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# Electron Interactions in Rashba Materials

Yasha Gindikin  Alex Kamenev  W.I. Fine Theoretical Physics Institute,
University of Minnesota, Minneapolis, MN 55455, USA

###### Abstract

We show that Rashba spin-orbit coupling modifies electron-electron interaction
vertex leading to a number of novel phenomena. First, the spin-orbit-modified
Coulomb interactions induce $p$-wave superconducting order, without any other
mediators of attraction. Second, a sufficiently strong spin-orbit coupling
results in a ferromagnetic order, associated with a Lifshitz transition from a
spherical to a toroidal Fermi surface. Such topology-changing transition leads
to distinct experimentally observable consequences. Finally, in sufficiently
clean Rashba materials, the ferromagnetism may coexist with the $p$-wave
superconductivity.

##  I Introduction

The study of spin-orbit interaction has been a fascinating area of research in
condensed matter physics for several decades [1, 2, 3]. Rashba spin-orbit
interaction (RSOI) offers a new way to control spin states using purely
electrical means rather than magnetic fields. This presents a massive
advantage for spintronics, a field of study that aims to exploit the quantum
spin degree of freedom [4, 5] to develop new computational and data storage
technologies [6, 7, 8].

In recent years, the discovery of materials with giant RSOI has opened up new
avenues for investigating exotic quantum phenomena and exploring potential
applications in spintronic devices [9, 10, 11, 12, 13, 14, 15, 16]. These
include the engineered $\mathrm{LaAlO_{3}/SrTiO_{3}}$ heterostructures [17,
18, 19] and graphene on the two-dimensional (2D) layers of the transition
metal dichalcogenides [20, 21], to name a few.

The Rashba effect [22], characterized by locking of spin and momenta
directions, results in lifting the spin-degeneracy of conduction band
electrons away from the center of a Brillouin zone. It originates from a
momentum dependent Berry phase, emerging due to a mixing between the
conduction and valence bands. If the valence bands are further split by intra-
atomic relativistic spin-orbit coupling, the Berry phase acquire a spin-
dependent component. The origin of the Berry phase is in the non-commutativity
of a momentum-dependent unitary transformation, $U({\bm{p}})$, from atomic
orbitals to Bloch bands basis and a coordinate-dependent scalar potential,
$\varphi({\bm{r}})$. The Moyal expansion of these non-commutative operators
leads to gradient commutators of the form
$\varepsilon_{\lambda\mu\nu}\grad^{\nu}_{\bm{r}}\varphi({\bm{r}})\grad^{\mu}_{\bm{p}}U(\bm{p})\,\hat{\sigma}^{\lambda}$,
where $\mu,\nu,\lambda$ are Cartesian coordinates and $\hat{\sigma}$ are spin
Pauli matrices. In this way Rashba spin-orbit effects are inextricably
proportional to a local electric field
$\bm{\mathcal{E}}(\bm{r})=-\bm{\nabla}_{\\!\bm{r}}\varphi({\bm{r}})$.

For conduction band electrons close to the Brillouin zone center the resulting
coupling can be described via a commonly used phenomenological single-particle
Rashba Hamiltonian, which is linear in the wave vector, $\bm{p}$, [23]

$H=\alpha\left[\bm{\mathcal{E}}\times{\bm{p}}\right]\cdot\hat{\bm{\sigma}},$
(1)

where $\alpha$ is a material-dependent Rashba constant. The presence of the
local electric field, $\bm{\mathcal{E}}$, here is usually associated with a
broken inversion symmetry. In 2D the normal electric field originates either
from a polar substrate, or from an externally applied gate voltage. In 3D
metallic crystals uniform fields are screened. However, for an inversion
symmetry broken unit cell, local fields may alternate along the polar
direction.

An advent of pair spin-orbit interactions (PSOI) [24] comes from a simple
realization that the electric field, $\bm{\mathcal{E}}$, in Eq. (1) may as
well come from the Coulomb electron-electron interactions. Accepting this
fact, immediately leads to a number of remarkable consequences. First, no
inversion symmetry breaking is required for pair-RSOI effects. Second, the
emerging spin order is qualitatively different from the more familiar single-
particle RSOI. Take, e.g., a 2D case: the normal electric field of the single-
particle RSOI (1) results in a chiral locking of in-plane spins and momenta.
On the other hand, for the pair-RSOI the electric field (as well as momentum)
are purely in-plane, forcing spins to be ordered along the normal direction.
Despite their potential to give rise to intriguing collective phenomena [25],
manifestations of the Rashba effect in the two-body interactions remain
relatively unexplored.

The PSOI was coined to trigger instabilities in the charge sector of 1D and 2D
electron systems. These instabilities in the particle-particle channel may
facilitate formation of bound electron pairs with non-trivial orbital and spin
structure, including $p$-wave pairing [26, 27, 28, 29, 30, 31, 32].
Furthermore, the instabilities observed in the particle-hole channel act as a
precursor for phase separation, assisting the development of highly
unconventional correlated states [33, 34, 35].

In this work, we focus on superconducting and magnetic instabilities in 3D and
2D Rashba materials [36, 37]. The former manifests itself in a particular
nodal $p$-wave superconducting state, consistent with the two-body bound
states discussed in Refs. [26, 27, 28, 29, 30]. Stability of such
superconducting state does not require phonon (or any other) mediator of
attraction and relies solely on the Coulomb repulsion and PSOI. It does
impose, however, certain requirements on the purity of the material. The
ferromagnetic (FM) ordering arises from the Stoner-like mechanism for a
sufficiently large SOI constant, $\alpha$. Due to PSOI the FM breaking of the
rotation symmetry in the spin space is associated with a deformation and
eventual topology-changing Lifshitz transition of the Fermi surface in the
momentum space. Such a transition from genus-zero to genus-one Fermi surface
leads to experimentally detectable consequences in thermodynamic and transport
properties, as well as in spectra of Shubnikov-de Haas and Friedel
oscillations. Finally, the FM and the $p$-wave superconducting orders may
coexist. Interestingly, the toroidal Fermi surface allows for a nodeless fully
spin-polarized order parameter.

##  II Electron-electron interactions in Rashba Materials

In this section we work out consequences of the phenomenological Rashba spin-
orbit coupling, Eq. (1), for electron-electron interaction Hamiltonian. To
this end consider a 3D Rashba system without the inversion symmetry breaking.
A 2D case is a simple particular case of this consideration, which is briefly
mentioned in the end. Since a 3D electronic system with the inversion symmetry
does not allow for a bulk electric field, its tempting to conclude that the
RSOI term (1) is inconsequential. However, such conclusion is premature
because of the fluctuating electric fields created by electron-electron
interactions.

To describe the physics originating from this observation one deals with a
two-component spinor electron creation operator
$\hat{\psi}_{\bm{p}}^{\dagger}\equiv\spinor{\hat{c}_{\bm{p},\uparrow}^{\dagger}\hat{c}_{\bm{p},\downarrow}^{\dagger}}$,
describing conduction band electrons in a spin-up/spin-down basis. The
corresponding electron creation operator in the coordinate basis is denoted
$\hat{\psi}^{\dagger}(\bm{r})$.

It is convenient to define the spin current tensor as

$\hat{\jmath}^{\lambda\mu}(\bm{r})=\frac{1}{2mi}\left(\hat{\psi}^{\dagger}(\bm{r})\hat{\sigma}^{\lambda}\grad_{\bm{r}}^{\mu}{\hat{\psi}(\bm{r})}-(\grad_{\bm{r}}^{\mu}{\hat{\psi}^{\dagger}(\bm{r})})\hat{\sigma}^{\lambda}\hat{\psi}(\bm{r})\right),$
(2)

with Greek indices $\lambda,\mu,\ldots$ standing for the Cartesian components.
111Throughout the text we set $\hbar=1$, and suppress the ${(volume)}^{-1}$
factors before the sums over momenta. This object describes a flow of the
$\lambda$ component of the spin angular momentum in the $\mu$ spatial
direction.

The Hamiltonian for electrons interacting with an external scalar electric
potential, $\varphi(\bm{r},t)$, which includes RSOI, Eq. (1), is

$\begin{split}\hat{H}_{\mathrm{ext}}&=-\int
d\bm{r}\,\left(e\varphi(\bm{r},t)\hat{\rho}(\bm{r})+\alpha
m\,\varepsilon_{\lambda\mu\nu}\mathcal{E}^{\nu}(\bm{r})\hat{\jmath}^{\lambda\mu}(\bm{r})\right)\\\
&=-\int d\bm{r}\,\varphi(\bm{r},t)\left(e\hat{\rho}(\bm{r})+\alpha
m\,\varepsilon_{\lambda\mu\nu}\grad_{\bm{r}}^{\nu}\hat{\jmath}^{\lambda\mu}(\bm{r})\right)\\\
&=-\int
d\bm{r}\,\varphi(\bm{r},t)\left(e\hat{\psi}^{\dagger}\hat{\psi}+i\alpha\,\varepsilon_{\lambda\mu\nu}\grad_{\bm{r}}^{\mu}\hat{\psi}^{\dagger}\hat{\sigma}^{\lambda}\grad_{\bm{r}}^{\nu}\hat{\psi}\right),\end{split}$
(3)

where $\hat{\rho}(\bm{r})\equiv\hat{\psi}^{\dagger}(\bm{r})\hat{\psi}(\bm{r})$
is the electron density, and $\varepsilon_{\lambda\mu\nu}$ is the Levi-Civita
symbol. In the second line we used that
$\mathcal{E}^{\nu}(\bm{r})=-\grad_{\bm{r}}^{\nu}\varphi(\bm{r})$ and performed
integration by parts. The last terms in the integrands here are the RSOI,
which can be viewed as the renormalization of the electron density operator by
the gradient corrections. Its appearance warrants some elaboration. The very
notion of the SOI is an attribute of a simplified single-band portrayal of the
electron dynamics, accomplished via the projection of a microscopic multi-band
Hamiltonian onto the conduction band. Through this projection, an effective
Hamiltonian emerges with the SOI term, reflecting the contribution of the
valence band(s) to the conduction band electrons dynamics. In particular, the
local scalar potential disturbs both valence and conduction bands. Due to the
momentum-dependent nature of the rotation from local orbitals to the band
representation, it leads to the gradient correction to the conduction-band
density response. The SOI term in Eq. (3) reflects the spin-dependent part of
this gradient correction. The $\sim\hat{\sigma}_{0}$ gradient terms may also
exist, but they do not lead to qualitatively new effects.

In the momentum representation Eq. (3) results in the following interaction
vertex between the electrons and the scalar potential

$\hat{H}_{\mathrm{ext}}=\sum_{\bm{p}_{1},\bm{p}_{2}}\varphi(\bm{p}_{2}-\bm{p}_{1},t)\,\hat{\psi}^{\dagger}_{\bm{p}_{1}}\hat{\Gamma}_{\bm{p}_{1}\bm{p}_{2}}\hat{\psi}_{\bm{p}_{2}}\,,$
(4)

where the vertex $\hat{\Gamma}_{\bm{p}_{1}\bm{p}_{2}}$ is a matrix in the spin
as well as in the momentum spaces given by 222By its nature, the anomalous
contribution affects only the local distribution of the electron density, but
keeps the total number of electrons in the conduction band constant. This is
evident from the fact that the extra term is zero in the long-wave limit of
$\bm{q}=\bm{p}_{1}-\bm{p}_{2}\to 0$.

$\hat{\Gamma}_{\bm{p}_{1}\bm{p}_{2}}=-e\hat{\sigma}_{0}-i\alpha_{p_{1}p_{2}}[\bm{p}_{1}\times\bm{p}_{2}]\cdot\hat{\bm{\sigma}}\,.$
(5)

In writing this expression, we accommodated the possibility that the SOI
“constant”, $\alpha$, can in fact exhibit momentum dependence,
$\alpha\to\alpha_{p_{1}p_{2}}$, making $\alpha_{p_{1}p_{2}}$ to decrease if
either of the two momenta exceeds a certain cutoff. Such dependence indeed
arises from a more detailed microscopic analysis within a multi-band model, as
outlined in Appendix A.

In interacting systems the scalar potential is a quantized Gaussian
fluctuating field with the correlation function given by
$\langle\varphi(\bm{q},t)\varphi(-\bm{q},t^{\prime})\rangle\equiv\mathcal{U}_{\bm{q}}\,\delta(t-t^{\prime})$.
In 3D materials the latter is the usual Coulomb potential,
$\mathcal{U}_{\bm{q}}=4\pi/q^{2}$, while in 2D it may include effects of
dielectric constant $\epsilon$ of a surrounding media and, possibly, screening
by external metallic gates. This leads to the e-e interaction Hamiltonian of
the form:

$\hat{H}_{\mathrm{int}}=\frac{1}{2}\sum_{\begin{subarray}{c}\bm{p}_{1},\bm{p}_{2}\\\
\bm{p}^{\prime}_{1},\bm{p}^{\prime}_{2}\end{subarray}}\mathcal{U}_{\bm{p}_{2}-\bm{p}_{1}}\\!\normord{\hat{\psi}^{\dagger}_{\bm{p}_{1}}\hat{\Gamma}_{\bm{p}_{1}\bm{p}_{2}}\hat{\psi}_{\bm{p}_{2}}\hat{\psi}^{\dagger}_{\bm{p}^{\prime}_{1}}\hat{\Gamma}_{\bm{p}^{\prime}_{1}\bm{p}^{\prime}_{2}}\hat{\psi}_{\bm{p}^{\prime}_{2}}}\,,$
(6)

where the momentum summation is limited to momentum conserving processes with
$\bm{p}_{1}+\bm{p}^{\prime}_{1}=\bm{p}_{2}+\bm{p}^{\prime}_{2}$ and the colon
signs denote the normal ordering of the electron creation/annihilation
operators.

The total Hamiltonian
$\hat{H}_{\mathrm{tot}}=\hat{H}_{\mathrm{kin}}+\hat{H}_{\mathrm{int}}$ also
includes the kinetic energy

$\hat{H}_{\mathrm{kin}}=\sum_{\bm{p}}\hat{\psi}^{\dagger}_{\bm{p}}\,\xi_{\bm{p}}\hat{\sigma}_{0}\,\hat{\psi}_{\bm{p}}\,.$
(7)

To be specific, we assume quadratic energy dispersion,
$\xi_{\bm{p}}=\bm{p}^{2}/2m-\mu$. In 3D case in the absence of the bulk
electric field, $\bm{\mathcal{E}}$, RSOI materials do not have a single-
particle spin-orbit term.

The situation is different in 2D where the normal electric field
$\mathcal{E}_{z}$ may be present due to external gate, or other inversion
symmetry breaking mechanism. In this case the single-particle Rashba term is
expected, in general, making the kinetic part to be

$\hat{H}_{\mathrm{kin}}=\sum_{\bm{p}}\hat{\psi}^{\dagger}_{\bm{p}}\left(\xi_{\bm{p}}\hat{\sigma}_{0}+\alpha\,\mathcal{E}_{z}{[\bm{p}\times\hat{\sigma}]}_{z}\right)\hat{\psi}_{\bm{p}}\,,$
(8)

where both $\bm{p}$ and $\hat{\sigma}$ are 2D vectors in $x-y$ plane. The
corresponding interaction vertex acquires the spin-diagonal form:
$\hat{\Gamma}_{\bm{p}_{1}\bm{p}_{2}}=-e\hat{\sigma}_{0}-i\alpha_{p_{1}p_{2}}{[\bm{p}_{1}\times\bm{p}_{2}]}_{z}\hat{\sigma}_{z}$,
as both the initial and final momenta are in the $x-y$ plane.

##  III $P$-wave Superconductivity

Here, we show that the interplay of PSOI and Coulomb repulsion favors $p$-wave
superconducting order, without the need for phonons or other mediators of
attraction. The mechanism bears similarities to the Kohn-Luttinger higher
angular momentum superconductivity [38, 39]. Notably, the key distinction lies
in the source of the effective attraction in odd orbital channels. In the
present case it originates from the PSOI, while the interaction potential,
$\mathcal{U}_{\bm{q}}$, may be considered as a structureless short-range
potential (either repulsive or attractive).

###  III.1 Variational Considerations

Consider an approximate Bardeen–Cooper–Schrieffer (BCS) Hamiltonian that
describes the pairing interaction between electrons with zero total momentum.
Making the spin indexes explicit, the interaction Hamiltonian (6) acquires the
form

$\hat{H}_{\mathrm{BCS}}\propto\mathcal{U}_{\bm{p}_{2}-\bm{p}_{1}}\hat{\Gamma}_{\bm{p}_{1}\bm{p}_{2}}^{s_{1}s_{2}}\hat{\Gamma}_{-\bm{p}_{1}-\bm{p}_{2}}^{r_{1}r_{2}}\,{\psi}^{\dagger}_{\bm{p}_{1}s_{1}}{\psi}^{\dagger}_{-\bm{p}_{1}r_{1}}{\psi}_{-\bm{p}_{2}r_{2}}{\psi}_{\bm{p}_{2}s_{2}},$
(9)

where we omitted momentum and spin summation symbols for brevity. Anticipating
the spin-triplet superconductivity we introduce anomalous expectation values
as

$\left\langle{\psi}_{-\bm{p}r}{\psi}_{\bm{p}s}\right\rangle=\sum_{m=-1}^{1}b_{\bm{p}}^{m}\,\hat{\gamma}^{m}_{rs}\,,$
(10)

where $m=-1,0,1$ labels $z$-components of the spin angular momentum for
triplet anomalous average. Their spin structure is encoded by the three
symmetric matrices

$\displaystyle\hat{\gamma}^{-1}=\begin{pmatrix}0&0\\\ 0&1\end{pmatrix};$
$\displaystyle\hat{\gamma}^{0}=\frac{1}{\sqrt{2}}\begin{pmatrix}0&1\\\
1&0\end{pmatrix};$ $\displaystyle\hat{\gamma}^{1}=\begin{pmatrix}1&0\\\
0&0\end{pmatrix}.$ (11)

Accordingly, the anomalous averages, $b_{\bm{p}}^{m}$, are antisymmetric
functions of the orbital momentum:

$b_{-\bm{p}}^{m}=-b_{\bm{p}}^{m}\,.$ (12)

In the spin-orbit interacting system the orbital character of the anomalous
expectations appears to be rigidly coupled to its spin content. In particular,
the lowest harmonic of the odd orbital function is the $p$-wave with

$b_{\bm{p}}^{m}=b_{p}^{m}\,Y^{-m\operatorname{sign}(\alpha)}_{1}(\theta_{\bm{p}},\phi_{\bm{p}})\,,$
(13)

where even parts $b_{p}^{m}$ are functions of the absolute value of the
momentum, $Y_{1}^{m}(\theta,\phi)$, are $l=1$ (odd) spherical harmonics and
$\theta_{\bm{p}},\phi_{\bm{p}}$ are spherical angles of the momentum vector
$\bm{p}$.

More generally one could write
$b_{\bm{p}}^{m}=\sum_{m^{\prime}}b_{p}^{mm^{\prime}}\,Y^{m^{\prime}}_{1}$,
with a tensor orbital symmetric part, $b_{p}^{mm^{\prime}}$. The reason for
the particular assignment (13) between the spin projection index, $m$, and
orbital functions $Y_{1}^{m^{\prime}}$ is justified by the minimization of the
interaction energy, as explained below.

An alternative representation [40] of the spin-symmetric anomalous expectation
(10) is

$\langle{\psi}_{-\bm{p}r}{\psi}_{\bm{p}s}\rangle=\bm{d}_{\bm{p}}\cdot{(\bm{\sigma}\,i\sigma_{y})}_{rs}\,,$
(14)

with $\bm{d}_{-\bm{p}}=-\bm{d}_{\bm{p}}$. The relation between the two
notations are: $b^{\pm 1}=\pm(d^{x}\pm id^{y})$ and $b^{0}=\sqrt{2}\,d^{z}$.
The $p$-wave orbital part (13) translates into
$d^{\mu}_{\bm{p}}=d^{\mu}_{p}\,p^{\mu}$ with $\mu=x,y,z$.

The mean-field expectation value of the BCS Hamiltonian (9) is given by

$\langle\hat{H}_{\mathrm{BCS}}\rangle\propto\mathcal{U}_{\bm{p}_{2}-\bm{p}_{1}}\\!\\!\\!\sum\limits_{m_{1},m_{2}}\\!\bar{b}_{\bm{p}_{1}}^{m_{1}}b_{\bm{p}_{2}}^{m_{2}}\,\operatorname{Tr}\left\\{\hat{\Gamma}_{\bm{p}_{1}\bm{p}_{2}}\hat{\gamma}^{m_{1}}\hat{\Gamma}_{\bm{p}_{1}\bm{p}_{2}}\hat{\gamma}^{m_{2}}\right\\}\,,$
(15)

where the trace runs in the spin space, and we used that
$\hat{\Gamma}_{-\bm{p}_{1}-\bm{p}_{2}}=\hat{\Gamma}_{\bm{p}_{1}\bm{p}_{2}}$.
One may show that, with the harmonic content of Eq. (13), only $m_{1}=m_{2}$
terms are important. It is thus useful to employ that

$\displaystyle\operatorname{Tr}\left\\{\hat{\gamma}^{m}\hat{\gamma}^{m}\right\\}=1\,;$
$\displaystyle\operatorname{Tr}\left\\{\hat{\gamma}^{m}\hat{\sigma}_{\mu}\hat{\gamma}^{m}\right\\}=m\,\delta_{\mu
z}\,;$ (16) $\displaystyle\operatorname{Tr}\left\\{\hat{\gamma}^{\pm
1}\hat{\sigma}_{\mu}\hat{\gamma}^{\pm
1}\hat{\sigma}_{\nu}\right\\}=\delta_{\mu z}\delta_{\nu z}\,,$

and for completeness:
$\operatorname{Tr}\left\\{\hat{\gamma}^{0}\hat{\sigma}_{\mu}\hat{\gamma}^{0}\hat{\sigma}_{\mu}\right\\}=(1,-1,-1)$,
with $\mu=x,y,z$, while all other traces vanish.

To simplify the discussion of Eq. (15) let us first assume short range
repulsive interaction (e.g., screened Coulomb) with
$\mathcal{U}_{\bm{p}_{2}-\bm{p}_{1}}=\mathcal{U}_{0}>0$. This may seem to be
in contradiction with the announced $p$-wave nature of the anomalous averages.
Indeed, being taken at the same spatial point the triplet component (10) seems
to be zero. This is not the case due to the spin-orbit part of the
interactions, which adds factors $[\bm{p}_{1}\times\bm{p}_{2}]$ to the
interaction vertexes. This makes the effective interaction to be momentum-
dependent, even if $\mathcal{U}_{\bm{p}_{2}-\bm{p}_{1}}=\operatorname{const}$.
In other words, factors $[\bm{p}_{1}\times\bm{p}_{2}]$ mean derivatives of the
$\psi$ operators in Eq. (10):
$\langle\grad^{\mu}\hat{\psi}\grad^{\nu}\hat{\psi}\rangle$, cf. Eq. (3), which
does not have to vanish, even if taken at the same spatial point.

With $\mathcal{U}_{\bm{p}_{2}-\bm{p}_{1}}=\mathcal{U}_{0}$ and with the help
of Eqs. (5) and (13), one may perform an explicit angular integration over
$\theta_{\bm{p}_{1}},\phi_{\bm{p}_{1}},\theta_{\bm{p}_{2}},\phi_{\bm{p}_{2}}$
in Eq. (15). As expected, only spin-orbit part of the interactions
contributes. Moreover, the only non-vanishing term is linear in SOI constant,
$\alpha$. According to (16), only
${[\bm{p}_{1}\times\bm{p}_{2}]}_{z}=p_{1}p_{2}\sin\theta_{\bm{p}_{1}}\sin\theta_{\bm{p}_{2}}\sin(\phi_{\bm{p}_{1}}-\phi_{\bm{p}_{2}})$,
as well as $m=\pm 1$ contribute. Recalling that $Y_{1}^{\pm
1}(\theta,\phi)\propto\sin\theta\,e^{\pm i\phi}$, one finds for the
corresponding angular integral:

$\iint\\!\\!d\phi_{\bm{p}_{1}}d\phi_{\bm{p}_{2}}\sin(\phi_{\bm{p}_{1}}-\phi_{\bm{p}_{2}})\,e^{im\operatorname{sign}(\alpha)(\phi_{\bm{p}_{1}}-\phi_{\bm{p}_{2}})}\int\\!\\!d\theta_{\bm{p}_{1}}\\!\sin^{3}\theta_{\bm{p}_{1}}\int\\!\\!d\theta_{\bm{p}_{2}}\\!\sin^{3}\theta_{\bm{p}_{2}}=im\operatorname{sign}(\alpha)\,\frac{\pi^{2}}{2}\left(\frac{4}{3}\right)^{2}\,.$
(17)

Using this along with Eq. (16), one obtains for the expectation value of the
BCS interaction energy (15):

$\langle\hat{H}_{\mathrm{BCS}}\rangle=-\,\frac{|\alpha|\,e\,\mathcal{U}_{0}}{3{(2\pi)}^{5}}\,\sum_{m}\,m^{2}\left\lvert\int\limits_{0}^{\infty}p^{3}dp\,b_{p}^{m}\right\rvert^{2}\,.$
(18)

The amplitudes $b_{p}^{\pm 1}$ should be determined from the self-consistency
conditions. Before doing it, let us discuss some lessons learned from Eq.
(18).

The first order in SOI constant, $\alpha$, contribution to the interaction
energy appears to be non-analytic, i.e. $\sim-\lvert\alpha\rvert$. This is
achieved by assigning $Y_{1}^{-\operatorname{sign}(\alpha)}$ orbital component
to the $m=1$ triplet and $Y_{1}^{\operatorname{sign}(\alpha)}$ orbital
component to $m=-1$ triplet component. This way the Cooper channel interaction
energy is negative, even for the repulsive short range interactions. Clearly
for attractive interactions, one can reverse the assignment of spin and
orbital components to again achieve the energy decrease. Also notice that, in
this approximation, the $m=0$ component of the triplet does not affect the
energy.

If the interactions are long(er) range, one can expand the interaction
potential in spherical harmonics as

$\mathcal{U}_{\bm{p}_{2}-\bm{p}_{1}}=\mathcal{U}_{0}+\mathcal{U}_{1}\sin\theta_{\bm{p}_{1}}\sin\theta_{\bm{p}_{2}}\cos(\phi_{\bm{p}_{1}}-\phi_{\bm{p}_{2}})+\ldots$
(19)

Calculating the angular integrals one finds an extra term in the BCS energy
$\sim(e^{2}-\tfrac{9}{64}\alpha^{2}p_{1}^{2}p_{2}^{2})\mathcal{U}_{1}$. For
the typical case of positive $\mathcal{U}_{1}>0$, the SOI tend to decrease the
energy, while the Coulomb part increases the energy. For a strong screening,
$\mathcal{U}_{1}\ll\mathcal{U}_{0}$, the energy decrease (18) leads to a non-
zero anomalous average $b_{p}^{\pm 1}$ and thus to the $p$-wave
superconductivity. Notice that the pairing mechanism is a combination of PSOI
and the screened Coulomb interactions, with no need for phonons or any other
mediators of attractive interactions.

Another lesson is that $m=\pm 1$ components of the spin triplet are
essentially independent and equivalent, whereas there is no $m=0$ condensate.
This implies that spin up electrons condense into $b_{p}^{1}$ condensate
wavefunction, while spin down ones condense into $b_{p}^{-1}$ condensate and
$b_{p}^{0}=0$. Therefore the self-consistency condition maybe considered
separately for spin up and down electrons.

###  III.2 Self-Consistency Condition

Following the last remark, consider the all spins up(down) part of the BCS
Hamiltonian (9). The relevant vertexes are
$\Gamma_{\bm{p}_{1}\bm{p}_{2}}^{\pm}=-e\mp
i\alpha{[\bm{p}_{1}\times\bm{p}_{2}]}_{z}$, where (and for the remainder of
this section) we omit all spin indexes. The corresponding Bogoliubov-de-Gennes
(BdG) Hamiltonians take the form

$\hat{H}_{\mathrm{BdG}}=\sum_{\bm{p}}\left[\xi_{\bm{p}}\psi_{\bm{p}}^{\dagger}\psi_{\bm{p}}+\Delta_{\bm{p}}^{\pm}\,\,{\psi}^{\dagger}_{\bm{p}}{\psi}^{\dagger}_{-\bm{p}}+\bar{\Delta}_{\bm{p}}^{\pm}\,\,{\psi}_{-\bm{p}}{\psi}_{\bm{p}}\right]\,,$
(20)

where

$\Delta_{\bm{p}}^{\pm}=-\sum_{\bm{p}_{2}}\,\mathcal{U}_{\bm{p}-\bm{p}_{2}}\left(\Gamma_{\bm{p}\bm{p}_{2}}^{\pm}\right)^{2}b_{\bm{p}_{2}}^{\pm
1}.$ (21)

Thanks to $\mathcal{U}_{-\bm{q}}=\mathcal{U}_{\bm{q}}$,
$\Gamma_{-\bm{p}-\bm{p}_{2}}=\Gamma_{\bm{p}\bm{p}_{2}}$, and
$b_{-\bm{p}_{2}}^{\pm 1}=-b_{\bm{p}_{2}}^{\pm 1}$, one finds that the order
parameter is odd: $\Delta_{-\bm{p}}^{\pm}=-\Delta_{\bm{p}}^{\pm}$, as expected
for a spinless BdG Hamiltonian. Performing Bogoliubov transformation and
evaluating the anomalous average (10) in a standard way [41], one obtains the
self-consistency condition

$\Delta_{\bm{p}}^{\pm}=-\sum\limits_{\bm{p}_{2}}\,\mathcal{U}_{\bm{p}-\bm{p}_{2}}\left(\Gamma_{\bm{p}\bm{p}_{2}}^{\pm}\right)^{2}\frac{\Delta_{\bm{p}_{2}}^{\pm}}{2E_{\bm{p}_{2}}}\,\tanh\frac{E_{\bm{p}_{2}}}{2T}\,,$
(22)

where
$E_{\bm{p}}=\sqrt{\xi_{\bm{p}}^{2}+{\left\lvert\Delta_{\bm{p}}^{\pm}\right\rvert}^{2}}\,$.

Let us analyze this relation for the case of the short range repulsive
interactions, $\mathcal{U}_{\bm{q}}=\mathcal{U}_{0}>0$. Following Eq. (13),
one may look for a $p$-wave solution of the form

$\Delta_{\bm{p}}^{\pm}=\Delta_{p}\sin\theta_{\bm{p}}\,e^{\mp
i\operatorname{sign}(\alpha)\phi_{\bm{p}}}.$ (23)

The angular $\phi_{\bm{p}_{2}}$ integration selects the $\sim
ei\alpha[\bm{p}_{1}\times\bm{p}_{2}]_{z}$ term out of
$\left(\Gamma_{\bm{p}\bm{p}_{2}}^{\pm}\right)^{2}$, leading to:

$1=\frac{\lvert\alpha\rvert e\mathcal{U}_{0}\nu p_{F}^{2}}{4}\int
d\xi\,\int_{0}^{\pi}\\!\\!d\theta\,\sin^{3}\theta\,\,\frac{\tanh\frac{\sqrt{\xi^{2}+\Delta^{2}\sin^{2}\theta}}{2T}}{\sqrt{\xi^{2}+\Delta^{2}\sin^{2}\theta}},$
(24)

where $\nu$ is a density of states at the Fermi level, and we anticipated that
the $\xi$ integration is limited to a narrow vicinity of the Fermi energy,
leading to $\Delta_{p}\approx\Delta_{p_{F}}=\Delta$. Employing that
$\int_{0}^{\pi}d\theta\sin^{3}\theta=4/3$, one finds

$T_{c}\approx\Lambda\exp\left\\{-\frac{3}{2\lvert\alpha\rvert
e\mathcal{U}_{0}\nu p_{F}^{2}}\right\\},$ (25)

where $\Lambda$ is an energy scale over which the SOI “constant”, $\alpha$
decreases. From Eq. (24) one also finds that

$\Delta(T=0)=2.03\,T_{c},$ (26)

valid for $T_{c}\ll\Lambda$.

To estimate the critical temperature in Eq. (25), consider the screened
Coulomb interactions in the Thomas-Fermi (i.e. static, long-wavelength)
approximation, with the interaction strength given by
$e^{2}\mathcal{U}_{0}=\nu^{-1}$. Equation (25) reads:

$T_{c}\approx\Lambda\exp\left\\{-\beta\,\frac{r_{s}^{2}}{\tilde{\alpha}}\right\\};\qquad\tilde{\alpha}=\frac{\lvert\alpha\rvert}{ea_{B}^{2}}\,,$
(27)

where we introduced dimensionless parameters $\tilde{\alpha}$ for the SOI
magnitude, and $r_{s}$ for the average distance between electrons, expressed
in units of Bohr radius, $a_{B}$. The factor $\beta$ is given by $\beta\approx
0.41$ in 3D and $\beta=0.75$ in 2D.

Within the Kane model of the band structure, the dimensionless SOI magnitude
is estimated as $\tilde{\alpha}\approx 1/\epsilon^{2}$, where $\epsilon$ is
the bulk dielectric constant [34]. Assuming $\Lambda$ to be of the order of
the band gap, $\Lambda\approx$500\text{\,}\mathrm{m}\mathrm{e}\mathrm{V}$$,
the electron concentration corresponding to $r_{s}=0.5$, and a dielectric
constant $\epsilon=10$, our estimation yields a critical temperature
$T_{c}\approx$0.3\text{\,}\mathrm{K}$$ for 3D systems. Higher critical
temperatures can be anticipated for materials with a giant Rashba effect, e.g.
$\mathrm{BiTeI}$ [10], the $\mathrm{BiSb}$ monolayers [11], 2D transition
metal dichalcogenides [12], graphene with adsorbed heavy elements [14, 15],
perovskites [16] and oxides [13].

###  III.3 Discussion of the Superconducting Order

We found that the screened Coulomb interactions in 3D Rashba materials lead to
the tendency towards the $p$-wave superconducting order with the order
parameter of the form (23), for up and down spins correspondingly. The
condition for emergence of such an order is rather mild $e\lvert\alpha\rvert
p_{F}^{2}\mathcal{U}_{0}+\alpha^{2}p_{F}^{4}\mathcal{U}_{1}>e^{2}\mathcal{U}_{1}$,
where we omitted numerical factors for brevity. Since the Anderson theorem
does not apply to such momentum-dependent order parameter, one expects that
disorder suppresses the $p$-wave order, in case elastic scattering rate
exceeds the order parameter, $\Delta$, Eq. (26).

The order parameter (23), spontaneously breaks the global $U(1)$ and
rotational symmetry. It does not break the time reversal symmetry, however,
since the spin up condensate maps onto spin down one upon time reversal
transformation. (The time reversal is broken, of course, if in addition a
certain spin polarization develops, as discussed in the next section.) This
puts the corresponding superconductor into the DIII symmetry class. It is
interesting to notice that DIII is topological in 3D with the $Z$ topological
index.

Figure 1: Density of states of the nodal $p$-wave superconductor.

In the model with RSOI and the short range repulsive interactions the
$\Delta^{0}$ condensate, corresponding to $m=0$, is absent. This is different
from 3He phase B, where the order parameter is isotropic with equal
$\Delta^{\pm}$ and $\Delta^{0}$ [42]. In our case the $\Delta^{\pm}$ order
parameters have two nodes along the $p_{z}$ direction, leading to the absence
of the hard gap in the quasiparticle spectrum.

The corresponding density of states of a superconductor with the nodal order
parameter (23) is given by

$\nu(\epsilon)=\sum_{\bm{p}}\,\delta\left(\epsilon-\sqrt{\xi_{\bm{p}}^{2}+\Delta^{2}\sin^{2}\theta_{\bm{p}}}\right),$
(28)

where $\sum_{\bm{p}}\ldots=\nu\int d\xi_{\bm{p}}\,\tfrac{1}{2}\\!\int
d\theta_{\bm{p}}\sin\theta_{\bm{p}}\ldots$. This way one finds, see Fig. 1,

$\nu(\epsilon)=\nu\,\frac{\epsilon}{\Delta}\begin{dcases}\operatorname{arctanh}(\epsilon/\Delta);&\epsilon<\Delta\,,\\\
\operatorname{arccoth}(\epsilon/\Delta);&\epsilon>\Delta\,.\end{dcases}$ (29)

Unlike the s-wave case, the gap is soft with
$\nu(\epsilon)\approx\nu{(\epsilon/\Delta)}^{2}$ at $\epsilon\ll\Delta$. The
singularity at $\epsilon=\Delta$ is not a square root one, but rather
logarithmic, $\nu(\epsilon)\approx-\nu\log\lvert\epsilon/\Delta-1\rvert$.

In view of the gapless nature of the quasiparticle spectrum in 3D the notion
of the topological index is somewhat mute. The situation is rather different
in 2D. There $\Delta^{\pm}$ spin up and down condensates exhibit $p_{x}\mp
i\operatorname{sign}(\alpha)p_{y}$ orbital structure, which is gaped in 2D
(the $z$ axis is normal to the 2D material). The two decoupled condensates
belong to the symmetry class $D$ and exhibit
$Z=\mp\operatorname{sign}(\alpha)$ Chern numbers. If their vortexes are
spatially separated, they harbor Majorana states. Stability of such zero-
energy states depends on a nature of mutual interactions between vortexes in
$\Delta^{\pm}$ condensates. In case of attraction the Majorana states tend to
annihilate each other, while the repulsion indicates their stability. A more
careful consideration is required to find the nature of interactions between
the two condensates.

An interesting consequence of having two (almost non-interacting) condensates,
formed by electrons with the opposite spins, is that the superconductor can
carry a spin-polarized supercurrent. Such a current may be induced by either
injecting from a ferromagnet, or by creating a disbalance between $\Delta^{+}$
and $\Delta^{-}$ with an external magnetic field. One may thus think of a
superconducting spin-valve with, e.g., two ferromagnetic leads with colinear
or anti-colinear magnetization directions.

##  IV Ferromagnetism

In this section, we show that PSOI-mediated exchange interaction may lead to a
ferromagnetism. Due to the spin-orbit coupling, it breaks rotational symmetry
both in the spin and momentum spaces, resulting in an anisotropic Fermi
surface. For a sufficiently large SOI constant, $\alpha$, the Fermi surface of
the ferromagnet undergoes the topological Lifshitz transition from an
ellipsoidal to a toroidal shape.

###  IV.1 Mean-field theory

Figure 2: The first order diagram for the exchange energy of the electron gas.
The momentum-dependent vertices $\hat{\Gamma}_{\bm{p}\bm{p}^{\prime}}$ of Eq.
(5) reflect the PSOI contribution to the energy.

We treat the e-e interaction of Eq. (6) in a Hartree-Fock (HF) approximation.
The Hartree (direct) contribution is not modified by the PSOI. Indeed, for the
Hartree part of Eq. (6) $\bm{p}_{1}=\bm{p}_{2}$ and
$\bm{p}_{1}^{\prime}=\bm{p}_{2}^{\prime}$, nullifying the SOI parts of
interaction vertexes, $\hat{\Gamma}$, Eq. (5), due to the vector products of
identical momenta. As a result, the Hartree term is compensated by the
positive charge of the lattice and is absent in charge neutral systems. The
Fock (exchange) contribution comes from the “oyster” diagram of Fig. 2. It is
given by

$E_{\mathrm{xc}}=-\frac{1}{2}\sum_{\bm{p}\bm{p}^{\prime}}\mathcal{U}_{\bm{p}-\bm{p}^{\prime}}\operatorname{Tr}\\{\hat{G}_{\bm{p}}\hat{\Gamma}_{\bm{p}\bm{p}^{\prime}}\hat{G}_{\bm{p}^{\prime}}\hat{\Gamma}_{\bm{p}^{\prime}\bm{p}}\\}\,,$
(30)

where $\hat{G}_{\bm{p}}$ is the equal-time electron Green function.

To reach a spin-polarized state, one may assume a symmetry-breaking in any
given direction, say along the $z$-axis. Under this assumption, the Green
function takes the form 333In principle the Green function does not have to be
diagonal for all values of momenta in the same basis. For example, one may
think of the Green functions with the off-diagonal elements, e.g., of the form
$\propto(\sigma_{x}p_{x}+\sigma_{y}p_{y})$, signifying an interaction-induced
single-particle RSOI. One may show, however, that such states have higher
energy, than that given by the diagonal ansatz.

$\hat{G}_{\bm{p}}=\mathcal{G}_{0}(\bm{p})\hat{\sigma}_{0}+\mathcal{G}_{3}(\bm{p})\hat{\sigma}_{3}\;,$
(31)

or explicitly,

$\hat{G}_{\bm{p}}=\begin{pmatrix}n_{+}(\bm{p})&0\\\
0&n_{-}(\bm{p})\end{pmatrix}\,,$ (32)

where $n_{\pm}(\bm{p})$ are occupation numbers of a state $\bm{p}$ by
electrons with the two spin projections. Computing the trace in Eq. (30) one
obtains

$\begin{split}E_{\mathrm{xc}}=&{}-\sum_{\bm{p}\bm{p}^{\prime}}e^{2}\mathcal{U}_{\bm{p}-\bm{p}^{\prime}}\left\\{\mathcal{G}_{0}(\bm{p})\mathcal{G}_{0}(\bm{p}^{\prime})+\mathcal{G}_{3}(\bm{p})\mathcal{G}_{3}(\bm{p}^{\prime})\right\\}\\\
&{}-\sum_{\bm{p}\bm{p}^{\prime}}\alpha_{pp^{\prime}}^{2}\mathcal{U}_{\bm{p}-\bm{p}^{\prime}}\left\\{\mathcal{G}_{0}(\bm{p})\mathcal{G}_{0}(\bm{p}^{\prime})(\bm{\mathcal{M}}\cdot\bm{\mathcal{M}})+\mathcal{G}_{3}(\bm{p})\mathcal{G}_{3}(\bm{p}^{\prime})(\mathcal{M}_{z}^{2}-\mathcal{M}_{x}^{2}-\mathcal{M}_{y}^{2})\right\\}\,,\end{split}$
(33)

with
$\bm{\mathcal{M}}\equiv\bm{\mathcal{M}}(\bm{p},\bm{p}^{\prime})=[\bm{p}\times\bm{p}^{\prime}]$.
The first line of this equation is a common exchange energy of the Coulomb-
interacting electron gas [43]. The second line is the PSOI contribution to the
exchange energy. Notice that, unlike the Cooper channel above, the linear in
$\alpha$ term is absent.

###  IV.2 Paramagnetic phase

In the paramagnetic electron state $n_{-}(\bm{p})=n_{+}(\bm{p})\equiv
n(\bm{p})$ and Eq. (33) takes the form

$\begin{split}E_{\mathrm{xc}}=&{}-\sum_{\bm{p}\bm{p}^{\prime}}e^{2}\mathcal{U}_{\bm{p}-\bm{p}^{\prime}}n(\bm{p})n(\bm{p}^{\prime})\\\
&{}-\sum_{\bm{p}\bm{p}^{\prime}}\alpha_{pp^{\prime}}^{2}\mathcal{U}_{\bm{p}-\bm{p}^{\prime}}n(\bm{p})n(\bm{p}^{\prime})\bm{\mathcal{M}}^{2}(\bm{p},\bm{p}^{\prime})\,.\end{split}$
(34)

Taking variation with respect to $n(\bm{p})$, one finds the HF effective
dispersion of the form

$\begin{split}\hat{V}_{\bm{p}}^{\mathrm{HF}}={}&\frac{p^{2}}{2m}-2\sum_{\bm{p}^{\prime}}e^{2}\mathcal{U}_{\bm{p}-\bm{p}^{\prime}}n(\bm{p}^{\prime})\\\
&{}-2\sum_{\bm{p}^{\prime}}\alpha_{pp^{\prime}}^{2}\mathcal{U}_{\bm{p}-\bm{p}^{\prime}}n(\bm{p}^{\prime})\bm{\mathcal{M}}^{2}(\bm{p},\bm{p}^{\prime})\,.\end{split}$
(35)

In what follows, we assume a short-ranged, momentum-independent e-e
interaction potential,
$\mathcal{U}_{\bm{p}-\bm{p}^{\prime}}\equiv\mathcal{U}_{0}$, and adopt a
specific momentum dependence for the SOI magnitude, 444Subsequently, these
model restrictions will be relaxed to account for realistic long-ranged
interactions with dielectric screening and the precise microscopic momentum-
dependence of the SOI. A numerical treatment of 2D Rashba systems
incorporating these aspects can be found in Appendix C.

$\alpha_{pp^{\prime}}^{2}=\alpha^{2}\frac{\Lambda^{2}}{(\Lambda+p^{2})(\Lambda+{p^{\prime}}^{2})}\,,$
(36)

where $\Lambda/2m$ is an energy scale of the order of the band gap. This is a
simplest functional form that captures the decreasing magnitude of SOI as
momentum increases. We stress that by neglecting the momentum dependence of
$\alpha$ one could infer that the interaction parameter of the PSOI increases
indefinitely as the electron concentration grows [25]. However, an extremely
dense electron gas should essentially behave as non-interacting. To avoid the
contradiction, the theory of the PSOI must be generalized to account for the
momentum dependence of the SOI magnitude.

The ground state is constructed by filling the quantum states of the lowest
energy. These states have the least momentum when the HF potential
$\hat{V}_{\bm{p}}^{\mathrm{HF}}$ increases monotonically with $p$. However,
this is not the case with the potential of Eq. (35). The momentum dependence
of $\bm{\mathcal{M}}(\bm{p},\bm{p}^{\prime})$ makes the effective dispersion
non-monotonous, creating a moat-like dispersion. Consequently, it becomes
beneficial for electrons to populate states with higher momentum — a distinct
feature of the PSOI. This fact eventually leads to the topological Lifshitz
transition.

To see this, first assume that electrons in a 3D gas with concentration $n$
form a Fermi sea enclosed by a spherical Fermi surface of radius
$k_{F}={(3\pi^{2}n)^{1/3}}$, with
$n(\bm{p})=\theta(k_{F}-\lvert\bm{p}\rvert)$. Such distribution function
generates a spherically symmetric effective dispersion of the form

$\hat{V}_{\bm{p}}^{\mathrm{HF}}=\frac{p^{2}}{2m}-e^{2}\mathcal{U}_{0}n-\frac{2\alpha^{2}}{3\pi^{2}}\mathcal{U}_{0}\frac{\Lambda^{2}p^{2}}{\Lambda+p^{2}}\,\Xi(k_{F})\,.$
(37)

The factor $\Xi(k_{F})$ is given by

$\Xi(k_{F})=\frac{k_{F}^{3}}{3}-k_{F}\Lambda+\Lambda^{\frac{3}{2}}\arctan\frac{k_{F}}{\sqrt{\Lambda}}\,,$
(38)

and interpolates between $\Xi(k_{F})\approx k_{F}^{5}/5\Lambda$ for
$k_{F}\ll\sqrt{\Lambda}$ and $\Xi(k_{F})\approx k_{F}^{3}/3$ for
$k_{F}\gg\sqrt{\Lambda}$.

Figure 3: The effective dispersion $\hat{V}_{\bm{p}}^{\mathrm{HF}}$ as a
function of momentum for two values of the SOI magnitude. The position of the
Fermi momentum is shown by a vertical dotted line. The occupied states are
shown in color. The Fermi surface is a sphere, as shown in the inset.

The effective mass of the electron diverges when the SOI magnitude $\alpha$
equals to

$\alpha_{0}=\sqrt{\frac{3\pi^{2}}{4m\mathcal{U}_{0}\Lambda\Xi(k_{F})}}\,,$
(39)

with a flat band forming around the Brillouin zone center. The dependence of
the effective dispersion on the momentum becomes non-monotonous as soon as
$\alpha>\alpha_{0}$, see Fig. 3. A minimum appears for a finite momentum,
which depends on $\alpha$. For a given electron concentration, there exist a
critical $\alpha_{1}$ such that for $\alpha\leq\alpha_{1}$ all states with
$\lvert\bm{p}\rvert\leq k_{F}$ are occupied. This renders the Fermi surface as
a sphere, in agreement with our initial assumption.

The critical value of $\alpha_{1}$ is inferred from

$\hat{V}_{\bm{p}}^{\mathrm{HF}}\big{|}_{p=0}=\hat{V}_{\bm{p}}^{\mathrm{HF}}\big{|}_{p=k_{F}}$
(40)

to give $\alpha_{1}=\alpha_{0}\sqrt{1+k_{F}^{2}/\Lambda}$. For
$\alpha>\alpha_{1}$ states with the lowest energy no longer correspond to the
smallest momenta. Instead, they are found in a range of momenta
$k_{F_{1}}\leq\lvert\bm{p}\rvert\leq k_{F_{2}}$, forming a hollow spherical
shell. This behavior is illustrated in Fig. 4. Such a transformation signals a
Lifshitz transition, changing a topology of the Fermi surface.

Figure 4: The effective dispersion $\hat{V}_{\bm{p}}^{\mathrm{HF}}$ as a
function of momentum for the larger-than-critical value of the SOI magnitude.
The populated states are enclosed by a shell-like Fermi surface.

The corresponding distribution function takes the form

$n(\bm{p})=\theta(\lvert\bm{p}\rvert-
k_{F_{1}})\theta(k_{F_{2}}-\lvert\bm{p}\rvert)\,,$ (41)

where the two Fermi momenta are related by a requirement of the fixed electron
concentration, $k_{F_{2}}^{3}-k_{F_{1}}^{3}=3\pi^{2}n$. Such a distribution
gives rise to a self-consistent solution provided that

$\hat{V}_{\bm{p}}^{\mathrm{HF}}\big{|}_{p=k_{F_{1}}}=\hat{V}_{\bm{p}}^{\mathrm{HF}}\big{|}_{p=k_{F_{2}}}\,.$
(42)

The effective dispersion calculated with the distribution function (41) is

$\hat{V}_{\bm{p}}^{\mathrm{HF}}=\frac{p^{2}}{2m}-e^{2}\mathcal{U}_{0}n-\frac{2\alpha^{2}\mathcal{U}_{0}}{3\pi^{2}}\frac{\Lambda^{2}p^{2}}{\Lambda+p^{2}}\left[\Xi(k_{F_{2}})-\Xi(k_{F_{1}})\right]\,.$
(43)

Equations (42)-(43) lead to the following relation that allows one to find the
two Fermi momenta,

$\alpha^{2}[\Xi(k_{F_{2}})-\Xi(k_{F_{1}})]=\left(1+\frac{k_{F_{1}}^{2}}{\Lambda}\right)\left(1+\frac{k_{F_{2}}^{2}}{\Lambda}\right)\frac{3\pi^{2}}{4m\mathcal{U}_{0}\Lambda}\,.$
(44)

As soon as $\alpha>\alpha_{1}$, the solution of this equation emerges with
$k_{F_{1}}>0$, which determines the position of the shell boundaries in the
momentum space.

The total energy per one electron, calculated self-consistently from Eq. (34)
and taking into account the kinetic energy of the electrons filling the shell
in the momentum space, equals

$\begin{split}\varepsilon=&{}\frac{1}{10\pi^{2}n}\frac{k_{F_{2}}^{5}-k_{F_{1}}^{5}}{m}-e^{2}\mathcal{U}_{0}\frac{n}{4}\\\
&{}-\frac{\alpha^{2}}{6\pi^{4}n}\mathcal{U}_{0}\Lambda^{2}{\left[\Xi(k_{F_{2}})-\Xi(k_{F_{1}})\right]}^{2}\,.\end{split}$
(45)

This expression is valid for both shell-like ($\alpha>\alpha_{1}$) and
spherical ($\alpha\leq\alpha_{1}$) Fermi surfaces, with $k_{F_{1}}=0$ in the
latter case.

###  IV.3 Ferromagnetic phase

In a fully spin-polarized case of $n_{-}(\bm{p})=0$, the Eq. (33) reduces to

$\begin{split}E_{\mathrm{xc}}=&{}-\frac{1}{2}\sum_{\bm{p}\bm{p}^{\prime}}e^{2}\mathcal{U}_{\bm{p}-\bm{p}^{\prime}}n(\bm{p})n(\bm{p}^{\prime})\\\
&{}-\frac{1}{2}\sum_{\bm{p}\bm{p}^{\prime}}\alpha_{pp^{\prime}}^{2}\mathcal{U}_{\bm{p}-\bm{p}^{\prime}}n(\bm{p})n(\bm{p}^{\prime})\mathcal{M}_{z}^{2}(\bm{p},\bm{p}^{\prime})\,,\end{split}$
(46)

with the corresponding HF dispersion given by

$\begin{split}\hat{V}_{\bm{p}}^{\mathrm{HF}}={}&\frac{p^{2}}{2m}-\sum_{\bm{p}^{\prime}}e^{2}\mathcal{U}_{\bm{p}-\bm{p}^{\prime}}n(\bm{p}^{\prime})\\\
&{}-\sum_{\bm{p}^{\prime}}\alpha_{pp^{\prime}}^{2}\mathcal{U}_{\bm{p}-\bm{p}^{\prime}}n(\bm{p}^{\prime})\mathcal{M}_{z}^{2}(\bm{p},\bm{p}^{\prime})\,.\end{split}$
(47)

Assuming for a moment a fully spin-polarized spherical Fermi surface with

$K_{F}=\sqrt[3]{2}k_{F}={(6\pi^{2}n)}^{\frac{1}{3}},$ (48)

results in an anisotropic HF effective dispersion

$\hat{V}_{\bm{p}}^{\mathrm{HF}}=\frac{p^{2}}{2m}-e^{2}\mathcal{U}_{0}n-\frac{\alpha^{2}\mathcal{U}_{0}}{6\pi^{2}}\frac{\Lambda^{2}p^{2}_{\parallel}}{\Lambda+p^{2}}\,\Xi(K_{F})\,,$
(49)

where $\bm{p}_{\parallel}\equiv(p_{x},p_{y},0)$ is the in-plane momentum. This
indicates an increased in-plane effective mass and therefore an anisotropic
pancake shape of the Fermi surface.

Figure 5: The effective dispersion $\hat{V}_{\bm{p}}^{\mathrm{HF}}$ as a
function of the in-plane momentum $p_{\parallel}$ at $p_{z}=0$ for weak SOI.

According to Eq. (49), surfaces of a constant energy may be parameterized by

$(p^{2}+\Lambda)(p^{2}+\zeta)=4K^{2}p_{\parallel}^{2}\,,$ (50)

with some constants $\zeta$ and $K$ to be determined self-consistently. For a
small $\alpha$ the Fermi surface has a flattened ellipsoidal shape. However,
upon the increase of $\alpha$ it undergoes a qualitative change. The in-plane
effective mass diverges for the SOI magnitude of

$\alpha_{2}=\sqrt{\frac{3\pi^{2}}{m\mathcal{U}_{0}\Lambda\Xi(K_{F})}}\,.$ (51)

For $\alpha>\alpha_{3}$, where
$\alpha_{3}=\alpha_{2}\sqrt{1+K_{F}^{2}/\Lambda}$, the Fermi surface acquires
a toroidal shape, with the in-plane momenta limited to a range $K_{F_{1}}\leq
p_{\parallel}\leq K_{F_{2}}$, while the normal momenta are concentrating near
the center, $\lvert p_{z}\rvert<K_{F_{z}}$. As a result, the system undergoes
the topological Lifshitz transition, with the Fermi surface genus changing
from zero to one.

Figure 6: The effective dispersion $\hat{V}_{\bm{p}}^{\mathrm{HF}}$ as a
function of the in-plane momentum $p_{\parallel}$ at $p_{z}=0$, for the
larger-than-critical value of the SOI magnitude. The Fermi surface is shaped
like a torus.

One may show (see Appendix B) that the Fermi surface of Eq. (50) provides a
self-consistent solution to Eq. (47). The total energy per one electron in the
toroidal phase is equal to

$\varepsilon=\frac{1}{4m}\left(4K^{2}-\Lambda-\zeta\right)-e^{2}\mathcal{U}_{0}\frac{n}{2}-\frac{\alpha^{2}\mathcal{U}_{0}\Lambda^{2}\Xi^{2}}{4n}\,,$
(52)

with parameters $K$, $\zeta$ and $\Xi$ detailed in the Appenidix B. The
results for the elliptical phase are quite similar; however, we have omitted
them for brevity.

###  IV.4 Ferromagnetic Transitions

Figure 7: The energy per particle of a 3D electron gas, in units of
$K_{F}^{2}/2m$, as a function of the dimensionless SOI magnitude
$\tilde{\alpha}=\alpha K_{F}^{2}/e$, for $\tilde{\mathcal{U}}=1$ and
$\Lambda/K_{F}^{2}=1$. The red curve corresponds to the ferromagnetic (FM)
phase, and the blue curve to the paramagnetic (PM) one. The Stoner transition
occurs at $\tilde{\alpha}=\tilde{\alpha}_{\mathrm{FMT}}$.

Figure 7 depicts the total HF energy per electron for both the paramagnetic
and spin-polarized phases, plotted against the magnitude of the PSOI. This is
represented for $\tilde{\mathcal{U}}=1$, where the e-e interaction parameter
is normalized by the density of states as:

$\tilde{\mathcal{U}}=e^{2}\mathcal{U}_{0}\frac{mK_{F}}{2\pi^{2}\hbar^{2}}\,.$
(53)

At $\tilde{\alpha}=0$ the system is in the paramagnetic phase characterized by
a spherical Fermi surface. As PSOI increases, the first order ferromagnetic
transition occurs at a critical
$\tilde{\alpha}=\tilde{\alpha}_{\mathrm{FMT}}$, which is accompanied by a
Lifshitz transition to a state with an elliptical Fermi surface. In other
words, both spin symmetry and the Fermi-surface symmetry simultaneously
undergo a change at this point. As PSOI continues to increase, it induces a
Lifshitz transition to a toroidal spin-polarized phase at
$\tilde{\alpha}=\tilde{\alpha}_{3}$.

Figure 8: The energy per particle of a 3D electron gas as a function of the
dimensionless SOI magnitude $\tilde{\alpha}=\alpha K_{F}^{2}/e$ for
$\tilde{\mathcal{U}}=0.5$. The red curve corresponds to the ferromagnetic
phase, and the blue curve to the paramagnetic one.

For a smaller interaction parameter of $\tilde{\mathcal{U}}=0.5$, the system
can transition directly from the spherical paramagnetic phase to the toroidal
spin-polarized phase, bypassing the elliptical phase, as depicted in Fig. 8.
Conversely, with increased interaction, the system defaults to a ferromagnetic
state at zero SOI due to the conventional Stoner instability. Yet, any non-
zero value of $\alpha$ ensues breaking of the Fermi surface’s spherical
symmetry. A further increase of $\alpha$ results into the Lifshitz transition
from the ellipsoidal to the toroidal phase at
$\tilde{\alpha}=\tilde{\alpha}_{3}$, as visualized in Fig. 9.

Figure 9: The energy per particle of a 3D electron gas as a function of the
dimensionless SOI magnitude $\tilde{\alpha}=\alpha K_{F}^{2}/e$ for
$\tilde{\mathcal{U}}=2$. The red curve corresponds to the ferromagnetic phase,
and the blue curve to the paramagnetic one.

These findings are summarized in Fig. 10, showing the phase diagram in the
coordinates of dimensionless SOI coupling $\tilde{\alpha}$ vs. the interaction
strength $\tilde{\mathcal{U}}$. The paramagnet phase exhibits the first order
transition to the fully polarized ferromagnet. The latter may come with either
flat ellipsoidal or toroidal Fermi surfaces with the topological Lifshitz
transition between them. The hollow spherical shell paramagnetic phase,
discussed above, is not realized (at least within the present model), since
the transition to the FM phase takes place before it becomes favorable. It is
worth mentioning that the critical values of the parameters for the Stoner and
Lifshitz transitions, obtained from the HF energy computations, are not
expected to be quantitatively accurate. Correlation effects renormalize them,
as indicated by the Monte Carlo simulations. Nevertheless, one may expect that
the qualitative features of the phase diagram are still present, even with the
account of the correlations.

Figure 10: The phase diagram of the 3D Rashba system. Three horizontal cross-
sections correspond to Figs. 7–9.

###  IV.5 Ferromagnetic Superconductors

Both ellipsoidal and toroidal fully polarized ferromagnets are susceptible to
the development of the $p$-wave superconducting order, discussed in Section
III. In this case only $\Delta^{+}_{\bm{p}}$ order parameter is realized,
while $\Delta^{0}$ and $\Delta^{-}$ are absent. Since the toroidal
ferromagnetic phase requires a critical SOI strength, it is always formally
within the parameter region with the superconducting ground state. In reality,
if the disorder is sufficiently strong, the toroidal FM may not develop
superconductivity down to zero temperature.

The coexisting FM and superconducting orders break rotational, time-reversal
and global $U(1)$ symmetries. This puts the model into symmetry class D. The
latter is not topological in 3D, though it does admit an integer Chern number
in 2D. It is interesting to notice that the topological Lifshitz transition
from elliptical to the toroidal Fermi surface has direct consequences for the
superconducting order. Indeed, the $p$-wave superconductor with the order
parameter $\Delta^{+}$ and the ellipsoidal Fermi surface is nodal with the two
nodes on the flat parts of the ellipsoid (due to the $\sin\theta_{\bm{p}}$
factor in Eq. (13)). Once the Fermi surface transitions to the toroidal shape
the nodes disappear and the $p$-wave state $\Delta^{+}$ is fully gaped
(similarly to its 2D counterpart). This may lead to a weak first order
transition between the nodal and gaped groundstates, but its details require a
further investigation.

###  IV.6 Manifestations of the ferromagnetic transitions

The FM state is, of course, associated with a net magnetic moment. This may be
hard to detect due to, e.g., a domain structure (in presence of some uniaxial
magnetic anisotropy). Moreover, the magnetic moment itself does not reveal the
nature of the state and properties of its Fermi surface. Here we briefly
discuss other observable manifestations of the FM transitions.

The first order transition between the isotropic PM state and an anisotropic
FM state is associated with the jump in the electronic density of states at
the Fermi surface. It may be detected thus through a measurement of the
electronic specific heat. The onset of the anisotropy in the orbital space may
lead to anisotropic behavior of transport coefficients. This issue requires a
separate study, since its involves specific scattering mechanisms. The
Lifshitz transition between the ellipsoidal and toroidal Fermi surfaces does
not lead to a discontinuity in the density of states. It leads, however, to
its nonanalitic behavior,
$\nu(\tilde{\alpha})=\nu-\nu_{1}\sqrt{\tilde{\alpha}_{3}-\tilde{\alpha}}\,\theta(\tilde{\alpha}_{3}-\tilde{\alpha})$,
on the elipsoidal side of the transition. This may also lead to detectable
features in the electronic specific heat.

Besides thermodynamic or transport measurements, one may detect Friedel
oscillations of the electronic density, created by an impurity. Specifically,
the emergence of the two Fermi momenta, $k_{F_{1}}$ and $k_{F_{2}}$, in the
toroidal phase is reflected in a non-analytic behavior of a static charge
susceptibility $\chi(q)$ at points $q=2k_{F_{1}}$, $q=2k_{F_{2}}$, and
$q=k_{F_{1}}\pm k_{F_{2}}$. Consequently, the spatial profile of the Friedel
oscillations contains the corresponding harmonics, which can be observed using
STM. The reconstruction of the Fermi surface may be also visible in the
periods of the Shubnikov-de Haas magnetoresistance oscillations [44]. Indeed,
the oscillation periods are associated with the areas of extremal cross
sections of the Fermi surface, which change abruptly at all FM transitions,
considered above. Finaly, the more delicate probes such as ARPES or polarized
neutron scattering can reveal information about the shape of the Fermi surface
and a local magnetic moment.

##  V Concluding remarks

Rashba spin-orbit coupling of conduction band electrons is a legacy of a
multi-band nature of the material and presence of (spin-orbit split) valence
bands. Projection of a multi-band model onto the conduction band degrees of
freedom leads to a spin-dependent renormalization of the electron density, as
seen in Eq. (3). Besides more familiar manifestations in the single-particle
dispersion relation (for a broken inversion symmetry), RSOI modifies the
electron-electron interaction vertex. The latter effect, known as PSOI, does
not require an external electric field, or the inversion symmetry breaking. It
relies on internal fluctuating electric fields created by Coulomb interactions
between (renormalized) conduction band electrons. As a result, it has
implications for symmetric and isotropic 3D materials, where the single-
particle Rashba effects are absent.

Here we considered two consequences of PSOI-modified electron-electron
interaction vertex. In a way reminiscent to Kohn-Luttinger effect, it leads to
the $p$-wave superconducting pairing, mediated solely by the Coulomb
repulsion. In sufficiently clean materials it results in a particular $p$-wave
order, where spin up/down electrons form triplet pairs with $m=\pm 1$
projection of the angular momentum on a spontaneously chosen direction ($m=0$
component is disfavored). In 3D such order parameter has two nodes along the
chosen direction, while in 2D this direction is the normal one and the state
is fully gaped. The PSOI induced $p$-wave superconductor exhibits several
unique features such as spin-polarized supercurrents and the spin-valve effect
as well as peculiar structure of the magnetic vortexes.

Another consequence is that the Stoner FM transition, mediated by the exchange
Coulomb interaction, is facilitated by a strong enough PSOI. Moreover, it is
associated with the rotation symmetry breaking not only in the spin, but also
in the orbital spaces. The latter effect leads to deformation of the Fermi
surface of the polarized itinerant electrons into the ellipsoidal or toroidal
shape. It may undergo genus-changing topological Lifshitz transition between
these shapes as a function of the SOI magnitude or interactions strength. The
transition may have a number of observable manifestations in thermodynamic and
transport properties, as well as in the spectra of Friedel and Shubnikov-de
Haas oscillations. Finally, the anisotropic Stoner ferromagnet may coexist
with the $p$-wave superconductor. If this is the case, the Lifshitz
ellipsoidal to toroidal transition manifest itself as the transition between
nodal and fully gaped superconducting states.

###### Acknowledgements.

Helpful discussions with A. Chubukov, R. Fernandes, M. Glazov, D. Gutman, D.
Maslov, Yu. Oreg, V. Sablikov, B. Shklovskii and G. Vignale are gratefully
acknowledged. The work of Y.G. was supported by the Simons Foundation Grant
No. 1249376; A.K. was supported by the NSF Grant No. DMR-2037654.

##  Appendix A Microscopic model of the RSOI

Here a microscopic derivation of the RSOI Hamiltonian is outlined, with a
focus on a momentum dependence of the SOI coupling for a particularly simple
case of a 2D electron system with the inversion symmetry. The approach is
based on the symmetric Bernevig-Hughes-Zhang (BHZ) model [45]. This model is
widely recognized for its capacity to describe various 2D systems with strong
RSOI, across both topological and trivial phases.

The BHZ model is expressed in a four-band basis, denoted as
$\mathcal{B}={\\{\lvert\operatorname{E\\!\uparrow}\rangle,\lvert\operatorname{H\\!\uparrow}\rangle,\lvert\operatorname{E\\!\downarrow}\rangle,\lvert\operatorname{H\\!\downarrow}\rangle\\}}^{\intercal}$.
Within this basis, the states $\lvert\operatorname{E\\!\uparrow}\rangle$ and
$\lvert\operatorname{E\\!\downarrow}\rangle$ consist of the electron- and
light-hole band states, each carrying an angular momentum projection of
$m_{J}=\pm 1/2$. Conversely, the states
$\lvert\operatorname{H\\!\uparrow}\rangle$ and
$\lvert\operatorname{H\\!\downarrow}\rangle$ correspond to the heavy-hole
states, with an angular momentum projection of $m_{J}=\pm 3/2$. The
Hamiltonian governing the envelope wave-function in this context is as follows

$\hat{H}_{\mathrm{BHZ}}=\begin{bmatrix}m(\hat{p})&A\hat{p}_{+}&0&0\\\
A\hat{p}_{-}&-m(\hat{p})&0&0\\\ 0&0&m(\hat{p})&-A\hat{p}_{-}\\\
0&0&-A\hat{p}_{+}&-m(\hat{p})\end{bmatrix}\,.$ (54)

Here

$m(\hat{p})=m+B\hat{p}^{2}\,,$ (55)

with $m$ the band gap, $B$ the bare dispersion curvature, and $A$ the band
hybridization parameter. Then, $\hat{p}$ is the momentum operator,
$\hat{p}_{\pm}=\hat{p}_{x}\pm i\hat{p}_{y}$.

The BHZ model inherently incorporates a strong SOI, premised on the assumption
of an infinitely large spin-splitting in the valence band. Since the spin
sectors are decoupled it is possible to treat them separately by considering a
$2\times 2$ Hamiltonian

$\hat{H}_{0}(\hat{p})=m(\hat{p})\tau_{3}+A(\hat{\bm{p}}\cdot\hat{\bm{\tau}})\,,$
(56)

with $\hat{\tau}_{i}$ the pseudospin (subband) Pauli matrices. The Hamiltonian
can be diagonalized

$\hat{\mathcal{H}}_{0}(\hat{p})=\hat{U}(\hat{p})\hat{H}_{0}(\hat{p})\hat{U}^{+}(\hat{p})$
(57)

by a unitary Foldy-Wouthuysen transformation

$\hat{U}(\hat{p})=\cos\theta(\hat{p})+\sin\theta(\hat{p})\frac{(\hat{\bm{p}}\cdot\hat{\bm{\tau}})}{p}\tau_{3}$
(58)

to acquire the form

$\hat{\mathcal{H}}_{0}(\hat{p})=\varepsilon(\hat{p})\tau_{3}\,,$ (59)

with the band dispersion

$\varepsilon(\hat{p})=\sqrt{m^{2}(\hat{p})+A^{2}\hat{p}^{2}}\,,$ (60)

and

$\tan 2\theta(\hat{p})=-A\frac{\hat{p}}{m(\hat{p})}\,.$ (61)

For a sufficiently small hybridization parameter the unitary operator (58) may
be approximated as

$\hat{U}(\hat{p})\approx
1-\frac{A}{2m(\hat{p})}(\hat{\bm{p}}\cdot\hat{\bm{\tau}})\tau_{3}-\frac{A^{2}\hat{p}^{2}}{8m^{2}(\hat{p})}+\mathcal{O}(A^{3})\,,$
(62)

under the condition that $Ap\ll m(p)$.

In the context of the Dirac equation, this condition holds true only for
limited momentum values near the minimum of the electron band. Within this
regime, it gives rise to the conventional formula for the relativistic SOI,
featuring a constant magnitude for $\alpha$, equal to
$\alpha=e\lambdabar^{2}/4$, $\lambdabar$ being the Compton wavelength of the
electron [46].

In contrast, the BHZ model presents a notable departure from this picture. By
virtue of Eq. (55), the above expansion applies to both the small ($p\to 0$)
and large ($p\gtrsim A/B$) values of momenta, provided $B\neq 0$. This broader
applicability range allows for a generalized RSOI Hamiltonian capable of
describing electron dynamics in the quantum states with large momenta.

Applying the unitary transformation to the Hamiltonian

$\hat{H}(\hat{p})=\hat{H}_{0}(\hat{p})-e\varphi(\hat{\bm{r}}),$ (63)

that describes the interaction with a scalar potential, one should take into
account that the momentum-dependent operators $\hat{U}(\hat{p})$ do not
commute with a position-dependent $\varphi(\hat{\bm{r}})$. The commutator
terms generate the operator

$\hat{\mathcal{V}}(\hat{p})=\frac{A}{2m(\hat{p})}\hat{\bm{\tau}}\cdot(-e\grad_{\bm{r}}\varphi\times\hat{\bm{p}})\frac{A}{2m(\hat{p})}\,.$
(64)

Within the low-energy approximation one focuses on the electron dynamics in
the conduction band by declaring the spinor components in the valence band to
be small and keeping only the $\hat{\mathcal{V}}_{11}$ matrix element
pertaining to the conduction band. Combining the terms
$\hat{\mathcal{V}}_{11}^{(s)}$ from both spin sectors the following
generalized RSOI Hamiltonian in the reduced conduction band basis is obtained,

$\hat{H}_{\mathrm{RSOI}}(\hat{p})=\frac{A}{2m(\hat{p})}\hat{\bm{\sigma}}\cdot(-e\grad_{\bm{r}}\varphi\times\hat{\bm{p}})\frac{A}{2m(\hat{p})}\,,$
(65)

with $\hat{\bm{\sigma}}$ the real spin Pauli vector. Equation (65) is
consistent with earlier result of Ref. [47] in the limit of small momenta.
This gives rise to the interaction vertex of Eq. (5) with a momentum-dependent
SO coupling of the form

$\alpha_{p_{1}p_{2}}=\frac{A^{2}}{4m(p_{1})m(p_{2})}\,,$ (66)

which incorporates the decrease of SOI magnitude at large momenta.

##  Appendix B Self-consistent HF solution

Here, the toroidal Fermi surface is demonstrated to be a self-consistent
solution of Eq. (47). The derivation proceeds in two steps. First, the
effective dispersion in Eq. (47) is found assuming that the electrons occupy
the toric region in the momentum space limited by the Fermi surface of Eq.
(50). Second, the isoenergetic surface of the resulting HF potential is
verified to be that of Eq. (50) under the proper choice of parameters, which
results in self-consistency conditions.

A change of variables $q_{\parallel}^{2}=p^{\prime 2}_{\parallel}+\delta^{2}$,
$q_{z}=p^{\prime}_{z}$ with $\delta=(\Lambda-\zeta)/4K$ transforms Eq. (50) to
the canonical equation of a torus,

${(q^{2}+K^{2}-k^{2})}^{2}=4K^{2}q_{\parallel}^{2}\,,$ (67)

where

$k^{2}={(K-\delta)}^{2}-\zeta\,.$ (68)

The torus interior, $\mathcal{D}$, can be parametrized by

$\begin{split}q_{x}&=(K+\eta\cos\theta)\cos\phi\,,\\\
q_{y}&=(K+\eta\cos\theta)\sin\phi\,,\\\ q_{z}&=\eta\sin\theta\,,\\\
\end{split}$ (69)

with $\eta\in(0,k)$, $\phi\in(0,2\pi)$, and $\theta\in(0,2\pi)$. The Jacobian
of the transformation is $J(\eta,\theta)=\eta(K+\eta\cos\theta)$.

The momentum-dependent factor that enters Eq. (47) is

$\mathcal{M}_{z}^{2}(\bm{p},\bm{p}^{\prime})={[\bm{p}\times\bm{p^{\prime}}]}^{2}_{z}=p^{2}_{\parallel}p^{\prime
2}_{\parallel}\sin^{2}\widehat{\bm{p}_{\parallel}\bm{p}^{\prime}_{\parallel}}\,.$
(70)

Due to the axial symmetry of the problem, the
$\sin^{2}\widehat{\bm{p}_{\parallel}\bm{p}^{\prime}_{\parallel}}$ averages to
$1/2$ when integrated over the polar angle. The PSOI contribution to the HF
potential of Eq. (47) takes the form

$\begin{split}\hat{V}_{\mathrm{PSOI}}(\bm{p})=&{}-\frac{\alpha^{2}\mathcal{U}_{0}\Lambda^{2}}{\Lambda+p^{2}}\int\frac{d^{3}p^{\prime}}{{(2\pi)}^{3}}\frac{{[\bm{p}\times\bm{p^{\prime}}]}^{2}_{z}}{\Lambda+p^{\prime
2}}\\\
=&{}-\frac{\alpha^{2}\mathcal{U}_{0}\Lambda^{2}p_{\parallel}^{2}}{\Lambda+p^{2}}\frac{\Xi}{2}\,,\end{split}$
(71)

where

$\begin{split}\Xi=&{}\int\frac{d^{3}p^{\prime}}{{(2\pi)}^{3}}\frac{p^{\prime
2}_{\parallel}}{\Lambda+p^{\prime 2}}=\int_{\mathcal{D}}\frac{d\eta d\theta
d\phi}{{(2\pi)}^{3}}J(\eta,\theta)\frac{{(K+\eta\cos\theta)}^{2}-\delta^{2}}{\eta^{2}+2K\eta\cos\theta+K^{2}+\Lambda-\delta^{2}}\\\
={}&\frac{1}{96\pi
K^{3}}\Big{[}k^{6}-k^{4}\left(3\delta^{2}+\varkappa+3K^{2}-3\Lambda\right)+k^{2}\left(\left(\delta^{2}-\Lambda\right)\left(3\delta^{2}+2\varkappa-3\Lambda\right)+21K^{4}+2K^{2}(\varkappa-6\Lambda)\right)\\\
&{}+\left({\left(\delta^{2}-\Lambda\right)}^{2}+K^{4}-2K^{2}\left(\delta^{2}+5\Lambda\right)\right)\left(-\delta^{2}-\varkappa+K^{2}+\Lambda\right)\Big{]}\,,\end{split}$
(72)

with

$\varkappa\equiv\sqrt{\left[{(K+k)}^{2}+\Lambda-\delta^{2}\right]\left[{(K-k)}^{2}+\Lambda-\delta^{2}\right]}\,.$
(73)

Consequently, the total HF potential of Eq. (47) acquires the form

$\hat{V}_{\bm{p}}^{\mathrm{HF}}=\frac{p^{2}}{2m}-e^{2}\mathcal{U}_{0}n-\frac{\alpha^{2}\mathcal{U}_{0}\Lambda^{2}p_{\parallel}^{2}}{\Lambda+p^{2}}\frac{\Xi}{2}\,.$
(74)

The Fermi surface of $\hat{V}_{\bm{p}}^{\mathrm{HF}}=E_{F}$ for the electron
dispersion of Eq. (74) has indeed the assumed form of Eq. (50), provided that
the following self-consistency conditions hold:

$\displaystyle K^{2}$
$\displaystyle=\frac{m\alpha^{2}\mathcal{U}_{0}\Lambda^{2}}{4}\Xi\,,$ (75)
$\displaystyle\zeta$ $\displaystyle=-2m(E_{F}+e^{2}\mathcal{U}_{0}n)\,,$ (76)

which should be supplemented by Eq. (68) and the relation $k^{2}K=4\pi n$
between the Fermi sea volume and electron concentration. The system of these
four equations allows one to find the three quantities: $K$, $k$, and $\zeta$,
which determine the geometry of the Fermi surface, as well as the Fermi energy
$E_{F}$.

##  Appendix C 2D Rashba Ferromagnets

Here the effects of PSOI are explored in 2D electron systems. To begin,
consider a particular case of a system symmetric with respect to the inversion
of the normal $z\to-z$. Within this framework, the PSOI emerges solely due to
the in-plane Coulomb fields, created by interacting electrons. The symmetric
2D structures represent a fascinating object of research in their own right,
as they are realized through freely suspended 2D layers, with a focus on
amplifying the effects of e-e interactions [48].

The expressions for the effective electron dispersion and exchange energy
presented in Section IV also apply here, with the integration running over the
2D momentum space. The analysis is performed within the BHZ model [45], with
the specific form of the spin-orbit coupling as given by Eq. (66), and kinetic
energy of Eq. (60).

The interaction in 2D layers of Rashba materials is governed by the Rytova-
Keldysh potential [49, 50]

$\mathcal{U}_{\bm{q}}=\frac{2\pi}{q(1+ql)}\,,$ (77)

where the length $l$ is a phenomenological parameter of the theory, which
generally speaking should be determined from the first-principles calculations
[51]. It can be estimated from the layer polarizability as $l=\epsilon d/2$,
where $d$ stands for the layer thickness, and $\epsilon$ is the in-plane
component of the dielectric tensor of the corresponding bulk material.

Spin polarization is controlled by the balance between the kinetic and
exchange interaction (both Coulomb and PSOI) energies. The degree of the spin
polarization is found by identifying the minimum of the total HF energy. As
the three contributions display distinct density dependencies [25], it is
anticipated that the lowest energy state will be FM in two regions: at a low
density as a result of Coulomb exchange and at a high density due to PSOI.

Figure 11: The phase diagram for the spin polarization of a 2D electron gas
with PSOI. The system parameters are those typically found in 2D Rashba
materials [2]: $m=$0.01\text{\,}\mathrm{e}\mathrm{V}$$,
$B=$50\text{\,}{\mathrm{\text{\r{A}}}^{2}}\mathrm{e}\mathrm{V}$$,
$d=$50\text{\,}\mathrm{\text{\r{A}}}$$, $\epsilon=20$.

These expectations are confirmed by Fig. 11, which presents the phase diagram
of system magnetization vs. BHZ velocity $A$, which serves as the magnitude of
the PSOI, and $r_{s}$ values 555The parameter $r_{s}$, defined through the
Bohr radius, typically describes a Galilean-invariant electron system that
interacts via a power-law Coulomb potential. It is clear that neither of these
characteristics apply to our system. However, we still need to establish some
scale to investigate the dependence of the HF energy on the electron
concentration. Thus we formally define the Bohr radius $a_{B}$ for the Coulomb
interaction within the bulk material, characterized by the dielectric constant
$\epsilon$. This definition is made under the assumption of an effective
electron mass, as determined by the bare band curvature $B$..

One identifies two distinct, fully spin-polarized phases. A new region of the
FM phase emerges at lower $r_{s}$ values when the PSOI magnitude surpasses a
certain critical threshold. Its Fermi surface exhibits a shape of annulus
$k_{F_{1}}\leq k\leq k_{F_{2}}$, which can be viewed as a 2D projection of the
toroidal Fermi-surface discussed in Section IV.3.

This result is seemingly at odds with the conventional perspective on Stoner
ferromagnetism, which is expected to only occur at sufficiently low electron
densities. Such transition, driven by the Coulomb exchange, is indeed present
as FM2 phase in Fig. 11 in a FM region of high $r_{s}$. It exhibits a circular
Fermi surface. Conversely, the annular FM1 phase emerges in the regime of a
dense electron gas, where the PSOI interactions are strong due to their strong
momentum dependence. Both FM phases exhibit spin polarization normal to the
plane, with the easy axis set by the PSOI. This is also different from the
conventional 2D Stoner ferromagnetism, where the spin rotation symmetry can
spontaneously break in any direction.

The emergence of the annular FM phase is due to the moat-band dispersion
created by PSOI. Notice that the single-particle Rashba effect, due to a
normal electric field $\mathcal{E}_{z}$, also leads to a similar moat-band. In
the latter scenario, the electron system displays a momentum-locked chiral
spin texture. In the annular FM phase, on the other hand, electrons exhibit
complete spin polarization in the $z$-direction.

In both FM1 and FM2 phases, electrons remain unaffected by the single-particle
RSOI,
$H_{\mathrm{RSOI}}=\alpha(\bm{\sigma}\times\bm{\mathcal{E}}_{z})\cdot\bm{p}$,
since the magnetization, $\bm{\sigma}$, is colinear with the external field.
Only when the electric field surpass a certain critical magnitude of the order
of $\mathcal{E}_{c}\simeq v_{F}/\alpha$, the $z$-polarized FM phases lose
their stability. A transition to an alternative spin-polarized phase,
potentially exhibiting nematic order, is plausible [52, 53, 54] at a larger
external field.

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