# Baryon topology in hypersphere soliton model

Soon-Tae Hong<EMAIL_ADDRESS>Center for Quantum Spacetime and
Department of Physics, Sogang University, Seoul 04107, Korea

###### Abstract

Exploiting a topological soliton on a hypersphere, we construct nucleon charge
profile functions and find the density distributions for proton and neutron
plotted versus the hypersphere third angle $\mu$. The neutron charge density
is shown to possess a nontrivial $\mu$ dependence, consisting of both positive
and negative charge density fractions. We next investigate the inner topology
of the hypersphere soliton, by making use of the schematic Möbius strips which
are related with the tubular neighborhood of half-twist circle in the manifold
$S^{3}$. In particular, we find that in the hypersphere soliton the nucleons
are delineated in terms of a knot structure of two Möbius strip type circles
in $S^{3}$. Moreover, the two Hopf-linked Möbius strip type circles in the
hypersphere soliton are shown to correspond to (uu, d) in proton and (dd, u)
in neutron, respectively, in the quark model.

hypersphere soliton, baryon, charge density profiles, knot, twist circle

###### pacs:

12.39.Dc, 13.40.Em, 14.20.-c

##  I Introduction

It is well known that the Dirac Hamiltonian scheme has been employed di ;
pvn82 ; pvn89 ; hong15 , to convert the second class constraints into the
first class ones. Moreover, making use of the Dirac quantization, there have
been many attempts for quantizing the constrained systems, in order to obtain
rigorous energy spectrum and investigate BRST brst1 ; brst2 ; brst3
symmetries involved in the systems. In particular, the first order tetrad
gravity has been investigated in the Dirac Hamiltonian scheme to see that the
second class constraints reduce the theory to second order tetrad gravity and
the first class constraints are different from those in the second order
formalism, but satisfy the same gauge algebra if one uses Dirac brackets pvn82
. There have been attempts to study the quantum BRST charge for quadratically
nonlinear Lie algebras, and to derive a condition which is necessary and
sufficient for the existence of a nilpotent BRST charge pvn89 . In a (1+1)
dimensional supersymmetric soliton, topological boundary conditions and the
BPS bound have also been investigated to fix ambiguities in the quantum mass
of the soliton pvn99 . Moreover the Hamiltonian constraint approach has been
applied to the cosmological perturbation theory grav1 ; grav2 , and to the
supersymmetric self-dual loop quantum cosmology eder21 .

On the other hand, it is well known that the standard Skyrmion model skyrme61
; anw83 ; hong98plb ; hong15  is a constrained Hamiltonian system. In
particular, the Skyrmion has been analyzed via the canonical quantization in
the second class formalism, to study the baryon kinematics and its static
properties anw83 . Later, the hypersphere soliton model (HSM) has been
proposed on the curved manifold manton1  to obtain a topological lower bound
on the baryon soliton energy and a set of equations of motion. Next we have
newly predicted the baryon physical properties, such as masses, charge radii
and magnetic moments in the HSM hong98plb .

Recently we have studied a mechanism of baryon mass genesis by exploiting a
pulsating soliton in HSM hong21 . To do this, first we have exploited the
first class Hamiltonian to quantize the hypersphere soliton, and then have
predicted the axial coupling constant whose value is comparable to its
corresponding experimental data. Next we have formulated the intrinsic
frequencies of the pulsating baryons. Explicitly, we have evaluated the
intrinsic frequencies $\omega_{N}$ and $\omega_{\Delta}$ of the nucleon and
delta hyperon, respectively, to obtain the identity
$\omega_{\Delta}=2\omega_{N}$.

In this paper we will consider the HSM, to study the baryon phenomenology and
profile functions inside baryons. We will also construct nucleon charge
profile functions and depict the density distributions for proton and neutron
plotted versus the hypersphere third angle $\mu$. As a result, we will find
that the neutron charge density has a nontrivial $\mu$ dependence. Next, in
the HSM, we will investigate the knot structure of the Möbius strips
associated with the tubular neighborhood of the half-twist circle inside the
topological soliton. Moreover, we will consider two Hopf-linked kauffman ;
baez  Möbius strip type circles in the HSM, to show that they correspond to
(uu, d) and (dd, u) combinations for the proton and neutron, respectively, in
the quark model.

In Sec. II, we will briefly recapitulate baryon formalism to study the profile
function inside the baryons. In Sec. III, we will analyze the charge profile
functions of nucleons. In Sec. IV, we will investigate the knot structures of
the Möbius strip type half-twist circles inside the nucleons. Sec. V includes
conclusions. In Appendix A, hypersphere geometry is summarized, and the Hopf
fibration hopf ; manton2 ; bott  is discussed on the hypersphere.

##  II Baryon characteristics in HSM

In this section, we exploit the HSM manton1 ; hong98plb ; hong21  to briefly
study the baryon phenomenology and the profile function in Fig. 1(a) in the
identity map associated with the BPS bound in the baryon soliton energy. To do
this, we start with the Skyrmion Lagrangian density

${\cal L}=\frac{f_{\pi}^{2}}{4}{\rm
tr}(\partial_{\mu}U^{\dagger}\partial^{\mu}U)+\frac{1}{32e^{2}}{\rm
tr}[U^{\dagger}\partial_{\mu}U,U^{\dagger}\partial_{\nu}U]^{2},$ (2.1)

where $U$ is an SU(2) chiral field, and $f_{\pi}$ and $e$ are a pion decay
constant and a dimensionless Skyrme parameter, respectively. Next the quartic
term is necessary to stabilize the soliton in the baryon sector. Now we
proceed to investigate baryon phenomenology by using the three-sphere metric
given in Appendix A

$ds^{2}=\lambda^{2}d\mu^{2}+\lambda^{2}\sin^{2}\mu~{}(d\theta^{2}+\sin^{2}\theta~{}d\phi^{2}).$
(2.2)

Here the ranges of the hyperspherical coordinates are given by
$0\leq\mu\leq\pi$, $0\leq\theta\leq\pi$ and $0\leq\phi\leq 2\pi$, and
$\lambda$ ($0\leq\lambda<\infty$) is the radius parameter of the hypersphere
$S^{3}$. Note that $S^{3}$ is described in terms of the three dimensional
hypersphere coordinates $(\mu,\theta,\phi)$ and the radius parameter
$\lambda$.

In the hypersphere soliton on $S^{3}$, we obtain the soliton energy manton1 ;
hong98plb ; hong21

$E=\frac{f_{\pi}}{e}\left[2\pi
L\int_{0}^{\pi}d\mu\sin^{2}\mu\left(\left(\frac{df}{d\mu}+\frac{1}{L}\frac{\sin^{2}f}{\sin^{2}\mu}\right)^{2}+2\left(\frac{1}{L}\frac{df}{d\mu}+1\right)^{2}\frac{\sin^{2}f}{\sin^{2}\mu}\right)+6\pi^{2}\right],$
(2.3)

where $L=ef_{\pi}\lambda$ ($0\leq L<\infty$) is a radius parameter expressed
in dimensionless units. Here $f(\mu)$ is the profile function for the
hypersphere soliton, and satisfies $f(0)=\pi$ and $f(\pi)=0$ for unit baryon
number. The above soliton energy has the BPS bound in the soliton model
manton1 ; hong98plb ; hong21 ; faddeev ; manton2  to yield the inequality

$E\geq\frac{6\pi^{2}f_{\pi}}{e}|B|,$ (2.4)

where $B$ is given by hong98plb

$B=-\frac{2}{\pi}\int_{0}^{\pi}d\mu\sin^{2}f\frac{df}{d\mu}.$ (2.5)

Note that $B$ is a conserved topological charge which is a baryon number in
the HSM. For a single soliton, the topological charge is one manton1 ;
hong98plb ; hong21 ; faddeev ; manton2 . In the cases of multi-Skyrmion
solutions with $|B|>1$, the soliton energies will not satisfy the bound in Eq.
(2.4) and possess much higher energy. The hedgehog solutions for the multi-
Skyrmions then have been shown to be almost certainly all unstable saddle-
points of the energy manton2 .

The profile function $f$ in the soliton energy lower bound satisfies equations
of motion manton1 ; hong98plb ,

$\frac{df}{d\mu}+\frac{1}{L}\frac{\sin^{2}f}{\sin^{2}\mu}=0,~{}~{}~{}\frac{1}{L}\frac{df}{d\mu}+1=0.$
(2.6)

One of the simplest solutions of (2.6) is the identity map manton1 ; hong98plb
; hong21

$f(\mu)=\pi-\mu$ (2.7)

with the condition $L=L_{B}$, where $L_{B}\equiv ef_{\pi}\lambda_{B}=1$. The
profile function $f(\mu)$ in the identity map, associated with the BPS bound
in the soliton energy, is depicted in terms of the angle $\mu$ in Fig. 1(a).
The fixed value $L=L_{B}$ can be used to obtain the soliton energy lower bound
manton1 ; hong98plb ; hong21 111From now on we will use the condition
$L_{B}=1$ to predict the physical quantities such as moment of inertia, baryon
masses and charge radii in the HSM. We will then write the results for the
identity map after the integral or definition expressions in (2.10), (2.12)
and (3.2).

$E=\frac{6\pi^{2}f_{\pi}}{e}.$ (2.8)

After performing the Dirac quantization in the first class formalism di ;
hong15 , we obtain the modified Hamiltonian spectrum hong15 ; hong21

$\langle H\rangle=E+\frac{1}{2{\cal I}}\left[I(I+1)+\frac{1}{4}\right],$ (2.9)

where $I$ $(=1/2,~{}3/2,...)$ are isospin quantum numbers of baryons. Here $E$
is given by (2.8) and ${\cal I}$ is the moment of inertia hong98plb

${\cal
I}=\frac{8\pi}{3e^{3}f_{\pi}}\int_{0}^{\pi}d\mu\sin^{2}\mu\sin^{2}f\left(1+\left(\frac{df}{d\mu}\right)^{2}+\frac{\sin^{2}f}{\sin^{2}\mu}\right)=\frac{3\pi^{2}}{e^{3}f_{\pi}}.$
(2.10)

As a result, exploiting (2.9) we construct the Weyl ordering corrected nucleon
and delta hyperon masses, respectively hong21

$M_{N}=ef_{\pi}\left(\frac{6\pi^{2}}{e^{2}}+\frac{e^{2}}{6\pi^{2}}\right),~{}~{}~{}M_{\Delta}=ef_{\pi}\left(\frac{6\pi^{2}}{e^{2}}+\frac{2e^{2}}{3\pi^{2}}\right).$
(2.11)

Next, in the HSM we obtain the electric isovector mean square charge radius
$\langle r^{2}\rangle_{E,I=1}$ of the form hong98plb

$\langle r^{2}\rangle_{E,I=1}=\frac{8\pi}{3e^{2}f_{\pi}^{2}{\tilde{\cal
I}}}\int_{0}^{\pi}d\mu\sin^{4}\mu\sin^{2}f\left(1+\left(\frac{df}{d\mu}\right)^{2}+\frac{\sin^{2}f}{\sin^{2}\mu}\right)=\frac{5}{6e^{2}f_{\pi}^{2}},$
(2.12)

where $\tilde{{\cal I}}$ is the dimensionless moment of inertia defined as
$\tilde{{\cal I}}=e^{3}f_{\pi}{\cal I}$. Similarly, we find the other charge
radii in the HSM to yield hong98plb ; hong21

$\langle r^{2}\rangle^{1/2}_{E,I=1}=\langle r^{2}\rangle^{1/2}_{M,I=0}=\langle
r^{2}\rangle^{1/2}_{M,I=1}=\langle r^{2}\rangle^{1/2}_{M,p}=\langle
r^{2}\rangle^{1/2}_{M,n}=\sqrt{\frac{5}{6}}\frac{1}{ef_{\pi}},\\\ $ (2.13)

where the subscripts $E$ and $M$ stand for the electric and magnetic radii,
respectively.

Making use of the charge radii in (2.13) we fix the value of $\langle
r^{2}\rangle_{M,p}^{1/2}$ with the corresponding experimental data $\langle
r^{2}\rangle_{M,p}^{1/2}=0.80$ fm. We then have

$ef_{\pi}=225.23~{}{\rm MeV}=(0.876~{}{\rm fm})^{-1}.$ (2.14)

Note that, inserting $ef_{\pi}=(0.876~{}{\rm fm})^{-1}$ into the condition
$L_{B}=1$, we obtain $\lambda_{B}=0.876~{}{\rm fm}$. The hypersphere soliton
is thus defined in terms of the fixed radius parameter $\lambda_{B}$ which is
comparable to the predictions for the charge radii in (2.13). In other words,
in the HSM the universe ${\cal U}$ is given by a sum of two subspaces, namely
${\cal U}=V_{B}(\lambda\leq\lambda_{B})\oplus\bar{V}_{B}(\lambda>\lambda_{B})$
where $V_{B}(\lambda\leq\lambda_{B})$ and $\bar{V}_{B}(\lambda>\lambda_{B})$
are the volume subspaces inside and outside the hypersphere soliton,
respectively. Moreover, in HSM the baryons are described in terms of $V_{B}$
which is a three dimensional volume defined inside $S^{3}$ having the volume
element in (A.4). Note that most of baryon charges are located inside the
volume $V_{B}$ of radius parameter $\lambda_{B}$ as shown in (2.13).
Explicitly $\langle
r^{2}\rangle^{1/2}_{M,p}=\sqrt{\frac{5}{6}}\lambda_{B}=0.91~{}\lambda_{B}$,
for instance.

Finally note also that the hypersphere coordinates $(\mu,\theta,\phi)$ are
integrated out in (2.3), and $E$ in (2.8) is a function of
$\lambda_{B}=\frac{1}{ef_{\pi}}$ or equivalently $f_{\pi}$ and $e$ only.
Moreover, similar to $E$ in (2.3) and $B$ in (2.5), the above physical
quantities such as the moment of inertia and charge radii are also given by
integral expressions, as in (2.10) and (2.12) for instance. These physical
quantities are thus obtained in terms of $f_{\pi}$ and $e$ only.

##  III Charge density profiles in HSM

In this section, we investigate the charge density profiles of nucleons in
Fig. 1(b) and Fig. 1(c). Note that, as in the the profile function in the
identity map in Fig. 1(a), these charge density profiles are formulated in
terms of the angle $\mu$. Moreover the charge density profiles $\rho_{p}(\mu)$
and $\rho_{n}(\mu)$ are associated with the charges inside the nucleons, while
the profile function $f(\mu)$ in the identity map is related with the BPS
bound in the soliton energy.

Now, using the three-sphere metric in (2.2), we construct the charge density
profiles of proton and neutron

$\rho_{p}(\mu)=\frac{1}{2}\left[\rho_{I=0}(\mu)+\rho_{I=1}(\mu)\right],~{}~{}~{}\rho_{n}(\mu)=\frac{1}{2}\left[\rho_{I=0}(\mu)-\rho_{I=1}(\mu)\right],$
(3.1)

where the isoscalar and isovector electric charge densities are given by
hong98plb

$\displaystyle\rho_{I=0}(\mu)$ $\displaystyle=$
$\displaystyle-\frac{2}{\pi}\sin^{2}f\frac{df}{d\mu}=\frac{2}{\pi}\sin^{2}\mu,$
$\displaystyle\rho_{I=1}(\mu)$ $\displaystyle=$
$\displaystyle\frac{8\pi}{3{\tilde{\cal
I}}}\sin^{2}\mu\sin^{2}f\left(1+\left(\frac{df}{d\mu}\right)^{2}+\frac{\sin^{2}f}{\sin^{2}\mu}\right)=\frac{8}{3\pi}\sin^{4}\mu,$
(3.2)

respectively, to produce

$\rho_{p}(\mu)=\frac{1}{\pi}\sin^{2}\mu\left(1+\frac{4}{3}\sin^{2}\mu\right),~{}~{}~{}\rho_{n}(\mu)=\frac{1}{\pi}\sin^{2}\mu\left(1-\frac{4}{3}\sin^{2}\mu\right).$
(3.3)

We then find the proton and neutron charges as follows

$Q_{p}=\int_{0}^{\pi}d\mu~{}\rho_{p}(\mu),~{}~{}~{}Q_{n}=\int_{0}^{\pi}d\mu~{}\rho_{n}(\mu),$
(3.4)

which yield $Q_{p}=1$ and $Q_{n}=0$ as expected.

Figure 1: (a) The profile function $f(\mu)$ in the identity map, and the
charge density profiles (b) $\rho_{p}(\mu)$ and (c) $\rho_{n}(\mu)$, for
proton and neutron respectively, are plotted versus $\mu$. Here $f(\mu)$ is
related with the BPS bound in the soliton energy, and $\rho_{p}(\mu)$ and
$\rho_{n}(\mu)$ are associated with the charges inside the nucleons.

The proton and neutron charge densities in (3.3) are depicted in Fig. 1(b) and
Fig. 1(c), respectively. From (3.3) we find two root values of
$\rho_{n}(\mu)$=0: $\mu=\frac{\pi}{3}$ and $\mu=\frac{2\pi}{3}$. We now have
three regions I, II and III in the neutron charge density as shown in Fig.
1(c) and for each region, we obtain the charge fractions as follows

$Q_{n}^{\rm I}=\frac{\sqrt{3}}{16\pi},~{}~{}~{}Q_{n}^{\rm
II}=-\frac{\sqrt{3}}{8\pi},~{}~{}~{}Q_{n}^{\rm III}=\frac{\sqrt{3}}{16\pi},$
(3.5)

which are in good agreement with the result $Q_{n}=0$. Note that the charge
density profiles of proton and neutron have no dependence on the coordinates
$\theta$ and $\phi$ and they have dependence only on the coordinate $\mu$ on
$S^{1}$.222Note that for a given fixed value of $\mu=\mu_{0}$, the three-
sphere metric in (A.1) becomes
$ds^{2}=\lambda^{2}\sin^{2}\mu_{0}~{}(d\theta^{2}+\sin^{2}\theta~{}d\phi^{2})$.
This two-sphere metric is given by the coordinates ($\theta$, $\phi$) and the
fixed two-sphere radius parameter $\lambda\sin\mu_{0}$. The charge density
profiles of the nucleons thus do not depend on ($\theta$, $\phi$), but depend
on $\mu$ for the fixed value of $\mu=\mu_{0}$ corresponding to the two-sphere
radius parameter $\lambda\sin\mu_{0}$. Here and in Sec. IV and Appendix A, we
use $\lambda$ instead of $\lambda_{B}$ for the radius parameter of
hypersphere, for simplicity.

##  IV Topology and knot structure in HSM

Now we investigate the topological aspects and knot structure in the
hypersphere soliton. To to this, we first consider the (1+1) dimensional sine-
Gordon soliton whose Lagrangian is given in appropriate length and energy
units as follows skyrme61b ; raja ; proceeding ; manton2

$L_{SG}=\int
dx\left[\frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi+\cos\phi-1\right],$
(4.1)

from which we obtain the field equation of the form

$\Box\phi+\sin\phi=0.$ (4.2)

Note that the sine-Gordon field equation satisfies the discrete symmetry
$\phi(x,t)\rightarrow\phi(x,t)+2N\pi$, $N=0,\pm 1,\pm 2,\cdots$. The
topological charge for a static solution of the sine-Gordon kink is then given
by $Q_{top}=\frac{1}{2\pi}[\phi(+\infty)-\phi(-\infty)]$. We now have the
static soliton solutions of the form: $\phi(x)=+4\tan^{-1}\exp x$ and
$\phi(x)=-4\tan^{-1}\exp x$, for the cases of $Q_{top}=+1$ for a kink and
$Q_{top}=-1$ for an anti-kink, respectively. These sine-Gordon kink solutions
represent infinite Möbius strip feature proceeding . Moreover the sine-Gordon
kink state is identifiable as a fermion raja . Note in the hypersphere soliton
that, exploiting $B$ in (2.5), we have the profile function $f(\mu)=\pi-\mu$
for the hypersphere soliton with the topological charge $B=+1$ hong98plb ;
manton2 . We also obtain $f(\mu)=-(\pi-\mu)$ for the hypersphere anti-soliton
with $B=-1$ manton2 . In this sense, in the hypersphere soliton, we have the
Möbius strip feature similar to that in the sine-Gordon kink.

Next we consider a half-twist circle $S^{1}$ inside $S^{3}$ which will be
discussed in Appendix A, in order to briefly investigate the relations between
the three dimensional tubular neighborhood of the half-twist circle and the
corresponding Möbius strip. We now assume that the point $o$ is the center of
the two dimensional disc $D$ representing the cross section of the tubular
neighborhood of the half-twist circle as shown in Fig. 2. Here the line
segment $ad$ represents the cross section of the Möbius strip. Next we cut the
disc $D$ of the tubular neighborhood into three pieces by using the line
segments $ef$ and $gh$, and then we let the points $b$ and $c$ denote the
corresponding cross sectional cutting lines on the Möbius strip. Here $\psi$
represents the rotation angle of the disc $D$ (or the line segment $ad$) with
respect to the vertical line which is the initial position of $D$. As the disc
$D$ rotates along the circle $S^{1}$ of radius parameter $\lambda$, the angle
$\psi$ increases from $0$ to $\pi$, to complete the cycle of the half-twist.
In other words, after the complete rotation the final disc $D$ arrives at the
initial disc position, with the angle shift $\pi$. By gluing these two discs
together, we obtain a Möbius strip type tubular neighborhood of the half-twist
circle $S^{1}$. In particular, the tubular neighborhood is splitted into three
pieces with nontrivial volume shapes along the half-twist circle. Moreover the
line segment $ad$ inside the tubular neighborhood makes the Möbius strip. In
this respect without loss of generality, instead of exploiting the tubular
neighborhood of the half-twist circle $S^{1}$ inside $S^{3}$, for visual
clarity we will use the schematic Möbius strips in Fig. 3(a) and Fig. 3(b) in
the further discussions for the topology and knot structure of the baryons
below. We reemphasize that, in Fig. 3(a) and Fig. 3(b), the Möbius strips
represent schematically the tubular neighborhoods of the half-twist circles.

Figure 2: The cross section of tubular neighborhood of the half-twist circle
is delineated by the schematic Möbius strip.

Figure 3: (a) The schematic Möbius strip in the HSM corresponds to the
fermion, and (b) the splitted Möbius strip explains the knot structure of the
two different Möbius strip type circles in the hypersphere soliton.

Note that, to describe a fermion such as a nucleon, we need a Möbius strip
with a half-twist, associated with a half-twist circle $S^{1}$ inside
$S^{3}(=S^{2}\times S^{1})$, as shown in Fig. 3(a). For more details about the
Möbius strip related with the fermion, see the last paragraph in this section.
Here the half-twist circle $S^{1}$ plays a major role in the topological
structure of the fermion. To relate the HSM to the quark model where, in the
fermions such as proton and neutron, there exist three quarks, we separate the
Möbius strip into three pieces as shown in Fig. 3(a) to yield two nontrivial
Möbius strips which are Hopf-linked kauffman ; baez  to each other, as shown
in Fig. 3(b). More specifically, we have the Möbius strip with the
circumference length equal to that of the original Möbius strip in Fig. 3(a)
and with a half-twist. This small Möbius strip originates from the middle
piece of the Möbius strip in Fig. 3(a). Next we have the other Möbius strip
with the circumference length of two times that of the original Möbius strip
in Fig. 3(a) and with two twists. This large Möbius strip consists of the
first and third pieces of the Möbius strip in Fig. 3(a). Note that the two
twists of the large Möbius strip are equal to two times addition of two half-
twists of the first and third pieces of the original Möbius strip. Moreover
the large strip corresponds to uu in uud and dd in udd for proton and neutron
defined in the quark model, respectively. Now we emphasize that two Möbius
strips in Fig. 3(b) are Hopf-linked to produce a knot structure, which will be
also discussed in Appendix A.

Now we return to exploiting the tubular neighborhood of the half-twist circle
in the HSM defined on the hypersphere $S^{3}$, instead of using the schematic
Möbius strip. As a result, we find that there exists inside the nucleon the
half-twist circle $S^{1}(\subset S^{3})$ possessing the Möbius strip type
tubular neighborhood as shown in Fig. 3(a). The tubular neighborhood is now
splitted into three pieces along the half-twist circle to yield two Hopf-
linked sub-manifolds due to the Möbius strip characteristic of the tubular
neighborhood as shown in Fig. 3(b). Indeed, the two Hopf-linked Möbius strip
type circles in Fig. 3(b) explains the knot structure of the two different
Möbius strip type circles, which correspond to (uu, d) in proton and (dd, u)
in neutron, respectively, in the quark model. For more details about the two
Hopf-linked circles inside $S^{3}$, see Appendix A.

Finally we classify the boson and fermion in terms of the strip structures.
The boson performs $2\pi$ rotation to return its starting situation. The
feature is explained by exploiting $2\pi$ rotation on the ordinary strip. Next
the fermion performs $4\pi$ rotation to return its initial situation according
to the relativistic quantum mechanics. The characteristic is elucidated by
using $4\pi$ rotation on the Möbius strip shown in Fig. 3(a). The baryonic
inner structure is thus successfully explained on the hypersphere $S^{3}$. In
contrast, in the case of bosonic inner structure, we do not need to consider
such a hypersphere geometry.

##  V Conclusions

In summary, in the HSM we have performed the first class Dirac quantization,
to predict more rigorously the physical quantities such as the baryon masses
and charge radii. To do this, we have exploited the identity map associated
with the BPS bound in the baryon soliton energy. Moreover we have investigated
the charge density profile functions of nucleons, and then depicted the
density distributions for proton and neutron plotted versus the hypersphere
third angle $\mu$. In particular, the neutron charge density has been shown to
have the nontrivial $\mu$ dependence, consisting of both the positive and
negative charge density parts. Next we have investigated the inner topological
structures of the hypersphere soliton, by using the Möbius strip type tubular
neighborhood of the half-twist circle in $S^{3}$. In particular, we have shown
that, in the HSM, the nucleons are described by the knot of two Hopf-linked
Möbius strip type circles inside the nucleons. As a result, we have found that
the two Hopf-linked Möbius strip type circles in the HSM correspond to the
(uu, d) in proton and (dd, u) in neutron, respectively, in the quark model.

###### Acknowledgements.

The author would like to thank Professor Peter van Nieuwenhuizen for helpful
correspondence and kind encouragement. He was supported by Basic Science
Research Program through the National Research Foundation of Korea funded by
the Ministry of Education, NRF-2019R1I1A1A01058449.

##  Appendix A Hypersphere geometry

In order to investigate a hypersphere of radius parameter $\lambda$, we
consider the three-sphere metric on $S^{3}$

$ds^{2}=\lambda^{2}d\mu^{2}+\lambda^{2}\sin^{2}\mu~{}(d\theta^{2}+\sin^{2}\theta~{}d\phi^{2}).$
(A.1)

The ranges of the hyperspherical coordinates are given by $0\leq\mu\leq\pi$,
$0\leq\theta\leq\pi$ and $0\leq\phi\leq 2\pi$, and $\lambda$
($0\leq\lambda<\infty$) is the radius parameter of $S^{3}$. On the
hypersphere, the three-sphere metric is given by the line element

$d\vec{l}=\hat{e}_{\mu}\lambda~{}d\mu+\hat{e}_{\theta}\lambda\sin\mu~{}d\theta+\hat{e}_{\phi}\lambda\sin\mu\sin\theta~{}d\phi,$
(A.2)

where $(\hat{e}_{\mu},\hat{e}_{\theta},\hat{e}_{\phi})$ are the unit vectors
along the three directions.

The area elements are given by

$d\vec{a}_{\mu}=\hat{e}_{\mu}\lambda^{2}\sin^{2}\mu\sin\theta~{}d\theta~{}d\phi,~{}~{}~{}d\vec{a}_{\theta}=\hat{e}_{\theta}\lambda^{2}\sin\mu\sin\theta~{}d\mu~{}d\phi,~{}~{}~{}d\vec{a}_{\phi}=\hat{e}_{\phi}\lambda^{2}\sin\mu~{}d\mu~{}d\theta,$
(A.3)

and the three dimensional volume element is defined as

$dV=\lambda^{3}\sin^{2}\mu\sin\theta~{}d\mu~{}d\theta~{}d\phi.$ (A.4)

In the hyperspherical coordinates system, the gradient operator is given by

$\nabla=\hat{e}_{\mu}\frac{1}{\lambda}\frac{\partial}{\partial\mu}+\hat{e}_{\theta}\frac{1}{\lambda\sin\mu}\frac{\partial}{\partial\theta}+\hat{e}_{\phi}\frac{1}{\lambda\sin\mu\sin\theta}\frac{\partial}{\partial\phi}.$
(A.5)

Next, we investigate a half-twist circle $S^{1}(\subset S^{3})$ on the
hypersphere. For the space manifolds $S^{3}$ and $S^{2}$, we have the homotopy
group toda ; manton2 ; schwarz  $\Pi_{3}(S^{2})={\mathbf{Z}}$. Note that in
the homotopy group, the target space $S^{2}$ is a geometrical manifold. Due to
the above homotopy group, we have an associated integer topological charge,
namely the Hopf charge manton2 . Here the Hopf charge is intrinsically
different from the topological charge $B$ associated with
$\Pi_{3}(S^{3})={\mathbf{Z}}$. In other words, one cannot construct any Hopf
charge having a form similar to $B$ in (2.5). Instead, we have the preimage of
a point on the target manifold $S^{2}$ which is a closed loop inside $S^{3}$
manton2 . Moreover it is shown in the Hopf fibration hopf ; manton2 ; bott
that the closed loop is a half-twist circle $S^{1}(\subset S^{3})$. To be
specific, noting that the homology groups for $S^{3}$ and $S^{2}\times S^{1}$
are given by $H_{1}(S^{3})=0$ and $H_{1}(S^{2}\times S^{1})={\mathbf{Z}}$
derham , one finds that the hypersphere $S^{3}$ is locally a product of two
manifolds $S^{3}=S^{2}\times S^{1}$ ryder , implying that the $S^{1}(\subset
S^{3})$ is the half-twist circle of radius parameter $\lambda$. It is also
interesting to note that the preimages inside $S^{3}$ decompose the
hypersphere into a continuous family of circles, and two distinct circles are
linked inside $S^{3}$ manton2 ; bott  and form the Hopf link kauffman ; baez .

## References

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  * (19) H. Hopf, Über die abbildungen der dreidimensionalen sphäre auf die kugelfläche, Math. Ann. 104, 637 (1931).
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#  A compact formula for the quantum fluctuations of energy

Rajeev Singh  Institute of Nuclear Physics Polish Academy of Sciences,
PL-31-342 Kraków, Poland

###### Abstract

A formula to calculate the quantum fluctuations of energy in small subsystems
of a hot and relativistic gas is derived. We find an increase in fluctuations
for subsystems of small sizes, but we agrees with the energy fluctuations in
the canonical ensemble if the size is large enough. Not only one can use our
expression to find the limit of the concepts of energy density or fluid
element in connection to relativistic heavy-ion collisions, but also in other
areas of physics where one studies matter with high temperature and velocity.

###### pacs:

25.75.q, 24.10.Nz, 24.70.+s

day month yearday month year Quantum fluctuations, energy density,
relativistic heavy-ion collisions

##  1 Introduction

Quantum fluctuations of physical observables, which commonly arise from
quantum nature of the system [Huang:1987asp], play a crucial role because they
give out information about phase transitions [Smoluchowski,
PhysRevLett.85.2076, Gross:1980br, Huang:1987asp, Haussler:2008gx,
Herzog:2016kno, Vovchenko:2016dix, Steinheimer:2016cir, Lohr-Robles:2021ihl,
Bai:2021wrh, Fortunato:2021fby, Li:2021qcb, Yang:2021hmg, Sami:2021ufn,
deBrito:2021pmw], and large scale structure formation [Lifshitz:1963ps,
PhysRevLett.49.1110, Choudhury:2013woa, Choudhury:2016cso, Choudhury:2016pfr,
Choudhury:2021tuu, Graef:2021tid] as well as phenomena which are dissipative
in nature [Kubo1, Berges:1998rc, Halasz:1998qr, Stephanov:1998dy,
Stephanov:1999zu, Hatta:2002sj, Son:2004iv, Stephanov:2008qz,
Berdnikov:1999ph, CaronHuot:2011dr, Kitazawa:2013bta, Goswami:2020yez,
Fu:2021wyc, Pradeep:2021opj, Goswami:2021opr, Bollweg:2021vqf,
Schmidt:2021pey, Guo:2021qtf]. This work studies the quantum fluctuations of
energy in small systems (in other words, subsystems) of a hot and relativistic
gas [Das:2020ddr, Das:2021aar, Das:2021rck, Singh:2021tih, Singh:2021dbu].
These fluctuations increase for small size of the subsystem, but agrees with
the thermal fluctuations in the canonical ensemble if the size of the
subsystem is large enough. Our results can be very helpful in determining the
size of the subsystem (for specific particle mass and temperature) for which
quantum energy fluctuations reach classical limit and may be ignored in such
studies. We are specifically interested in the hot matter system which is
believed to be produced during the relativistic heavy-ion collisions, and
relativistic hydrodynamics has became a widely accepted theory to describe the
physics of heavy-ion collisions [Fukushima:2013rx, Jaiswal:2016hex,
Florkowski:2017olj, Fukushima:2018grm, Romatschke:2017ejr, Bhadury:2021oat].
Within the theory of relativistic hydrodynamics, we need to use the concepts
of energy density and pressure in order to characterize the fluid cells
locally, which give rise the question that how the definition of energy
density of such a small system can be defined properly having the size of
about 1 fm. In this article, we calculate and derive a formula to calculate
the quantum fluctuations and then use this expression for some particular
physical situations in the context of relativistic heavy-ion collisions. We
use the metric convention as $g_{\mu\nu}=\hbox{diag}(+1,-1,-1,-1)$. Bold font
is used to denote three-vectors and the scalar product of both four- and
three-vectors is represented by a dot, i.e., $a^{\mu}b_{\mu}=a\cdot
b=a^{0}b^{0}-\bm{a}\cdot\bm{b}$.

##  2 Basic definitions

We assume a subsystem $S_{a}$ of the larger thermodynamic system $S_{V}$ which
contain the spinless boson particles having mass $m$. This system $S_{V}$ is
defined by the canonical ensemble and specified by the temperature $T$, where
$\beta=1/T$. Note that, volume $V$ of $S_{V}$ is larger than the volume of
$S_{a}$, and $V$ is large enough for doing integrals over the momentum of the
particles. We characterize our bosonic system by a quantum scalar field which
is in thermal equilibrium as [Chen:2018cts]

$\phi(t,{\bm{x}})=\int\frac{d^{3}k}{\sqrt{(2\pi)^{3}~{}2\omega_{\bm{k}}}}\left(a_{\bm{k}}^{{\phantom{\dagger}}}e^{-ik\cdot
x}+a_{\bm{k}}^{\dagger}e^{ik\cdot x}\right),$ (1)

with $a_{\bm{k}}^{{\phantom{\dagger}}}$ and $a_{\bm{k}}^{\dagger}$ being the
annihilation and creation operators, respectively, and
$\omega_{\bm{k}}=\sqrt{{\bm{k}}^{2}+m^{2}}$ is the particle’s energy. For the
calculation of thermal averaging we need to know the following expectation
values of the products of two and four creation and/or annihilation operators
[Itzykson:1980rh, Evans:1996bha]. Other combinations of creation and
annihilation operators can be obtained easily from the following expression

$\displaystyle\langle
a^{\dagger}_{{\bm{k}}}a_{{\bm{k}}^{\prime}}^{{\phantom{\dagger}}}\rangle$
$\displaystyle=\delta^{(3)}({\bm{k}}-{\bm{k}}^{\prime})f(\omega_{{\bm{k}}}),$
(2) $\displaystyle\langle
a^{\dagger}_{{\bm{k}}}a^{\dagger}_{{\bm{k}}^{\prime}}a_{{\bm{p}}}^{{\phantom{\dagger}}}a_{{\bm{p}}^{\prime}}^{{\phantom{\dagger}}}\rangle$
$\displaystyle=\bigg{(}\delta^{(3)}({\bm{k}}-{\bm{p}})~{}\delta^{(3)}({\bm{k}}^{\prime}-{\bm{p}}^{\prime})+\delta^{(3)}({\bm{k}}-{\bm{p^{\prime}}})~{}\delta^{(3)}({\bm{k}}^{\prime}-{\bm{p}})\bigg{)}f(\omega_{{\bm{k}}})f(\omega_{{\bm{k}}^{\prime}}).$
(3)

where $f(\omega_{{\bm{k}}})$ is the Bose–Einstein distribution function,
$f(\omega_{{\bm{k}}})=1/(\exp[\beta~{}\omega_{{\bm{k}}}]-1)$.

Now we define [Chen:2018cts] the following operator $\mathcal{H}_{a}$ which
characterizes the energy density of a finite size subsystem $S_{a}$ which is
placed at the origin of the coordinate system where
$\mathcal{H}(x)\equiv(\dot{\phi}^{2}+({\bf{\nabla}\phi)}^{2}+m^{2}\phi^{2})/2$
is the Hamiltonian density for the free real scalar field

$\displaystyle\mathcal{H}_{a}=\frac{1}{(a\sqrt{\pi})^{3}}\int
d^{3}{\bm{x}}~{}\mathcal{H}(x)~{}\exp\left(-\frac{{\bm{x}}^{2}}{a^{2}}\right).$
(4)

We used above the Gaussian profile in order to remove boundary effects which
might come from the sharp boundaries.

Now one can easily obtain the thermal average of $\mathcal{H}_{a}$ as
[Huang:1987asp]

$\displaystyle\langle:\mathcal{H}_{a}:\rangle=\int\frac{d^{3}{{k}}}{(2\pi)^{3}}~{}\omega_{{\bm{k}}}~{}f\left(\omega_{{\bm{k}}}\right)\equiv\varepsilon(T).$
(5)

To arrive at the Eq. (5), we used normal ordering in order to remove the
infinite vacuum contribution coming from the zero-point energy term. Eq. (5)
is also independent of the size $a$ reflecting the spatial uniformity of the
thermodynamic system $S_{V}$. In order to calculate the quantum fluctuation of
energy of the subsystem $S_{a}$ we use the formula for variation and the
normalized standard deviation, given respectively as

$\displaystyle\sigma^{2}(a,m,T)$
$\displaystyle=\langle:\mathcal{H}_{a}::\mathcal{H}_{a}:\rangle-\langle:\mathcal{H}_{a}:\rangle^{2}\,,$
(6) $\displaystyle\sigma_{n}(a,m,T)$
$\displaystyle=\frac{(\langle:\mathcal{H}_{a}::\mathcal{H}_{a}:\rangle-\langle:\mathcal{H}_{a}:\rangle^{2})^{1/2}}{\langle:\mathcal{H}_{a}:\rangle}\,.$
(7)

Using Eqs. (2) and (3), we find

$\displaystyle\sigma^{2}(a,m,T)$ $\displaystyle=\int
dK~{}dK^{\prime}f(\omega_{{\bm{k}}})(1+f(\omega_{{\bm{k}}^{\prime}}))\bigg{[}(\omega_{{\bm{k}}}\omega_{{\bm{k}}^{\prime}}+{\bm{k}}\cdot{\bm{k}}^{\prime}+m^{2})^{2}e^{-\frac{a^{2}}{2}({\bm{k}}-{\bm{k}}^{\prime})^{2}}$
$\displaystyle+(\omega_{{\bm{k}}}\omega_{{\bm{k}}^{\prime}}+{\bm{k}}\cdot{\bm{k}}^{\prime}-m^{2})^{2}e^{-\frac{a^{2}}{2}({\bm{k}}+{\bm{k}}^{\prime})^{2}}\bigg{]}.$
(8)

where $dK=d^{3}{{k}}/((2\pi)^{3}2\omega_{{\bm{k}}})$ and neglected a
temperature independent divergent term which may be related to the pure vacuum
energy fluctuation [Phillips:2000jm].

Eq. (8) is the main result of ours which determine the quantum fluctuations of
energy of the “Gaussian” subsystems $S_{a}$ of the larger thermodynamic system
$S_{V}$. Numerically we can calculate the results for any system of size $a$,
temperature $T$, and mass $m$, the plots of which will be shown below.

One should also keep in mind the factor of degeneracy which is important when
we study the thermodynamic properties of particles. This degeneracy factor is
related to the internal degrees of freedom, for instance, spin, isospin or
color charge. Therefore to take into consideration in our framework, the $g$
copies of particles, one needs to consider the $g$ copies of scalar field
which is given by both creation and annihilation operators which commute for
different species of particles. Hence, the following replacements are required
for the calculation of the quantum fluctuations

$\varepsilon\rightarrow g\varepsilon,\quad\sigma^{2}\rightarrow g\sigma^{2}.$
(9)

##  3 Thermodynamic limit

We believe that in the limit of large system size, that is,
$a\rightarrow\infty$ (but still with $a^{3}\ll V$), our quantum fluctuation
expression should agree with classical statistical mechanics [Huang:1987asp].
Hence, we use the following representation of the Dirac delta function in Eq.
(8)

$\displaystyle\delta^{(3)}({\bm{k}}-{\bm{p}})=\lim_{a\to\infty}\frac{a^{3}}{(2\pi)^{3/2}}e^{-\frac{a^{2}}{2}({\bm{k}}-{\bm{p}})^{2}}.$
(10)

which give rise to the formula valid in the large size $a$ limit

$\displaystyle\sigma^{2}$
$\displaystyle\sim\frac{g}{(2\pi)^{3/2}a^{3}}\int\frac{d^{3}{{k}}}{(2\pi)^{3}}~{}\omega_{{\bm{k}}}^{2}~{}f(\omega_{{\bm{k}}})(1+f(\omega_{{\bm{k}}})).$

Note that we calculate the quantum fluctuation for massless particles as
$\sigma^{2}\sim T^{5}/a^{3}$, therefore RHS of the last equation can be
written as

$c_{V}=\frac{d\varepsilon}{dT}=\frac{g}{T^{2}}\int\frac{d^{3}{{k}}}{(2\pi)^{3}}~{}\omega_{{\bm{k}}}^{2}~{}f(\omega_{{\bm{k}}})(1+f(\omega_{{\bm{k}}})).$
(11)

where $c_{V}$ is the specific heat at constant volume. Hence, in large $a$
limit [Mrowczynski:1997mj]

$\displaystyle
V_{a}\sigma_{n}^{2}=\frac{T^{2}c_{V}}{\varepsilon^{2}}=V\frac{\langle
H^{2}\rangle-\langle H\rangle^{2}}{\langle H\rangle^{2}}\equiv
V\sigma^{2}_{H},$ (12)

with $V_{a}=a^{3}(2\pi)^{3/2}$ and $H$ being the Hamiltonian of $S_{V}$, and
RHS of Eq. (12) is the normalized energy fluctuation in the thermodynamical
system $S_{V}$ [Huang:1987asp]. Here, $V_{a}=a^{3}(2\pi)^{3/2}$ is the volume
of $S_{a}$.

Figure 1: (left) Variation of the normalized energy density fluctuation
$\sigma_{n}$ in the subsystem $S_{a}$ with size $a$ for various values of the
temperature $T$ but with fixed particle mass $m=0.3$ GeV. (right) Same as left
figure but for fixed temperature $T=0.15$ GeV and different particles masses.

##  4 Numerical results

Since we correctly reproduced thermodynamic limit for $a\rightarrow\infty$, we
can do some numerical calculations. Fig. (1) show the variation of the
normalized energy density fluctuation $\sigma_{n}$ having the size $a$ for the
subsystem with various values of temperature and mass of the system. We use
degeneracy factor to be $g=40$ in order to be consistent with the heavy-ion
experiments [Kisiel:2005hn]. Fig. (1) indicate that $\sigma_{n}$ decreases
with increasing the system size $a$ as expected from the behavior of
fluctuations. Since, from Eq. (8) we see that $\sigma_{n}$ explodes in the
limit $a\to 0$ hence we start our numerical calculations at $a=0.3$ fm. Having
fixed mass of the particle we find the decrease in the normalized fluctuations
with the increase in temperature. On the other hand, at fixed $T$ the
fluctuations grow with $m$.

Figure 2: (left) Variation of the normalized energy fluctuation in the
subsystem $S_{a}$ with the size $a$ for various values of the temperature at
$m=0.3$ GeV. (right) Same as left figure but for different values of mass at
$T=0.15$ GeV.

Fig. (2) presents the variation of $V_{a}\sigma_{n}^{2}/V\sigma^{2}_{H}$ with
the subsystem size $a$ for particles with mass $m$ obeying Bose-Einstein
statistics. From Eq. (12) we see that it approaches the thermodynamic limit
[Kapusta:2011gt] and hence $V_{a}\sigma_{n}^{2}/V\sigma^{2}_{H}$ should go to
unity, which is seen from Fig. (2).

##  5 Conclusions

In this work we have derived the expression for the quantum fluctuation of
energy for the subsystems of a hot and relativistic gas which beautifully
agrees with the thermodynamic fluctuation expression for the large system
size. We also explained and outlined the possible consequences of our results
for the description of relativistic heavy-ion systems.

I am very grateful and thankful to A. Das, W. Florkowski, and R. Ryblewski for
their beautiful collaboration. This research was supported in part by the
Polish National Science Centre Grants No. 2016/23/B/ST2/00717 and No.
2018/30/E/ST2/00432, and IFJ PAN.


# Holstein polarons, Rashba-like spin splitting and Ising superconductivity in
electron-doped MoSe2

Sung Won Jung<EMAIL_ADDRESS>Diamond Light Source, Harwell Campus,
Didcot, OX11 0DE, United Kingdom  Department of Physics and Research Institute
of Molecular Alchemy, Gyeongsang National University, Jinju 52828, Republic of
Korea  Saumya Mukherjee  Diamond Light Source, Harwell Campus, Didcot, OX11
0DE, United Kingdom  Van der Waals–Zeeman Institute, Institute of Physics,
University of Amsterdam, 1098 XH, Amsterdam, Netherlands  Matthew D. Watson
Diamond Light Source, Harwell Campus, Didcot, OX11 0DE, United Kingdom  Daniil
V. Evtushinsky  École Polytechnique Fédérale de Lausanne, CH-1015 Lausanne,
Switzerland  Cephise Cacho  Diamond Light Source, Harwell Campus, Didcot, OX11
0DE, United Kingdom  Edoardo Martino  École Polytechnique Fédérale de
Lausanne, CH-1015 Lausanne, Switzerland  Helmut Berger  École Polytechnique
Fédérale de Lausanne, CH-1015 Lausanne, Switzerland  Timur K. Kim
<EMAIL_ADDRESS>Diamond Light Source, Harwell Campus, Didcot, OX11
0DE, United Kingdom

###### Abstract

Interaction between electrons and phonons in solids is a key effect defining
physical properties of materials such as electrical and thermal conductivity.
In transitional metal dichalcogenides (TMDCs) the electron-phonon coupling
results in the creation of polarons, quasiparticles that manifest themselves
as discrete features in the electronic spectral function. In this study, we
report the formation of polarons at the alkali-dosed MoSe2 surface, where
Rashba-like spin splitting of the conduction band states is caused by an
inversion-symmetry breaking electric field. In addition, we observe the
crossover from phonon-like to plasmon-like polaronic spectral features at
MoSe2 surface with increasing doping. Our findings support the concept of
electron-phonon coupling mediated superconductivity in electron-doped layered
TMDC materials, observed using ionic liquid gating technology. Furthermore,
the discovered spin-splitting at the Fermi level could offer crucial
experimental validation for theoretical models of Ising-type superconductivity
in these materials.

dichalcogenides, polarons, superconductivity

##  I Introduction

Polarons are collective excitation of electrons interacting with the lattice.
These quasiparticles describe an electron moving in a crystal with positively
charged ions displacing from their equilibrium positions to effectively screen
negative electron charge Landau (1933); Landau and Pekar (1948). This coupling
between the electron and lattice vibrations (phonon cloud) lowers the electron
mobility and increases the electron’s effective mass.

The resulting polaronic bound state can be described by Fröhlich or Holstein
Hamiltonians. The Fröhlich model describes the interaction of electrons with
optical phonons, which leads to the formation of large polarons, whose radii
are much larger than the lattice constant. The Holstein model, on the other
hand, describes the interaction with acoustic phonons, which leads to the
formation of small polarons, whose radii are of the same order of magnitude as
the lattice constant Emin (2013).

Formation of the polarons due to a momentum dependent electron-phonon coupling
leads to new features in the electronic spectral function at specific electron
valleys, e.g. the emergence of discrete energy levels and discontinuities in
the density of states Franchini  _et al._ (2021). In a recent study of alkali-
metal dosing on MoS2 surface such discontinuities in density of states have
been experimentally observed at the ${\rm K}$ points of the Brillouin zone
using angle-resolved photoemission spectroscopy (ARPES) Kang  _et al._ (2018).
Understanding the changing nature of the quasiparticles as a function of
doping is an important prerequisite for understanding the onset of
superconductivity observed in highly electron-doped MoS2 under liquid ionic
gating Ye  _et al._ (2012); Lu  _et al._ (2015).

Theoretical considerations point to two key novelties in the case of multi-
valley electron-doped TDMCs. First, polaron modes should be observed around
both ${\rm K}$ and $\Sigma$ valleys in MoS2 Garcia-Goiricelaya  _et al._
(2019).

Figure 1: Phonon-mediated polarons in electron doped MoSe2: (a) Multi-valley
scattering of conduction electrons in the first Brillouin zone of MoSe2; (b)
Electron-like parabolic band at ${\rm K}$-valley with Holstein polarons
simulated by momentum-average model Goodvin  _et al._ (2006).

Figure 1(a) shows the multi-valley scattering between $\Sigma$ and ${\rm K}$
valleys in the Brillouin zone. The electron scattering vectors $q_{1}$ and
$q_{2}$ correspond to either $\Gamma$-${\rm K}$ or $\Gamma$-${\rm M}$ phonon
wave vectors. For both MoS2 and MoSe2 the acoustic phonon modes have similar
energies for ${\rm K}$ and ${\rm M}$ and the dominant contribution to
electron-phonon coupling arises from the highest energy acoustic phonon mode.
This behaviour has been demonstrated both experimentally Zeng  _et al._ (2012)
and theoretically Kaasbjerg  _et al._ (2012) for MoS2. However, only ${\rm K}$
valley Holstein polarons (Fig. 1(b)) have been observed in the ARPES study of
Rb-dosed MoS2 surface Kang  _et al._ (2018). Second, the spin-orbit coupling
(SOC) should induce the splitting of the electron bands at $\Sigma$ and
$\Sigma$’ valleys due to inversion symmetry breaking on the surface with
alkali metal dosing. However, in all previous ARPES studies of alkali-dosed
semiconducting TMDCs the spin-split features at $\Sigma$ valley have not been
resolved Zhang  _et al._ (2014); Eknapakul  _et al._ (2014); Alidoust  _et
al._ (2014); Riley  _et al._ (2015); Miwa  _et al._ (2015); Zhang  _et al._
(2016); Kim  _et al._ (2016, 2017); Kang  _et al._ (2017, 2018); Han  _et al._
(2021).

Figure 2: Evolution of the MoSe2 electronic structure with Rb dosing: (a-a5)
ARPES spectra of $\Gamma$-${\rm K}$ conduction band dispersions measured near
the Fermi energy for dosing 0, 20, 60, 100, 220, 580s. (b-b5) Corresponding
valence band spectra in a wide energy range.

The isostructural MoSe2 compound is the perfect model system for probing the
multi-valley physics induced by electron-doping. Density functional theory
(DFT) calculations show that in contrast to MoS2, the conduction band minimum
(CBM) in bulk MoSe2 is at the $\Sigma$, rather than at ${\rm K}$ valley (see
Supplementary Fig. S1). For the alkali-metal dosing on MoSe2 surface one
expects to have larger SOC band-splitting as in the case of MoS2. The
accessibility of the $\Sigma$ valley, as well as the ${\rm K}$ valley expected
at higher doping Kim  _et al._ (2016, 2017), makes MoSe2 the best candidate to
study physics of multi-valley polaron modes using high-resolution ARPES.

##  II Results and Discussion

###  II.1 Evolution of the electronic band structure with dosing

In our ARPES data shown in Fig. 2 the spectral intensity appears at the Fermi
level, marking the occupation of the conduction band and corresponding
semiconductor to conductor transition with the Rb dosing on the MoSe2 surface.
After initial dosing the maximum spectral weight at the Fermi level is
observed around $\Sigma$ valley, showing an indirect band gap
$E_{g}^{\Gamma-\Sigma}$ $\sim$1.4 eV. However, with increasing amount of dosed
Rb the relative intensity of $\Sigma$ valley decreases, whereas the intensity
of ${\rm K}$ valley increases. Correspondingly, it appears that the conduction
band minimum shifts from $\Sigma$ to ${\rm K}$ valley as in case of monolayer
MoSe2 (see Fig. 2(a1-a3)). Withing observed energy broadening the band minima
at $\Sigma$ and ${\rm K}$ are very close (see Supplementary Fig. S5),
incidentally making electron doped MoSe2 perfect system for studying multi-
valley physics. Simultaneously with the shift of the CBM intensity, the
intensity of the valence band compared to the conduction band decreases (see
Fig. 2 (b1-b3)). This first crossover at around 40 s of Rb dosing marks the
transition from the electron doped bulk to the more “monolayer-like” doped
surface conduction band structure Eknapakul  _et al._ (2014); Riley  _et al._
(2015); Kang  _et al._ (2017); Kim  _et al._ (2017).

The accumulation of alkali atoms on the surface and increased electron doping
of the top-layer causes a corresponding increase of the electric field, and
results in the amplification of the inversion symmetry-breaking at the
surface. At dosing above this first crossover the measured electronic
structure is a superposition of the bands from both the bulk material and the
doped surface. This observation is supported by our Se 3d core level and ${\rm
K}$-valley valence band data (see Supplementary Fig. S4 and Fig. S5
correspondingly).

At Rb dosing times above 120 s the relative intensity of the valence bands
increases due to the interlayer intercalation of Rb atoms Eknapakul  _et al._
(2014). At the same time the energy difference between bulk and surface Se 3d
core levels or “effective chemical shift” saturates and does not increase with
Rb deposition time (see Supplementary Fig. S4). This corresponds to the second
crossover from surface doping to interlayer intercalation regime. For dosing
times above 340 s one start to observe build up of the photoemission intensity
at the Fermi level around $\Gamma$-point as seen in Fig. 2(a5). This
corresponds to the formation of the metallic Rb states on the MoSe2 surface
Kim  _et al._ (2017) and the third crossover from interlayer intercalation
regime to growth of ordered alkali metal layer on the surface.

###  II.2 Holstein polarons in “surface doping” regime

The most intriguing effects in the electronic structure are observed in the
“surface doping” regime, corresponding to dosing times between first and
second crossover. Figure 3(a,b) shows the electronic band structure of MoSe2
at Fermi level at 120 s Rb-dosing on the surface.

Figure 3: Formation of polarons in the electron-doped MoSe2: (a) Fermi surface
and (b) $\Sigma$-${\rm K}$ band dispersion measured after 120 s Rb dosing;
Corresponding second derivative of ARPES intensity at $\Sigma$ (c) and ${\rm
K}$ (d) valleys.

The measured Fermi surface shows two electron-like pockets at $\Sigma$ valley
and single electron-like pocket at ${\rm K}$ valley. The second derivative
images of $\Sigma$ and ${\rm K}$ valleys in Fig. 3(c,d) both show well-defined
discontinuities of the spectral function separated by 20 meV. Moreover, the
photoemission spectra integrated the momentum range between $\Sigma$ and ${\rm
K}$ valleys in Fig. 5(a), show discrete non-dispersing energy levels with the
same energy separation (dashed vertical black lines in Fig. 5(b)).

The energy of these spectral features is consistent with the calculated
acoustic phonon mode, and their non-dispersing character in an extended
momentum space, all point toward formation of small range polarons.Therefore,
for the electron doped MoSe2 Holstein type polarons are observed at both ${\rm
K}$ and $\Sigma$ valleys.

###  II.3 Spin-splitting at the $\Sigma$ valley in “surface doping” regime

While polarons at the spin-degenerate ${\rm K}$-valley has been observed
previously for MoS2 Kang  _et al._ (2018), polarons at the spin-polarised
$\Sigma$ valleys are discovered for the first time. The spin-orbit splitting
of the electron band dispersion at $\Sigma$ valley is apparent in ARPES data
measured along the $\Gamma$-${\rm K}$ direction. Figure 4 shows band
dispersions at the $\Sigma$ valley together with the Rashba-like model fit
constrained by the band positions from the corresponding momentum distribution
curves (MDCs) at the Fermi level.

Figure 4: Evolution of the spin-orbit coupling with Rb dosing: (a-e) Band
dispersion at $\Sigma$ valley with Rb dosing. White doted lines are fits with
Rashba-like model with $\Gamma$ been time-reversal invariant momentum (TRIM)
point. (f) Extracted spin-split energy at $\Sigma$ valley.

The increase of the Fermi surface size indicates the gradual electron doping
of MoSe2 with Rb dosing. From the same data one could see that the magnitude
of band-splitting is also increasing with electron doping.This Rashba-like
spin-splitting of the bands comes from inversion symmetry breaking due to the
electric field created by electron charge transfer from dosed alkali atoms at
the surface. Such inversion symmetry breaking on the surface has been seen at
the valence band of semiconducting Kang  _et al._ (2017); Kim  _et al._ (2017)
as well as at the conduction band of semimetallic Clark  _et al._ (2019)
alkali dosed TMDC materials. Figure 4(f) shows how the magnitude of the
observed splitting consistently and monotonically becomes larger as the
surface charge increases. This observation aligns perfectly with our
understanding that the band splitting is Rashba-like and arises from the
combination of spin-orbit coupling with the increasing inversion symmetry-
breaking electric field at the surface.

###  II.4 Plasmonic flat states in intercalation regime

At dosing times above 340 s one observes a formation of the flat states in the
vicinity of the CBM with much higher binding energies than for Holstein
polaronic modes (white arrow at Fig. 5(a)).

Figure 5: Crossover from polarons to plasmons in the MoSe2 electronic
structure with Rb dosing: (a) Wide energy range $\Gamma$-${\rm K}$ band
dispersion measured after 340 s Rb dosing. A white arrow indicates a flat
state between $\Sigma$ and ${\rm K}$ valleys; (b) Integrated Energy
Distribution Curves (EDCs) between $\Sigma$ and ${\rm K}$ valleys (yellow box
in (a)).

To emphasize the difference between these new flat states and Holstein
polarons, we analysed momentum integrated EDCs between $\Sigma$ and ${\rm K}$
valleys in Fig.5(a). Fig. 5(b) shows that indeed two different features are
visible in the integrated EDCs. The characteristic binding energies of the
polaronic states in the “surface doping” regime are constant and correspond to
the acoustic phonon frequency $\simeq$20 meV, but the higher binding energies
of the flat states in the “interlayer intercalation” regime significantly
increase with dosing. While binding energies of these flat states are an order
of magnitude larger that of Holstein polarons, they are also an order of
magnitude smaller then the 3.4 eV bulk plasmon frequency of metallic Rb Vom
Felde  _et al._ (1987). Binding energies of the flat states as a function 2D
electron carrier density can be fitted with plasmon-like dispersion
$\hbar\omega_{p}=\sqrt{(n_{e}^{\Sigma}+n_{e}^{K})e^{2}/m\epsilon}$ . The
obtained dielectric constant $\epsilon$$\simeq$14.3 is comparable with the
calculated in-plane dielectric constant for monolayer MoSe2 Laturia  _et al._
(2018). This suggests that these flat states may indeed be plasmonic polarons,
similar to the case of EuO Riley  _et al._ (2018) and anatase TiO2 Ma  _et
al._ (2020). Plasmonic polarons have been observed by ARPES in electron doped
MoS2 Caruso  _et al._ (2021), MoS2/TiO2 Xiang  _et al._ (2023), HfS2 Emeis
_et al._ (2023) and WS2 Ulstrup  _et al._ (2023).

##  III Conclusion

By studying electronic structure evolution of MoSe2 with Rb surface dosing we
have identified three different regimes: surface doping, inter-layer
intercalation and ordered alkali metal layer formation. Figure 6(a) shows the
two dimensional (2D) electron carrier density as a function of the alkali
metal dosing time, calculated using the Luttinger’s theorem and experimentally
determined sizes of the Fermi surface. In the “surface doping” regime we
observe continuous electron doping of MoSe2 with corresponding increase of the
Fermi surface size and value of the Rashba-like spin-orbit splitting.

Figure 6: Phase diagram and Fermi surface of MoSe2 with electron doping: (a)
2D electron carrier density as a function of the Rb-dosing time; (b) Measured
Fermi surface of the electron doped MoSe2 after 140 s Rb dosing. (c) Fermi
surface extrapolated for maximum Tc $\sim$7 K carrier density doping level Shi
_et al._ (2015); (d,e, f) Schematic of the crystal structure for surface
dosing, intercalation and metallic Rb regimes. L1 and L2 notes topmost and
second topmost layers, correspondingly.

The changing nature of the quasiparticles at the Fermi level due to electron-
phonon coupling, as well as the evolution of the Fermi surface topology and
the electronic density of states, plays a crucial role in understanding
superconductivity in electron-doped TMDC semiconductors discovered using
liquid gating devices Taniguchi  _et al._ (2012); Ye  _et al._ (2012); Shi
_et al._ (2015); Lu  _et al._ (2015). In the case of MoSe2 the
superconductivity occurs at high 2D electron carrier densities
$\geq$8$\cdot$1013 cm-2 and maximum Tc $\sim$7 K is achieved at electron
density $\simeq$16.9$\cdot$1013 cm-2, correspondingly Shi  _et al._ (2015).

While our results show that the increasing surface electric field description
is no longer applicable when we dose Rb atoms above 120 s, we could
extrapolate a gradual increase of the 2D electron carrier density with the
alkali metal dosing time within the ”surface doping” regime up to the
superconducting transition. In this case, using our ARPES data we discover
that maximum Tc doping level corresponds to the Fermi surface topology where
the electron-like pocket at ${\rm K}$ valley touches the outer electron pocket
at $\Sigma$ valley (see Fig. 6(b)). In another words, the maximum Tc
corresponds to the Fermi level being located exactly at the conduction band
saddle point between ${\rm K}$ and $\Sigma$ valleys. This can lead to a
dramatic enhancement of pairing strength, due to strong increase of the
momentum resolved electron-phonon coupling constant $\lambda$, like in case of
Rb intercalated PtTe2 Wu  _et al._ (2021). It can be assumed that with a
further increase in doping, the electronic structure will undergo a Lifshitz
transition, significantly reducing the density of states at the Fermi level
and, correspondingly, suppressing electronic pairing. This can naturally
explain the observed superconducting dome for Tc vs. doping phase diagram Ye
_et al._ (2012); Shi  _et al._ (2015).

Using ionic liquid gating devices, it was found that for electric-field-
induced superconductivity in MoS2 the observed upper critical field is four-
five times exceeds the Pauli limit Lu  _et al._ (2015); Saito  _et al._
(2016). This violation of the paramagnetic limit and significant spin-orbit
coupling at the Fermi level suggest a non-spin-singlet pairing that is robust
against an external magnetic field. Ising pairing for the valley alternating
spin-split Fermi surface has been suggested in a several theoretical works for
non-centrosymmetric TMDC superconductors Zhou  _et al._ (2016); Ilić  _et al._
(2017); Wang  _et al._ (2019); Liu  _et al._ (2020); Wickramaratne  _et al._
(2020); Semenov (2024). However, all these models consider basic Fermi surface
topology with spin-degenerate states at ${\rm K}$ valleys only. At the same
time our study shows the direct evidence of Rashba-like splitting of the bands
at $\Sigma$ valley for such in-situ electron doped materials.

Important consequence of this discovery is that the significant part of the
total Fermi surface will be spin split at the optimal doping. This agrees with
the theoretical prediction Zhao  _et al._ (2020) that with anti-symmetric
field effect doping the resulting Fermi surface is dominantly formed by the
pockets around $\Sigma$ valleys (only $\sim$30% of its area is around ${\rm
K}$ valleys as shown in Fig.6(b)). Thus any realistic pairing model of Ising
superconductivity in electron doped TMDCs should include such composite spin-
split multi-valley Fermi surface topology and corresponding interband
scattering processes.

To summarise, we observed spin-polarised Holstein type polaronic
quasiparticles at Rb dosed MoSe2 surfaces. This is a direct evidence of the
predicted Rashba-like spin-orbit split polarons Covaci and Berciu (2009). We
found that beyond the ”surface doping” regime threshold, when dosed alkali
atoms intercalate into the van der Waals (vdW) gap between the layers, new
plasmon-like quasiparticles emerge. Our findings not only suggest the crucial
role of polaron formation for the superconductivity in electron-doped
transition-metal dichalcogenides, but also provide experimental foundation for
theoretical models of Ising-type superconductivity in these materials.

##  IV Methods

High quality single crystals of MoSe2 were grown by the chemical vapor
transport technique Legma  _et al._ (1993). ARPES measurements were performed
at the I05 beamline at the Diamond Light Source, UK Hoesch  _et al._ (2017).
The photoelectron’s energy and angular distributions were analysed with a
SCIENTA R4000 hemispherical analyser. The angular resolution was 0.2∘, and the
overall energy resolution was better than 5 meV. P-polarised 60 eV photons
corresponding to the $\Gamma$ point of the bulk Brillouin zone were used for
high-resolution ARPES, and 120 eV photons for Se 3d core level measurements.
Single crystal samples were cleaved and measured in ultra high-vacuum at
temperatures below 10 K and pressures below 1$\cdot$10-10 mbar. Rb atoms were
dosed in several sequences on a cold MoSe2 surface using a commercial SAES
alkali-metal dispenser. The density functional theory (DFT) calculations
including spin-orbit coupling with generalized gradient approximation (GGA)
were performed using WEIN2K package Blaha  _et al._ (2020). Spectral function
simulation for polarons has been done by using momentum average (MA)
approximation Goodvin  _et al._ (2006) with 20 meV acoustic phonons.

##  V Acknowledgements

We acknowledge Diamond Light Source for time on Beamline I05-ARPES under
Proposal NT26631. S. W. Jung was supported by the National Research Foundation
of Korea (NRF) grant funded by the Korea government (MSIT) (No.
NRF-2022R1C1C1004978) and supported by Learning & Academic research
institution for Master’s·PhD students, and Postdocs (LAMP) Program of the
National Research Foundation of Korea (NRF) grant funded by the Ministry of
Education (No. RS-2023-00301974).

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## Supplementary

### Density Functional Theory (DFT) calculations

Band structure DFT calculations including spin-orbit coupling show that bottom
of the conduction band at $\Sigma$-valley is lower for bulk MoSe2 then for
bulk MoS2 and lower then bottom of the conduction band at ${\rm K}$-valley.
This makes MoSe2 a much better choice to study polaron formation at
$\Sigma$-valley. Importantly, in case of broken inversion symmetry, due to
electric field built up on the surface caused by the alkali metal dosing, one
expect corresponding spin-splitting at $\Sigma$-valley, like in the case of
monolayer MoSe2.

Figure S1: DFT+SOC calculated electronic structure of bulk MoS2, bulk MoSe2
and monolayer MoSe2.

### The evolution of MoSe2 electronic structure with Rb dosing.

Photoemission data for conduction (Fig. S2) and valence (Fig. S3) bands
together with corresponding Se 3d core levels data (Fig. S4) were measured for
all Rb dosing sequences.

Figure S2: Conduction band electronic structure at the Fermi-level of Rb dosed
MoSe2 surface. Figure S3: Valence band electronic structure of Rb dosed MoSe2
surface. Figure S4: Se 3p core-levels with Rb dosing. (a) Se 3d5/2/3d3/2 XPS
data with Voight profile doublets fit for various dosing times. Red and blue
curves indicate bulk (P1) and surface (P2) components; (b) Ratio of the bulk
and surface components versus Rb dosing time; (c) Energy difference between
bulk and surface components versus Rb dosing time; The background colour in
(b) and (c) indicates surface dosing, intercalation and Rb metallisation
regimes.

A single 3d5/2/3d3/2 doublet for a pristine MoSe2 crystal splits into two
components with Rb dosing as shown in Fig. S4(a). Similar to the case of
HfTe2El Youbi  _et al._ (2020) one could fit these core level data assuming
bulk (P1) and surface (P2) components due to the electron doping of the MoSe2
surface. Obtained peak area ratio and energy differences (or effective
chemical shift) are shown in Fig. S4(b,c). Below critical dosing of 120 s one
observes increase of the difference between bulk and surface doublets binding
energies with dosing time. This corresponds to the gradual accumulation of Rb
atoms, increased electron doping of the MoSe2 surface, and corresponding
increase of effective electric filed.

### Doping evolution of the band gap in MoSe2

Similar to the previously published ARPES data of alkali-dosed MoSe2 Kim  _et
al._ (2017); Kang  _et al._ (2017) we observe superposition of the bulk and
surface electronic states. This is especially evident at top of valence band
at ${\rm K}$-valley where one detects weak surface related features at lower
binding energies than bulk ones. At higher doping levels this might lead to a
shift of the valence band maxima (VBM) from $\Gamma$ to ${\rm K}$ valley and
corresponding transition from indirect to direct band gap electronic
structure. This has been clearly observed by ARPES in the case of MoTe2 Kang
_et al._ (2017). However, in the case of MoSe2 this is less obvious from the
ARPES data Kim  _et al._ (2017); Kang  _et al._ (2017).

In Fig. S5(b,c) photoemission peak maxima of the conduction band for both
$\Sigma$ to ${\rm K}$ valleys shift towards higher binding energies, away from
the Fermi level. However withing experimental broadening of corresponding
spectra, one could not with absolute certainty conclude, that conduction band
minima is shifting from $\Sigma$ to ${\rm K}$ valley. Therefore, despite
obvious spectral intensity shift from $\Sigma$ to ${\rm K}$ valley in Fig. S2,
we could not presume the crossover to a monolayer conduction band dispersion
in alkali-metal dosed MoSe2. Correspondingly, in Fig. S5(e,f) photoemission
peak maxima of the valence band for both $\Gamma$ to ${\rm K}$ valleys shift
towards Fermi level with dosing. Withing experimental broadening of
corresponding spectra, one could not determine, that valence band maxima is
shifting from $\Gamma$ to ${\rm K}$ valley. The lower intensity surface
related features might be especially well masked by the bulk component at
$\Gamma$ valley as compared to ${\rm K}$ valley as seen in Fig. S7. Additional
broadening of the spectra due to significant kz dispersion at the centre of
the Brillouin zone make distinguishing bulk and surface components there even
more challenging. We can not exclude electron doping induced indirect to
direct band gap transition in MoSe2, however this might be beyond ”surface
doping” regime achievable with alkali metal dosing at the surface.

Figure S5: Evolution of the conduction band minima and valence band maxima
with Rb dosing: (a) Conduction bands dispersion along $\Gamma$-${\rm K}$
direction; (b) Dispersion of conduction band minima with surface doping
obtained from EDC at $\Sigma$ point; (c) Dispersion of conduction band minima
with surface doping obtained from EDC at ${\rm K}$ point; (d) Valence bands
dispersion along $\Gamma$-${\rm K}$ direction. L1 and L2 notes topmost and
second topmost layers, correspondingly; (e) Dispersion of Valence band maxima
with surface doping obtained from EDC at $\Sigma$ point; (f) Dispersion of
Valence band maxima with surface doping obtained from EDC at ${\rm K}$ point;
Figure S6: Evolution of conduction band minima with Rb dosing: (a) Energy
distribution curves (EDCs) at $\Sigma$ point; (b) Energy distribution curves
(EDCs) at ${\rm K}$ point. Figure S7: Evolution of valence band maxima with Rb
dosing: (a) Energy distribution curves (EDCs) with fits at $\Gamma$ point; (b)
Energy distribution curves (EDCs) with fits at ${\rm K}$ point.

Figure S8: Constant energy maps (hv=180eV) of MoSe2 surface after 140 s Rb-
dosing:

(a) Momentum distribution map at Fermi level and (b) Momentum distribution map
at the top of the valence band.

### Rashba-like splitting at $\Sigma$ valley

To analyse spin-splitting at $\Sigma$ valley as a function of surface doping
we used Rashba-like model with $\Gamma$ as time-reversal invariant momenta
(TRIM) point shown in Fig. S10(a).

The effective Hamiltonian can be written as:

$H(\vec{k})=\frac{\hbar^{2}}{2m^{*}}(\vec{k}-\vec{k_{0}})^{2}\pm\alpha_{R}(\hat{n}\times\vec{\sigma})\cdot\vec{k}+E_{offset}$
(1)

where k0 is the momentum of the conduction band minimum near $\Sigma$ without
spin-orbit coupling, $\sigma$ is Pauli matrices, $E_{offset}$ is the chemical
shift due to the electron doping via Rb dosing and m∗ is the effective mass.

From equation (1), one can make the system of equations with four $k_{F}$
(S1,S2,S3 and S4) shown in Fig. S10.

$\displaystyle\frac{\hbar^{2}}{2m^{*}}(k_{S1}-k_{0})^{2}-\alpha_{R}k_{S1}+E_{offset}=0$
(2)
$\displaystyle\frac{\hbar^{2}}{2m^{*}}(k_{S2}-k_{0})^{2}+\alpha_{R}k_{S2}+E_{offset}=0$
(3)
$\displaystyle\frac{\hbar^{2}}{2m^{*}}(k_{S3}-k_{0})^{2}+\alpha_{R}k_{S3}+E_{offset}=0$
(4)
$\displaystyle\frac{\hbar^{2}}{2m^{*}}(k_{S4}-k_{0})^{2}-\alpha_{R}k_{S4}+E_{offset}=0$
(5)

By subtracting the equation (5) with the equation (2), one can make

$\frac{\hbar^{2}}{2m^{*}}(k_{S1}+k_{S4}-2k_{0})-\alpha_{R}=0$ (6)

Likewise, one can subtract equation (4) from equation (3).

$\frac{\hbar^{2}}{2m^{*}}(k_{S2}+k_{S3}-2k_{0})+\alpha_{R}=0$ (7)

From equation (6) and (7), one can derive $k_{0}$ from four $k_{F}$ values

$k_{0}=\frac{k_{S1}+k_{S2}+k_{S3}+k_{S4}}{4}$ (8)

By replacing $k_{0}$ in equation (6) and (7), one can derive effective Rashba
parameter $\alpha_{R}$

$\alpha_{R}=\frac{\hbar^{2}}{2m^{*}}\frac{(k_{S1}+k_{S4}-k_{S2}-k_{S3})}{2}$
(9)

From equations (1), (8) and (9), the splitting energy $E_{splitting}$ can be
derived as

$E_{splitting}=2\alpha_{R}k_{0}-\frac{2m^{*}\alpha_{R}^{2}}{\hbar^{2}}$ (10)

To fit ARPES data, we used constant DFT value for the effective mass
$m^{*}=0.54m_{e}$ at $\Sigma$ valley and $k_{F}$ values obtained from MDC fits
shown in Fig. S9.

Figure S9: Momentum distribution curves (MDCs) of $\Sigma$ valleys at Fermi-
level.

The effective Rashba parameter $\alpha_{R}$ versus Rb dosing time is shown in
Fig. S10(b). Corresponding extracted spin-split energy $E_{splitting}$ at
$\Sigma$ valley as a function of Rb dosing time is shown in the main text Fig.
4.

Figure S10: Spin-splitting at $\Sigma$ valley: (a) Rashba-like model with
$\Gamma$ as time-reversal invariant momentum (TRIM) point; (b) Extracted
Rashba parameter ($\alpha_{R}$).

.

For comparison, for Au(111) and Cu(111) surface states measured Rashba
parameter $\alpha_{R}$ = 330 and 38 meVÅ give energy splitting $E_{splitting}$
= 110 and 16 meV, correspondingly.

…