et al._
(2011); Ortix (2015); Gentile  _et al._ (2015); Ying  _et al._ (2016); Pandey
_et al._ (2018); Salamone  _et al._ (2022) seems to have been overlooked.

On the other hand, as part of the geometrical potential $\mathcal{R}$, we find
a novel scalar geometrical potential depending on the Rashba SOC strength,
$\bar{\mathcal{W}}^{B}_{3}\bar{\mathcal{W}}^{B3}$, which has been unnoticed in
previous studies. This SOC-induced geometrical potential representing the
remaining contribution from the normal component of the non-abelian gauge
field does not distinguish between spins. Curiously, this scalar potential is
also present in one-dimensional systems, as can be found in the appendix B.

##  V Controlability of the SDC

We now turn the attention to the possibility of a induced SDC based on the
analogy with an electromagnetic field in a deformed surface. For a spinless
particle confined to a (flat) quasi-two-dimensional in the presence of a
tilted magnetic field, one would expect that a tilted magnetic field affects
the in-plane motion only via its perpendicular component ($B_{\perp}$) and no
for the in-plane component ($B_{||}$), as can be verified experimentally. This
must be so while the magnetic length $\ell_{B_{||}}=\sqrt{\hbar/(eB_{||})}$ be
much greater than the confinement width
$\ell_{c}=\epsilon\sqrt{2\hbar/(m\omega)}$. Nevertheless, if
$\ell_{B_{||}}\sim\ell_{c}$ then $B_{||}$ can modified the surface energy
spectrum because the spatial confinement and the magnetic field compete.
Indeed, the in-plane magnetic field can lead to fictitious effects if one does
not deal with terms analogous to $\mathcal{H}^{(1)}_{e}$ properly (see Brandt
and Sánchez-Monroy (2018) for more details).

On the other hand, in a deformed surface, only when there is a strong surface
confinement, i) the normal and surface states are decoupled and ii) the SOC is
unable to promote transition between normal states. The premise of strong
confinement in the surface typically assumed to investigate the SDC Chang  _et
al._ (2013); Wang  _et al._ (2017); Kosugi (2011); Siu  _et al._ (2018); Zhang
_et al._ (2007); Liu  _et al._ (2011); Ortix (2015); Gentile  _et al._ (2015);
Ying  _et al._ (2016); Pandey  _et al._ (2018); Salamone  _et al._ (2022),
together with the proper description of the invariance gauge leads to a
vanished extrinsic SDC. However, the absence of terms proportional to
$\xi\text{H}\tilde{\sigma}^{i}$ does not imply that there is no Rashba effect
on a curved 2D surface. The Rashba SOC is intrinsically included in covariant
derivative $\mathcal{D}_{\mu}$. Therefore, this is a potential route to use
the intrinsic SDC. Additionally, our description suggests that an extrinsic
induced SDC could be obtained in materials with a large enough deformation or
weak confinement, where the thin-layer approximation should be enhanced.
However, in such a scenario, transitions between normal states are allowed,
and consequently, the Hilbert space of the 2D states is mixed with normal
dynamics. In order to illustrate these results in the context of an specific
Hamiltonian, we discuss the example of the Gaussian bumps deformation in 2D
materials.

##  VI The Gaussian bumps deformation

Figure 1: An illustrative example showcasing a specific geometry: the Gaussian
bump.

One of the most common deformation in two-dimensional materials is the
Gaussian bumps located in the center of the surface Debrégeas  _et al._
(2001); de Juan  _et al._ (2007); Tran  _et al._ (2020). Based on the
previously describe approach, we now focus on the study of the SDC for the
curved Gaussian surfaces in typical nanostructures. A point in the surface can
be represented by

$\bm{r}(r,\theta)=r\cos\varphi\bm{\hat{i}}+r\sin\varphi\bm{\hat{j}}+f(r)\bm{\hat{k}},$
(47)

where $r$ is the radius, $\varphi$ is the angle, and $f(r)$ is the height. The
surface is curved if and only if $\dot{f}=df/dr\neq 0$. For this geometry, the
metric tensor is given by
$g_{\mu\nu}=\text{diag}\left(1+\dot{f}^{2},r^{2}\right)$. For detailed
calculations of this geometry see the appendix C. Assuming the electric field
$\mathbf{E}=-\nabla V$ has only a $z$ component, the non-zero components of
the non-Abelian field are

$\mathcal{W}_{x}=-\frac{m\alpha}{\hbar}\sigma^{y},\ \ \ \
\mathcal{W}_{y}=\frac{m\alpha}{\hbar}\sigma^{x}.$ (48)

Using Eqs. (91), (92), and (93), the effective Hamiltonian can be obtained

$\mathcal{H}_{e}=-\frac{\hbar^{2}}{2mg_{m}}\left[\Omega_{0}+\Omega_{r}\bar{\mathcal{W}}_{r}+\Omega_{\varphi}\bar{\mathcal{W}}_{\varphi}\right]+\mathcal{R}.$
(49)

Where we write the kinetic energy as

$\Omega_{0}=\partial_{r}\left(g_{d}^{-1}\partial_{r}\right)+\partial_{\varphi}\left(g_{d}\partial_{\varphi}\right).$
(50)

Here, the notation $g_{1}=1+\dot{f}^{2}$, $g_{d}=\sqrt{g_{1}}/r$, and
$g_{m}=\sqrt{g_{1}}r$ is introduced. Additionally, considering the non-abelian
gauge field as

$\displaystyle\bar{\mathcal{W}}_{r}$ $\displaystyle=$
$\displaystyle-\frac{m\alpha}{\hbar}\cos(\varphi)\sigma^{y}+\frac{m\alpha}{\hbar}\sin(\varphi)\sigma^{x},$
(51) $\displaystyle\bar{\mathcal{W}}_{\varphi}$ $\displaystyle=$
$\displaystyle\frac{m\alpha}{\hbar}r\sin(\varphi)\sigma^{y}+\frac{m\alpha}{\hbar}r\cos(\varphi)\sigma^{x},$
(52)

with $\Omega_{\varphi}=\frac{2i}{\hbar r^{2}}\partial_{\varphi}$ and

$\Omega_{r}=\frac{i}{\hbar
rg_{1}}(2r\partial_{r}-g_{1}+1-r\dot{f}\ddot{f}/g_{1}).$ (53)

Finally, the term $\mathcal{R}$ is given by

$\mathcal{R}=-\frac{h^{2}\left(-r\ddot{f}+\dot{f}^{3}+\dot{f}\right)^{2}}{8mr^{2}g_{1}^{3}}-\frac{m\alpha^{2}}{2\hbar^{2}}\frac{\dot{f}^{2}}{g_{1}},$
(54)

The first term on the right side corresponds to the geometric potential, which
depends on both the intrinsic and extrinsic geometry, whereas the second term
corresponds to a (curvature-induced) scalar potential. This example
illustrates how the curvature can influence the spin dynamics of particles
through derivatives of the metric tensor, indicating the presence of intrinsic
spin-dependent coupling (SDC).

##  VII Conclusion

Based on the unitary transformation usually used in many-body physics for the
separation of variables and asymptotic behavior, we derive a general
methodology for the separation of normal and surface degrees of freedom. We
demonstrate that, due to gauge invariance, the SDC depends only on the
intrinsic geometry of the surface. Therefore, a surface with zero Gaussian
curvature does not modify the spin dynamics. We have also find previously
unnoticed scalar geometrical potential proportional to the Rashba SOC
strength. In the proposed methodology, it is intrinsically assumed that the
non-abelian gauge field describing the SOC acts as an external field, implying
that the components of $\mathcal{W}_{i}$ ($i=1,2,3$) remain unaltered when the
system suffers deformations. This assumption deserves future considerations.
In the appendix B, we provide a detailed general deduction applicable to both
quantum surfaces and wires. Additionally, our work suggests that a route to
induce SDC is to consider distortions in the crystalographic point group of
three-dimensional materials, as usually demonstrated by density functional
calculations. The use of the proposed formalism for the study of SDC in the
context of other possible deformations can open a path for future work in this
field.

##  Appendix A Perturbative expansion in a general $m$-dimensional submanifold

In this appendix, we address the most general scenario wherein the particle is
confined to an $m$-dimensional submanifold –a low-dimensional space
($\mathcal{M}$)–within an $n$-dimensional manifold ($\mathcal{N}$). The pa
rticle interacts with an external (abelian) electromagnetic field, which
propagates into an $n$-dimensional Euclidean space, along with a non-abelian
gauge field (Rashba spin-orbit coupling). The procedure extends the results
presented in Brandt and Sánchez-Monroy (2015, 2017); Liang  _et al._ (2020).

### Geometrical preliminaries

We assume that $\mathcal{M}$ has a tubular neighborhood, such that Sánchez-
Monroy (2016)

$\displaystyle\bm{R}(x^{\mu},y^{i})=\bm{r}(q^{\mu})+q^{j}\hat{\bm{n}}_{j}(q^{\mu}),$
(55)

defines an arbitrary point in the vicinity of $\mathcal{M}$, where
$\hat{\bm{n}}_{j}$ are orthonormal vectors to $\mathcal{M}$. If
$\bm{t}_{\mu}=\frac{\partial\bm{r}}{\partial q^{\mu}}$ are $m$ tangent vectors
on $\mathcal{M}$, the first fundamental form (metric), the second fundamental
form (extrinsic curvature) and the normal fundamental form (geometric or
extrinsic torsion) are respectively,

$\displaystyle g_{\mu\nu}\equiv{\bm{t}}_{\mu}\cdot{\bm{t}}_{\nu},$ (56)
$\displaystyle\alpha_{i\mu\nu}\equiv-{\bm{t}}_{\mu}\cdot\partial_{\nu}\hat{\bm{n}}_{i},$
(57) $\displaystyle\beta_{\mu i}^{\ \
k}\eta_{kj}\equiv\hat{\bm{n}}_{i}\cdot\partial_{\mu}\hat{\bm{n}}_{j}$ . (58)

Note that extrinsic torsion arises only when the codimension is greater than
one, $n-m>1$. Previously, it was shown that any torsion contributes solely to
coupling with angular momentum operators in normal space Schuster and Jaffe
(2003); Sánchez-Monroy (2016). For confining potentials exhibiting $SO(p)$
symmetry, the eigenstates of the effective Hamiltonian $\mathcal{H}^{(0)}_{e}$
can be decomposed into irreducible $SO(p)$ multiplets: Consequently, if the
system is initially prepared in the ground state of $\mathcal{H}^{(0)}_{e}$,
its angular momentum in the normal coordinates vanishes, and considering
torsion will add nothing. Thus, for the sake of simplicity, in the subsequent
discussion, we assume that the manifold is devoid of extrinsic torsion.
Therefore, the metric for the Euclidean space is given by:

$\displaystyle G_{AB}=\left(\begin{array}[]{ll}\gamma_{\mu\nu}&0\\\
0&\delta_{ij}\end{array}\right)$ (61)

where $\gamma_{\mu\nu}$ is given by

$\displaystyle\gamma_{\mu\nu}=g_{\mu\nu}-2q^{k}\alpha_{k\mu\nu}+q^{k}q^{l}\alpha_{k\mu\rho}g^{\rho\sigma}\alpha_{l\sigma\nu}.$
(62)

Here, we employ Latin indices $A$, $B$ running from $1$ to $n$ for the
coordinates in the Euclidean space, Latin indices $i$, $j,\dots$ run from $1$
to $n-m$ and the Greek indices $\mu$, $\nu,\dots$ run from $1$ to $m$.
Denoting the determinants of $G_{AB}$ and $\gamma_{\mu\nu}$ as $|G|$ and
$|\gamma|$, respectively, it is possible to show that $|G|=|\gamma|$. The
inverse of the metric tensor $G_{AB}$ can be calculated exactly, and is given
by

$\displaystyle G^{AB}=\left(\begin{array}[]{ll}\lambda^{\mu\nu}&0\\\
0&\delta^{ij}\end{array}\right),$ (65)

where $\lambda^{\mu\nu}\equiv(\gamma^{-1})_{\mu\nu}$ is the inverse of
$\gamma_{\mu\nu}$ Schuster and Jaffe (2003).

### Perturbative expansion

We rescale the normal coordinates by $q\rightarrow\epsilon q$. Expanding
$\mathcal{H}_{e}$ in powers of $\epsilon$ yields

$\displaystyle\epsilon^{2}\mathcal{H}_{e}=\mathcal{H}^{(0)}_{e}+\epsilon\mathcal{H}_{e}^{(1)}+\epsilon^{2}\mathcal{H}^{(2)}_{e}+\ldots.$
(66)

As $\bm{\mathcal{Z}}(q^{\mu},\epsilon q^{i})$ and
$\bm{\mathcal{W}}(q^{\mu},\epsilon q^{i})$ are now dependent on $\epsilon$
their Taylor expansions are given by Eqs. (41) and (42), respectively.
Therefore, the contributions to the zero, first and second order terms are

$\displaystyle\mathcal{H}^{(0)}_{e}$ $\displaystyle=$
$\displaystyle-\frac{\hbar^{2}}{2m}\partial_{i}\partial^{i}+V_{c},$ (67)
$\displaystyle\mathcal{H}^{(1)}_{e}$ $\displaystyle=$
$\displaystyle-\frac{i}{m}\hbar\bar{\mathcal{Z}}_{i}\partial^{i},$ (68)

$\displaystyle 2m\mathcal{H}^{(2)}_{e}\chi$ $\displaystyle=$
$\displaystyle-\frac{\hbar^{2}}{|g|^{1/2}}\bigg{[}\partial_{\mu}g^{\mu\nu}|g|^{1/2}\partial_{\nu}\bigg{]}\chi+2\mathcal{V}_{g}m\chi-2i\hbar\bar{\mathcal{Z}}_{\mu}g^{\mu\nu}\partial_{\nu}\chi-\frac{i\hbar}{|g|^{1/2}}\partial_{\mu}\left(|g|^{1/2}g^{\mu\nu}\bar{\mathcal{Z}}_{\nu}\right)\chi$
(69)
$\displaystyle-(\bar{\mathcal{Z}}_{i}\bar{\mathcal{Z}}^{i}+\bar{\mathcal{Z}}_{\mu}g^{\mu\nu}\bar{\mathcal{Z}}_{\nu})\chi-2i\hbar\bar{\mathcal{Z}}_{i}q^{i}H\chi-i\hbar\frac{\partial\bar{\mathcal{Z}}_{i}}{\partial
q_{i}}\chi-2i\hbar\frac{\partial\bar{\mathcal{Z}}_{i}}{\partial
q^{j}}q^{j}\partial^{i}\chi$
$\displaystyle-\left[\bar{\mathcal{W}}_{i}\bar{\mathcal{W}}^{i}+\bar{\mathcal{W}}_{\mu}g^{\mu\nu}\bar{\mathcal{W}}_{\nu}\right]\chi+2m\mathcal{Z}_{0}\chi,$

where $\mathcal{V}_{g}$ is a geometric potential given by

$\displaystyle
V_{G}=-\frac{\hbar^{2}}{8m}g^{\nu\mu}g^{\rho\sigma}\left(\alpha_{k\mu\nu}\alpha^{k}_{\
\rho\sigma}-2\alpha_{k\mu\sigma}\alpha^{k}_{\ \rho\nu}\right).$ (70)

##  Appendix B Decoupling in the abelian case

The dynamics of the tangent and normal degrees of freedom are coupled by the
first-order term (Eq. (68)), whereas the curvature is coupled with the
components of fields $\bar{\mathcal{W}}_{i}$ and $\bar{\mathcal{A}}_{i}$ in
the sixth term of (Eq. (69)). In addition, the in-plane and out-of-plane modes
are coupled by the seventh and eighth terms of (Eq. (69)).

To obtain the effective dynamics, it is necessary to “freeze” the normal
degrees of freedom Maraner (1995); Schuster and Jaffe (2003). However, the
first-order term (Eq. (68)) couples the dynamics of the tangent and normal
degrees of freedom through the normal components of the $U(1)\times SU(2)$
gauge potential ($\bm{\mathcal{Z}}$) evaluated at $q^{j}$ equal to zero
($\bar{\mathcal{Z}}_{i}=\mathcal{Z}_{i}(q^{\mu},0)$). In the case that there
was only an abelian field, the dynamics could be decoupled by performing a
gauge transformation Brandt and Sánchez-Monroy (2015); Sánchez-Monroy (2016);
Brandt and Sánchez-Monroy (2017)

$\gamma(x^{\mu},\epsilon
q^{i})=-\epsilon\bar{\mathcal{A}}_{i}q^{i}-\frac{\epsilon^{2}}{2}\frac{\partial\bar{\mathcal{A}}_{i}}{\partial
q^{j}}q^{i}q^{j}+\mathcal{O}((\epsilon q)^{3}),$ (71)

for which it is satisfied that

$\displaystyle\bar{\mathcal{A}}^{\prime}_{i}=\bar{\mathcal{A}}_{i}+\frac{1}{\epsilon}\partial_{i}\gamma^{\prime}=-\epsilon\frac{q^{j}}{2}\bar{\mathcal{F}}_{ij}+\mathcal{O}((\epsilon
q)^{2}),$ (72)

where $\bar{\mathcal{F}}_{ij}=\left.\mathcal{F}_{ij}\right|_{q=0}$ is the
(abelian) electromagnetic field tensor.

For the non-Abelian case, we proceed similarly, using the gauge transformation
(32) to decouple the dynamics. After performing both Abelian and non-Abelian
gauge transformations, the zero, first, and second order terms are as follows:

$\mathcal{H}^{(0)}_{e}=-\frac{\hbar^{2}}{2m}\partial_{3}\partial^{3}+\mathcal{V}_{c},$
(73) $\mathcal{H}^{(1)}_{e}=0,$ (74)

and

$\displaystyle 2m\mathcal{H}^{(2)}_{e}\chi$ $\displaystyle=$
$\displaystyle-\frac{\hbar^{2}}{|g|^{1/2}}\bigg{[}\partial_{\mu}g^{\mu\nu}|g|^{1/2}\partial_{\nu}\bigg{]}\chi+2\mathcal{V}_{g}m\chi-2i\hbar\bar{\mathcal{Z}}_{\mu}g^{\mu\nu}\partial_{\nu}\chi-\frac{i\hbar}{|g|^{1/2}}\partial_{\mu}\left(|g|^{1/2}g^{\mu\nu}\bar{\mathcal{Z}}_{\nu}\right)\chi$
(76)
$\displaystyle-\bar{\mathcal{Z}}_{\mu}g^{\mu\nu}\bar{\mathcal{Z}}_{\nu}\chi-\left[\bar{\mathcal{W}}^{B}_{i}\bar{\mathcal{W}}^{Bi}+\bar{\mathcal{W}}_{\mu}g^{\mu\nu}\bar{\mathcal{W}}_{\nu}\right]\chi+2m\mathcal{Z}_{0}\chi+e\bar{\mathcal{F}}^{ij}L_{ij}\chi+\bar{\mathcal{G}}^{ij}L_{ij}\chi.$

Here, $L_{ij}=i\hbar(y_{j}\partial_{i}-y_{i}\partial_{j})$ are the angular
momentum operators in the space normal to $\mathcal{M}$. As before, if the
system is initially prepared in the ground state of $\mathcal{H}^{(0)}_{e}$,
its angular momentum in the normal coordinates vanishes, and the last two
terms do not contribute. Considering the above and in the case of a surface,
this directly leads to Eq. (45).

##  Appendix C Details of the Gaussian bump

To accurately represent a surface in a three-dimensional space, we employ
Monge parametrization, where the surface is described by a single height
function relative to a reference plane. In our case, the two coordinates used
to describe the surface are the $r$ and $\varphi$ coordinates of the
cylindrical coordinate system, so $\bm{r}_{||}=\bm{r}(r,\varphi)$. A point on
the surface is given by by
$\bm{r}(r,\theta)=r\cos\varphi\bm{\hat{i}}+r\sin\varphi\bm{\hat{j}}+f(r)\bm{\hat{k}}$,
where $r$ is the radius, $\varphi$ is the angle, and $z$ is the height. The
tangent vectors are given by ${t}_{\mu}=d\bm{r}_{||}/dq^{\mu}$, so

$\bm{t}_{\varphi}=\frac{\partial\vec{r}}{\partial\varphi}=\left(\begin{array}[]{c}-r\sin\varphi\\\
r\cos\varphi\\\ 0\end{array}\right)\quad\text{ and
}\quad\bm{t}_{r}=\frac{\partial\vec{r}}{\partial
r}=\left(\begin{array}[]{c}\cos\varphi\\\ \sin\varphi\\\
\dot{f}\end{array}\right),$

with $\dot{f}=\frac{df}{dr}$, the metric, or first fundamental form, on the
surface $S$ is defined by Eq. (56), in this case is given by
$g_{\mu\nu}=\text{diag}\left(1+\dot{f}^{2},r^{2}\right)$. The unit normal
vector field can be determined as follows

$\hat{\bm{n}}=\frac{\bm{t}_{\varphi}\times\bm{t}_{r}}{\left|\bm{t}_{\varphi}\times\bm{t}_{r}\right|}=\frac{1}{\sqrt{1+\dot{f}^{2}}}\left(\begin{array}[]{c}\dot{f}\cos\varphi\\\
\dot{f}\sin\varphi\\\ -\dot{r}\end{array}\right)$ (77)

We can then calculate the Weingarten curvature matrix
$\alpha_{\mu\nu}\equiv-\bm{t}_{\mu}\cdot\partial_{\nu}\hat{\bm{n}}=\partial_{\nu}\bm{t}_{\mu}\cdot\hat{\bm{n}}$
is

$\displaystyle\alpha_{\mu\nu}=\frac{1}{\sqrt{1+\dot{f}^{2}}}\left(\begin{array}[]{cc}-\ddot{f}&0\\\
0&-r\dot{f}\end{array}\right)\text{, }$ (80)

so,

$\displaystyle\alpha_{\mu}^{\nu}$ $\displaystyle=$
$\displaystyle-\alpha_{\mu\nu}g^{\mu\nu}=\left(\begin{array}[]{cc}\frac{\ddot{f}}{\left(1+\dot{f}^{2}\right)^{3/2}}&0\\\
0&\frac{\dot{f}}{r\sqrt{1+\dot{f}^{2}}}\end{array}\right)$ (83)
$\displaystyle=$ $\displaystyle\left(\begin{array}[]{cc}-\kappa_{1}&0\\\
0&-\kappa_{2}\end{array}\right),$ (86)

where $\kappa_{1}$ and $\kappa_{2}$ are the principal curvatures. In general,
for the Monge parameterization, we have $\bm{R}=\bm{r}_{||}+q^{3}\hat{\bm{n}}$
and
$\hat{\bm{n}}=\frac{1}{\sqrt{1+\dot{f}^{2}}}(\dot{f}\cos\varphi\bm{\hat{i}}+\dot{f}\sin\varphi\bm{\hat{j}}-\dot{r}\bm{\hat{k}})$,
then

$\bm{R}(r,\theta,q^{3})=\cos\varphi\left(r+q^{3}\frac{\dot{f}}{\sqrt{1+\dot{f}^{2}}}\right)\bm{\hat{i}}+\sin\varphi\left(r+q^{3}\frac{\dot{f}}{\sqrt{1+\dot{f}^{2}}}\right)\bm{\hat{j}}+\left(f(r)-q^{3}\frac{\dot{r}}{\sqrt{1+\dot{f}^{2}}}\right)\bm{\hat{k}}.$
(87)

To obtain the $SU(2)$ non-Abelian gauge field in the adapted coordinate frame
evaluated at $q^{3}=0$, we need to compute:

$\displaystyle\left.\left\\{\frac{\partial x}{\partial r},\frac{\partial
x}{\partial\varphi},\frac{\partial x}{\partial
q^{3}}\right\\}\right|_{q^{3}=0}$ $\displaystyle=$
$\displaystyle\left\\{\cos(\varphi),-r\sin(\varphi),\frac{\dot{f}(r)\cos(\varphi)}{\sqrt{\dot{f}(r)^{2}+1}}\right\\},$
(88) $\displaystyle\left.\left\\{\frac{\partial y}{\partial r},\frac{\partial
y}{\partial\varphi},\frac{\partial y}{\partial
q^{3}}\right\\}\right|_{q^{3}=0}$ $\displaystyle=$
$\displaystyle\left\\{\sin(\varphi),r\cos(\varphi),\frac{\dot{f}(r)\sin(\varphi)}{\sqrt{\dot{f}(r)^{2}+1}}\right\\},$
(89) $\displaystyle\left.\left\\{\frac{\partial z}{\partial r},\frac{\partial
z}{\partial\varphi},\frac{\partial z}{\partial
q^{3}}\right\\}\right|_{q^{3}=0}$ $\displaystyle=$
$\displaystyle\left\\{\dot{f}(r)\sin(\varphi),f(r)\cos(\varphi),-\frac{\dot{f}(r)\sin(\varphi)}{\sqrt{\dot{f}(r)^{2}+1}}\right\\}.$
(90)

Assuming that the electric field $\mathbf{E}=-\nabla V$ has only a $z$
component, the non-zero components of the non-Abelian field are
$\mathcal{W}_{x}=-\frac{m\alpha}{\hbar}\sigma^{y}$, and
$\mathcal{W}_{y}=\frac{m\alpha}{\hbar}\sigma^{x}$. Therefore, the field in the
adapted coordinate frame, $q^{i}=(r,\varphi,q^{3})$, can be calculated using
$\mathcal{W}_{q^{i}}=\frac{\partial x^{\mu}}{\partial
q^{i}}\mathcal{W}_{x^{\mu}}$, so

$\displaystyle\bar{\mathcal{W}}_{r}$ $\displaystyle=$
$\displaystyle\bar{\mathcal{W}}_{x}\cos(\varphi)+\bar{\mathcal{W}}_{y}\sin(\varphi)$
(91) $\displaystyle=$
$\displaystyle-\frac{m\alpha}{\hbar}\cos(\varphi)\sigma^{y}+\frac{m\alpha}{\hbar}\sin(\varphi)\sigma^{x},$
$\displaystyle\bar{\mathcal{W}}_{\varphi}$ $\displaystyle=$
$\displaystyle-\bar{\mathcal{W}}_{x}r\sin(\varphi)+\bar{\mathcal{W}}_{y}r\cos(\varphi)$
(92) $\displaystyle=$
$\displaystyle\frac{m\alpha}{\hbar}r\sin(\varphi)\sigma^{y}+\frac{m\alpha}{\hbar}r\cos(\varphi)\sigma^{x},$
$\displaystyle\bar{\mathcal{W}}_{q^{3}}$ $\displaystyle=$
$\displaystyle\bar{\mathcal{W}}_{x}\frac{\dot{f}(r)\cos(\varphi)}{\sqrt{\dot{f}(r)^{2}+1}}+\bar{\mathcal{W}}_{y}\frac{\dot{f}(r)\sin(\varphi)}{\sqrt{\dot{f}(r)^{2}+1}}$
(93) $\displaystyle=$
$\displaystyle\frac{-m\alpha\dot{f}(r)}{\hbar\sqrt{\dot{f}(r)^{2}+1}}\left(\cos(\varphi)\sigma^{y}+\sin(\varphi)\sigma^{x}\right).$

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