# Electron correlations due to pair spin-orbit interaction in 2D electron
systems

Yasha Gindikin and Vladimir A. Sablikov  Kotelnikov Institute of Radio
Engineering and Electronics, Russian Academy of Sciences, Fryazino branch,
Fryazino, 141190, Russia

###### Abstract

We show that the pair spin-orbit interaction (PSOI) created by the Coulomb
fields of interacting electrons leads to the appearance of an unconventional
correlated electronic state in two-dimensional materials with the large Rashba
effect. The effect manifests itself at sufficiently strong PSOI as a sharp
peak in the structure factor, indicating a tendency to form a striped
structure with a spatial scale determined by the competition between the
Coulomb repulsion and the PSOI-induced attraction of electrons. Above the
critical value of PSOI, the system becomes unstable with respect to the charge
density fluctuations on this scale.

##  I Introduction

The interplay of the strong spin-orbit interaction (SOI) in Rashba materials
Manchon  _et al._ (2015) and Coulomb electron-electron (e-e) repulsion leads
to a bunch of qualitatively new physical effects, including the emergence of
new correlated states Berg  _et al._ (2012); Silvestrov and Entin-Wohlman
(2014); Ashrafi  _et al._ (2013); Fu (2015), unusual collective modes Maiti
_et al._ (2015); D’Amico  _et al._ (2019); Verma  _et al._ (2020), and even
bound electron pairs (BEPs) Damanet  _et al._ (2021).

The Rashba SOI is produced by the electric fields external to the crystal
lattice. In quantum structures the common sources of this field include the
confining potential, the charged impurities and structure defects. It is well
known that the Coulomb fields of interacting electrons also produce the SOI
which manifests directly in the e-e interaction Hamiltonian Bethe and Salpeter
(1957). As a result the interaction Hamiltonian gains the contribution that
depends on the electron spins and momenta. Effects of this so-called pair
spin-orbit interaction (PSOI) were until very recently considered only as a
small perturbation in such problems as spin dynamics, spin-spin interaction,
spin current generation, etc Boguslawski (1980); Bădescu  _et al._ (2005);
Glazov and Kulakovskii (2009); Glazov  _et al._ (2011). However, similarly to
the Rashba SOI, the PSOI is strongly enhanced in Rashba materials and
therefore can produce strong changes in electronic states Gindikin and
Sablikov (2020).

The strong PSOI can generate a plethora of non-trivial effects due to the
effective attraction that this interaction creates. The attraction mechanism
is quite clear Gindikin and Sablikov (2020). The PSOI created by the Coulomb
electric field of a given electron decreases the energy of another electron
possessing a particular spin orientation relative to its momentum. This effect
increases with decreasing the distance between electrons, which exactly
implies the attraction. The attraction can lead to the emergence of BEPs with
highly unusual configuration of the charge and spin density Gindikin and
Sablikov (2018a, b); Gindikin  _et al._ (2020).

Of greatest interest is, of course, the collective behavior of a many-electron
system with a strong PSOI, but this problem has been studied extremely poorly
to date. In Ref. Gindikin and Sablikov (2017) we considered a specific
situation of a gated one-dimensional quantum wire with the PSOI produced by
means of the image charges on the gate. In this case the PSOI leads to the
appearance of a correlated state with unusual collective excitations. One of
the two collective modes strongly softens in the long-wavelength part of the
spectrum with increasing the PSOI strength, and becomes unstable when the PSOI
exceeds a critical value.

In this paper, we turn to a more general statement of the problem by
considering a two-dimensional (2D) electron system with a richer configuration
of Coulomb fields generating PSOI. The electron dynamics is described using
the $k\cdot p$ method Voon and Willatzen (2009), assuming that the Coulomb
electric fields are sufficiently smooth.

To begin with, we note that in the presence of the PSOI the effective strength
of the e-e interaction is determined by two parameters. This is in stark
contrast to the conventional case of an electron gas with Coulomb interaction
only, where the interaction strength is characterized by the parameter
$r_{s}$, which is the ratio of the inter-electron distance to the Bohr radius
$a_{B}$. The e-e interaction Hamiltonian contains the PSOI component
$H_{\mathrm{PSOI}}$ in addition to the usual Coulomb term $H_{\mathrm{Coul}}$,
so that the system Hamiltonian is

$\begin{split}H={}&H_{\mathrm{kin}}+H_{\mathrm{Coul}}+H_{\mathrm{PSOI}}\\\
={}&\sum_{i}\frac{\hat{\bm{p}}_{i}^{2}}{2m}+\frac{1}{2}\sum_{i\neq
j}\mathcal{U}(\bm{r}_{i}-\bm{r}_{j})\\\ &{}+\frac{\alpha}{\hbar}\sum_{i\neq
j}\left(\hat{\bm{p}}_{i}\times\bm{\mathcal{E}}(\bm{r}_{i}-\bm{r}_{j})\right)\cdot\bm{\sigma}_{i}\,.\end{split}$
(1)

Here $\mathcal{U}(\bm{r})=e^{2}/\epsilon r$ is the Coulomb interaction
potential,
$\bm{\mathcal{E}}(\bm{r})=\frac{1}{e}\nabla_{\\!\bm{r}}\mathcal{U}(\bm{r})$ is
the pair Coulomb field that produces PSOI, $\hat{\bm{p}}_{i}$ is the momentum
operator of the $i$-th electron, $m$ is the effective mass,
$\bm{\sigma}\equiv(\sigma_{x},\sigma_{y},\sigma_{z})$ is the Pauli vector, and
$\alpha$ stands for the Rashba constant of the material. It is convenient to
introduce the dimensionless Rashba constant
$\tilde{\alpha}=\alpha/ea_{B}^{2}$.

The e-e interaction strength is characterized by the ratio of the interaction
energy to the Fermi energy. The parameter $r_{s}$ relates only to the Coulomb
term. The contribution of the PSOI term is described by another parameter
$\tilde{\alpha}/r_{s}$. It is remarkable that both parameters depend
differently on the parameters of the electronic system. In particular, while
the parameter $r_{s}$ decreases with increasing the electron density, the
parameter $\tilde{\alpha}/r_{s}$, on the contrary, increases. Therefore the
PSOI correlations can dominate when the density is high enough. While the
effect of Coulomb correlations is largely understood Giuliani and Vignale
(2005), the role of the PSOI-induced correlations and the conditions under
which they lead to a radical rearrangement of the electronic system remain to
be elucidated.

This paper aims to find out whether the PSOI creates characteristic
correlations, under what conditions they become significant, and how this
manifests itself in the spectrum of collective excitations. To this end, we
study the collective excitations and charge correlations in a 2D electron
system with the in-plane reflection symmetry, where the PSOI is produced by
the in-plane pair Coulomb field. The calculations are carried out in the
framework of the random phase approximation (RPA).

We have found that the static structure factor $S(q)$ as a function of the
wave vector $q$ acquires a sharp peak around a certain value of $q=q_{c}$ when
the PSOI parameter is large enough $\tilde{\alpha}/r_{s}\gtrsim 1/4$, which
indicates that the PSOI component of the e-e interaction is comparable in
magnitude to the Fermi energy. The peak clearly shows the appearance of strong
electron correlations on the $q_{c}$ scale, which are specific for the PSOI.
They arise owing to the competition between the Coulomb repulsion of electrons
and their attraction caused by the PSOI, which determines this characteristic
spatial scale. Interestingly, the PSOI correlated state appears at rather high
density of electrons and, correspondingly, at small $r_{s}$, when the usual
Coulomb interaction is small.

When $\tilde{\alpha}/r_{s}$ exceeds a critical value, a new mode due to PSOI
arises in the spectrum of collective excitations of the system in addition to
common long-wave plasmons. The mode exists only in a finite band of wave
vectors around $q_{c}$, the band width growing with $\tilde{\alpha}/r_{s}$.
The mode frequency is purely imaginary. It is interesting that the electron
density fluctuations growing with time are not polarized in spin. Thus the
spatially uniform paramagnetic state of the electron system becomes unstable
with respect to the charge density fluctuations on the $q_{c}$ scale. For
realistic values of the SOI parameter $\tilde{\alpha}\ll 1$ in Rashba
materials, the critical value $q_{c}\propto\tilde{\alpha}^{1/3}k_{F}$ lies in
the long-wave part of the spectrum.

##  II Model and results

In this section we consider the linear response of the 2D electron gas with
PSOI to the external electric potential, the dynamic charge susceptibility,
the static structure factor, and the spectrum of the collective modes.

The 2D electron system is assumed to be symmetric with respect to the
inversion of the normal to the plane. In this case the PSOI is produced by the
in-plane pair Coulomb field in contrast to the gated one-dimensional quantum
wire where only the normal component of the Coulomb field is important
Gindikin and Sablikov (2017). It is worth noting that the PSOI crucially
depends on the geometry of the generating electric fields and momenta of
interacting electrons. In the situation under consideration, both of these
quantities are 2D vectors, the topology of which is determined self-
consistently.

The results are obtained using the equation of motion for the quantum Wigner
function, which we derive and solve in the RPA, following Ref. Gindikin and
Sablikov (2017). The details of the calculation are presented in Appendix A.

###  II.1 Charge susceptibility

The density $n^{(s)}_{q\omega}$ of the electrons with the $z$-component of the
spin equal to $s=\pm 1$, in units of $\tfrac{\hbar}{2}$, satisfies the
following system of linear equations

$\displaystyle\chi_{0}^{-1}n^{(s)}_{q\omega}-V_{q\omega}\sum_{\varsigma=\pm}n^{(\varsigma)}_{q\omega}=\varphi_{q\omega}\,,$
(2)

with the external potential $\varphi_{q\omega}$, and the interaction potential

$V_{q\omega}=\mathcal{U}_{q}+8\frac{\alpha^{2}}{e^{2}}\mathcal{U}^{2}_{q}\chi_{j}\,.$
(3)

The first term of the interaction potential is due to the Coulomb e-e
repulsion. For the 2D electron gas formed in a uniform system with a bulk
dielectric constant $\epsilon$ the e-e repulsion is governed by the pure
Coulomb potential $\mathcal{U}_{q}=2\pi e^{2}/\epsilon q$. The second term of
the interaction potential is exactly due to the PSOI. The dynamic
susceptibilities $\chi_{0}$ and $\chi_{j}$ are given by Eq. (30) and Eq. (35).

Since $V_{q\omega}$ is spin-independent, the solutions of Eq. (2) correspond
to the equal response of up- and down-spin densities,
$n^{(+)}_{q\omega}=n^{(-)}_{q\omega}$. The dynamic charge susceptibility is

$\chi_{n}(q,\omega)=\frac{1}{{(2\chi_{0})}^{-1}-V_{q\omega}}\,.$ (4)

###  II.2 Static structure factor

Figure 1: The static structure factor $S(q)$ as a function of $q$ for three
values of the $r_{s}$. The PSOI magnitude is $\tilde{\alpha}=0.1$, which
corresponds to $r_{s}^{*}=0.3$.

Consider the static structure factor $S(q)$, which is related to the charge
susceptibility of Eq. (4) via

$S(q)=-\frac{\hbar}{\pi
n}\int_{0}^{\infty}d\omega\,\operatorname{Im}\chi_{n}(q,\omega)\,,$ (5)

$n$ being the mean electron density. It is of interest to study the structure
factor as a function of $q$ for different values of the e-e interaction
parameters. Since there are two such parameters, it is convenient to fix the
value of the PSOI constant $\tilde{\alpha}$ and change the parameter $r_{s}$
in such a way that both interaction parameters, $r_{s}$ and
$\tilde{\alpha}/r_{s}$, are varied. The result is plotted in Fig. 1.

First of all, we found that the structure factor has a strong singularity at a
certain value of the parameter $r_{s}=r_{s}^{*}$,

$r_{s}^{*}=\frac{2^{\frac{13}{6}}\tilde{\alpha}}{\sqrt{2^{\frac{1}{3}}+2{\tilde{\alpha}}^{\frac{2}{3}}3^{\frac{2}{3}}}}\,.$
(6)

As $r_{s}$ lowers down to this critical value, the spectral weight is shifted
towards the long-wave part of the spectrum, and eventually a sharp peak is
formed in the structure factor at the critical value $q_{c}$ of the wave-
vector, given by

$\frac{q_{c}}{k_{F}}=\frac{2{\tilde{\alpha}}^{\frac{1}{3}}3^{\frac{1}{3}}}{\sqrt{2^{\frac{1}{3}}+2{\tilde{\alpha}}^{\frac{2}{3}}3^{\frac{2}{3}}}}\,,$
(7)

which indicates the appearance of strong electron correlations due to PSOI.

The characteristic spatial scale $q_{c}$ arises as a result of the competition
between the Coulomb repulsion of electrons and their attraction caused by the
PSOI. Its dependence on $\tilde{\alpha}$ is displayed in Fig. 2. When PSOI is
extremely strong, $\tilde{\alpha}\gg 1$, the critical value tends to
$q_{c}=\sqrt{2}k_{F}$. For small SOI parameter $\tilde{\alpha}\ll 1$,
typically found in common Rashba materials, the critical value
$q_{c}\propto\tilde{\alpha}^{1/3}k_{F}$ lies in the long-wave part of the
spectrum. The dependence of the critical value of $r_{s}^{*}$ on the PSOI
strength is plotted in Fig. 3.

Figure 2: The critical value $q_{c}$ as a function of $\tilde{\alpha}$. Figure
3: The critical value $r_{s}^{*}$ as a function of $\tilde{\alpha}$.

###  II.3 Collective modes

The collective modes are given by zeroes of the denominator of Eq. (4). This
equation has two types of solutions defining two collective modes.

In the long-wave region the collective modes are common plasmons, the spectrum
of which gets a correction from the PSOI. At $q\ll k_{F}$,

$\omega_{pl}=\omega_{2D}\sqrt{1+\frac{\alpha^{2}}{e^{2}}q^{2}k_{F}^{2}}\,,$
(8)

with

$\omega_{2D}=\sqrt{\frac{2\pi e^{2}n}{\epsilon m}q}$ (9)

being the standard plasmon dispersion in a 2D electron gas Stern (1967). The
correction, albeit small, can be accessible to experiment, since the high-
accuracy measurements of the 2D plasmon spectra become available such as
detecting the microwave absorption from the recombination photoluminescence
spectrum Muravev  _et al._ (2020); Zarezin  _et al._ (2022). The plasmon
spectrum is shown in Fig. 4.

Figure 4: The plasmon frequency $\omega$ as a function of wave vector for
three values of the PSOI constant, with $r_{s}=1$. The frequency is normalized
at $\omega_{0}=v_{F}k_{F}$. The dashed line shows the boundary of the
particle-hole continuum.

Most importantly, a new collective mode due to PSOI arises in addition to
plasmons as soon as $r_{s}\leq r_{s}^{*}$. At $r_{s}=r_{s}^{*}$ the mode
appears to exist at a single critical value $q_{c}$ of the wave-vector,
whereas at $r_{s}<r_{s}^{*}$ the region where the mode exists expands to a
finite band of wave vectors $[q_{1},q_{2}]$, the band width growing with
lowering $r_{s}$. The spectrum of this mode is illustrated by Fig. 5.

Figure 5: The imaginary part of the frequency of a new collective mode due to
PSOI as a function of wave vector. The mode dispersion is shown for three
values of $r_{s}$ to trace how the instability develops in the system with
increasing the PSOI interaction parameter of $\tilde{\alpha}/r_{s}$. The PSOI
magnitude is fixed to be $\tilde{\alpha}=0.1$, which corresponds to
$r_{s}^{*}=0.3$.

The mode frequency is purely imaginary. For every $q$ within the allowed band
of $q\in(q_{1},q_{2})$ there are two branches, both with
$\omega^{\prime\prime}>0$ and $\omega^{\prime\prime}<0$, forming together the
petal-like shape. The frequencies of the two branches give, respectively, the
increment and decrement of the time-dependent fluctuations in the system. The
mode is characterized by the oscillations of up- and down-spin densities with
equal amplitudes $n^{(+)}_{q\omega}=n^{(-)}_{q\omega}$, which corresponds to
the excitation in the charge sector. Hence the electron density fluctuations
growing with time as $\propto\exp(\omega^{\prime\prime}t)$ are not polarized
in spin. This indicates that an instability of the spatially uniform
paramagnetic ground state of the 2D electron gas with PSOI develops with
respect to the charge density fluctuations on the $q_{c}$ scale. However, at
$r_{s}>r_{s}^{*}$ the system is stable.

The effects of PSOI are enhanced in 2D atomically thin layers, either freely
suspended Pokhabov  _et al._ (2018); Rössler  _et al._ (2010) or immersed in a
weak dielectric. The e-e repulsion is governed there by the Rytova-Keldysh
potential

$\mathcal{U}_{q}=\frac{2\pi e^{2}}{q(1+ql)}\,,$ (10)

where $l$ is a characteristic length that can be roughly estimated as
$\epsilon_{\parallel}d/2$, with $d$ being the layer thickness,
$\epsilon_{\parallel}$ the in-plane dielectric constant of the layer material
Wang  _et al._ (2018).

The spectrum of the new collective mode for this case is displayed in Fig. 6.
Because of the decreased dielectric screening the critical value $r_{s}^{*}$
increases as compared to the purely Coulomb case considered above.

Figure 6: The imaginary part of the frequency of the collective mode due to
PSOI as a function of wave vector for three values of $r_{s}$. Here a 2D layer
in vacuum is considered, its thickness being $d=0.02a_{B}$, the in-plane
dielectric constant $\epsilon_{\parallel}=15$. The PSOI magnitude is
$\tilde{\alpha}=0.1$, which corresponds to $r_{s}^{*}=1.054$.

##  III Concluding remarks

We studied electron correlations and collective modes in a 2D electron system
with PSOI produced by the in-plane Coulomb electric fields of interacting
electrons. The peculiarity of this system is that the e-e interaction strength
is described by two interaction parameters, $r_{s}$ and
$\tilde{\alpha}/r_{s}$, related to the Coulomb interaction and the PSOI. They
are characterized by the opposite dependence on the electron density. The
attractive PSOI prevails over the Coulomb repulsion when the density is high
enough.

This gives rise to strong electron correlations on a certain spatial scale
that manifest themselves as a sharp peak in the static structure factor at
$q=q_{c}$. Moreover, as soon as $\tilde{\alpha}/r_{s}$ exceeds a critical
value, a new mode with purely imaginary frequency appears in a band of wave
vectors around $q_{c}$. In other words, the spatially uniform paramagnetic
ground state becomes unstable with respect to the charge density fluctuations
on the $q_{c}$ scale.

This altogether indicates a tendency for an electron state to form a striped
structure of some sort on this spatial scale. Within the linear analysis
undertaken in the present paper, it is impossible to predict a specific type
of the electron state that would correspond to the true energy minimum.
Instead of the perturbative RPA, the self-consistent approach, like the
Hartree-Fock Berg  _et al._ (2012); Fogler  _et al._ (1996), is necessary to
attack this problem.

The obtained results, of course, do not imply that a sufficiently dense 2D
electron gas with arbitrarily small PSOI is always unstable towards the
density fluctuations. For very large electron densities the considerations
based on the one-band model and $k\cdot p$ approximation lose their validity.

On the other hand, large values of $\tilde{\alpha}\gtrsim 1$ are not
attainable at least in classical semiconductors with $sp^{3}$ band
hybridization, where the upper limit for the Rashba constant is of the order
of $\tilde{\alpha}\approx\epsilon^{-2}$, with $\epsilon$ being the bulk
dielectric constant 111We are grateful to Sergei Tarasenko for pointing this
out (S. Tarasenko, private communication).

Nonetheless, the recent rise of 2D systems with giant SOI Chen  _et al._
(2021); Varignon  _et al._ (2018) gives us some hope for the realization of
the extremely strong Rashba SOI by means of other physical mechanisms. Thus,
in oxide heterostructures and films Pai  _et al._ (2018); Stemmer and Allen
(2018) the strong indications were found for the electronic nematicity Božović
and Levy (2020), and for the formation of BEPs Cheng  _et al._ (2015); Annadi
_et al._ (2018); Mikheev  _et al._ (2021) well beyond the superconducting
phase. While the exact mechanisms behind these effects still remain unclear,
at least some of them are likely due to the interplay of the giant SOI and
collective effects Damanet  _et al._ (2021). That being said, the quest for a
particular system with a giant SOI where the effects predicted in the present
paper could develop is still a challenge for the future.

###### Acknowledgements.

This work was carried out in the framework of the state task and supported by
the Russian Foundation for Basic Research, Project No. 20–02–00126.

## Appendix A

In this section we obtain the linear response functions of the 2D electron gas
with PSOI within RPA. The microscopic derivation of the RPA is based on the
equations of motion for the quantum Wigner distribution functions (WDFs)
Hasegawa and Shimizu (1975); Gindikin and Sablikov (2017).

We start with the second quantized Hamiltonian

$\displaystyle H={}$ $\displaystyle-\frac{\hbar^{2}}{2m}\sum_{s}\int
d\bm{r}\,\psi^{+}_{s}(\bm{r})\nabla_{\\!\bm{r}}^{2}\psi_{s}(\bm{r})$ (11)
$\displaystyle{}+\frac{1}{2}\sum_{s_{1}s_{2}}\int
d\bm{r}_{1}d\bm{r}_{2}\,\psi^{+}_{s_{1}}(\bm{r}_{1})\psi^{+}_{s_{2}}(\bm{r}_{2})\mathcal{U}(\bm{r}_{1}-\bm{r}_{2})\psi_{s_{2}}(\bm{r}_{2})\psi_{s_{1}}(\bm{r}_{1})$
(12) $\displaystyle{}+\sum_{s}\int
d\bm{r}\,\psi^{+}_{s}(\bm{r})\varphi(\bm{r},t)\psi_{s}(\bm{r})$ (13)
$\displaystyle{}-i\alpha\sum_{s_{1}s_{2}}\int
d\bm{r}_{1}d\bm{r}_{2}\,\psi^{+}_{s_{1}}(\bm{r}_{1})\psi^{+}_{s_{2}}(\bm{r}_{2})\left[(s_{1}\partial_{x_{1}}-s_{2}\partial_{x_{2}})\mathcal{E}_{y}(\bm{r}_{1}-\bm{r}_{2})-(s_{1}\partial_{y_{1}}-s_{2}\partial_{y_{2}})\mathcal{E}_{x}(\bm{r}_{1}-\bm{r}_{2})\right]\psi_{s_{2}}(\bm{r}_{2})\psi_{s_{1}}(\bm{r}_{1})\,,$
(14)

where $\psi_{s}(\bm{r})$ is the electron field operator in the $s_{z}$
representation ($s=\pm 1$), and $\bm{r}\equiv(x,y,0)$ stands for the in-plane
position. Eq. (11) is the kinetic energy operator, Eq. (12) is the Coulomb
interaction, Eq. (13) is the external perturbation, and Eq. (14) is the PSOI.

Let us introduce the single-particle and two-particle Klimontovich operators
Klimontovich (1958) as

$\hat{f}^{(s)}(\bm{r},\bm{p},t)=\frac{1}{{(2\pi)}^{2}}\int
d\bm{R}\,e^{i\bm{p}\cdot\bm{R}}\psi_{s}^{+}\left(\bm{r}+\tfrac{\bm{R}}{2},t\right)\psi_{s}\left(\bm{r}-\tfrac{\bm{R}}{2},t\right)$
(15)

and

$\hat{f}^{(s_{1},s_{2})}(\bm{r}_{1},\bm{p}_{1},\bm{r}_{2},\bm{p}_{2},t)=\frac{1}{{(2\pi)}^{4}}\int
d\bm{R}_{1}d\bm{R}_{2}\,e^{i(\bm{p}_{1}\cdot\bm{R}_{1}+\bm{p}_{2}\cdot\bm{R}_{2})}\psi_{s_{1}}^{+}\left(\bm{r}_{1}+\tfrac{\bm{R}_{1}}{2},t\right)\psi_{s_{2}}^{+}\left(\bm{r}_{2}+\tfrac{\bm{R}_{2}}{2},t\right)\psi_{s_{2}}\left(\bm{r}_{2}-\tfrac{\bm{R}_{2}}{2},t\right)\psi_{s_{1}}\left(\bm{r}_{1}-\tfrac{\bm{R}_{1}}{2},t\right)\,.$
(16)

The expectation value of the Klimontovich operators w.r.t. the ground state of
$H$ are exactly the WDFs, i.e.
$\left\langle\hat{f}^{(s)}(\bm{r},\bm{p},t)\right\rangle\equiv{f}^{(s)}(\bm{r},\bm{p},t)$.
By commuting $\hat{f}^{(s)}(\bm{r},\bm{p},t)$ with $H$ and taking the
expectation value we obtain the first equation of motion in the Bogoliubov-
Born-Green-Kirkwood-Yvon hierarchy Bonitz (2016):

$\displaystyle i\hbar\partial_{t}f^{(s)}(\bm{r},\bm{p},t)={}$
$\displaystyle-\frac{i\hbar^{2}}{m}\bm{p}\cdot\nabla_{\\!\bm{r}}f^{(s)}(\bm{r},\bm{p},t)+\frac{1}{{(2\pi)}^{2}}\int
d\bm{R}d\bm{p}_{1}\,e^{i(\bm{p}-\bm{p}_{1})\cdot\bm{R}}\left\\{\varphi\left(\bm{r}-\tfrac{\bm{R}}{2},t\right)-\varphi\left(\bm{r}+\tfrac{\bm{R}}{2},t\right)\right\\}f^{(s)}(\bm{r},\bm{p}_{1},t)$
(17) $\displaystyle{}+\frac{1}{{(2\pi)}^{2}}\sum_{\varsigma}\int
d\bm{r}_{1}d\bm{R}\,d\bm{p}_{1}d\bm{p}_{2}\,e^{i(\bm{p}-\bm{p}_{1})\cdot\bm{R}}\left\\{\mathcal{U}\left(\bm{r}-\bm{r}_{1}-\tfrac{\bm{R}}{2}\right)-\mathcal{U}\left(\bm{r}-\bm{r}_{1}+\tfrac{\bm{R}}{2}\right)\right\\}f^{(s,\varsigma)}(\bm{r},\bm{p}_{1},\bm{r}_{1},\bm{p}_{2},t)$
$\displaystyle{}-\frac{i\alpha}{2\pi^{2}}\sum_{\varsigma}\int
d\bm{r}_{1}d\bm{R}\,d\bm{p}_{1}d\bm{p}_{2}\,e^{i(\bm{p}-\bm{p}_{1})\cdot\bm{R}}\mathcal{E}_{y}\left(\bm{r}_{1}-\bm{r}+\tfrac{\bm{R}}{2}\right)\left[\varsigma(\tfrac{1}{2}\partial_{x_{1}}+ip_{2x})-s(\tfrac{1}{2}\partial_{x}+ip_{1x})\right]f^{(s,\varsigma)}(\bm{r},\bm{p}_{1},\bm{r}_{1},\bm{p}_{2},t)$
$\displaystyle{}+\frac{i\alpha}{2\pi^{2}}\sum_{\varsigma}\int
d\bm{r}_{1}d\bm{R}\,d\bm{p}_{1}d\bm{p}_{2}\,e^{i(\bm{p}-\bm{p}_{1})\cdot\bm{R}}\mathcal{E}_{x}\left(\bm{r}_{1}-\bm{r}+\tfrac{\bm{R}}{2}\right)\left[\varsigma(\tfrac{1}{2}\partial_{y_{1}}+ip_{2y})-s(\tfrac{1}{2}\partial_{y}+ip_{1y})\right]f^{(s,\varsigma)}(\bm{r},\bm{p}_{1},\bm{r}_{1},\bm{p}_{2},t)$
$\displaystyle{}-\frac{i\alpha}{2\pi^{2}}\sum_{\varsigma}\int
d\bm{r}_{1}d\bm{R}\,d\bm{p}_{1}d\bm{p}_{2}\,e^{i(\bm{p}-\bm{p}_{1})\cdot\bm{R}}\mathcal{E}_{y}\left(\bm{r}_{1}-\bm{r}-\tfrac{\bm{R}}{2}\right)\left[\varsigma(\tfrac{1}{2}\partial_{x_{1}}-ip_{2x})-s(\tfrac{1}{2}\partial_{x}-ip_{1x})\right]f^{(s,\varsigma)}(\bm{r},\bm{p}_{1},\bm{r}_{1},\bm{p}_{2},t)$
$\displaystyle{}+\frac{i\alpha}{2\pi^{2}}\sum_{\varsigma}\int
d\bm{r}_{1}d\bm{R}\,d\bm{p}_{1}d\bm{p}_{2}\,e^{i(\bm{p}-\bm{p}_{1})\cdot\bm{R}}\mathcal{E}_{x}\left(\bm{r}_{1}-\bm{r}-\tfrac{\bm{R}}{2}\right)\left[\varsigma(\tfrac{1}{2}\partial_{y_{1}}-ip_{2y})-s(\tfrac{1}{2}\partial_{y}-ip_{1y})\right]f^{(s,\varsigma)}(\bm{r},\bm{p}_{1},\bm{r}_{1},\bm{p}_{2},t)\,.$

RPA consists in truncating the hierarchy by factorizing the two-particle WDF
according to

$f^{(s_{1},s_{2})}(\bm{r}_{1},\bm{p}_{1},\bm{r}_{2},\bm{p}_{2},t)=f^{(s_{1})}(\bm{r}_{1},\bm{p}_{1},t)f^{(s_{2})}(\bm{r}_{2},\bm{p}_{2},t)\,.$
(18)

Let us define the deviation $f^{(s)}_{1}(\bm{r},\bm{p},t)$ of
$f^{(s)}(\bm{r},\bm{p},t)$ from its equilibrium value $f_{0}^{(s)}(\bm{p})$ as
a result of the external perturbation,

$f^{(s)}_{1}(\bm{r},\bm{p},t)=f^{(s)}(\bm{r},\bm{p},t)-f^{(s)}_{0}(\bm{p})\,.$
(19)

The equation of motion for $f^{(s)}_{1}(\bm{r},\bm{p},t)$, linearized with
respect to the external potential, reads in Fourier representation as

$\begin{split}-\hbar\omega
f^{(s)}_{1}(\bm{q},\bm{p},\omega)={}&-\frac{\hbar^{2}\bm{p}\cdot\bm{q}}{m}f^{(s)}_{1}(\bm{q},\bm{p},\omega)+\left[f^{(s)}_{0}(\bm{p}+\tfrac{\bm{q}}{2})-f^{(s)}_{0}(\bm{p}-\tfrac{\bm{q}}{2})\right]\times\\\
&\times\left\\{\varphi_{q\omega}+\mathcal{U}_{q}\sum_{\varsigma}n^{(\varsigma)}_{\bm{q}\omega}+2\alpha
s{(\bm{p}\times\bm{\mathcal{E}}_{\bm{q}})}_{z}\sum_{\varsigma}n^{(\varsigma)}_{\bm{q}\omega}+2\alpha\frac{m}{e\hbar}\sum_{\varsigma}\varsigma{(\bm{j}^{(\varsigma)}_{\bm{q}\omega}\times\bm{\mathcal{E}}_{\bm{q}})}_{z}\right\\}\,.\end{split}$
(20)

Here $n^{(s)}_{q\omega}$ and $\bm{j}^{(s)}_{\bm{q}\omega}$ are the $s$-spin
components of the electron density and paramagnetic current response, related
to the WDF, respectively, by

$n^{(s)}_{q\omega}=\int d\bm{p}\,f^{(s)}_{1}(\bm{q},\bm{p},\omega)$ (21)

and

$\bm{j}^{(s)}_{\bm{q}\omega}=-\frac{e\hbar}{m}\int
d\bm{p}\,\bm{p}f^{(s)}_{1}(\bm{q},\bm{p},\omega)\,.$ (22)

The Fourier component of the in-plane electric field is given by
$\bm{\mathcal{E}}_{\bm{q}}=\frac{\bm{q}}{ie}\mathcal{U}_{q}$. Two specific
forms of the e-e repulsion potential $\mathcal{U}_{q}$ are considered in the
main text.

The first two moments of the Eq. (20) are

$\displaystyle n^{(s)}_{q\omega}={}$
$\displaystyle\chi_{0}\left(\varphi_{q\omega}+\mathcal{U}_{q}\sum_{\varsigma}n^{(\varsigma)}_{q\omega}+2\alpha\frac{m}{e\hbar}\sum_{\varsigma}\varsigma{(\bm{j}^{(\varsigma)}_{\bm{q}\omega}\times\bm{\mathcal{E}}_{\bm{q}})}_{z}\right)$
$\displaystyle{}+\chi_{1}\frac{2\alpha
s}{ie}\mathcal{U}_{q}\sum_{\varsigma}n^{(\varsigma)}_{q\omega}\,,$ (23)

and

$\displaystyle-\frac{m}{e\hbar}\bm{j}^{(s)}_{\bm{q}\omega}={}$
$\displaystyle\bm{\upchi}_{2}\left(\varphi_{q\omega}+\mathcal{U}_{q}\sum_{\varsigma}n^{(\varsigma)}_{q\omega}+2\alpha\frac{m}{e\hbar}\sum_{\varsigma}\varsigma{(\bm{j}^{(\varsigma)}_{\bm{q}\omega}\times\bm{\mathcal{E}}_{\bm{q}})}_{z}\right)$
$\displaystyle{}+\bm{\upchi}_{3}\frac{2\alpha
s}{ie}\mathcal{U}_{q}\sum_{\varsigma}n^{(\varsigma)}_{q\omega}\,.$ (24)

Four susceptibilities entering the Eqs. (23–24) are as follows,

$\displaystyle\chi_{0}$ $\displaystyle=\int
d\bm{p}\,\frac{f^{(s)}_{0}(\bm{p}+\frac{\bm{q}}{2})-f^{(s)}_{0}(\bm{p}-\frac{\bm{q}}{2})}{\frac{\hbar^{2}}{m}\bm{p}\cdot\bm{q}-\hbar(\omega+i0)}\,,$
(25) $\displaystyle\chi_{1}$ $\displaystyle=\int
d\bm{p}\,\frac{f^{(s)}_{0}(\bm{p}+\frac{\bm{q}}{2})-f^{(s)}_{0}(\bm{p}-\frac{\bm{q}}{2})}{\frac{\hbar^{2}}{m}\bm{p}\cdot\bm{q}-\hbar(\omega+i0)}{(\bm{p}\times\bm{q})}_{z}\,,$
(26) $\displaystyle\bm{\upchi}_{2}$ $\displaystyle=\int
d\bm{p}\,\frac{f^{(s)}_{0}(\bm{p}+\frac{\bm{q}}{2})-f^{(s)}_{0}(\bm{p}-\frac{\bm{q}}{2})}{\frac{\hbar^{2}}{m}\bm{p}\cdot\bm{q}-\hbar(\omega+i0)}\bm{p}\,,$
(27) $\displaystyle\bm{\upchi}_{3}$ $\displaystyle=\int
d\bm{p}\,\frac{f^{(s)}_{0}(\bm{p}+\frac{\bm{q}}{2})-f^{(s)}_{0}(\bm{p}-\frac{\bm{q}}{2})}{\frac{\hbar^{2}}{m}\bm{p}\cdot\bm{q}-\hbar(\omega+i0)}\bm{p}{(\bm{p}\times\bm{q})}_{z}\,.$
(28)

In a non-self-consistent RPA, $f^{(s)}_{0}(\bm{p})$ is the momentum
distribution of the free 2D paramagnetic electron gas,

$f^{(s)}_{0}(\bm{p})=\frac{1}{{(2\pi)}^{2}}\theta(k_{F}-|\bm{p}|)\,,$ (29)

$k_{F}$ being the Fermi wave vector. In this case

$\displaystyle\chi_{0}={}$
$\displaystyle\frac{N_{\sigma}}{(q/k_{F})}\left(\nu_{-}-\operatorname{sign}(\operatorname{Re}\nu_{-})\sqrt{\nu_{-}^{2}-1}-\nu_{+}+\operatorname{sign}(\operatorname{Re}\nu_{+})\sqrt{\nu_{+}^{2}-1}\right)\,,$
(30) $\displaystyle\chi_{1}={}$ $\displaystyle 0\,,$ (31)
$\displaystyle\bm{\upchi}_{2}={}$
$\displaystyle\frac{\bm{q}N_{\sigma}}{2(q/k_{F})}\left(\nu_{-}-\operatorname{sign}(\operatorname{Re}\nu_{-})\sqrt{\nu_{-}^{2}-1}+\nu_{+}-\operatorname{sign}(\operatorname{Re}\nu_{+})\sqrt{\nu_{+}^{2}-1}\right)$
(32)
$\displaystyle{}+\frac{\bm{q}N_{\sigma}}{{(q/k_{F})}^{2}}\left(\nu_{-}^{2}-\nu_{-}\operatorname{sign}(\operatorname{Re}\nu_{-})\sqrt{\nu_{-}^{2}-1}-\nu_{+}^{2}+\nu_{+}\operatorname{sign}(\operatorname{Re}\nu_{+})\sqrt{\nu_{+}^{2}-1}\right)\,,$
(33) $\displaystyle\bm{\upchi}_{3}={}$ $\displaystyle
N_{\sigma}k_{F}^{2}\frac{\bm{q}_{\perp}}{(q/k_{F})}\left(\frac{\nu_{-}}{2}-\frac{1}{3}\operatorname{sign}(\operatorname{Re}\nu_{-})\left[{(\nu_{-}^{2})}^{\frac{3}{2}}-{(\nu_{-}^{2}-1)}^{\frac{3}{2}}\right]-\frac{\nu_{+}}{2}+\frac{1}{3}\operatorname{sign}(\operatorname{Re}\nu_{+})\left[{(\nu_{+}^{2})}^{\frac{3}{2}}-{(\nu_{+}^{2}-1)}^{\frac{3}{2}}\right]\right)\,.$
(34)

Here $\nu_{\pm}=\frac{\omega+i0}{qv_{F}}\pm\frac{q}{2k_{F}}$,
$N_{\sigma}=\frac{m}{2\pi\hbar^{2}}$ is the 2D density of states of spin
$\sigma$ per unit area, $\bm{q}_{\perp}\equiv(q_{y},-q_{x})$, the branch cut
lies along the negative real axis. Eq. (30) is due to Stern Stern (1967).

Expressing $n^{(s)}_{q\omega}$ from a coupled system of equations Eqs. (23–24)
gives Eq. (2) of the main text with

$\displaystyle\chi_{j}\equiv{}{(\bm{\upchi}_{3}\times\bm{q})}_{z}=N_{\sigma}k_{F}^{4}\frac{q}{k_{F}}\Bigg{(}\frac{\nu_{-}}{2}-\frac{1}{3}\operatorname{sign}(\operatorname{Re}\nu_{-})\left[{(\nu_{-}^{2})}^{\frac{3}{2}}-{(\nu_{-}^{2}-1)}^{\frac{3}{2}}\right]-\frac{\nu_{+}}{2}+\frac{1}{3}\operatorname{sign}(\operatorname{Re}\nu_{+})\left[{(\nu_{+}^{2})}^{\frac{3}{2}}-{(\nu_{+}^{2}-1)}^{\frac{3}{2}}\right]\Bigg{)}\,.$
(35)

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