# Nonlinear dynamics of ferroelectrics with three-well local potential

R. Yevych, M. Medulych, Yu. Vysochanskii

(Received March 22, 2018, in final form April 26, 2018)

###### Abstract

For Sn2P2S6 ferroelectrics, the appearance of spontaneous polarization is
related to stereoactivity of tin cations and valence fluctuations of
phosphorous cations. Here, the continuous phase transition and its behavior
under pressure is determined by thee-well local potential and can be described
in an anharmonic quantum oscillator model. For such a model, the spectrum of
pseudospin fluctuations at different temperatures and pressures has been
calculated and compared with the data of Raman spectroscopy. It was revealed
that the ferroelectric lattice instability is related to several low energy
optic modes.

Key words: phase diagrams, ferroelectrics, lattice models in statistical
physics

PACS: 05.50.+q, 05.70.Fh, 63.70.+h, 77.84.-s

###### Abstract

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ñòåðåîàêòèâíñòþ êàòîíâ îëîâà òà âàëåíòíèìè ôëóêòóàöÿìè êàòîíâ ôîñôîðó.
Íåïåðåðâíèé ôàçîâèé ïåðåõä òà éîãî ïîâåäíêà ç òèñêîì âèçíààòüñÿ ëîêàëüíèì
òðèÿìíèì ïîòåíöàëîì îïèñóòüñÿ â ìîäåë àíãàðìîííîãî êâàíòîâîãî îñöèëÿòîðà. Äëÿ
òàêî¿ ìîäåë ðîçðàõîâàí ñïåêòðè ôëóêòóàöé ïñåâäîñïíà ïðè ðçíèõ òåìïåðàòóðàõ
òèñêàõ, ÿê ïîðâíþþòüñÿ ç äàíèìè ðàìàíâñüêî¿ ñïåêòðîñêîï¿. Ïîêàçàíî, ùî
íåñòéêñòü ñåãíåòîåëåêòðèíî¿ ðàòêè ïîâ’ÿçàíà ç äåêëüêîìà íèçüêîåíåðãåòèíèìè
îïòèíèìè ìîäàìè.

Ключов слова: ôàçîâ äàãðàìè, ñåãíåòîåëåêòðèê, ðàòêîâ ìîäåë â ñòàòèñòèíé ôçèö

##  1 Introduction

For Sn2P2S6 ferroelectrics, the second order phase transition at $T_{0}\approx
337$ K having a mixed displacive-order/disorder character takes place [1]. The
soft optic mode was observed by Raman scattering in ferroelectric phase, but
temperature evolution of the soft mode spectrum is rather complicated —
several low energy optic branches linearly interact at heating, and the
relaxational central peak grows in the vicinity of the phase transition
temperature [2]. A complex transformation of Sn2P2S6 lattice dynamics was also
confirmed by inelastic neutron scattering [3] as well as by diffusive X-ray
scattering [4].

Calorimetric investigation [5] reveals a high entropy of continuous
ferroelectric transition in Sn2P2S6 — about 8.6 J$\cdot$K-1mol-1 which is
typical of an order/disorder transition. In addition, according to dielectric
data for Sn2P2S6 [6], the Curie-Weiss constant is approximately equal to
$0.6\cdot 10^{5}$ K which is typical of displacive transitions.

Crossover character (i.e., displacive-ordering) of the phase transition in
Sn2P2S6 was theoretically analyzed in a discrete model of a two-well (Ising-
like) local potential [7]. It was found that the energy barrier between the
wells and intercell interaction is equal to $U\approx 4\cdot 10^{-21}$ J.
Closeness of these values well agrees with the mixed displacive-ordering
character of the considered transition.

At further investigations using DFT techniques, it was found [8] that the
local potential in Sn2P2S6 ferroelectrics has a three-well shape.
Peculiarities of the chemical bounding in this compound, which are related to
electron lone pair stereoactivity of Sn2+ cations [8, 9] and to valence
fluctuations
$\text{P}^{4+}+\text{P}^{4+}\rightarrow\text{P}^{3+}+\text{P}^{5+}$ inside
(P2S6)4- anions, determine the three-well shape of the local potential for
spontaneous polarization fluctuations [10]. Microscopic origin of
ferroelectric lattice instability in Sn2P2S6 can be described as the second-
order Jahn-Teller (SOJT) effect [11] and their thermodynamics can be
considered within the framework of Blume-Emery-Griffiths model [12, 13]. By
taking into account the electronic correlations within the known Hubbard
model, the presentation of Anderson’s electron pairs flipping [14, 15, 16] can
be included into consideration of Sn2P2S6 family ferroelectrics [10]. More
generalized Hubbard-Holstein-like models treat electronic correlations
together with their mixing with phonon excitations [17]. In this
approximation, having changed the local three-well potential by flattening the
side wells, the calculated continuous phase transition temperature decreases
and tricritical point (TCP) is reached [14, 15, 16]. Below TCP temperature,
the first order ferroelectric phase transition line further drops down to 0 K.
In case of Sn2P2S6 family ferroelectric crystals, such an evolution can be
induced at substitution of tin for lead or under hydrostatic compression [10].

Electronic recharging and lattice instability can be presented as pseudospin
fluctuations in anharmonic potential of a complicated shape. An anharmonic
quantum oscillator (AQO) model was proposed earlier for a description of a
temperature-pressure diagram of Sn2P2S6 and of a temperature-composition
diagram of (PbySn1-y)2P2S6 ferroelectric mixed crystals [10].

The description of anharmonic dynamics for ferroelectric lattices is a very
complicated task. Early anharmonic splitting of the soft optic mode was
theoretically predicted [18] and experimentally observed [19] for SbSJ
ferroelectrics with a two-well local potential. In this paper we use the AQO
model having a three-well local potential for the purpose of analysing the
Sn2P2S6 ferroelectric dynamics at variation of temperature and pressure.

##  2 Short description of a model

In the present work we used the AQO model for studying the dynamic and
dielectric properties of Sn2P2S6 ferroelectric crystal. Such a model was
successfully used earlier for investigations of structural phase transitions
of model and real systems [20, 21, 18, 22]. In the AQO model, the real crystal
lattice is represented as a system of one-dimensional interacting quantum
anharmonic oscillators. In this approach, the model Hamiltonian is as follows:

$H=\sum_{i}\left[T(p_{i})+V(x_{i})\right]-\sum_{ij}J_{ij}x_{i}x_{j}\,,$ (1)

where $T(p_{i})$ and $V(x_{i})$ are kinetic and local potential energies,
respectively, as functions of the momentum $p_{i}$ and position $x_{i}$
operators of the $i$-th oscillator; $J_{ij}$ are the coupling constants
between $i$-th and $j$-th oscillators. In order to simplify the equation (1),
a mean-field approach was used. It is assumed that each oscillator is
influenced by a self-consistent linear symmetry-breaking field which depends
on the average displacement of a system of oscillators. Thus, the last term in
Hamiltonian (1) which describes an interaction can be replaced by

$\sum_{ij}J_{ij}x_{i}x_{j}\rightarrow\sum_{i}Jx_{i}\langle x\rangle,$ (2)

where $J$ is the coupling constant between the field and oscillators, $\langle
x\rangle$ is the average displacement of oscillator. This suggestion makes it
possible to represent a model Hamiltonian as a sum of one-particle non-
interacting Hamiltonians:

$\displaystyle H$ $\displaystyle=\sum_{i}H_{i}^{\textrm{eff}},$ $\displaystyle
H_{i}^{\textrm{eff}}$ $\displaystyle=T(p_{i})+V(x_{i})-Jx_{i}\langle
x\rangle.$ (3)

Having self-consistently solved the Schrödinger equation with an effective
Hamiltonian (2), it is easy to obtain a full set of eigen-energies
$\\{E_{n}\\}$ and wave functions $\\{\Psi_{n}(x)\\}$. The average displacement
$\langle x\rangle$ is determined at each temperature $T$ by:

$\displaystyle\langle x\rangle=\sum_{n}P_{n}\langle x_{n}\rangle,\qquad
P_{n}=\frac{\mathrm{e}^{-E_{n}/kT}}{\sum_{n}\mathrm{e}^{-E_{n}/kT}}\,,\qquad\langle
x_{n}\rangle=\frac{\int\Psi_{n}^{*}x\Psi_{n}\mathrm{d}x}{\int\Psi_{n}^{*}\Psi_{n}\mathrm{d}x}\,,$
(4)

where $P_{n}$ is a probability to find the oscillator at the $n$-th level with
the energy $E_{n}$, $\langle x_{n}\rangle$ is the average displacement for the
$n$-th level, $k$ is the Boltzmann constant. On numerical calculations, there
was used a matrix representation form for position and momentum operators
[23]. Eigenfunctions of anharmonic oscillator have been found as a series of
eigenfunctions of a harmonic oscillator. The lowest 200 energy levels were
accounted for in the present work to obtain a satisfying accuracy on
calculations.

To study the influence of an external field $E$ on the system under
consideration, the term proportional to $Ex_{i}$ should be included in the
equation (2). To investigate the phonon-like excitations within the framework
of AQO model, the phonon spectral function was calculated, given by [24]

$\rho_{\textrm{ph}}(\omega)=\sum_{ij}A_{ij}\delta(\omega+E_{i}-E_{j}).$ (5)

The spectral weight $A_{ij}$ is given by

$A_{ij}=\frac{\mathrm{e}^{-E_{i}/kT}-\mathrm{e}^{-E_{j}/kT}}{\sum_{i}\mathrm{e}^{-E_{i}/kT}}\left|\int\Psi_{i}^{*}x\Psi_{j}\mathrm{d}x\right|^{2}.$

To simulate a broadening, each spectral line has been replaced by Lorentzian
profiles with the same half-width of 3 cm-1 and intensities proportional to
their spectral weight.

Since an order parameter is proportional to the average displacement $\langle
x\rangle$, one can investigate a different type of a phase diagram by changing
the temperature, a coupling constant or the shape of the local potential.
These simulations may correspond to the applied pressure or chemical
substitution, for example, in real systems.

##  3 Discussion of results

In the present work we use a three-well local potential for order parameter
fluctuations of Sn2P2S6 crystal [8]. The evolution of this potential with
pressure is presented in figure 1. As can be seen in figure 1, transformation
of the shape of the potential under compression has a interesting feature: at
a pressure equal to the characteristic one $p_{\textrm{c}}\approx 0.5$ GPa,
all three wells are of the same depth (at $p<p_{\textrm{c}}$, the side-wells
are deeper than the central one, and at $p>p_{\textrm{c}}$, the central well
becomes the deepest). Typical temperature dependencies of the order parameter
depending on the shape of the potential and coupling constant $J$ are shown in
figure 2. It should be noted that the coupling constant has mainly an effect
on temperature of the phase transition. However, the shape of the potential
determines a possibility of realizing a certain type of transition in the
system. Based on this potential, the AQO model describes the temperature-
pressure-coupling constant phase diagram for Sn2P2S6 -like crystals as shown
in figure 3 [10]. Here, the characteristic pressure $p_{\textrm{c}}$ is a
lower bound of a region on the pressure scale where the first-order phase
transitions and metastable solutions (with higher energy of a ground state)
can be realized. The case of Sn2P2S6 crystal corresponds to the value of the
coupling constant about 8.82 J/m2 on the represented diagram.

Figure 1: (Colour online) The shape of the local potential for Sn2P2S6 crystal
at different pressures [25].

Figure 2: (Colour online) Typical temperature dependencies of the order
parameter at a different shape of three-well local potential (on the inset):
(a) — continuous transition at $T=T_{\text{C}}$ and $J_{1}=9$ J/m2; (b) —
continuous transition at $J_{2}=11$ J/m2 and $T=T_{\textrm{C}1}$ (metastable
solution exists at $T\leqslant T_{2}$), discontinuous transition at $J_{3}=9$
J/m2 and $T=T_{\textrm{C}2}$ (metastable and stable solutions coexist in the
same temperature range). Figure 3: (Colour online) The temperature-pressure-
coupling constant phase diagram of the systems having a local three-well
potential [10]. The position of Sn2P2S6 crystal on the diagram is marked with
a black circle.

The fluctuations of spontaneous polarization are determined by three values of
pseudospin (“$-1$”, “0”, and “$+1$”), and the thermodynamics of such a system
is described by BEG model [12, 13]. The pseudospin is composed of ion and
electron components which are examined in Hubbard-Holstein models [17].
Corresponding contributions to spontaneous polarization relate to normal
coordinates of polar phonons with $B_{u}$ symmetry and to a recharging within
SnPS3 structural groups [10]. The ion component of spontaneous polarization
mainly relates to low-frequency phonon modes — the translational vibrations of
a crystal lattice with frequencies below 150 cm-1. Being determined by charge
disproportion, the electron recharging can be generally associated with
internal vibrations of (P2S6)4- anion structural groups, that is, can be
realized through the changes of P-P and P-S bond length, that belong to
375–600 cm-1 frequency range [1]. It can be supposed that variation of the
energy levels of a quantum anharmonic oscillator qualitatively reflects the
temperature evolution of nonlinear dynamics of a ferroelectric system having a
local many-well potential.

Figure 4: (Colour online) Local three-well potential at $T=0$ K (a) and
$T=350$ K (b) (thick black line) and their energy levels with corresponding
squared wave functions (thin black and blue lines). According to probabilities
by equation (4) at a given temperature, the average energy levels and wave
functions were calculated (dashed red lines). (c) — temperature dependence of
a few low-energy levels (black solid lines) and order parameter (blue dashed
line).

Being calculated within the mean-field approach, the temperature evolution of
the energy spectrum (see figure 4) correlates with the temperature dependence
of spontaneous polarization which disappears at heating and is accompanied by
a transition from ferroelectric phase to the paraelectric one. Theoretical
pressure-temperature-coupling constant phase diagram (see figure 3) shows that
the temperature of phase transition drops down to 0 K at a pressure about 2.2
GPa for Sn2P2S6 crystal. Moreover, tricritical point is indicated near the
temperature of 250 K and within 0.4–0.6 GPa pressure range. Using AQO model it
is possible to analyze the temperature transformation of phonon spectra
associated with spontaneous polarization for Sn2P2S6 crystal.

The temperature transformation of the spectrum corresponding to transitions
between energy levels of quantum oscillator having a three-well potential at
an ambient pressure is shown in figure 5 (a). At low temperatures, the
transitions are excited between the levels with the energy difference about
120 cm-1. At a temperature increase, the probability of an occupancy at higher
energy non-equidistant levels grows which becomes evident due to the
appearance of spectral lines of a lower energy. Approaching the phase
transition point ($T_{0}=337$ K), the spectral weight passes to a group of
overlapping lines in the frequency range of 25–80 cm-1. A separate line
becomes apparent about 12 cm-1 which can be related to the experimentally
observed soft optic mode. There are low frequency (from 10 to 30 cm-1)
spectral lines in the paraelectric phase that do not show a significant
behavior while temperature increases.

Figure 5: The calculated spectra for a three-well potential (a) and the
observed Raman spectra (b) of Sn2P2S6 at different temperatures for a normal
pressure.

The calculated temperature evolution of the energy spectrum for a system of
anharmonic quantum oscillators generally agrees with the tendency of
temperature changes in experimentally observed Raman spectra in Sn2P2S6
crystal [see figure 5 (b)]. At heating in a ferroelectric phase, low energy
spectral lines slightly decrease its frequency and significantly change the
width and the shape asymmetry. The wide spectral line which is identified as a
soft optic mode [1, 2] appears below 40 cm-1. It spreads with the temperature
increase and its maximum shifts to lower frequencies. In addition, there
appears a central peak of quasielastic light scattering which can be related
to the relaxation dynamics of pseudospins between central and side wells of
the local potential. According to the experimental Raman spectra temperature
evolution [2] in the ferroelectric phase of Sn2P2S6 crystal, the product of
squared frequencies for several lowest energy optic modes is the most
satisfactory only regarding the expected linear temperature dependence at
heating to the second order phase transition temperature. Such a manifestation
of the lattice instability related to the structural transition agrees with
the temperature dependence of the energy distance between the levels
calculated in the AQO model (see figure 6).

Figure 6: (Colour online) The calculated temperature dependencies of the
product of squared frequencies for $n$ lower modes (lines). Experimental data
from work [2] are shown by open circles.

Figure 7: The calculated spectra for a three-well potential at different
temperatures for pressures: (a) — 0.4 GPa, (b) — 2.2 GPa.

It is interesting to analyse the influence of pressure on the calculated
temperature transformation of the energy spectrum of a system of AQOs. At a
tricritical pressure, a broad peak caused by the merging of a group of
spectral lines appears below 50 cm-1 at a lower temperature — about 250 K (see
figure 7). However, the lowest energy band is found at a frequency 25 cm-1
here. In the paraelectric phase at $T=300$ K, the lowest frequency line is
located near 5 cm-1 which indicates a rapid temperature evolution of the
spectrum of pseudospin excitations at heating in the vicinity of tricritical
point. Under the pressure of 1 GPa and at temperature of 200 K in the
paraelectric phase, the lowest band in the spectrum lies at 20 cm-1. By
applying a pressure of 1.5 GPa, the paraelectric phase becomes stable at
cooling to low temperatures (see figure 3). In this case, a group of lines
below 50 cm-1 manifests itself already at $T=50$ K. At a further temperature
increase, there is almost a continuous spectrum in the range from 25 to 75
cm-1. While the pressure grows up to 2.2 GPa, the calculated spectrum contains
a more contrast set of lines, which is shifted to higher frequencies (see
figure 7) in accordance with the experimental data regarding the pressure
influence on the Raman spectra in a paraelectric phase of Sn2P2S6 [1].
Besides, the pressure transformation of experimentally observed Raman spectra
at room temperature qualitatively agreers with the calculated one (see figure
8). Approaching the critical point from both low or high pressures, the low
energy spectral lines obviously decrease their frequencies.

Figure 8: The calculated spectra for a three-well potential (a) and the
observed Raman spectra [1] (b) of Sn2P2S6 at different pressures for room
temperature.

##  4 Conclusions

For Sn2P2S6 proper uniaxial ferroelectrics with a local three-well potential
for spontaneous polarization fluctuations, an anharmonic quantum oscillator
model has been developed and used for a description of a lattice instability
related to a ferroelectric phase transition. The calculated temperature
dependence of pseudospin fluctuations agrees with the observed changes of
Raman spectra at heating in a ferroelectric phase. The hydrostatic pressure
influence on the local potential shape is clearly observed by the variation of
the spectra of pseudospin fluctuations. The lattice instability related to a
continuous phase transition is manifested as a linear temperature dependence
for a product of squared frequencies of several low energy optical modes. A
complicated “soft mode” spectrum is a characteristic peculiarity of a mixed
displacive-order/disorder transition with a rearrangement of crystal chemical
bounding by Sn2+ cation stereoactivity and by the valence fluctuations
$\text{P}^{4+}+\text{P}^{4+}\rightarrow\text{P}^{3+}+\text{P}^{5+}$ for
phosphorous cations.

## Acknowledgements

Authors acknowledge helpful discussions with Prof. Ihor V. Stasyuk.

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# Quantum mechanics of fermion confined to a curved surface in Foldy-
Wouthuysen representation

Hao Zhao1,2 Yong-Long Wang2,3<EMAIL_ADDRESS>Cheng-Zhi Ye2 Run
Cheng1,2 Guo-Hua Liang1 Hui Liu1

###### Abstract

In Foldy-Wouthuysen representation, we deduce the effective quantum mechanics
for a particle confined to a curved surface by using the thin-layer
quantization scheme. We find that the spin effect caused by confined potential
as the results of relativistic correction in the non-relativistic limit.
Furthermore, the spin connection appeared in curved surface which depends on
curvature contributes a Zeeman-like gap in the relativistic correction term.
In addition, the confined potential also induces a curvature-independent
energy shift, which is from the zitterbewegung effect. As an example, we apply
the effective Hamiltonian to torus surface, in which we obtain expectantly the
spin effects related to confined potential. Those results directly demonstrate
the scaling of the uncommutation of the non-relativistic limit and the thin-
layer quantization formalism.

##  I Introduction

With the rapid development of nano-technology, more and more nanostructures
can be fabricated with complex geometrical and topological structuresEbbesen
and Ajayan (1992); Chopra  _et al._ (1995); Kosynkin  _et al._ (2009); Jiao
_et al._ (2009). The presence of geometrical and topological deformations
leads those deformed nanosystems with various novel quantum properties.
Specifically, the curvature effects were widely investigated in thin magnetic
shellsGaididei  _et al._ (2014); Meng  _et al._ (2014), nematic shellsNapoli
and Vergori (2012), titania single crystals Yang  _et al._ (2014), smectic
liquid crystals Kim  _et al._ (2016), quantum spin Hall system Huang  _et al._
(2017), photonic crystal fiber Beravat  _et al._ (2016), domain wall pinning
Yershov  _et al._ (2015), domain wall motion Yershov  _et al._ (2016),
antiferromagnets Pylypovskyi  _et al._ (2020) and so on. For two-dimensional
(2D) curved surfaces, the thin-layer quantization scheme Jensen and Koppe
(1971); da Costa (1981); Wang and Zong (2016) is an effective and suitable
method, which has been successfully employed to deduce effective Maxwell’s
equationBatz and Peschel (2008); Lai  _et al._ (2018, 2019), effective
Schrödinger equationFerrari and Cuoghi (2008), effective Pauli equationKosugi
(2011); Shikakhwa and Chair (2016); Wang  _et al._ (2014) and effective Dirac
equationBrandt and Sánchez-Monroy (2016); Liang  _et al._ (2018, 2020);
Maraner and Pachos (2008); Ouyang  _et al._ (1999); OLPAK (2012); Burgess and
Jensen (1993) for electromagnetic field and electrons confined to a 2D curved
surface. There are two important results, geometric potential Jensen and Koppe
(1971); da Costa (1981) and geometric momentum Liu  _et al._ (2011); Wang  _et
al._ (2017), which have been proved experimentally in topological
crystalSzameit  _et al._ (2010) and plasmon polarizationSpittel  _et al._
(2015), respectively.

The thin-layer quantization formalism is valid and successful, but there are
still some difficulties that have to be reconsidered, such as the
uncommutation of the non-relativistic limit and the thin-layer quantization
process Wang  _et al._ (2021). Specifically, for relativistic fermions in the
external field, the non-relativistic limit contributes a magnetic moment
remained in the effective Pauli equation, surprisingly without the effective
spin-orbit coupling and corresponding energy correction induced by external
field. In superconducting materials and valley electronics devices, the
effects of field-induced Darwin termCapelle and Gross (1999); Capelle (2001)
and spin precessionWu  _et al._ (2011); Lee  _et al._ (2012); Wu and Lue
(2012) have been considered. In order to describe these two effects in the
non-relativistic limit, we will reconsider the Dirac equation in curved
spacetime by introducing the Foldy-Wouthuysen representation (FWR)Foldy and
Wouthuysen (1950); Eriksen (1958); Silenko (2003); Neznamov and Silenko
(2009); Silenko (2008).

In this paper, in the spirit of the thin-layer quantization scheme we will
discuss the effective dynamics of Dirac fermion confined to a curved surface
in FWR. This paper is organized as follows. In Sec. II, we briefly review the
Dirac equation in curved spacetime and the Foldy-Wouthuysen transformation
(FWT). In Sec. III, in the presence of an external field, we first consider a
Dirac fermion confined to a curved surface $S$, and then we deduce the
corresponding effective Hamiltonian in FWR by using the thin-layer
quantization approach. Interestingly, there appears a Zeeman effect induced by
the coupling of spin connection and the external field. As an example, a torus
surface is considered, in which the curvature-induced Zeeman splitting is
demonstrated. Finally, we present our conclusions in Sec. IV.

##  II Dirac equation and its Foldy-Wouthuysen representation

In this section, we will briefly review the Dirac equation in a curved
spacetime and FWT in the presence of an external field. A relativistic
electron in a curved space-time can be described by the Dirac equation, that
is

$\displaystyle(i{\gamma}^{\mu}\nabla_{\mu}-{m})\Phi=0,$ (1)

where $\nabla_{\mu}$ denotes a covariant derivative with
$\nabla_{\mu}=\partial_{\mu}+\Gamma_{\mu}$, wherein $\Gamma_{\mu}$ is a spin
connection and $\mu=0,1,2,3$ describe the four coordinate variables of a
curved space-time, and ${\gamma}^{\mu}$ stands for a Dirac matrix in a curved
space-time, which can be expressed as

$\displaystyle{\gamma}^{\mu}={E^{\mu}}_{\alpha}{\gamma}^{\alpha},$

here $\gamma^{\alpha}$ stands for a Dirac matrix in a flat space-time, wherein
$\alpha=0,1,2,3$ describes the four coordinate variables of flat space-time,
and ${E^{\mu}}_{\alpha}$ are vierbeins that can be defined by

$\displaystyle{E^{\mu}}_{\alpha}=\frac{\partial q^{\mu}}{\partial
x^{\alpha}},$

where $q^{\mu}$ stands for a coordinate variable of a curved space-time, and
$x^{\alpha}$ stands for a coordinate variable of a flat space-time. With the
vierbeins ${E^{\mu}}_{\alpha}$, the inverse of metric tensor defined in a
curved space-time $G^{\mu\nu}$ can be given by

$G^{\mu\nu}={E^{\mu}}_{\alpha}{E^{\nu}}_{\beta}\eta^{\alpha\beta}$

where $\eta^{\alpha\beta}$ is the inverse metric tensor in a flat space-time
with $\eta^{\alpha\beta}=diag(-1,1,1,1)$. And the spin connection
$\Gamma_{\mu}$ can be expressed as

$\Gamma_{\mu}=\frac{1}{4}{E_{\nu}}^{\alpha}(\partial_{\mu}E^{\nu\beta}+{\Gamma^{\nu}}_{\kappa\mu}E^{\kappa\beta})\Sigma_{\alpha\beta},$

with ${\Gamma^{\nu}}_{\kappa\mu}$ being the Christoffel symbol and
$\Sigma_{\alpha\beta}=[\gamma_{\alpha},\gamma_{\beta}]/2$. In the present
paper, the space is curved and the time is taken as a constant, and thus
$ds^{2}$ can be described as

$ds^{2}=-G_{00}dq^{0}dq^{0}+G_{AB}dq^{A}dq^{B},$

where $G_{00}=1$ and $A,B=1,2,3$ stand for the three coordinate variables of a
curved space. Therefore, ${E_{\nu}}^{\alpha}$ can be simplified as

$E^{\alpha}_{\nu}=\left(\begin{array}[]{cc}1&0\\\ 0&{E_{A}}^{I}\\\
\end{array}\right),$

where $I=1,2,3$ stand for the three coordinate variables of a flat space. And
the space components of Dirac matrices $\gamma^{A}$ equal to
${E^{A}}_{I}\gamma^{I}$. By introducing a Dirac matrix $\beta=\gamma^{0}$, the
Dirac equation can be rewritten as

$i\frac{\partial\Phi}{\partial t}=H\Phi,\ \ $

where $H$ is a Dirac Hamiltonian as

$H=-i\beta{\gamma}^{A}\nabla_{A}+\beta m.$

In view of the uncommutation of non-relativistic limit and thin-layer
quantization scheme, the influence of non-relativistic limit on the geometric
effects, which are obtained in the thin-layer quantization formalism, is
worthy of discussion further. Without loss of generality, we boldly assume
Dirac fermion in an external electric field and specifically discuss in FWT.
In the presence of external field, Dirac fermion can be described by a Dirac
Hamiltonian, that is

$H=-i{\gamma}^{A}\nabla_{A}+m+V,$ (2)

where the the $\nabla_{A}=\partial_{A}+\Gamma_{A}+A_{A}$, within $A_{A}$ is a
gauge potential such as magnetic vector potential, $V$ is the potential of
Dirac fermion in an external field. By introducing the matrix $\beta$, the
Dirac Hamiltonian Eq. (2) can be reepressed as

$\displaystyle H=-i\beta{\gamma}^{A}\nabla_{A}+\beta m+\beta V,$ (3)

With FWT, the Dirac Hamiltonian Eq. (2) can be simplified in FWR as

$\displaystyle H=\beta m+\mathcal{E}+\mathcal{O},$ (4)

where $\mathcal{E}=\beta V$ is even form that commutes with $\beta$, and
$\mathcal{O}=-i\beta{\gamma}^{A}\nabla_{A}$ is odd form that anti-commutes
with $\beta$. As $S$ does not depend on time with
$S=\frac{-i\beta\mathcal{O}}{2m}$, the first unitary operator can be directly
described by $e^{iS}$, we can obtain a new Dirac Hamiltonian as

$\displaystyle\begin{split}H_{FW}=e^{iS}He^{-iS}=&\beta
m+\beta\mathcal{E}+\frac{\beta}{2m}\mathcal{O}^{2}-\frac{1}{8m^{2}}[\mathcal{O},[\mathcal{O},\mathcal{E}]]\\\
&+\frac{\beta}{2m}[\mathcal{O},\mathcal{E}]-\frac{1}{3m^{2}}\mathcal{O}^{3}+...,\end{split}$
(5)

where it is worthwhile to notice that the even part $\mathcal{E}$ is
invariant, but the odd part $\mathcal{O}$ is expanded into a new even part and
an odd part. The final goal of FWT is to transform the original Dirac
Hamiltonian into a block diagonal one, in other words, to eliminate the off-
diagonal part that is odd part. In terms of the odd part in Eq. (5), the
second unitary operator can be expressed as $e^{iS_{2}}$ with
$S_{2}=\frac{-i\beta\mathcal{O}_{2}}{2m}$ and
$\mathcal{O}_{2}=\frac{\beta}{2m}[\mathcal{O},\mathcal{E}]$. For the rest odd
part, the third unitary operator can be given by $e^{iS_{3}}$, with
$S_{3}=\frac{-i\beta\mathcal{O}_{3}}{2m}$ and
$\mathcal{O}_{3}=-\frac{1}{3m^{2}}\mathcal{O}^{3}$. It is clear that the
transformation only converges if the $m$ is much larger than the others in Eq.
2. In the case of the non-relativistic limit, the first four terms of $H_{FW}$
are a valid approximation due to the rapid decay of the power series of
$\frac{1}{m}$, and we will consider only them next in FWR.

##  III Foldy-Wouthuysen representation of Dirac equation on curved surface

In this section, the effective quantum mechanics will be discussed in FWR for
the Dirac fermion confined to a 2D curved surface. Without loss of generality,
the 2D curved surface $S$ is originally embedded in a three-dimensional
Euclidean space. For the sake of convenience, we can adapt a curvilinear
coordinate system with $(q_{1},q_{2},q_{3})$, where $q_{1}$ and $q_{2}$ are
two tangent coordinate variables of $S$ and $q_{3}$ is a normal one.
Therefore, in the small neighborhood of $S$ denoted as $\Xi S$, the position
of point can be parametrized by

$\vec{R}(q_{1},q_{2},q_{3})=\vec{r}(q_{1},q_{2})+q_{3}\vec{n}(q_{1},q_{2})$
(6)

where $\vec{n}(q_{1},q_{2})$ is the normal unit basis of $S$, which is a
function of $q_{1}$ and $q_{2}$. With Eq. 6, the covariant elements of the
metric tensor $G_{AB}$ in $\Xi S$ can be defined as

$G_{AB}=\frac{\partial\vec{R}}{\partial
q^{A}}\cdot\frac{\partial\vec{R}}{\partial q^{B}}.$

And on $S$ described by $\vec{r}(q_{1},q_{2})$, the covariant elements of the
metric tensor $g_{AB}$ can be given by

$g_{ab}=\frac{\partial\vec{r}}{\partial
q^{a}}\cdot\frac{\partial\vec{r}}{\partial q^{b}},$

where $a,b=1,2$ denote the two tangent coordinate variables of $S$. According
to the two definitions of $G_{AB}$ and $g_{ab}$, it is easy to prove that the
relationship between $G_{ab}$ and $g_{ab}$ satisfies the following equation

$\begin{split}&G_{ab}=g_{ab}+q_{3}[\alpha g+(\alpha
g)^{T}]_{ab}+q^{2}_{3}(\alpha g\alpha^{T})_{ab},\end{split}$ (7)

and $G_{a3}=G_{3a}=0$, $G_{33}=1$, where $\alpha$ is the Weingarten curvature
tensor that can be expressed as

$\alpha_{ab}=\frac{\partial\vec{r}}{\partial
q^{a}}\cdot\frac{\partial\vec{n}}{\partial q^{b}}.$

In $\Xi S$ and with the adapted coordinate system of $(q_{1},q_{2},q_{3})$,
the vierbeins can be expressed as

$E_{A}^{I}=\left(\begin{array}[]{cc}E_{a}^{i}&0\\\ 0&1\\\ \end{array}\right)$
(8)

where

$\begin{split}&E_{a}^{i}=e_{a}^{i}+q_{3}\alpha^{b}_{a}e^{i}_{b},\\\
&E_{i}^{a}=e_{i}^{a}-q_{3}\alpha^{b}_{a}e^{b}_{i}+O[q^{2}_{3}].\end{split}$
(9)

With these vierbeins and their inverses, the reduced Dirac matrices
$\gamma^{a}$ can be expressed as $\gamma^{a}=E^{a}_{i}\gamma^{i}$ and
$\gamma^{3}$ is invariant.

###  III.1 Effective Hamiltonian on surface in FWR

In the spirit of the thin-layer quantization scheme, we can introduce a
confining potential $V(q_{3})$ to confine Dirac fermion to the curved surface
$S$. For simplicity, the confining potential can be taken as a square well
that vanishes on $S$ and infinity off $S$. Therefore, the Dirac Hamiltonian
should be replaced by

$\displaystyle H=-i\beta{\gamma}^{A}\nabla_{A}+{\beta m}+{\beta V(q_{3})},$
(10)

where $\nabla_{A}=\partial_{A}+\Gamma_{A}$, the term
$-i\beta{\gamma}^{A}\nabla_{A}$ is odd. Therefore, the first unitary operator
for FWT can be given by $U_{1}=e^{iS_{1}}$, here
$S_{1}=\frac{-{\gamma}^{A}\nabla_{A}}{2m}$. The first transformed Hamiltonian
can be expressed as

$\displaystyle\begin{split}H_{FW}=&e^{iS_{1}}He^{-iS_{1}}\\\ =&\beta m+\beta
V(q_{3})+\frac{\beta}{2m}({{\gamma}^{A}\nabla_{A}})^{2}\\\
&+\frac{1}{8m^{2}}[\beta{\gamma}^{A}\nabla_{A},[\beta{\gamma}^{A}\nabla_{A},\beta
V(q_{3})]]\\\ &+\frac{-i\beta}{2m}[\beta{\gamma}^{A}\nabla_{A},\beta
V(q_{3})]-\frac{i}{3m^{2}}({\beta{\gamma}^{A}\nabla_{A}})^{3}+...,\end{split}$
(11)

in the right hand side, the former four terms are block and even, but the
latter two terms are odd, which can provide the second unitary operator
$U_{2}=e^{iS_{2}}$ and the third unitary operator $U_{3}=e^{iS_{3}}$, here
$S_{2}=\frac{-1}{4m^{2}}[\beta{\gamma}^{A}\nabla_{A},\beta V(q_{3})]$,
$S_{3}=\frac{-\beta}{6m^{3}}({\beta{\gamma}^{A}\nabla_{A}})^{3}$. Transformed
by $e^{iS_{2}}$ and $e^{iS_{3}}$, the new Hamiltonian can be described by

$\displaystyle\begin{split}H^{1}_{FW}=&e^{iS_{3}}e^{iS_{2}}H_{FW}e^{-iS_{2}}e^{-iS_{3}}\\\
=&e^{iS_{3}}e^{iS_{2}}e^{iS_{1}}He^{-iS_{1}}e^{-iS_{2}}e^{iS_{3}}\\\ =&\beta
m+\beta V(q_{3})+\frac{\beta}{2m}({{\gamma}^{A}\nabla_{A}})^{2}\\\
&+\frac{1}{8m^{2}}[\beta{\gamma}^{A}\nabla_{A},[\beta{\gamma}^{A}\nabla_{A},\beta
V(q_{3})]]+....\end{split}$ (12)

Obviously, $H^{1}_{FW}$ is a power series of $\frac{1}{m}$, which would
converge when $m$ is dominant in Hamiltonian. In other words, the Hamiltonian
is effect in non-relativistic limit. Through three transformations, in Eq.
(12) the former four terms are very greater than the rest terms for a certain
mass. Without loss of generality, in the present paper the former four terms
is taken as the Dirac Hamiltonian denoted as $H_{EFW}$.

According to the thin-layer quantization approach, we can confine a fermion to
$S$ by introducing a confining potential $V(q_{3})$. In the squeezing process,
the differential homeomorphism transformation induced by curvature have to be
considered. Specifically, the curvature-induced transformation can be
described by the relationship, $G=f^{2}g$, where $G$ is the determinant of the
metric tensor matrix $G_{AB}$ defined in $\Xi S$, $g$ is that of $g_{ab}$ on
$S$, and $f$ is the rescaling factor with
$f=1+Tr(\alpha)q_{3}+det(\alpha)q_{3}^{2}$. Under the rescaling
transformation, the relationship between an introduced new wave function
$\psi$ and the initial wave function $\Phi$ can be expressed in the following
form

$\psi=f^{1/2}\Phi,$ (13)

and the transformed Dirac Hamiltonian satisfies the following transformation

$\bar{H}_{FW}\rightarrow f^{\frac{1}{2}}H_{FW}f^{-\frac{1}{2}}.$ (14)

And then we can expand the Hamiltonian $\bar{H}_{FWT}$ as a power series of
$q_{3}$ and take the limit $q_{3}=0$, the effective Dirac Hamiltonian confined
to $S$ can be obtained as

$\bar{H}_{EFW}=\beta m+\beta V(q_{3})+H^{\prime}_{FW}+H^{\prime\prime}_{FW}$
(15)

where

$\begin{split}H^{\prime}_{FW}=&\frac{\beta}{2m}\\{[\bar{\gamma^{a}}(\partial_{a}+\Omega_{a})][\bar{\gamma^{b}}(\partial_{b}+\Omega_{b})]+\partial_{3}^{2}\\\
&-\gamma^{3}\alpha_{b}^{a}e_{i}^{b}\gamma^{i}\partial_{a}+\gamma^{3}\alpha_{b}^{a}e_{i}^{b}\gamma^{i}(\Omega_{a}+\frac{1}{2}\epsilon^{a}_{b}\bar{\gamma_{a}}\gamma^{3}\alpha_{a}^{a})\\\
&+\gamma^{3}\bar{\gamma^{a}}\partial_{3}\Gamma_{a}+\partial_{3}\Gamma_{3}+|\epsilon^{a}_{b}|\alpha^{a}_{b}\alpha^{b}_{a}/2\\},\end{split}$
(16)

wherein $\bar{\gamma^{a}}=e^{a}_{i}\gamma^{i}$ is the reduced Dirac matrix
defined on $S$, $\Omega_{a}$ is the the normal part of spin connection
$\Gamma_{a}$, which does not depend on $q_{3}$. And $\epsilon$ is a second
order matrix whose anti-diagonal elements are one and the others are zero. And
$H^{\prime\prime}_{FW}$ is

$\begin{split}H^{\prime\prime}_{FW}=&-\frac{V(q_{3})}{m}H^{\prime}_{FW}-\frac{\beta}{8m^{2}}[2\gamma^{3}(\partial_{3}V(q_{3}))\bar{\gamma^{a}}(\partial_{a}+\Omega_{a})\\\
&+4(\partial_{3}V(q_{3}))\partial_{3}+(\partial_{3}^{2}V(q_{3}))].\end{split}$
(17)

In the above discussion, $H^{\prime}_{FW}$ is the effective result of
$\frac{\beta}{2m}({{\gamma}^{A}\nabla_{A}})^{2}$ in $\bar{H}_{EFW}$ by using
the thin-layer quantization approach, and $H^{\prime\prime}_{FW}$ is that of
$\frac{1}{8m^{2}}[\beta{\gamma}^{A}\nabla_{A},[\beta{\gamma}^{A}\nabla_{A},\beta
V(q_{3})]]$. $H^{\prime}_{FW}$ and $H^{\prime\prime}_{FW}$ are both block
diagonal Hamiltonians which are proportional to $\beta$, so the two block in
them are opposites of each other.

In the thin-layer quantization formalism, the last step is to separate
analytically the surface component from the normal one for $\bar{H}_{EFW}$. It
is easy to accomplish for the $\beta m+\beta V(q_{3})+H^{\prime}_{FW}$, but it
is invalid for $H^{\prime\prime}_{FW}$ that results from the coupling of
$V(q_{3})$ and the tangent momentum. Fortunately, the magnitude of
$H^{\prime\prime}_{FW}$ is very smaller than other terms, and it can be simply
taken as perturbation. In $\beta m+\beta V(q_{3})+H^{\prime}_{FW}$, the normal
Hamiltonian is

$\begin{split}H_{n}=\beta V(q_{3})+\beta\partial^{2}_{3}\end{split}$ (18)

and the surface effective Hamiltonian is

$\begin{split}H_{s}=&\beta
m+\frac{\beta}{2m}\\{[\bar{\gamma^{a}}(\partial_{a}+\Omega_{a})][\bar{\gamma^{b}}(\partial_{b}+\Omega_{b})]+\\\
&-\gamma^{3}\alpha_{b}^{a}e_{i}^{b}\gamma^{i}\partial_{a}+\gamma^{3}\alpha_{b}^{a}e_{i}^{b}\gamma^{i}(\Omega_{a}+\frac{1}{2}\epsilon^{a}_{b}\bar{\gamma_{a}}\gamma^{3}\alpha_{a}^{a})\\\
&+\gamma^{3}\bar{\gamma^{a}}\partial_{3}\Gamma_{a}+\partial_{3}\Gamma_{3}+|\epsilon^{a}_{b}|\alpha^{a}_{b}\alpha^{b}_{a}/2\\}.\end{split}$
(19)

As a block diagonal Hamiltonian, the two block of $H_{s}$ are opposite each
other, the positive energy block corresponds to Pauli equation and the other
does to the negative state. In $H_{s}$, $\Omega_{a}$ plays the role of pseudo-
magnetic field that is given in Ref. Liang  _et al._ (2018), which is induced
by Gauss curvature. And the spin-orbit coupling term
$-\gamma^{3}\alpha_{b}^{a}e_{i}^{b}\gamma^{i}\partial_{a}$ is in agreement
with that in Ref. Liang  _et al._ (2018). The rest terms
$\gamma^{3}\alpha_{b}^{a}e_{i}^{b}\gamma^{i}(\Omega_{a}+\frac{1}{2}\epsilon^{a}_{b}\bar{\gamma_{a}}\gamma^{3}\alpha_{a}^{a})+\gamma^{3}\bar{\gamma^{a}}\partial_{3}\Gamma_{a}$
play the role of Zeeman-like splitting that is contributed by the gradient of
vierbein fields and the normal spin connection together. In the presence of
curvature, the SOC is deformed. As a result, there is the Zeeman-like
splitting in the considered system. The last term
$|\epsilon^{a}_{b}|\alpha^{a}_{b}\alpha^{b}_{a}/2$ is the known geometric
potential da Costa (1981) that depends on the anti-diagonal elements of
Wengarten matrix.

###  III.2 Effect of confined potential

In comparison with the Pauli equation, the non-relativistic limit of Dirac
equation, the effective Dirac equation contains additional high power terms of
$\frac{1}{m}$. It is straightforward that $H^{\prime\prime}_{FW}$ is a
function of $\frac{1}{m^{2}}$, which is a relativistic effect added by the
presence of external field. When a fermion is moving in an external field,
which will feel a force given by the external field. In other words, the
momentum is coupled with the spin in Dirac equation, and the coupling can be
modified by the external field. As a result, the spin precession will occur in
the external field. In Eq. (10), the confining potential plays a same role of
mass. It is easy to check that in the non-relativistic limit the confining
potential can provide a mass correction that is proportional to
$\frac{V(q_{3})}{m}$. More importantly, the confining potential $V(q_{3})$ is
a function of $q_{3}$ not a constant as $m$, the normal gradient
$\partial_{3}V$ can provide a force to modify the fermion by deforming SOC. It
is worthwhile to notice that on a curved surface the curvature will couple
with the spin of Dirac fermion that will deform the original SOC. The
deformation can induce a Zeeman-like splitting. Furthermore, the second-order
$\partial^{2}_{3}V(q_{3})$ can only fluctuate the normal position of fermion
that is caused by the interference between positive energy state and the
negative energy one, which is called zitterbewegung. As a result, the
zitterbewegung will contribute a constant energy shift to the fermion energy
level.

To illustrate the actions of confining potential on the effective Dirac
dynamics in FWR, we consider three simple examples for the confining potential
as: (a) $V(q_{3})=m\omega|q_{3}|$; (b) $V(q_{3})=m\omega q^{2}_{3}$ with
$\omega$ is a constant; (c) a deep square well with a certain width $L$. In
the case of (a), the $\partial_{3}V(q_{3})$ is a constant that depends on the
frequency $\omega$, which will induce a certain spin precession correction and
Zeeman-like splitting, no zitterbewegung for vanishing
$\partial^{2}_{3}V(q_{3})$. In the spirit of the thin-layer quantization
scheme, the effective Dirac Hamiltonian should be replaced by the following
form

$H_{eff}=H_{s}-\frac{\beta}{4m}\gamma^{3}\omega\bar{\gamma^{a}}(\partial_{a}+\Omega_{a})$
(20)

and the normal Hamiltonian would be corrected to

$\begin{split}H_{n}=\beta
V(q_{3})+\beta\partial^{2}_{3}-\frac{\beta}{2m}\omega\partial_{3}\end{split}$
(21)

where the $q_{3}$-dependent terms are ignored. In the case of (b),
$\partial_{3}V(q_{3})=2m\omega q_{3}$ can not significantly affect the spin
precession and Zeeman-like splitting, because of the dependence of $q_{3}$. In
contrast to case (a), the difference is the zitterbewegung in the case of (b).
Therefore, the effective Dirac Hamiltonian should be replaced by

$H_{eff}=H_{s}-\frac{\beta}{4m}\omega.$ (22)

In the case of (c), $H^{\prime\prime}_{FW}$ vanishes on $S$, the confined
potential does not affect the effective Dirac Hamiltonian.

According to the above discussions, what is considerably different from the
known conclusion given by R. C. T. de Costa da Costa (1981) is that the
details of confining potential does not affect the effective quantum dynamics
on a curved surface. In FWR, the specific form of confining potential plays an
important role in the effective Dirac Hamiltonian. Specifically, the effects
of the details of confining potential can measure the scaling of uncommutation
of the non-relativistic limit and the thin-layer quantization process.

###  III.3 An example: torus surface

In this section, the previous discussions will be applied to a torus that can
be parametrized by $(\theta,\varphi)$ shown in Fig. 1. Therefore, the
effective Hamiltonian in Eq. (19) is

$\begin{split}H_{s}=&\beta
m+\frac{\beta}{2m}\\{[\bar{\gamma^{\theta}}\partial_{\theta}+\bar{\gamma^{\varphi}}(\partial_{\varphi}+i\frac{\sin\theta}{2}\Sigma_{3})]^{2}\\\
&-\gamma^{3}(\frac{1}{r}\bar{\gamma^{\theta}}\partial_{\theta}+\frac{\cos\theta}{(R+r\cos\theta)}\bar{\gamma^{\varphi}}\partial_{\varphi})\\\
&-i\frac{\cos\theta}{(R+r\cos\theta)}\frac{\sin\theta}{2}\Sigma^{\varphi},\end{split}$
(23)

and the contributions given by the confining potential is

$\begin{split}&H^{\prime\prime}_{FW}=-\frac{V(q_{3})}{m}H^{\prime}_{FW}-\frac{\beta}{8m^{2}}\\{2\gamma^{3}(\partial_{3}V(q_{3}))[\bar{\gamma^{\theta}}\partial_{\theta}\\\
&+\bar{\gamma^{\varphi}}(\partial_{\varphi}+i\frac{\sin\theta}{2}\Sigma_{3})]+4(\partial_{3}V(q_{3}))\partial_{3}+(\partial_{3}^{2}V(q_{3}))\\},\end{split}$
(24)

where $\bar{\gamma^{\theta}}=e^{\theta}_{i}\gamma^{i}$ and
$\bar{\gamma^{\theta}}=e^{\varphi}_{i}\gamma^{i}$ are tangent Dirac matrices,
$\Sigma_{3}=i\gamma_{1}\gamma_{2}$ is the normal one, and $\Sigma^{\varphi}$
is the $\varphi$ Dirac matrix.

On the torus surface, the spin connection contained in the operator
$\nabla_{\theta}$ vanishes, and that in $\nabla_{\varphi}$ is
$\frac{i\sin\theta\Sigma_{3}}{2}$. The novanishing term plays a role of
pseudo-magnetic field that leads to the appearance of Zeeman-like gap
depending on position. In addition, in $H^{\prime\prime}_{FW}$ the couplings
depending on curvature can enlarge the Zeeman-like gap in the $\varphi$
direction. When $\theta=0$ or $\pi$, the spin connection vanishes, and the
Zeeman-like gap also vanishes. And when $\theta=\frac{\pi}{2}$ or
$\frac{3\pi}{2}$, the coefficient of Zeeman-like gap,
$\frac{\cos\theta}{(R+r\cos\theta)}$, will vanishes, the spin gap in $H_{s}$
will vanish, too. Therefore, we can experimentally observe the spin degeneracy
of particles that is determined by the confining potential.

Figure 1: A torus surface with curvilinear coordinate variables $\theta$ and
$\varphi$.

The spin-orbit interaction terms in $H_{s}$ depend on
$\gamma^{\theta}\partial_{\theta}$ and
$\bar{\gamma^{\varphi}}\partial_{\varphi}$ by the coefficients being functions
of curvature. In other words, the Zeeman-like splitting can be controlled by
modifying the curvature of curved surface. Due to the Wengarten matrix
$\alpha$ is diagonal, the known geometric potential vanishes on the torus
surface.

##  IV CONCLUSION

In this paper, we have briefly reviewed the Dirac equation describing the
relativistic electron confined to a curved surface, and the Foldy-Wouthuysen
transformation. In FWR, by using the thin-layer quantization approach we have
deduced the effective Dirac Hamiltonian for the relativistic particle confined
to a curved surface. Strikingly, there are two contributions in the final
effective Hamiltonian, which are provided by the introduced confining
potential. As the results of FWT, the two contributions are spin-orbit couple
and zittebewegung, respectively. Furthermore, the spin connection determined
by the geometry of curved surface is coupled with spin, the coupling
contributes a Zeeman-like splitting. These results directly demonstrate the
uncommutation of the non-relativistic limit and the thin-layer quantization
process which can be employed to quantum measure.

The above discussions have been applied to a torus surface, which are valid.
The spin connection plays the role of pseudo-magnetic field, and couples with
spin as a SOC, which can induce the Zeeman-like gap. These results can be
employed to precisely control the particular properties of spintronic devices
and valley electronics by designing their geometries.

##  V ACKNOWLEDGEMENTS

This work is jointly supported by the Natural Science Foundation of Shandong
Province of China (Grant No. ZR2020MA091), the National Major state Basic
Research and Development of China (Grant No. 2016YFE0129300), the National
Natural Science Foundation of China (Grants No. 11690030, No. 11690033, No.
11535005, No. 61425018).

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