# Hydrodynamic theory of vorticity-induced spin transport

Gen Tatara  RIKEN Center for Emergent Matter Science (CEMS) and RIKEN Cluster
for Pioneering Research (CPR), 2-1 Hirosawa, Wako, Saitama, 351-0198 Japan

###### Abstract

Electron spin transport in a disordered metal is theoretically studied from
the hydrodynamic viewpoint focusing on the role of electron vorticity. The
spin-resolved momentum flux density of electrons is calculated taking account
of the spin-orbit interaction and uniform magnetization, and the expression
for the spin motive force is obtained as the linear response to a driving
electric field. It is shown that the spin-resolved momentum flux density and
motive force are characterized by troidal moments expressed as vector products
of the applied external electric field and the spin polarization and/or
magnetization. The spin-vorticity and magnetization-vorticity couplings
studied recently are shown to arise from the toridal moments contribution to
the momentum flux density. Spin motive force turns out to have a
nonconservative contribution besides the conventional conservative one due to
the spin-vorticity coupling. Spin accumulation induced by an electric field is
calculated to demonstrate the direct relation between vorticity and induced
spin, and the spin Hall effect is interpreted as due to the spin-vorticity
coupling. The spin-vorticity coupling is shown to give rise to a vorticity-
induced torque and a spin relaxation. The vorticity-induced torque is a linear
effect of the spin-orbit interaction and is expected to be larger than the
second-order torques such as nonadiabatic ($\beta$) current-induced torque due
to magnetization structure. The intrinsic inverse spin Hall effect is argued
to correspond to the antisymmetric components of the momentum flux density in
the hydrodynamic context.

##  I Introduction

Electron transport in magnetic metals has been studied in various systems.
Technologically most successful effect would be the giant magnetoresistance
(GMR) effect discovered in magnetic layers [1]. GMR effects in multilayers
were theoretically described based on a two-current electron model derived
from the Boltzmann equation for the conduction electrons with spin up and down
[2]. Effects of magnetic structures have been argued in the context of
magnetoresistance and spin-transfer torque [3, 4, 5]. In the previous works on
the spin transport, effects of nonuniform magnetization were focused on, while
current density was treated as uniform. In reality, however, the current is
generally inhomogeneous at edges, resulting in local vorticity of electron
flow. Vorticity of current density $\bm{j}$, $\nabla\times\bm{j}$, carries
angular momentum and may couple to the magnetic field and spin, affecting
local spin transport. In fact, electron’s vorticity was shown theoretically to
couple to electron spin via an quantum relativistic effect [6], and various
phenomena induced by the spin-vorticity coupling have been discussed recently
[7, 8, 9].

Vorticity arises generally near surfaces and interfaces where longitudinal
flow is suppressed, and vorticity-induced effects are expected to dominate
various transport properties in thin films and mesoscopic systems. For
discussing vorticity effects, hydrodynamic equations are useful to identify
the forces due to inhomogeneity of the flow. Electron transport in metals with
disorder is governed by relaxation force instead of viscosity force in
conventional fluids. Coarse grained behaviors of such disordered metals are
described in terms of an ohmic fluid [10]. Hydrodynamic description of ohmic
electron fluid was employed to describe angular momentum generation in chiral
electron systems and anomalous Hall effect [11, 12]. Hydrodynamic equations
are conventionally represented in terms of current density $\bm{j}$. In ohmic
fluids, electric current density is locally related to a driving electric
field ${\bm{E}}$ as $\bm{j}=\sigma_{\rm e}{\bm{E}}$, where $\sigma_{\rm e}$ is
a conductivity tensor. Using the local relation, the hydrodynamic equations
can thus be represented in terms of the driving field. In the driving-field
representation, hydrodynamic coefficients are directly related to microscopic
response functions to the applied field, and are systematically calculable by
use of a microscopic linear response theory [13, 11].

In this paper, we study spin transport effects from the hydrodynamic
viewpoints focusing on the effects of vorticity. We calculate hydrodynamic
coefficients as a linear response to the applied field taking account of its
inhomogenuity to the lowest order of spatial derivative.

###  I.1 Spin-vorticity coupling

As was argued in Refs. [6, 7], spin-vorticity coupling is an interaction
proportional to ${{\bm{s}}}\cdot(\nabla\times\bm{j})$, the scalar product of
spin density ${{\bm{s}}}$ and vorticity. The coupling induces a force on
electron spin proportional to the gradient of vorticity and generates spin
current, resulting in a spin accumulation at edges where spin current
terminates. The vorticity-induced spin current can be detected by use of the
inverse spin Hall effect [7].

It was demonstrated recently [14] that this spin-vorticity coupling represents
the spin Hall effect [15, 16]. In fact, the spin Hall effect was shown to be
represented by a single equation

$\displaystyle{{\bm{s}}}=\lambda_{\rm sh}{\bm{\omega}}$ (1)

where ${\bm{\omega}}\equiv\nabla\times{\bm{E}}$ is a vorticity of the applied
field and $\lambda_{\rm sh}$ is a constant arising from the spin-orbit
interaction [14]. This relation is equivalent to the spin-vorticity coupling
at the lowest order in the spin-orbit interaction due to the local relation
between $\bm{j}$ and ${\bm{E}}$. In the disordered case, spin density
propagates by diffusion and the formula (1) is modified to include a diffusion
propagator (See Eq. (65)). The expression is then equivalent to the
conventional representation of spin accumulation in terms of spin diffusion
equation.

The spin-vorticity coupling potential induces a force proportional to its
gradient, $\nabla({{\bm{s}}}\cdot{\bm{\omega}})$. From the symmetry, another
form of the force $({{\bm{s}}}\cdot\nabla){\bm{\omega}}\equiv{\bm{f}}_{\rm
nc}$ is allowed. Noting $\nabla\cdot{\bm{\omega}}=0$, ${\bm{f}}_{\rm
nc}=\nabla\times({\bm{\omega}}\times{{\bm{s}}})$ is a nonconservative force,
which is allowed if there are viscosity or friction. We shall demonstrate that
such a nonconservative force indeed exists in the present electron spin fluid.
In the context of magnetization-vorticity coupling, the same form of a
nonconservative force with ${{\bm{s}}}$ replaced by magnetization was
identified in Ref. [12].

The spin-vorticity coupling indicates that there are current-induced torques
arising from vorticity of current in ferromagnets. The torque arises from the
inhomogeneity of current density instead of inhomogeneity of magnetization for
conventional current-induced torques. The effect would be localized near
surfaces and interfaces and is expected to be enhanced by introducing
artificial roughness or inhomogeneous structures. We shall demonstrate that
the vorticity-induced torque arises theoretically at the linear order of the
spin-orbit interaction, while conventional current-induced nonadiabatic torque
($\beta$ torque) is the second-order effect [5]. The vorticity-induced torque
is thus expected to have larger effects than conventional spin-orbit driven
torques. The vorticity-induced torque has a vanishing bulk component, while it
acts in thin films as an alternating torque, resulting in a spin relaxation.

##  II Anomalous Hall fluid

We first argue electron transport in a ferromagnetic metal with a uniform
magnetization, i.e., the anomalous Hall system, studied from the hydrodynamic
viewpoint in Ref. [12]. In that paper, the electron momentum flux density was
calculated diagrammatically in the presence of a uniform magnetization at the
linear response to the applied electric field treated as non uniform. It was
shown that there is a contribution to the viscosity coefficient arising from a
coupling between the magnetization and vorticity of electron velocity, i.e.,
the magnetization-vorticity coupling. In terms of applied field ${\bm{E}}$,
the coupling potential is $-\tilde{\zeta}_{M}^{\rm
f}{\bm{M}}\cdot{\bm{\omega}}$, where ${\bm{M}}$ is the magnetization vector
and ${\bm{\omega}}\equiv\nabla\times{\bm{E}}$ is a vorticity of the applied
field and $\tilde{\zeta}_{M}^{\rm f}$ is a coefficient. The force density due
to the coupling is

$\displaystyle{\bm{f}}_{M\omega}=\tilde{\zeta}_{M}^{\rm
f}\nabla({\bm{M}}\cdot{\bm{\omega}})$ (2)

Such a coupling has been argued from the phenomenological ground [17] and
derived microscopically [12]. It is regarded as a limit of spin-vorticity
coupling where spin has a finite expectation value. Electron fluid in a
uniform external magnetic field was studied in Ref. [18].

Phenomenologically, the magnetization-vorticity coupling is understood simply
taking account of the anomalous Hall effect in the conventional nonmagnetic
fluid. Fluid dynamics is described by the momentum flux density, $\pi_{ij}$,
which is an expectation value
$\left\langle{\hat{p}_{i}\hat{v}_{j}}\right\rangle$ of momentum and velocity
operators, $\hat{p}$ and $\hat{v}$, respectively ($i,j$ are spatial
directions). Its divergence is the force density for the fluid,
$f_{i}=-\nabla_{j}\pi_{ij}$. In nonmagnetic fluids with high symmetry,
$\pi_{ij}$ is written in terms of current density $\bm{j}$ as
$\pi_{ij}={\zeta}_{0}\delta_{ij}(\nabla\cdot\bm{j})+{\eta}_{0}[\nabla_{i}j_{j}+\nabla_{j}j_{i}]$,
where ${\zeta}_{0}$ and ${\eta}_{0}$ are viscosity constants [19]. The
momentum flux density may have antisymmetric component, $\pi_{ij}^{\rm a}$,
which is written generally as $\pi_{ij}^{\rm a}=\epsilon_{ijk}a_{k}$, where
${\bm{a}}$ is a vector invariant by the parity inversion
(${\bm{r}}\rightarrow-{\bm{r}}$). Without broken symmetry, vorticity
$\nabla\times\bm{j}$ is allowed as the vector ${\bm{a}}$, resulting in an
antisymmetric component $\pi_{ij}^{\rm
a}=\frac{\xi_{0}}{2}(\nabla_{i}j_{j}-\nabla_{j}j_{i})$ with a constant
$\xi_{0}$ [20]. The electron current density is proportional to the applied
field as $\bm{j}=\sigma_{\rm e}{\bm{E}}$, where $\sigma_{\rm e}$ is the
diagonal conductivity in the case of high symmetry. The momentum flux density
in this case is therefore

$\displaystyle\pi_{ij}=\zeta\delta_{ij}(\nabla\cdot{\bm{E}})+{\eta}[\nabla_{i}E_{j}+\nabla_{j}E_{i}]+\epsilon_{ijk}a_{k}$
(3)

with $\zeta\equiv{\zeta}_{0}\sigma_{\rm e}$ and
$\eta\equiv{\eta}_{0}\sigma_{\rm e}$ and ${\bm{a}}={\xi}{\bm{\omega}}$
($\xi\equiv\xi_{0}\sigma_{\rm e}$).

In the presence of magnetization, the anomalous Hall effect tends to distort
electron motion towards a perpendicular direction, i.e., $\bm{j}$ is modified
to be $\bm{j}+\alpha_{M}(\hat{{\bm{M}}}\times\bm{j})$, where $\alpha_{M}$ is a
constant and $\hat{{\bm{M}}}\equiv{\bm{M}}/|M|$. This corresponds to emergence
of a perpendicular driving field $\alpha_{M}(\hat{{\bm{M}}}\times{\bm{E}})$.
The anomalous Hall effect thus induces new components of the momentum flux
density, written in terms of a vector called a troidal moment

$\displaystyle\hat{{\bm{M}}}\times{\bm{E}}\equiv{\bm{T}}_{\hat{{\bm{M}}}},$
(4)

as

$\displaystyle\pi^{M}_{ij}=\zeta_{M}\delta_{ij}(\nabla\cdot{\bm{T}}_{\hat{{\bm{M}}}})+{\eta_{M}}[\nabla_{i}T_{{\hat{{\bm{M}}}},j}+\nabla_{j}T_{{\hat{{\bm{M}}}},i}]+\epsilon_{ijk}a_{M,k}.$
(5)

A possible vector ${\bm{a}}_{M}$ in the anomalous Hall fluid is

$\displaystyle{\bm{a}}_{M}$
$\displaystyle=\xi_{M}^{\omega}(\hat{{\bm{M}}}\times{\bm{\omega}})+\xi_{M}^{{\rm
v}}\hat{{\bm{M}}}(\nabla\cdot{\bm{E}})+\xi_{M}^{T}(\nabla\times{\bm{T}}_{\hat{{\bm{M}}}})$
(6)

where $\xi_{M}^{\omega}$, $\xi_{M}^{{\rm v}}$ and $\xi_{M}^{T}$ are
coefficients, representing the anomalous Hall component of vorticity, volume
change and troidal moment, respectively. In the microscopic calculation in the
ohmic regime in Ref. [12], $\xi_{M}^{\omega}$ and $\xi_{M}^{{\rm v}}$ arise
from the side-jump process, while $\xi_{M}^{T}=0$. Using
$\nabla\times{\bm{T}}_{\hat{{\bm{M}}}}=\hat{{\bm{M}}}(\nabla\cdot{\bm{E}})-(\hat{{\bm{M}}}\cdot\nabla){\bm{E}}$
for the present case of uniform $\hat{{\bm{M}}}$, different representations of
${\bm{a}}_{M}$ are possible. In metals, $\nabla\cdot{\bm{E}}=0$, and
$\nabla\times{\bm{T}}_{\hat{{\bm{M}}}}=-(\hat{{\bm{M}}}\cdot\nabla){\bm{E}}$,
resulting in

$\displaystyle{\bm{a}}_{M}$
$\displaystyle=\xi_{M}^{\omega}\nabla(\hat{{\bm{M}}}\cdot{\bm{E}})+\tilde{\xi}_{M}^{T}(\nabla\times{\bm{T}}_{\hat{{\bm{M}}}})$
(7)

where $\tilde{\xi}_{M}^{T}\equiv{\xi}_{M}^{T}+{\xi}_{M}^{\omega}$. The first
term $\xi_{M}^{\omega}$ has no physical effect because its force density
vanishes ($\nabla\times\nabla=0$).

The force density calculated from Eq. (5) is

$\displaystyle f_{M,i}$ $\displaystyle\equiv-\nabla_{j}\pi^{M}_{ij}$
$\displaystyle=-\zeta_{M}^{f}\nabla_{i}(\nabla\cdot{\bm{T}}_{\hat{{\bm{M}}}})-\eta_{M}^{f}\nabla^{2}T_{{\hat{{\bm{M}}}},i}+\xi_{M}^{\omega}(\hat{{\bm{M}}}\cdot\nabla){{\bm{\omega}}}_{i}+\xi_{M}^{\rm
v}(\hat{{\bm{M}}}\times\nabla)_{i}(\nabla\cdot{\bm{E}})$ (8)

where $\zeta_{M}^{f}\equiv\zeta_{M}+\eta_{M}+\xi_{M}^{T}$,
$\eta_{M}^{f}\equiv{\eta_{M}}-\xi_{M}^{T}$. The results were presented in Ref.
[12] without using ${\bm{T}}_{\hat{{\bm{M}}}}$. In the case of uniform
magnetization,

$\displaystyle\nabla\cdot{\bm{T}}_{\hat{{\bm{M}}}}=-(\hat{{\bm{M}}}\cdot{\bm{\omega}})$
(9)

and thus the first term of Eq. (8)
($\nabla_{i}(\nabla\cdot{\bm{T}}_{\hat{{\bm{M}}}}))$ represents the
magnetization-vorticity coupling (Eq. (2)). In metals, using incompressibility
$\nabla\cdot{\bm{E}}=0$,
$\nabla^{2}{\bm{T}}_{{\hat{{\bm{M}}}}}=-\nabla(\hat{{\bm{M}}}\cdot{\bm{\omega}})+(\hat{{\bm{M}}}\cdot\nabla){\bm{\omega}}$
and the total force density is written in terms of vorticity as

$\displaystyle{\bm{f}}_{M}$
$\displaystyle=\tilde{\zeta}_{M}^{f}\nabla(\hat{{\bm{M}}}\cdot{\bm{\omega}})+\tilde{\xi}_{M}^{\omega}(\hat{{\bm{M}}}\cdot\nabla){{\bm{\omega}}}$
(10)

where $\tilde{\zeta}_{M}^{f}\equiv\zeta_{M}^{f}+\eta_{M}^{f}$ and
$\tilde{\xi}_{M}^{\omega}\equiv\xi_{M}^{\omega}-\eta_{M}^{f}$. The first term
on the right-hand side of Eq. (10) is a conservative force arising from the
magnetization-vorticity coupling, while the second term is a nonconservative
force,
$\tilde{\xi}_{M}^{\omega}[\nabla\times({{\bm{\omega}}}\times\hat{{\bm{M}}})]$.
This result, Eq. (10), was presented by a microscopic calculation in Ref.
[12]. The two contributions, conservative and nonconservative, to the
vorticity-induced motive force leads to an anisotropic motive force with
respect to the direction of $\hat{{\bm{M}}}$ (see Sec. III.1).

##  III Spin-resolved momentum flux density and spin motive force

In this section, we consider the spin transport, i.e., the spin-resolved
hydrodynamic equation for electrons. Including a spin direction of electron,
the hydrodynamic coefficients have more possibilities [17]. The momentum flux
density is defined spin-dependent as
$\pi_{ij}^{s,\alpha}\equiv\left\langle{\hat{p}_{i}\hat{v}_{j}\sigma_{\alpha}}\right\rangle$,
where $\sigma_{\alpha}$ is the Pauli matrix ($\alpha$ denotes spin direction).
The time-derivative of the spin-resolved momentum density

$\displaystyle
p_{i}^{\alpha}\equiv\left\langle{\hat{p}_{i}\sigma_{\alpha}}\right\rangle$
(11)

is represented in terms of $\pi_{ij}^{s,\alpha}$ as

$\displaystyle\dot{p}_{i}^{\alpha}=-\nabla_{j}\pi_{ij}^{s,\alpha}.$ (12)

We introduce a unit vector $\hat{{{\bm{s}}}}$ to represent the spin direction
as

$\displaystyle\bm{\pi}^{s}_{ij}\equiv(\pi_{ij}^{s,x},\pi_{ij}^{s,y},\pi_{ij}^{s,z})\equiv\hat{{{\bm{s}}}}\overline{\pi}_{ij}^{s}$
(13)

where
$\overline{\pi}_{ij}^{s}\equiv((\pi_{ij}^{s,x})^{2}+(\pi_{ij}^{s,y})^{2}+(\pi_{ij}^{s,z})^{2})^{1/2}$.

Effect of spin (called the internal angular momentum in Ref. [17]) on the
electron fluid was theoretically discussed on a symmetry basis in Ref. [17].
The possible contributions of the total (spin-neutral) momentum flux density
when spin is present were argued.

Here we represent the momentum flux density as spin-polarized introducing a
spin direction $\hat{{{\bm{s}}}}$. The spatial derivative of the spin-resolved
momentum flux density represents a force acting on spin polarization, i.e.,
the spin motive force. The spin motive force is a convenient quantity to
discuss spin current generation.

One needs, however, to understand that spin-resolved quantities
$p_{i}^{\alpha}$ and $\bm{\pi}^{s}_{ij}$ are not directly measurable and are
not physical observables. In other words, the flux density $\bm{\pi}^{s}_{ij}$
is not conserved and cannot be defined uniquely, as is in the case of spin
current. Explicitly, ${\pi}^{s,\alpha}_{ij}$ is essentially an expectation
value of $\left\langle{\hat{p}_{i}\hat{v}_{j}\sigma_{\alpha}}\right\rangle$,
i.e., the velocity ($\hat{v}$) of momentum and spin (See Eqs. (29)(30)), but
its expression depends on the ordering of the three operators $\hat{p}_{i}$,
$\hat{v}_{j}$ and $\sigma_{\alpha}$ and is thus not unique. Like in the case
of spin Hall effect, predictions for experiments need to be provided in terms
of physical spin accumulation or charge current by solving the hydrodynamic
equation Eq. (12). Here in this paper, we discuss spin-resolved momentum flux
density for understanding roles of vorticity on the spin transport and leave
explicit calculations of hydrodynamic equations as a future work. Spin density
calculation is carried out independently from the momentum flux density
analysis in Sec. IV

###  III.1 Phenomenological argument

Figure 1: Parameterization of spin vectors,
${{\bm{s}}}_{\perp}\equiv{{{\bm{s}}}}-\hat{{\bm{M}}}(\hat{{\bm{M}}}\cdot{{{\bm{s}}}})$
and $\tilde{{{\bm{s}}}}\equiv{{\bm{s}}}\times\hat{{\bm{M}}}$.

Let us first proceed phenomenologically. We first focus on the symmetric (s)
component, $\bm{\pi}_{ij}^{s{\rm(s)}}$. Based on the expression for the
anomalous Hall system [12] (Eq. (5)), we expect an adiabatic component having
a spin polarization $\hat{{{\bm{s}}}}$ along the magnetization as
($\eta_{s}^{\parallel}$ and $\zeta_{s}^{\parallel}$ are coefficients)

$\displaystyle\overline{\pi}_{ij}^{s{\rm(s)}{\parallel}}$
$\displaystyle=(\hat{{{\bm{s}}}}\cdot\hat{{\bm{M}}})\left(\zeta_{s}^{\parallel}(\hat{{\bm{M}}}\cdot{\bm{\omega}})\delta_{ij}+\eta_{s}^{\parallel}(\nabla_{i}T_{{\hat{{\bm{M}}}},j}+\nabla_{j}T_{{\hat{{\bm{M}}}},i})\right)$
(14)

As for the nonadiabatic contribution due to perpendicular spin polarization,
we introduce

$\displaystyle{\hat{{{\bm{s}}}}}_{\perp}\equiv{\hat{{{\bm{s}}}}}-\hat{{\bm{M}}}(\hat{{\bm{M}}}\cdot{\hat{{{\bm{s}}}}})$
(15)

whose contribution is

$\displaystyle\overline{\pi}_{ij}^{s{\rm(s)}{\perp}}$
$\displaystyle=\zeta_{s}^{\perp}(\hat{{{\bm{s}}}}_{\perp}\cdot{\bm{\omega}})\delta_{ij}+\eta_{s}^{\perp}(\nabla_{i}T_{\hat{{{\bm{s}}}}_{\perp},j}+\nabla_{j}T_{\hat{{{\bm{s}}}}_{\perp},i})$
(16)

where

$\displaystyle{\bm{T}}_{{\hat{{{\bm{s}}}}_{\perp}}}\equiv\hat{{{\bm{s}}}}_{\perp}\times{\bm{E}}$
(17)

is a troidal moment due to the perpendicular spin polarization. Another spin
polarization perpendicular to $\hat{{\bm{M}}}$ is

$\displaystyle\tilde{{{{\bm{s}}}}}\equiv\hat{{{\bm{s}}}}\times\hat{{\bm{M}}}.$
(18)

It is a spin polarization induced by an anomalous Hall effect in the direction
perpendicular to ${\bm{M}}$, and its contribution would be

$\displaystyle\overline{\pi}_{ij}^{s{\rm(s)}{\perp^{\prime}}}$
$\displaystyle=\zeta_{s}^{\perp^{\prime}}(\tilde{{{{\bm{s}}}}}\cdot{\bm{\omega}})\delta_{ij}+\eta_{s}^{\perp^{\prime}}(\nabla_{i}T_{\tilde{{{{\bm{s}}}}},j}+\nabla_{j}T_{\tilde{{{{\bm{s}}}}},i})$
(19)

where

$\displaystyle{\bm{T}}_{\tilde{{{{\bm{s}}}}}}\equiv\tilde{{{{\bm{s}}}}}\times{\bm{E}}$
(20)

and $\zeta_{s}^{\perp^{\prime}}$ and $\eta_{s}^{\perp^{\prime}}$ are
constants.

Antisymmetric contributions with spin direction $\alpha$,
$\pi_{ij}^{s,\alpha}\equiv\epsilon_{ijk}a_{k}^{s,\alpha}$, are similarly
argued. We consider the case of $\nabla\cdot{\bm{E}}=0$. As we saw in Eq. (7),
only the troidal moment remains in this case. Besides the adiabatic component
proportional to Eq. (7), we therefore have (with coefficients
$\xi_{s}^{T_{\hat{{{\bm{s}}}}}}$ and $\xi_{s}^{T_{\tilde{{{\bm{s}}}}}}$)

$\displaystyle a_{k}^{s,\alpha}$
$\displaystyle=\xi_{s}^{T_{\hat{{{\bm{s}}}}}}(\nabla\times{\bm{T}}_{\hat{{{\bm{s}}}}})+\xi_{s}^{T_{\tilde{{{\bm{s}}}}}}(\nabla\times{\bm{T}}_{\tilde{{{\bm{s}}}}})$
(21)

(Here coefficients are defined using $\hat{{{\bm{s}}}}$ instead of
$\hat{{{\bm{s}}}}_{\perp}$, i.e., neglecting the adiabatic component.)

Due to Eq. (12), a spin-resolved force (a spin motive force) acting on the
electron spin polarized along direction $\alpha$ is

$\displaystyle f_{s,i}^{\alpha}$
$\displaystyle\equiv-\nabla_{j}\pi_{ij}^{s,\alpha}\equiv\hat{{{\bm{s}}}}\overline{f}_{s,i}$
(22)

As was argued in Sec. II for the case of anomalous Hall fluid, the motive
force when $\nabla\cdot{\bm{E}}=0$ is written in terms of vorticity. The
adiabatic contribution parallel to $\hat{{\bm{M}}}$ is essentially the same as
the anomalous Hall case (Eq. (10)), i.e.,

$\displaystyle\overline{{\bm{f}}}_{s}^{{\parallel}}$
$\displaystyle=(\hat{{{\bm{s}}}}\cdot\hat{{\bm{M}}})[\tilde{\zeta}^{f,{\parallel}}_{s}\nabla(\hat{{\bm{M}}}\cdot{\bm{\omega}})+\tilde{\xi}_{s}^{\omega,{\parallel}}(\hat{{\bm{M}}}\cdot\nabla){{\bm{\omega}}}]$
(23)

with constants $\tilde{\zeta}^{f,{\parallel}}_{s}$ and
$\tilde{\xi}_{s}^{\omega,{\parallel}}$, while other contributions lead to
motive force depending on the spin polarization as

$\displaystyle\overline{{\bm{f}}}_{s}^{{\perp}}$
$\displaystyle=\tilde{\zeta}^{f,\perp}_{s}\nabla(\hat{{{\bm{s}}}}\cdot{\bm{\omega}})+\tilde{\xi}_{s}^{\omega,\perp}(\hat{{{\bm{s}}}}\cdot\nabla){{\bm{\omega}}}+\tilde{\zeta}^{f,\perp^{\prime}}_{s}\nabla(\tilde{{{\bm{s}}}}\cdot{\bm{\omega}})+\tilde{\xi}_{s}^{\omega,\perp^{\prime}}(\tilde{{{\bm{s}}}}\cdot\nabla){{\bm{\omega}}}$
(24)

The contributions represented by coefficients $\tilde{\zeta}^{f,\perp}_{s}$
and $\tilde{\zeta}^{f,\perp^{\prime}}_{s}$ are conservative forces due to
coupling potential to vorticity, while $\tilde{\xi}_{s}^{\omega,\perp}$ and
$\tilde{\xi}_{s}^{\omega,\perp^{\prime}}$ represent nonconservative forces.
Interestingly, conservative forces induce spin polarization
($\hat{{{\bm{s}}}}$ or $\tilde{{{\bm{s}}}}$) along ${\bm{\omega}}$ and force
along the gradient of ${\bm{\omega}}$, while the noncoservative ones induce
force and spin in the direction of ${\bm{\omega}}$ and the gradient of
${\bm{\omega}}$, respectively.

Figure 2:  A schematic picture showing spin motive forces of Eq. (24) in a
thin film in the $xy$-plane with applied current along the $x$-direction. The
contributions $\tilde{\zeta}^{f,\perp}_{s}$ and
$\tilde{\xi}_{s}^{\omega,\perp}$ connecting vorticity and $\hat{{{\bm{s}}}}$
are depicted. At the boundaries, the flow vanishes, meaning that total
effective field ${\bm{E}}$ including the applied field and friction force from
the boundary vanishes due to proportionality of $\bm{j}$ and ${\bm{E}}$. The
derivative $\partial_{z}E_{x}$ is therefore finite, namely, vorticity $\omega$
emerges near the boundary along the $y$ direction. The vorticity has opposite
signs on the upper and lower plane, resulting in $\partial_{z}\omega_{y}$. The
contribution of conventional conservative spin-vorticity coupling, the term
$\tilde{\zeta}^{f,\perp}_{s}$, thus induces a spin polarization (short arrows
with sphere) along the $y$ direction, and motive force along the $z$ axis
(blue arrows). The nonconservative contribution,
$\tilde{\xi}^{\omega,\perp}_{s}$, in contrast induces a spin polarization
along the $z$ direction, and motive force along the $y$ axis (magenta arrows).

Let us consider electron fluid in a thin film as in Fig. 2 and see the
contributions of $\tilde{\zeta}^{f,\perp}_{s}$ and
$\tilde{\xi}_{s}^{\omega,\perp}$ connecting vorticity and $\hat{{{\bm{s}}}}$.
The driving field ${\bm{E}}$ includes all the forces acting on the electrons,
such as friction force from the boundary, besides the applied external
electric field. The field ${\bm{E}}$ is therefore locally proportional to the
local fluid current $\bm{j}$ neglecting the higher order contributions of the
spin-orbit interaction, and is suppressed near the boundary as in Fig. 2 [12].
This suppression of fluid velocity in the $z$-direction leads to a vorticity
along the $y$-direction, $\omega_{y}=\partial_{z}E_{x}$. The vorticity changes
sign on the upper and lower boundaries and changes along the $z$-direction.
For electron spin along $+y$ direction, the conservative spin motive force is
along the $z$ axis as
$f^{\zeta}_{z}=\tilde{\zeta}^{f,\perp}_{s}\nabla_{z}{\bm{\omega}}_{y}$, i.e.,
is in the $-z$ direction (if $\tilde{\zeta}^{f,\perp}_{s}>0$), while it is
$+z$ direction for the electron spin pointing $-y$ direction. Thus spin
current along the $z$ axis is induced by the spin-vorticity coupling as argued
in previous works [7]. The nonconservative motive force described by the term
$\tilde{\xi}_{s}^{\omega,\perp}$, which is mainly from the asymmetric
viscosity, acts in the $-y$ ($+y$) direction for spin polarization along $+z$
($-z$) direction, inducing a spin current in the $y$ direction (Fig. 2). The
contribution $\tilde{\zeta}^{f,\perp^{\prime}}_{s}$ and
$\tilde{\xi}_{s}^{\omega,\perp^{\prime}}$ induce orthogonal spin polarizations
to $\tilde{\zeta}^{f,\perp}_{s}$ and $\tilde{\xi}_{s}^{\omega,\perp}$,
respectively. The adiabatic contributions and anomalous Hall contributions are
discussed by replasing $\hat{{{\bm{s}}}}$ by $\hat{{\bm{M}}}$ in the above
argument.

In the next section, we carry out microscopic calculation on a simplified
model to confirm the above argument.

###  III.2 Microscopic derivation

We first derive the definition of the spin-resolved momentum flux density by
deriving a hydrodynamic equation for the time-derivative of momentum density,
$\dot{{\bm{p}}}$. In the present case with spin, spin-resolved momentum
density, defined as

$\displaystyle
p_{i}^{\alpha}\equiv\left\langle{c^{\dagger}\hat{p}_{i}\sigma_{\alpha}c}\right\rangle,$
(25)

is considered, where $\hat{p}_{i}$ and $\sigma_{\alpha}$ denote the operators
for momentum and spin ($i$ and $\alpha$ represents direction), respectively,
$\left\langle{\ }\right\rangle$ denotes quantum average and $c$ and
$c^{\dagger}$ are electron field operators. Its time-derivative is derived by
use of the Heisenberg equation of motion,
$\dot{p}_{i}^{\alpha}=i\left\langle{[H,c^{\dagger}\hat{p}_{i}\sigma_{\alpha}c]}\right\rangle$,
where $[A,B]\equiv AB-BA$ is a commutator and $H$ is the total Hamiltonian, as
was done in the spinless cases [11, 12]. The Hamiltonian we consider is

$\displaystyle H$
$\displaystyle=\int\\!{d^{3}r}c^{\dagger}\left(\frac{-\nabla^{2}}{2m}-({\bm{M}}\cdot{\bm{\sigma}})\right)c+H_{\rm
so}+H_{\rm i}$ (26)

where ${\bm{M}}$ is a vector representing the magnetization including the
exchange coupling constant. The spin-orbit interaction by impurities is
represented by

$\displaystyle H_{\rm so}$ $\displaystyle=\lambda\int
d^{3}rc^{\dagger}({\bm{r}})[(\nabla
v({\bm{r}})\times\hat{{\bm{p}}})\cdot{\bm{\sigma}}]c({\bm{r}})$
$\displaystyle=i\lambda\sum_{{\bm{k}}{\bm{k}}^{\prime}}v_{{\bm{k}}^{\prime}-{\bm{k}}}({\bm{k}}^{\prime}\times{\bm{k}})\cdot
c^{\dagger}_{{\bm{k}}^{\prime}}{\bm{\sigma}}c_{{\bm{k}}}$ (27)

where $\lambda$ is a coupling constant, $v({\bm{r}})=v_{\rm
i}\sum_{{\bm{R}}_{n}}\delta({\bm{r}}-{\bm{R}}_{n})$ is an impurity potential,
where $v_{\rm i}$ and ${\bm{R}}_{n}$ are the strength of the impurity
potential and the position of $n$-th impurity, respectively, and the impurity
scattering potential is $H_{\rm i}=\int
d^{3}rv({\bm{r}})c^{\dagger}({\bm{r}})c({\bm{r}})$. The impurity scattering
induces an electron lifetime of elastic scattering, $\tau$, given by
$\tau^{-1}=2\pi\nu n_{\rm i}{v_{\rm i}}^{2}$, where $n_{\rm i}$ is the
impurity concentration and $\nu$ is the density of states of electron.

A driving electric field for electron flow, ${\bm{E}}$, is included using a
vector potential ${\bm{A}}$ satisfying ${\bm{E}}=-\dot{{\bm{A}}}$. As is known
generally for transport theories, dominant nonequilibrium contributions for
the case of time-independent ${\bm{E}}$ are those containing both retarded and
advanced Green’s functions [11], and we shall focus on these contributions.
Spatial inhomogeneity is taken into account by expanding response functions
with respect to the wave vector ${\bm{q}}$ of ${\bm{E}}$ to the linear order.

In the present model, the driving field is not only the applied external
field, but is an effective one that includes other extrinsic forces such as
those introduced by boundaries. At the boundary, fluid velocity vanishes and
this fact is imposed usually as a boundary condition in solving fluid
dynamics. In the present microscopic modeling of the ohmic fluid, such
boundary effects are taken account by assuming that the total driving field
vanishes at the boundary as was done in Refs. [11, 12]. In fact, vanishing
fluid velocity indicates that the force due to the applied field and the
friction from the boundary cancel each other. In practice, our total effective
field is related to the actual current density $\bm{j}$ via a local relation
${\bm{E}}=(\sigma_{\rm e})^{-1}\bm{j}$, where $\sigma_{\rm e}$ is a
conductivity tensor. In the present analysis focusing on the lowest order of
the spin-orbit interaction, $\sigma_{\rm e}$ can be treated as diagonal. It
would be an interesting future work to solve numerically the whole
hydrodynamic equation taking account of off-diagonal conductivity.

The hydrodynamic equation for spin-resolved momentum density is (see Sec. A
for derivation)

$\displaystyle\dot{p}_{i}^{\alpha}$
$\displaystyle=-\nabla_{j}\pi_{ij}^{s,\alpha}+f_{s,i}^{\alpha}$ (28)

where $\pi^{s,\alpha}_{ij}\equiv\pi^{0\alpha}_{ij}+\delta\pi^{\alpha}_{ij}$
with

$\displaystyle\pi^{0\alpha}_{ij}({\bm{r}},t)$ $\displaystyle\equiv-i{\rm
tr}\left[\hat{p}_{i}\\{\hat{v}_{j},\sigma_{\alpha}\\}G^{<}({\bm{r}},t,{\bm{r}},t)\right]$
(29)

is a conventional contribution in the form of momentum and velocity, while

$\displaystyle\delta\pi^{\alpha}_{ij}({\bm{r}},t)$
$\displaystyle\equiv-i\frac{\lambda}{2}{\rm
tr}\left[(\nabla_{i}\nabla_{k}V)(\delta_{j\alpha}\delta_{k\beta}-\delta_{j\beta}\delta_{k\alpha})\sigma_{\beta}G^{<}({\bm{r}},t,{\bm{r}},t)\right]$
(30)

is an anomalous contribution from the spin-orbit potential. (tr is a trace
over spin and $\\{A,B\\}\equiv AB+BA$.) Here $G^{<}$ is the full lesser
Green’s function including the external field and

$\displaystyle\hat{v}_{j}$
$\displaystyle=\frac{\hat{p}_{j}}{m}+\delta\hat{v}_{j}$ (31)

is the total velocity operator with the anomalous velocity

$\displaystyle\delta\hat{v}_{j}=-\lambda(\nabla V\times{\bm{\sigma}})_{j}$
(32)

due to the spin-orbit interaction. The term $f_{s,i}^{\alpha}$ in Eq. (28) is
the one which cannot be written as a divergence of flow, interpreted purely as
a force. The spin-resolved momentum, which is essentially the spin current, is
not conserved in the presence of spin relaxation processes, and this is why we
have force density besides the momentum flux density. This fact means that the
definition of the spin-dependent momentum flux density is not unique.
Nevertheless it is a useful quantity to study the roles of vorticity on the
spin transport, like spin motive force (spin gauge field) in spintronics [21].

We calculate the momentum flux density focus on the contribution from the
spin-orbit interaction, as it is essential to couple electron flow and its
spin. In the spin-orbit contribution of $\pi^{0\alpha}_{ij}$, there are two
processes, one arising from the normal velocity $\frac{{\bm{p}}}{m}$, the
contribution historically called skew scattering contribution, and the other
arising from the anomalous velocity $\delta\bm{v}$, called the side-jump
contribution.

The linear response contribution to the external field ${\bm{E}}$ is written
as $\pi_{ij}^{s,\alpha}=\pi_{ijk}^{s,\alpha}E_{k}$, where
$\pi_{ijk}^{s,\alpha}$ is a correlation function with one more current vertex
for the applied field [11]. Below, we evaluate dominant contributions to
$\pi_{ijk}^{s,\alpha}$, which are those including both retarded and advanced
Green’s functions [11].

####  III.2.1 Skew-scattering contribution

Figure 3:  Feynman diagrams for the dominant contribution to the spin-resolved
momentum flux density $\pi^{s,\alpha}_{ijk}$ at the linear order in the spin-
orbit interaction and linear response to the applied field, denoted by
$\times$ at the right end (suffix $k$ represents the direction of the applied
field). Solid lines represent free electron Green’s functions, where upper and
lower lines denote retarded ($g_{{\bm{k}}}^{\rm r}$) and advanced
($g_{{\bm{k}}}^{\rm a}$) Green’s functions, respectively, and ${\bm{k}}$ and
${\bm{k}}^{\prime}$ are the electron wave vectors. The wave vector ${\bm{q}}$
is that of the external field and represents the inhomogeneity of flow.
Complex conjugate processes (turned upside down) are also taken into account.
(a): Skew-scattering contribution $\pi^{{\rm(ss)}\alpha}_{ijk}$. The left
vertex with $k_{i}k_{j}$ represents the vertex for the momentum flux density,
and a vertex with $\lambda$ denotes the spin-orbit interaction. (b): A small
contribution that is neglected. (c): The contribution arising from the
anomalous velocity $\delta v$ in $\pi^{0\alpha}_{ijk}$, called the side-jump
contribution. (d): The anomalous contribution $\delta\pi^{\alpha}_{ijk}$.

We first consider the process of Fig. 3(a) (skew scattering), which is

$\displaystyle\pi_{ijk}^{{\rm(ss)}\alpha}({\bm{q}})$
$\displaystyle=\frac{e}{V^{3}}\frac{i}{\pi}\frac{\lambda n_{\rm i}{v_{\rm
i}}^{3}}{m^{2}}{\rm
Re}\sum_{{\bm{k}}{\bm{k}}^{\prime}}k_{i}k_{j}k^{\prime}_{k}\left[\left({\bm{k}}^{\prime}+\frac{{\bm{q}}}{2}\right)\times\left({\bm{k}}+\frac{{\bm{q}}}{2}\right)\right]_{\beta}$
$\displaystyle\times{\rm
tr}[\sigma_{\alpha}g_{{\bm{k}}+\frac{{\bm{q}}}{2}}^{\rm
r}\sigma_{\beta}g_{{\bm{k}}^{\prime}+\frac{{\bm{q}}}{2}}^{\rm
r}g_{{\bm{k}}^{\prime}-\frac{{\bm{q}}}{2}}^{\rm
a}g_{{\bm{k}}^{\prime\prime}}^{\rm a}g_{{\bm{k}}-\frac{{\bm{q}}}{2}}^{\rm a}]$
(33)

where ${\bm{q}}$ is the wave vector of the external field, $g_{{\bm{k}}}^{\rm
r}\equiv[-\frac{k^{2}}{2m}+{\epsilon_{F}}+{\bm{M}}\cdot{\bm{\sigma}}+\frac{i}{2\tau}]^{-1}$
is the free retarded green’s function with elastic lifetime $\tau$ arising
from the impurities and $g_{{\bm{k}}}^{\rm a}\equiv(g_{{\bm{k}}}^{\rm
r})^{*}$, ${\epsilon_{F}}$ being the Fermi energy. The real (imaginary) part
is denoted by Re (Im).

We calculate the response function to the lowest order in the external wave
vector ${\bm{q}}$ and neglecting contributions smaller by a factor of
$({\epsilon_{F}}\tau)^{-1}$. Noting that

$\displaystyle{\rm tr}[\sigma^{\alpha}A\sigma^{\beta}B]$
$\displaystyle=(\delta_{\alpha\beta}-\delta_{\alpha z}\delta_{\beta
z})\sum_{\sigma}A_{\sigma}B_{-\sigma}+\delta_{\alpha z}\delta_{\beta
z}\sum_{\sigma}A_{\sigma}B_{\sigma}-i\epsilon_{\alpha\beta
z}\sum_{\sigma}\sigma A_{\sigma}B_{-\sigma}$ (34)

for diagonal matrices $A=\left(\begin{array}[]{cc}A_{+}&0\\\
0&A_{-}\end{array}\right)$ and $B$, the trace over the spin is calculated to
obtain (neglecting $O(q^{2})$)

$\displaystyle\pi_{ijk}^{{\rm(ss)}\alpha}({\bm{q}})$
$\displaystyle=i\sum_{\sigma}{\rm
Re}[I_{ijk\alpha}^{-\sigma,\sigma}+\delta_{\alpha
z}(I_{ijkz}^{\sigma,\sigma}-I_{ijkz}^{-\sigma,\sigma})+i\epsilon_{\alpha\beta
z}\sigma I_{ijk\beta}^{-\sigma,\sigma}]$ (35)

where $I_{ijk\alpha}^{\sigma^{\prime}\sigma}\equiv\frac{1}{\pi}\frac{\lambda
n_{\rm i}{v_{\rm
i}}^{3}}{m^{2}}\tilde{I}_{ijk\alpha}^{\sigma^{\prime}\sigma}$, with

$\displaystyle\tilde{I}_{ijk\alpha}^{\sigma^{\prime}\sigma}$
$\displaystyle=\frac{1}{V^{3}}\sum_{{\bm{k}}{\bm{k}}^{\prime}{\bm{k}}^{\prime\prime}}k_{i}k_{j}k^{\prime}_{k}({\bm{k}}^{\prime}\times{\bm{k}})_{\alpha}g_{{\bm{k}}+\frac{{\bm{q}}}{2},\sigma^{\prime}}^{\rm
r}g_{{\bm{k}}^{\prime}+\frac{{\bm{q}}}{2},\sigma}^{\rm
r}g_{{\bm{k}}^{\prime}-\frac{{\bm{q}}}{2},\sigma}^{\rm
a}g_{{\bm{k}}^{\prime\prime},\sigma}^{\rm
a}g_{{\bm{k}}-\frac{{\bm{q}}}{2},\sigma}^{\rm a}$ (36)

The summation over the wave vectors are carried out using

$\displaystyle\frac{1}{V}\sum_{{\bm{k}}}g_{{\bm{k}},\sigma}^{\rm a}$
$\displaystyle=i\pi\nu_{\sigma},\,\,\,\frac{1}{V}\sum_{{\bm{k}}}k_{i}k_{j}g_{{\bm{k}},\sigma}^{\rm
r}g_{{\bm{k}},\sigma}^{\rm
a}=\frac{2\pi}{3}\nu_{\sigma}k_{F\sigma}^{2}\tau\delta_{ij}$
$\displaystyle\frac{1}{V}\sum_{{\bm{k}}}k_{i}k_{j}k_{k}g_{{\bm{k}}+\frac{{\bm{q}}}{2},\sigma}^{\rm
r}g_{{\bm{k}}-\frac{{\bm{q}}}{2},\sigma^{\prime}}^{\rm a}$
$\displaystyle=i\frac{\pi}{15m}(\delta_{ij}q_{k}+\delta_{ik}q_{j}+\delta_{jk}q_{i})\frac{{\nu_{\sigma}}k_{F\sigma}^{4}+\nu_{\sigma^{\prime}}k_{F\sigma^{\prime}}^{4}}{((\sigma-\sigma^{\prime})M+\frac{i}{\tau})^{2}}$

where $\nu_{\sigma}$, $k_{F\sigma}$ are density of states at the Fermi energy
and Fermi wave vector of electron with spin $\sigma=\pm$, respectively, as

$\displaystyle\tilde{I}_{ijk\alpha}^{\sigma^{\prime}\sigma}$
$\displaystyle=-\frac{2\pi^{3}}{45m}(\nu_{\sigma})^{2}(k_{F\sigma})^{2}\tau\epsilon_{\alpha
km}(\delta_{ij}q_{m}+\delta_{im}q_{j}+\delta_{jm}q_{i})\frac{{\nu_{\sigma^{\prime}}}k_{F\sigma^{\prime}}^{4}+\nu_{\sigma}k_{F\sigma}^{4}}{((\sigma^{\prime}-\sigma)M+\frac{i}{\tau})^{2}}$
(38)

Thus

$\displaystyle\pi_{ijk}^{{\rm(ss)}\alpha}({\bm{q}})$
$\displaystyle=i\biggl{[}[(\hat{{\bm{\alpha}}}\times{\bm{q}})_{k}\delta_{ij}+\epsilon_{ik\alpha}q_{j}+\epsilon_{jk\alpha}q_{i}]{\rm
Re}\sum_{\sigma}J_{-}^{\sigma}$ $\displaystyle+\delta_{\alpha
z}[(\hat{\bm{z}}\times{\bm{q}})_{k}\delta_{ij}+\epsilon_{ikz}q_{j}+\epsilon_{jkz}q_{i}]{\rm
Re}\sum_{\sigma}(J_{+}^{\sigma}-J_{-}^{\sigma})$
$\displaystyle-\epsilon_{\alpha\beta z}[-\epsilon_{\beta
kl}q_{l}\delta_{ij}+\epsilon_{ik\beta}q_{j}+\epsilon_{jk\beta}q_{i}]{\rm
Im}\sum_{\sigma}\sigma J_{-}^{\sigma}\biggr{]}$ (39)

where

$\displaystyle J_{-}^{\sigma}$ $\displaystyle=\frac{2\pi^{2}}{45m^{3}}\lambda
n_{\rm i}{v_{\rm
i}}^{3}\left(\sum_{\sigma^{\prime}}{\nu_{\sigma^{\prime}}}k_{F\sigma^{\prime}}^{4}\right)(\nu_{\sigma})^{2}(k_{F\sigma})^{2}\tau\frac{4M^{2}-\frac{1}{\tau^{2}}+4i\sigma\frac{M}{\tau}}{(4M^{2}+\frac{1}{\tau^{2}})^{2}}$
$\displaystyle J_{+}^{\sigma}$ $\displaystyle=-\frac{2\pi^{2}}{45m^{3}}\lambda
n_{\rm i}{v_{\rm i}}^{3}(\nu_{\sigma})^{3}(k_{F\sigma})^{6}2\tau^{3}$ (40)

We thus find that the result is consistent with the phenomenological argument
of Sec. III.1; We have

$\displaystyle\pi_{ij}^{{\rm(ss)}\alpha}$
$\displaystyle=\eta_{\parallel}^{\rm(ss)}\hat{{\bm{M}}}_{\alpha}\left((\hat{{\bm{M}}}\cdot{\bm{\omega}})\delta_{ij}+\nabla_{i}T_{j}+\nabla_{j}T_{i}\right)$
$\displaystyle+\eta_{\perp}^{\rm(ss)}\left((\hat{{\bm{\alpha}}}_{\perp}\cdot{\bm{\omega}})\delta_{ij}+\nabla_{i}T^{\alpha_{\perp}}_{j}+\nabla_{j}T^{\alpha_{\perp}}_{i}\right)$
$\displaystyle+\eta_{\perp^{\prime}}^{\rm(ss)}\left((\hat{{\bm{\alpha}}}_{\perp^{\prime}}\cdot{\bm{\omega}})\delta_{ij}+\nabla_{i}T^{\alpha_{\perp^{\prime}}}_{j}+\nabla_{j}T^{\alpha_{\perp^{\prime}}}_{i}\right)$
(41)

where $\hat{{\bm{\alpha}}}$ is the unit vector along the spin direction
$\alpha$,
$\hat{{\bm{\alpha}}}_{\perp}\equiv\hat{{\bm{\alpha}}}-\hat{{\bm{M}}}(\hat{{\bm{M}}}\cdot\hat{{\bm{\alpha}}})$,
$\hat{{\bm{\alpha}}}_{\perp^{\prime}}\equiv\hat{{\bm{M}}}\times\hat{{\bm{\alpha}}}$,
and

$\displaystyle\eta_{\parallel}^{\rm(ss)}$ $\displaystyle\equiv-{\rm
Re}\sum_{\sigma}J_{+}^{\sigma}$ $\displaystyle\eta_{\perp}^{\rm(ss)}$
$\displaystyle\equiv-{\rm Re}\sum_{\sigma}J_{-}^{\sigma}$
$\displaystyle\eta_{\perp^{\prime}}^{\rm(ss)}$ $\displaystyle\equiv-{\rm
Im}\sum_{\sigma}\sigma J_{-}^{\sigma}$ (42)

In terms of spin polarization vector $\hat{{{\bm{s}}}}$ (Eq. (13)), the
amplitude of the flux density is

$\displaystyle\overline{\pi}_{ij}^{{\rm(ss)}}$
$\displaystyle=\eta_{\parallel}^{\rm(ss)}(\hat{{{\bm{s}}}}\cdot\hat{{\bm{M}}})\left((\hat{{\bm{M}}}\cdot{\bm{\omega}})\delta_{ij}+\nabla_{i}T_{\hat{{\bm{M}}},j}+\nabla_{j}T_{\hat{{\bm{M}}},i}\right)$
$\displaystyle+\eta_{\perp}^{\rm(ss)}\left((\hat{{{\bm{s}}}}_{\perp}\cdot{\bm{\omega}})\delta_{ij}+\nabla_{i}T_{{{\bm{s}}}_{\perp},j}+\nabla_{j}T_{{{\bm{s}}}_{\perp},i}\right)$
$\displaystyle+\eta_{\perp^{\prime}}^{\rm(ss)}\left((\hat{{{\bm{s}}}}_{\perp^{\prime}}\cdot{\bm{\omega}})\delta_{ij}+\nabla_{i}T_{{{\bm{s}}}_{\perp^{\prime}},j}+\nabla_{j}T_{{{\bm{s}}}_{\perp^{\prime}},i}\right)$
(43)

Let us estimate the magnitude of the coefficients. Our analysis in the ohmic
regime (dirty metal) assumes ${\epsilon_{F}}\tau\gg 1$. We first consider
strong ferromagnet with $M\tau\gg 1$. We simplify $\nu_{+}\sim\nu_{-}\sim
1/{\epsilon_{F}}$ and $k_{F\sigma}\sim{k_{F}}$ for order of magnitude estimate
and neglect numerical factors. We then have

$\displaystyle J_{+}^{\sigma}$ $\displaystyle\sim\frac{\epsilon_{\rm
so}}{m}\tau^{2},$ $\displaystyle J_{-}^{\sigma}$
$\displaystyle\sim\frac{\epsilon_{\rm
so}}{m}\frac{1}{M^{2}}\left(1+i\sigma\frac{1}{M\tau}\right)$ (44)

and thus

$\displaystyle\eta_{\parallel}^{\rm(ss)}$
$\displaystyle\sim\frac{\epsilon_{\rm so}}{m}\tau^{2},$
$\displaystyle\eta_{\perp}^{\rm(ss)}$ $\displaystyle\sim\frac{\epsilon_{\rm
so}}{m}\frac{1}{M^{2}},$ $\displaystyle\eta_{\perp^{\prime}}^{\rm(ss)}$
$\displaystyle\sim\frac{\epsilon_{\rm so}}{m}\frac{1}{M^{3}\tau}$ (45)

where $\epsilon_{\rm so}\equiv\lambda v_{\rm i}{k_{F}}^{2}$ is the energy
scale of the spin-orbit interaction. Namely, the adiabatic component is
dominant and $\eta_{\perp}^{\rm(ss)}$ contribution is dominant in the
nonadiabatic contributions.

In the limit of $M=0$, $J_{+}^{\sigma}=J_{-}^{\sigma}\sim\frac{\epsilon_{\rm
so}}{m}\tau^{2}$ and we have

$\displaystyle\eta_{\parallel}^{\rm(ss)}$
$\displaystyle=\eta_{\perp}^{\rm(ss)}\sim\frac{\epsilon_{\rm so}}{m}\tau^{2},$
$\displaystyle\eta_{\perp^{\prime}}^{\rm(ss)}=0,$ (46)

which is natural from the rotational symmetry when $M=0$.

There is a similar process with less impurity scattering, shown in Fig. 3(b).
Compared to the contribution $\pi_{ij}^{{\rm(ss)}\alpha}$, it is without a
factor of $i\pi\nu v_{\rm i}$. Due to the extra factor of $i$, the imaginary
and real parts are replaced, resulting in a vanishing contribution for $M=0$
and a reduction factor of $(M\tau)^{-1}$ in the case of $M\tau\gg 1$.
Considering a factor of $\nu v_{\rm i}\sim\sqrt{\nu\tau}$, the contribution
vanishes ($M=0$) or smaller by a factor of $(M\tau)^{-1/2}$. We thus neglect
this process.

####  III.2.2 Side-jump and other contributions

The contributions arising from the anomalous velocity $\delta\bm{v}$ (the
side-jump contribution, depicted in Fig. 3(c)) is

$\displaystyle\pi_{ijk}^{{\rm(sj)}\alpha}({\bm{q}})$
$\displaystyle=\frac{-i}{4\pi}\frac{1}{V^{2}}\frac{\lambda n_{\rm i}v_{\rm
i}^{2}}{2m}\epsilon_{\alpha
jl}\sum_{{\bm{k}}{\bm{k}}^{\prime}}{(k^{\prime}+k+q)_{i}}\left(k+\frac{q}{2}\right)_{k}(k^{\prime}-k)_{l}{\rm
Re}{\rm tr}[g_{{\bm{k}}^{\prime}}^{\rm r}g_{{\bm{k}}}^{\rm
r}g_{{\bm{k}}+{\bm{q}}}^{\rm a}]$ (47)

It turns out to be

$\displaystyle\pi_{ijk}^{{\rm(sj)}\alpha}({\bm{q}})$
$\displaystyle=i\eta^{\rm(sj)}_{0}\left(\frac{2}{3}\epsilon_{\alpha
ij}q_{k}+\delta_{ik}\epsilon_{\alpha jl}q_{l}+\epsilon_{\alpha
jk}q_{i}\right)$ (48)

where

$\displaystyle\eta^{\rm(sj)}_{0}\equiv\frac{\pi}{15}\frac{\lambda n_{\rm
i}v_{\rm i}^{2}}{m^{2}}\sum_{\sigma}(\nu_{\sigma})^{2}k_{{\rm
F}\sigma}^{4}\tau^{2}.$ (49)

Thus, neglecting the term that vanishes in the force, we obtain

$\displaystyle\pi_{ij}^{{\rm(sj)}}$
$\displaystyle=\eta^{{\rm(sj)},T_{\hat{{{\bm{s}}}}}}(\nabla_{i}T_{\hat{{{\bm{s}}}},j}+\nabla_{j}T_{\hat{{{\bm{s}}}},i})+\epsilon_{ijk}a_{k}^{{\rm(sj)}}$
(50)

where

$\displaystyle{\bm{a}}^{{\rm(sj)}}$ $\displaystyle=\xi^{{\rm(sj)},{\rm
v}}\hat{{{\bm{s}}}}(\nabla\cdot{\bm{E}})+\xi^{{\rm(sj)},T_{\hat{{{\bm{s}}}}}}(\nabla\times{\bm{T}}_{\hat{{{\bm{s}}}}})$
(51)

with $\eta^{{\rm(sj)},T_{\hat{{{\bm{s}}}}}}=-\frac{1}{2}\eta^{\rm(sj)}_{0}$,
$\xi^{{\rm(sj)},{\rm v}}=\frac{2}{3}\eta^{\rm(sj)}_{0}$,
$\xi^{{\rm(sj)},T_{\hat{{{\bm{s}}}}}}=-\frac{1}{2}\eta^{\rm(sj)}_{0}$. A
unique feature of the side-jump contribution is that it does not vanish for
$M=0$, in contrast to the skew scattering contribution. In the same order of
estimate as in Eq. (45), we have

$\displaystyle\eta^{\rm(sj)}_{0}\sim\frac{\epsilon_{\rm
so}}{m}\sqrt{\frac{\tau^{3}}{{\epsilon_{F}}}}$ (52)

The nonadiabatic contribution with spin polarized perpendicular to
$\hat{{\bm{M}}}$ is therefore dominated by the side-jump contribution instead
of skew-scattering contribution represented by $\eta_{\perp}^{\rm(ss)}$ in the
disordered metal with $M\tau\gg 1$.

Finally, the anomalous contribution to the momentum flux density is (Fig.
3(d))

$\displaystyle\delta\pi^{\alpha}_{ij}({\bm{q}})=\frac{(-i)^{2}}{2V^{3}}\frac{\lambda
n_{\rm i}v_{\rm
i}^{3}}{m}\sum_{{\bm{k}}{\bm{k}}^{\prime}{\bm{k}}^{\prime\prime}}(k^{\prime}-k)_{i}(k^{\prime}-k)_{l}\left(k^{\prime}+\frac{q}{2}\right)_{k}(\delta_{j\alpha}\delta_{l\beta}-\delta_{j\beta}\delta_{l\alpha}){\rm
tr}[\sigma_{\beta}(g_{{\bm{k}}}^{\rm r}g_{{\bm{k}}^{\prime\prime}}^{\rm
r}+g_{{\bm{k}}}^{\rm a}g_{{\bm{k}}^{\prime\prime}}^{\rm
a})g_{{\bm{k}}^{\prime}}^{\rm r}g_{{\bm{k}}^{\prime}+{\bm{q}}}^{\rm a}]$ (53)

After some calculation, we obtain

$\displaystyle\delta\pi_{ijk}^{\alpha}({\bm{q}})$
$\displaystyle=i\delta\pi\left(\frac{8}{3}(\delta_{iz}\delta_{j\alpha}-\delta_{i\alpha}\delta_{jz})q_{k}+\delta_{ik}(q_{z}\delta_{j\alpha}-q_{\alpha}\delta_{jz})-(\delta_{jz}\delta_{k\alpha}-\delta_{j\alpha}\delta_{kz})q_{i}\right)$
(54)

where

$\displaystyle\delta\pi\equiv-\frac{2\pi^{3}}{15m^{2}}\lambda{n_{\rm i}v_{\rm
i}^{3}}\tau^{2}\sum_{\sigma}\sigma\nu_{\sigma}^{3}k_{F\sigma}^{4}$ (55)

Neglecting unphysical contribution that vanishes in the force,

$\displaystyle\overline{\delta\pi}_{ij}$
$\displaystyle=\delta\eta_{T_{\perp^{\prime}}}(\nabla_{i}T_{\tilde{{{\bm{s}}}},j}+\nabla_{j}T_{\tilde{{{\bm{s}}}},i})+\epsilon_{ijk}\delta
a_{k}^{\tilde{{{\bm{s}}}}}$ (56)

where $\delta\eta_{T_{\perp^{\prime}}}=\delta\pi$ and

$\displaystyle\delta{\bm{a}}^{\tilde{{{\bm{s}}}}}$
$\displaystyle=\delta\xi^{{\rm
v}}\tilde{{{{\bm{s}}}}}(\nabla\cdot{\bm{E}})+\delta\xi^{T_{\tilde{{{\bm{s}}}}}}(\nabla\times{\bm{T}}_{\tilde{{{\bm{s}}}}})$
(57)

with $\delta\xi^{{\rm v}}=\frac{8}{3}\delta\pi$,
$\delta\xi^{T_{\tilde{{{\bm{s}}}}}}=\delta\pi$. The order of magnitude for
$M\tau\gg 1$ is $\delta\pi\sim\frac{\epsilon_{\rm
so}}{m}\frac{\tau}{{\epsilon_{F}}}$ and is small than the side-jump
contribution but is larger than the skew scattering nonadiabatic contribution.

In the limit of $M=0$, the order of magnitudes of $\eta^{\rm(sj)}_{0}$ and
$\delta\pi$ are the same as in the $M\tau\gg 1$ case. Thus these contributions
are smaller than the skew scattering contribution for $M=0$, Eq. (46). The
side-jump contribution being smaller than the skew scattering one is due to
the fact that the contribution involves less Green’s functions than the skew
scattering, resulting in smaller order of ${\epsilon_{F}}\tau(\gg 1)$
considering the fact that the peak magnitude of the Green’s function is
$\sim\tau$. When $M\tau\gg 1$, in contrast, side-jump contribution dominates
over the skew scattering as for the perpendicular spin polarization, as the
Green’s functions for the spin-flip processes are suppressed to be $\sim
M^{-1}$, while side-jump process contains a contribution without spin flip
(Eq. 47).

Phenomenological results in Sec. III.1 are therefore confirmed by microscopic
calculations in this subsection.

##  IV Vorticity-induced spin

Spin-resolved force density represents a force acting on spin components,
namely, spin motive force driving spin current. Although it would be
interesting to explore spin transport phenomena solving the hydrodynamic
equation with spin, we leave it as a future work and study the result of the
spin current generation, namely, the spin accumulation induced by the
vorticity-induced spin motive forces.

Like spin current, spin-resolved force density is not a clear physical
observable, as the force acting on electrons with a particular spin
polarization is not detectable. In contrast, spin density which we are going
to calculate here is a physical observable. In fact, spin Hall effect
originally argued as a relation between the spin density and an applied
electric field by Dyakonov and Perel [15] is free from the ambiguity of
definition of spin current [14]. Here we discuss spin density taking account
of inhomogeneous electric field to support physical consequence of spin
hydrodynamic equations.

###  IV.1 Spin Hall effect in the viewpoint of spin-vorticity coupling

As was pointed out [14], an inhomogeneous electric field applied to a metal
with spin-orbit interaction induces a spin density as in Eq. (1). This
relation in fact is a representation of spin Hall effect written in terms of
spin density instead of spin current. The spin accumulation given by Eq. (1)
in fact represents the spin accumulation formed at edges as a result of spin
current generated by the applied electric field. The expression corresponds to
the clean system where electron diffusion is not relevant and in the
disordered case, the expression is multiplied by a diffusion propagator [14]
(see eq. (65)). It indicates that spin Hall effect is a consequence of spin-
vorticity coupling. The spin generation by the spin Hall effect has been
argued mostly in the absence of magnetization, to discuss the pure spin
current. When a magnetization is present, spin polarization along the
magnetization $\hat{{\bm{M}}}$ (the adiabatic component) and orthogonal
components $\hat{{\bm{M}}}\times{\bm{\omega}}$ arise. We can therefore write

$\displaystyle{{\bm{s}}}({\bm{r}})$ $\displaystyle=\lambda_{\rm
sh}{\bm{\omega}}+\lambda_{\parallel}\hat{{\bm{M}}}(\hat{{\bm{M}}}\cdot{\bm{\omega}})+\lambda_{\perp}(\hat{{\bm{M}}}\times{\bm{\omega}})$
(58)

with coefficients $\lambda_{\parallel}$ and $\lambda_{\perp}$. The term
$\lambda_{\parallel}$ represents the adiabatic component (along ${\bm{M}}$) of
the spin polarization and $\lambda_{\perp}$ represents the anomalous Hall
effect for the vorticity.

###  IV.2 Miscroscopic calculation

Here we calculate the coefficients based on the model of spin-orbit
interaction due to heavy impurities of Sec. III.2. The dominant contribution
arises from the same diagram as in Fig. 3(a) with the left vertex replaced by
a Pauli matrix. The spin density polarized along direction $\alpha$ induced by
the electric field along the $k$-direction is

$\displaystyle s_{k}^{\alpha}({\bm{q}})$
$\displaystyle=\frac{e}{V^{3}}\frac{i}{\pi}\frac{\lambda n_{\rm i}{v_{\rm
i}}^{3}}{m}{\rm
Re}\sum_{{\bm{k}}{\bm{k}}^{\prime}}k^{\prime}_{k}\left[\left({\bm{k}}^{\prime}+\frac{{\bm{q}}}{2}\right)\times\left({\bm{k}}+\frac{{\bm{q}}}{2}\right)\right]_{\beta}{\rm
tr}[\sigma_{\alpha}g_{{\bm{k}}+\frac{{\bm{q}}}{2}}^{\rm
r}\sigma_{\beta}g_{{\bm{k}}^{\prime}+\frac{{\bm{q}}}{2}}^{\rm
r}g_{{\bm{k}}^{\prime}-\frac{{\bm{q}}}{2}}^{\rm
a}g_{{\bm{k}}^{\prime\prime}}^{\rm a}g_{{\bm{k}}-\frac{{\bm{q}}}{2}}^{\rm a}]$
(59)

which leads using Eq. (34) to

$\displaystyle s_{k}^{\alpha}({\bm{q}})$ $\displaystyle=s_{k}^{{\rm
sh},\alpha}({\bm{q}})+\delta_{\alpha,z}s_{k}^{{\parallel}}({\bm{q}})+\epsilon_{\alpha\beta
z}s_{k}^{{\perp}}({\bm{q}})$ $\displaystyle s_{k}^{{\rm sh},\alpha}({\bm{q}})$
$\displaystyle=\frac{e}{V^{3}}\frac{i}{\pi}\frac{\lambda n_{\rm i}{v_{\rm
i}}^{3}}{m}{\rm
Re}\sum_{{\bm{k}}{\bm{k}}^{\prime}{\bm{k}}^{\prime\prime}}k^{\prime}_{k}\left[{\bm{k}}^{\prime}\times{\bm{k}}\right]_{\alpha}\sum_{\sigma=\pm}g_{{\bm{k}}+\frac{{\bm{q}}}{2},\sigma}^{\rm
r}g_{{\bm{k}}^{\prime}+\frac{{\bm{q}}}{2},-\sigma}^{\rm
r}g_{{\bm{k}}^{\prime}-\frac{{\bm{q}}}{2},-\sigma}^{\rm
a}g_{{\bm{k}}^{\prime\prime},-\sigma}^{\rm
a}g_{{\bm{k}}-\frac{{\bm{q}}}{2},-\sigma}^{\rm a}$ $\displaystyle
s_{k}^{{\parallel}}({\bm{q}})$
$\displaystyle=\frac{e}{V^{3}}\frac{i}{\pi}\frac{\lambda n_{\rm i}{v_{\rm
i}}^{3}}{m}{\rm
Re}\sum_{{\bm{k}}{\bm{k}}^{\prime}{\bm{k}}^{\prime\prime}}k^{\prime}_{k}\left[{\bm{k}}^{\prime}\times{\bm{k}}\right]_{z}\sum_{\sigma=\pm}(g_{{\bm{k}}+\frac{{\bm{q}}}{2},\sigma}^{\rm
r}-g_{{\bm{k}}+\frac{{\bm{q}}}{2},-\sigma}^{\rm
r})g_{{\bm{k}}^{\prime}+\frac{{\bm{q}}}{2},\sigma}^{\rm
r}g_{{\bm{k}}^{\prime}-\frac{{\bm{q}}}{2},\sigma}^{\rm
a}g_{{\bm{k}}^{\prime\prime},\sigma}^{\rm
a}g_{{\bm{k}}-\frac{{\bm{q}}}{2},\sigma}^{\rm a}$ $\displaystyle
s_{k}^{\perp}({\bm{q}})$
$\displaystyle=\frac{e}{V^{3}}\frac{i}{\pi}\frac{\lambda n_{\rm i}{v_{\rm
i}}^{3}}{m}{\rm
Re}(-i)\sum_{{\bm{k}}{\bm{k}}^{\prime}{\bm{k}}^{\prime\prime}}k^{\prime}_{k}\left[{\bm{k}}^{\prime}\times{\bm{k}}\right]_{\beta}\sum_{\sigma=\pm}\sigma
g_{{\bm{k}}+\frac{{\bm{q}}}{2},\sigma}^{\rm
r}g_{{\bm{k}}^{\prime}+\frac{{\bm{q}}}{2},-\sigma}^{\rm
r}g_{{\bm{k}}^{\prime}-\frac{{\bm{q}}}{2},-\sigma}^{\rm
a}g_{{\bm{k}}^{\prime\prime},-\sigma}^{\rm
a}g_{{\bm{k}}-\frac{{\bm{q}}}{2},-\sigma}^{\rm a}$ (60)

After summation over the wave vectors, the coefficients in Eq. (58) are
obtained as

$\displaystyle\lambda_{\rm sh}$ $\displaystyle=e\frac{\pi^{2}}{3}\frac{\lambda
n_{\rm i}{v_{\rm
i}}^{3}}{m^{2}}\tau\frac{4M^{2}-\frac{1}{\tau^{2}}}{\left(4M^{2}+\frac{1}{\tau^{2}}\right)^{2}}\sum_{\sigma\sigma^{\prime}}\nu_{\sigma}k_{F\sigma}^{2}\nu_{\sigma^{\prime}}^{2}k_{F\sigma^{\prime}}^{2}$
$\displaystyle\lambda_{\parallel}$
$\displaystyle=e\frac{\pi^{2}}{3}\frac{\lambda n_{\rm i}{v_{\rm
i}}^{3}}{m^{2}}(-2\tau)\sum_{\sigma}\left[\nu_{\sigma}^{3}k_{F\sigma}^{4}\tau^{2}+\nu_{\sigma}^{2}k_{F\sigma}^{2}\sum_{\sigma^{\prime}}\nu_{\sigma^{\prime}}k_{F\sigma^{\prime}}^{2}\frac{\left(4M^{2}-\frac{1}{\tau^{2}}\right)}{\left(4M^{2}+\frac{1}{\tau^{2}}\right)^{2}}\right]$
$\displaystyle\lambda_{\perp}$ $\displaystyle=e\frac{\pi^{2}}{3}\frac{\lambda
n_{\rm i}{v_{\rm
i}}^{3}}{m^{2}}\frac{-4M}{\left(4M^{2}+\frac{1}{\tau^{2}}\right)^{2}}\sum_{\sigma\sigma^{\prime}}\nu_{\sigma}k_{F\sigma}^{2}\nu_{\sigma^{\prime}}^{2}k_{F\sigma^{\prime}}^{2}\sigma^{\prime}$
(61)

In the disordered case with $M\tau\gg 1$, the order of magnitude of each term
is

$\displaystyle\lambda_{\rm sh}$ $\displaystyle\sim\frac{\epsilon_{\rm
so}}{{k_{F}}^{2}}\frac{1}{M^{2}},$ $\displaystyle\lambda_{\parallel}$
$\displaystyle\sim\frac{\epsilon_{\rm so}}{{k_{F}}^{2}}\tau^{2},$
$\displaystyle\lambda_{\perp}$ $\displaystyle\sim\frac{\epsilon_{\rm
so}}{{k_{F}}^{2}}\frac{1}{M^{3}\tau}$ (62)

meaning that the adiabatic spin polarization is dominant, while the spin Hall
effect dominates the perpendicular component. In the limit of $M=0$,
$\lambda_{\rm sh}$ representing the spin-vorticity coupling and spin Hall
effect remains finite, while $\lambda_{\parallel}$ and $\lambda_{\perp}$
vanish.

###  IV.3 Vorticity-induced torque

In ferromagnets, the vorticity-induced spin, Eq. (58), generates a current-
induced torque near surfaces given by

$\displaystyle{\bm{T}}_{\omega}$ $\displaystyle=\lambda_{\rm
sh}[\hat{{\bm{M}}}\times(\nabla\times{\bm{E}})]+\lambda_{\perp}[\hat{{\bm{M}}}\times[\hat{{\bm{M}}}\times(\nabla\times{\bm{E}})]]$
(63)

In ferromagnets with $M\tau\gg 1$ and $M/{\epsilon_{F}}=O(1)$, the first term
dominates as is seen from Eq. (62). Interestingly the coefficient
$\lambda_{\rm sh}$ does not depend on $\tau$ in this regime, resulting in a
universal torque when written in terms of the applied field ${\bm{E}}$.
Although current-induced torques on magnetic textures have been discussed
intensively, effect of vorticity and inhomogeneous current density has not
been focused on. It is of interest to confirm experimentally the correlation
between the vorticity-induced torque and the spin Hall effect, both determined
by the coefficient $\lambda_{\rm sh}$.

Let us compare the vorticity-induced torque to the so-called $\beta$-torque
induced by spin relaxation and magnetization structure,

$\displaystyle{\bm{T}}_{\beta}=\beta
a^{3}[\hat{{\bm{M}}}\times(\bm{j}\cdot\nabla)\hat{{\bm{M}}}],$ (64)

where $\beta$ is a small constant representing the rate of spin relaxation and
$a$ is the lattice constant. Its magnitude in common ferromagnets is typically
of the order of $\beta\sim(M\tau_{\rm sf})^{-1}$, where $\tau_{\rm
sf}^{-1}\sim\epsilon_{\rm so}^{2}/{\epsilon_{F}}$ in the case of spin-orbit
interaction [22, 23]. In terms of the field $E$,
${\bm{T}}_{\beta}=\frac{e}{{k_{F}}^{2}}\widetilde{\beta}[\hat{{\bm{M}}}\times({\bm{E}}\cdot\nabla)\hat{{\bm{M}}}]$,
where $\widetilde{\beta}\equiv\frac{{k_{F}}^{2}}{e}\beta a^{3}\sigma_{\rm
e}\sim\widetilde{\epsilon_{\rm so}}^{2}{\epsilon_{F}}\tau$, with
$\widetilde{\epsilon_{\rm so}}\equiv\epsilon_{\rm so}/{\epsilon_{F}}$. Thus
the ratio of the torque and the field is
${\bm{T}}_{\beta}/E\sim\frac{e}{{k_{F}}^{2}\ell_{M}}\widetilde{\epsilon_{\rm
so}}^{2}{\epsilon_{F}}\tau$, while for the vorticity-induced torque it is
${\bm{T}}_{\omega}/E\sim\frac{e}{{k_{F}}^{2}\ell_{\omega}}\widetilde{\epsilon_{\rm
so}}$, where $\ell_{M}$ is the length scale of magnetization structure and
$\ell_{\omega}$ is the length scale of surface vorticity in the case of ohmic
fluid. Considering the fact that the $\beta$ torque is second order in the
spin-orbit interaction, while the vorticity torque is linear, surface
vorticity may have a crucial role in current-induced torques. Even stronger
effect is expected when surface roughness is considered, as roughness would
henhance generation of vorticity. Nucleation of magnetic domain wall and
skyrmion by artificial notches [24] would be carried out by the vorticity-
induced torque as was argued in Ref. [9].

Figure 4: (a) Schematic figure showing the current profile near the boundary
(chosen as $z=0$). Vorticity is finite in the length scale of $\ell_{\omega}$,
where the current density changes. (b) Schematic figure showing vorticity-
induced spin relaxation mechanism due to electron diffusion. The sign denotes
the direction of vorticity ${\bm{\omega}}$. Electron diffusion is represented
by a zigzag line.

Taking account of diffusion of electron, the spin density of Eq. (58) is
modified to be long-ranged as [14]

$\displaystyle{{\bm{s}}}({\bm{r}})$ $\displaystyle=\int
d^{3}r^{\prime}D_{s}({\bm{r}}-{\bm{r}}^{\prime})[\lambda_{\rm
sh}{\bm{\omega}}({\bm{r}}^{\prime})+\lambda_{\parallel}\hat{{\bm{M}}}(\hat{{\bm{M}}}\cdot{\bm{\omega}}({\bm{r}}^{\prime}))+\lambda_{\perp}(\hat{{\bm{M}}}\times{\bm{\omega}}({\bm{r}}^{\prime}))]$
(65)

where
$D_{s}({\bm{r}})\equiv\frac{1}{\tau}\sum_{{\bm{q}}}\frac{e^{i{\bm{q}}\cdot{\bm{r}}}}{Dq^{2}+\frac{1}{\tau_{s}}}$
is a diffusion propagator of electron spin, $D$ and $\tau_{s}$ are the
diffusion constant and spin relaxation time of electron, respectively. The
electron spin diffusion length is $\ell_{s}\equiv\sqrt{D\tau_{s}}$. In the
disordered case, therefore, the vorticity-induced torque is determined by
thevorticity average over the spin diffusion length,
$\overline{{\bm{\omega}}}$. Let us consider the case where $\ell_{s}$ is
larger than the surface depth where the vorticity is finite, $\ell_{\omega}$.
Choosing the axis as in Fig. 4(a), local vorticity of current density $\bm{j}$
is $\omega^{(j)}_{y}=\partial_{z}j_{x}$ and its average is

$\displaystyle\overline{\omega}^{(j)}_{y}=\frac{1}{\ell_{\omega}}\int_{0}^{\ell_{\omega}}dz\partial_{z}j_{x}=\frac{\overline{j}_{x}}{\ell_{\omega}}$
(66)

where $\overline{j}_{x}$ is the current far from the surface and $j_{x}$ at
the surface ($z=0$) is assumed to vanish. The averaged torque near the
surface,

$\displaystyle\overline{{\bm{T}}}_{\omega}$ $\displaystyle=\lambda_{\rm
sh}[\hat{{\bm{M}}}\times\overline{{\bm{\omega}}}]+\lambda_{\perp}[\hat{{\bm{M}}}\times[\hat{{\bm{M}}}\times\overline{{\bm{\omega}}}]]$
(67)

is therefore

$\displaystyle\overline{{\bm{T}}}_{\omega}$
$\displaystyle=\frac{1}{\sigma_{\rm e}}\left[\frac{\lambda_{\rm
sh}}{\ell_{\omega}}[\hat{{\bm{M}}}\times(\hat{\bm{z}}\times\overline{\bm{j}})]+\lambda_{\perp}[\hat{{\bm{M}}}\times[\hat{{\bm{M}}}\times(\hat{\bm{z}}\times\overline{\bm{j}})]]\right]$
(68)

where $\hat{\bm{z}}$ denotes the direction normal to the surface. The torque
represented by $\lambda_{\rm sh}$ has the same form as the one due to the
surface Rashba-Edelstein effect, which induces spin polarization proportional
to $\hat{\bm{z}}\times\bm{j}$ [25].

For thin film like in Fig. 2, the vorticity-induced torque points opposite on
the upper and lower plane and has no bulk effects.

###  IV.4 Vorticity-induced spin relaxation

The vorticity-induced torque is naturally inhomogeneous, and would lead to
spin relaxation effects. Let us consider a thin film (Figs. 2,4(b)) of
thickness $d$ with electric field parallel to the magnetization. The vorticity
is then perpendicular to ${\bm{M}}$. The vorticity-induced torque drives the
electron spin as

$\displaystyle\dot{{{\bm{s}}}}=\gamma\left[\lambda_{\rm
sh}({{{\bm{s}}}}\times{\bm{\omega}})+\lambda_{\perp}({{{\bm{s}}}}\times\hat{{\bm{M}}})(\hat{{\bm{M}}}\cdot{\bm{\omega}})+\lambda_{\perp}[{{{\bm{s}}}}\times(\hat{{\bm{M}}}\times{\bm{\omega}})]\right],$
(69)

where $\gamma\equiv\frac{e}{m}$ is the electron gyromagnetoratio and thus the
electron spin polarized along ${\bm{M}}$ gets flipped in a timescale of
$(\lambda_{\rm sh}\omega)^{-1}$ (neglecting $\lambda_{\perp}(\ll\lambda_{\rm
sh})$). In a thin film, vorticity ${\bm{\omega}}$ arising from inhomogeneous
flow points opposite on the upper and lower surfaces (Fig. 4(b)). If call the
length scale where vorticity is finite measured from the surface as
$\ell_{\omega}$, (in the ohmnic fluid, $\ell_{\omega}\simeq\ell$ (mean free
path)), $2\ell_{\omega}/d$ of the total electron spins are disturbed by the
vorticity-induced torque. The vorticity-induced spin relaxation time is
therefore estimated as

$\displaystyle{\tau_{\omega}}^{-1}$
$\displaystyle=\frac{\ell_{\omega}}{d}\lambda_{\rm sh}\omega$ (70)

assuming that the spin diffusion length is longer than $d$. Its magnitude is
as ${\tau_{\omega}}^{-1}\sim\widetilde{\epsilon_{\rm
so}}\frac{eE}{{k_{F}}}\frac{1}{{k_{F}}d}$ assuming $\omega\sim
E/\ell_{\omega}$. For $E=10^{-2}$ V/$\mu$m and ${k_{F}}^{-1}\sim a=10^{-10}$m,
$\frac{eE}{{k_{F}}}=10^{-6}$ eV=$2\times 10^{8}$ Hz. Let us define a damping
constant as $\alpha_{\omega}\equiv{\tau_{\omega}}^{-1}t_{M}$, where $t_{M}$ is
the time-scale of magnetization dynamics. For ${t_{M}}^{-1}\sim 1$ GHz, the
above estimate leads to $\alpha_{\omega}\sim
0.2\times\frac{\widetilde{\epsilon_{\rm so}}}{{k_{F}}d}$. For a large spin-
orbit metal of a thin film, in particular with surface roughness, the
vorticity-induced relaxation may dominate over the intrinsic Gilbert damping.
Experimentally, vorticity-induced damping would be separable from other
isotropic intrinsic origins by using the anisotropic nature, i.e., vorticity-
induced damping is suppressed if ${\bm{E}}\perp{\bm{M}}$.

##  V Vorticity-induced inverse spin Hall effect

Spin current generation by fluid vorticity in a pipe was discussed in Ref.
[26], and the inverse spin Hall voltage due to the spin current and measured
on heavy metal leads was argued. In the case of the Hagen-Poiseuille flow
considered there, the generated spin current is in the radial direction with
spin polarization perpendicular to both radian and pipe directions and the
inverse spin Hall voltage is along the flow.

The vorticity-induced inverse spin Hall effect is studied in the present
context by calculating the spin-neutral force density (motive force for
electric charge) taking account of the spin-orbit interaction to the second
order in the absence of $M$. The inverse spin Hall voltage here is an
intrinsic one without leads of heavy metals, and does not apply to
experimental situations with heavy metal leads.

The spin Hall coefficient ${\lambda}_{\rm sh}$ is a coefficient of the
correlation function of spin and charge current density linear in the external
wave vector [14]. The inverse spin Hall effect is represented as

$\displaystyle\bm{j}={\lambda}_{\rm ish}(\nabla\times{\bm{B}}_{\omega}),$ (71)

where ${\lambda}_{\rm ish}$ is a coefficient proportional to $\lambda_{\rm
sh}$ and ${\bm{B}}_{\omega}$ is an effective magnetic field that drives spin
density [14]. In the vorticity-driven case, the effective field is induced by
the vorticity as

$\displaystyle{\bm{B}}_{\omega}=\frac{\lambda_{\rm sh}}{\chi}{\bm{\omega}},$
(72)

where $\chi$ is spin susceptibility. In the ohmic regime, the relation (71)
indicates that there is a motive force ${\bm{f}}^{\rm a}$ acting on electron
charge that is proportional to the rotation of vorticity,
$\nabla\times{\bm{\omega}}$. The inverse spin Hall motive force is therefore
represented by an antisymmetric component of the momentum flux density for
charge (spin neutral), $\pi^{\rm a}_{ij}=\xi^{\rm a}\epsilon_{ijk}a_{k}$ with
a vector ${\bm{a}}$, as ${\bm{f}}^{\rm a}=-\xi^{\rm a}(\nabla\times{\bm{a}})$,
resulting in an inverse spin Hall current
$\bm{j}\propto(\nabla\times{\bm{a}})$. As has been argued [20] (See also Eq.
(3)), an antisymmetric component arises from vorticity,
${\bm{a}}\propto{\bm{\omega}}$. We thus obtain the inverse spin Hall current
$\bm{j}\propto\nabla\times{\bm{\omega}}$, consistent with Eqs. (71)(72).

Figure 5: Feynman diagram representing the vorticity-induced inverse spin Hall
effect at the second order of the spin-orbit interaction ($\lambda$). The
anomalous velocity vertex gives rise to an antisymmetric component of the
momentum flux density that describes inverse spin Hall current, Eq. (71).

Let us confirm this fact by a microscopic calculation of spin-neutral momentum
flux density
$\pi_{ij}\equiv\left\langle{\hat{p}_{i}\hat{v}_{j}}\right\rangle\equiv\pi_{ijk}E_{k}$
in the case of $M=0$ including the spin-orbit interaction to the second order.
Obviously, the skew scattering contribution is symmetric with respect to $i$
and $j$ [12], and antisymmetic contribution arises from the side-jump
contribution, whose dominant contribution is (shown in Fig. 5) ((sj)(2)
denotes the side-jump process at the second order of the spin-orbit
interaction)

$\displaystyle\pi_{ijk}^{{\rm(sj)}(2)}({\bm{q}})$ $\displaystyle=\frac{1}{4\pi
V^{4}}\frac{\lambda^{2}v_{\rm i}^{5}n_{\rm i}^{2}}{2m}i\epsilon_{\alpha
jl}\epsilon_{mn\beta}$
$\displaystyle\times\sum_{{\bm{k}}{\bm{k}}^{\prime}{\bm{k}}^{\prime\prime}{\bm{k}}^{\prime\prime\prime}}{(k^{\prime}+k+q)_{i}}\left(k^{\prime\prime}+\frac{q}{2}\right)_{k}(k^{\prime}-k)_{l}k_{n}k^{\prime\prime}_{m}i{\rm
Im}{\rm tr}[g_{{\bm{k}}^{\prime}}^{\rm
r}\sigma_{\alpha}g_{{\bm{k}}+{\bm{q}}}^{\rm
a}\sigma_{\beta}g_{{\bm{k}}^{\prime\prime}+{\bm{q}}}^{\rm
a}g_{{\bm{k}}^{\prime\prime}}^{\rm r}g_{{\bm{k}}^{\prime\prime\prime}}^{\rm
r}g_{{\bm{k}}}^{\rm r}]$ (73)

The antisymmetric component of $\pi_{ij}^{(2)}$ is written in terms of a
vector ${\bm{a}}_{\omega}$ as $\pi_{ij}^{(2){\rm
a}}=\epsilon_{ijk}a_{\omega,k}$, where

$\displaystyle{\bm{a}}_{\omega}$
$\displaystyle=\xi^{(2)}_{\omega}{\bm{\omega}}$
$\displaystyle\xi^{(2)}_{\omega}$
$\displaystyle=\frac{\pi}{12}\left(\frac{\epsilon_{\rm
so}}{{\epsilon_{F}}}\right)^{2}D\tau^{2}v_{\rm i}=\lambda_{\rm
sh}\left(\frac{\epsilon_{\rm so}}{{\epsilon_{F}}}\right)(v_{\rm
i}\tau){\epsilon_{F}}$ (74)

The inverse spin Hall field induced by vorticity is therefore

$\displaystyle{\bm{E}}_{\omega}=-\nabla\times{\bm{a}}_{\omega}=-\xi^{(2)}_{\omega}\nabla\times{\bm{\omega}}$
(75)

The inverse spin Hall voltage due to the vorticity-induced spin current is
therefore along the applied field ${\bm{E}}$ [6]. We note that the inverse
spin Hall voltage in reality would be mixed with contributions from the
symmetric components of the momentum flux density (see argument in Sec. II).

##  VI Summary and discussion

We have explored the effects of vorticity of electron flow in metals on the
spin transport based on a hydrodynamic viewpoint. The spin-resolved momentum
flux density was discussed extending the results for anomalous Hall fluid
studied previously in Ref. [12]. When represented by an external electric
field ${\bm{E}}$, the anomalous Hall contributions to the momentum flux
density are written in terms of a troidal moment,
${\bm{T}}_{{\bm{M}}}={\bm{M}}\times{\bm{E}}$. When electron spin is taken into
account, the spin-resolved momentum flux density $\pi_{ij}^{\alpha}$ ($i$ and
$j$ represent the spatial direction and $\alpha$ denotes spin direction) are
characterized by the three contributions, namely, the adiabatic contribution
where spin ${{\bm{s}}}$ is parallel to ${\bm{M}}$, the nonadiabatic
contribution written by
${{\bm{s}}}_{\perp}\equiv{{\bm{s}}}-\hat{{\bm{M}}}(\hat{{\bm{M}}}\cdot{{\bm{s}}})$
and a contribution written by
$\tilde{{{\bm{s}}}}\equiv{{\bm{s}}}\times{\bm{M}}$. The troidal moment
${\bm{T}}_{{\bm{M}}}$ is therefore extended to include two nonadiabatic
contributions,
${\bm{T}}_{{{\bm{s}}}_{\perp}}\equiv{{\bm{s}}}_{\perp}\times{\bm{E}}$ and
${\bm{T}}_{\tilde{{{\bm{s}}}}}\equiv\tilde{{{\bm{s}}}}\times{\bm{E}}$. Those
results were confirmed by linear response calculations in a model with the
impurity-induced spin-orbit interaction.

The spin-resolved force density $-\nabla_{j}\pi_{ij}^{\alpha}$, or spin motive
force, was argued and roles of electron vorticity on the motive force was
discussed. Besides the conventional conservative force due to spin-vorticity
coupling, proportional to $\nabla({{\bm{s}}}\cdot{\bm{\omega}})$
(${\bm{\omega}}$ is the vorticity), there is a nonconservative force,
proportional to
$\nabla\times({\bm{\omega}}\times{{\bm{s}}})=({{\bm{s}}}\cdot\nabla){\bm{\omega}}$.
Like spin current, the spin-resolved momentum flux density and force density
are physical conserved current, and their definition are not unique.

Spin density induced by vorticity was studied. It was argued that the spin-
vorticity coupling represents the spin Hall effect. The vorticity-induced spin
density was discussed to give rise to a damping localized near the surface and
interface where vorticity is induced. The torque arises from the homogeneity
of the current (or total field), in contrast to conventional current-induced
relaxation torques such as $\beta$ torque arising from inhomogeneous
magnetization. The vorticity-induced torque is a linear effect of spin-orbit
interaction, while the conventional relaxation torque is second-order, meaning
that the vorticity effects may dominate in thin films. The effect leads to
torque and relaxation localized near surfaces and interfaces. The inverse spin
Hall effect due to the vorticity-induced spin current was shown to be
described by the antisymmetric part of the momentum flux density evaluated at
the second order of the spin-orbit interaction.

We considered the case of uniform magnetization with inhomogenuity of the
applied electric field. For a complete discussion of the troidal moment,
inhomogeneous magnetization needs to be taken into account. For this future
work, an effective gauge field approach [21] is expected to be useful.

###### Acknowledgements.

The author thank H. Funaki and R. Toshio for valuable discussion. This study
was supported by a Grant-in-Aid for Scientific Research (B) (No. 21H01034)
from the Japan Society for the Promotion of Science.

##  Appendix A Derivation of momentum flux density of Eqs. (29) (30)

Here we derive the expression for the spin-resolved momentum flux density by
evaluating the time-derivative of the spin-resolved momentum,
$\dot{p}_{i}^{\alpha}=i\left\langle{[H,c^{\dagger}\hat{p}_{i}\sigma_{\alpha}c]}\right\rangle$.
The contribution from the spin-orbit coupling is focused. The commutator is
calculated using
$\nabla^{{\bm{r}}}_{i}\delta({\bm{r}}-{\bm{r}}^{\prime})=-\nabla^{{\bm{r}}^{\prime}}_{i}\delta({\bm{r}}-{\bm{r}}^{\prime})$
and integral by parts as

$\displaystyle i$ $\displaystyle[H_{\rm
so},(c^{\dagger}\hat{p}_{i}\sigma_{\alpha}c)_{{\bm{r}}}]=-\frac{i}{4}\lambda\int
d{\bm{r}}^{\prime}\epsilon_{jkl}(\nabla^{{\bm{r}}^{\prime}}_{j}V({\bm{r}}^{\prime}))$
$\displaystyle\times\left[c^{\dagger}({\bm{r}}^{\prime})\sigma_{l}(\nabla_{k}c({\bm{r}}^{\prime}))-(\nabla_{k}c^{\dagger}({\bm{r}}^{\prime}))\sigma_{l}c({\bm{r}}^{\prime}),c^{\dagger}({\bm{r}})\sigma_{\alpha}(\nabla_{i}c({\bm{r}}))-(\nabla_{i}c^{\dagger}({\bm{r}}))\sigma_{\alpha}c({\bm{r}})\right]$
$\displaystyle=$
$\displaystyle-\frac{i}{2}\lambda\biggl{[}(\nabla_{j}V)\biggl{[}-\epsilon_{jk\alpha}\nabla_{k}[c^{\dagger}\stackrel{{\scriptstyle\leftrightarrow}}{{\nabla_{i}}}c]+i\epsilon_{jkl}\epsilon_{l\alpha\beta}[c^{\dagger}\sigma_{\beta}\stackrel{{\scriptstyle\leftrightarrow}}{{\nabla_{i}}}(\nabla_{k}c)-(\nabla_{k}c^{\dagger})\sigma_{\beta}\stackrel{{\scriptstyle\leftrightarrow}}{{\nabla_{i}}}c]\biggr{]}$
$\displaystyle+(\nabla_{i}\nabla_{j}V)\biggl{[}-\epsilon_{jk\alpha}(c^{\dagger}\stackrel{{\scriptstyle\leftrightarrow}}{{\nabla_{k}}}c)+i\epsilon_{jkl}\epsilon_{l\alpha\beta}\nabla_{k}[c^{\dagger}\sigma_{\beta}c]\biggr{]}\biggr{]}$
$\displaystyle=-\nabla_{j}\hat{\pi}^{{\rm so},\alpha}_{ij}+\hat{f}^{{\rm
so},\alpha}_{i}$ (76)

where

$\displaystyle\hat{\pi}^{{\rm so},\alpha}_{ij}$
$\displaystyle=\frac{i}{2}\lambda\left[(\nabla_{k}V)\epsilon_{jk\alpha}[c^{\dagger}\stackrel{{\scriptstyle\leftrightarrow}}{{\nabla_{i}}}c]-i(\nabla_{i}\nabla_{k}V)\epsilon_{jkl}\epsilon_{l\alpha\beta}(c^{\dagger}\sigma_{\beta}c)\right]$
(77) $\displaystyle\hat{f}^{{\rm so},\alpha}_{i}$
$\displaystyle=\frac{i}{2}\lambda\biggl{[}(\nabla_{i}\nabla_{j}V)\epsilon_{jk\alpha}(c^{\dagger}\stackrel{{\scriptstyle\leftrightarrow}}{{\nabla_{k}}}c)-i(\nabla_{j}V)\epsilon_{jkl}\epsilon_{l\alpha\beta}[c^{\dagger}\sigma_{\beta}\stackrel{{\scriptstyle\leftrightarrow}}{{\nabla_{i}}}(\nabla_{k}c)-(\nabla_{k}c^{\dagger})\sigma_{\beta}\stackrel{{\scriptstyle\leftrightarrow}}{{\nabla_{i}}}c]\biggr{]}$
(78)

are the spin-orbit contributions to the momentum flux density and force
density, respectively.

The contribution of the kinetic term, $\hat{\pi}^{K,\alpha}_{ij}$, is
similarly calculated, and the total momentum flux density operator is

$\displaystyle\hat{\pi}^{K,\alpha}_{ij}+\hat{\pi}^{{\rm
so},\alpha}_{ij}({\bm{r}},t)$
$\displaystyle=c^{\dagger}\biggl{[}\frac{1}{m}\hat{p}_{i}\hat{p}_{j}\sigma_{\alpha}-\lambda(\nabla_{k}V)\epsilon_{jk\alpha}\hat{p}_{i}+\frac{1}{2}\lambda(\nabla_{i}\nabla_{k}V)(\delta_{j\alpha}\delta_{k\beta}-\delta_{j\beta}\delta_{k\alpha})\sigma_{\beta}\biggr{]}c$
$\displaystyle\equiv\hat{\pi}^{s,\alpha}_{ij}$ (79)

whose expectation value is Eqs. (29)(30).

The fact that the time-derivative of the spin-resolved momentum density is not
written in terms of conserved current in the presence of spin relaxation, in
the same way as the case of spin current. This means that the momentum flux
density can not be defined uniquely. In other words, spin-resolved momentum
density is not physical observable. In this paper, we use the form of Eq.
(77). Different definitions result in different values of viscosity constants.
We note, however, that physical quantity like spin density are uniquely
defined like in the spin current case [14].

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