#  Spin and Valley Hall effects induced by asymmetric interparticle scattering

M. M. Glazov  L. E. Golub  Ioffe Institute, 194021 St. Petersburg, Russia

###### Abstract

We develop the theory of the spin and valley Hall effects in two-dimensional
systems caused by asymmetric – skew – scattering of the quasiparticles. The
collision integral is derived in the third order in the particle-particle
interaction with account for the spin-orbit coupling both for bosons and
fermions. It is shown that the scattering asymmetry appears only in the
processes where the interaction between the particles in the initial and
intermediate state is present. We show that for degenerate electrons or
nondegenerate particles the spin and valley currents induced by interparticle
collisions are suppressed with their steady-state values being proportional to
the squared temperature or density, respectively. Our results imply non-Fermi
liquid properties of electrons in the presence of electron-electron skew
scattering. Strong deviations from conventional picture of interparticle
scattering are also demonstrated for the skew scattering of two-dimensional
degenerate bosons, e.g. excitons or exciton polaritons: The spin or valley
current of degenerate bosons contains the enhancement factor exponentially
growing with increase in the particle density.

##  I Introduction

Spin-dependent effects in condensed matter result from a coupling between the
electron orbital and spin degrees of freedom caused by the spin-orbit
interaction. Spin currents [1] take a special place among of the most exciting
phenomena in the field both from fundamental and applied physics viewpoints.
An interesting issue here is the possibility to have a spin current in
thermodynamic equilibrium with the most prominent example being the Rashba
medium – a system with linear in the momentum spin-dependent terms in the
electron or hole Hamiltonian [2, 3, 4] – where the spin can flow even if the
system is not perturbed [5, 6]. However, in a uniform media these currents do
not result in any spin accumulation. The spin transport and spin accumulation
arises in the presence of an external electric field where the spin Hall
effect (SHE) arises [7, 8], see Ref. [1] for review. It results in a
conversion of the particle flux to the transverse spin current and subsequent
accumulation of the spin polarization at the sample’s edges [9].
Generalization of this phenomenon to the multivalley systems is the valley
Hall effect (VHE), i.e., the conversion of the particle flux to the
perpendicular opposite flows of particles belonging to two different valleys
[10, 11, 12, 13].

The microscopic mechanisms of SHE and VHE are naturally related to the spin-
orbit coupling and can be both intrinsic and extrinsic [14], and the dominant
one in many cases is the skew-scattering mechanism associated with the spin-
or valley-dependent asymmetry of the particle scattering by a defect [15].
This skew scattering has been studied in both bulk semiconductors and
heterostructures as well as in metals for the scattering by both impurities
and acoustic phonons [16, 17, 11]. The possibility of an asymmetric – skew –
electron-electron scattering has been briefly discussed in the literature in
the context of the spin Hall drag in double quantum well structures [18] and
recently in Ref. [13] for the spin accumulation in the hydrodynamic electron
transport regime, see also Ref. [19]. However, the theory of the electron-
electron skew scattering is far from being complete.

In this work, we consider the SHE and VHE caused by the inter-particle
collisions only. This situation is highly relevant, e.g. for two-dimensional
(2D) high-mobility semiconductors and graphene where electron-electron
scattering dominates over the impurity and phonon scattering and controls the
transport effects [20, 21, 22, 23, 24, 25, 26, 27], see Ref. [28] for review.
We argue that the skew scattering and, hence, the SHE and VHE are possible at
the inter-particle scattering. We derive the kinetic equation with allowance
for the skew scattering processes both for fermions and bosons and demonstrate
asymmetric in spin and momenta terms in the collision integral. They appear
beyond the Born approximation for the scattering amplitude. We show that for
the spin or valley Hall effect to take place, the scattering should occur due
to the interaction of the particles in the initial and intermediate states. It
results in unconventional behavior of SHE and VHE in both degenerate Fermi and
Bose as well as in nondegenerate gases.

The paper is organized as follows. In Sec. II we present the kinetic equation
and derive the collision integral with allowance for the asymmetric scattering
in the third-order perturbation theory. We demonstrate derivation by two
methods, using the scattering matrix and using the Keldysh diagram technique.
Section III contains the theory of the SHE and VHE at the interparticle
collisions and particular results for the Fermi, Boltzmann and Bose systems.
General implications of the obtained results and conclusion are presented in
the end of the paper, Sec. IV. Details of calculations are given in
Appendices.

##  II Interparticle skew collision integral

We consider a two-dimensional system in the $(xy)$ plane and assume that the
quasiparticles are characterized, in addition to the wavevector $\bm{k}$, by a
spin or valley index $s=\pm$. In conventional semiconductor quantum wells with
the two-dimensional electron gas $s$ is the $z$ component of the electron
spin, while in the transition-metal dichalcogenide monolayers $s$
distinguishes two valleys $\bm{K}_{+}$ ($s=+$) and $\bm{K}_{-}$ ($s=-$) [11,
29, 13]. In these systems, the two valleys are related by the time-reversal
symmetry and share the same properties with the spin component. In the case of
Bose quasiparticles, for instance, excitons in quantum wells or transition-
metal dichalcogenide monolayers or exciton-polaritons in microcavities,
$s=\pm$ denote the polarization of exciton ($\sigma^{+}$ or $\sigma^{-}$) also
known as the exciton pseudospin and some times termed as a valley index in
two-dimensional transition-metal dichalcogenides [30, 31, 32, 33, 12, 34].

In standard conditions of the SHE or VHE the quasiparticles are driven by an
external force, real electric field in the case of electrons or synthetic
fields for excitons and exciton-polaritons [11, 12, 35] and the spin- or
valley current is detected in the transversal direction. Hereafter we take
into account the spin- and valley-dependent contributions to the scattering
matrix elements that are related to the $\bm{k}\cdot\bm{p}$-mixing with the
remote bands, see Refs. [11, 12, 13] for details. Moreover, we focus on the
asymmetric or skew scattering contributions to the SHE and VHE which can be
considered independently of the anomalous contributions, side-jump and
anomalous velocity. The side-jump effect in the case of electron-electron
scattering has been studied in detail in Refs. [36, 13]. Thus, we can describe
SHE and VHE within the kinetic equation approach and introduce the
distribution functions $f_{\bm{k},\pm}$ for the quasiparticles with a given
spin or valley index $s$, while off-diagonal in $s$ elements of the density
matrix are unimportant for the present study.

Figure 1: Sketch of the scattering processes contributing to the skew effect.
(a) First-order contribution. (b) and (c) Second-order contributions. Red and
blue arrows denote the electrons with $s=+$ and $-$, respectively, green
dotted line shows the interaction.

We present the steady-state kinetic equation for the distribution functions
$f_{\bm{k},s}$ in the following form

$\frac{\hbar}{m}(\bm{F}\cdot\bm{k})\derivative{f^{0}_{k}}{\varepsilon_{k}}+\frac{\delta
f_{\bm{k},s}}{\tau_{p}}=\mathop{\rm St}{[}f_{\bm{k},s}].$ (1)

Here $\bm{F}$ is the force acting on the quasiparticles due to the real or
synthetic electric field, $f^{0}_{k}$ is the equilibrium distribution
function, $\delta f_{\bm{k},s}=f_{\bm{k},s}-\langle f_{\bm{k},s}\rangle$ is
the anisotropic correction and the angular brackets denote the angular
averaging, the energy $\varepsilon_{k}=\hbar^{2}k^{2}/(2m)$ with $m$ being the
effective mass of the quasiparticle, $\tau_{p}$ is the momentum relaxation
time caused by the scattering off the static disorder or phonons,111We assume
that the disorder is Gaussian and neglect two-phonon processes, in this case
the disorder and phonon scattering do not have asymmetric component. and
$\mathop{\rm St}{[}f_{\bm{k},s}]$ is the interparticle collision integral. It
contains the symmetric and antisymmetric parts. The symmetric part of the
collision integral is responsible for thermalization of quasiparticles, their
viscosity and, importantly, relaxation of the spin or valley current [37, 38,
39, 40, 41]. The asymmetric part is responsible for the spin or valley Hall
effect.

The scattering matrix element
${\bm{k}^{\prime},s_{1};\bm{p}^{\prime},s_{1}^{\prime}}\to{\bm{k},s;\bm{p},s^{\prime}}$
by a (long-range) interaction potential $V(\bm{r}_{1}-\bm{r}_{2})$ reads

$\left<\bm{k},s;\bm{p},s^{\prime}\right|V\left|\bm{k}^{\prime},s_{1};\bm{p}^{\prime},s_{1}^{\prime}\right>=\delta_{\bm{k}+\bm{p},\bm{k}^{\prime}+\bm{p}^{\prime}}\\\
\times\left[V(|\bm{k}-\bm{k}^{\prime}|)\langle{\bm{k},s}|{\bm{k}^{\prime},s_{1}}\rangle\langle{\bm{p},s^{\prime}}|{\bm{p}^{\prime},s_{1}^{\prime}}\rangle\right.\\\
\pm\left.(\bm{k}^{\prime}\leftrightarrow\bm{p}^{\prime},s_{1}\leftrightarrow
s_{1}^{\prime})\right].$ (2)

Here the top sign ($+$) refers to the bosons and the bottom sign ($-$) refers
to the fermions, we introduced the 2D Fourier-image of the interaction
potential, $V(q)$, and used the bra-ket notation
$\langle{\bm{k},s}|{\bm{k}^{\prime},s_{1}}\rangle$ for the overlap integral of
the Bloch amplitudes
$v_{0}^{-1}\int_{v_{0}}u_{\bm{k},s}^{*}(\bm{\rho})u_{\bm{k}^{\prime},s_{1}}(\bm{\rho})\differential\bm{\rho}$
where the integration is performed over the unit cell volume $v_{0}$.

Here and in what follows we consider collisions of particles (bosons or
fermions) with different spin or valley indices $s$ and $s^{\prime}\neq s$.
The collisions of the particles with the same $s=s^{\prime}$ do not produce
any spin or valley current owing to the momentum conservation, and do not
significantly contribute to the relaxation of the spin and valley currents
[13]; and we disregard spin- or valley-flip processes. Taking $s^{\prime}=-s$
in Eq. (2) we have:

$\left<\bm{k},s;\bm{p},-s\right|V\left|\bm{k}^{\prime},s_{1};\bm{p}^{\prime},s_{1}^{\prime}\right>=\delta_{\bm{k}+\bm{p},\bm{k}^{\prime}+\bm{p}^{\prime}}\\\
\times\left[V(|\bm{k}-\bm{k}^{\prime}|)\langle{\bm{k},s}|{\bm{k}^{\prime},s}\rangle\langle{\bm{p},-s}|{\bm{p}^{\prime},-s}\rangle\right.\delta_{s_{1},s}\delta_{s_{1}^{\prime},-s}\\\
\pm\left.(\bm{k}^{\prime}\leftrightarrow\bm{p}^{\prime},s_{1}\leftrightarrow
s_{1}^{\prime})\right].$ (3)

We see that the matrix element in Eq. (3) is non-zero only if (i) $s_{1}=s$
and $s_{1}^{\prime}=-s$ or (ii) $s_{1}^{\prime}=s$ and $s_{1}=-s$. Unlike
interparticle scattering with the same spins or valley indices, the processes
(i) and (ii) do not interfere in any order of perturbation series because, in
Eq. (3), either the first line is non-zero [process (i) is active] and the
second line is zero [process (ii) is inactive] or vice versa. Hence, we can
write for the amplitude of scattering with $s=+$:

$T_{\bm{k}+,\bm{p}-;\bm{k}^{\prime}s_{1},\bm{p}^{\prime}s_{1}^{\prime}}=T_{\bm{k}\bm{p},\bm{k}^{\prime}\bm{p}^{\prime}}\delta_{s_{1},+}\delta_{s_{1}^{\prime},-}\pm(\bm{k}^{\prime}\leftrightarrow\bm{p}^{\prime},s_{1}\leftrightarrow
s_{1}^{\prime}),$ (4)

where $T_{\bm{k}\bm{p},\bm{k}^{\prime}\bm{p}^{\prime}}\equiv
T_{\bm{k}+,\bm{p}-;\bm{k}^{\prime}+,\bm{p}^{\prime}-}$. As a result we obtain
the general expression for the collision integral

$\text{St}[f_{\bm{k}+}]={4\pi\over\hbar}\sum_{\bm{p},\bm{k}^{\prime},\bm{p}^{\prime}}\delta(\varepsilon_{k}+\varepsilon_{p}-\varepsilon_{k^{\prime}}-\varepsilon_{p^{\prime}})\delta_{\bm{k}+\bm{p},\bm{k}^{\prime}+\bm{p}^{\prime}}\\\
\times\bigl{[}\absolutevalue{T_{\bm{k}\bm{p},\bm{k}^{\prime}\bm{p}^{\prime}}}^{2}f_{\bm{k}^{\prime}+}f_{\bm{p}^{\prime}-}(1\pm
f_{\bm{k}+})(1\pm f_{\bm{p}-})\\\
-(\bm{k},\bm{p}\leftrightarrow\bm{k}^{\prime},\bm{p}^{\prime})\bigr{]},$ (5)

where the factor $4\pi=2\times 2\pi$ comes from the account of the second term
in Eq. (4).

In the first order in $V$, the scattering amplitude
$T_{\bm{k}\bm{p},\bm{k}^{\prime}\bm{p}^{\prime}}$ coincides with the matrix
element:

$V(\absolutevalue{\bm{k}-\bm{k}^{\prime}})\innerproduct{{\bm{k}},+}{{\bm{k}^{\prime}},+}\innerproduct{{\bm{p}},-}{{\bm{p}^{\prime}},-}\equiv
U_{\bm{k}\bm{p},\bm{k}^{\prime}\bm{p}^{\prime}}.$ (6)

The collision integrals can be derived within the lowest Born approximation
for the scattering matrix (we assume that the standard criteria for the
smallness of the scattering potential are fulfilled, and for spin polarized
fermions and bosons they are given, respectively, in Refs. [40] and [42], see
also Refs. [43, 44]). First, we present the expressions for the collision
integrals responsible for the spin and valley current relaxation. Following
Refs. [40, 42, 13] we have from Eqs. (5) and (6)

$\text{St}_{\rm
rel}[f_{\bm{k}+}]={4\pi\over\hbar}\sum_{\bm{p},\bm{k}^{\prime},\bm{p}^{\prime}}\delta(\varepsilon_{k}+\varepsilon_{p}-\varepsilon_{k^{\prime}}-\varepsilon_{p^{\prime}})\delta_{\bm{k}+\bm{p},\bm{k}^{\prime}+\bm{p}^{\prime}}\\\
\times\bigl{[}\absolutevalue{U_{\bm{k}\bm{p},\bm{k}^{\prime}\bm{p}^{\prime}}}^{2}f_{\bm{k}^{\prime}+}f_{\bm{p}^{\prime}-}(1\pm
f_{\bm{k}+})(1\pm f_{\bm{p}-})\\\
-(\bm{k},\bm{p}\leftrightarrow\bm{k}^{\prime},\bm{p}^{\prime})\bigr{]}.$ (7)

Here and in what follows we are interested in the spin current generation due
to the skew scattering. Since the force itself due to the first term in the
kinetic equation (1) produces the same anisotropic contribution to the
distribution function for both spin or valley states, we omit subscripts
$s,s^{\prime}$ in the distribution functions in Eq. (5). Our aim is to derive
the asymmetric contribution to the collision integral responsible for the skew
scattering. Similarly to the case of the impurity scattering,
$|U_{\bm{k}\bm{p},\bm{k}^{\prime}\bm{p}^{\prime}}|^{2}$ does not contain
asymmetric part. The skew scattering probability can be obtained in the next
to the first Born approximation [45]. This means that in the standard
collision integral (5) one should take the squared modulus of the scattering
amplitude, $\absolutevalue{T}^{2}$, in the third order in the scattering
potential $U$, i.e. include the processes of scattering via intermediate
states as well.

The relevant processes are shown schematically in Fig. 1: the diagram (a)
demonstrates the first Born approximation, while the diagrams (b) and (c) show
the processes where the transition
$\bm{k}^{\prime},\bm{p}^{\prime}\to\bm{k},\bm{p}$ takes place via intermediate
states $\bm{k}^{\prime\prime}$ and $\bm{p}^{\prime\prime}$. Note that the
process (b) can be interpreted as a result of consecutive interactions: Two
particles occupying the initial states $\bm{k}^{\prime}$ and $\bm{p}^{\prime}$
interact and scatter to the intermediate states $\bm{k}^{\prime\prime}$ and
$\bm{p}^{\prime\prime}$, respectively, then interact again and scatter to the
final states $\bm{k}$ and $\bm{p}$. The process (c) requires interaction of
the particle in any of initial state with the particle in the intermediate
state, e.g., the particle in the initial $\bm{k}^{\prime}$ state interacts
with the particle in the intermediate $\bm{p}^{\prime\prime}$ state. As a
result they scatter to $\bm{k}^{\prime\prime}$ and $\bm{p}$, respectively,
then the particle that arrived in the intermediate state
$\bm{k}^{\prime\prime}$ interacts with the one in the initial state
$\bm{p}^{\prime}$ and they scatter to $\bm{k}$ and $\bm{p}^{\prime\prime}$.
Accordingly, this process is possible only in the presence of particles in
intermediate states. The interference of the processes (a) and (b,c) results
in the skew-scattering contributions we are looking for. We stress that the
presence of the processes depicted in Fig. 1(c), where the interactions with
intermediate states play a role, does not allow one to use the standard form
of the interparticle collision integral with the replacement of the Born
scattering amplitude by the total scattering amplitude. To be specific, we
assume that one particle with final, intermediate and initial wavevectors
$\bm{k}$, $\bm{k}^{\prime\prime}$ and $\bm{k}^{\prime}$ has $s=+$ and another
particle scattered $\bm{p}^{\prime}\to\bm{p}$ or
$\bm{p}^{\prime}\to\bm{p}^{\prime\prime}\to\bm{p}$ has $s^{\prime}=-$, and
bear in mind that the final expressions for the scattering rates should
contain corresponding permutations of the wavevectors
$\bm{k}^{\prime},\bm{k}^{\prime\prime}$ with
$\bm{p}^{\prime},\bm{p}^{\prime\prime}$. Under these assumptions we can write
the Born perturbation series for
$T_{\bm{k}\bm{p},\bm{k}^{\prime}\bm{p}^{\prime}}$ only. Importantly, in
derivation of the $T_{\bm{k}\bm{p},\bm{k}^{\prime}\bm{p}^{\prime}}$ a special
care should be taken to account the occupancies of the intermediate states: As
it is well known for the case of elastic scattering, only a proper account for
all processes, including the ones where the intermediate state is occupied,
results in the correct form of asymmetric terms in the collision integral
[45]. In our case we need, therefore, to consider all possibilities for the
intermediate states $\bm{k}^{\prime\prime},\bm{p}^{\prime\prime}$: both can be
empty or occupied, or one of these states can be empty and another one can be
occupied.

Up to the second order in the interaction potential, the scattering amplitude
has the form

$T_{\bm{k}\bm{p},\bm{k}^{\prime}\bm{p}^{\prime}}=U_{\bm{k}\bm{p},\bm{k}^{\prime}\bm{p}^{\prime}}+\sum_{\bm{k}^{\prime\prime}\bm{p}^{\prime\prime}}\biggl{\\{}{U_{\bm{k}\bm{p},\bm{k}^{\prime\prime}\bm{p}^{\prime\prime}}U_{\bm{k}^{\prime\prime}\bm{p}^{\prime\prime},\bm{k}^{\prime}\bm{p}^{\prime}}\over\varepsilon_{k}+\varepsilon_{p}-\varepsilon_{k^{\prime\prime}}-\varepsilon_{p^{\prime\prime}}+i0}\\\
\times\delta_{\bm{k}+\bm{p},\bm{k}^{\prime\prime}+\bm{p}^{\prime\prime}}\quantity[f_{\bm{k}^{\prime\prime}}f_{\bm{p}^{\prime\prime}}+(1\pm
f_{\bm{k}^{\prime\prime}})(1\pm f_{\bm{p}^{\prime\prime}})]\\\
+{U_{\bm{k}\bm{p}^{\prime\prime},\bm{k}^{\prime\prime}\bm{p}^{\prime}}U_{\bm{k}^{\prime\prime}\bm{p},\bm{k}^{\prime}\bm{p}^{\prime\prime}}\over\varepsilon_{k}+\varepsilon_{p^{\prime\prime}}-\varepsilon_{k^{\prime\prime}}-\varepsilon_{p^{\prime}}+i0}\\\
\times\delta_{\bm{k}-\bm{p}^{\prime},\bm{k}^{\prime\prime}-\bm{p}^{\prime\prime}}\quantity[f_{\bm{k}^{\prime\prime}}(1\pm
f_{\bm{p}^{\prime\prime}})+(1\pm
f_{\bm{k}^{\prime\prime}})f_{\bm{p}^{\prime\prime}}]\biggr{\\}}.$ (8)

Here the first term in the sum describes scattering via two occupied or two
empty states while the second term describes scattering via one occupied and
one empty state.

The skew scattering term in the lowest (3rd) order in $U$ is obtained from the
interference of the 1st and 2nd order terms in
$\absolutevalue{T_{\bm{k}\bm{p},\bm{k}^{\prime}\bm{p}^{\prime}}}^{2}$ where
the $\delta$-functions are taken from the energy denominators:

$\absolutevalue{T_{\bm{k}\bm{p},\bm{k}^{\prime}\bm{p}^{\prime}}}^{2}_{\rm
sk}=4\pi\sum_{\bm{k}^{\prime\prime}\bm{p}^{\prime\prime}}\Bigl{\\{}\text{Im}\quantity[U_{\bm{k}\bm{p},\bm{k}^{\prime}\bm{p}^{\prime}}^{*}U_{\bm{k}\bm{p},\bm{k}^{\prime\prime}\bm{p}^{\prime\prime}}U_{\bm{k}^{\prime\prime}\bm{p}^{\prime\prime},\bm{k}^{\prime}\bm{p}^{\prime}}]\\\
\times\delta(\varepsilon_{k}+\varepsilon_{p}-\varepsilon_{k^{\prime\prime}}-\varepsilon_{p^{\prime\prime}})\delta_{\bm{k}+\bm{p},\bm{k}^{\prime\prime}+\bm{p}^{\prime\prime}}\\\
\times\quantity[f_{\bm{k}^{\prime\prime}}f_{\bm{p}^{\prime\prime}}+(1\pm
f_{\bm{k}^{\prime\prime}})(1\pm f_{\bm{p}^{\prime\prime}})]\\\
+\text{Im}\quantity[U_{\bm{k}\bm{p},\bm{k}^{\prime}\bm{p}^{\prime}}^{*}U_{\bm{k}\bm{p}^{\prime\prime},\bm{k}^{\prime\prime}\bm{p}^{\prime}}U_{\bm{k}^{\prime\prime}\bm{p},\bm{k}^{\prime}\bm{p}^{\prime\prime}}]\\\
\times\delta(\varepsilon_{k}+\varepsilon_{p^{\prime\prime}}-\varepsilon_{k^{\prime\prime}}-\varepsilon_{p^{\prime}})\delta_{\bm{k}-\bm{p}^{\prime},\bm{k}^{\prime\prime}-\bm{p}^{\prime\prime}}\\\
\times\quantity[f_{\bm{k}^{\prime\prime}}(1\pm
f_{\bm{p}^{\prime\prime}})+(1\pm
f_{\bm{k}^{\prime\prime}})f_{\bm{p}^{\prime\prime}}]\Bigr{\\}}.$ (9)

Here the factor ‘4’ accounts for the permutations
$\bm{k}^{\prime\prime}\leftrightarrow\bm{p}^{\prime\prime}$ in the
intermediate states while the permutation
$(\bm{k}^{\prime}\leftrightarrow\bm{p}^{\prime})$ is taken account in the
prefactor in Eq. (5).

Within the minimal model the overlap of the Bloch amplitudes with allowance
for the spin-orbit coupling can be written as

$\langle{\bm{k},s}|{\bm{k}^{\prime},s}\rangle=1+s\xi[\bm{k}\times\bm{k}^{\prime}]_{z},$
(10)

where the real parameter $\xi$ describes the strength of the spin-orbit or
valley-orbit coupling yielding [46, 47, 48, 18, 13]

$U_{\bm{k}\bm{p},\bm{k}^{\prime}\bm{p}^{\prime}}=V(\absolutevalue{\bm{k}-\bm{k}^{\prime}})\quantity(1+i\xi[\bm{k}\times\bm{k}^{\prime}-\bm{p}\times\bm{p}^{\prime}]_{z}).$
(11)

In this work we disregard any spin-splitting of the energy spectrum, caused,
e.g., by the Rashba effect. The electron-electron collision integral with
allowance for the $\bm{k}$-linear terms in the spectrum is presented in Ref.
[49] and it does not contain asymmetric contributions.

Using Eq. (11) and the general momentum-conservation condition
$\bm{k}+\bm{p}=\bm{k}^{\prime}+\bm{p}^{\prime}$ we obtain that both the first
and the second $\text{Im}[\ldots]$ in Eq. (9) are given by

$\xi
V(\absolutevalue{\bm{k}-\bm{k}^{\prime}})V(\absolutevalue{\bm{k}-\bm{k}^{\prime\prime}})V(\absolutevalue{\bm{k}^{\prime\prime}-\bm{k}^{\prime}})\mathcal{S}_{z},$
(12)

where

$\bm{\mathcal{S}}=(\bm{k}^{\prime\prime}+\bm{p}^{\prime\prime}-\bm{k}-\bm{p})\times(\bm{k}^{\prime}-\bm{k}).$
(13)

We observe that, due to the momentum-conservation factor
$\delta_{\bm{k}+\bm{p},\bm{k}^{\prime\prime}+\bm{p}^{\prime\prime}}$, the
first $\text{Im}[\ldots]$ in Eq. (9) does not contribute to the skew
scattering probability. Thus, transitions described by the diagram in Fig.
1(b) do not contribute to the interparticle skew scattering rate in agreement
with analysis in Ref. [13]. In the case of fermions it means that the
scattering processes where the two-particle intermediate states are empty or
fully occupied play no role. It already demonstrates significant difference of
the asymmetric interparticle scattering as compared to the skew scattering by
static impurities.

Nevertheless, the momentum conservation for the second $\text{Im}[\ldots]$ in
Eq. (9) differs, therefore we obtain a nonzero result in general. We stress
that an asymmetry of the quasiparticle collisions occurs at scattering
processes described by Fig. 1(c) where the particle in the initial state
interacts with the particle in the intermediate state. For instance, for
fermions it means that the skew scattering occurs only via such the two-
particle states, where one single particle state is empty and another one is
occupied. As we demonstrate below it results in unconventional density and
temperature dependence of the SHE and VHE under interparticle collisions.

Assuming that the scattering is sufficiently short-range: $V(q)=V_{0}$, as it
is the case in the gated two-dimensional electron systems and also for
excitons, we finally obtain the asymmetric contribution to the interparticle
collision integral:

$\text{St}_{\rm sk}[f_{\bm{k}}]\\\
={16\pi^{2}\over\hbar}V_{0}^{3}\xi\sum_{\bm{k}^{\prime}\bm{p}^{\prime},\bm{k}^{\prime\prime}\bm{p}^{\prime\prime},\bm{p}}\delta(\varepsilon_{k}+\varepsilon_{p}-\varepsilon_{k^{\prime}}-\varepsilon_{p^{\prime}})\delta_{\bm{k}+\bm{p},\bm{k}^{\prime}+\bm{p}^{\prime}}\\\
\times[(1\pm f_{\bm{k}})(1\pm
f_{\bm{p}})f_{\bm{k}^{\prime}}f_{\bm{p}^{\prime}}+(\bm{k},\bm{p}\leftrightarrow\bm{k}^{\prime},\bm{p}^{\prime})]\\\
\times{\mathcal{S}}_{z}\delta(\varepsilon_{k}+\varepsilon_{p^{\prime\prime}}-\varepsilon_{k^{\prime\prime}}-\varepsilon_{p^{\prime}})\delta_{\bm{k}+\bm{p}^{\prime\prime},\bm{k}^{\prime\prime}+\bm{p}^{\prime}}\\\
\times\quantity[f_{\bm{k}^{\prime\prime}}(1\pm
f_{\bm{p}^{\prime\prime}})+(\bm{k}^{\prime\prime}\leftrightarrow\bm{p}^{\prime\prime})].$
(14)

Here we took into account that the interchange
$\bm{k},\bm{p}\leftrightarrow\bm{k}^{\prime},\bm{p}^{\prime}$ changes the sign
of $\bm{\mathcal{S}}$ while both energy and both momentum conservation laws in
Eq. (9) are intact. Therefore the ‘out-scattering’ term is added to the ‘in-
scattering’ one with the only modification in the occupation factors of the
initial and final states.

Figure 2: Examples of the diagrams in the Keldysh technique describing the in-
scattering processes. (a) Scattering via two empty or two occupied
intermediate states, cf. Fig. 1(b). (b) Scattering with the intermediate
states with one being empty and another being occupied, cf. Fig. 1(c).

Equation (14) can be also derived using the Keldysh diagram technique. The in-
and out-scattering terms can be recast as

$\mathop{\rm
St}{[}f_{\bm{k}}]=\mathrm{i}\Sigma^{-+}(1-f_{\bm{k}})+\mathrm{i}\Sigma^{+-}f_{\bm{k}},$
(15)

where $\Sigma^{\alpha\beta}$ are the corresponding self-energies with
$\alpha,\beta=\pm$ enumerating branches at the Keldysh contour, Fig. 2. Two
basic third order processes are shown in Fig. 2 with the diagram (a) and its
flipped counterpart describing the transitions via the empty or occupied
intermediate states [interaction lines do not cross, cf. Fig. 1(b)], while the
diagram (b) and its flipped counterpart describe the transitions where we take
into account interaction of the particle in the initial and in one of the
$\bm{k}^{\prime\prime}$, $\bm{p}^{\prime\prime}$ intermediate states
[interaction lines cross, cf. Fig. 1(c)]. Using the explicit expressions for
the Green’s functions and performing corresponding integrations (see Appendix
A) we obtain the collision integral exactly in the same form as in Eq. (14).

The skew-scattering part of the collision integral $\text{St}_{\rm sk}$ is
zero at the equilibrium distribution function $f_{k}^{0}$. This fact follows
from the general arguments: for the spin-orbit coupling in the form of Eq.
(11) any equilibrium spin currents are forbidden. One can also demonstrate it
explicitly making use of the fact that the equilibrium distribution functions
depend on the energy only and using the energy and momentum conservation laws
in Eq. (14) and explicit form of $\mathcal{S}_{z}$, see Appendix B for
details.

##  III Spin and Valley Hall effects

Following Ref. [11] we define the spin/valley current as

$\bm{j}_{\rm SHE/VHE}\equiv\bm{j}^{\rm
s}=\frac{1}{2}\sum_{s,\bm{k}}s\frac{\hbar\bm{k}}{m}f_{\bm{k},s}.$ (16)

In the studied situation the electrons are unpolarized, thus it is sufficient
to calculate the current for $s=+$ branch, while the current in the $s=-$
branch has the same magnitude and opposite direction. We now turn to the
calculation of the SHE and VHE from the kinetic Eq. (1) with the collision
integrals (14) and (7). We treat the spin-orbit coupling as a small
perturbation, accordingly, we solve Eq. (1) by iterations.

We calculate the velocity generation rate:

$\left<\dot{\bm{V}}\right>=\sum_{\bm{k}}{\hbar\bm{k}\over m}\text{St}_{\rm
sk}[f_{\bm{k}}].$ (17)

Analysis shows (see Appendix C) that this generation rate is nonzero only if
the following conditions are met in Eq. (14) for $\text{St}_{\rm sk}$

$\bm{p}^{\prime}=\bm{k},\quad\bm{k}^{\prime}=\bm{p},\quad\bm{p}^{\prime\prime}=\bm{k}^{\prime\prime}.$
(18)

In other words, the only relevant scattering processes are
$\bm{k}+,\bm{p}-\rightarrow\bm{k}^{\prime\prime}+,\bm{k}^{\prime\prime}-\rightarrow\bm{p}+,\bm{k}-$.

Therefore we simplify the velocity generation rate applying Eq. (18)
everywhere except for the $\delta$-functions:

$\left<\dot{\bm{V}}\right>={64\pi^{2}\over m}V_{0}^{3}\xi\\\
\times\sum_{\bm{k}^{\prime}\bm{p}^{\prime},\bm{k}^{\prime\prime}\bm{p}^{\prime\prime},\bm{k}\bm{p}}\delta(\varepsilon_{k}+\varepsilon_{p}-\varepsilon_{k^{\prime}}-\varepsilon_{p^{\prime}})\delta_{\bm{k}+\bm{p},\bm{k}^{\prime}+\bm{p}^{\prime}}\\\
\times\delta(\varepsilon_{k}+\varepsilon_{p^{\prime\prime}}-\varepsilon_{k^{\prime\prime}}-\varepsilon_{p^{\prime}})\delta_{\bm{k}+\bm{p}^{\prime\prime},\bm{k}^{\prime\prime}+\bm{p}^{\prime}}\\\
\times\bm{k}{\mathcal{S}}_{z}(1\pm f_{\bm{k}})(1\pm
f_{\bm{p}})f_{\bm{k}}f_{\bm{p}}f_{\bm{k}^{\prime\prime}}(1\pm
f_{\bm{k}^{\prime\prime}}).$ (19)

Since generation of the spin and valley Hall current is a nonequilibrium
response to the external force $\bm{F}$, substituting the distributions
$f_{\bm{k}}=f_{k}^{0}-\hbar(\bm{v}_{dr}\cdot\bm{k})\differential
f_{k}^{0}/\differential\varepsilon_{k}$ with the drift velocity
$\bm{v}_{dr}=\bm{F}\tau_{p}/m$ and linearizing Eq. (19) in $\bm{v}_{dr}$ we
obtain

$\left<\dot{V}_{y}\right>=v_{dr,x}{16\pi^{2}\hbar\over
mT}V_{0}^{3}\xi\sum_{\bm{k}^{\prime}\bm{p}^{\prime},\bm{k}^{\prime\prime}\bm{p}^{\prime\prime},\bm{k}\bm{p}}\delta(\varepsilon_{k}+\varepsilon_{p}-\varepsilon_{k^{\prime}}-\varepsilon_{p^{\prime}})\delta_{\bm{k}+\bm{p},\bm{k}^{\prime}+\bm{p}^{\prime}}(1\pm
f_{k}^{0})(1\pm f_{p}^{0})f_{k}^{0}f_{p}^{0}f_{k^{\prime\prime}}^{0}(1\pm
f_{k^{\prime\prime}}^{0})\\\
\times\delta(\varepsilon_{k}+\varepsilon_{p^{\prime\prime}}-\varepsilon_{k^{\prime\prime}}-\varepsilon_{p^{\prime}})\delta_{\bm{k}+\bm{p}^{\prime\prime},\bm{k}^{\prime\prime}+\bm{p}^{\prime}}\bm{\mathcal{S}}\cdot\quantity{\quantity[\bm{k}\quantity(1\pm
2f_{k}^{0})+\bm{p}\quantity(1\pm
2f_{p}^{0})+\bm{k}^{\prime\prime}\quantity(1\pm
2f_{k^{\prime\prime}}^{0})]\times(\bm{k}-\bm{p})}.$ (20)

Here we assume $\bm{v}_{dr}\parallel x$ and $\tau_{p}$ to be energy
independent, accordingly we calculate the generation rate of the transverse to
$\bm{F}$ velocity,
$\bm{\mathcal{S}}=(\bm{k}+\bm{p}-2\bm{k}^{\prime\prime})\times(\bm{k}-\bm{p})$,
and $T$ is the temperature in energy units.

In order to find the spin or valley current in the steady state one has to
take into account its relaxation. If the dominant relaxation mechanism is the
elastic scattering by impurities or quasielastic scattering by phonons, then

$j^{s}_{y}=\tau_{p}\left<\dot{V}_{y}\right>.$ (21a) However, we are mainly
interested in the situation where the interparticle collisions dominate the
relaxation. Conserving the total momentum of the colliding quasiparticles they
result in the relaxation of the spin and valley currents. Generally, these
collisions are inelastic and the solution of the corresponding kinetic
equation is quite involved [37, 40]. To illustrate the effect and provide an
estimate of its magnitude it is convenient to introduce a relaxation time
approximation and present the current in the form similar to Eq. (21a)
$j^{s}_{y}=\tau_{\rm sc}\left<\dot{V}_{y}\right>$ (21b)

with $\tau_{\rm sc}$ being the corresponding spin current relaxation time, it
is derived in Appendix D and defined by Eq. (39). Note that by contrast to
relatively slow relaxation of odd harmonics of spin-averaged distribution (or
lack of relaxation in the case of the first harmonic) [50], the interpartcle
collisions result in the efficient relaxation of both first and third angular
harmonics of the spin and valley distrubtion [40].

Still, the expressions for the SHE and VHE currents for arbitrary statistics
of quasiparticles are quite involved. Below we consider the most illustrative
and experimentally relevant limits of Fermi, Boltzmann and Bose gases.

###  III.1 Fermi gas

In the case of degenerate electrons, a dramatic suppression of the SHE and VHE
at the electron-electron collisions occurs. We recall that the electron-
electron scattering time responsible for the spin current relaxation at
$E_{\rm F}\gg T$ behaves as

$\frac{1}{\tau_{\rm sc}}=C\frac{T^{2}}{E_{F}},$ (22)

where the constant $C$ weakly depends on the temperature [40]: The Pauli
exclusion principle blocks electron-electron collisions at low temperatures.
Inspecting general Eq. (20) we note that the factors
$f^{0}_{k^{\prime\prime}}(1-f^{0}_{k^{\prime\prime}})$ should provide
additional reduction of the phase space and one can expect
$\left\langle\dot{V}_{y}\right\rangle\propto T^{3}$.

However the analysis shows that it is not the case, and
$\left\langle\dot{V}_{y}\right\rangle$ is expected to behave as $T^{4}$.
Indeed, to obtain a result in the lowest order in $T/E_{F}$ we can set all
absolute values of the wavevectors equal to the Fermi wavevector
$k=k^{\prime}=k^{\prime\prime}=p=p^{\prime}=p^{\prime\prime}=k_{\rm F}$
everywhere except for the equilibrium occupations. Then we come to the energy
integrals ($q=k,p$ or $k^{\prime\prime}$)

$\int_{0}^{\infty}\differential\varepsilon_{q}(1-f_{q}^{0})f_{q}^{0}(1-2f_{q}^{0})\propto\int_{0}^{\infty}\differential\varepsilon_{q}\derivative[2]{f_{q}^{0}}{\varepsilon_{q}},$
(23)

which are exponentially small at $E_{\rm F}\gg T$.

Hence, the non-zero result for the scattering rate can be obtained in the next
order in $T/E_{\rm F}$ where one takes into account deviations of the absolute
values of the wavevectors from $k_{\rm F}$. Consequently, we expect for
degenerate electrons

$j_{y}^{s}\sim N_{1}v_{dr,x}\times\xi k_{\text{F}}^{2}\times
gV_{0}\times\quantity({T\over E_{\text{F}}})^{2}.$ (24)

As a result, the generation of the spin and valley currents due to the
electron-electron scattering is suppressed. By contrast, interparticle
scattering results in relatively efficient relaxation of the spin and valley
currents.

###  III.2 Boltzmann gas

For Boltzmann statistics we have for both bosons and fermions
$f_{k}^{0}=N_{1}/(gT)\exp(-\varepsilon_{k}/T){\ll 1}$ where $N_{1}$ and
$g=m/(2\pi\hbar^{2})$ are the two-dimensional particle density and the density
of states per one spin or valley.

Calculations presented in Appendix E (see also Appendix D.1) yield the steady-
state spin current value in the form

$j_{y}^{s}=-{256\sqrt{3}\over\pi}N_{1}v_{dr,x}\times\xi k_{T}^{2}\times
gV_{0}\times{E_{\text{F}}\over T}.$ (25)

Here $N_{1}v_{dr,x}$ is the particle current in one spin subband, $\xi
k_{T}^{2}$ is the dimensionless spin-orbit coupling strength
($k_{T}=\sqrt{2mT}/\hbar$ is the thermal wavevector), $gV_{0}$ is the non-Born
parameter, and the factor $E_{\text{F}}/T=4\pi N_{1}/k_{T}^{2}\ll 1$ stems
from occupations of intermediate states. In derivation we used the spin-
current relaxation time for the non-degenerate case, Eq. (44):

${1\over\tau_{\text{sc}}}={2m\over\hbar^{3}}V_{0}^{2}N_{1}.$ (26)

It is instructive to compare the result (25) with the spin current induced by
the scattering by the static short-range impurities with the same density
$N_{1}$ in the absence of electron-electron collisions. In these situations
the relaxation rates for the spin currents are about the same. From Eq. (22)
of Ref. [11] we obtain

$j_{y}^{s}\sim N_{1}v_{dr,x}\times\xi k_{T}^{2}\times gV_{0},$ (27)

where the numerical coefficient on the order of unity is omitted. Comparing
Eqs. (25) and (27) we observe that the spin and valley currents induced by the
electron-electron collisions have additional smallness $\sim E_{\rm F}/T$
related, as discussed above, to the fact that the skew scattering requires the
intermediate state $\bm{k}^{\prime\prime}$ to be occupied. In the case of
impurity scattering the intermediate state occupancies play no role thus
parametrically enhancing the generation rate. However, while for reasonable
electron densities $N_{1}\sim 10^{11}\text{cm}^{-2}$,
$m=0.1\leavevmode\nobreak\ m_{0}$ and at room temperature the small parameter
$E_{\rm F}/T$ is about $0.1$, the numerical prefactor in Eq. (25)
$256\sqrt{3}/\pi\approx 141.1$ provides significant enhancement of the effect.
Thus, in the state of the art samples the interparticle scattering effect on
the spin current can be pronounced.

###  III.3 Degenerate Bose gas

While degeneracy in Fermi systems results in suppression of the interparticle
scattering effects, in degenerate Bose gas stimulated scattering processed
dominate. As a result, for degenerate bosons the occupancy of the intermediate
states is expected to significantly enhance the skew scattering effect
compared to the enhancement of the spin current relaxation rate.

To illustrate the enhancement we observe that for the degenerate Bose gas, the
distribution function can be represented as
$f_{k}^{0}=T/(\varepsilon_{k}-\mu)\gg 1$ with the chemical potential $\mu<0$,
$\absolutevalue{\mu}\ll T$. For energies $\varepsilon_{k}-\mu\gtrsim T$ the
distribution function exponential decays, such energy range is irrelevant for
the following. Calculations presented in Appendix F demonstrate the
exponentially large steady-state value of the spin/valley current:

$j_{y}^{s}=-\mathcal{C}\times v_{dr,x}gT\times\xi k_{T}^{2}\times
gV_{0}\times\exp({2N_{1}\over gT}).$ (28)

Here $gT$ is the characteristic concentration of particles where the gas
becomes degenerate, consequently $2N_{1}/(gT)>1$ for degenerate bosons, and
the numerical coefficient calculated in Appendix F is
$\mathcal{C}\approx{1340.4}$. In deriving Eq. (28) we used the value of the
spin current relaxation rate (49):

$\frac{1}{\tau_{\rm sc}}=\mathcal{C}_{\tau}\times
2^{11}{g^{2}TV_{0}^{2}\over\hbar}\exp({N_{1}\over gT}),$ (29)

where $\mathcal{C}_{\tau}\approx 0.36$ is the numerical coefficient, for
calculations see Appendix D.2.

Equation (28) shows a possibility of a dominant role of the skew scattering at
low enough temperatures in Bose systems. However, the analysis presented above
is valid for temperatures exceeding the temperature of the Berezinskii-
Kosterlitz-Thouless transitions to the superfluid state. The analysis of the
collective properties of bosons with the skew scattering processes included
goes beyond the scope of the present work.

##  IV Conclusion

In this work we have developed a theory of the spin and valley Hall effects in
two-dimensional systems in the presence of frequent collisions between the
quasiparticles. We have addressed both fermions, electrons in quantum wells or
two-dimensional semiconductors, and bosons, e.g., excitons or exciton-
polaritons. We have focused on the skew scattering effect where the colliding
particles with the opposite spin or valley indices scatter in the opposite
directions. The skew scattering results in the generation of a spin or valley
current in the transverse direction to the particle current induced, e.g., by
a real or synthetic electric field. The relaxation of the spin and valley
current also assumed to occur due to the interparticle scattering.

We have derived the asymmetric – skew – contribution to the interparticle
collision integral in the third order of perturbation theory in the
interaction potential. We have demonstrated that occupancies of intermediate
states play a crucial role in the effect: the skew scattering occurs in the
processes where the particle in the initial state interacts with the particle
in the intermediate state. For fermions, for example, the two-particle
intermediate states which are empty of fully occupied play no role. It results
in a specific behavior of the spin and valley current generation rates. In a
non-degenerate gas the generation rate is proportional to the squared density
of the particles, while in a degenerate Fermi gas the generation rate
demonstrates $T^{4}$ temperature dependence. By contrast, for degenerate Bose
gas this results in exponential enhancement of the spin and valley currents
with increasing of particle concentration or decreasing the temperature.
Moreover, in the Boltzmann and Fermi gases, the parametrically small spin or
valley current is additionally increased because of numerical factors on the
order $10^{3}\ldots 10^{4}$ that appear in the resulting expressions.

Hence, an asymmetry of the interparticle scattering results in deviations of a
classical Fermi-liquid picture where in the processes of quasiparticle
scattering occupancies of intermediate states play no role. By contrast, for
bosons, the role of the skew scattering turns out to be significant when the
Bose gas becomes degenerate. The analysis of the non-Fermi liquid effects for
degenerate electrons and effects of asymmetry in interactions of bosons on
collective phenomena is a separate problem for further studies.

###### Acknowledgements.

We are grateful to I. S. Burmistrov for interest to this work and useful
comments. The financial support of the Russian Science Foundation (Project No.
22-12-00211) is acknowledged. The work of L. E. G. was supported by the
Foundation for the Advancement of Theoretical Physics and Mathematics
‘‘BASIS’’.

##  Appendix A Derivation of the skew scattering integral within the Keldysh
technique

It is instructive to derive the collision integral using the diagram technique
for non-equilibrium processes. According to the general rules [51, 52], the
evolution of the distribution function is governed by the self-energies
$\Sigma^{-+}$ and $\Sigma^{+-}$ which describe the in- and out-scattering
processes, see Eq. (15). In the case of the skew scattering, the self-energies
are given by the diagrams containing three interaction lines. Making use of
the fact that, in the absence of retardation, the interaction line connect the
same parts of the Keldysh contour, we obtain that $\Sigma^{-+}$ is determined
by the diagrams in Fig. 2 and their counterparts where the interaction occurs
on the ‘$-$’ line of the Keldysh contour (also, in agreement with the
description of the interaction matrix element (3) the summation should be
performed of the diagrams with permuted
$\bm{k}^{\prime}\leftrightarrow\bm{p}^{\prime}$,
$\bm{k}^{\prime\prime}\leftrightarrow\bm{p}^{\prime\prime}$). Using the
explicit expressions for the Greens functions (top signs refer to the fermions
and bottom signs to the bosons)

$\displaystyle G^{-+}(\varepsilon,\bm{k})=\pm
2\pi\mathrm{i}f_{\bm{k}}\delta(\varepsilon-\varepsilon_{k}),$ (30a)
$\displaystyle G^{+-}(\varepsilon,\bm{k})=-2\pi\mathrm{i}(1\mp
f_{\bm{k}})\delta(\varepsilon-\varepsilon_{k}),$ (30b) $\displaystyle
G^{--}(\varepsilon,\bm{k})=\frac{1\mp
f_{\bm{k}}}{\varepsilon-\varepsilon_{k}+\mathrm{i}0}\pm\frac{f_{\bm{k}}}{\varepsilon-\varepsilon_{k}-\mathrm{i}0},$
(30c) $\displaystyle
G^{++}(\varepsilon,\bm{k})=-[G^{--}(\varepsilon,\bm{k})]^{*},$ (30d)

we obtain the following contributions $\mathrm{i}\Sigma^{-+}$:

$\mbox{Fig.\leavevmode\nobreak\
\ref{fig:keldysh}(a):}\quad\mathrm{i}\Sigma^{-+}_{a}(\varepsilon,\bm{k})\\\
=4\sum_{\bm{k}^{\prime}\bm{k}^{\prime\prime}\bm{p}^{\prime\prime}}U_{\bm{k}^{\prime}\bm{p}^{\prime},\bm{k}\bm{p}}U_{\bm{k}\bm{p},\bm{k}^{\prime\prime}\bm{p}^{\prime\prime}}U_{\bm{k}^{\prime\prime}\bm{p}^{\prime\prime},\bm{k}^{\prime}\bm{p}^{\prime}}\\\
\int\frac{d\Omega_{1}}{2\pi\mathrm{i}}\int\frac{d\Omega_{2}}{2\pi\mathrm{i}}G^{-+}(\varepsilon-\Omega_{2},\bm{k}^{\prime})G^{++}(\varepsilon-\Omega_{2}+\Omega_{1},\bm{k}^{\prime\prime})\\\
\times\int\frac{d\omega}{2\pi\mathrm{i}}G^{+-}(\omega,\bm{p})G^{++}(\omega+\Omega_{2}-\Omega_{1},\bm{p}^{\prime\prime})G^{-+}(\omega+\Omega_{2},\bm{p}^{\prime}),$
(31a) $\mbox{Fig.\leavevmode\nobreak\
\ref{fig:keldysh}(b):}\quad\mathrm{i}\Sigma^{-+}_{b}(\varepsilon,\bm{k})\\\
=4\sum_{\bm{k}^{\prime}\bm{k}^{\prime\prime}\bm{p}^{\prime\prime}}U_{\bm{k}^{\prime}\bm{p}^{\prime},\bm{k}\bm{p}}U_{\bm{k}\bm{p},\bm{k}^{\prime\prime}\bm{p}^{\prime\prime}}U_{\bm{k}^{\prime\prime}\bm{p}^{\prime\prime},\bm{k}^{\prime}\bm{p}^{\prime}}\\\
\int\frac{d\Omega_{1}}{2\pi\mathrm{i}}\int\frac{d\Omega_{2}}{2\pi\mathrm{i}}G^{-+}(\varepsilon-\Omega_{1},\bm{k}^{\prime})G^{++}(\varepsilon+\Omega_{1}-\Omega_{2},\bm{k}^{\prime\prime})\\\
\times\int\frac{d\omega}{2\pi\mathrm{i}}G^{+-}(\omega,\bm{p})G^{++}(\omega+\Omega_{1},\bm{p}^{\prime\prime})G^{-+}(\omega+\Omega_{2},\bm{p}^{\prime}),$
(31b) Here the factor $4$ arises from the permutations of the initial and
intermediate wavevectors. The inclusion of the diagrams with interaction on
the ‘$-$’ contour results in taking twice the real part of Eqs. (31).

In Eq. (31a) we can integrate over $\omega$ using the $\delta$-function in
$G^{+-}(\omega,\bm{p})$ and over $\Omega_{2}$ using any of $\delta$-functions
in $G^{-+}(\varepsilon-\Omega_{2},\bm{k}^{\prime})$ or
$G^{-+}(\omega+\Omega_{2},\bm{p}^{\prime})$. As a result, we obtain

$\mathrm{i}\Sigma_{a}^{-+}(\varepsilon,\bm{k})\propto\delta(\varepsilon+\varepsilon_{\bm{p}}-\varepsilon_{\bm{k}^{\prime}}-\varepsilon_{\bm{p}^{\prime}})(1-f_{\bm{p}})f_{\bm{k}^{\prime}}f_{\bm{p}^{\prime}}\\\
\times\int\frac{d\Omega_{1}}{2\pi\mathrm{i}}G^{++}(\varepsilon_{\bm{k}^{\prime}}+\Omega_{1},\bm{k}^{\prime\prime})G^{++}(\varepsilon_{\bm{p}^{\prime}}-\Omega_{1},\bm{p}^{\prime\prime}).$
(32)

To be non-zero, the poles in the integral over $\Omega_{1}$ should have
different imaginary parts. Hence, in both $G^{++}$ Green’s functions only the
terms with similar occupation factors $\propto
f_{\bm{k}^{\prime\prime}}f_{\bm{p}^{\prime\prime}}$ and $(1\mp
f_{\bm{k}^{\prime\prime}})(1\mp f_{\bm{p}^{\prime\prime}})$ do not vanish.
Integrating using the residue theorem and adding the conjugate term we arrive
at the contribution to the in-scattering rate given by the first three lines
in Eq. (9) of the main text.

Analogous calculation of $\Sigma_{b}^{-+}$ in (31b) yields

$\mathrm{i}\Sigma_{b}^{-+}(\varepsilon,\bm{k})\propto\delta(\varepsilon+\varepsilon_{\bm{p}}-\varepsilon_{\bm{k}^{\prime}}-\varepsilon_{\bm{p}^{\prime}})(1-f_{\bm{p}})f_{\bm{k}^{\prime}}f_{\bm{p}^{\prime}}\\\
\times\int\frac{d\Omega_{1}}{2\pi\mathrm{i}}G^{++}(\varepsilon_{\bm{k}^{\prime}}+\Omega_{1},\bm{k}^{\prime\prime})G^{++}(\varepsilon_{\bm{p}}+\Omega_{1},\bm{p}^{\prime\prime}).$
(33)

The integral over $\Omega_{1}$ in this case does not vanish only if unlike
terms in the product of Greens functions is taken: either
$f_{\bm{k}^{\prime\prime}}(1\mp f_{\bm{p}^{\prime\prime}})$ or
$f_{\bm{p}^{\prime\prime}}(1\mp f_{\bm{k}^{\prime\prime}})$. Corresponding
integration results in the contribution to the in-scattering rate given by the
last three lines in Eq. (9) of the main text. Summing Eqs. (32) and (33) we
arrive at the in-scattering term of Eq. (14) of the main text. The evaluation
of the out-scattering terms is similar. Total collision integral fully agrees
with (14) of the main text.

##  Appendix B Absence of the spin current in equilibrium

Let us take an isotropic distribution function $f_{k}$ (i.e., in the absence
of external field) and consider the angular Fourier harmonics of the skew
scattering integral, Eq. (14),

$\displaystyle S_{n}=\sum_{\bm{k}}g_{k}\sin{(n\varphi_{\bm{k}})}\text{St}_{\rm
sk}[f_{k}],$ $\displaystyle
C_{n}=\sum_{\bm{k}}g_{k}\sin{(n\varphi_{\bm{k}})}\text{St}_{\rm sk}[f_{k}].$

Here $g_{k}$ is an arbitrary function of electron energy (that vanishes at
$k\to\infty$), and $n=0,1,\ldots$ Performing the change of the variables
$(k_{x},k_{y})\to(-k_{x},k_{y})$, $(p_{x},p_{y})\to(-p_{x},p_{y})$ (and
similar for
$\bm{k}^{\prime},\bm{p}^{\prime},\bm{k}^{\prime\prime},\bm{p}^{\prime\prime}$)
which corresponds to the reflection in $(yz)$ plane we see that
${\mathcal{S}}_{z}$ changes its sign while all other multipliers are intact.
As a result, $S_{n}\equiv 0$. Similar analysis with the reflection in the
$(xz)$ plane shows that $C_{n}\equiv 0$. As a result, $\text{St}_{\rm
sk}[f_{k}]$ vanishes for any angular-independent distribution function
$f_{k}$. In particular, it is zero for equilibrium distribution function. As a
result, no spin and valley current can flow in equilibrium conditions for the
spin-orbit interaction in the form of Eq. (10).

##  Appendix C Kinematics of the skew scattering

In order to prove the relations (18) we introduce

$\bm{\varkappa}=\bm{k}-\bm{p},\quad\bm{\varkappa}^{\prime}=\bm{k}^{\prime}-\bm{p}^{\prime},\quad\bm{Q}=\bm{k}+\bm{p}-(\bm{k}^{\prime\prime}+\bm{p}^{\prime\prime}).$
(34)

Then we have

$\bm{\mathcal{S}}\delta(\varepsilon_{k}+\varepsilon_{p}-\varepsilon_{k^{\prime}}-\varepsilon_{p^{\prime}})\delta(\varepsilon_{k}+\varepsilon_{p^{\prime\prime}}-\varepsilon_{k^{\prime\prime}}-\varepsilon_{p^{\prime}})\\\
={\bm{Q}\times(\bm{\varkappa}-\bm{\varkappa}^{\prime})\over
2}\quantity({4m\over\hbar^{2}})^{2}\delta(\varkappa^{2}-\varkappa^{\prime
2})\delta[\bm{Q}\cdot(\bm{\varkappa}{+}\bm{\varkappa}^{\prime})].$ (35)

The second $\delta$-function implies one of 3 possibilities: i)
$\bm{Q}=\bm{0}$, ii) $\bm{Q}\perp(\bm{\varkappa}{+}\bm{\varkappa}^{\prime})$,
and iii) $(\bm{\varkappa}{+}\bm{\varkappa}^{\prime})=0$. In the cases i) and
ii) $\bm{\mathcal{S}}$ vanishes, therefore we obtain a nonzero result in the
case iii) only. This yields the relations (18).

Taking in Eq. (20) $\bm{\varkappa}^{\prime}=-\bm{\varkappa}$ everywhere except
for the $\delta$-function, we perform summation over
$\bm{\varkappa}^{\prime}$:

$\sum_{\bm{\varkappa}^{\prime}}\delta(\varkappa^{2}-\varkappa^{\prime
2})\delta[\bm{Q}\cdot(\bm{\varkappa}{+}\bm{\varkappa}^{\prime})]\\\ ={1\over
4\pi}\left<\delta[Q\varkappa(\cos\varphi+\cos\varphi^{\prime})]\right>_{\varphi^{\prime}}={1\over
8\pi^{2}Q\varkappa\absolutevalue{\sin\varphi}},$ (36)

where $\varphi$ is an angle between $\bm{Q}$ and $\bm{\varkappa}$. Here only
one root $\varphi^{\prime}=\varphi+\pi$ was used because the other,
$\varphi^{\prime}=\pi-\varphi$, corresponds to
$\bm{Q}\perp(\bm{\varkappa}+\bm{\varkappa}^{\prime})$ which yields zero, see
above. Then we have:

$\left<\dot{V}_{y}\right>=v_{dr,x}{16m\over
T\hbar^{3}}V_{0}^{3}\xi\sum_{\bm{K},\bm{\varkappa},\bm{Q}}{[\bm{Q}\times\bm{\varkappa}]_{z}\over\absolutevalue{\bm{Q}\times\bm{\varkappa}}}\\\
\times(1\pm f_{k}^{0})(1\pm
f_{p}^{0})f_{k}^{0}f_{p}^{0}f_{k^{\prime\prime}}^{0}(1\pm
f_{k^{\prime\prime}}^{0})\\\ \times\quantity[\bm{K}\times\bm{\varkappa}(3\pm
2f_{k}^{0}\pm 2f_{p}^{0}\pm 2f_{k^{\prime\prime}}^{0})\mp
2\bm{Q}\times\bm{\varkappa}f_{k^{\prime\prime}}^{0}]_{z}.$ (37)

Here we introduced $\bm{K}=\bm{k}+\bm{p}$.

##  Appendix D Relaxation of spin and valley currents due to interparticle
collisions

It is convenient to introduce the pseudospin component

$s_{\bm{k}}=\frac{1}{2}\sum_{s}sf_{\bm{k},s}$

and obtain from the collision integrals (7) the contribution responsible for
the relaxation of the pseudospin [cf. Refs. [40, 42]]:

$\mathop{\rm St}{[s_{\bm{k}}]}=-\frac{4\pi
V_{0}^{2}}{\hbar}\sum_{\bm{k}^{\prime}\bm{p}\bm{p}^{\prime}}\delta_{\bm{k}+\bm{p},\bm{k}^{\prime}+\bm{p}^{\prime}}\delta(\varepsilon_{k}+\varepsilon_{p}-\varepsilon_{k^{\prime}}-\varepsilon_{p^{\prime}})\\\
\times\left[s_{\bm{k}}F(\bm{p};\bm{k}^{\prime},\bm{p}^{\prime})-s_{\bm{k}^{\prime}}F(\bm{p}^{\prime};\bm{k},\bm{p})\right],$
(38)

where

$F(\bm{p};\bm{k}^{\prime},\bm{p}^{\prime})=f_{\bm{p}}(1\pm
f_{\bm{k}^{\prime}}\pm f_{\bm{p}^{\prime}})\mp
f_{\bm{p}^{\prime}}f_{\bm{k}^{\prime}}.$

To determine the relaxation time $\tau_{\rm sc}$ we take $s_{\bm{k}}$ in the
form $s_{\bm{k}}=Ak_{x}f_{k}^{0}(1\pm f_{k}^{0})$ which corresponds to a
variational solution of kinetic equation ($A$ is a variational parameter) with
the generation rate $Gk_{x}f_{k}^{0}(1\pm f_{k}^{0})$. The relaxation time is
determined from the relation $A=\tau_{\rm sc}G$. To find $A$ we substitute
variational solution into the kinetic equation, multiply it by
$\cos{\varphi_{\bm{k}}}$ where $\varphi_{\bm{k}}$ is the angle between the
$x$-axis and $\bm{k}$ and sum over $\bm{k}$ with the result

$\frac{1}{\tau_{\rm sc}}=-\frac{\sum_{\bm{k}}\cos{\varphi}_{\bm{k}}\mathop{\rm
St}{[k_{x}f_{k}^{0}(1\pm f_{k}^{0})]}}{\frac{1}{2}\sum_{\bm{k}}kf_{k}^{0}(1\pm
f_{k}^{0})}.$ (39)

Substitution of Eq. (38) yields

$\frac{1}{\tau_{\rm sc}}=\frac{4\pi
V_{0}^{2}/\hbar}{\sum_{\bm{k}}kf_{k}^{0}(1\pm f_{k}^{0})}\\\
\times\sum\limits_{\bm{k}\bm{p}\bm{k}^{\prime}\bm{p}^{\prime}}\delta_{\bm{k}+\bm{p},\bm{k}^{\prime}+\bm{p}^{\prime}}\delta(\varepsilon_{k}+\varepsilon_{p}-\varepsilon_{k^{\prime}}-\varepsilon_{p^{\prime}})\\\
\times\quantity(k-\bm{k}\cdot\bm{k}^{\prime}/k)f_{k}^{0}(1\pm
f_{k}^{0})F(\bm{p};\bm{k}^{\prime},\bm{p}^{\prime}),$ (40)

where we made substitutions $\cos{\varphi}_{\bm{k}}k_{x}\to{k/2}$,
$\cos{\varphi}_{\bm{k}}k_{x}^{\prime}\to{\bm{k}\cdot\bm{k}^{\prime}/(2k)}$
taking into account the isotropy in the $xy$ plane. Then using the notations
$\bm{\varkappa}$, $\bm{\varkappa}^{\prime}$ introduced in Eq. (34) we obtain

${1\over\tau_{\text{sc}}}={2mV_{0}^{2}\over\hbar^{3}}{\sum\limits_{\bm{k},\bm{\varkappa},\bm{\varkappa}^{\prime}}\left<{\bm{k}\cdot(\bm{\varkappa}-\bm{\varkappa}^{\prime})\over
k}f_{k}^{0}(1\pm
f_{k}^{0})F(\bm{p};\bm{k}^{\prime},\bm{p}^{\prime})\right>\over\sum\limits_{\bm{k}}kf_{k}^{0}(1\pm
f_{k}^{0})}.$ (41)

Here angular brackets mean averaging over directions of the vector
$\bm{\varkappa}^{\prime}$ at $\varkappa^{\prime}=\varkappa$, and
$\bm{p}=\bm{k}-\bm{\varkappa}$,
$\bm{k}^{\prime}=\bm{k}+(\bm{\varkappa}^{\prime}-\bm{\varkappa})/2$,
$\bm{p}^{\prime}=\bm{k}-(\bm{\varkappa}^{\prime}+\bm{\varkappa})/2$.

###  D.1 Relaxation rate at Boltzmann statistics

For non-degenerate particles (Boltzmann statistics) we take in Eq. (41)
$f^{0}_{k}=(N_{1}/gT)\exp(-\varepsilon_{k}/T)\ll 1$. Then we obtain

${1\over\tau_{\text{sc}}}={2mV_{0}^{2}\over\hbar^{3}}{\sum\limits_{\bm{k},\bm{\varkappa}}{\bm{k}\cdot\bm{\varkappa}\over
k}f_{k}^{0}f_{\bm{k}-\bm{\varkappa}}^{0}\over\sum\limits_{\bm{k}}kf_{k}^{0}}.$
(42)

Performing angular averaging with help of
$\left<\cos{\theta}\exp(a\cos{\theta})\right>_{\theta}=I_{1}(a)$, we get

${1\over\tau_{\text{sc}}}={4\pi V_{0}^{2}N_{1}\over\hbar
Tk_{T}^{3}/(8\sqrt{\pi})}\sum\limits_{\bm{k},\bm{\varkappa}}\varkappa
I_{1}\quantity({2k\varkappa\over k_{T}^{2}})\exp(-{2k^{2}+\varkappa^{2}\over
k_{T}^{2}}).$ (43)

Then integrating over absolute values of $k$ and $\varkappa$ we obtain

${1\over\tau_{\text{sc}}}={2m\over\hbar^{3}}V_{0}^{2}N_{1}.$ (44)

###  D.2 Relaxation rate at Bose-Einstein statistics

For degenerate Bose particles we take in Eq. (41)

$f^{0}_{k}=T/(\varepsilon_{k}+\absolutevalue{\mu})\gg 1,$ (45)

where $\mu<0$ is related to the particle density with a fixed spin/ in a given
valley via $N_{1}=gT\ln(T/\absolutevalue{\mu})$. Then we obtain

$\frac{1}{\tau_{\rm
sc}}=\frac{2^{9}mV_{0}^{2}g^{2}T^{4}}{\hbar^{3}\mu^{2}}\sqrt{2m\absolutevalue{\mu}\over\hbar^{2}}{1\over\sum_{\bm{k}}k\quantity(f_{k}^{0})^{2}}\mathcal{C}_{\tau},$
(46)

where

$\mathcal{C}_{\tau}=\\\ \int\limits_{0}^{\infty}\differential
X\int\limits_{0}^{\infty}\differential
Y\Biggl{<}{Y(1-\cos\varphi)+\sqrt{XY}[\cos(\theta-\varphi)-\cos\varphi]\over\sqrt{X+Y+2\sqrt{XY}\cos\theta}}\\\
\times{1\over(X+Y+1)^{2}-4XY\cos^{2}(\theta-\varphi)}\\\
\times{1\over(X+Y+1)^{2}-4XY\cos^{2}\theta}\Biggr{>}_{\varphi,\theta}.$ (47)

Here we used $X=\varepsilon_{\bm{k}+\bm{p}}/\absolutevalue{\mu}$,
$Y=\varepsilon_{\varkappa}/\absolutevalue{\mu}$. Note that while the
simplified form of the distribution function (45) is valid for
$\varepsilon_{k}\ll T$, the integrals in Eq. (47) converge at $X,Y\to\infty$,
thus we can extend the integration up to $+\infty$.

Since

$\sum_{\bm{k}}k\quantity(f_{k}^{0})^{2}={\pi gT^{2}\over
2\absolutevalue{\mu}}\sqrt{2m\absolutevalue{\mu}\over\hbar^{2}},$ (48)

we obtain

$\frac{1}{\tau_{\rm
sc}}=2^{11}{(gT)^{2}V_{0}^{2}\over\hbar\absolutevalue{\mu}}\mathcal{C}_{\tau}.$
(49)

For numerical evaluation of the 4-fold integral (47) it is convenient to
change the variables formally introducing $R$ and $\alpha$ as
$X=R\cos{\alpha}$, $Y=R\sin{\alpha}$. Correspondingly, the integration is
carried out over a sector in the plane $0\leqslant\alpha\leqslant\pi/2$, and
$R\in[0,\infty)$. Numerical calculation shows that

$\mathcal{C}_{\tau}\approx 0.36.$ (50)

##  Appendix E Spin and valley current generation rate at Boltzmann statistics

At Boltzmann statistics we have

$f_{k^{\prime}}^{0}f_{p^{\prime}}^{0}f_{k^{\prime\prime}}^{0}=f_{{\absolutevalue{\bm{K}-\bm{\varkappa}}\over
2}}^{0}f_{{\absolutevalue{\bm{K}+\bm{\varkappa}}\over
2}}^{0}f_{{\absolutevalue{\bm{K}-\bm{Q}}\over 2}}^{0}=\quantity({N_{1}\over
gT})^{3}\exp[-{\hbar^{2}\over 4mT}\quantity({3\over
2}K^{2}+\varkappa^{2}+{1\over 2}Q^{2}-\bm{K}\cdot\bm{Q})],$ (51)

and obtain from Eq. (37):

$\left<\dot{V}_{y}\right>=v_{dr,x}{16m\over
T\hbar^{3}}V_{0}^{3}\xi\quantity({N_{1}\over
gT})^{3}\sum_{\bm{K},\bm{\varkappa},\bm{Q}}\exp[-{\hbar^{2}\over
4mT}\quantity({3\over 2}K^{2}+\varkappa^{2}+{1\over
2}Q^{2}-KQ\cos\theta)]{\sin\varphi\over\absolutevalue{\sin\varphi}}\quantity[3K\sin(\theta-\varphi)-Q\sin\varphi]\varkappa.$
(52)

Here $\theta$ is an angle between $\bm{K}$ and $\bm{Q}$, and $\varphi$ is the
angle between $\bm{Q}$ and $\bm{\varkappa}$.

Averaging over $\theta$ yields:

$\left<\exp(a\cos\theta)\right>_{\theta}=I_{0}(a),\qquad\left<\exp(a\cos\theta)\sin(\theta-\varphi)\right>_{\theta}=-\sin\varphi
I_{1}(a),$ (53)

and we get

$\left<\dot{V}_{y}\right>=-v_{dr,x}{{16}m\over
T\hbar^{3}}V_{0}^{3}\xi\quantity({N_{1}\over
gT})^{3}\sum_{\bm{K},\bm{\varkappa},\bm{Q}}\exp[-{\hbar^{2}\over
4mT}\quantity({3\over 2}K^{2}+\varkappa^{2}+{1\over
2}Q^{2})]{\sin^{2}\varphi\over\absolutevalue{\sin\varphi}}\quantity[3KI_{1}\quantity({\hbar^{2}KQ\over
4mT})+QI_{0}\quantity({\hbar^{2}KQ\over 4mT})]\varkappa.$ (54)

Averaging over $\varphi$ and summation over $\bm{\varkappa}$ yield:

$\left<{\sin^{2}\varphi\over\absolutevalue{\sin\varphi}}\right>_{\varphi}={2\over\pi},\qquad\sum_{\bm{\varkappa}}\varkappa\exp(-{\hbar^{2}\varkappa^{2}\over
4mT})={1\over\sqrt{\pi}}\quantity({mT\over\pi\hbar^{2}})^{3/2},$ (55)

therefore we get

$\left<\dot{V}_{y}\right>=-v_{dr,x}{{32}m\over
T\hbar^{3}\sqrt{\pi}}V_{0}^{3}\xi\quantity({N_{1}\over
gT})^{3}\quantity({mT\over\pi\hbar^{2}})^{3/2}\sum_{\bm{K},\bm{Q}}\exp[-{\hbar^{2}\over
4mT}\quantity({3\over 2}K^{2}+{1\over
2}Q^{2})]\quantity[3KI_{1}\quantity({\hbar^{2}KQ\over
4mT})+QI_{0}\quantity({\hbar^{2}KQ\over 4mT})].$ (56)

Calculations show:

$\sum_{\bm{K},\bm{Q}}\exp[-{\hbar^{2}\over 4mT}\quantity({3\over
2}K^{2}+{1\over 2}Q^{2})]3KI_{1}\quantity({\hbar^{2}KQ\over
4mT})=\sum_{\bm{K},\bm{Q}}\exp[-{\hbar^{2}\over 4mT}\quantity({3\over
2}K^{2}+{1\over 2}Q^{2})]QI_{0}\quantity({\hbar^{2}KQ\over 4mT})\\\
=\quantity({mT\over\hbar^{2}})^{5/2}{2\sqrt{3}\over\pi^{3/2}}.$ (57)

Therefore we finally get for the generation rate:

$\left<\dot{V}_{y}\right>=-N_{1}v_{dr,x}{256\sqrt{3}\over\pi}gV_{0}\xi
k_{T}^{2}{E_{\rm F}\over T}\times{1\over\tau_{\rm sc}}.$ (58)

##  Appendix F Calculation of the spin current at Bose statistics

At $f^{0}_{k}=T/(\varepsilon_{k}+\absolutevalue{\mu})\gg 1$ we obtain from Eq.
(37):

$\left<\dot{V}_{y}\right>=-v_{dr,x}2^{20}\pi g^{4}V_{0}^{3}\xi
k_{T}^{2}{T^{5}\over\hbar\absolutevalue{\mu}^{3}}\mathcal{C}_{g},$ (59)

where

$\mathcal{C}_{g}=\int\limits_{0}^{\infty}\differential
X_{K}\int\limits_{0}^{\infty}\differential
X_{Q}\int\limits_{0}^{\infty}\differential
X_{\varkappa}\Biggl{<}{\absolutevalue{\sin\varphi}\sqrt{X_{\varkappa}}\over\quantity[\quantity(X_{K}+X_{\varkappa}+1)^{2}-4X_{K}X_{\varkappa}\cos^{2}(\theta-\varphi))^{2}]^{2}\quantity(X_{K}+X_{Q}-2\sqrt{X_{K}X_{Q}}\cos\theta+1)^{2}}\\\
\times\quantity[{2\quantity(X_{K}+X_{\varkappa}+1)\sqrt{X_{K}}\cos\theta\over\quantity(X_{K}+X_{\varkappa}+1)^{2}-4X_{K}X_{\varkappa}\cos^{2}(\theta-\varphi)}+{\sqrt{X_{K}}\cos\theta+\sqrt{X_{Q}}\over
X_{K}+X_{Q}-2\sqrt{X_{K}X_{Q}}\cos\theta+1}]\Biggr{>}_{\varphi,\theta}.$ (60)

Here we introduced dimensionless energies
$X_{q}=\varepsilon_{q}/\absolutevalue{\mu}$ ($q=K,Q,\varkappa$), and, as
before, the upper limits for integration were extended to $+\infty$ due to the
convergence of integrals.

Numerical calculation yields $\mathcal{C}_{g}\approx 0.3$.

Using Eq. (49) we can rewrite the velocity generation rate in the form

$\left<\dot{V}_{y}\right>=-v_{dr,x}gT\xi
k_{T}^{2}gV_{0}{T^{2}\over\absolutevalue{\mu}^{2}}\times{\mathcal{C}\over\tau_{\rm
sc}},$ (61)

where
$\mathcal{C}={2^{9}\pi\mathcal{C}_{g}/\mathcal{C}_{\tau}}\approx{1340.4}$.

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