#  Calculation of magnon drag force induced by an electric current in
ferromagnetic metals

Hiroshi Funaki  Kavli Institute for Theoretical Sciences, University of
Chinese Academy of Sciences, Beijing, 100190, China  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

Magnon drag effect induced by an applied electric field in ferromagnetic
metals is theoretically studied by a microscopic calculation of the force on
magnons arising from magnon emission/absorption and scattering due to driven
electrons. It is shown that magnon scattering contribution dominates over the
emission/absorption one in a wide temperature regime in good metals with long
elastic lifetime $\tau$, as the latter has a relative suppression factor of
$(\Delta\tau)^{-2}$ due to the electron spin flip by the magnon, where
$\Delta$ is the $sd$ exchange interaction energy. Spin-transfer efficiency is
discussed including the magnon drag effect.

##  I Introduction

Magnon or spin wave is an elementary excitation in magnetic systems, which are
expected to be useful in spintronics for information transport and processing
different from the conventional electronics [1]. An obvious advantage of using
magnons is the fact that they exist even in insulators, where conduction
electrons are absent. Spin-transfer effect due to magnon current was
theoretically argued in the context of domain wall motion in Ref. [2]. Magnons
are, however, not easy to control and to detect, as it does not have direct
coupling to an electric field. Thermal driving by applying a temperature
gradient is therefore a common method for inducing magnon flow like in the
spin Seebeck effect [3]. Unlike conduction electrons, magnon effects are
generally temperature-dependent owing to a Bose distribution function for
magnon excitation, but the effects are not easy to separate from other bosonic
origins such as phonons with similar character. Responses to an external
magnetic field were used to identify the magnon contribution [4]. Magnon drag
effect was experimentally identified by use of a thermopile structure to
cancel nonmagnetic origins in Ref. [5].

In ferromagnetic metals, conduction electron spin is polarized due to strong
$sd$ exchange interaction to the magnetization, as suggested by large
magnetoresistance in magnetic multilayers and high efficiency of the spin-
transfer effect for domain walls. Strong $sd$ exchange interaction indicates
that the system is a strongly correlated fluid of magnon and electron with
spin up and down, where a significant magnon drag effect by the electron flow
and vice versa are expected. In 1976, magnon-drag Peltier effect, i.e., the
energy current due to magnon flow when an electric field is applied, was
phenomenologically argued and magnon drift velocity was found from
experimental data to be $v_{\rm m}=\mu_{\rm m/e}v_{\rm e}$ with a constant
$\mu_{\rm m/e}=2-3$, where $v_{\rm e}$ is the electron drift velocity [4]. It
is striking that the magnons are driven so efficiently by an electric field
even without a direct coupling. Magnon-drag effect was studied by use of a
microscopic diagrammatic method in Ref. [6]. The magnon velocity was
calculated taking account of the magnon emission and absorption due to the sd
exchange interaction and an applied electric field, and an expression
$\mu_{\rm m/e}\propto P\eta/(\alpha_{\rm G}\Delta)$ was obtained for the
velocity ratio, where $P$ denotes the spin polarization of conduction
electron, $\eta$ is the inverse electron elastic lifetime, $\Delta$ is the
electron spin polarization energy and $\alpha_{\rm G}$ is the (Gilbert)
damping constant for magnon. The electric current induced by the magnon
current driven by a temperature gradient was calculated and argued in the
context of magnonic spin-motive force in Ref. [7].

The mechanism considered in Refs. [4, 6], the magnon emission and absorption,
is the following process. When the conduction electron has a drift velocity
opposite to the applied electric field, a forward emission of a magnon and an
absorption of a backward propagating magnon occurs (Fig. 1). The transferred
momentum from or to the magnon contributes to a drag force. Magnon carries a
spin of $-1$, and thus an absorption and emission flips the electron spin. The
absorption/emission event therefore results in a high energy excitation of the
order of $2\Delta$ for the electron at the Fermi energy, unless the magnon
wave vector matches the difference of the Fermi wave vectors, $k_{F+}-k_{F-}$
for spin $\pm$. The electron amplitude for the force has therefore a
suppression factor of $(\Delta\tau)^{-n}$ ($n\geq 1$), where $n$ turns out to
be 1 (Eq. (27)), $\tau$ being the electron elastic lifetime. Being a single
magnon process, the magnon amplitude for the emission/absorption turns out to
depend on the temperature $T$ as $T^{\frac{5}{2}}$. Besides the emission and
absorption, electron drift causes a magnon scattering. The scattering
conserves the electron spin, and the electron stays near the Fermi energy, and
thus there is no electron suppression factor for the scattering process. The
scattering is a two magnon process, whose amplitude turns out to be $\propto
T^{4}$ and is weaker at low temperatures. The relative strength of the
emission/absorption and scattering contribution to the force is given by
$\Gamma_{{\rm ea}/{\rm sc}}\propto
S^{-1}(\Delta\tau)^{-2}({k_{B}}T/\tilde{J})^{-\frac{3}{2}}$ (Eq. (43)), where
$\tilde{J}$ is the ferromagnetic exchange energy, $S$ is the magnitude of
localized spin and ${k_{B}}$ is the Boltzmann constant. The factor of $S$ is
due to the fact that emission/absorption and absorption processes corresponds
to the first and the second order of the $1/S$-expansion for the magnons. The
emission/absorption process therefore dominates for ${k_{B}}T\lesssim
S^{-\frac{2}{3}}(\Delta\tau)^{-\frac{4}{3}}\tilde{J}$, which correspond to a
very low temperature for good metals with a long elastic lifetime.

Figure 1: Diagrammatic representation of the magnon emission, absorption and
scattering. Solid line is the electron with spin labeled by $\sigma=\pm$ and
wavy line is the magnon.

Besides the Peltier effect, magnon drag effect contributes to other transport
effects. In fact, magnon velocity contributes to a temperature-dependent spin-
transfer effect, and magnon drag force is directly detected as a contribution
to electric conductivity. Close analysis of experimental data on various
transport effects would make possible the separation of the magnon drag effect
from other effects. The aim of the paper is to provide a comprehensive theory
of magnon drag effects on spin-transfer effect and resistivity by calculating
the drag force microscopically using the approach of Ref. [8, 9] taking
account of the magnon scattering effect. It turns out that the magnon
scattering effect for the magnon drag force and spin-transfer effect is larger
than the emission/absorption effect in the wide temperature range as we argued
above.

Our result for the magnon emission/absorption contribution is $\mu_{\rm
m/e}\propto\eta/(\alpha_{\rm G}\Delta)$ (the first term of Eq. (54)), and is
different from the result of Ref. [6], which is proportional to $P$. Comparing
the analyses of Ref. [6] and ours, the difference appears to arose from
perhaps an insufficient treatment in Ref. [6] of electron (lesser) and hole
(greater) contributions of the vertex function $\Lambda$ (Eq. (99)).
Physically, the absence of $P$ in the magnon velocity appears natural as
driven electrons with both spin contribute to the forward motion of magnons.
In fact, up spin electron, which can only absorb a magnon, transfers positive
momentum to magnons by absorbing magnons moving backward relative to the
electron, while down spin electron pushes magnons forward by emitting magnons
forward. The transferred momentum and force in both cases are positive and
would not change sign by a reversal of spin polarization of the electrons.

##  II Model

The system we study is the conduction electron coupling to the localized spin
(magnetization) by the $sd$ exchange interaction. The localized spin
Hamiltonian we consider is the one with an exchange interaction and an easy-
axis anisotropy energy, represented by the strength $J$ and $K$, respectively:

$\displaystyle H_{S}$
$\displaystyle=\frac{1}{2a^{3}}\int\\!{d^{3}r}\left[J(\nabla{{\bm{S}}})^{2}-KS_{z}^{2}\right]$
(1)

where $a$ is the lattice constant. We consider the case where the localized
spin ${{\bm{S}}}$ is polarized along the $z$ axis. The fluctuation around the
average, the magnon, is taken into account using the Holstein-Primakov
expansion as

$\displaystyle{{\bm{S}}}=\left(\begin{array}[]{c}\sqrt{\frac{S}{2}}(b+b^{\dagger})\\\
\sqrt{\frac{S}{2}}(-i)(b-b^{\dagger})\\\
S-b^{\dagger}b\end{array}\right)+O(b,b^{\dagger})^{3}$ (5)

The magnon energy for a wave vector ${\bm{p}}$ reads

$\displaystyle\omega_{p}\equiv JSp^{2}+KS$ (6)

The current density of magnon is

$\displaystyle\bm{j}_{\rm m}$
$\displaystyle=-i{JS}b^{\dagger}\stackrel{{\scriptstyle\leftrightarrow}}{{\nabla}}b$
(7)

and momentum density is ${\bm{p}}_{\rm
m}=-\frac{i}{2}b^{\dagger}\stackrel{{\scriptstyle\leftrightarrow}}{{\nabla}}b$.

The $sd$ exchange interaction Hamiltonian is

$\displaystyle H_{sd}$
$\displaystyle=-\frac{\Delta}{S}\int\\!{d^{3}r}{{\bm{S}}}\cdot(c^{\dagger}{\bm{\sigma}}c)$
(8)

where $\Delta$ is the $sd$ coupling energy and $S\equiv|{{\bm{S}}}|$. The
interaction in terms of magnon field reads $H_{sd}=H_{sd}^{(1)}+H_{sd}^{(2)}$,
where

$\displaystyle H_{sd}^{(1)}$
$\displaystyle=-\frac{\Delta}{\sqrt{2S}}\int\\!{d^{3}r}\left[bc^{\dagger}\sigma_{-}c+b^{\dagger}c^{\dagger}\sigma_{+}c\right]$
$\displaystyle H_{sd}^{(2)}$
$\displaystyle=\frac{\Delta}{S}\int\\!{d^{3}r}b^{\dagger}bc^{\dagger}\sigma_{z}c$
(9)

represent magnon emission/absorption and scattering, respectively.

Force on magnon induced by the electron is calculated by evaluating
$\dot{{\bm{p}}}_{\rm m}$ taking account of $H_{sd}$. The force due to the
linear order of the $sd$ exchange interaction,
${F}^{(1)}_{i}\equiv\frac{1}{2}[H_{sd}^{(1)},b^{\dagger}\stackrel{{\scriptstyle\leftrightarrow}}{{\nabla}}_{i}b]$
is

$\displaystyle{F}^{(1)}_{i}({\bm{r}})$
$\displaystyle=-\frac{\Delta}{2\sqrt{2S}}\left[(\nabla_{i}b^{\dagger})(c^{\dagger}\sigma_{+}c)-b^{\dagger}\nabla_{i}(c^{\dagger}\sigma_{+}c)+(\nabla_{i}b)(c^{\dagger}\sigma_{-}c)-b\nabla_{i}(c^{\dagger}\sigma_{-}c)\right]$
(10)

Defining Fourier transform as
$c({\bm{r}})=\sum_{{\bm{k}}}e^{i{\bm{k}}\cdot{\bm{r}}}c_{\bm{k}}$,
$b({\bm{r}})=\sum_{{\bm{p}}}e^{i{\bm{p}}\cdot{\bm{r}}}b_{\bm{p}}$,
${F}^{(1)}_{i}({\bm{r}})\equiv\sum_{\bm{q}}e^{i{\bm{q}}\cdot{\bm{r}}}{F}^{(1)}_{i}({\bm{q}})$,
the momentum representation is

$\displaystyle{F}^{(1)}_{i}({\bm{q}})$
$\displaystyle=-i\frac{\Delta}{2\sqrt{2S}}\sum_{{\bm{k}}{\bm{k}}^{\prime}{\bm{p}}}(k^{\prime}-k+p)_{i}\left[b_{-{\bm{p}}}^{\dagger}(c_{{\bm{k}}^{\prime}}^{\dagger}\sigma_{+}c_{{\bm{k}}})+b_{{\bm{p}}}(c_{{\bm{k}}^{\prime}}^{\dagger}\sigma_{-}c_{{\bm{k}}})\right]_{{\bm{q}}=-{\bm{k}}^{\prime}+{\bm{k}}+{\bm{p}}}$
(11)

The uniform component of the force is

$\displaystyle{F}^{(1)}_{i}(0)$
$\displaystyle=i\frac{\Delta}{\sqrt{2S}}\sum_{{\bm{k}}{\bm{p}}}p_{i}\left[b_{{\bm{p}}}^{\dagger}(c_{{\bm{k}}-{\bm{p}}}^{\dagger}\sigma_{+}c_{{\bm{k}}})+b_{-{\bm{p}}}(c_{{\bm{k}}-{\bm{p}}}^{\dagger}\sigma_{-}c_{{\bm{k}}})\right]$
(12)

showing that spin down electron emitting a magnon of momentum ${\bm{p}}$ and
spin up electron absorbing magnon of $-{\bm{p}}$ result in a ’positive’ force
on magnon (Fig. 1).

The force due to the magnon scattering contribution is

$\displaystyle{F}^{(2)}_{i}({\bm{r}})$
$\displaystyle=-\frac{\Delta}{S}\nabla_{i}(c^{\dagger}\sigma_{z}c)b^{\dagger}b$
(13)

indicating that the force arises from the compression of the electron spin
density. The uniform component in the Fourier representation is

$\displaystyle{F}^{(2)}_{i}({\bm{q}}=0)$
$\displaystyle=-i{J_{sd}}\sum_{{\bm{k}}{\bm{p}}{\bm{p}}^{\prime}}(p^{\prime}-p)_{i}b_{{\bm{p}}^{\prime}}^{\dagger}b_{{\bm{p}}}(c_{{\bm{k}}^{\prime}}^{\dagger}\sigma_{z}c_{{\bm{k}}})|_{{\bm{k}}^{\prime}={\bm{k}}+{\bm{p}}-{\bm{p}}^{\prime}}$
(14)

##  III Linear response calculation of magnon force

We calculate the expectation value of magnon force as a linear response to an
applied electric field, represented by use of a vector potential, ${\bm{A}}$,
where ${\bm{E}}=-\dot{{\bm{A}}}$. The coupling to the electric current is
represented by an interaction Hamiltonian

$\displaystyle H_{\rm em}$
$\displaystyle=-\sum_{{\bm{k}}{\bm{q}}}\frac{e}{m}\left({\bm{A}}({\bm{q}},t)\cdot{\bm{k}}\right)c_{{\bm{k}}+\frac{{\bm{q}}}{2}}^{\dagger}c_{{\bm{k}}-\frac{{\bm{q}}}{2}}$
(15)

where $e$ is the electron charge.

###  III.1 Magnon emission/absorption contribution

In this subsection, the magnon force due to emission and absorption is
calculated to the lowest (the second) order in the $sd$ exchange interaction.
We focus on the uniform component of the force. Using the path-ordered (non-
equilibrium or Keldysh) Green’s function, $g_{{\bm{k}}}(t,t^{\prime})$
(${\bm{k}}$ is the electron wave vector), defined for time on a path
$C=C_{\rightarrow}+C_{\leftarrow}$ (See Appendix A), the expectation value of
the force is written in terms of the lesser component for the electron
contribution as (diagrams shown in Fig. 2)

$\displaystyle{F}^{(1)}_{i}(t,{\bm{q}})=$
$\displaystyle\frac{e{\Delta}^{2}}{2Sm}\sum_{{\bm{k}}{\bm{k}}^{\prime}{\bm{p}}}\int\\!\frac{d\Omega}{2\pi}A_{j}(\Omega)p_{i}k_{j}\int_{C}dt_{1}\int_{C}dt_{2}{\rm
tr}$ $\displaystyle\biggl{[}e^{-i\Omega
t_{2}}\sigma_{-}\Pi_{{\bm{k}}}^{(+),({\bm{k}}-{\bm{p}},{\bm{p}})}(t,t_{1})\sigma_{+}g_{{\bm{k}}}(t_{1},t_{2})g_{{\bm{k}}}(t_{2},t^{\prime})$
$\displaystyle-e^{-i\Omega
t_{1}}\sigma_{+}g_{{\bm{k}}}(t,t_{1})g_{{\bm{k}}}(t_{1},t_{2})\sigma_{-}\Pi_{{\bm{k}}}^{(+),({\bm{k}}-{\bm{p}},{\bm{p}})}(t_{2},t^{\prime})$
$\displaystyle+e^{-i\Omega
t_{1}}\sigma_{-}g_{{\bm{k}}}(t,t_{1})g_{{\bm{k}}}(t_{1},t_{2})\sigma_{+}\Pi_{{\bm{k}}}^{(-),({\bm{k}}+{\bm{p}},{\bm{p}})}(t_{2},t^{\prime})$
$\displaystyle-e^{-i\Omega
t_{2}}\sigma_{+}\Pi_{k}^{(-),({\bm{k}}+{\bm{p}},{\bm{p}})}(t,t_{1})\sigma_{-}g_{{\bm{k}}}(t_{1},t_{2})g_{{\bm{k}}}(t_{2},t^{\prime})\biggr{]}$
(16)

Here the time $t$ is on the path $C_{\rightarrow}$ while $t^{\prime}$ is on
the path $C_{\leftarrow}$, and they correspond to the same real time $t$. We
used the fact that the magnon operator at the force vertex can be at either
$t$ or $t^{\prime}$. The magnon-electron composite propagators are defined as
(Fig. 3)

$\displaystyle\Pi_{k}^{(+),(k-p,p)}(t,t^{\prime})$ $\displaystyle\equiv
ig_{k-p}(t,t^{\prime})d_{p}(t,t^{\prime})$
$\displaystyle\Pi_{k}^{(-),(k+p,p)}(t,t^{\prime})$ $\displaystyle\equiv
ig_{k+p}(t,t^{\prime})d_{p}(t^{\prime},t)$ (17)

corresponding to the propagation in the same ($+$) and opposite ($-$)time
direction, respectively, where $d_{p}(t,t^{\prime})$ is the magnon Green’s
function. The imaginary factor $i$ is for $\Pi_{k}^{(\pm),(k-p,p)}$ to have
the same behavior like a single Green’s function, such as the lesser and
greater components are pure imaginary.

Figure 2:  Feynman diagrams for the force on magnon due to emission and
absorption. The external wave vector, ${\bm{q}}$, is zero for the uniform
component we consider.

Figure 3: Diagrammatic representation of the two magnon-electron pair
propagators. Electron spin is $+$ for a magnon-electron pair and $-$ for a
magnon hole-electron pair.

Writing in terms of the real-time Green’s functions using properties of the
Green’s functions and composite propagators described in Appendix A, the force
reads

$\displaystyle{F}^{(1)}_{i}=$
$\displaystyle\frac{e\Delta^{2}}{2Sm}\sum_{{\bm{k}}{\bm{k}}^{\prime}{\bm{p}}}\int\\!\frac{d\Omega}{2\pi}\int\\!\frac{d\omega}{2\pi}A_{j}(\Omega)p_{i}k_{j}{\rm
tr}\biggl{[}$ $\displaystyle[\sigma_{-}\Pi_{k\omega}^{(+),(k-p,p),{\rm
r}}\sigma_{+}g_{k\omega}^{\rm r}g^{<}_{k,\omega-\Omega}+({\rm r}<{\rm
a})+(<{\rm a}{\rm a})]$ $\displaystyle-[\sigma_{+}g_{k,\omega+\Omega}^{\rm
r}g_{k\omega}^{\rm r}\sigma_{-}\Pi_{k\omega}^{(+),(k-p,p),<}+({\rm r}<{\rm
a})+(<{\rm a}{\rm a})]$
$\displaystyle-[\sigma_{+}\Pi_{k\omega}^{(-),(k+p,p),{\rm
r}}\sigma_{-}g_{k\omega}^{\rm r}g_{k,\omega-\Omega}^{<}+({\rm r}<{\rm
a})+(<{\rm a}{\rm a})]$ $\displaystyle+[\sigma_{-}g_{k,\omega+\Omega}^{\rm
r}g_{k\omega}^{\rm r}\sigma_{+}\Pi_{k\omega}^{(-),(k+p,p),<}+({\rm r}<{\rm
a})+(<{\rm a}{\rm a})]\biggr{]}$ (18)

In the case of static electric field ($\Omega\rightarrow 0$), the Fermi
surface contribution, which is dominant, is obtained using the relation (70)
as

$\displaystyle{F}^{(1)}_{i}=$
$\displaystyle\frac{e\Delta^{2}}{2Sm}\sum_{{\bm{k}}{\bm{k}}^{\prime}{\bm{p}}}\int\\!\frac{d\Omega}{2\pi}\int\\!\frac{d\omega}{2\pi}\Omega
A_{j}(\Omega)f^{\prime}(\omega)p_{i}k_{j}{\rm tr}\biggl{[}$
$\displaystyle\left[\sigma_{-}\Pi_{k\omega}^{(+),(k-p,p),{\rm
r}}\sigma_{+}g_{k\omega}^{\rm r}g^{\rm
a}_{k,\omega}-\sigma_{+}g_{k,\omega}^{\rm r}g_{k,\omega}^{\rm
a}\sigma_{-}\Pi_{k\omega}^{(+),(k-p,p),{\rm a}}\right]$
$\displaystyle-\left[\sigma_{+}\Pi_{k\omega}^{(-),(k+p,p),{\rm
r}}\sigma_{-}g_{k\omega}^{\rm r}g_{k,\omega}^{\rm
a}-\sigma_{-}g_{k,\omega}^{\rm r}g_{k\omega}^{\rm
a}\sigma_{+}\Pi_{k\omega}^{(-),(k+p,p),{\rm a}}\right]\biggr{]}$ (19)

where $f(\omega)\equiv[e^{\beta\omega}+1]^{-1}$,
$\beta\equiv({\bm{k}}T)^{-1}$, ${k_{B}}$ and $T$ being the Boltzmann constant
and temperature, respectively. At low temperature,
$f^{\prime}(\omega)\simeq-\delta(\omega)$, and we obtain uniform component as
${F}^{(1)}_{i}={\cal F}^{(1)}_{ij}eE_{j}$
($E=\int\frac{d\Omega}{2\pi}e^{-i\Omega t}i\Omega A_{\Omega}$), where the
coefficient is ($g_{{\bm{k}}}\equiv g_{{\bm{k}},\omega=0}$)

$\displaystyle{\cal F}^{(1)}_{ij}=$ $\displaystyle-\frac{\Delta^{2}}{4\pi
Sm}\sum_{{\bm{k}}{\bm{k}}^{\prime}{\bm{p}}}p_{i}k_{j}{\rm Im}{\rm
tr}\biggl{[}\sigma_{-}|g_{k}^{\rm
a}|^{2}\sigma_{+}\Pi_{k\omega=0}^{(-),(k+p,p),{\rm a}}-\sigma_{+}|g_{k}^{\rm
a}|^{2}\sigma_{-}\Pi_{k\omega=0}^{(+),(k-p,p),{\rm a}}\biggr{]}$
$\displaystyle=$ $\displaystyle-\frac{\Delta^{2}}{2\pi
Sm}\sum_{{\bm{k}}{\bm{k}}^{\prime}{\bm{p}}}p_{i}k_{j}\sum_{\pm}(\pm)|g_{k\pm}^{\rm
a}|^{2}{\rm Im}[\Pi_{k,\mp}^{(\mp),(k\pm p,p),{\rm a}}]$ (20)

where $\Pi_{k,\mp}$ denotes $\Pi_{k,\omega=0}$ with electron spin $\mp$.
Writing the imaginary part of the pair propagator by use of magnon propagator
(Eq. (73)), we obtain

$\displaystyle{\cal F}^{(1)}_{ij}=$ $\displaystyle\frac{2\Delta^{2}}{\pi
Sm}\sum_{{\bm{k}}{\bm{k}}^{\prime}{\bm{p}}}p_{i}k_{j}\sum_{\nu}n_{\nu}(1-f_{\nu})\sum_{\pm}{\rm
Im}[d^{\rm a}_{p,\mp\nu}]|g_{k\pm}^{\rm a}|^{2}{\rm Im}[g_{k\pm
p,-\nu,\mp}^{\rm a}]$ (21)

where $\nu$ is the magnon frequency, $\sum_{\nu}=\int\frac{d\nu}{2\pi}$ and
$n_{\nu}\equiv[e^{\beta\nu}-1]^{-1}$ is the Bose distribution function. The
magnon Green’s function at weak damping is

$\displaystyle{\rm Im}[d^{\rm a}_{p,\mp\nu}]=$ $\displaystyle{\rm
Im}\frac{1}{\mp\nu-\omega_{p}-i\eta_{\rm
m}}\simeq\pi\delta(\mp\nu-\omega_{p})$ (22)

where $\eta_{\rm m}$ represents damping of magnon, which is $\eta_{\rm
m}=\alpha_{\rm G}\omega_{p}$ in terms of the Gilbert damping constant
$\alpha_{\rm G}$. To proceed, we consider the case where magnon dynamics is
slow compared to the electron one, namely, frequency $\nu$ in electron Green’s
functions is treated as zero. The approximation assumes therefore that
$\omega_{\rm m}\tau\ll 1$, where $\omega_{\rm m}$ is typical magnon energy. In
this case, we obtain the sum of $\nu$ as $\sum_{\nu}n_{\nu}(1-f_{\nu}){\rm
Im}[d^{\rm a}_{p,\mp\nu}]=\mp\frac{1}{2}n_{\omega_{p}}(1-f_{\omega_{p}})$,
where we used $n_{-\nu}+f_{-\nu}=-(n_{\nu}+f_{\nu})=-2n_{\nu}(1-f_{\nu})$. We
thus have

$\displaystyle{\cal F}^{(1)}_{ij}=$ $\displaystyle-\frac{\Delta^{2}}{\pi
Sm}\sum_{{\bm{k}}{\bm{k}}^{\prime}{\bm{p}}}p_{i}k_{j}n_{\omega_{p}}(1-f_{\omega_{p}})\sum_{\pm}(\pm)|g_{k\pm}^{\rm
a}|^{2}{\rm Im}[g_{k\pm p,0,\mp}^{\rm a}]$ (23)

The wave vector for magnon is also assumed to be small, $p\ll{k_{F}}$. The
expression in this case becomes, using $g_{k\pm p,-\nu,\mp}^{\rm a}\simeq
g_{k\pm p,\mp}^{\rm a}\simeq g_{k\mp}^{\rm
a}\pm\frac{{\bm{k}}\cdot{\bm{p}}}{m}(g_{k\mp}^{\rm a})^{2}$,

$\displaystyle{\cal F}^{(1)}_{ij}=$ $\displaystyle-\frac{\Delta^{2}}{\pi
Sm^{2}}\sum_{{\bm{k}}{\bm{p}}}p_{i}k_{j}({\bm{k}}\cdot{\bm{p}})n_{\omega_{p}}(1-f_{\omega_{p}})\sum_{\pm}|g_{k\pm}^{\rm
a}|^{2}{\rm Im}(g_{k\mp}^{\rm a})^{2}$ $\displaystyle=$
$\displaystyle\delta_{ij}\overline{{\cal F}}^{(1)}$
$\displaystyle\overline{{\cal F}}^{(1)}=$
$\displaystyle-\frac{\Delta^{2}}{3\pi
Sm^{2}}\sum_{{\bm{p}}}p^{2}n_{\omega_{p}}(1-f_{\omega_{p}})\sum_{{\bm{k}}}\sum_{\pm}k^{2}|g_{k\pm}^{\rm
a}|^{2}{\rm Im}(g_{k\mp}^{\rm a})^{2}$ (24)

The electron part is estimated by use of contour integration with respect to
electron energy as (derivation in Appendix B)

$\displaystyle{\rm Im}\sum_{{\bm{k}}\sigma}k^{2}|g_{k\sigma}^{\rm
a}|^{2}(g_{k,-\sigma}^{\rm a})^{2}$
$\displaystyle=-\frac{9\pi}{8}\frac{ma^{3}\overline{n_{\rm
e}}}{\Delta^{2}{\epsilon_{F}}}$ (25)

where $\overline{n_{\rm e}}=\sum_{\sigma}{n_{\rm e}}_{\sigma}$ and ${n_{\rm
e}}_{\sigma}\equiv\frac{1}{3ma^{3}}\sum_{\sigma}\sigma{{\nu_{\rm
e}}_{\sigma}k_{\sigma}^{2}}$ are the total and spin-resolved electron density,
respectively, ${\nu_{\rm e}}_{\sigma}$ and $k_{\sigma}$ being the electron
density of states at the Fermi energy and the Fermi wave length for spin
$\sigma$, respectively. The force coefficient is

$\displaystyle\overline{{\cal F}}^{(1)}=$
$\displaystyle\frac{3}{8Sma^{2}}\frac{a^{3}\overline{n_{\rm
e}}}{{\epsilon_{F}}}W^{(1)}(T)$ (26) $\displaystyle W^{(1)}(T)\equiv$
$\displaystyle a^{2}\sum_{{\bm{p}}}p^{2}n_{\omega_{p}}(1-f_{\omega_{p}})$ (27)

###  III.2 Magnon scattering contribution

The force due to the magnon scattering, ${F}^{(2)}_{i}$, is similarly
calculated. The uniform component of the linear response at the second order
in the $sd$ exchange interaction reads

$\displaystyle{F}^{(2{\rm a})}_{i}(t)=$
$\displaystyle-\frac{e\Delta^{2}}{S^{2}m}\int\\!\frac{d\Omega}{2\pi}{\bm{A}}_{j}(q,\Omega)\sum_{{\bm{k}}{\bm{k}}^{\prime}{\bm{p}}}(p-p^{\prime})_{i}k_{j}\int_{C}dt_{1}\int_{C}dt_{2}{\rm
tr}$ $\displaystyle\biggl{[}e^{i\Omega
t_{2}}\sigma_{z}\Pi^{(2),k^{\prime},p,p^{\prime}}_{k}(t,t_{1})\sigma_{z}g_{{\bm{k}}}(t_{1},t_{2})g_{{\bm{k}}}(t_{2},t^{\prime})+e^{i\Omega
t_{2}}\sigma_{z}g_{{\bm{k}}}(t,t_{1})g_{{\bm{k}}}(t_{1},t_{2})\sigma_{z}\Pi^{(2),k^{\prime},p^{\prime},p}_{{\bm{k}}}(t_{2},t^{\prime})\biggr{]}_{k^{\prime}=k+p-p^{\prime}}$
(28)

where a composite propagator of electron and two magnons is defined as (Fig.
4)

$\displaystyle\Pi^{(2),k,p,p^{\prime}}_{k+p-p^{\prime}}(t,t^{\prime})$
$\displaystyle=g_{{\bm{k}}}(t,t^{\prime})d_{{\bm{p}}}(t,t^{\prime})d_{{\bm{p}}^{\prime}}(t^{\prime},t)$
(29)

Figure 4:  Feynman diagrams for the two magnon and electron composite
propagator.

The leading Fermi surface term in the limit of $\Omega\rightarrow 0$ is

$\displaystyle{F}^{(2{\rm a})}_{i}=$
$\displaystyle-\frac{e\Delta^{2}}{S^{2}m}\int\\!\frac{d\Omega}{2\pi}\int\\!\frac{d\omega}{2\pi}\Omega{\bm{A}}_{j}(\Omega)\sum_{{\bm{k}}{\bm{k}}^{\prime}{\bm{p}}}(p-p^{\prime})_{i}k_{j}f^{\prime}(\omega){\rm
tr}$ $\displaystyle\biggl{[}\sigma_{z}\Pi^{(2),k^{\prime},p,p^{\prime},{\rm
r}}_{k}(\omega)\sigma_{z}g_{{\bm{k}}}^{\rm r}(\omega)g_{{\bm{k}}}^{\rm
a}(\omega)+\sigma_{z}g_{{\bm{k}}}^{\rm r}(\omega)g_{{\bm{k}}}^{\rm
a}(\omega)\sigma_{z}\Pi^{(2),k^{\prime},p^{\prime},p,{\rm
a}}_{{\bm{k}}}(\omega)\biggr{]}_{k^{\prime}=k+p-p^{\prime}}$ (30)

As the Green’s functions are diagonal in spin, the two Pauli matrix
$\sigma_{z}$ becomes irrelevant. Namely, the scattering force is a response of
electron charge sector. The effect is similar to the $z$-component of the
electron magnetic susceptibility in the presence of a magnetization along $z$
direction. We obtain finally (see Eq. (82))

$\displaystyle{F}^{(2{\rm a})}_{i}$ $\displaystyle={\cal F}^{(2{\rm
a})}_{ij}eE_{j}$ $\displaystyle{\cal F}^{(2{\rm a})}_{ij}$
$\displaystyle=\frac{\Delta^{2}}{\pi
S^{2}m}\sum_{{\bm{p}}{\bm{p}}^{\prime}}\sum_{{\bm{k}}\sigma}(p-p^{\prime})_{i}k_{j}{\rm
tr}\biggl{[}{\rm Im}(\Pi^{(2),k+p-p^{\prime},p,p^{\prime},{\rm
a}}_{k,\sigma,\omega=0})|g_{{\bm{k}}\sigma}^{\rm a}|^{2}\biggr{]}$
$\displaystyle=-2\frac{\Delta^{2}}{\pi
S^{2}m}\sum_{{\bm{p}}{\bm{p}}^{\prime}}\sum_{{\bm{k}}\sigma}(p-p^{\prime})_{i}k_{j}{\rm
Im}[d_{p\nu}^{\rm a}]{\rm Im}[d_{p^{\prime}\nu^{\prime}}^{\rm
a}]n_{\nu}(1+n_{\nu^{\prime}})f_{\nu^{\prime}-\nu}{\rm
Im}[g_{k+p-p^{\prime}}^{\rm a}]|g_{{\bm{k}}\sigma}^{\rm a}|^{2}$ (31)

Figure 5:  Feynman diagrams for the two magnon and electron composite
propagator with magnon absorption and emission at different times. Electron
spin (denoted by $\pm$) is fixed for emission and absorption processes.

The scattering force has other contributions of higher-order in the $sd$
exchange interaction, with the emission and absorption vertices at different
time as in Fig. 5. This contribution is characterized by the composite
propagator

$\displaystyle\Pi_{k+p-p^{\prime}}^{(2+),(k,p,p^{\prime})}(t,t^{\prime})$
$\displaystyle=-i\frac{\Delta}{2}\int_{C}dt_{1}\Pi_{k+p}^{(+),(k,p)}(t,t_{1})g_{k+p,-}(t_{1},t^{\prime})d_{p^{\prime}}(t^{\prime},t)$
$\displaystyle\Pi_{k+p-p^{\prime}}^{(2-),(k,p,p^{\prime})}(t,t^{\prime})$
$\displaystyle=-i\frac{\Delta}{2}\int_{C}dt_{1}\Pi_{k-p^{\prime}}^{(-),(k,p^{\prime})}(t,t_{1})g_{k-p^{\prime},+}(t_{1},t^{\prime})d_{p}(t,t^{\prime})$
(32)

where factors are included to have the same normalization as $\Pi^{(2)}$. The
electron spin ($\pm$) is fixed for the two propagators, as the magnon
absorption and emission causes lowering and highering of electron spin,
respectively. Considering a strong $sd$ splitting case, electron spin flip
costs energy of $2\Delta$, and thus contributions with more electron Green’s
functions with flipped spin are suppressed and neglected. The electron
propagators with flipped spin inside the composite propagator of Eq. (32) is
approximated as
$g_{k+p,\pm}(t,t^{\prime})\sim\mp\delta(t-t^{\prime})(2\Delta)^{-1}$,
resulting in

$\displaystyle\Pi_{k+p-p^{\prime}}^{(2+),(k,p,p^{\prime})}(t,t^{\prime})$
$\displaystyle\simeq\frac{-i}{4}\Pi_{k+p,+}^{(+),(k,p)}(t,t^{\prime})d_{p^{\prime}}(t^{\prime},t)=\frac{1}{4}\Pi^{(2),k,p,p^{\prime}}_{k+p-p^{\prime},+}(t,t^{\prime})$
$\displaystyle\Pi_{k+p-p^{\prime}}^{(2-),(k,p,p^{\prime})}(t,t^{\prime})$
$\displaystyle\simeq\frac{i}{4}\Pi_{k-p^{\prime},-}^{(-),(k,p^{\prime})}(t,t^{\prime})d_{p}(t,t^{\prime})=-\frac{1}{4}\Pi^{(2),k,p,p^{\prime}}_{k+p-p^{\prime},-}(t,t^{\prime})$
(33)

The contribution to the force coefficient from the propagators of Fig. 5 is
therefore

$\displaystyle{\cal F}^{(2{\rm b})}_{ij}$
$\displaystyle=i\frac{\Delta^{2}}{\pi
S^{2}m}\sum_{{\bm{p}}{\bm{p}}^{\prime}}\sum_{{\bm{k}}\sigma}(p-p^{\prime})_{i}k_{j}\frac{\sigma}{4}{\rm
tr}\biggl{[}{\rm Im}(\Pi^{(2),k+p-p^{\prime},p,p^{\prime},{\rm
a}}_{k,\sigma,\omega=0})|g_{{\bm{k}}\sigma}^{\rm a}|^{2}\biggr{]}$ (34)

Using Eq. (82) and assuming $p,p^{\prime}\ll k$ and low frequency magnons, we
obtain the sum of the two contributions as

$\displaystyle{\cal F}^{(2)}_{ij}$ $\displaystyle\equiv{\cal F}^{(2{\rm
a})}_{ij}+{\cal F}^{(2{\rm b})}_{ij}=\delta_{ij}\overline{{\cal F}}^{(2)}$
$\displaystyle\overline{{\cal F}}^{(2)}$
$\displaystyle=-\frac{\Delta^{2}}{3S^{2}m^{2}}\sum_{{\bm{p}}{\bm{p}}^{\prime}}({\bm{p}}-{\bm{p}}^{\prime})^{2}n_{\omega_{p}}(1+n_{\omega_{p^{\prime}}})f_{\omega_{p^{\prime}}-\omega_{p}}\sum_{{\bm{k}}\sigma}k^{2}\left(1+\frac{\sigma}{4}\right)|g_{{\bm{k}}\sigma}^{\rm
a}|^{2}{\rm Im}[(g_{{\bm{k}}\sigma}^{\rm a})^{2}]$ (35)

The summation over ${\bm{k}}$ and $\sigma$ is (using
$k_{i}(g_{{\bm{k}}\sigma}^{\rm
a})^{3}=\frac{m}{2}\partial_{k_{i}}(g_{{\bm{k}}\sigma}^{\rm a})^{2}$ and
integral by parts)

$\displaystyle\sum_{{\bm{k}}\sigma}k^{2}|g_{{\bm{k}}\sigma}^{\rm a}|^{2}{\rm
Im}[(g_{{\bm{k}}\sigma}^{\rm a})^{2}]$ $\displaystyle=-\pi
m\tau^{2}\sum_{\sigma}{\nu_{\rm e}}_{\sigma}$ (36)

We therefore obtain ($\overline{{\nu_{\rm e}}}\equiv\sum_{\sigma}{\nu_{\rm
e}}_{\sigma}$ and $P_{\nu}\equiv\frac{\sum_{\sigma}\sigma{\nu_{\rm
e}}_{\sigma}}{\sum_{\sigma}{\nu_{\rm e}}_{\sigma}}$)

$\displaystyle\overline{{\cal F}}^{(2)}=$
$\displaystyle\frac{\pi}{3S^{2}ma^{2}}{(\Delta\tau)}^{2}\overline{{\nu_{\rm
e}}}\left(1+\frac{P_{\nu}}{4}\right)W^{(2)}(T)$ (37) $\displaystyle
W^{(2)}(T)\equiv$ $\displaystyle
a^{2}\sum_{{\bm{p}}{\bm{p}}^{\prime}}({\bm{p}}-{\bm{p}}^{\prime})^{2}n_{\omega_{p}}(1+n_{\omega_{p^{\prime}}})f_{\omega_{p^{\prime}}-\omega_{p}}$
(38)

###  III.3 Force on electron

The recoil force on the electron with spin $\sigma$ arising from the magnon
emission/absorption due to the $sd$ exchange interaction is (in the field
operator form)

$\displaystyle{F}^{(1)}_{{\rm e}+,i}({\bm{q}}=0)$
$\displaystyle=i\frac{\Delta}{\sqrt{2S}}\sum_{{\bm{k}}{\bm{k}}^{\prime}{\bm{p}}}p_{i}b_{{\bm{p}}}(c_{{\bm{k}}^{\prime}}^{\dagger}\sigma_{-}c_{{\bm{k}}})_{{\bm{k}}^{\prime}={\bm{k}}+{\bm{p}}}$
$\displaystyle{F}^{(1)}_{{\rm e}-,i}({\bm{q}}=0)$
$\displaystyle=i\frac{\Delta}{\sqrt{2S}}\sum_{{\bm{k}}{\bm{k}}^{\prime}{\bm{p}}}p_{i}b_{-{\bm{p}}}^{\dagger}(c_{{\bm{k}}^{\prime}}^{\dagger}\sigma_{+}c_{{\bm{k}}})_{{\bm{k}}^{\prime}={\bm{k}}+{\bm{p}}}$
(39)

whose sum is opposite to the force on magnons; $\sum_{\sigma}{F}^{(1)}_{{\rm
e}\sigma,i}=-{F}^{(1)}_{{\rm m},i}$. It turns out that the force is spin-
independent, i.e.,

$\displaystyle{F}^{(1)}_{{\rm e}+{i}}={F}^{(1)}_{{\rm e}-{i}}=$
$\displaystyle-\frac{1}{2}{F}^{(1)}_{i}$ (40)

This result indicates that magnon emission and absorption induced by an
electric field does not acts as spin motive force but drives only charge
sector.

The force on electron spin arising from magnon scattering is calculated using

$\displaystyle{F}^{(2)}_{{\rm e}\sigma,i}$
$\displaystyle=\frac{\Delta}{S}[\nabla(b^{\dagger}b)]\sigma
c^{\dagger}_{\sigma}c_{\sigma}$ (41)

and its coefficient (${F}^{(2)}_{{\rm e}\sigma}\equiv eE{\cal F}^{(2)}_{{\rm
e}\sigma}$) is (from Eq. (35)),

$\displaystyle{\cal F}^{(2)}_{{\rm e}\sigma}$
$\displaystyle=\frac{{\Delta}^{2}}{6S^{2}m^{2}}\sum_{{\bm{p}}{\bm{p}}^{\prime}}({\bm{p}}-{\bm{p}}^{\prime})^{2}n_{\omega_{p}}(1+n_{\omega_{p^{\prime}}})f_{\omega_{p^{\prime}}-\omega_{p}}\left(1+\frac{\sigma}{4}\right)\sum_{{\bm{k}}}k^{2}|g_{{\bm{k}}\sigma}^{\rm
a}|^{2}{\rm Im}[(g_{{\bm{k}}\sigma}^{\rm a})^{2}]$ (42)

This force is generally spin-dependent.

Those forces on the electron are different from the driven-magnon contribution
to the motive force due to smooth magnetization structures discussed in Refs.
[10, 7]. In fact, the motive force in the adiabatic (slowly varying) limit are
proportional to the magnon energy current linear in the magnon momentum, while
the force argued in the present analysis are the second order of $p$ and
$p^{\prime}$, corresponding to nonadiabatic contributions.

##  IV Total force on magnon

From Eqs. (26) (38), the total force on magnon lowest order in the $sd$
exchange interaction, ${F}_{\rm m}\equiv{F}^{(1)}+{F}^{(2)}={\cal F}_{\rm
m}eE$, where the coefficient ${\cal F}_{\rm m}\equiv\overline{{\cal
F}}^{(1)}+\overline{{\cal F}}^{(2)}$ is

$\displaystyle{\cal F}_{\rm m}=$
$\displaystyle\gamma_{1}W^{(1)}(T)+\gamma_{2}(\Delta\tau)^{2}W^{(2)}(T)$ (43)

where

$\displaystyle\gamma_{1}$
$\displaystyle\equiv\frac{3}{8S}\gamma_{f}\overline{n_{\rm e}}a^{3}$
$\displaystyle\gamma_{2}$
$\displaystyle\equiv\frac{\pi}{3S^{2}}\gamma_{f}{\epsilon_{F}}\overline{{\nu_{\rm
e}}}(1+\tfrac{P_{\nu}}{4})$ $\displaystyle\gamma_{f}$
$\displaystyle\equiv\frac{1}{ma^{2}{\epsilon_{F}}}$ (44)

where the magnon weight factors are

$\displaystyle W^{(1)}(T)=$
$\displaystyle\frac{a^{5}}{2\pi^{2}}\int_{0}^{\infty}dpp^{4}\frac{1}{e^{2\beta\omega_{p}}-1}$
$\displaystyle W^{(2)}(T)=$
$\displaystyle\frac{a^{8}}{(2\pi^{2})^{2}}\int_{0}^{\infty}dp\int_{0}^{\infty}dp^{\prime}p^{2}(p^{\prime})^{2}(p^{2}+(p^{\prime})^{2})\frac{1}{e^{\beta\omega_{p}}-1}\frac{1}{1-e^{-\beta\omega_{p^{\prime}}}}\frac{1}{e^{\beta(\omega_{p^{\prime}}-\omega_{p})}+1}$
(45)

Considering strong spin polarization in 3$d$ ferromagnets, we may approximate
$\gamma_{1}\sim\gamma_{2}$, and then the magnon scattering contribution has a
larger coefficient by a factor of $(\Delta\tau)^{2}$ compared to the
emission/absorption contribution. Considering temperatures higher than the
magnon gap (${k_{B}}T\gg KS$) the weight factors in three dimensions are
($x\equiv\beta\omega_{p}$)

$\displaystyle W^{(1)}(T)=$ $\displaystyle
I_{f}^{(1)}\left(\frac{{k_{B}}T}{\tilde{J}}\right)^{\frac{5}{2}}$
$\displaystyle W^{(2)}(T)=$ $\displaystyle
I_{f}^{(2)}\left(\frac{{k_{B}}T}{\tilde{J}}\right)^{4}$ (46)

where $\tilde{J}\equiv JS/a^{2}$ and

$\displaystyle I_{f}^{(1)}$
$\displaystyle\equiv\frac{1}{4\pi^{2}}\int_{0}^{\infty}dx\frac{x^{\frac{3}{2}}}{e^{2x}-1}=7.985\times
10^{-3}$ $\displaystyle I_{f}^{(2)}$
$\displaystyle\equiv\frac{1}{(4\pi^{2})^{2}}\int\int_{0}^{\infty}dxdx^{\prime}\frac{x^{\frac{1}{2}}(x^{\prime})^{\frac{1}{2}}(x+x^{\prime})}{(e^{x}-1)(1-e^{-x^{\prime}})(e^{x^{\prime}-x}+1)}=1.069\times
10^{-2}$ (47)

The total force coefficient ${\cal F}_{\rm m}$ is plotted as function of
temperature in Fig. 6. The temperature is normalized by $\tilde{J}$, which is
related to the mean-field ferromagnetic transition temperature in three
dimensions as ${k_{B}}T_{\rm c}=2(S+1)\tilde{J}$ [11]. The temperature regime
in Fig. 6 thus corresponds to low temperature ($T\lesssim T_{\rm c}/4$ for
$S=1$). This is confirmed from the plot of the magnon number per cite (Fig.
6),

$\displaystyle n_{\rm
m}=a^{3}\int\frac{p^{2}dp}{2\pi^{2}}\frac{1}{e^{\beta\omega_{p}}-1}=I_{n}\left(\frac{{k_{B}}T}{\tilde{J}}\right)^{\frac{3}{2}}$
(48)

where

$\displaystyle
I_{n}\equiv\frac{1}{4\pi^{2}}\int_{0}^{\infty}dx\frac{\sqrt{x}}{e^{x}-1}=0.05864$
(49)

As is seen in Fig. 6, the scattering contribution (${F}^{(2)}$) dominates in
the wide temperature region, as a result of large factor $(\Delta\tau)^{2}$.
The crossover temperature from emission/absorption to scattering regime is
${k_{B}}T_{\rm ea-
sc}=\tilde{J}\left(\frac{\gamma_{1}I^{(1)}}{\gamma_{2}I^{(2)}}\right)^{\frac{2}{3}}(\Delta\tau)^{-\frac{4}{3}}$.
The crossover temperatures shall be discussed in Sec. IV.3.

Figure 6: The total force coefficient ${\cal F}_{\rm m}$ as function of
$\tilde{T}\equiv{k_{B}}T/\tilde{J}$ (denoted by $T$ in the $x$ axis) for
$\Delta\tau=2,10,20$ and $\gamma_{1}=\gamma_{2}=1$. The contribution ${\cal
F}^{(1)}$ is shown by a dashed line. The magnon number per cite $n$, plotted
on the right axis, indicates that the temperature regime is dilute magnon
regime.

The force on magnon resulting in the $sd$ exchange interaction means that the
opposite force acts on electrons. It reads,

$\displaystyle{\bm{{F}}}_{e}$
$\displaystyle\equiv-({\bm{{F}}}^{(1)}+{\bm{{F}}}^{(2)})=-e{{\cal F}}_{\rm
m}{\bm{E}}$ (50)

The coefficient ${\cal F}_{\rm m}$ thus corresponds to a reduction of the
applied electric field as ${\bm{E}}\rightarrow(1-{\cal F}_{\rm m}){\bm{E}}$.
The effect represents a resistance due to magnon scattering and emission. The
electric current taking account magnons is $\bm{j}=\sigma_{\rm e}(1-{\cal
F}_{\rm m}){\bm{E}}$ ($\sigma_{\rm e}$ is the Boltzmann conductivity). The
magnon contribution to the resistivity in the case of ${\cal F}_{\rm m}\ll 1$
is therefore

$\displaystyle\delta\rho_{\rm m}$ $\displaystyle=\rho_{0}{\cal F}_{\rm m}$
(51)

where $\rho_{0}=1/\sigma_{\rm e}$. The force on magnon is therefore directly
accessible by the resistivity measurement.

###  IV.1 Magnon velocity

When a force ${F}_{\rm m}$ acts on a magnon, the magnon is driven at a
velocity ${F}_{\rm m}\frac{\tau_{\rm m}}{m_{\rm m}}$, where $\tau_{\rm m}$ and
$m_{\rm m}$ are the magnon lifetime and mass. The lifetime is written in terms
of the Gilbert damping parameter $\alpha_{\rm G}$ and the frequency of the
magnon $\omega$ as $1/\tau_{\rm m}=\alpha_{\rm G}\omega$ and magnon mass in
the present case is $m_{\rm m}=(2JS)^{-1}$. The correctness of the above
argument is supported by a direct linear response calculation of the magnon
velocity (Appendix C). From our results of the forces, Eq. (43), the magnon
current induced by each force reads $\bm{j}_{\rm m}^{(1)}=\sigma_{\rm
m}^{(1)}e{\bm{E}}$ and $\bm{j}_{\rm m}^{(2)}=\sigma_{\rm m}^{(2)}e{\bm{E}}$,
where

$\displaystyle\sigma_{\rm m}^{(1)}=$
$\displaystyle\frac{2\gamma_{1}}{\alpha_{\rm
G}a}\left(\frac{{k_{B}}T}{\tilde{J}}\right)^{\frac{3}{2}}I_{v}^{(1)}$
$\displaystyle\sigma_{\rm m}^{(2)}=$
$\displaystyle\frac{2\gamma_{2}}{\alpha_{\rm
G}a}(\Delta\tau)^{2}\left(\frac{{k_{B}}T}{\tilde{J}}\right)^{3}I_{v}^{(2)}$
(52)

correspond to conductivity for magnons (without the factor of $e^{2}$), with
magnon integrals

$\displaystyle I_{v}^{(1)}$
$\displaystyle\equiv\frac{1}{4\pi^{2}}\int_{0}^{\infty}dx\frac{x^{\frac{1}{2}}}{e^{2x}-1}=2.073\times
10^{-2}$ $\displaystyle I_{v}^{(2)}$
$\displaystyle\equiv\frac{1}{(4\pi^{2})^{2}}\int_{x_{0}}^{\infty}dx\int_{0}^{\infty}dx^{\prime}\frac{x^{-\frac{1}{2}}(x^{\prime})^{\frac{1}{2}}(x+x^{\prime})}{(e^{x}-1)(1-e^{-x^{\prime}})(e^{x^{\prime}-x}+1)}$
(53)

. The expressions for $I_{v}^{(2)}$ diverges at low energy due to insufficient
phenomenological treatment of magnon lifetime in terms of the Gilbert damping
in the low frequency limit. Here we avoid the problem by introducing a low
energy cutoff $\omega_{0}$ ($x_{0}=\omega_{0}^{2}/({k_{B}}T)$). $I_{v}^{(2)}$
is of the order of 0.01 for $x_{0}\gtrsim 0.02$ and logarithmically diverges
as $x_{0}\rightarrow 0$. The electric conductivity in the same approximation
($1/(2ma^{2})\sim{\epsilon_{F}}$, ${k_{F}}a\sim 1$, $\overline{{\nu_{\rm
e}}}\sim 1/{\epsilon_{F}}$) is $\sigma_{\rm
e}/e^{2}\simeq\frac{1}{a}{\epsilon_{F}}\tau$. The ratio of the magnon
conductivity to the electron one thus is

$\displaystyle\mu_{\rm m/e}=\frac{\sigma_{\rm m}}{\sigma_{\rm e}}$
$\displaystyle\sim\frac{2}{\alpha_{\rm
G}{\epsilon_{F}}\tau}\left[\gamma_{1}I_{v}^{(1)}\tilde{T}^{\frac{3}{2}}+\gamma_{2}I_{v}^{(2)}({\Delta\tau})^{2}\tilde{T}^{{3}}\right]$
(54)

where $\tilde{T}\equiv\frac{{k_{B}}T}{\tilde{J}}$. As seen from Fig. 7, in
good metals with large $\Delta\tau$, magnon conductivity is larger than the
electric conductivity even for a low temperature (e.g., $\tilde{T}\gtrsim 0.4$
for ${\epsilon_{F}}\tau=20$). The result is qualitatively consistent with
seminal work of Grannemann and Berger [4], where it was argued that average
drift velocity of magnon is 2-3 times larger than that of electron in Ni66Cu34
and Ni69Fe31. Our result for $\sigma_{\rm m}^{(1)}$ is consistent with
previous analysis [6], where energy current driven by an electric field was
microscopically calculated taking account of the magnon emission/absorption
($f^{(1)}$) . Our result, however, indicates that more efficient magnon-drag
effect occurs due to the magnon scattering ($f^{(2)}$) in strongly spin-
polarized good metals. The reason is that the scattering contribution is a
response of the electron charge sector (spin summed), while the
emission/absorption is a spin response containing electron propagators with
opposite spins, resulting in a relative suppression factor of
$(\Delta\tau)^{-2}$. As the electron elastic lifetime is long,
${\epsilon_{F}}\tau\gg 1$, except for extremely dirty metals, the enhancement
factor of $(\Delta\tau)^{2}$ for scattering contribution makes the
contribution larger even for temperatures of
${k_{B}}T\gtrsim(\Delta\tau)^{-\frac{4}{3}}\tilde{J}$ (See Sec. IV.3).

The expression for the emission/absorption, $\sigma_{\rm m}^{(1)}$, of Eq.
(52) is consistent with previous analysis indicating $\sigma_{\rm
m}^{(1)}\propto\frac{1}{\alpha_{\rm G}\tau}T^{\frac{3}{2}}$ [4, 6], although
the result of Ref. [6] is proportional to electron spin polarization,
$\sigma_{\rm m}^{(1)}\propto\frac{P}{\alpha_{\rm G}\tau}T^{\frac{3}{2}}$,
probably due to an insufficient treatment of magnon and hole contributions in
Ref. [6].

The result for $\sigma_{\rm m}^{(2)}$ suggests that the magnon damping effect
for low energy magnons are critical for estimation of the magnon conductivity.
Further theoretical and experimental investigations are expected in this
direction.

Figure 7:  The ratio $\frac{\sigma_{\rm m}}{\sigma_{\rm e}}$ of magnon and
electron conductivity plotted for $\tilde{T}={k_{B}}T/\tilde{J}$ for
$\Delta\tau=2,10,20$. Solid and dashed lines corresponds to the total magnon
conductivity and the contribution from the emission/absorption ($\sigma_{\rm
m}^{(1)}$), respectively. Parameters used are $\gamma_{1}=\gamma_{2}=1$ and
$\Delta={\epsilon_{F}}$, $I_{v}^{(2)}=0.01$ and $\alpha_{\rm G}=0.01$. For
good metal (${\epsilon_{F}}\tau\gtrsim 10$), the magnon conductivity is larger
than the electric conductivity even at low temperature of $\tilde{T}\gtrsim
0.5$. The emission/absorption contribution is smaller than the scattering
contribution in this temperature range as a result of a relative suppression
factor of $(\Delta\tau)^{-2}$.

###  IV.2 Magnon spin-transfer effect

The magnon current can be estimated by observing magnetization dynamics
induced by the magnon spin-transfer effect. In the case of electron spin-
transfer effect, the flow of magnetization structures is at the velocity of

$\displaystyle v_{\rm st,e}$ $\displaystyle=\frac{a^{3}P}{2eS}j$ (55)

in the direction of spin current $Pj$
($P=\frac{n_{\uparrow}-n_{\downarrow}}{n_{\uparrow}+n_{\downarrow}}$) in the
adiabatic limit [12]. The current is written in terms of the current in the
low temperature limit (i.e., without magnons) $j^{(0)}$ as $j=j^{(0)}(1-{\cal
F}_{\rm m})$, where ${\cal F}_{\rm m}$ represents the resistivity effect due
to magnons. Assuming the adiabatic limit for magnons, the magnon spin transfer
effect drives magnetization structures at the velocity of

$\displaystyle v_{\rm st,m}$ $\displaystyle=-\frac{a^{3}}{S}j_{\rm m}$ (56)

in the opposite direction to the magnon current. Considering the two
mechanisms for magnon current, the velocity of the magnetization structure for
an applied electric current $\bm{j}$ is

$\displaystyle v_{\rm st}$ $\displaystyle=\frac{a^{3}}{2eS}P_{\rm eff}j^{(0)}$
(57)

where

$\displaystyle P_{\rm eff}$ $\displaystyle\equiv P\left[1-{\cal F}_{\rm
m}\right]-\frac{4}{\alpha_{\rm
G}{\epsilon_{F}}\tau}\left[\gamma_{1}I_{v}^{(1)}\tilde{T}^{\frac{3}{2}}+\gamma_{2}I_{v}^{(2)}({\Delta\tau})^{2}\tilde{T}^{3}\right]$
(58)

represents the effective spin-transfer efficiency including the magnon
effects, plotted in Fig. 8. In the weak damping regime $\alpha_{\rm
G}\lesssim({\epsilon_{F}}\tau)^{-1}$, the contribution of ${\cal F}_{\rm m}$
(magnon resistivity) is negligible compared to the conductivity correction
(the last square bracket) in the temperature regime in Fig. 8. Crossover from
electron-dominated to the magnon-dominated regime occurs at
${k_{B}}T/\tilde{J}\lesssim 0.2$ in the present case, with a significant
negative enhancement in the high temperature regime. The magnon spin-transfer
effect is correlated with the behavior of magnon contribution to the
resistance, that is proportional to the force plotted in Fig. 6.
Identification of magnon drag effects would be carried out by careful analyses
of temperature dependence of the experimental data.

Figure 8:  The effective spin-transfer efficiency $P_{\rm eff}$ plotted for
$\tilde{T}={k_{B}}T/\tilde{J}$ for $\Delta\tau=2,10,20$ and $P=1$ (solid
lines) and $P=-1$ (dashed lines). Parameters used are
$\gamma_{1}=\gamma_{2}=1$ and $\Delta={\epsilon_{F}}$, $I_{v}^{(2)}=0.01$ and
$\alpha_{\rm G}=0.01$.

###  IV.3 Crossover temperatures

Let us look into the crossover temperatures based on the results Eqs. (43)
(58). We consider the case $\gamma_{1}\sim\gamma_{2}\sim 1$,
$\Delta/{\epsilon_{F}}\sim 1$ and neglect the contribution from ${\cal F}_{\rm
m}$ in $P_{\rm eff}$ for simplicity. As for the magnons, emission/absorption
effect is dominant at low temperature, and the crossover to the scattering-
dominated temperature is read from Eq. (43) as

$\displaystyle T_{{\rm ea-
sc}}\simeq(\Delta\tau)^{-\frac{4}{3}}\tilde{J}/{k_{B}}$ (59)

The crossover would be seen in the magnon resistivity, assuming that drag
force is not directly observable.

The spin-transfer efficiency, $P_{\rm eff}$, at zero temperature reduces to
the electron origin, $P$. The magnon emission/absorption contribution becomes
larger than the electron contribution above

$\displaystyle T_{{\rm e-ea}}\simeq(12\times\alpha_{\rm
G})^{\frac{2}{3}}(\Delta\tau)^{\frac{2}{3}}\tilde{J}/{k_{B}}$ (60)

which corresponds to high temperature $T_{{\rm e-ea}}\gtrsim\tilde{J}/{k_{B}}$
unless in an extremely low damping materials with $\alpha_{\rm G}\ll
0.08\times(\Delta\tau)^{-1}$. In contrast, magnon scattering effect overcomes
the electron contribution at lower temperature of

$\displaystyle T_{{\rm e-sc}}\simeq(25\times\alpha_{\rm
G})^{\frac{1}{3}}(\Delta\tau)^{-\frac{1}{3}}\tilde{J}/{k_{B}}$ (61)

for $I_{v}^{(2)}=0.01$, meaning that the magnon emission/absorption regime
emerges only in very dirty metals with $\Delta\tau\sim O(1)$ (Fig. 8). The
crossover behavior is summarized in Fig. 9.

Figure 9:  Schematic figure showing the crossover temperatures. In the magnon
force $F_{\rm m}$ and magnon contribution to the electric resistivity
($\Delta\rho_{\rm m}$), crossover from magnon emission/absorption (e/a) to
scattering is at $T_{\rm ea-sc}$. The effective spin polarization (spin
transfer efficiency) $P_{\rm eff}$ is dominated by the electron contribution
at low temperature while a crossover to magnon spin-transfer dominated regime
driven by magnon scattering occurs at $T_{\rm e-sc}$. The order of magnitude
of $P_{\rm eff}$ in terms of powers of $\Delta\tau$ is shown. The magnon
emission/absorption process, relevant above $T_{\rm e-ea}$,d would not be
dominant in $P_{\rm eff}$ in the dilute magnon regime
${k_{B}}T/\tilde{J}\lesssim 1$ unless in very dirty metal with $\Delta\tau\sim
O(1)$.

##  V Summary

We have calculated the force between the magnon and conduction electron when
an electric field is applied to a ferromagnetic metal based on a microscopic
approach. The force due to magnon emission/absorption and scattering were
considered and the latter turned out to dominate in a wide temperature regime
in good metals with long elastic mean free path. The magnon contribution to
the resistivity and total spin-transfer efficiency were discussed.

###### Acknowledgements.

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 Properties of magnon-electron composite propagators

The path-ordered Green’s functions, defined on a time contour $C$ (Fig. 10),
are written in terms of real-time Green’s functions as

$\displaystyle G^{--}(t,t^{\prime})$ $\displaystyle\equiv G(t\in
C_{\rightarrow},t^{\prime}\in
C_{\rightarrow})=\theta(t-t^{\prime})G^{>}(t,t^{\prime})+\theta(t^{\prime}-t)G^{<}(t,t^{\prime})=G^{\rm
t}(t,t^{\prime})$ $\displaystyle G^{-+}(t,t^{\prime})$ $\displaystyle\equiv
G(t\in C_{\rightarrow},t^{\prime}\in C_{\leftarrow})=G^{<}(t,t^{\prime})$
$\displaystyle G^{+-}(t,t^{\prime})$ $\displaystyle\equiv G(t\in
C_{\leftarrow},t^{\prime}\in C_{\rightarrow})=G^{>}(t,t^{\prime})$
$\displaystyle G^{++}(t,t^{\prime})$ $\displaystyle\equiv G(t\in
C_{\leftarrow},t^{\prime}\in
C_{\leftarrow})=\theta(t-t^{\prime})G^{<}(t,t^{\prime})+\theta(t^{\prime}-t)G^{>}(t,t^{\prime})=G^{\rm\bar{t}}(t,t^{\prime})$
(62)

where the time on the path $C_{\rightarrow}$ and $C_{\leftarrow}$ are denoted
as $-$ and $+$, respectively, and $G^{\rm t}$ and $G^{\rm\bar{t}}$ are the
time-ordered and anti-time-ordered Green’s functions. These expressions are
direct consequence of the definition of path ordering on $C$. A
straightforward relation derived from Eq. (62) is

$\displaystyle
G^{--}(t,t^{\prime})+G^{++}(t,t^{\prime})=G^{-+}(t,t^{\prime})+G^{+-}(t,t^{\prime})=G^{\rm
K}(t,t^{\prime})\equiv[G^{<}+G^{>}](t,t^{\prime})$ (63)

Noting the relations

$\displaystyle G^{\rm t}$ $\displaystyle=G^{<}+G^{\rm r}=G^{>}+G^{\rm a}$
$\displaystyle G^{\rm\bar{t}}$ $\displaystyle=G^{<}-G^{\rm a}=G^{>}-G^{\rm
r},$ (64)

which read $G^{\rm a}=G^{--}-G^{+-}$ and $G^{\rm r}=G^{+-}-G^{++}$, the
definition (62) is summarized as

$\displaystyle G^{\alpha\beta}(t,t^{\prime})$
$\displaystyle=\frac{1}{2}[G^{\rm K}(t,t^{\prime})-\alpha G^{\rm
a}(t,t^{\prime})-\beta G^{\rm r}(t,t^{\prime})]$ (65)

where $\alpha,\beta=\pm$ are labels representing the path for the time,
$G^{\rm a}$ and $G^{\rm r}$ are the advanced and retarded Green’s functions.

Figure 10: The time contour the path-ordered Green’s functions are defined.
The upper (lower) path is $C_{\rightarrow}$ ($C_{\leftarrow}$).

###  A.1 Magnon electron pair propagator

An advantage to define the pair propagators defined in Eq. (17) is that the
definition has the same structure with respect to time as the single particle
case (62) and thus the propagators satisfy the same relation as Eqs. (63)(64).
They therefore satisfy in parallel to Eq. (65)

$\displaystyle\Pi_{k}^{(\pm),(k\mp p,p),{\alpha\beta}}$
$\displaystyle=\frac{1}{2}\left(\Pi_{k}^{(\pm),(k\mp p,p),{\rm
K}}-\alpha\Pi_{k}^{(\pm),(k\mp p,p),{{\rm a}}}-\beta\Pi_{k}^{(\pm),(k\mp
p,p),{{\rm r}}}\right)$ (66)

where

$\displaystyle\Pi_{k}^{(+),(k-p,p),\mu}(t,t^{\prime})$
$\displaystyle=i[g_{k-p}^{\mu}(t,t^{\prime})d_{p}^{>}(t,t^{\prime})+g_{k-p}^{<}(t,t^{\prime})d_{p}^{\mu}(t,t^{\prime})]$
$\displaystyle\Pi_{k}^{(+),(k-p,p),\nu}(t,t^{\prime})$
$\displaystyle=i[g_{k-p}^{\nu}(t,t^{\prime})d_{p}^{\nu}(t,t^{\prime})]$ (67)

for $\mu={\rm r},{\rm a}$ and $\nu=>,<$ (with $\Pi^{\rm
K}\equiv\Pi^{<}+\Pi^{>}$) and

$\displaystyle\Pi_{k}^{(-),(k+p,p),{\rm a}}(t,t^{\prime})$
$\displaystyle=i[g_{k+p}^{\rm
a}(t,t^{\prime})d_{p}^{<}(t^{\prime},t)+g_{k+p}^{<}(t,t^{\prime})d_{p}^{\rm
r}(t^{\prime},t)]$ $\displaystyle\Pi_{k}^{(-),(k+p,p),{\rm r}}(t,t^{\prime})$
$\displaystyle=i[g_{k+p}^{\rm
r}(t,t^{\prime})d_{p}^{<}(t^{\prime},t)+g_{k+p}^{<}(t,t^{\prime})d_{p}^{\rm
a}(t^{\prime},t)]$ $\displaystyle\Pi_{k}^{(-),(k+p,p),<}(t,t^{\prime})$
$\displaystyle=i[g_{k+p}^{<}(t,t^{\prime})d_{p}^{>}(t^{\prime},t)]$
$\displaystyle\Pi_{k}^{(-),(k+p,p),>}(t,t^{\prime})$
$\displaystyle=i[g_{k+p}^{>}(t,t^{\prime})d_{p}^{<}(t^{\prime},t)]$ (68)

This fact indicates that the pair propagator indeed behaves as a propagator of
a composite particle. In the frequency representation,

$\displaystyle\Pi_{k}^{(+),(k+p,p),>}(\omega)$ $\displaystyle\equiv\int
dte^{i\omega t}\Pi_{k}^{(+),(k+p,p),>}(t)=2\pi
i\sum_{\omega_{1}}g_{k+p}^{>}(\omega_{1})d_{p}^{>}(-\omega_{1}-\omega)$
$\displaystyle\Pi_{k}^{(-),(k+p,p),>}(\omega)$ $\displaystyle\equiv\int
dte^{i\omega t}\Pi_{k}^{(-),(k+p,p),>}(t)=2\pi
i\sum_{\omega_{1}}g_{k+p}^{>}(\omega_{1})d_{p}^{<}(\omega_{1}-\omega)$ (69)

Moreover, the pair propagator satisfy the the same relation as single particle
Green’s function without dynamic interaction, i.e.,

$\displaystyle\Pi_{k}^{(\pm),(k\mp p,p),<}(\omega)$
$\displaystyle=f(\omega)[\Pi_{k}^{(\pm),(k\mp p,p),{\rm
a}}(\omega)-\Pi_{k}^{(\pm),(k\mp p,p),{\rm r}}(\omega)]$ (70)

which is useful to extract the low energy contributions.

For the $\omega=0$ component of the advanced pair propagator in Eq. (20), we
have, using Eqs. (67)(68),

$\displaystyle\Pi_{k\sigma}^{(+),(k-p,p),{\rm a}}$
$\displaystyle=i\sum_{\nu}[g_{k-p,\nu,\sigma}^{\rm
a}d^{>}_{p,-\nu}+g_{k-p,\nu,\sigma}^{<}d^{\rm a}_{p,-\nu}]$ (71)

where $\nu$ is the frequency for electron Green’s function. The frequency of
magnon is $-\nu$ as the total frequency of the pair propagator $\pi^{(+)}$ is
zero and

$\displaystyle\Pi_{k\sigma}^{(-),(k+p,p),{\rm a}}$
$\displaystyle=i\sum_{\nu}[g_{k+p,\nu,\sigma}^{\rm
a}d^{<}_{p,\nu}+g_{k+p,\nu,\sigma}^{<}d^{\rm r}_{p,\nu}]$ (72)

where magnon frequency is equal to the electron one for $\Pi^{(-)}$. We thus
obtain

$\displaystyle{\rm Im}\Pi_{k\sigma}^{(+),(k-p,p),{\rm a}}$
$\displaystyle=4\sum_{\nu}n_{\nu}(1-f_{\nu}){\rm Im}[g_{k-p,\nu,\sigma}^{\rm
a}]{\rm Im}[d^{\rm a}_{p,-\nu}]]$ $\displaystyle{\rm
Im}\Pi_{k\sigma}^{(-),(k+p,p),{\rm a}}$
$\displaystyle=-4\sum_{\nu}n_{\nu}(1-f_{\nu}){\rm Im}[g_{k+p,\nu,\sigma}^{\rm
a}]{\rm Im}[d^{\rm a}_{p,\nu}]]$ (73)

where we used $f_{-\nu}=1-f_{\nu}$ and
$n_{\nu}+f_{\nu}=2n_{\nu}(1-f_{\nu})=2(1+n_{\nu})f_{\nu}$

###  A.2 Two-magnon electron composite propagator

Two-magnon electron composite propagator defined by Eq. (29) has the same
mathematical structure as the magnon-electron pair propagator and single
particle Green’s functions. Namely, by definitions ($\pm$ denotes the time
contour $C_{\rightarrow}$ and $C_{\leftarrow}$)

$\displaystyle\Pi^{(2),k^{\prime},p,p^{\prime}}_{k\sigma}(t,t^{\prime})$
$\displaystyle\equiv
g_{{\bm{k}}^{\prime}\sigma}(t,t^{\prime})d_{{\bm{p}}}(t,t^{\prime})d_{{\bm{p}}^{\prime}}(t^{\prime},t)$
$\displaystyle\Pi^{(2),k^{\prime},p,p^{\prime}(--)}_{k\sigma}(t,t^{\prime})$
$\displaystyle=\theta(t^{\prime}-t)g_{{\bm{k}}^{\prime}\sigma}^{<}(t,t^{\prime})d_{{\bm{p}}}^{<}(t,t^{\prime})d_{{\bm{p}}^{\prime}}^{>}(t^{\prime},t)+\theta(t-t^{\prime})g_{{\bm{k}}^{\prime}\sigma}^{>}(t,t^{\prime})d_{{\bm{p}}}^{>}(t,t^{\prime})d_{{\bm{p}}^{\prime}}^{<}(t^{\prime},t)$
$\displaystyle\Pi^{(2),k^{\prime},p,p^{\prime}(++)}_{k\sigma}(t,t^{\prime})$
$\displaystyle=\theta(t-t^{\prime})g_{{\bm{k}}^{\prime}\sigma}^{<}(t,t^{\prime})d_{{\bm{p}}}^{<}(t,t^{\prime})d_{{\bm{p}}^{\prime}}^{>}(t^{\prime},t)+\theta(t^{\prime}-t)g_{{\bm{k}}^{\prime}\sigma}^{>}(t,t^{\prime})d_{{\bm{p}}}^{>}(t,t^{\prime})d_{{\bm{p}}^{\prime}}^{<}(t^{\prime},t)$
(74)

we have

$\displaystyle\Pi^{(2),k^{\prime},p,p^{\prime}(--)}_{k\sigma}(t,t^{\prime})+\Pi^{(2),k^{\prime},p,p^{\prime}(++)}_{k\sigma}(t,t^{\prime})$
$\displaystyle=\Pi^{(2),k^{\prime},p,p^{\prime}(-+)}_{k\sigma}(t,t^{\prime})+\Pi^{(2),k^{\prime},p,p^{\prime}(+-)}_{k\sigma}(t,t^{\prime})$
(75)

and this relation allows us to write ($\alpha,\beta=\pm$)

$\displaystyle\Pi^{(2),k^{\prime},p,p^{\prime}(\alpha\beta)}_{k\sigma}(t,t^{\prime})$
$\displaystyle=\frac{1}{2}\left[\Pi^{(2),k^{\prime},p,p^{\prime},K}_{k\sigma}(t,t^{\prime})-\alpha\Pi^{(2),k^{\prime},p,p^{\prime},{\rm
a}}_{k\sigma}(t,t^{\prime})-\beta\Pi^{(2),k^{\prime},p,p^{\prime},{\rm
r}}_{k\sigma}(t,t^{\prime})\right]$ (76)

where

$\displaystyle\Pi^{(2),k^{\prime},p,p^{\prime},K}_{k\sigma}$
$\displaystyle\equiv(\Pi^{(2),k^{\prime},p,p^{\prime}(++)}_{k\sigma}+\Pi^{(2),k^{\prime},p,p^{\prime}(--)}_{k\sigma})$
$\displaystyle\Pi^{(2),k^{\prime},p,p^{\prime},{\rm a}}_{k\sigma}$
$\displaystyle\equiv(\Pi^{(2),k^{\prime},p,p^{\prime}(-+)}_{k\sigma}-\Pi^{(2),k^{\prime},p,p^{\prime}(++)}_{k\sigma})=(\Pi^{(2),k^{\prime},p,p^{\prime}(--)}_{k\sigma}-\Pi^{(2),k^{\prime},p,p^{\prime}(+-)}_{k\sigma})$
$\displaystyle\Pi^{(2),k^{\prime},p,p^{\prime},{\rm r}}_{k\sigma}$
$\displaystyle\equiv(\Pi^{(2),k^{\prime},p,p^{\prime}(+-)}_{k\sigma}-\Pi^{(2),k^{\prime},p,p^{\prime}(++)}_{k\sigma})=(\Pi^{(2),k^{\prime},p,p^{\prime}(--)}_{k\sigma}-\Pi^{(2),k^{\prime},p,p^{\prime}(-+)}_{k\sigma})$
(77)

Note that here retarded, advanced components are not in the original sense for
the single particle Green’s functions, written in terms of (anti)commutators
of field operators. Nevertheless, the relation (76) indicates that multi
particle propagators behaves mathematically the same as single particle
Green’s functions, and it simplifies the calculation greatly.

Using explicit expressions

$\displaystyle\Pi^{(2),k^{\prime},p,p^{\prime}(--)}_{k\sigma}(t,t^{\prime})$
$\displaystyle=g_{{\bm{k}}^{\prime}\sigma}^{<}(t,t^{\prime})d_{{\bm{p}}}^{<}(t,t^{\prime})d_{{\bm{p}}^{\prime}}^{<}(t^{\prime},t)+g_{{\bm{k}}^{\prime}\sigma}^{\rm
r}(t,t^{\prime})d_{{\bm{p}}}^{<}(t,t^{\prime})d_{{\bm{p}}^{\prime}}^{<}(t^{\prime},t)$
$\displaystyle+g_{{\bm{k}}^{\prime}\sigma}^{<}(t,t^{\prime})d_{{\bm{p}}}^{<}(t,t^{\prime})d_{{\bm{p}}^{\prime}}^{\rm
r}(t^{\prime},t)+g_{{\bm{k}}^{\prime}\sigma}^{>}(t,t^{\prime})d_{{\bm{p}}}^{\rm
r}(t,t^{\prime})d_{{\bm{p}}^{\prime}}^{<}(t^{\prime},t)$
$\displaystyle\Pi^{(2),k^{\prime},p,p^{\prime}(-+)}_{k\sigma}(t,t^{\prime})$
$\displaystyle=g_{{\bm{k}}^{\prime}\sigma}^{<}(t,t^{\prime})d_{{\bm{p}}}^{<}(t,t^{\prime})d_{{\bm{p}}^{\prime}}^{>}(t^{\prime},t)$
$\displaystyle\Pi^{(2),k^{\prime},p,p^{\prime}(+-)}_{k\sigma}(t,t^{\prime})$
$\displaystyle=g_{{\bm{k}}^{\prime}\sigma}^{>}(t,t^{\prime})d_{{\bm{p}}}^{>}(t,t^{\prime})d_{{\bm{p}}^{\prime}}^{<}(t^{\prime},t)$
(78)

we obtain

$\displaystyle\Pi^{(2),k^{\prime},p,p^{\prime},K}_{k\sigma}(t,t^{\prime})$
$\displaystyle=\left[g_{{\bm{k}}^{\prime}\sigma}^{<}(t,t^{\prime})d_{{\bm{p}}}^{<}(t,t^{\prime})d_{{\bm{p}}^{\prime}}^{>}(t^{\prime},t)+g_{{\bm{k}}^{\prime}\sigma}^{>}(t,t^{\prime})d_{{\bm{p}}}^{>}(t,t^{\prime})d_{{\bm{p}}^{\prime}}^{<}(t^{\prime},t)\right]$
$\displaystyle\Pi^{(2),k^{\prime},p,p^{\prime},{\rm
a}}_{k\sigma}(t,t^{\prime})$
$\displaystyle=\left[g_{{\bm{k}}^{\prime}\sigma}^{<}(t,t^{\prime})d_{{\bm{p}}}^{<}(t,t^{\prime})d_{{\bm{p}}^{\prime}}^{\rm
r}(t^{\prime},t)+g_{{\bm{k}}^{\prime}\sigma}^{<}(t,t^{\prime})d_{{\bm{p}}}^{\rm
a}(t,t^{\prime})d_{{\bm{p}}^{\prime}}^{<}(t^{\prime},t)+g_{{\bm{k}}^{\prime}\sigma}^{\rm
a}(t,t^{\prime})d_{{\bm{p}}}^{>}(t,t^{\prime})d_{{\bm{p}}^{\prime}}^{<}(t^{\prime},t)\right]$
$\displaystyle\Pi^{(2),k^{\prime},p,p^{\prime},{\rm
r}}_{k\sigma}(t,t^{\prime})$
$\displaystyle=\left[g_{{\bm{k}}^{\prime}\sigma}^{<}(t,t^{\prime})d_{{\bm{p}}}^{\rm
r}(t,t^{\prime})d_{{\bm{p}}^{\prime}}^{<}(t^{\prime},t)+g_{{\bm{k}}^{\prime}\sigma}^{<}(t,t^{\prime})d_{{\bm{p}}}^{<}(t,t^{\prime})d_{{\bm{p}}^{\prime}}^{\rm
a}(t^{\prime},t)+g_{{\bm{k}}^{\prime}\sigma}^{\rm
r}(t,t^{\prime})d_{{\bm{p}}}^{>}(t,t^{\prime})d_{{\bm{p}}^{\prime}}^{<}(t^{\prime},t)\right]$
(79)

The advanced component can be derived also by use of decompose relation and
Eq. (67) as

$\displaystyle\Pi^{(2),k^{\prime},p,p^{\prime},{\rm
a}}_{k\sigma}(t,t^{\prime})$
$\displaystyle=-i[\Pi_{k+p}^{(+)(k,p)}(t,t^{\prime})d_{p^{\prime}}(t^{\prime},t)]^{\rm
a}$ $\displaystyle=-i[\Pi_{k+p}^{(+)(k,p),{\rm
a}}(t,t^{\prime})d^{<}_{p^{\prime}}(t^{\prime},t)+\Pi_{k+p}^{(+)(k,p),<}(t,t^{\prime})d^{\rm
r}_{p^{\prime}}(t^{\prime},t)]$
$\displaystyle=(g_{{\bm{k}}\sigma}^{<}(t,t^{\prime})d_{{\bm{p}}}^{<}(t,t^{\prime})d_{{\bm{p}}^{\prime}}^{\rm
r}(t^{\prime},t)+g_{{\bm{k}}\sigma}^{<}(t,t^{\prime})d_{{\bm{p}}}^{\rm
a}(t,t^{\prime})d_{{\bm{p}}^{\prime}}^{<}(t^{\prime},t)+g_{{\bm{k}}\sigma}^{\rm
a}(t,t^{\prime})d_{{\bm{p}}}^{>}(t,t^{\prime})d_{{\bm{p}}^{\prime}}^{<}(t^{\prime},t))$
(80)

indicating consistency of composite propagator representation. The advanced
component of the composite propagator in the Fourier representation is

$\displaystyle\Pi^{(2),k^{\prime},p,p^{\prime},{\rm a}}_{k\sigma}$
$\displaystyle=\sum_{\nu\nu^{\prime}}[g_{{\bm{k}}^{\prime},\nu^{\prime}-\nu,\sigma}^{\rm
a}d_{{\bm{p}}\nu}^{<}d_{{\bm{p}}^{\prime}\nu^{\prime}}^{>}+g_{{\bm{k}}^{\prime},\nu^{\prime}-\nu,\sigma}^{>}d_{{\bm{p}}\nu}^{\rm
a}d_{{\bm{p}}^{\prime}\nu^{\prime}}^{<}+g_{{\bm{k}}^{\prime},\nu^{\prime}-\nu,\sigma}^{>}d_{{\bm{p}}\nu}^{<}d_{{\bm{p}}^{\prime}\nu^{\prime}}^{\rm
r}]$ (81)

Using ${\rm Im}[d_{p\nu}^{\rm a}]=\pi\delta(\nu-\omega_{p})$, we have

$\displaystyle{\rm Im}[\Pi^{(2),k^{\prime},p,p^{\prime},{\rm a}}_{k\sigma}]$
$\displaystyle=-[n_{\omega_{p}}(1+n_{\omega_{p}^{\prime}})f_{\omega_{p^{\prime}}-{\omega_{p}}}+n_{\omega_{p^{\prime}}}(1+n_{\omega_{p}})f_{{\omega_{p}}-{\omega_{p}^{\prime}}}]{\rm
Im}[g_{{\bm{k}}^{\prime},\nu^{\prime}-\nu,\sigma}^{\rm a}]$ (82)

##  Appendix B Calculation of electron contribution of Eq. (25)

Figure 11: energy contour $C_{\epsilon_{F}}$ for calculating the $k$-summation
of the electron Green’s function.

The electron contribution, Eq. (25), to the emission/absorption force
$F^{(1)}$ is calculated here by use of a contour integration. We first write
$I_{\sigma}\equiv\sum_{{\bm{k}}}k^{2}|g_{k\sigma}^{\rm
a}|^{2}(g_{k,-\sigma}^{\rm a})^{2}$ by use of the energy
($\epsilon\equiv\epsilon_{k}$) integral as

$\displaystyle I_{\sigma}$
$\displaystyle=\int_{-{\epsilon_{F}}}^{\infty}d\epsilon\frac{{\nu_{\rm
e}}(\epsilon)k^{2}(\epsilon)}{(\epsilon-\sigma\Delta)^{2}+\eta^{2}}\frac{1}{(\epsilon+\sigma\Delta+i\eta)^{2}}$
(83)

where ${\nu_{\rm e}}(\epsilon)=\frac{mk(\epsilon)a^{3}}{2\pi^{2}}$ is the
electron density of states,
$k(\epsilon)\equiv\sqrt{2m(\epsilon+{\epsilon_{F}})}$. The integration is
written as an integration over a contour $C_{{\epsilon_{F}}}$ avoiding a cut
along the real axis (due to ${\nu_{\rm
e}}(\epsilon)\sqrt{\epsilon+{\epsilon_{F}}}$) shown in Fig. 11as

$\displaystyle I_{\sigma}$
$\displaystyle=\frac{1}{2}\int_{C_{{\epsilon_{F}}}}d\epsilon\frac{{\nu_{\rm
e}}(\epsilon)k^{2}(\epsilon)}{(\epsilon-\sigma\Delta)^{2}+\eta^{2}}\frac{1}{(\epsilon+\sigma\Delta+i\eta)^{2}}$
(84)

The residues at poles $\epsilon=\sigma\Delta\pm i\eta$ and
$\epsilon=-\sigma\Delta-i\eta$ are calculated, paying attention to the fact
that ${\nu_{\rm e}}(\epsilon-i\eta)=-{\nu_{\rm e}}(\epsilon+i\eta)$ due to the
cut, to obtain

$\displaystyle I_{\sigma}$ $\displaystyle=\frac{\pi}{2\eta}{\nu_{\rm
e}}_{\sigma}\frac{1}{4}\left[\frac{1}{\Delta^{2}}+\frac{1}{(\Delta+i\sigma\eta)^{2}}\right]+i\pi\frac{d}{d\epsilon}\left[\frac{{\nu_{\rm
e}}(\epsilon)k^{2}(\epsilon)}{\epsilon-\sigma\Delta+i\eta}\frac{1}{\epsilon-\sigma\Delta-i\eta}\right]_{\epsilon=-\sigma\Delta-i\eta}$
(85)

It turns out that the lowest order contribution in the limit of
$\eta/{\epsilon_{F}}\ll 1$ arises from the derivative of
$\nu(\epsilon)k^{2}(\epsilon)$, i.e.,

$\displaystyle I_{\sigma}$
$\displaystyle=-i\pi\frac{3}{8{\epsilon_{F}}}{\nu_{\rm
e}}_{-\sigma}k^{2}_{-\sigma}\frac{1}{\Delta(\Delta+i\sigma\eta)}$ (86)

We therefore obtain

$\displaystyle{\rm Im}\sum_{\sigma}I_{\sigma}$
$\displaystyle=-\frac{3\pi}{8}\frac{1}{\Delta^{2}{\epsilon_{F}}}\sum_{\sigma}\sigma{{\nu_{\rm
e}}_{\sigma}k_{\sigma}^{2}}=-\frac{9\pi}{8}\frac{ma^{3}n_{\rm
e}}{\Delta^{2}{\epsilon_{F}}}$ (87)

where $n_{\rm e}=\frac{1}{3ma^{3}}\sum_{\sigma}\sigma{{\nu_{\rm
e}}_{\sigma}k_{\sigma}^{2}}$ is the total electron density.

##  Appendix C Linear response calculation of magnon current

Here we calculate the magnon current driven by an electric field directly
diagrammatically to show the correctness of the argument in Sec. IV.1 of the
velocity based on the calculation result of the force on magnon. The
calculation of the emission/absorption contribution is essentially the same as
the one for the magnon energy current in Ref. [6], while scattering
contribution was not argued there. Magnon current is

$\displaystyle\bm{j}_{\rm m}$
$\displaystyle=-i{JS}b^{\dagger}\stackrel{{\scriptstyle\leftrightarrow}}{{\nabla}}b$
$\displaystyle=JS(\nabla_{r}-\nabla_{r^{\prime}})D^{<}({\bm{r}},t,{\bm{r}}^{\prime},t)|_{{\bm{r}}={\bm{r}}^{\prime}}$
(88)

where
$D^{<}({\bm{r}},t,{\bm{r}}^{\prime},t^{\prime})=-i\left\langle{b^{\dagger}({\bm{r}}^{\prime},t^{\prime})b({\bm{r}},t)}\right\rangle$
is the full Greens function of magnon including interactions (the sign of $-$
is for boson lesser Green’s function). The full Green’s function at the linear
response is perturbatively expanded including the $sd$ exchange interaction to
the second order and the electric field

###  C.1 Magnon emission/absorption contribution

Figure 12:  Feynman diagrams representing the magnon current (left vertex)
induced by an electric field (denoted by a dotted line) by the magnon
emission/absorption due to the $sd$ exchange interaction. Wavy and solid lines
denote magnon and electron propagators, respectively. The sign $\pm$ denotes
electron spin.

Here magnon current $j_{\rm m}^{(1)}$ due to the linear magnon coupling to
electrons arising from the $sd$ exchange interaction is calculated. The
corresponding contribution to the Green’s function defined on the time
contour,$D^{(1)}({\bm{r}},t,{\bm{r}}^{\prime},t^{\prime})$, is (diagrams shown
in Fig. 12)

$\displaystyle D^{(1)}({\bm{r}},t,{\bm{r}}^{\prime},t^{\prime})$
$\displaystyle=i\frac{e}{m}\frac{\Delta^{2}}{2S}\int_{C}dt_{1}\int_{C}dt_{2}\int_{C}dt_{3}\sum_{r_{1}r_{2}r_{3}}d({\bm{r}}_{2},t_{2},{\bm{r}}^{\prime},t^{\prime})d({\bm{r}},t,{\bm{r}}_{1},t_{1})A_{j}(t_{3})$
$\displaystyle\times{\rm
tr}[\sigma_{+}g({{\bm{r}}_{1},t_{1}},{\bm{r}}_{2},t_{2})\sigma_{-}g({\bm{r}}_{2},t_{2},{\bm{r}}_{3},t_{3})\hat{p}_{j}g({\bm{r}}_{3},t_{3},{\bm{r}}_{1},t_{1})$
$\displaystyle+\sigma_{+}g({{\bm{r}}_{1},t_{1}},{\bm{r}}_{3},t_{3})\hat{p}_{j}g({\bm{r}}_{3},t_{3},{\bm{r}}_{2},t_{2})\sigma_{-}g({\bm{r}}_{2},t_{2},{\bm{r}}_{1},t_{1})]$
(89)

Denoting $({\bm{r}},t)$ by $x$, the Green’s function part is

$\displaystyle\int_{C}dt_{3}$ $\displaystyle d(x,x_{1}){\rm
tr}[\sigma_{+}g(x_{1},x_{2})\sigma_{-}g(x_{2},x_{3})({\bm{A}}(t_{3})\cdot\hat{{\bm{p}}})g(x_{3},x_{1})$
$\displaystyle+g(x_{1},x_{3})({\bm{A}}(t_{3})\cdot\hat{{\bm{p}}})g(x_{3},x_{2})\sigma_{-}g(x_{2},x_{1})\sigma_{+}]d(x_{2},x^{\prime})$
$\displaystyle=d(x,x_{1})[\tilde{\Lambda}_{-}(x_{1},x_{2})+\tilde{\Lambda}_{+}(x_{2},x_{1})]d(x_{2},x^{\prime})$
(90)

where (subscripts $\pm$ denotes electron spin)

$\displaystyle\tilde{\Lambda}_{-}(x_{1},x_{2})$ $\displaystyle\equiv
g_{-}(x_{1},x_{2})(gAg)_{+}(x_{2},x_{1})$
$\displaystyle\tilde{\Lambda}_{+}(x_{2},x_{1})$ $\displaystyle\equiv
g_{+}(x_{2},x_{1})(gAg)_{-}(x_{1},x_{2})$ (91)

and
$gAg(x_{2},x_{1})\equiv\int_{C}dt_{3}g(x_{2},x_{3})({\bm{A}}(t_{3})\cdot\hat{{\bm{p}}})g(x_{3},x_{1})$
behaves as composite propagators. The lesser component, $D^{(1)<}$, for Eq.
(88) is calculated using

$\displaystyle\int_{C}dt_{1}$
$\displaystyle\int_{C}dt_{2}[d(x,x_{1})[\tilde{\Lambda}_{-}(x_{1},x_{2})+\tilde{\Lambda}_{+}(x_{2},x_{1})]d(x_{2},x^{\prime})]^{<}$
$\displaystyle=\int_{-\infty}^{\infty}dt_{1}\int_{-\infty}^{\infty}dt_{2}\sum_{\alpha\beta}(-)^{2}\alpha\beta\left[d^{-\alpha}(x,x_{1})[\tilde{\Lambda}_{-}^{\alpha\beta}(x_{1},x_{2})+\tilde{\Lambda}_{+}^{\beta\alpha}(x_{2},x_{1})\right]d^{\beta+}(x_{2},x^{\prime})$
(92)

Here superscripts such as in $d^{\alpha\beta}$ ($\alpha,\beta=\pm$) denotes
the time contour: $d^{--}(t,t_{1})=d(t\in C_{\rightarrow},t_{1}\in
C_{\rightarrow})$, $d^{-+}(t,t_{1})=d(t\in C_{\rightarrow},t_{1}\in
C_{\leftarrow})$, etc. The result is (suppressing the time integration)

$\displaystyle[\int_{C}dt_{1}\int_{C}dt_{2}d(x,x_{1})\tilde{\Lambda}_{-}(x_{1},x_{2})d(x_{2},x^{\prime})]^{<}$
$\displaystyle=\frac{1}{2}(d^{\rm r}\tilde{\Lambda}_{-}^{\rm K}d^{\rm
a}+d^{\rm r}\tilde{\Lambda}_{-}^{\rm r}d^{\rm K}+d^{\rm
K}\tilde{\Lambda}_{-}^{\rm a}d^{\rm a})$
$\displaystyle[\int_{C}dt_{1}\int_{C}dt_{2}d(x,x_{1})\tilde{\Lambda}_{+}(x_{2},x_{1})d(x_{2},x^{\prime})]^{<}$
$\displaystyle=\frac{1}{2}(d^{\rm r}\tilde{\Lambda}_{+}^{\rm K}d^{\rm
a}+d^{\rm r}\tilde{\Lambda}_{+}^{\rm a}d^{\rm K}+d^{\rm
K}\tilde{\Lambda}_{+}^{\rm r}d^{\rm a})$ (93)

Fourier representation of $\tilde{\Lambda}$ is (using $A(t)\propto e^{-i\Omega
t}$)

$\displaystyle\tilde{\Lambda}_{\sigma}(x_{1},x_{2})$
$\displaystyle=\sum_{{\bm{k}}{\bm{k}}^{\prime}}\sum_{\Omega\omega\omega^{\prime}}e^{-i(\omega-\omega^{\prime})(t_{1}-t_{2})}e^{-i({\bm{k}}-{\bm{k}}^{\prime})\cdot({\bm{r}}_{1}-{\bm{r}}_{2})}e^{-i\Omega
t_{2}}({\bm{A}}(\Omega)\cdot{{\bm{k}}})\Lambda_{k,\omega+\Omega,\omega,-\sigma}^{k^{\prime},\omega^{\prime},\sigma}$
(94)

where

$\displaystyle\Lambda_{k,\omega+\Omega,\omega,-\sigma}^{k^{\prime},\omega^{\prime},\sigma}\equiv
g_{{\bm{k}}^{\prime},\omega^{\prime},\sigma}g_{{\bm{k}},\omega+\Omega,-\sigma}g_{{\bm{k}},\omega,-\sigma}$
(95)

The magnon current therefore reads

$\displaystyle j_{{\rm m},i}^{(1)}$
$\displaystyle=-\frac{e}{2m}J\Delta^{2}\sum_{p\nu}\sum_{k\omega}\sum_{\Omega}p_{i}k_{j}A_{j}(\Omega)\biggl{[}\left\\{d_{p\nu}^{\rm
r}d_{p\nu}^{\rm a}[\Lambda^{k+p,\omega+\nu,-}_{k,\omega+\Omega,\omega,+}]^{\rm
K}+(rKr)+(Kaa)\right\\}$ $\displaystyle+\left\\{d_{p\nu}^{\rm r}d_{p\nu}^{\rm
a}[\Lambda^{k-p,\omega-\nu,+}_{k,\omega+\Omega,\omega,-}]^{\rm
K}+(rKa)+(Kar)\right\\}\biggr{]}$ (96)

Here the external frequency $\Omega$ is neglected in the magnon frequencies
($\nu+\Omega\sim\nu$) focusing the Fermi surface (excitation) contribution.

In Eq. (96), (rKr)+(Kaa) terms are, using $d_{p\nu}^{\rm
K}=-(1+2n_{\nu})(d_{p\nu}^{{\rm a}}-d_{p\nu}^{{\rm r}})$,

$\displaystyle d_{p\nu}^{\rm r}d_{p\nu}^{\rm
K}[\Lambda^{k+p,\omega+\nu,-}_{k,\omega+\Omega,\omega,+}]^{\rm r}$
$\displaystyle+d_{p\nu}^{\rm K}d_{p\nu}^{{\rm
a}}[\Lambda^{k+p,\omega+\nu,-}_{k,\omega+\Omega,\omega,+}]^{\rm a}$
$\displaystyle=-(1+2n_{\nu})[d_{p\nu}^{\rm r}(d_{p\nu}^{{\rm
a}}-d_{p\nu}^{{\rm
r}})[\Lambda^{k+p,\omega+\nu,-}_{k,\omega+\Omega,\omega,+}]^{\rm
r}+(d_{p\nu}^{{\rm a}}-d_{p\nu}^{{\rm r}})d_{p\nu}^{{\rm
a}}[\Lambda^{k+p,\omega+\nu,-}_{k,\omega+\Omega,\omega,+}]^{\rm a}]$
$\displaystyle\simeq(1+2n_{\nu})d_{p\nu}^{\rm r}d_{p\nu}^{{\rm
a}}\left[[\Lambda^{k+p,\omega+\nu,-}_{k,\omega+\Omega,\omega,+}]^{\rm
a}-[\Lambda^{k+p,\omega+\nu,-}_{k,\omega+\Omega,\omega,+}]^{\rm r}\right]$
(97)

where contributions $(d_{p\nu}^{\rm r})^{2}$ and $(d_{p\nu}^{\rm a})^{2}$ are
neglected as they are smaller compared to $d_{p\nu}^{\rm r}d_{p\nu}^{{\rm a}}$
after $\nu$ summation. The magnon current in this approximation reads

$\displaystyle j_{{\rm m},i}^{(1)}$
$\displaystyle=-\frac{e}{2m}J\Delta^{2}\sum_{p\nu}\sum_{k\omega}\sum_{\Omega}p_{i}k_{j}A_{j}(\Omega)d_{p\nu}^{\rm
r}d_{p\nu}^{\rm
a}\left[[\Lambda^{k+p,\omega+\nu,-}_{k+,\omega+\Omega,\omega}]^{\rm
K}+(1+2n_{\nu})\left[[\Lambda^{k+p,\omega+\nu,-}_{k+,\omega+\Omega,\omega}]^{{\rm
a}}-[\Lambda^{k+p,\omega+\nu,-}_{k+,\omega+\Omega,\omega}]^{{\rm
r}}\right]\right.$
$\displaystyle\left.+[\Lambda^{k-p,\omega-\nu,+}_{k-,\omega+\Omega,\omega}]^{\rm
K}-(1+2n_{\nu})\left[[\Lambda^{k-p,\omega-\nu,+}_{k-,\omega+\Omega,\omega}]^{{\rm
a}}-[\Lambda^{k-p,\omega-\nu,+}_{k-,\omega+\Omega,\omega}]^{{\rm
r}}\right]\right]$ (98)

The Keldysh, advanced and retarded components of $\Lambda$ are (in the
suppressed notation)

$\displaystyle\Lambda^{\rm K}$ $\displaystyle=g^{<}(gg)^{>}+g^{>}(gg)^{<}$
$\displaystyle\Lambda^{\rm a}$ $\displaystyle=g^{{\rm
a}}(gg)^{<}+g^{<}(gg)^{\rm r}=g^{{\rm a}}(gg)^{>}+g^{>}(gg)^{\rm r}$
$\displaystyle\Lambda^{\rm r}$ $\displaystyle=g^{{\rm
r}}(gg)^{<}+g^{<}(gg)^{\rm a}=g^{{\rm r}}(gg)^{>}+g^{>}(gg)^{\rm a}$ (99)

Focusing on the Fermi surface (excitation) contribution, we neglect terms
containing only the retarded or the advanced Green’s functions, to obtain

$\displaystyle\Lambda^{\rm K}$
$\displaystyle\simeq(f_{\omega}-f_{\omega^{\prime}})(2f_{\omega^{\prime\prime}}-1)(g^{\rm
a}-g^{\rm r})g^{\rm r}g^{\rm a}$
$\displaystyle-[f_{\omega^{\prime\prime}}(f_{\omega^{\prime}}-1)+f_{\omega^{\prime}}(f_{\omega^{\prime\prime}}-1)]g^{\rm
r}(gg)^{\rm
a}-[f_{\omega^{\prime\prime}}(f_{\omega}-1)+f_{\omega}(f_{\omega^{\prime\prime}}-1)]g^{\rm
a}(gg)^{\rm r}$ $\displaystyle\Lambda^{{\rm a}}$ $\displaystyle\simeq
f_{\omega^{\prime\prime}}g^{\rm a}(gg)^{\rm r}+g^{\rm
a}\left[(f_{\omega}-f_{\omega^{\prime}})g^{\rm r}g^{\rm a}-f_{\omega}(gg)^{\rm
r}\right]$ $\displaystyle\Lambda^{{\rm r}}$ $\displaystyle\simeq-
f_{\omega^{\prime\prime}}g^{\rm r}(gg)^{\rm a}+g^{\rm
r}\left[(f_{\omega}-f_{\omega^{\prime}})g^{\rm r}g^{\rm
a}+f_{\omega^{\prime}}(gg)^{\rm a}\right]$ (100)

where $\Lambda$ of Eq. (95) is simply denoted by
$g_{\omega^{\prime\prime}}g_{\omega^{\prime}}g_{\omega}$ (with
$\omega^{\prime\prime}=\omega\pm\nu$, $\omega^{\prime}-\omega+\Omega$). Using
the Fermi surface approximation for the two Greens function,
$(gg)^{<}\simeq(f_{\omega}-f_{\omega^{\prime}})g^{\rm r}g^{\rm a}$, we obtain

$\displaystyle\Lambda^{\rm K}$
$\displaystyle\simeq(2f_{\pm\nu}-1)(\Lambda^{{\rm a}}-\Lambda^{{\rm r}})$
$\displaystyle\Lambda^{{\rm a}}$ $\displaystyle\simeq-\Omega
f^{\prime}(\omega)g^{\rm a}g^{\rm r}g^{\rm a}$ $\displaystyle\Lambda^{{\rm
r}}$ $\displaystyle\simeq-\Omega f^{\prime}(\omega)g^{\rm r}g^{\rm r}g^{\rm
a}$ (101)

Considering low temperature, $f^{\prime}(\omega)\simeq-\delta(\omega)$, we
obtain (using $2f_{-\nu}-1=1-2f_{\nu}$)

$\displaystyle j_{{\rm m},i}^{(1)}$ $\displaystyle\simeq-\frac{e}{2\pi
m}J\Delta^{2}\sum_{p\nu}\sum_{k\omega}\sum_{\Omega}p_{i}k_{j}\Omega
A_{j}(\Omega)|d_{p\nu}^{\rm
a}|^{2}\sum_{\pm}(\pm)(f_{\nu}+n_{\nu})\left[[\Lambda^{k\pm
p,\pm\nu,\mp}_{k,\pm,0,0}]^{{\rm a}}-[\Lambda^{k\pm
p,\pm\nu,\mp}_{k,\pm,0,0}]^{{\rm r}}\right]$ (102)

Using $f_{\nu}+n_{\nu}=2n_{\nu}(1-f_{\nu})$, and $i\Omega A=E$, we obtain the
final result of

$\displaystyle j_{{\rm m},i}^{(1)}$ $\displaystyle\simeq-\frac{2e}{\pi
m}{J\Delta^{2}}{\tau_{\rm
m}}\sum_{p\nu}\sum_{k\omega}\sum_{\Omega}p_{i}k_{j}E_{j}{\rm Im}[d_{p\nu}^{\rm
a}]\sum_{\pm}(\pm)n_{\nu}(1-f_{\nu}){\rm Im}[g_{k\pm p,\pm\nu,\mp}^{\rm
a}]|g_{k,\pm}^{{\rm a}}|^{2}$ $\displaystyle\equiv\sigma_{{\rm
m},i,j}^{(1)}E_{j}$ (103)

We therefore confirm the correct relation between the force and velocity of
magnon, $\sigma_{{\rm m},ij}^{(1)}={\cal F}_{ij}^{(1)}\frac{\tau_{\rm
m}}{m_{\rm m}}$, where $m_{\rm m}=(2JS)^{-1}$ is the effective magnon mass.

###  C.2 Magnon scattering contribution

Figure 13:  Feynman diagrams representing the magnon current (left vertex)
induced by an electric field (denoted by a dotted line) due to the magnon
scattering by the $sd$ exchange interaction. Wavy and solid lines denote
magnon and electron propagators, respectively. $\sigma$ denotes electron spin.

Here the second-order magnon coupling to electrons representing the magnon
scattering arising from the $sd$ exchange interaction is studied. This
contribution was not studied in Ref. [6]. The contribution to the magnon
Green’s function defined on the time contour,
$D^{(2)}({\bm{r}},t,{\bm{r}}^{\prime},t^{\prime})$, is (Fig. 13)

$\displaystyle D^{(2)}$
$\displaystyle({\bm{r}},t,{\bm{r}}^{\prime},t^{\prime})=i\frac{e}{m}\frac{\Delta^{2}}{S^{2}}\int_{C}dt_{1}\int_{C}dt_{2}\int_{C}dt_{3}\sum_{r_{1}r_{2}r_{3}}d({\bm{r}}_{2},t_{2},{\bm{r}}^{\prime},t^{\prime})d({\bm{r}},t,{\bm{r}}_{1},t_{1})A_{j}(t_{3})$
$\displaystyle\times{\rm
tr}[\sigma_{z}\Pi^{(+)}({{\bm{r}}_{1},t_{1}},{\bm{r}}_{2},t_{2})\sigma_{z}g({\bm{r}}_{2},t_{2},{\bm{r}}_{3},t_{3})\hat{p}_{j}g({\bm{r}}_{3},t_{3},{\bm{r}}_{1},t_{1})$
$\displaystyle+\sigma_{z}g({{\bm{r}}_{1},t_{1}},{\bm{r}}_{3},t_{3})\hat{p}_{j}g({\bm{r}}_{3},t_{3},{\bm{r}}_{2},t_{2})\sigma_{z}\Pi^{(-)}({\bm{r}}_{2},t_{2},{\bm{r}}_{1},t_{1})]$
(104)

where $\Pi^{(\pm)}$ are magnon-electron pair propagators defined in Eq. (17)
(suffixes are suppressed in Eq. (104)). As the expression of Eq. (104) has the
same structure as Eq. (89), we immediately obtain the magnon current as (see
Eq. (96))

$\displaystyle j_{{\rm m},i}^{(2)}$
$\displaystyle=-\frac{e}{2mS}J\Delta^{2}\sum_{p\nu}\sum_{k\omega\Omega}p_{i}k_{j}A_{j}(\Omega)$
$\displaystyle\times\left[\left\\{d_{p\nu}^{\rm r}d_{p\nu}^{\rm
a}[\Lambda^{(2+),k+p,\omega+\nu}_{k,\omega+\Omega,\omega}]^{\rm
K}+(rKr)+(Kaa)\right\\}+\left\\{d_{p\nu}^{\rm r}d_{p\nu}^{\rm
a}[\Lambda^{(2-),k-p,\omega-\nu}_{k,\omega+\Omega,\omega}]^{\rm
K}+(rKa)+(Kar)\right\\}\right]$ (105)

where ((${\bm{p}}^{\prime}$ is the wave vector of the magnon in the pair
propagators)

$\displaystyle\Lambda^{(2+),k+p,\omega+\nu}_{k,\omega+\Omega,\omega}$
$\displaystyle=\sum_{\sigma}\Pi_{{\bm{k}}+{\bm{p}},\omega+\nu}^{(+),({\bm{k}}+{\bm{p}}-{\bm{p}}^{\prime},\sigma,{\bm{p}}^{\prime})}g_{{\bm{k}},\omega+\Omega,\sigma}g_{{\bm{k}},\omega,\sigma}$
$\displaystyle\Lambda^{(2-),k-p,\omega-\nu}_{k,\omega+\Omega,\omega}$
$\displaystyle=\sum_{\sigma}\Pi_{{\bm{k}}-{\bm{p}},\omega-\nu}^{(-),({\bm{k}}-{\bm{p}}+{\bm{p}}^{\prime},\sigma,{\bm{p}}^{\prime})}g_{{\bm{k}},\omega+\Omega,\sigma}g_{{\bm{k}},\omega,\sigma}$
(106)

Using the Fermi surface approximation as in Eq. (101),

$\displaystyle\Lambda^{(2){\rm K}}$
$\displaystyle\simeq(2f_{\pm\nu}-1)(\Lambda^{(2){\rm a}}-\Lambda^{(2){\rm
r}})$ $\displaystyle\Lambda^{(2){\rm a}}$
$\displaystyle\simeq(f_{\omega}-f_{\omega^{\prime}})\Pi^{\rm a}g^{\rm r}g^{\rm
a}\simeq-\Omega f^{\prime}(\omega)\Pi^{\rm a}g^{\rm r}g^{\rm a}$
$\displaystyle\Lambda^{(2){\rm r}}$ $\displaystyle\simeq-\Omega
f^{\prime}(\omega)\Pi^{\rm r}g^{\rm r}g^{\rm a}$ (107)

we obtain

$\displaystyle j_{{\rm m},i}^{(2)}$ $\displaystyle\simeq-\frac{2e}{\pi
mS}J\Delta^{2}\sum_{pp^{\prime}k}\sum_{\nu}p_{i}k_{j}E_{j}|d_{p\nu}^{\rm
a}|^{2}\sum_{\pm}(\pm)\sum_{\sigma}n_{\nu}(1-f_{\nu}){\rm
Im}[{\Pi_{{\bm{k}}\pm{\bm{p}},\pm\nu}^{(\pm),({\bm{k}}\pm{\bm{p}}-{\bm{p}}^{\prime},\sigma,{\bm{p}}^{\prime})}}^{\rm
a}]|g_{k,\sigma}^{{\rm a}}|^{2}$ (108)

Here the magnon-electron pair propagator has an external frequency of $\pm\nu$
(external magnon frequency), and its imaginary part reads (see Eq. (73) for
the case of $\nu=0$)

$\displaystyle{\rm Im}\Pi_{k\pm
p,\sigma,\pm\nu}^{(\pm),(k\pm(p-p^{\prime}),\pm(\nu-\nu^{\prime});p^{\prime},\nu^{\prime}),{\rm
a}}$
$\displaystyle=2\sum_{\nu^{\prime}}(n_{\nu^{\prime}}+f_{\nu^{\prime}-\nu}){\rm
Im}[g_{k\pm(p-p^{\prime}),\pm(\nu-\nu^{\prime}),\sigma}^{\rm a}]{\rm
Im}[d^{\rm a}_{p^{\prime},\nu^{\prime}}]$ (109)

Using
$(n_{\nu^{\prime}}+f_{\nu^{\prime}-\nu})=(1+n_{\nu^{\prime}})f_{\nu^{\prime}-\nu}(1-f_{\nu})^{-1}$
and ${\rm Im}[d_{p^{\prime}\nu^{\prime}}^{\rm a}]=2\eta_{\rm
m}|d_{p^{\prime}\nu^{\prime}}^{\rm a}|^{2}=(\tau_{\rm
m})^{-1}|d_{p^{\prime}\nu^{\prime}}^{\rm a}|^{2}$ ($\tau_{\rm m}$ is the
relaxation time for magnon), we obtain

$\displaystyle j_{{\rm m},i}^{(2)}$ $\displaystyle=-\frac{2e}{\pi
mS}\frac{J\Delta^{2}}{\tau_{\rm m}}\sum_{pp^{\prime}k}p_{i}k_{j}E_{j}{\rm
Im}[d_{p\nu}^{\rm a}]{\rm Im}[d_{p^{\prime}\nu^{\prime}}^{\rm
a}]\sum_{\pm}(\pm)\sum_{\nu\nu^{\prime}}\sum_{\sigma}n_{\nu}(1+n_{\nu^{\prime}})f_{\nu^{\prime}-\nu}{\rm
Im}[g_{k\pm(p-p^{\prime}),\pm(\nu-\nu^{\prime}),\sigma}^{\rm
a}]|g_{k,\sigma}^{{\rm a}}|^{2}$ (110)

confirming the relation between the corresponding force density and current
density, $j_{{\rm m},i}^{(2)}={\cal F}^{(2)}\frac{\tau_{\rm m}}{m_{\rm m}}$.

## References

  * Chumak  _et al._ [2015] A. V. Chumak, V. I. Vasyuchka, A. A. Serga, and B. Hillebrands, Magnon spintronics, Nature Physics 11, 453 (2015).
  * Yan  _et al._ [2011] P. Yan, X. S. Wang, and X. R. Wang, All-magnonic spin-transfer torque and domain wall propagation, Phys. Rev. Lett. 107, 177207 (2011).
  * Uchida  _et al._ [2010] K. Uchida, J. Xiao, H. Adachi, J. Ohe, S. Takahashi, J. Ieda, T. Ota, Y. Kajiwara, H. Umezawa, H. Kawai, G. E. W. Bauer, S. Maekawa, and E. Saitoh, Spin seebeck insulator, Nat. Mater. 9, 894–897 (2010).
  * Grannemann and Berger [1976] G. N. Grannemann and L. Berger, Magnon-drag peltier effect in a ni-cu alloy, Phys. Rev. B 13, 2072 (1976).
  * Costache  _et al._ [2012] M. V. Costache, G. Bridoux, I. Neumann, and S. O. Valenzuela, Magnon-drag thermopile, Nature Materials 11, 199 (2012).
  * Miura and Sakuma [2012] D. Miura and A. Sakuma, Microscopic theory of magnon-drag thermoelectric transport in ferromagnetic metals, Journal of the Physical Society of Japan 81, 113602 (2012), http://dx.doi.org/10.1143/JPSJ.81.113602 .
  * Yamaguchi  _et al._ [2019] T. Yamaguchi, H. Kohno, and R. A. Duine, Microscopic theory of magnon-drag electron flow in ferromagnetic metals, Phys. Rev. B 99, 094425 (2019).
  * Funaki and Tatara [2021] H. Funaki and G. Tatara, Hydrodynamic theory of chiral angular momentum generation in metals, Phys. Rev. Research 3, 023160 (2021).
  * Funaki  _et al._ [2021] H. Funaki, R. Toshio, and G. Tatara, Vorticity-induced anomalous hall effect in an electron fluid, Phys. Rev. Research 3, 033075 (2021).
  * Lucassen  _et al._ [2011] M. E. Lucassen, C. H. Wong, R. A. Duine, and Y. Tserkovnyak, Spin-transfer mechanism for magnon-drag thermopower, Applied Physics Letters 99, 262506 (2011).
  * Kittel [1996] C. Kittel,  _Introduction to Solid State Physics_ (John Wiley & Sons, 1996).
  * Tatara  _et al._ [2008] G. Tatara, H. Kohno, and J. Shibata, Microscopic approach to current-driven domain wall dynamics, Physics Reports 468, 213 (2008).