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# Quantum Scattering Theory of Spin Transfer Torque, Spin Pumping and
Fluctuations

Arne Brataas  Center for Quantum Spintronics, Department of Physics, Norwegian
University of Science and Technology, NO-7491 Trondheim, Norway

###### Abstract

Spin transfer torque and spin pumping are central reciprocal phenomena in
spintronics. These phenomena occur in hybrid systems of normal metals and
magnets. Spin transfer is the conversion of spin currents in metals to a
torque on the magnetization of magnets. Spin pumping is the emission of spin
currents from precessing magnets. Here, we demonstrate a general way to
understand these effects within a quantum out-of-equilibrium path-integral
model. Our results agree with known expressions for spin transfer and spin
pumping in terms of transverse (mixing) conductances when there are no
fluctuations. However, at a finite temperature, frequency or spin
accumulation, the magnet also experiences fluctuating torques. In the
classical regime, when the thermal energy is larger than the bias voltage and
precession frequency, we reproduce the classical Brownian-Langevin forces
associated with spin transfer and spin pumping. At low temperatures, in the
quantum regime, we demonstrate that magnetization fluctuations differ in the
elastic and inelastic electron transport regimes. Furthermore, we show how
additional transport coefficients beyond the mixing conductance govern the
fluctuations. Some of these coefficients are related to electron shot noise
because of the discrete spin angular momentum of electrons. We estimate the
fluctuation coefficients of clean, tunnel, and disordered junctions and in the
case of an insulating magnet. Our results open a path for exploring low-
temperature magnetization dynamics and spin caloritronics.

##  I Introduction

Spin transfer and spin pumping are central phenomena in spintronics. Berger
proposed the pioneering concept of an interaction between magnetic textures
and electric currents in the late 1970sBerger (1979); Freitas and Berger
(1985). The concept of spin transfer torqueBerger (1996); Slonczewski (1996)
revitalized the fields of spintronics and magnetoelectronics. The seminal
experiments in Refs. Tsoi  _et al._ , 1998; Myers  _et al._ , 1999; Katine
_et al._ , 2000; Kiselev  _et al._ , 2003; Krivorotov  _et al._ , 2005
amplified its early recognition. Spin transfer causes a torque on magnets from
spin-polarized currents. This phenomenon excites magnets, changes the
magnetization orientation, and creates dynamic motion or moving magnetization
textures; see the reviews and concepts in Refs. Silva and Rippard, 2008; Ralph
and Stiles, 2008; Parkin  _et al._ , 2008; Brataas  _et al._ , 2012; Manchon
_et al._ , 2019.

In the 1970s, researchers also understood that dynamic magnets emit spin
currentsMonod  _et al._ (1972); Silsbee  _et al._ (1979). Similar to spin
transfer torque, this phenomenon has become essential in spintronics, with
improved measurementsMizukami  _et al._ (2001); Urban  _et al._ (2001) and the
development of a quantitative theory and wider applicability of the effect,
called spin pumpingTserkovnyak  _et al._ (2002); Heinrich  _et al._ (2003);
Tserkovnyak  _et al._ (2005); Mosendz  _et al._ (2010); Ando  _et al._ (2011);
Sandweg  _et al._ (2011); Shikoh  _et al._ (2013); Cheng  _et al._ (2014);
Kamra and Belzig (2017); Johansen and Brataas (2017); Li  _et al._ (2020);
Vaidya  _et al._ (2020).

Most theories describing spin transfer and spin pumping treat magnetization as
a classical variable and use conservation of angular momentum to deduce the
spin transfer torque or spin pumpingWaintal  _et al._ (2000); Brataas  _et
al._ (2000, 2001); Stiles and Zangwill (2002); Tserkovnyak  _et al._ (2002).
An alternative but also semiclassical approach is to view the spin transfer
torque as arising from an exchange potential outside of equilibriumNunez and
MacDonald (2006). Finally, a few papers use quantum mechanical language to
discuss spin pumping and spin transfer but are often limited to tunnel
junctions and no fluctuationsBender  _et al._ (2012, 2014).

Foros et al. realized that there are also fluctuating forces related to spin
transfer and spin pumpingForos  _et al._ (2005). For example, thermal
fluctuations of the spin currents in normal metal-ferromagnet junctions give
rise to Brownian stochastic forces on the magnet. The strengths of these
forces are exactly in agreement with the fluctuation-dissipation theorem for
spin pumping-enhanced Gilbert damping, as should be expected. These
fluctuations are essential ingredients in the field of ambient temperature
spin caloritronicsXiao  _et al._ (2009, 2010); Bauer  _et al._ (2012).
However, when the spin accumulation in the normal metal exceeds the thermal
energy, the fluctuation-dissipation theorem no longer holdsForos  _et al._
(2005). In this quantum regime, shot noise caused by the discrete nature of
the spin angular momentum carried by individual electrons enhances magnetic
fluctuationsForos  _et al._ (2005). A similar picture emerges in ferromagnet-
ferromagnet tunnel junctionsChudnovskiy  _et al._ (2008); Swiebodzinski  _et
al._ (2010). The latter article introduced path integrals to compute spin
transfer, spin pumping, and fluctuations.

Traditionally, spin transfer and spin pumping are explored at ambient
temperatures. The thermal energy is larger than the magnetic resonance
frequency and the bias charge or spin accumulation. However, new developments
in measurements and the prospect of achieving an improved understanding
motivate lower-temperature measurements and theory, where quantum fluctuations
come into play. In ferromagnet-normal metal-ferromagnet spin valves, Zholud et
al. pioneered the observation of quantum effects in the anomalous behavior of
magnetization fluctuations at low temperaturesZholud  _et al._ (2017). In such
spin valves, there are quantum fluctuations that reduce the giant
magnetoresistance in the presence of a voltage biasZholud  _et al._ (2017);
Qaiumzadeh and Brataas (2018).

We find rich and complex quantum fluctuations in normal metal-ferromagnet
systems. Our results are relevant to measurements of magnetization
fluctuations in such systems. Furthermore, they may be essential in spin
transfer-controlled magnonics and Bose-Einstein condensation of
magnonsDivinskiy  _et al._ (2021). More generally, our results will be central
to the development of an understanding of low-temperature spin
caloritronicsBauer  _et al._ (2012). Experimental probing of our results for
the quantum fluctuations requires low-temperature measurements such as in Ref.
Zholud  _et al._ , 2017 carried out at 3.4 K.

This paper describes spin transfer and spin pumping in arbitrary junctions
using an out-of-equilibrium quantum picture that considers quantum and
classical fluctuations. We describe arbitrary contacts between ferromagnets
and normal metals. To this end, we use the scattering theory of electron
transportBüttiker (1992). We aim to describe the temporal spin dynamics. To do
so, we use the out-of-equilibrium path-integral formulation. By integrating
out the fermions that may be out of equilibrium with spin and charge
accumulations, we find the effective action for the magnons that we can in
turn express as an effective Landau-Lifshitz-Gilbert-Slonczewski equation with
fluctuating forces.

Our contribution to the field is twofold. We demonstrate that earlier
semiclassical approaches to describing spin transfer and spin pumping in
arbitrary junctions agree with a complete quantum treatment of magnons and
electrons to leading order in the total spin. More importantly, we derive
expressions for the quantum and classical fluctuations that the magnet
experiences in terms of the reflection and transmission coefficient of the
junctions. Each electron carries a discrete amount of spin angular momentum
$\hbar/2$. The discreteness of the spin angular momentum causes shot noise, as
experienced by the magnet when the bias voltage is larger than the temperature
and precession frequency. In our endeavor, we merge earlier semiclassical
approaches that describe any junction in terms of the scattering matrixBrataas
_et al._ (2000, 2001); Tserkovnyak  _et al._ (2002, 2005) with the out-of-
equilibrium path-integral formalism introduced in Refs. Duine  _et al._ ,
2007; Chudnovskiy  _et al._ , 2008; Swiebodzinski  _et al._ , 2010

We consider systems in which the spin angular momentum is conserved for
electron transport at the normal metal-ferrromagnet junction and within the
ferromagnet. Nevertheless, we include spin-orbit interactions within the
normal metals for the electron transport. Additionally, spin-orbit induced
magnetic anisotropy energies can be included in straightforward manners. We
believe that generalization of our methods and calculations to spin-orbit
torques generated at junctionsManchon and Zhang (2008, 2009); Miron  _et al._
(2010, 2011); Liu  _et al._ (2012); Garello  _et al._ (2013); Mellnik  _et
al._ (2014); MacNeill  _et al._ (2017); Zhu  _et al._ (2019); Manchon  _et
al._ (2019) and magnonic charge pumping is feasible and deserves our
attention.

We organize the manuscript as follows. First, section II presents the system
and the assumptions. Then, section III describes the normal metals with the
charge and spin accumulations that may exist therein. We present our main
results for the spin dynamics in section IV. Section V describes the model of
the system introduced in section II. We use the closed contour path-integral
method in section VI to compute the main results in section IV resulting from
the model in section V. The derivation of the stochastic Langevin forces is in
section VII. We derive a valuable connection to the scattering formulation of
the electron-magnon interaction in section VIII. Finally, we conclude the
paper in section IX.

##  II System and Assumptions

We consider the system shown in Fig. 1. Normal metals surround a ferromagnet.
The accumulation of the out-of-equilibrium charge ($\mu_{c}$) and spin
(${\bm{\mu}}_{s}$) potentials in normal metals can drive the spin dynamics in
the ferromagnet. Furthermore, the spin dynamics can induce spin currents in
the metals. The magnet can be insulating or conducting. In either case, the
electron transport in the normal metals can influence the spin dynamics.

Figure 1: Schematic depiction of a normal metal-magnet-normal metal system.
Spin accumulations in the normal metals affect the spin dynamics described by
precession of the temporal unit vector along the order parameter ${\bf m}(t)$.
The figure shows two normal metals. Our results can be straightforwardly
generalized to an arbitrary number of normal metals attached to the
ferromagnet.

We will derive how the semiclassical magnetization dynamics respond to spin
and charge accumulations. We assume that the total spin of the ferromagnet is
large, $S\gg 1$. In this sense, our calculation is semiclassical. However, we
will also consider fluctuations in the magnetization dynamics. An often-used
assumption is that the thermal energy ($k_{B}T$) is much larger than all the
other energy scales related to the magnetization dynamics. This scenario
represents the classical regime. While we will cover this classical regime, we
will also consider the quantum regime in which the dynamic frequency
($\omega$) and charge and spin accumulations are larger than the thermal
energy. This generalization results in shot noise contributions to the
fluctuating forces acting on the magnet. Ref. Foros  _et al._ , 2005 pioneered
the identification of shot noise contributions to magnetization dynamics and
computed some of the terms in some regimes. Quantum fluctuations dominate when
the frequency or accumulation is larger than the thermal energy. We will
demonstrate that the fluctuations obey a rich and complex behavior that can
reveal the nature of the electron transport and how the magnet responds to it.

In all situations, we will assume that the Fermi energy is the largest energy
scale in the system. This assumption is valid in metallic systems since the
Fermi energy is approximately 1-10 eV, whereas the thermal energy,
magnetization frequency, and accumulation are in the meV range. Therefore,
consistent with the large Fermi energy assumption, we consider systems much
larger than the Fermi wavelength in metals. In other words, the system is much
larger than the lattice spacing.

In our explicit calculation, we assume the magnet to be homogeneous. However,
we assert that our results are also valid for magnetic textures as long as the
variation is larger than the Fermi wavelength, which is always the case except
in extreme cases. The underlying reason why the results have broader
applicability than only to homogeneous ferromagnets is the significant
difference in the length scales of the magnetic variations and the typical
wavelength of the electrons. Hence, the electrons only see the magnetization
in a range near the Fermi wavelength.

##  III Normal Metal Reservoirs

This section describes the normal metals. Out-of-equilibrium charge and spin
accumulations may exist in the normal metal reservoirs. These may arise from
currents from other ferromagnets or normal metals. Alternatively, they may
arise from currents in the normal metal reservoirs via the spin Hall
effectSinova  _et al._ (2015). We will not repeat such discussions, which are
not central to our development. Instead, we will assume that some of these
mechanisms within the circuit induce charge and spin accumulations.
Furthermore, we will assume that these accumulations are static and set by
external forces for clarity. Generalizations to consider the influence of the
ferromagnet on the spin and charge accumulations and temporal evolution can be
performed via magnetoelectronic circuit theory, as described in Refs. Brataas
_et al._ , 2000, 2002; Tserkovnyak  _et al._ , 2005; Brataas  _et al._ ,
2006a; Foros  _et al._ , 2007.

In the reservoirs, the operator $\hat{a}_{\sigma\beta}$ annihilates an
electron with spin $\sigma$ in state $\beta$. The statistical occupation of
the state is

$\langle\hat{a}^{\dagger}_{s\alpha}\hat{a}_{\sigma\beta}\rangle=\delta_{\alpha\beta}n_{s\sigma\alpha}\,,$
(1)

where $n_{s\sigma\alpha}$ is the $2\times 2$ out-of-equilibrium occupation of
the electrons in state $\alpha$ in spin space. The quantum number $\alpha$
contains lead $\kappa$, transverse waveguide mode $n$, and energy $\epsilon$.
In general, in isotropic systems, the $2\times 2$ out-of-equilibrium
occupation of the electrons only depends on the energy $\epsilon$ of the state
but can depend on lead $\kappa$:

$\displaystyle n_{s\sigma\kappa}(\epsilon)$
$\displaystyle=\left[f_{\uparrow\kappa}(\epsilon)+f_{\downarrow\kappa}(\epsilon)\right]1_{s\sigma}/2$
$\displaystyle+\left[f_{\uparrow\kappa}(\epsilon)-f_{\downarrow\kappa}(\epsilon)\right]{\bm{u}}_{\kappa}\cdot{\bm{\sigma}}_{s\sigma}/2\,,$
(2)

where $f_{\uparrow}$ ($f_{\downarrow}$) is the spin distribution aligned
(anti-aligned) with the spin accumulations directed along the unit vector
${\bm{u}}_{\kappa}$. ${\bm{\sigma}}$ is a vector of Pauli matrices.

Electron transport can be inelastic, elastic, or between these regimes. We
will demonstrate that the spin fluctuations in the magnet differ in these
transport regimes at low temperatures. In other words, measurements of the
spin fluctuations will reveal the electron transport regime.

####  III.0.1 Inelastic transport

In the inelastic transport regime, the electron distribution is analogous to
that in the equilibrium situation and is determined by Fermi-Dirac functions,
but with out-of-equilibrium chemical potentials. There is charge accumulation
$\mu_{\kappa c}^{in}$ and spin accumulation ${\bm{\mu}}_{\kappa s}^{in}$ in
reservoir $\kappa$. The spin accumulation ${\bm{\mu}}_{\kappa s}^{in}$ has a
direction along the unit vector ${\bm{u}}_{\kappa s}$ and an amplitude
$\mu_{\kappa s}^{in}$, with ${\bm{\mu}}_{\kappa s}^{in}={\bm{u}}_{\kappa
s}\mu_{s}^{in}$. The inelastic electron spin-up and spin-down distributions
are

$\displaystyle f_{\uparrow\kappa}^{in}(\epsilon)$
$\displaystyle=f(\epsilon-\mu_{\kappa c}^{in}-\mu_{\kappa s}^{in}/2)\,,$ (3a)
$\displaystyle f_{\downarrow\kappa}^{in}(\epsilon)$
$\displaystyle=f(\epsilon-\mu_{\kappa c}^{in}+\mu_{\kappa s}^{in}/2)\,,$ (3b)

where $f(\epsilon)$ is the Fermi-Dirac distribution function that depends on
the temperature $T$.

####  III.0.2 Elastic transport

In contrast to the inelastic transport regime, in elastic transport, energy is
conserved. Let us assume that the elastic transport is a result of transport
from two or more large reservoirs with equilibrium electron distributions
$f(\epsilon-\mu_{1})$, $f(\epsilon-\mu_{2})$, etc. Since the energy is
conserved, the components of the distributions aligned and anti-aligned with
the spin accumulation must be

$\displaystyle f_{\uparrow\kappa}^{el}(\epsilon)$
$\displaystyle=\sum_{i}R_{\uparrow\kappa i}f(\epsilon-\mu_{i})\,,$ (4a)
$\displaystyle f_{\downarrow\kappa}^{el}(\epsilon)$
$\displaystyle=\sum_{i}R_{\downarrow\kappa i}f(\epsilon-\mu_{i})\,,$ (4b)

where $R_{\uparrow\kappa i}$ and $R_{\downarrow\kappa i}$ are transport
coefficients that depend on the materials and geometry. We maintain that the
bias voltage is much smaller than the Fermi energy, so the transport
coefficients $R_{\uparrow\kappa i}$ and $R_{\downarrow\kappa i}$ are constant
in the energy range of the bias voltage. At equilibrium,
$f_{\uparrow\kappa}^{el}(\epsilon)=f_{\downarrow\kappa}^{el}(\epsilon)=f(\epsilon-\mu_{0})$,
where $\mu_{0}$ is the common equilibrium chemical potential. Therefore,

$\displaystyle\sum_{i}R_{s\kappa i}=1\,.$ (5)

In the elastic transport regime, for deterministic transport properties such
as the spin transfer torque, defining the effective elastic accumulation as
follows is convenient:

$\displaystyle\mu_{\kappa c}^{el}+\mu_{\kappa s}^{el}/2=\int
d\epsilon\left[f^{el}_{\uparrow\kappa}(\epsilon)-f(\epsilon-\mu_{0})\right]\,,$
$\displaystyle\mu_{\kappa c}^{el}-\mu_{\kappa s}^{el}/2=\int
d\epsilon\left[f^{el}_{\downarrow\kappa}(\epsilon)-f(\epsilon-\mu_{0})\right]\,.$
(6a)

We will demonstrate that the spin transfer torque is the same in the elastic
and inelastic transport regimes provided that the charge and spin
accumulations are the same, i.e., $\mu_{\kappa c}^{\text{el}}=\mu_{\kappa
c}^{\text{in}}$ and ${\bm{\mu}}_{\kappa s}^{\text{el}}={\bm{\mu}}_{\kappa
s}^{\text{in}}$. The reason is that spin transfer and spin pumping are
governed by electrons at the Fermi level, which is much larger than the
precession frequency and temperature. Then, how the electrons are distributed
within a small window close to the Fermi level does not matter. However, we
will show that the low-temperature spin fluctuations differ in the two
regimes. This dependence on the transport regime occurs because the spin
accumulation or precession frequency is much smaller than the Fermi energy and
can reveal the low-energy occupation of the electrons.

##  IV Main Results - Spin Dynamics

This section presents our main findings. Our main result is the derivation of
a generalized Landau-Lifshitz-Gilbert-Slonczewski (LLGS) equation with
stochastic forces that reflect electron transport fluctuations valid from the
quantum low-temperature regime to the classical, high-temperature regime. We
find that the spin dynamics consist of bulk contributions, spin transfer
torque, spin pumping, and electron transport-induced fluctuations:

$\partial_{t}{\bm{m}}={\bm{\tau}}_{\text{b}}+{\bm{\tau}}_{\text{stt}}+\tilde{\bm{\tau}}_{\text{stt}}+{\bm{\tau}}_{\text{sp}}+\tilde{\bm{\tau}}_{\text{sp}}+{\bm{\tau}}_{\text{fl}}\,.$
(7)

The bulk torque is conventional and independent of the electron reservoirs,

$\displaystyle{\bm{\tau}}_{\text{b}}=-\gamma{\bm{m}}\times\left[{\bm{H}}_{\text{eff}}+{\bm{h}}_{\text{b}}(t)\right]+\alpha_{\text{b}}{\bm{m}}\times\partial_{t}{\bm{m}}\,,$
(8)

where $\gamma=-g^{*}\mu_{B}/\hbar$ is the gyromagnetic ratio,
${\bm{H}}_{\text{eff}}=-\partial F/\partial{\bm{m}}M_{s}$ is the effective
magnetic field governed by the free energy $F$ of the magnet,
$\alpha_{\text{b}}$ is the bulk Gilbert damping coefficient, and
${\bm{h}}_{\text{b}}(t)$ is the fluctuating field. The average of the
fluctuating field vanishes, i.e., $\langle{\bm{h}}_{\text{b}}(t)\rangle=0$,
while the fluctuations are related to the Gilbert damping coefficient by the
fluctuation-dissipation theorem:

$\langle\gamma h_{\text{b}i}(t)\gamma
h_{\text{b}j}(\tau)\rangle_{\omega}=\alpha_{0}\frac{2k_{B}T}{S}\zeta\left(\frac{\omega}{2k_{B}T}\right)\,,$
(9)

where $S$ is the total spin of the magnet, the Fourier transform on the left-
hand side of Eq. (9) is defined in Eq. (197), and

$\zeta(x)=x/\tanh{x}\,.$ (10)

In the classical high-temperature limit, $\zeta(x\rightarrow 0)=1$.

With our alternative approach, we reproduce the established result for the
spin transfer torquesBrataas  _et al._ (2000, 2001, 2006b) arising from spin
accumulation in the left (${\bm{\mu}}_{s}$) and right ($\tilde{\bm{\mu}}_{s}$)
reservoirs:

$\displaystyle{\bm{\tau}}_{\text{stt}}$
$\displaystyle=\alpha^{r}{\bm{m}}\times{\bm{m}}\times{\bm{\mu}}_{s}-\alpha^{i}{\bf
m}\times{\bm{\mu}}_{s}\,,$ (11a)
$\displaystyle\tilde{{\bm{\tau}}}_{\text{stt}}$
$\displaystyle=\tilde{\alpha}^{r}{\bm{m}}\times{\bm{m}}\times\tilde{{\bm{\mu}}}_{s}-\tilde{\alpha}^{i}{\bf
m}\times\tilde{{\bm{\mu}}}_{s}\,,$ (11b)

where $\alpha$ ($\tilde{\alpha}$) is the enhanced Gilbert damping coefficient
associated with spin pumping in the left (right) reservoir. The superscript
$r$ ($i$) denotes the real (imaginary) part. As previously stated, the
expression for the spin transfer torque is similar in the elastic and
inelastic transport regimes and only depends on the total amount of out-of-
equilibrium spins. We find agreement with previous worksBrataas  _et al._
(2000, 2001, 2006b, 2011) in that the scattering matrix determines the spin
transfer (spin pumping) coefficients as follows:

$\displaystyle\alpha$ $\displaystyle=\frac{1}{4\pi
S}\text{Tr}\left[1-r^{\dagger}_{\uparrow}r_{\downarrow}-t^{\dagger}_{\uparrow}t_{\downarrow}\right]\,,$
(12a) $\displaystyle\tilde{\alpha}$ $\displaystyle=\frac{1}{4\pi
S}\text{Tr}\left[1-\tilde{r}^{\dagger}_{\uparrow}\tilde{r}_{\downarrow}-\tilde{t}^{\dagger}_{\uparrow}\tilde{t}_{\downarrow}\right]\,,$
(12b)

where $r_{s}$ ($\tilde{r}_{s}$) is the reflection amplitude matrix associated
with transport from the left (right) lead for electrons with spin $s$ along
the magnetization direction and $t_{s}$ ($\tilde{t}$) is the transmission
amplitude matrix associated with transport from the left (right) lead for
electrons with spin $s$ along the magnetization direction. $r$, $\tilde{r}$,
$t$, and $\tilde{t}$ are elements of the scattering matrix defined in Eq.
(141).

We also reproduce the established expression for spin pumping in the left and
right leadsTserkovnyak  _et al._ (2002); Brataas  _et al._ (2002); Tserkovnyak
_et al._ (2005):

$\displaystyle{\bm{\tau}}_{\text{sp}}$
$\displaystyle=\left(\alpha+\tilde{\alpha}\right)^{r}{\bf
m}\times\partial_{t}{\bm{m}}$
$\displaystyle-\left(\alpha+\tilde{\alpha}\right)^{r}{\bm{m}}\times{\bm{m}}\times\partial_{t}{\bm{m}}\,.$
(13a)

The correspondence between the spin transfer torque (11) and spin pumping (13)
is a result of Onsager reciprocity relationsBrataas  _et al._ (2011). Spin
pumping is the reciprocal phenomenon of spin transfer.

Our main result is electron transport-enhanced fluctuations. We find that
there is a fluctuating torque

${\bm{\tau}}=-\gamma{\bm{m}}\times{\bm{h}}(t)\,,$ (14)

where ${\bm{h}}(t)$ is the fluctuating field.

At equilibrium, in the absence of bias voltages, we find that the fluctuation-
dissipation theorem including the effects of spin pumping holds, so

$\langle\gamma h_{i}^{\text{e}}(t)\gamma
h_{j}^{\text{e}}(\tau)\rangle_{\omega}=\delta_{ij}\left(\alpha^{r}+\tilde{\alpha}^{r}\right)\frac{2k_{B}T}{S}\zeta\left(\frac{\omega}{2k_{B}T}\right)\,,$
(15)

where $\zeta(x)$ was defined in Eq. (10). Ref. Foros  _et al._ , 2005 first
obtained the classical limit, in the regime $k_{B}T\gg\omega$, of this result.
Eq. (15) generalizes this equilibrium result to finite frequencies. As
expected, the fluctuations in Eq. (15) are similar to the bulk fluctuations in
Eq. (9). As established in Ref. Foros  _et al._ , 2005, this implies that at
high temperatures, the LLGS equation holds with fluctuating forces governed by
a total Gilbert damping coefficient
$\alpha_{\text{b}}+\alpha^{r}+\tilde{\alpha}^{r}$.

Charge and spin accumulations in the normal metals modify the fluctuations.
Out of equilibrium, the fluctuating field ${\bm{h}}(t)$ has contributions
arising from longitudinal and transverse components of the spin accumulation
with respect to the magnetization.

In a right-handed coordinate system where the unit vector of the magnetization
is along the $z$ direction, we find the general result for the fluctuations:

$\displaystyle\langle\gamma h_{x}(t)\gamma h_{x}(\tau)\rangle_{\omega}$
$\displaystyle=-\frac{i}{4S}\left[\Sigma^{K}(\omega)+\Sigma^{K}(-\omega)\right]+\frac{1}{S}\text{Im}\left[\tilde{\Sigma}_{\uparrow\downarrow}^{K}\right]\,,$
(16a) $\displaystyle\langle\gamma h_{y}(t)\gamma h_{y}(\tau)\rangle_{\omega}$
$\displaystyle=-\frac{i}{4S}\left[\Sigma^{K}(\omega)+\Sigma^{K}(-\omega)\right]-\frac{1}{S}\text{Im}\left[\tilde{\Sigma}_{\uparrow\downarrow}^{K}\right]\,,$
(16b) $\displaystyle\langle\gamma h_{x}(t)\gamma h_{y}(\tau)\rangle_{\omega}$
$\displaystyle=-\frac{1}{4S}\left[\Sigma^{K}(\omega)-\Sigma^{K}(-\omega)\right]+\frac{1}{S}\text{Re}\left[\tilde{\Sigma}_{\uparrow\downarrow}(\omega)\right]\,,$
(16c)

where $\Sigma^{K}(\omega)$ and
$\tilde{\Sigma}^{K}_{\uparrow\downarrow}(\omega)$ are Keldysh components of
self-energies. $\Sigma^{K}(\omega)$ is driven by charge accumulation and spin
accumulation along the magnetization. In contrast,
$\tilde{\Sigma}^{K}_{\uparrow\downarrow}(\omega)$ is driven by spin
accumulation transverse to the magnetization.

The self-energy associated with charge accumulation and spin accumulation
along the magnetization is

$\displaystyle\Sigma^{K}(\omega)=i\sum_{\kappa\lambda}\sigma_{\kappa\lambda}\pi_{\kappa\lambda}(\omega)\,.$
(17)

The sums over $\kappa$ and $\lambda$ are over the left and right lead indices,
and the noise parameters in the left-right lead basis are

$\displaystyle\sigma=\left(\begin{array}[]{cc}\alpha^{r}-\beta^{r}&\beta^{r}\\\
\tilde{\beta}^{r}&\tilde{\alpha}^{r}-\tilde{\beta}^{r}\end{array}\right)\,.$
(20)

The spin transfer/spin pumping coefficients $\alpha$ and $\tilde{\alpha}$ are
introduced in Eq. (12). The shot noise parameters relevant when the charge
accumulation and longitudinal spin accumulation exceed the thermal energy and
resonance frequency are

$\displaystyle\beta_{\uparrow\downarrow}^{r}$ $\displaystyle=\frac{1}{8\pi
S}\text{Tr}\left[1-\left(r_{\uparrow}^{\dagger}r_{\downarrow}+t_{\uparrow}^{\dagger}t_{\downarrow}\right)\left(r_{\downarrow}^{\dagger}r_{\uparrow}+t_{\downarrow}^{\dagger}t_{\uparrow}\right)\right]\,,$
(21a) $\displaystyle\tilde{\beta}_{\uparrow\downarrow}^{r}$
$\displaystyle=\frac{1}{8\pi
S}\text{Tr}\left[1-\left(\tilde{r}_{\uparrow}^{\dagger}\tilde{r}_{\downarrow}+\tilde{t}_{\uparrow}^{\dagger}\tilde{t}_{\downarrow}\right)\left(\tilde{r}_{\downarrow}^{\dagger}\tilde{r}_{\uparrow}+\tilde{t}_{\downarrow}^{\dagger}\tilde{t}_{\uparrow}\right)\right]\,.$
(21b)

The fluctuations differ in the elastic and inelastic transport regimes. This
difference facilitates observation of the electron transport regime by
measuring the magnetic fluctuations. At equilibrium, in the inelastic
transport regime, and in the elastic transport regime, the contributions are
as follows:

$\displaystyle\pi_{\kappa\lambda}^{\text{eq}}$
$\displaystyle=4k_{B}T\zeta\left(\frac{\omega}{2k_{B}T}\right)\,,$ (22a)
$\displaystyle\pi_{\kappa\lambda}^{\text{in}}$
$\displaystyle=4k_{B}T\sum_{\sigma s}p_{\uparrow\sigma\kappa}p_{\downarrow
s\lambda}\zeta\left(\frac{\omega+\mu_{\sigma\kappa}-\mu_{s\lambda}}{2k_{B}T}\right)\,,$
(22b) $\displaystyle\pi_{\kappa\lambda}^{\text{el}}$
$\displaystyle=4k_{B}T\sum_{\sigma
s}\sum_{ij}p_{\uparrow\sigma\kappa}p_{\downarrow s\lambda}R_{\sigma\kappa
i}R_{s\lambda j}\zeta\left(\frac{\omega+\mu_{i}-\mu_{j}}{2k_{B}T}\right)\,,$
(22c)

where the function $\zeta$ was defined in Eq. (10) and we have introduced the
projection factor that depends on the longitudinal component of the spin
accumulation, $u_{z\kappa}$, in the distribution in Eq. (2):

$\displaystyle p_{s\sigma\kappa}$
$\displaystyle=(1-u_{z\kappa})/2+\delta_{s\sigma}u_{z\kappa}\,.$ (23)

The coefficient $R$ that governs the linear response distribution in the
elastic transport regime is defined in Eq. (4).

When spin accumulation is transverse to the magnetization, there are
contributions from the self-energy:

$\displaystyle\tilde{\Sigma}_{\uparrow\downarrow}^{K}(\omega)=-i\sum_{\kappa\lambda}\tilde{\sigma}_{\kappa\lambda}\tilde{\pi}_{\kappa\lambda}(\omega)\,,$
(24)

where the noise matrix is

$\displaystyle\tilde{\sigma}^{K}=\left(\begin{array}[]{cc}\alpha-\gamma_{\uparrow\downarrow}&\delta_{\uparrow\downarrow}\\\
\tilde{\delta_{\uparrow\downarrow}}&\tilde{\alpha}-\tilde{\gamma}_{\uparrow\downarrow}\end{array}\right)\,,$
(27)

where the shot noise parameters associated with the transverse spin
accumulation are

$\displaystyle\gamma_{\downarrow\uparrow}$ $\displaystyle=\frac{1}{16\pi
S}\text{Tr}\left[1-\left(r_{\downarrow}^{\dagger}r_{\uparrow}+t_{\downarrow}^{\dagger}t_{\uparrow}\right)^{2}\right]\,,$
(28) $\displaystyle\delta_{\downarrow\uparrow}$ $\displaystyle=\frac{1}{8\pi
S}\text{Tr}\left[\left(r_{\downarrow}^{\dagger}\tilde{t}_{\uparrow}+t_{\uparrow}^{\dagger}\tilde{r}_{\uparrow}\right)\left(\tilde{t}_{\downarrow}^{\dagger}r_{\uparrow}+\tilde{r}_{\downarrow}^{\dagger}t_{\uparrow}\right)\right]\,.$
(29)

At equilibrium, in the inelastic and elastic transport regimes, we find

$\displaystyle\tilde{\pi}^{\text{eq}}_{\kappa\lambda}$ $\displaystyle=0\,,$
(30a) $\displaystyle\tilde{\pi}^{\text{in}}_{\kappa\lambda}$
$\displaystyle=u_{\kappa s-}u_{\lambda
s-}\left[F(\mu_{\lambda\uparrow}-\mu_{\kappa\uparrow}-\omega)+F(\mu_{\lambda\downarrow}-\mu_{\kappa\downarrow}-\omega)-F(\mu_{\lambda\uparrow}-\mu_{\kappa\downarrow}-\omega)-F(\mu_{\lambda\downarrow}-\mu_{\kappa\uparrow}-\omega)\right]\,,$
(30b) $\displaystyle\tilde{\pi}_{\kappa\lambda}^{\text{el}}$
$\displaystyle=u_{\kappa s-}u_{\lambda s-}\sum_{ij}\left(R_{\uparrow\kappa
i}-R_{\downarrow\kappa i}\right)\left(R_{\uparrow\lambda
j}-R_{\downarrow\lambda j}\right)F(\mu_{j}-\mu_{i}-\omega)\,,$ (30c)

where

$\displaystyle F(\mu_{2}-\mu_{1})$
$\displaystyle=\int_{-\infty}^{\infty}d\epsilon
f(\epsilon-\mu_{1})\left[1-f(\epsilon-\mu_{2})\right]$
$\displaystyle=\left[\mu_{2}-\mu_{1}\right]n(\mu_{2}-\mu_{1})\,,$ (31)

$f(\epsilon)$ is the Fermi-Dirac distribution, and $n(\mu_{2}-\mu_{1})$ is the
Bose-Einstein distribution function at temperature $T$. At high temperatures,
$F\rightarrow k_{B}T$, and
$\tilde{\pi}^{\text{in}}=0=\tilde{\psi}^{\text{el}}$.

We discuss how the spin transfer/spin pumping and shot noise coefficients
behave in different junctions in Appendix B. In the following subsection, we
discuss three relevant scenarios for the fluctuations.

###  IV.1 Charge Accumulation-Driven Spin Fluctuations

Let us consider a scenario in which there is no spin accumulation in the
normal metals. In the inelastic transport regime, the out-of-equilibrium
chemical potentials are then independent of the spins,
$\mu_{\sigma\kappa}\rightarrow\mu_{\kappa}$. Hence,

$\displaystyle\pi^{\text{in}}_{\kappa\lambda}$
$\displaystyle=4k_{B}T\xi\left(\frac{\omega+\mu_{\kappa}-\mu_{\lambda}}{2k_{B}T}\right)\,,$
(32a) $\displaystyle\pi^{\text{el}}_{\kappa\lambda}$
$\displaystyle=4k_{B}T\sum_{ij}R_{\kappa i}R_{\lambda
j}\xi\left(\frac{\omega+\mu_{i}-\mu_{j}}{2k_{B}T}\right)\,,$ (32b)

where $R_{\kappa i}=R_{\sigma\kappa i}$ and is independent of the spin when
there is no spin accumulation. Furthermore,

$\tilde{\pi}_{\kappa\lambda}=0\,.$ (33)

Consequently, $\tilde{\Sigma}^{K}_{\uparrow\downarrow}=0$. In, e.g., the
inelastic transport regime,

$\displaystyle
i\Sigma^{K}=-4k_{B}T\left\\{\left[\left(\alpha^{r}+\tilde{\alpha}^{r}\right)-\left(\beta^{r}+\tilde{\beta}\right)\right]\xi\left(\frac{\omega}{2k_{B}T}\right)+\beta^{r}\xi\left(\frac{\omega+\mu_{L}-\mu_{R}}{2k_{B}T}\right)+\tilde{\beta}^{r}\xi\left(\frac{\omega-\mu_{L}+\mu_{R}}{2k_{B}T}\right)\right\\}\,.$
(34)

From Eq. (34), we see that measurements of the magnetic fluctuations reveal
the shot noise coefficients $\beta^{r}$ and $\tilde{\beta}$ at finite bias
voltages.

Simple limits of charge bias-driven magnetic fluctuations when there is strong
dephasing were discussed in Ref. Foros  _et al._ , 2007. The more general
coefficients $\beta^{r}$ and $\tilde{\beta}^{r}$ introduced here captures the
results in the earlier work and is more general and valid for any degree of
spin dephasing.

###  IV.2 One Reservoir

Let us consider a scenario in which there is only one reservoir, the left, and
a spin accumulation therein.

First, we consider a longitudinal spin accumulation, e.g., spin accumulation
directed along the magnetization. In this case,
$\tilde{\pi}_{\kappa\lambda}=0$, so $\tilde{\Sigma}^{K}=0$. Furthermore,
$\beta^{r}=0$, and $\tilde{\beta}^{r}=0$. We also have that
$\tilde{\alpha}=0$. The fluctuations are then determined by

$\displaystyle\Sigma^{K,\text{in}}$ $\displaystyle=i\alpha
4k_{B}T\xi\left(\frac{\omega+\mu_{s}}{2k_{B}T}\right)\,,$ (35)
$\displaystyle\Sigma^{K,\text{el}}$ $\displaystyle=i\alpha
4k_{B}T\sum_{ij}R_{\uparrow Li}R_{\downarrow
Lj}\xi\left(\frac{\omega+\mu_{i}-\mu_{j}}{2k_{B}T}\right)$ (36)

and differ in the inelastic and elastic transport regimes. When the spin
accumulation or precession frequency exceeds the energy, quantum fluctuations
exceed the classical fluctuations.

Second, we consider transverse spin accumulation. Then, fluctuations are
governed by the coefficient

$C_{\downarrow\downarrow}=\frac{1}{2}\text{Tr}\left[1-\left(r_{\downarrow}r_{\downarrow}^{\dagger}\right)^{2}\right]\,,$
(37)

and $\delta_{\downarrow\uparrow}=0$. $C_{\downarrow\uparrow}$ resembles the
spin transfer/spin pumping coefficient $A_{\uparrow\downarrow}$ of Eq. (12)
but is not identical to it. However, when there is strong dephasing,
$C_{\downarrow\uparrow}=N/2=A_{\uparrow\downarrow}$, where $N$ is the number
of propagating channels (transverse waveguide modes). Furthermore,
$\tilde{\pi}_{LR}=\tilde{\pi}_{RL}=\tilde{\pi}_{RR}=0$. In the inelastic and
elastic transport regimes, we then find in the case of pure transverse spin
accumulation ($u_{zL}=0$)

$\displaystyle-i\tilde{\Sigma}^{K,\text{in}}_{\uparrow\downarrow}$
$\displaystyle=-\left(\alpha-\gamma_{\uparrow\downarrow}\right)u_{L-}^{2}\left[2F(-\omega)-F(\mu_{s}-\omega)-F(-\mu_{s}-\omega)]\right]\,,$
(38a) $\displaystyle-i\tilde{\Sigma}^{K,\text{el}}_{\uparrow\downarrow}$
$\displaystyle=-\left(\alpha-\gamma_{\uparrow\downarrow}\right)u_{L-}^{2}\sum_{ij}\left(R_{\uparrow
Li}-R_{\downarrow Lj}\right)\left(R_{\uparrow Lj}-R_{\downarrow
Lj}\right)F(\mu_{j}-\mu_{i}-\omega)\,.$ (38b)

At zero temperature and frequency, we find that we can use $F(\omega)=0$ and
$F(\mu_{s}-\mu)+F(-\mu_{s}-\omega)=-\lvert\mu_{s}\rvert$ in Eq. (38a) and
$F(\mu_{j}-\mu_{i}-\omega)=(\mu_{i}-\mu_{j})\theta(\mu_{i}-\mu_{j})$ in Eq.
(38b), where $\Theta(x)$ is the Heaviside function.

##  V Model

This section details the model that we use to describe the normal metal-
ferromagnet-normal metal system introduced in section II. The Hamiltonian is

$\hat{H}=\int d{\bf
r}\hat{\psi}^{\dagger}\left[H_{0}+u_{s}\bm{\sigma}\cdot\hat{\bm{S}}\right]\hat{\psi}+\hat{H}_{s}(\hat{\bf
S})\,,$ (39)

where
$\hat{\psi}^{\dagger}=(\hat{\psi}_{\uparrow}^{\dagger},\hat{\psi}_{\downarrow}^{\dagger})$
is the 2-component itinerant electron field operator, ${\bm{\sigma}}$ is the
vector of Pauli matrices in the $2\times 2$ spin space, and the spin operator
$\hat{\bm{S}}$ has a total spin angular momentum $S$ such that
$\hat{\bm{S}}^{2}=S(S+1)$. We set $\hbar=1$. $\hat{H}_{s}(\hat{\bf S})$
describes the transverse dynamics of the magnetization when it is not
interacting with the itinerant electrons. To discuss the influence of the
electrons on the magnetization dynamics, we consider a simple example in which
the magnetization is subject to an external magnetic field ${\bf H}$,

$\hat{H}_{s}=-g^{*}\mu_{B}{\bf H}\cdot\hat{\bf S}\,,$ (40)

where $g^{*}$ is the effective g-factor and $\mu_{B}$ is the Bohr magneton.
Our calculations can be straightforwardly generalized to arbitrary forms of
$\hat{H}_{s}$ including magnetic anisotropy energy and demagnetization fields.
However, such complications are not the focus of our attention. Our target is
the influence of the electrons on the magnet. Therefore, we keep the
Hamiltonian $\hat{H}_{s}$ as simple as possible in our explicit calculations
but assert that the results are valid for arbitrary forms of $\hat{H}_{s}$.
Furthermore, we disregard the possible effects of an external magnetic field
on the itinerant electrons since these effects are typically small in metallic
systems, or we can include them in straightforward generalizations of the spin
transport equationsBrataas  _et al._ (2001). Such complications are irrelevant
to our main points.

The spin-independent part of the single-particle electron Hamiltonian is

$H_{0}=-\frac{1}{2m}\nabla^{2}+V_{c}({\bm{r}})\,,$ (41)

where $V_{c}({\bm{r}})$ is the spatially dependent charge potential. In the
Hamiltonian (39), $u_{s}(\bf r)$ represents the spatially dependent exchange
interaction. It is finite only inside the magnet. In the classical limit of
the magnet, the magnitude of the spatially dependent spin potential
experienced by the itinerant electrons is $V_{s}({\bf r})=Su_{s}(\bf r)$.

We will derive the semiclassical spin dynamics valid for any deviation of the
magnetization from the equilibrium orientation. To this end, considering the
magnon dynamics near the instantaneous direction of the spin $\hat{{\bm{S}}}$
and adiabatically adjusting the evolution of the small deviation of the large
spin from its instantaneous direction are sufficientChudnovskiy  _et al._
(2008); Swiebodzinski  _et al._ (2010). For simplicity, we consider an
instantaneous orientation of the spin along the $z$ direction. We carry out a
Holstein-Primakoff transformation of the localized spin operator to the second
order in the deviation from the classical ground state:

$\displaystyle\hat{S}_{+}$
$\displaystyle=\hat{S}_{x}+i\hat{S}_{y}\approx\sqrt{2S}\hat{b}\,,$ (42a)
$\displaystyle\hat{S}_{-}$
$\displaystyle=\hat{S}_{x}-i\hat{S}_{y}\approx\sqrt{2S}\hat{b}^{\dagger}\,,$
(42b) $\displaystyle\hat{S}_{z}$
$\displaystyle=\left[S-\hat{b}^{\dagger}\hat{b}\right]\,,$ (42c)

where the magnon annihilation and creation operators $\hat{b}$ and
$\hat{b}^{\dagger}$ satisfy the boson commutation relation
$\left[\hat{b},\hat{b}^{\dagger}\right]=1$. Employing
$\bm{\sigma}\cdot\hat{\bm{S}}=\sigma_{-}\hat{S}_{+}+\sigma_{+}\hat{S}_{-}+\sigma_{z}\hat{S}_{z}$,
where $\sigma_{\pm}=(\sigma_{x}\pm i\sigma_{y})/2$, by expanding to the second
order in the magnon operators, the Hamiltonian (39) becomes

$\hat{H}=\hat{H}_{e}+\hat{H}_{m}+\hat{H}_{em}\,.$ (43)

The magnon Hamiltonian $\hat{H}_{m}$ is directly related to the Hamiltonian
$\hat{H}_{s}$ (40) describing the transverse dynamics of the magnetic moments
via the Holstein-Primakoff transformation (42). Disregarding a constant energy
contribution, the magnon Hamiltonian up to the second order in the magnon
operators is

$\hat{H}_{m}=E_{z}\hat{b}^{\dagger}\hat{b}+\sqrt{S/2}\left(E_{-}\hat{b}+E_{+}\hat{b}^{\dagger}\right)\,,$
(44)

where the magnon energy is

$\displaystyle E_{z}=g^{*}\mu_{B}H_{z}\,.$ (45)

The magnons are excited via the transverse magnetic field, governed by

$\displaystyle E_{\pm}=g^{*}\mu_{B}H_{\pm}\,,$ (46)

where $H_{\pm}=H_{x}\pm iH_{y}$.

The considerable algebra outlined in section VIII demonstrates how the
electronic Hamiltonian $H_{e}$ and the electron-magnon Hamiltonian $H_{em}$
become particularly transparent in terms of the scattering states of the
itinerant electrons. We find

$\hat{H}_{e}=\sum_{s\alpha}\epsilon_{\alpha}\hat{a}_{s\alpha}^{\dagger}\hat{a}_{s\alpha}\,,$
(47)

where $\hat{a}_{s\alpha}$ annihilates an electron with spin $s$ ($s=\uparrow$
or $s=\downarrow$). The quantum number $\alpha=\kappa n\epsilon$ captures the
lead ($\kappa=L$ or $\kappa=R$), the transverse waveguide mode $n$ and the
electron energy $\epsilon$. The electron energy consists of a transverse
contribution $\epsilon_{n}$ and a longitudinal contribution
$\epsilon(k)=k^{2}/2m$, where $k$ is the longitudinal momentum such that
$\epsilon=\epsilon_{n}+\epsilon(k)$. The eigenenergy is spin degenerate since
the leads are paramagnetic. Furthermore, the eigenenergy is independent of the
lead index since we consider similar left and right leads. In Eq. (47) and
similar expressions to follow, the sum over the states implies that
$\sum_{\alpha}X_{s\alpha}$ = $\sum_{\kappa
n}\int_{\epsilon_{n}}^{\infty}d\epsilon X_{s\kappa n}(\epsilon)$. In the
scattering basis, the electron-magnon interaction becomes

$\hat{H}_{em}=\hat{H}_{1}+\hat{H}_{2}\,.$ (48)

The contribution that is to the first order in the magnon operators is

$\hat{H}_{1}=\sqrt{\frac{2}{S}}\sum_{\alpha\beta}\left[\hat{b}\hat{a}_{\downarrow\alpha}^{\dagger}W_{\downarrow\uparrow}^{\alpha\beta}\hat{a}_{\uparrow\beta}+\hat{b}^{\dagger}\hat{a}_{\uparrow\alpha}^{\dagger}W_{\uparrow\downarrow}^{\alpha\beta}\hat{a}_{\downarrow\beta}\right]\,.$
(49)

The matrices $W_{\uparrow\downarrow}$ and $W_{\downarrow\uparrow}$ are
related. They are governed by the exchange potential $u_{s}({\bm{r}})$ and the
scattering states. Importantly, we demonstrate in Appendix A that they are
proportional to a generalized _inelastic_ mode-dependent transverse ”mixing”
transport property. However, we will demonstrate that to the lowest order in
the large spin limit $S\gg 1$, only the conventional elastic ”mixing”
conductance will contribute to the deterministic part of the spin dynamics. We
will derive that new transport coefficients beyond the mixing conductance
govern the fluctuations of the spin dynamics. The contribution to the
electron-magnon interaction to the second order in the magnon operators is

$\hat{H}_{2}=-\frac{1}{S}\sum_{\alpha\beta}\hat{b}^{\dagger}\hat{b}\left[\hat{a}_{\uparrow\alpha}^{\dagger}W_{\uparrow\uparrow}^{\alpha\beta}\hat{a}_{\uparrow\beta}-\hat{a}_{\downarrow\alpha}^{\dagger}W^{\alpha\beta}_{\downarrow\downarrow}\hat{a}_{\downarrow\beta}\right]\,.$
(50)

We will show that $\hat{H}_{2}$ does not contribute to the spin dynamics when
$S\gg 1$ and hence can be disregarded.

##  VI Closed Contour Action

This section first presents the out-of-equilibrium path-integral method used
to find the influence of electron transport on spin dynamics. Subsequently, we
compute the specific form of the effective action from the electrons on the
spin dynamics. To this end, we introduce the closed contour action $S$ and
partition function $Z$,

$Z=\int
D[\bar{\psi}_{\uparrow}\psi_{\uparrow}\bar{\psi}_{\downarrow}\psi_{\downarrow}\bar{\phi}\phi]e^{iS}\,.$
(51)

In action $S$, the integral over time is along the closed contour forward from
$t=-\infty$ to $t=+\infty$ and backward from $t=+\infty$ to $t=-\infty$. From
the partition function, we can find how the electrons influence the spin
dynamics. Action $S$ consists of contributions from the electrons, magnons,
and first- and second-order electron-magnon interactions:

$S=S_{e}+S_{m}+S_{1}+S_{2}\,.$ (52)

The electron contribution corresponding to the Hamiltonian (47) is

$\displaystyle S_{e}$
$\displaystyle=\int_{-\infty}^{\infty}\\!\\!\\!\\!dt\left[\bar{\psi}^{+}_{\uparrow}\left(i\partial_{t}-\epsilon\right)\psi^{+}_{\uparrow}-\bar{\psi}^{-}_{\uparrow}\left(\partial_{t}-\epsilon\right)\psi^{-}_{\uparrow}\right]+$
$\displaystyle\int_{-\infty}^{\infty}\\!\\!\\!\\!dt\left[\bar{\psi}^{+}_{\downarrow}\left(i\partial_{t}-\epsilon\right)\psi^{+}_{\downarrow}-\bar{\psi}^{-}_{\downarrow}\left(\partial_{t}-\epsilon\right)\psi^{-}_{\downarrow}\right]\,,$
(53)

where $\psi_{\sigma}$ ($\sigma=\uparrow$ or $\sigma=\downarrow$) is a vector
consisting of all Grassmann variables representing the electronic states. The
lead, transverse waveguide mode, and electron energy for spin $\sigma$
electrons span the vector. The superscript $+$ ($-$) denotes the forward
(backward) path variables. In Eq. (53), $\epsilon$ is a diagonal matrix
containing all single-particle energies spanned by the electronic states.

The magnon contribution to the action corresponding to the magnon Hamiltonian
(44) is

$\displaystyle
S_{m}=\int_{-\infty}^{\infty}\\!\\!\\!\\!dt\left[\bar{\phi}^{+}\left(i\partial_{t}-E_{z}\right)\phi^{+}-\bar{\phi}^{-}\left(i\partial_{t}-E_{z}\right)\phi^{-}\right]-$
$\displaystyle\sqrt{S/2}\int_{-\infty}^{\infty}\\!\\!\\!\\!dt\left[E_{-}\left(\phi^{+}-\phi^{-}\right)+E_{+}\left(\bar{\phi}^{+}-\bar{\phi}^{-}\right)\right]\,,$
(54)

where $\phi$ is a complex number and $\bar{\phi}$ is its conjugate number. The
electron-magnon interaction to the first order in the magnon operators
corresponding to the Hamiltonian (49) is

$\displaystyle S_{1}$
$\displaystyle=-\sqrt{\frac{2}{S}}\int_{-\infty}^{\infty}\\!\\!\\!\\!dt\left[\bar{\psi}_{\downarrow}^{+}W_{\downarrow\uparrow}\psi^{+}_{\uparrow}\phi^{+}-\bar{\psi}_{\downarrow}^{-}W_{\downarrow\uparrow}\psi^{-}_{\uparrow}\phi^{-}\right]-$
$\displaystyle\sqrt{\frac{2}{S}}\int_{-\infty}^{\infty}dt\left[\bar{\psi}^{+}_{\uparrow}W_{\uparrow\downarrow}\psi^{+}_{\downarrow}\bar{\phi}^{+}-\bar{\psi}_{\uparrow}^{-}W_{\uparrow\downarrow}\psi^{-}_{\downarrow}\bar{\phi}^{-}\right]\,.$
(55)

The electron-magnon interaction to the second order in the magnon operators
representing the Hamiltonian (50) is

$\displaystyle S_{2}$
$\displaystyle=\frac{1}{S}\int_{-\infty}^{\infty}\\!\\!\\!\\!dt\left[\bar{\psi}^{+}_{\uparrow}W_{\uparrow\uparrow}\psi^{+}_{\uparrow}-\bar{\psi}^{+}_{\downarrow}W_{\downarrow\downarrow}\psi^{+}_{\downarrow}\right]\bar{\phi}^{+}\phi^{+}$
$\displaystyle-\frac{1}{S}\int_{-\infty}^{\infty}\\!\\!\\!\\!dt\left[\bar{\psi}^{-}_{\uparrow}W_{\uparrow\uparrow}\psi_{\uparrow}-\bar{\psi}^{-}_{\downarrow}W_{\downarrow\downarrow}\psi_{\downarrow}\right]\bar{\phi}^{-}\phi^{-}\,.$
(56)

Introducing classical and quantum fields for the bosons and $1-2$ fields for
the fermions instead of the $+$ and $-$ fields on the forward and backward
paths is convenient and conventionalKamenev (2011).

###  VI.1 Classical Quantum and 1-2 Quantum Fields

For the bosons, we use the classical and quantum fields

$\displaystyle\phi^{c,q}$
$\displaystyle=\left(\phi^{+}\pm\phi^{-}\right)/\sqrt{2}\,,$ (57a)
$\displaystyle\bar{\phi}^{c,q}$
$\displaystyle=\left(\bar{\phi}^{+}\pm\bar{\phi}^{-}\right)/\sqrt{2}\,.$ (57b)

The upper sign is for the classical ($c$) field, and the lower sign is for the
quantum field ($q$). The structure of the boson Green function is therefore

$D=\left(\begin{array}[]{cc}D^{K}&D^{R}\\\ D^{A}&0\end{array}\right)\,.$ (58)

In the representation of Eqs. (57) and (58), the magnon contribution to the
action in Eq. (54) becomes

$\displaystyle S_{m}=\int dt\int
d\tau(\bar{\phi}^{c},\bar{\phi}^{q})D_{0}^{-1}\left(\begin{array}[]{cc}\phi^{c}\\\
\phi^{q}\end{array}\right)$ (60) $\displaystyle-\sqrt{S}\int
dt\left[E_{-}\phi^{q}+E_{+}\bar{\phi}^{q}\right]\,,$ (61)

where $\left[D_{0}^{-1}\right]^{R(A)}=\delta(t-\tau)(i\partial_{t}-E_{z}\pm
i0^{+})$.

For the fermions, we use a different notation of 1-2 fields:

$\displaystyle\psi^{1,2}=\left(\psi^{+}\pm\psi^{-}\right)/\sqrt{2}\,,$ (62a)
$\displaystyle\bar{\psi}^{1,2}=\left(\bar{\psi}^{+}\mp\bar{\psi}^{-}\right)/\sqrt{2}\,.$
(62b)

The structure of the fermion Green function is then

$G=\left(\begin{array}[]{cc}G^{R}&G^{K}\\\ 0&G^{A}\end{array}\right)\,.$ (63)

The occupations at the initial time $t\rightarrow-\infty$ are included via a
standard regularization of the Green functions for bosons and fermions. This
procedure results in the Keldysh components for the bosons and fermions. In
the Keldysh (classical quantum for bosons and 1-2 for fermions) space, we use
the matrices

$\gamma^{c}=\left(\begin{array}[]{cc}1&0\\\ 0&1\end{array}\right)$ (64)

and

$\gamma^{q}=\left(\begin{array}[]{cc}0&1\\\ 1&0\end{array}\right)\,.$ (65)

The contributions to the action involving the electrons, i.e., the electron
contribution and the electron-magnon contributions,
$S_{e,tot}=S_{e}+S_{1}+S_{2}$, are then

$S_{e,tot}=\int dt\int
d\tau\left(\psi^{1},\psi^{2}\right)G^{-1}\left(\begin{array}[]{cc}\psi^{1}\\\
\psi^{2}\end{array}\right)\,,$ (66)

where the inverse Green function for the fermions is

$G^{-1}=G_{0}^{-1}+\left[\tilde{W}_{1}+\tilde{W}_{2}\right]\,,$ (67)

$\left[G_{0}^{-1}\right]^{R(A)}=\delta(t-\tau)(i\partial_{t}-\epsilon\pm
i0^{+})$, $\tilde{W}_{1}=\delta(t-\tau)W_{1}$,
$\tilde{W}_{2}=\delta(t-\tau)W_{2}$,

$\displaystyle W_{1}$
$\displaystyle=-\frac{1}{\sqrt{S}}\sum_{x}\left[\bar{\psi}_{\downarrow}W_{\downarrow\uparrow}\gamma^{x}\psi_{\uparrow}\phi^{x}+\bar{\psi}_{\uparrow}W_{\uparrow\downarrow}\gamma^{x}\psi_{\downarrow}\bar{\phi}^{x}\right]$
(68)

and

$\displaystyle W_{2}$
$\displaystyle=\frac{1}{2S}\sum_{x}\left[\bar{\psi}_{\uparrow}W_{\uparrow\uparrow}\gamma^{x}\psi_{\uparrow}-\bar{\psi}_{\downarrow}W_{\downarrow\downarrow}\gamma^{x}\psi_{\downarrow}\right]\bar{\phi}\gamma^{x}\phi\,.$
(69)

In Eqs. (68) and (69), $x=c,q$. In the partition function (51), we can now
integrate over the fermions to find the effective contribution to the boson
action

$S_{\text{eff}}=-i\text{Tr}\ln\left[1+G_{0}\left(\tilde{W}_{1}+\tilde{W}_{2}\right)\right]\,.$
(70)

In the following sections, we will expand (70) to the second order in the
magnon operators, but first, we need to discuss the unperturbed Green
functions of the electrons that contain information about the charge and spin
accumulations in the reservoirs discussed in section III.

###  VI.2 Unperturbed Electron Green Function

The unperturbed electron Green function is the Green function in the absence
of the electron-magnon interactions in Eqs. (49) and (50) or, alternatively,
the electron-magnon interactions in Eqs. (55) and (56). The retarded and
advanced parts of the zeroth-order Green function $G_{0}$ are diagonal in spin
space and all other quantum variables (lead, transverse mode, and energy). In
the Fourier representation (197),

$G_{0\alpha\beta}^{R/A}(\omega)=\frac{1}{\omega\pm
i0^{+}-\epsilon_{\alpha}}\delta_{\alpha\beta}\,.$ (71)

However, the zeroth-order Keldysh component of the Green function $G_{0}$ has
off-diagonal components in spin space since there is a noncollinear spin
accumulation in the leads. This out-of-equilibrium noncollinear spin
accumulation is what couples to the localized spins.

There are charge and spin accumulations in the reservoirs, as described in
section III. Consequently, the Keldysh component of the Green function is

$G^{K}_{0s\alpha\sigma\beta}(\omega)=-2\pi
i\delta_{\alpha\beta}\delta(\omega-\epsilon_{\alpha})\left[\delta_{s\sigma}-2n_{s\sigma\alpha}\right]\,,$
(72)

where $n_{s\sigma\alpha}$ is the $2\times 2$ out-of-equilibrium occupation of
the electrons introduced in Eq. (2).

At equilibrium, $n_{s\sigma\alpha}=\delta_{s\sigma}f(\epsilon-\mu_{0})$, where
$\mu_{0}$ is the equilibrium chemical potential, $\epsilon$ is the energy of
state $\alpha$ with spin s, and $f(\epsilon-\mu_{0})$ is the Fermi-Dirac
distribution function. The fluctuation-dissipation theorem for fermions holds:

$\displaystyle
G_{0s\alpha\sigma\beta}^{K,\text{eq}}=\tanh{\frac{\epsilon-\mu_{0}}{2k_{B}T}}\delta_{s\sigma}\left[G^{R}_{0\alpha\beta}-G^{A}_{0\alpha\beta}\right]\,.$
(73)

Naturally, we will subsequently find that an analogous relation holds for the
(boson) magnon self-energy at equilibrium with a magnon chemical potential of
zero.

###  VI.3 First-Order Corrections to the Magnon Actions

By expanding the effective action (70) in the electron-magnon coupling, we
find the contribution to the action of the fermions to the first order in the
magnon operators:

$\displaystyle S_{1}=$
$\displaystyle-i\text{Tr}\left[G_{0}\tilde{W}_{1}\right]$ $\displaystyle=$
$\displaystyle-\frac{2}{\sqrt{S}}\sum_{\alpha}W_{\downarrow\uparrow\alpha\alpha}n_{\uparrow\downarrow\alpha}\int
dt\phi^{q}$
$\displaystyle-\frac{2}{\sqrt{S}}\sum_{\alpha}W_{\uparrow\downarrow\alpha\alpha}n_{\downarrow\uparrow\alpha}\int
dt\bar{\phi}^{q}\,.$ (74)

By comparing the electron-induced action in Eq. (74) with the contribution to
the magnetic action from the transverse magnetic field in Eq. (61), we find
that the spin accumulation gives rise to an effective transverse magnetic
field represented by the effective energies
$E_{+e}=2\sum_{\alpha}W_{\uparrow\downarrow}n_{\downarrow\uparrow\alpha\alpha}/S$
and
$E_{-e}=2\sum_{\alpha}W_{\downarrow\uparrow}n_{\uparrow\downarrow\alpha\alpha}/S$.
In other words, the action (74) corresponds to a transverse magnetic field
where the $x$\- and $y$-components are $H_{xe}=(E_{+e}+E_{-e})/2g^{*}\mu_{B}$
and $H_{ye}=(E_{+e}-E_{-e})/2ig^{*}\mu_{B}$. We observe that the effective
spin transfer magnetic fields are inversely proportional to the total spin
$S$.

We now use Eqs. (151) and (155) to find

$\sum_{\alpha}W_{\uparrow\downarrow\alpha\alpha}n_{\downarrow\uparrow\alpha}=iS\int
d\epsilon\left[\alpha n_{\downarrow\uparrow
L}(\epsilon)+\tilde{\alpha}n_{\downarrow\uparrow R}(\epsilon)\right]\,,$ (75)

where $n_{\downarrow\uparrow L}$ ($n_{\downarrow\uparrow R}$) is a transverse
component of the $2\times 2$ spin distribution in the left (right) reservoir
in Eq. (2) and we have defined the energy-dependent spin transfer coefficients
$\alpha$ and $\tilde{\alpha}$ in Eq. (12). Furthermore, in metallic systems,
the out-of-equilibrium distribution peaks at the Fermi energy, and

$\sum_{\alpha}W_{\uparrow\downarrow\alpha\alpha}n_{\uparrow\downarrow\alpha}=iS\left[\alpha\mu_{\downarrow\uparrow}+\tilde{\alpha}\tilde{\mu}_{\downarrow\uparrow}\right]\,,$
(76)

where the spin transfer coefficients $\alpha$ and $\tilde{\alpha}$ should be
evaluated at the Fermi energy and the effective chemical potentials are

$\displaystyle\mu_{\downarrow\uparrow}$ $\displaystyle=\int
d\epsilon_{nk}n_{\downarrow\uparrow L}\,,$ (77a)
$\displaystyle\tilde{\mu}_{\downarrow\uparrow}$ $\displaystyle=\int
d\epsilon_{nk}n_{\downarrow\uparrow R}$ (77b)

in the left and right reservoirs. We find similar results for
$\sum_{\alpha}W_{\downarrow\uparrow\alpha\alpha}n_{\uparrow\downarrow\alpha}$.
With the definition of the effective chemical potential in Eq. (77), Eq. (6)
and Eq. (3), we see that
$\mu^{in}_{\downarrow\uparrow}=\mu_{\downarrow\uparrow}=\mu^{el}_{\downarrow\uparrow}$
and
$\tilde{\mu}^{in}_{\downarrow\uparrow}=\tilde{\mu}_{\downarrow\uparrow}=\tilde{\mu}^{el}_{\downarrow\uparrow}$.
Therefore, the spin transfer torques are the same in the elastic and inelastic
transport regimes. This is because spin transfer arises from the transport of
electrons around the Fermi energy, where the transport coefficients are
constant in a metallic system since the out-of-equilibrium accumulations are
much smaller than the Fermi energy.

A more transparent representation of the effect of action $S_{1}$ on the spin
dynamics is in terms of the charge accumulation
$\mu_{c}=\text{Tr}\left[\check{\mu}\right]/2$ and spin accumulation vector
${\bm{\mu}}_{s}=\text{Tr}\left[{\bm{\sigma}}\check{\mu}\right]/2$, where
$\check{\mu}$ is the $2\times 2$ distribution function in spin space. Then,
$\mu_{\uparrow\downarrow}=\mu_{s,x}-i\mu_{s,y}$, and
$\mu_{\downarrow\uparrow}=\mu_{s,x}+i\mu_{s,y}$; thus, the spin transfer
effective field is ${\bm{H}}_{\text{stt}}={\bm{H}}_{e}+\tilde{{\bm{H}}}_{e}$,
where

$\displaystyle{\bm{H}}_{e}$
$\displaystyle=\frac{1}{g^{*}\mu_{B}}\left[\alpha^{r}{\bm{m}}\times{\bm{\mu}}_{s}+\alpha^{i}{\bf
m}\times{\bf m}\times{\bm{\mu}}_{s}\right]\,.$ (78a)
$\displaystyle\tilde{{\bm{H}}}_{e}$
$\displaystyle=\frac{1}{g^{*}\mu_{B}}\left[\tilde{\alpha}^{r}{\bm{m}}\times\tilde{{\bm{\mu}}}_{s}+\tilde{\alpha}^{i}{\bf
m}\times{\bf m}\times\tilde{{\bm{\mu}}}_{s}\right]\,.$ (78b)

Using the gyromagnetic ratio $\gamma=-g^{*}\mu_{B}/\hbar$ while maintaining
$\hbar=1$, we can also express the action of the effective fields in Eq. (78a)
on the spin dynamics as the spin transfer torques in Eq. (11).

In Eqs. (78a) and (78) and what follows, the superscript $r$ denotes the real
part, and the superscript $i$ denotes the imaginary part. The spin transfer
and spin pumping efficiencies associated with the left and right reservoirs
introduced in Eq. (12) are related to the ”mixing” conductances
$A_{\uparrow\downarrow}$ and $\tilde{A}_{\uparrow\downarrow}$ defined in Eqs.
(155a) and (155b) as follows:

$\displaystyle\alpha$ $\displaystyle=\frac{A_{\uparrow\downarrow}}{4\pi S}\,,$
(79a) $\displaystyle\tilde{\alpha}$
$\displaystyle=\frac{\tilde{A}_{\uparrow\downarrow}}{4\pi S}\,.$ (79b)

Since the mixing conductances $A_{\uparrow\downarrow}$ and
$\tilde{A}_{\uparrow\downarrow}$ in Eqs. (155a) and (155b) are normal metal-
ferromagnet interface properties proportional to the cross section and the
total spin $S$ is proportional to the volume of the ferromagnet, the spin
transfer efficiencies (12) are inversely proportional to the ferromagnet
thickness.

Additionally, we will soon see the well-known fact that spin pumping gives
rise to enhanced damping that acquires contributions from the left and right
interfaces governed by the spin transfer/spin pumping efficiencies $\alpha$
and $\tilde{\alpha}$ in Eq. (12).

###  VI.4 Second-Order Corrections to the Magnon Action

The contribution to the action of the fermions to the second order in the
magnon operators is $S_{2}=S_{2a}+S_{2b}$, where

$\displaystyle S_{2a}$ $\displaystyle=-i\text{Tr}\left[G_{0}W_{2}\right]\,,$
(80a) $\displaystyle S_{2b}$
$\displaystyle=\frac{i}{2}\text{Tr}\left[G_{0}W_{1}G_{0}W_{1}\right]\,.$ (80b)

We will compute these contributions separately. First, we compute

$S_{2a}=-\frac{1}{2S}\sum_{\alpha}\left[(1-2n_{\uparrow\uparrow\alpha})W_{\uparrow\uparrow\alpha}-(1-2n_{\downarrow\downarrow\alpha})W_{\downarrow\downarrow\alpha}\right]\int
dt\bar{\phi}\gamma^{q}\phi\,.$ (81)

Action $S_{2a}$ therefore renormalizes the magnon energy $E_{z}$, as it
appears in the magnon action (61) and corresponds to a longitudinal magnetic
field driven by the longitudinal spin accumulation represented by
$n_{\uparrow\uparrow\alpha}$ and $n_{\downarrow\downarrow\alpha}$.
$E_{z}\rightarrow E_{z}+E_{ze}$. $E_{ze}$ is finite even at equilibrium due to
the exchange coupling between the itinerant electrons and the localized spins.
Furthermore, the renormalized magnon energy can change due to the out-of-
equilibrium longitudinal spin accumulation, $E_{ez}=E_{ez,eq}+E_{ez,ne}$.
However, longitudinal magnetic fields have no consequences for the spin
dynamics since, in the instantaneous reference frame, such contributions
correspond to a total free energy of the form $a\hat{\bm{S}}^{2}$, where $a$
is a constantChudnovskiy  _et al._ (2008); Swiebodzinski  _et al._ (2010).
Hence, we will not discuss action $S_{2a}$ in Eq. (81) further.

Next, we consider the second term $S_{2b}$ in Eq. (80b). There are four
contributions to this term:

$\displaystyle S_{2b}=$ $\displaystyle\frac{i}{2S}\int dt\int
d\tau\sum_{xy}\text{Tr}\left[G_{0\uparrow\downarrow}(\tau,t)W_{\downarrow\uparrow}\gamma^{x}G_{0\uparrow\downarrow}(t,\tau)W_{\downarrow\uparrow}\gamma^{y}\right]\phi^{x}(t)\phi^{y}(\tau)$
$\displaystyle+$ $\displaystyle\frac{i}{2S}\int dt\int
d\tau\sum_{xy}\text{Tr}\left[G_{0\uparrow\uparrow}(\tau,t)W_{\uparrow\downarrow}\gamma^{x}G_{0\downarrow\downarrow}(t,\tau)W_{\downarrow\uparrow}\gamma^{y}\right]\bar{\phi}^{x}(t)\phi^{y}(\tau)$
$\displaystyle+$ $\displaystyle\frac{i}{2S}\int dt\int
d\tau\sum_{xy}\text{Tr}\left[G_{0\downarrow\downarrow}(\tau,t)W_{\downarrow\uparrow}\gamma^{x}G_{0\uparrow\uparrow}(t,\tau)W_{\uparrow\downarrow}\gamma^{y}\right]\phi^{x}(t)\bar{\phi}^{y}(\tau)$
$\displaystyle+$ $\displaystyle\frac{i}{2S}\int dt\int
d\tau\sum_{xy}\text{Tr}\left[G_{o\downarrow\uparrow}(t,\tau)W_{\uparrow\downarrow}\gamma^{x}G_{o\downarrow\uparrow}(t,\tau)W_{\uparrow\downarrow}\gamma^{y}\right]\bar{\phi}^{x}(t)\bar{\phi}^{y}(\tau)\,.$
(82)

We will now discuss the contributions to $S_{2b}$ line by line. The first line
of Eq. (82) becomes

$S_{2b}^{qq}=\int dt\int
d\tau\phi^{q}(t)\tilde{\Sigma}_{\uparrow\downarrow}^{K}(t,\tau)\phi^{q}(\tau)\,,$
(83)

where we have introduced the out-of-equilibrium transverse spin accumulation-
induced Keldysh component

$\tilde{\Sigma}_{\uparrow\downarrow}^{K}(t,\tau)=\frac{i}{2S}\sum_{\alpha\beta}G_{0\uparrow\downarrow\alpha\alpha}^{K}(\tau,t)W_{\downarrow\uparrow\alpha\beta}G_{0\uparrow\downarrow\beta\beta}^{K}(t,\tau)W_{\downarrow\uparrow\beta\alpha}\,.$
(84)

Inserting the electron Green functions given by Eqs. (71) and (72), we compute

$\tilde{\Sigma}_{\uparrow\downarrow}^{K}(t-\tau)=-\frac{2i}{S}\sum_{\alpha\beta}n_{\uparrow\downarrow\alpha}W_{\downarrow\uparrow\alpha\beta}n_{\uparrow\downarrow\beta}W_{\downarrow\uparrow\beta\alpha}e^{-i(\epsilon_{\alpha}-\epsilon_{\beta})(\tau-t)}\,.$
(85)

Importantly, inspecting the self-energy
$\tilde{\Sigma}_{\uparrow\downarrow}^{K}(t-\tau)$ in Eq. (85) demonstrates
that it is an even function of the relative time $t-\tau$:

$\displaystyle\tilde{\Sigma}_{\uparrow\downarrow}^{K}(t-\tau)=\tilde{\Sigma}_{\uparrow\downarrow}^{K}(\tau-t)\,.$
(86)

This symmetry is essential in its contribution to the fluctuating field
experienced by the ferromagnet.

Similarly, the fourth line of Eq. (82) becomes

$S_{2b}^{\bar{q}\bar{q}}=\int dt\int
d\tau\bar{\phi}^{q}(t)\tilde{\Sigma}_{\downarrow\uparrow}^{K}(t,\tau)\bar{\phi}^{q}(\tau)\,,$
(87)

where $\tilde{\Sigma}_{\downarrow\uparrow}^{K}$ is similar to
$\tilde{\Sigma}_{\uparrow\downarrow}^{K}$ in Eq. (85) by flipping both spin
indices, i.e.,

$\displaystyle\tilde{\Sigma}^{K}_{\downarrow\uparrow}(t-\tau)=-\frac{2i}{S}\sum_{\alpha\beta}n_{\downarrow\uparrow\alpha}W_{\uparrow\downarrow\alpha}n_{\downarrow\uparrow\beta}W_{\downarrow\uparrow\beta\alpha}e^{-i(\epsilon_{\alpha}-\epsilon_{\beta})(t-\tau)}\,.$
(88)

By using the properties of the complex conjugate of the out-of-equilibrium
distribution in Eq. (2), $n_{\downarrow\uparrow}^{*}=n_{\uparrow\downarrow}$,
we can relate $\tilde{\Sigma}^{K}_{\uparrow\downarrow}$ in Eq. (88) to
$\tilde{\Sigma}^{K}_{\downarrow\uparrow}$ in Eq. (85):

$\displaystyle\tilde{\Sigma}_{\downarrow\uparrow}^{K}(t-\tau)=-\left[\tilde{\Sigma}^{K}_{\uparrow\downarrow}(t-\tau)\right]^{*}\,,$
(89)

where the superscript $*$ implies the complex conjugate. We will demonstrate
that this relation (89) is essential in ensuring that the fluctuations of the
effective transverse fields are real numbers.

From Eq. (85), we see that $\tilde{\Sigma}^{K}_{\uparrow\downarrow}$ vanishes
at equilibrium. $\tilde{\Sigma}^{K}_{\uparrow\downarrow}$ is quadratic in the
out-of-equilibrium transverse spin accumulation in the classical limit when
the thermal energy is much larger than the spin accumulation,
$k_{B}T\gg\mu_{s}$. Therefore, actions $S_{2b}^{qq}$ and
$S_{2b}^{\bar{q}\bar{q}}$ can be disregarded in the classical limit.

Interestingly, there are contributions from
$\tilde{\Sigma}^{K}_{\uparrow\downarrow}$ related to quantum shot noise in the
quantum limit at low temperatures when $k_{B}T\leq\mu_{s}$. We find the
general expression for $\tilde{\Sigma}_{\uparrow\downarrow}^{K}$ valid for
arbitrary ratios between the spin accumulation and thermal energy by Fourier
transforming Eq. (85):

$\tilde{\Sigma}_{\uparrow\downarrow}^{K}(\omega)=-\frac{4\pi
i}{S}\sum_{\alpha\beta}n_{\uparrow\downarrow\alpha}W_{\downarrow\uparrow\alpha\beta}n_{\uparrow\downarrow\beta}W_{\downarrow\uparrow\beta\alpha}\delta(\omega+(\epsilon_{\alpha}-\epsilon_{\beta}))\,.$
(90)

Subsequently, we disregard small terms of the order $\mu_{s}/\epsilon_{F}$ and
$k_{B}T/\epsilon_{F}$ and define the transverse spin accumulation-driven
parameters

$\tilde{\pi}_{\kappa\lambda}(\omega)=-4\int_{-\infty}^{\infty}d\epsilon
n_{\uparrow\downarrow\kappa}(\epsilon)n_{\uparrow\downarrow\lambda}(\epsilon+\omega)$
(91)

so that the Fourier transform of $\tilde{\Sigma}^{K}_{\uparrow\downarrow}$
becomes Eq. (24), where

$\tilde{\sigma}_{\uparrow\downarrow\kappa\lambda}=-\frac{\pi}{S}\sum_{nm}W_{\downarrow\uparrow\kappa
n\lambda m}W_{\downarrow\uparrow\lambda m\kappa n}$ (92)

and all energy arguments of the matrix elements are at the Fermi energy. To
evaluate $\tilde{\pi}_{\kappa\lambda}(\omega)$ in Eq. (91), we use
distribution (2) and
${\bm{u}}_{\kappa}\cdot{\bm{\sigma}}_{\uparrow\downarrow}/2=(u_{\kappa
x}-iu_{\kappa y})/2\equiv u_{\kappa-}/2$ to find

$\displaystyle\tilde{\pi}_{\kappa\lambda}(\omega)=-u_{\kappa s-}u_{\lambda
s-}$
$\displaystyle\int_{-\infty}^{\infty}d\epsilon\left[f_{\uparrow\kappa}(\epsilon)-f_{\downarrow\kappa}(\epsilon)\right]\left[f_{\uparrow\lambda}(\epsilon+\omega)-f_{\downarrow\lambda}(\epsilon+\omega)\right]\,.$
(93)

We compute $\tilde{\pi}^{K}$ at equilibrium and in the inelastic and elastic
transport regimes and find the result in Eq. (30).

When the system is at equilibrium, we find
$\tilde{\pi}_{\kappa\lambda}(\omega)=0$. Furthermore, we observe that
$\tilde{\Sigma}_{\uparrow\downarrow}^{K}=0$ in the classical limit when
$k_{B}T\gg\mu_{s}$, as anticipated above.

Next, we can see that the second and third lines in Eq. (82) are identical by
interchanging $x\leftrightarrow y$ and $t\leftrightarrow\tau$ and using the
cyclic properties of the trace. Furthermore, the $cc$ components vanish
because they are combinations of products of retarded Green functions with
positive and negative time and products of advanced Green functions with
positive and negative time. The second and third lines in Eq. (82) give a
contribution to the $qc$ component of

$S_{2b}^{qc}=\int dt\int
d\tau\bar{\phi}^{q}(t)\Sigma^{R}(t,\tau)\phi^{c}(\tau)\,,$ (94)

where the retarded self-energy is

$\displaystyle\Sigma^{R}(t,\tau)=$
$\displaystyle\frac{i}{S}\text{Tr}\left[G_{0\uparrow\uparrow}^{K}(\tau,t)W_{\uparrow\downarrow}G_{0\downarrow\downarrow}^{R}(t,\tau)W_{\downarrow\uparrow}\right]$
$\displaystyle+$
$\displaystyle\frac{i}{S}\text{Tr}\left[G_{0\uparrow\uparrow}^{A}(\tau,t)W_{\uparrow\downarrow}G_{0\downarrow\downarrow}^{K}(t,\tau)W_{\uparrow\downarrow}\right]\,.$
(95)

We then compute that the retarded self-energy is

$\Sigma^{R}(t-\tau)=i\theta(t-\tau)\frac{2}{S}\sum_{\alpha\beta}\left(n_{\uparrow\uparrow\alpha}-n_{\downarrow\downarrow\beta}\right)\lvert
W_{\uparrow\downarrow\alpha\beta}\rvert^{2}e^{i(\epsilon_{\alpha}-\epsilon_{\beta})(t-\tau)}\,,$
(96)

where
$W_{\uparrow\downarrow\alpha\beta}^{*}=W_{\downarrow\uparrow\beta\alpha}$.
Similarly, we find that the contribution to the $cq$ component from the second
and third terms of Eq. (82) is

$S_{2b}^{cq}=\int dt\int d\tau\phi^{c}(t)D^{A}(t,\tau)\phi^{q}(\tau),$ (97)

where the advanced self-energy is

$\Sigma^{A}(t-\tau)=-i\theta(\tau-t)\frac{2}{S}\sum_{\alpha\beta}\left(n_{\uparrow\uparrow\alpha}-n_{\downarrow\downarrow\beta}\right)\lvert
W_{\uparrow\downarrow\alpha\beta}\rvert^{2}e^{i(\epsilon_{\alpha}-\epsilon_{\beta})(t-\tau)}\,.$
(98)

Finally, the contribution to the $qq$ component from the second and third
terms of Eq. (82) is

$S_{2b}^{qq}=\int dt\int
d\tau\bar{\phi}^{q}(t)\Sigma^{K}(t,\tau)\phi^{q}(\tau)\,,$ (99)

where the Keldysh self-energy is

$\Sigma^{K}(t,\tau)=\frac{2i}{S}\sum_{\alpha\beta}\left(-2n_{\uparrow\uparrow\alpha}n_{\downarrow\downarrow\beta}+n_{\uparrow\uparrow\alpha}+n_{\downarrow\downarrow\beta}\right)\lvert
W_{\uparrow\downarrow\alpha\beta}\rvert^{2}e^{i(\epsilon_{\alpha}-\epsilon_{\beta})(t-\tau)}\,.$
(100)

We note that the sum $i\Sigma^{K}(t-\tau)+i\Sigma^{K}(\tau-t)$ and the
difference $\Sigma^{K}(t-\tau)-\Sigma^{K}(\tau-t)$ are real numbers. These
properties are crucial in ensuring that the fluctuating transverse fields
acting on the magnetization are real numbers.

Collecting the $qc$, $cq$ and $qq$ components of $S_{2b}$, we see that the
contributions from the second and third terms of Eq. (82) can be expressed as

$S_{2b}^{\bar{c}q}+S_{2b}^{\bar{q}c}+S_{2b}^{\bar{q}q}=\int dt\int
d\tau\left(\bar{\phi}^{c}\bar{\phi}^{q}\right)\Sigma\left(\begin{array}[]{cc}\phi^{c}\\\
\phi^{q}\end{array}\right)\,,$ (101)

where the self-energy is

$\Sigma=\left(\begin{array}[]{cc}0&\Sigma^{A}\\\
\Sigma^{R}&\Sigma^{K}\end{array}\right)\,.$ (102)

Using the Fourier transform of Eq. (197), we find that the Fourier transforms
of the retarded and advanced self-energies are

$\displaystyle\Sigma^{R/A}=-\frac{2}{S}\sum_{\alpha\beta}\frac{n_{\uparrow\uparrow\alpha}-n_{\downarrow\downarrow\beta}}{\omega+\epsilon_{\alpha}-\epsilon_{\beta}\pm
i0^{+}}\lvert W_{\uparrow\downarrow\alpha\beta}\rvert^{2}\,.$ (103)

The Fourier transform of the Keldysh component of the self-energy is

$\Sigma^{K}=i\frac{4\pi}{S}\sum_{\alpha\beta}\delta(\omega+\epsilon_{\alpha}-\epsilon_{\beta})\left(-2n_{\uparrow\uparrow\alpha}n_{\downarrow\downarrow\beta}+n_{\uparrow\uparrow\alpha}+n_{\downarrow\downarrow\beta}\right)\lvert
W_{\uparrow\downarrow\alpha\beta}\rvert^{2}\,.$ (104)

As we see, $\Sigma^{K}$ is purely imaginary:

$\displaystyle\Sigma^{K}(\omega)=i\text{Im}\left[\Sigma^{K}(\omega)\right]\,,$
(105)

which is important because $i\Sigma^{K}(\omega)$ controls magnetization
fluctuations. Furthermore, the imaginary part is positive definite. This
feature is important because the self-energy appears as
$\exp\left[-\text{Im}\Sigma(\omega)\lvert\phi^{q}(\omega)\rvert^{2}\right]$ in
the partition function of Eq. (51). Consequently, the partition function is
well defined.

Disregarding small terms of the order $\omega/\epsilon_{F}$, we compute from
Eq. (103) that

$\displaystyle\Sigma^{R}-\Sigma^{A}$ $\displaystyle=i\omega\frac{1}{4\pi
S}2\left[A_{\uparrow\downarrow}^{r}+\tilde{A}_{\uparrow\downarrow}^{r}\right]$
(106a) $\displaystyle=i\omega 2\left[\alpha^{r}+\tilde{\alpha}^{r}\right]\,,$
(106b)

where $A_{\uparrow\downarrow}$ and $\tilde{A}_{\uparrow\downarrow}$ are
defined in Eq. (155) and the spin transfer coefficients $\alpha$ and
$\tilde{\alpha}$ are defined in Eq. (12). Additionally, we have the symmetry

$\displaystyle\Sigma^{K}(\omega)=\left[\Sigma^{K}(-\omega)\right]_{\uparrow\uparrow\leftrightarrow\downarrow\downarrow}\,.$
(107)

In other words, Eq. (107) describes that flipping the spin accumulation along
the magnetization and reversing the frequency have the same effect on the
self-energy $\Sigma^{K}(\omega)$.

In general, we disregard small terms of the order $k_{B}T/\epsilon_{F}$ and
$(\mu_{\uparrow}-\mu_{\downarrow})/\epsilon_{F}$ and find from Eq. (104) that
the self-energy can be expressed as Eq. (17), where the noise matrix is
defined as

$\sigma_{\kappa\lambda}=\frac{2\pi}{S}\sum_{nm}\lvert
W_{\uparrow\downarrow\kappa n\lambda m}\rvert^{2}\,,$ (108)

and the matrix element $W_{\uparrow\downarrow\kappa n\lambda m}$ should be
evaluated at the Fermi energy. We can relate $\sigma_{\kappa\lambda}$ to the
scattering matrix, as shown in Appendix A. On the basis of the lead (left or
right), we find from Eq. (159) that $\sigma$ can be expressed as in Eq. (20),
where the spin transfer/spin pumping coefficients $\alpha$ and
$\tilde{\alpha}$ are defined in Eq. (12) and the shot noise parameters
$\beta^{r}$ and $\tilde{\beta}^{r}$ are defined in Eq. (21).

In the expression for the Keldysh component of the self-energy in Eq. (17),
the matrix elements of $\pi$ depend on the temperature, spin accumulation, and
frequency as follows:

$\displaystyle\pi_{\kappa\lambda}(\omega)=-2\int
d\epsilon\left[2n_{\uparrow\uparrow\kappa}(\epsilon)n_{\downarrow\downarrow\lambda}(\epsilon+\omega)-n_{\uparrow\uparrow\kappa}(\epsilon)-n_{\downarrow\downarrow\lambda}(\epsilon+\omega)\right]\,,$
(109)

where the distributions $n_{\uparrow\uparrow}$ and $n_{\downarrow\downarrow}$
are defined in Eq. (2). We compute $\pi^{K}$ at equilibrium, in the inelastic
transport regime, and in the elastic transport regime and find Eq. (22).

At high temperatures, $k_{B}T\gg\lvert\mu_{\uparrow}-\mu_{\downarrow}\rvert$
and $k_{B}T\gg\omega$,
$\pi_{\kappa\lambda}^{\text{eq}}=\pi_{\kappa\lambda}^{\text{in}}=\pi_{\kappa\lambda}^{\text{el}}=4k_{B}T$,
and we see from Eqs. (17) and (20) that the self-energy approaches its
classical value, $\Sigma^{K}\rightarrow\Sigma^{K\text{cl}}$,

$\Sigma^{K\text{cl}}=4k_{B}Ti\left(\alpha_{\uparrow\downarrow}^{r}+\tilde{\alpha}_{\uparrow\downarrow}^{r}\right)\,.$
(110)

The advanced and retarded components of the self-energy (102) can be expanded
in terms of frequency $\omega$. To the zeroth order in frequency, the retarded
and advanced components of the self-energy in (102) appearing in action (101)
only renormalize the magnon energy $E_{z}$, as discussed above for action
$S_{2a}$. We disregard such terms since they have no consequences for the spin
dynamicsChudnovskiy  _et al._ (2008); Swiebodzinski  _et al._ (2010).
Expanding the retarded and advanced self-energies in Eq. (103) to the first
order in frequency and maintaining the limit
$\mu_{\uparrow}-\mu_{\downarrow}\ll\epsilon_{F}$, we find

$\displaystyle\Sigma^{R/A}(\omega)-\Sigma^{R/A}(\omega=0)$
$\displaystyle\approx\pm
i\pi\omega\frac{2}{S}\sum_{\alpha\beta}\left(-\frac{\partial
f}{\partial\epsilon}\right)_{\epsilon=\epsilon_{\alpha}}\delta(\epsilon_{\alpha}-\epsilon_{\beta})\lvert
W_{\uparrow\downarrow\alpha\beta}\rvert^{2}$ $\displaystyle=\pm
i\omega\left[\alpha^{r}+\tilde{\alpha}^{r}\right],$ (111)

where we used the relation (165) and the definition of the spin transfer/spin
pumping efficiency (12).

Summarizing this section, we find that the effective action that determines
the spin dynamics is

$\tilde{S}_{m}=S_{m}+S_{1}+S_{2b}^{d}+S_{2b}^{f}\,,$ (112)

where $S_{1}$ governs the spin transfer torque,

$S_{2b}^{d}=S_{2b}^{\bar{q}c}+S_{2b}^{\bar{c}q}$ (113)

governs the spin pumping-enhanced dissipation (Gilbert damping), and

$S_{2b}^{f}=S_{2b}^{\bar{q}\bar{q}}+S_{2b}^{qq}+S_{2b}^{\bar{q}q}$ (114)

governs the fluctuations.

In the low-frequency limit, we use Eq. (111) and can collect the dissipative
contributions to the spin dynamics as follows:111We generalize to a complex
spin pumping coefficient that is consistent with our finding for the spin
transfer torque.

$\displaystyle S_{2b}^{\bar{c}q}+S_{2b}^{c\bar{q}}$
$\displaystyle=\left(\alpha+\tilde{\alpha}\right)\int
dt\phi^{q}\partial_{t}\bar{\phi}^{c}$ $\displaystyle+$
$\displaystyle\left(\alpha^{*}+\tilde{\alpha}^{*}\right)\int
dt\bar{\phi}^{q}\partial_{t}\phi^{c}\,.$ (115)

In analogy with the magnon action $S_{m}$ in Eq. (61), we see that actions
$S_{2b}^{\bar{c}q}+S_{2b}^{c\bar{q}}$ contribute to the spin dynamics similar
to an effective dissipative transverse field of the form

$\displaystyle E_{d}^{+}$
$\displaystyle=-\left(\alpha+\tilde{\alpha}\right)\partial_{t}S_{+}/S\,,$
(116a) $\displaystyle E_{d}^{-}$
$\displaystyle=-\left(\alpha+\tilde{\alpha}\right)\partial_{t}S^{-}/S\,,$
(116b)

where we have used $\phi^{c}=S_{+}/\sqrt{S}$ in the semiclassical limit.
Consequently, we find the spin pumping torques in Eq. (13).

##  VII Stochastic Langevin Forces

This section derives the stochastic Langevin Forces from the effective action
derived in the previous section VI. We will find that fluctuating magnetic
fields describe the Langevin forces.

To see the effects of the actions that are quadratic in the quantum components
of the boson fields, $\phi^{q}$ and $\bar{\phi}^{q}$, we use the Gaussian
integral over the real variables ${h_{i}}$:

$\exp{\left[-\frac{1}{2}\sum_{ij}\phi_{i}\left(a\right)_{ij}\phi_{j}\right]}=\frac{1}{\sqrt{\text{det}(a)}}\int\Pi_{j}\left(\frac{dh_{j}}{\sqrt{2\pi}}\right)\exp{\left[\sum_{i}ih_{i}\phi_{i}-\frac{1}{2}\sum_{ij}h_{i}\left(a^{-1}\right)_{ij}h_{j}\right]}\,,$
(117)

assuming that the integral converges. Additionally, we use the Gaussian
integral over the complex variable $z=z^{r}+iz^{i}$:

$\exp{\left[-\sum_{ij}\phi_{i}b_{ij}\phi_{j}\right]}=\frac{1}{\text{det}(b)}\int\Pi_{j}D[\bar{z}_{j},z_{j}]\exp{\left[\sum_{i}i\bar{z}_{i}\phi_{i}+z_{i}\bar{\phi}_{i}-\sum_{ij}\bar{z}_{i}\left(b^{-1}\right)_{ij}z_{j}\right]}\,,$
(118)

assuming that the integral converges and where $D[\bar{z},z]=dz^{R}dzI/\pi$.

We will also use the results for the expectation value and fluctuations of the
real variables $h$ with the distribution determined by matrix $a$ as in the
right-hand side of Eq. (117) with $\phi=0$, for example,

$\displaystyle\langle h_{l}\rangle$ $\displaystyle=0\,,$ (119a)
$\displaystyle\langle h_{l}h_{m}\rangle$ $\displaystyle=a_{ml}\,.$ (119b)

We note that although the variables $h_{i}$ appearing in Eq. (117) are real,
the variance in Eq. (119) governed by complex distribution $a$ in Eq. (117)
can be complex.

Similarly, the expectation value and fluctuations of the complex variables $z$
with the distribution as in the right-hand side of Eq. (118) with $\phi=0$ are

$\displaystyle\langle\bar{z}_{l}\rangle$ $\displaystyle=0\,,$ (120a)
$\displaystyle\langle z_{l}z_{m}\rangle$ $\displaystyle=0\,,$ (120b)
$\displaystyle\langle\bar{z}_{l}z_{m}\rangle$ $\displaystyle=b_{ml}\,.$ (120c)

By using the Gaussian integral over the real variables (117), which is a
Hubbard-Stratonovich decoupling scheme, we separately remove the quadratic
terms in actions $S_{sb}^{\bar{q}\bar{q}}$ and $S_{2b}^{qq}$ with two
independent and real auxiliary fields $h^{\bar{q}\bar{q}}(t)$ and $h^{qq}(t)$.
For example, we use

$\displaystyle\exp{iS_{2b}^{qq}}$ $\displaystyle=\exp{i\int dt\int
d\tau\phi^{q}(t)\tilde{\Sigma}^{K}_{\uparrow\downarrow}(t-\tau)\phi^{q}(\tau)}$
(121)
$\displaystyle=\frac{1}{\sqrt{\text{det}\left[-2i\tilde{\Sigma}_{\uparrow\downarrow}^{K}\right]}}\int\Pi_{j}\frac{dh_{qq}(t_{j})}{\sqrt{2\pi}}\exp{\sum_{i}ih_{qq}(t_{i})\phi^{q}(t_{i})-\frac{1}{2}\sum_{ij}h_{qq}(t_{i})(-2i\tilde{\Sigma}_{\uparrow\downarrow}^{K})^{-1}(t_{i},t_{j})h_{qq}(t_{j})}\,,$
(122)

and we treat the effect of the contribution $\exp{iS_{2b}^{\bar{q}\bar{q}}}$
to the partition function in a similar manner. Then, the expectation values
and correlations of the auxiliary fields in our action satisfy

$\displaystyle\langle h^{qq}(t)\rangle$ $\displaystyle=0\,,$ (123a)
$\displaystyle\langle h^{qq}(t)h^{qq}(\tau)\rangle$
$\displaystyle=-2i\tilde{\Sigma}^{K}_{\uparrow\downarrow}(t-\tau)$ (123b)

and

$\displaystyle\langle h^{\bar{q}\bar{q}}(t)\rangle$ $\displaystyle=0\,,$
(124a) $\displaystyle\langle
h^{\bar{q}\bar{q}}(t)h^{\bar{q}\bar{q}}(\tau)\rangle$
$\displaystyle=-2i\tilde{\Sigma}^{K}_{\downarrow\uparrow}(t-\tau)\,.$ (124b)

Similarly, by using the Gaussian integral over complex variables (118), we
remove the quadratic terms in action $S_{2b}^{\bar{q}q}$:

$\displaystyle\exp{iS_{2b}^{\bar{q}q}}$ $\displaystyle=\exp{i\int dt\int
d\tau\bar{\phi}^{q}(t)\Sigma^{K}_{\uparrow\downarrow}(t-\tau)\phi^{q}(\tau)}$
(125)
$\displaystyle=\frac{1}{\text{det}\left[-i\Sigma^{K}\right]}\int\Pi_{j}\frac{dh_{qq}(t_{j})}{\sqrt{2\pi}}\exp{\sum_{i}i\bar{h}_{\bar{q}q}(t_{i})\phi^{q}(t_{i})+ih_{\bar{q}q}(t_{i})\bar{\phi}^{q}(t_{i})-\sum_{ij}\bar{h}_{\bar{q}q}(t_{i})(-i\Sigma^{K})^{-1}(t_{i},t_{j})h_{\bar{q}q}(t_{j})}\,.$
(126)

The complex auxiliary field $h^{\bar{q}q}(t)$ obeys the statistics

$\displaystyle\langle h^{\bar{q}q}(t)\rangle$ $\displaystyle=0\,,$ (127a)
$\displaystyle\langle h^{\bar{q}q}(t)h^{\bar{q}q}(\tau)\rangle$
$\displaystyle=0\,,$ (127b)
$\displaystyle\langle\bar{h}^{\bar{q}q}(t)h^{\bar{q}q}(\tau)\rangle$
$\displaystyle=-i\Sigma^{K}(t-\tau)\,.$ (127c)

Note the essential difference of a factor of 2 in the correlations between Eq.
(124) and Eq. (127), which is essential in fulfilling the fluctuation-
dissipation theorem for the magnetic fluctuations at equilibrium.

As a result of the Hubbard-Stratonovich decoupling scheme, we obtain an
effective action that is linear in the boson fields $\phi^{q}$ and
$\bar{\phi}^{q}$ and depends on the auxiliary and stochastic temporal fields
$h^{\bar{q}\bar{q}}$, $h^{qq}$, and $h^{\bar{q}q}$.

The fluctuating fields then effectively contribute to the action as follows:

$\displaystyle S_{2b}^{f}$ $\displaystyle=\int
dt\left[\bar{h}^{\bar{q}q}(t)+h^{qq}(t)\right]\phi^{q}(t)$ $\displaystyle+\int
dt\left[h^{\bar{q}q}(t)+h^{\bar{q}\bar{q}}(t)\right]\bar{\phi}^{q}(t)\,.$
(128)

From Eq. (61), we see that the fluctuating fields correspond to transverse
magnetic fields represented by effective energies

$\displaystyle E_{-}^{f}$
$\displaystyle=-\frac{1}{\sqrt{S}}\left[\bar{h}^{\bar{q}q}(t)+h^{qq}(t)\right]\,,$
(129a) $\displaystyle E_{+}^{f}$
$\displaystyle=-\frac{1}{\sqrt{S}}\left[h^{\bar{q}q}(t)+h^{\bar{q}\bar{q}}(t)\right]\,.$
(129b)

Using the relation between effective energies and effective transverse
magnetic fields, as we introduced in Eq. (46), the effective transverse
fluctuating fields are

$\displaystyle H_{x}$
$\displaystyle=\frac{1}{2\sqrt{S}g^{*}\mu_{B}}\left[h^{\bar{q}q}+\bar{h}^{\bar{q}q}+h^{qq}+h^{\bar{q}\bar{q}}\right]\,,$
(130a) $\displaystyle H_{y}$
$\displaystyle=\frac{1}{i2\sqrt{S}g^{*}\mu_{B}}\left[h^{\bar{q}q}-\bar{h}^{\bar{q}q}-h^{qq}+h^{\bar{q}\bar{q}}\right]\,.$
(130b)

By using the correlations of the auxiliary fields in Eqs. (124) and (127) and
the symmetries of the self-energies in Eqs. (86) and (105), we find that the
correlations between the fluctuating fields are

$\displaystyle\langle\gamma H_{x}(t)\gamma H_{x}(\tau)\rangle$
$\displaystyle=-\frac{i}{4S}\left[\Sigma^{K}(t-\tau)+\Sigma^{K}(\tau-t)\right]+\frac{1}{S}\text{Im}\tilde{\Sigma}^{K}_{\uparrow\downarrow}(t-\tau)\,,$
(131a) $\displaystyle\langle\gamma H_{y}(t)\gamma H_{y}(\tau)\rangle$
$\displaystyle=-\frac{i}{4S}\left[\Sigma^{K}(t-\tau)+\Sigma^{K}(\tau-t)\right]-\frac{1}{S}\text{Im}\tilde{\Sigma}^{K}_{\uparrow\downarrow}(t-\tau)\,,$
(131b) $\displaystyle\langle\gamma H_{x}(t)\gamma H_{y}(\tau)\rangle$
$\displaystyle=-\frac{1}{4S}\left[\Sigma^{K}(t-\tau)-\Sigma^{K}(\tau-t)\right]+\frac{1}{S}\text{Re}\tilde{\Sigma}^{K}_{\uparrow\downarrow}(t-\tau).$
(131c)

From the earlier derived relations in this section that
$i(\Sigma^{K}(t-\tau)+\Sigma^{K}(\tau-t))$ and
$(\Sigma^{K}(t-\tau)-\Sigma^{K}(\tau-t))$ are real numbers and
$\tilde{\Sigma}^{K}_{\uparrow\downarrow}(t-\tau)$ is an even function of time,
we see that all correlations between the fluctuating fields are real numbers
and that the correlations (131) are even functions of the time difference
$t-\tau$, as should be expected. Furthermore, we see from Eq. (100) that
$-\Sigma^{K}(t-\tau)\geq 0$, as expected for the equal time correlations.

By Fourier transforming Eq. (131), we find the correlations introduced in Eq.
(16).

##  VIII Scattering Formulation of the Electron-Magnon Interaction

This section presents how the electron-magnon interaction can be expressed in
terms of scattering matrices of electron transport. In the classical ground
state of the magnet, the itinerant electron contribution to the Hamiltonian
(39) is

$\hat{H}_{e}=\int d{\bf
r}\hat{\psi}^{\dagger}\left[H_{0}+V_{s}\sigma_{z}\right]\hat{\psi}\,.$ (132)

From the Hamiltonian (39), by using the Holstein-Primakoff transformation
(42), the electron-magnon coupling is

$\displaystyle H_{em}$ $\displaystyle=b^{\dagger}\sqrt{\frac{2}{S}}\int d{\bf
r}\hat{\psi}_{\uparrow}V_{s}\hat{\psi}_{\downarrow}+b\sqrt{\frac{2}{S}}\int
d{\bf r}\hat{\psi}_{\downarrow}V_{s}\hat{\psi}_{\uparrow}$
$\displaystyle-\hat{b}^{\dagger}\hat{b}\frac{1}{S}\left[\int d{\bf
r}\hat{\psi}_{\uparrow}V_{s}\hat{\psi}_{\uparrow}-\int d{\bf
r}\hat{\psi}_{\downarrow}V_{s}\hat{\psi}_{\downarrow}\right]\,.$ (133)

We use scattering states to express the itinerant electron field operators. As
a first representation, the field operators can be expressed as

$\hat{\psi}_{s}=\int_{0}^{\infty}\frac{dk}{\sqrt{2\pi}}\sum_{\kappa
n}\psi_{s\kappa nk}(x{\bm{\rho}})\hat{a}_{s\kappa nk}\,,$ (134)

where $\hat{l}_{s\kappa nk}$ annihilates an electron with spin $s$ in
reservoir $\kappa$ ($\kappa=L$ or $\kappa=R$) with transverse waveguide mode
$n$ and longitudinal momentum $k$. The associated spatially dependent wave
function is $\psi_{s\kappa nk}(x{\bm{\rho}})$, where $x$ is the longitudinal
coordinate and ${\bm{\rho}}$ denotes the transverse coordinates. The field
operators satisfy the anti-commutation relations

$\displaystyle\left\\{\hat{a}_{s\kappa nk},\hat{a}^{\dagger}_{t\lambda
mp}\right\\}$
$\displaystyle=\delta_{st}\delta_{\kappa\lambda}\delta_{nm}\delta(k-p)\,.$
(135)

From Eq. (132), we identify the spin-dependent single-particle electron
Hamiltonian $H_{s}$, where $H_{\uparrow}=H_{0}+V_{s}$ and
$H_{\downarrow}=H_{0}-V_{s}$. In general, in the lead and the scattering
region, the Schrödinger equation associated with an incoming electron from the
left or right is $H_{s}\psi_{s\kappa
nk}(x{\bm{\rho}})=\epsilon_{nk}\psi_{s\kappa nk}(x{\bm{\rho}})$. In the
following, it is essential that the single-particle energy $\epsilon_{nk}$ is
spin degenerate because the leads are paramagnetic, i.e., the potential
$V_{s}$ vanishes in the leads.

Expressing the field operator (134) and the scattering states in terms of the
total energy of the states $\epsilon=\epsilon_{n}+\epsilon_{k}$ by
substituting $dk\rightarrow d\epsilon(\partial\epsilon/dk)^{-1}$,
$\psi_{s\kappa nk}\rightarrow\psi_{s\kappa
n}(\epsilon)(\partial\epsilon/dk)^{1/2}$, and $\hat{a}_{s\kappa
nk}\rightarrow\hat{a}_{s\kappa n}(\epsilon)(\partial\epsilon/dk)^{1/2}$ is
more convenient. A second representation of the field operator is therefore

$\hat{\psi}_{s}=\frac{1}{\sqrt{2\pi}}\sum_{\kappa
n}\int_{\epsilon_{n}}^{\infty}d\epsilon\psi_{s\kappa n}(\epsilon
x{\bm{\rho}})\hat{a}_{s\kappa n}(\epsilon)\,.$ (136)

On the basis of the scattering states in the total energy representation
(136), using the orthogonality of the single-particle states and assuming that
the system size is much larger than the Fermi wavelength, the electronic
Hamilton operator becomes Eq. (47).

In terms of scattering states (136), the electron-magnon interaction contains
first- and second-order terms in the magnon operators,
$\hat{H}_{em}=\hat{H}_{1}+\hat{H}_{2}$. Using Eq. (133), we see that the
first-order terms in the magnon operators are expressed as in Eq. (49) and the
second-order terms in the magnon operators are given by Eq. (50), where the
matrix elements are

$W_{\uparrow\downarrow\alpha\beta}=\frac{1}{2\pi}\int d{\bf
r}\psi_{\uparrow\alpha}^{*}V_{s}\psi_{\downarrow\beta}\,,$ (137)

$W_{\downarrow\uparrow\beta\alpha}=(W_{\uparrow\downarrow\alpha\beta})^{*}$,
and

$W_{\sigma\sigma\alpha\beta}=\frac{1}{2\pi}\int d{\bf
r}\psi_{\sigma\alpha}^{*}V_{s}\psi_{\sigma\beta}\,.$ (138)

We will demonstrate in Appendix A that the matrix elements
$W_{\uparrow\downarrow\alpha\beta}$ and $W_{\downarrow\uparrow\beta\alpha}$
are related to a generalized inelastic (since the energy is not conserved)
$2\times 2$ transverse (mixing) conductance. However, the final results for
the spin dynamics demonstrate that they nevertheless only relate to the
elastic $2\times 2$ transverse (mixing) conductance, as expected since the
magnon frequency is tiny compared to the Fermi energy. Furthermore, we have
demonstrated in the previous sections that the matrix elements
$W_{\sigma\sigma\alpha\beta}$ diagonal in the spin indices are irrelevant to
the spin dynamics. Only spin-flip events are relevant in the semiclassical
limit $S\gg 1$.

##  IX Conclusion

We have presented a quantum scattering theory for spin transport and
magnetization dynamics in normal metal-ferromagnet systems. The path-integral
formalism generalizes earlier semiclassical results to the quantum low-
temperature regime and provides a valuable connection to the scattering
formalism of electron transport. At low temperatures, the chemical potential
in the normal metals controls quantum contributions to the magnetic noise.
Some of these contributions are related to shot noise because of the discrete
spin angular momentum of electrons. These quantum contributions dominate when
the temperature is lower than the spin or charge chemical potentials.

##  Appendix A Evaluation of Spin-Flip Matrix Elements

In this section, we demonstrate how the matrix elements
$W_{\uparrow\downarrow\alpha\beta}$ (137) are governed by the scattering
matrix, i.e., the reflection and transmission amplitudes. This implies that
the matrix elements are proportional to the surface area between the
ferromagnet and the normal metal instead of the volume of the ferromagnet, as
one might naively expect from the definition (137).

In the lead, $V_{s}$ vanishes, and $H_{0}$ decomposes into longitudinal and
transverse parts as follows:

$H_{0}=-\frac{1}{2m}\frac{\partial^{2}}{\partial
x^{2}}+H_{\perp}({\bm{\rho}})\,,$ (139)

so the transverse wave function $\phi_{n}$ obeys the Schrödinger equation:

$H_{\perp}({\bm{\rho}})\phi_{n}({\bm{\rho}})=\epsilon_{n}\phi_{n}({\bm{\rho}})\,,$
(140)

where $\epsilon_{n}$ is the transverse component of the single-particle energy
$\epsilon_{nk}=\epsilon_{k}+\epsilon_{n}$, where $\epsilon_{k}=k^{2}/2m$.

We introduce the energy $\epsilon$-dependent reflection and transmission
matrices $r$, $t$, $\tilde{r}$, and $\tilde{t}$ that constitute the S-matrix:

$S=\left(\begin{array}[]{cc}r&\tilde{t}\\\ t&\tilde{r}\end{array}\right)\,,$
(141)

which should not be confused with the total spin of the magnet $S$. The
S-matrix obeys the unitarity relations

$\displaystyle SS^{\dagger}$ $\displaystyle=1\,,$ (142a) $\displaystyle
S^{\dagger}S$ $\displaystyle=1\,.$ (142b)

In the left lead, when $x<x_{L}$, the scattering states at energy $\epsilon$
are

$\displaystyle\psi_{sLn}$
$\displaystyle=\sum_{m}\frac{\phi_{m}({\bm{\rho}})}{\sqrt{v_{m}}}\left[\delta^{mn}e^{ik_{n}\tilde{x}_{L}}+r_{s}^{mn}e^{-ik_{m}\tilde{x}_{L}}\right]\,,$
(143a) $\displaystyle\psi_{sRn}$
$\displaystyle=\sum_{m}\frac{\phi_{m}({\bm{\rho}})}{\sqrt{k_{m}}}\tilde{t}_{s}^{mn}e^{-ik\tilde{x}_{L}}\,,$
(143b)

where $v_{m}=\hbar k_{m}/m$, the longitudinal wavevector $k_{m}$ is determined
by $\hbar^{2}k_{n}^{2}/2m+\epsilon_{n}=\epsilon$, and $\tilde{x}_{L}=x-x_{L}$
is a local coordinate relative to the scattering region containing the
ferromagnet that originates at $x=x_{L}$ and ends at $x=x_{R}$. Similarly, in
the right lead, when $x>x_{R}$, the scattering states are

$\displaystyle\psi_{sLn}$
$\displaystyle=\sum_{m}\frac{\phi_{m}({\bm{\rho}})}{\sqrt{k_{m}}}t_{s}^{mn}e^{ik\tilde{x}_{R}}\,,$
(144a) $\displaystyle\psi_{sRn}$
$\displaystyle=\sum_{m}\frac{\phi_{m}({\bm{\rho}})}{\sqrt{k_{m}}}\left[\delta^{mn}e^{-ik\tilde{x}_{R}}+\tilde{r}_{s}^{mn}e^{ik\tilde{x}_{R}}\right]\,,$
(144b)

where $\tilde{x}_{R}=x-x_{R}$ is the local coordinate in the right lead.

Using $V_{s}=(H_{\uparrow}-H_{\downarrow})/2$, partial integration of the
kinetic energy along the transport direction, and the fact that the single-
particle scattering state energies are spin degenerate, we find that

$\displaystyle W_{\uparrow\downarrow\alpha\beta}$
$\displaystyle=\frac{1}{4\pi}\int_{x_{L}}^{x_{R}}\\!\\!\\!\\!\\!\\!dx\int\\!\\!d{\bm{\rho}}\left(\psi_{\uparrow\alpha}^{*}H_{\uparrow}\psi_{\downarrow\beta}-\psi_{\uparrow\alpha}^{*}H_{\downarrow}\psi_{\downarrow\beta}\right)$
$\displaystyle=\frac{1}{4\pi
i}\left[j_{\uparrow\downarrow\alpha\beta}(x=x_{R})-j_{\uparrow\downarrow\alpha\beta}(x=x_{L})\right]\,.$
(145)

We have introduced the off-diagonal component of the spin current in the
transport direction:

$j_{\uparrow\downarrow\alpha\beta}(x)=\frac{1}{2mi}\int{d{\bm{\rho}}}\left[\psi_{\uparrow\alpha\beta}^{*}\frac{\partial\psi_{\downarrow\beta}}{\partial
x}-\frac{\partial\psi_{\uparrow\alpha}^{*}}{\partial
x}\psi_{\downarrow\beta}\right]\,.$ (146)

The transverse component of the spin current with respect to the equilibrium
orientation of the magnetization flowing in the transport direction (146) can
be computed from the asymptotic behavior of the wave functions in Eqs. (143)
and (144).

We can now evaluate the spin current in terms of the scattering matrix. In the
final results for the effective magnon action, only matrix elements at the
Fermi energy are needed. For $RR$ components, we find

$\displaystyle j_{\uparrow\downarrow RR}^{nnm}(x_{R})$
$\displaystyle=-\left(1-\tilde{r}_{\uparrow}^{{\dagger}}\tilde{r}_{\downarrow}\right)^{nm}\,,$
(147a) $\displaystyle j_{\uparrow\downarrow RR}^{nnm}(x_{L})$
$\displaystyle=-\left(\tilde{t}_{\uparrow
k}^{{\dagger}}\tilde{t}_{\downarrow}\right)^{nm}\,,$ (147b)

the $LL$ components are

$\displaystyle j_{\uparrow\downarrow LL}^{nm}(x_{R})$
$\displaystyle=\left(t_{\uparrow k}^{{\dagger}}t_{\downarrow
k}\right)^{nm}\,,$ (148a) $\displaystyle j_{\uparrow\downarrow
LL}^{nm}(x_{L})$
$\displaystyle=\left(1-r_{\uparrow}^{{\dagger}}r_{\downarrow}\right)^{nm}\,,$
(148b)

the $LR$ components are

$\displaystyle j_{\uparrow\downarrow LR}^{nm}(x_{R})$
$\displaystyle=\left(t_{\uparrow}^{\dagger}\tilde{r}_{\downarrow}\right)^{nm}\,,$
(149a) $\displaystyle j_{\uparrow\downarrow LR}^{nm}(x_{L})$
$\displaystyle=-\left(r_{\uparrow}^{\dagger}\tilde{t}_{\downarrow}\right)^{nm}\,,$
(149b)

and the $RL$ components are

$\displaystyle j_{\uparrow\downarrow RL}^{nn}(x_{R})$
$\displaystyle=\left(\tilde{r}_{\uparrow}^{\dagger}t_{\downarrow}\right)^{nm}\,,$
(150a) $\displaystyle j_{\uparrow\downarrow RL}^{nn}(x_{L})$
$\displaystyle=-\left(\tilde{t}_{\uparrow}^{\dagger}r_{\downarrow}\right)^{nm}\,.$
(150b)

In the expressions (147), (148), (149), and(150), the superscript $nm$ denotes
the matrix element, and matrix multiplication of the reflection and
transmission coefficients is implied.

Consequently, we find from Eqs. (145), (147), (148), (149), and (150)

$\displaystyle W_{\uparrow\downarrow LL}^{nm}$
$\displaystyle=i\frac{1}{4\pi}A_{\uparrow\downarrow}^{nm}\,,$ (151a)
$\displaystyle W_{\uparrow\downarrow RR}^{nm}$
$\displaystyle=i\frac{1}{4\pi}\tilde{A}_{\uparrow\downarrow}^{nm}\,,$ (151b)

where

$\displaystyle A_{\uparrow\downarrow}^{nm}$
$\displaystyle=g_{r}^{nm}-g_{t}^{nm}\,,$ (152)
$\displaystyle\tilde{A}_{\uparrow\downarrow}^{nm}$
$\displaystyle=g_{\tilde{r}}^{nm}-g_{\tilde{t}}^{nm}\,,$ (153)

and we have introduced the dimensionless partial conductance matrix elements

$\displaystyle g_{r}^{nm}$
$\displaystyle=\left(1-r_{\uparrow}^{\dagger}r_{\downarrow}\right)^{nm}\,,$
(154a) $\displaystyle g_{\tilde{r}}^{nm}$
$\displaystyle=\left(1-\tilde{r}_{\uparrow}^{{\dagger}}\tilde{r}_{\downarrow}\right)^{nm}\,,$
(154b) $\displaystyle g_{t}^{nm}$
$\displaystyle=\left(t_{\uparrow}^{{\dagger}}t_{\downarrow}\right)^{nm}\,,$
(154c) $\displaystyle g_{\tilde{t}}^{nm}$
$\displaystyle=\left(\tilde{t}_{\uparrow}^{{\dagger}}\tilde{t}_{\downarrow}\right)^{nm}\,.$
(154d)

We also defineTserkovnyak  _et al._ (2002)

$\displaystyle A_{\uparrow\downarrow}$
$\displaystyle=\sum_{nn}W_{\uparrow\downarrow LL}^{nn}=g_{r}-g_{t}\,,$ (155a)
$\displaystyle\ \tilde{A}_{\uparrow\downarrow}$
$\displaystyle=\sum_{nn}W_{\uparrow\downarrow
RR}^{nn}=g_{\tilde{r}}-g_{\tilde{t}}\,,$ (155b)

where the dimensionless transverse (mixing) conductances are

$\displaystyle g_{r}$
$\displaystyle=\text{Tr}\left(1-r_{\uparrow}^{\dagger}r_{\downarrow}\right)\,,$
(156a) $\displaystyle g_{\tilde{r}}$
$\displaystyle=\text{Tr}\left(1-\tilde{r}_{\uparrow}^{{\dagger}}\tilde{r}_{\downarrow}\right)\,,$
(156b) $\displaystyle g_{t}$
$\displaystyle=\text{Tr}\left(t_{\uparrow}^{{\dagger}}t_{\downarrow}\right)\,,$
(156c) $\displaystyle g_{\tilde{t}}$
$\displaystyle=\text{Tr}\left(\tilde{t}_{\uparrow}^{{\dagger}}\tilde{t}_{\downarrow}\right)\,.$
(156d)

The spin transfer/spin pumping coefficients are defined as

$\displaystyle\alpha$ $\displaystyle=\frac{A_{\uparrow\downarrow}}{4\pi S}\,,$
(157a) $\displaystyle\tilde{\alpha}$
$\displaystyle=\frac{\tilde{A}_{\uparrow\downarrow}}{4\pi S}\,,$ (157b)

which can be expressed as in Eq. (12).

Additionally, we find that

$\displaystyle W_{\uparrow\downarrow LR}^{nm}$
$\displaystyle=-i\frac{1}{4\pi}\left(r_{\uparrow}^{\dagger}\tilde{t}_{\downarrow}+t_{\uparrow}^{\dagger}\tilde{r}_{\downarrow}\right)^{nm}\,,$
(158a) $\displaystyle W_{\uparrow\downarrow RL}^{nm}$
$\displaystyle=-i\frac{1}{4\pi}\left(\tilde{t}_{\uparrow}^{\dagger}r_{\downarrow}+\tilde{r}_{\uparrow}^{\dagger}t_{\downarrow}\right)^{nm}\,.$
(158b)

By using Eq. (151), Eq. (158), and the unitarity of the S-matrix (142), we
also find that

$\displaystyle\sum_{nm}\lvert W_{\uparrow\downarrow LL}^{nm}\rvert^{2}$
$\displaystyle=\left(\frac{1}{4\pi}\right)^{2}2\left(A_{\uparrow\downarrow}^{r}-B_{\uparrow\downarrow}^{r}\right)\,,$
(159a) $\displaystyle\sum_{nm}\lvert W_{\uparrow\downarrow RR}^{nm}\rvert^{2}$
$\displaystyle=\left(\frac{1}{4\pi}\right)^{2}2\left(\tilde{A}_{\uparrow\downarrow}^{r}-\tilde{B}_{\uparrow\downarrow}^{r}\right)\,,$
(159b) $\displaystyle\sum_{nm}\lvert W_{\uparrow\downarrow LR}^{nm}\rvert^{2}$
$\displaystyle=\left(\frac{1}{4\pi}\right)^{2}2B_{\uparrow\downarrow}^{r}\,,$
(159c) $\displaystyle\sum_{nm}\lvert W_{\uparrow\downarrow RL}^{nm}\rvert^{2}$
$\displaystyle=\left(\frac{1}{4\pi}\right)^{2}2\tilde{B}_{\uparrow\downarrow}^{r}\,,$
(159d)

where the shot noise parameters are

$\displaystyle B_{\uparrow\downarrow}^{r}$
$\displaystyle=\frac{1}{2}\text{Tr}\left[1-\left(r_{\uparrow}^{\dagger}r_{\downarrow}+t_{\uparrow}^{\dagger}t_{\downarrow}\right)\left(r_{\downarrow}^{\dagger}r_{\uparrow}+t_{\downarrow}^{\dagger}t_{\uparrow}\right)\right]\,,$
(160a) $\displaystyle\tilde{B}_{\uparrow\downarrow}^{r}$
$\displaystyle=\frac{1}{2}\text{Tr}\left[1-\left(\tilde{r}_{\uparrow}^{\dagger}\tilde{r}_{\downarrow}+\tilde{t}_{\uparrow}^{\dagger}\tilde{t}_{\downarrow}\right)\left(\tilde{r}_{\downarrow}^{\dagger}\tilde{r}_{\uparrow}+\tilde{t}_{\downarrow}^{\dagger}\tilde{t}_{\uparrow}\right)\right]$
(160b)

and $A_{\uparrow\downarrow}$ was introduced in Eq. (153).

The noise matrix parameters are

$\displaystyle\sigma=\frac{1}{4\pi
S}\left(\begin{array}[]{cc}A_{\uparrow\downarrow}^{r}-B_{\uparrow\downarrow}^{r}&B_{\uparrow\downarrow}^{r}\\\
\tilde{B}_{\uparrow\downarrow}^{r}&\tilde{A}_{\uparrow\downarrow}^{r}-\tilde{B}_{\uparrow\downarrow}^{r}\end{array}\right)\,,$
(163)

which can be expressed as in Eq. (20).

From the definition of the absolute squares of the matrix elements (159), we
see that the following conditions are fulfilled:

$\displaystyle A_{\uparrow\downarrow}^{r}$ $\displaystyle\geq 0\text{ ,
}\tilde{A}_{\uparrow\downarrow}^{r}\geq 0\,,$ (164a) $\displaystyle
B_{\uparrow\downarrow}^{r}$ $\displaystyle\geq 0\text{ ,
}\tilde{B}_{\uparrow\downarrow}^{r}\geq 0\,,$ (164b) $\displaystyle
A_{\uparrow\downarrow}^{r}$ $\displaystyle\geq
B_{\uparrow\downarrow}^{r},\text{ ,
}\tilde{A}_{\uparrow\downarrow}^{r}\geq\tilde{B}_{\uparrow\downarrow}^{r}\,.$
(164c)

Hence, all the matrix elements in Eq. (21) are positive definite, as expected
for fluctuations.

We can also find the sum rule

$\displaystyle\sum_{nm}\left[\lvert W_{\uparrow\downarrow
LL}^{nm}\rvert^{2}+\lvert W_{\uparrow\downarrow LR}^{nm}\rvert^{2}\right]$
$\displaystyle=\left(\frac{1}{4\pi}\right)^{2}2A^{r}_{\uparrow\downarrow}\,,$
(165a) $\displaystyle\sum_{nm}\left[\lvert W_{\uparrow\downarrow
RR}^{nm}\rvert^{2}+\lvert W_{\uparrow\downarrow RL}^{nm}\rvert^{2}\right]$
$\displaystyle=\left(\frac{1}{4\pi}\right)^{2}2\tilde{A}^{R}_{\uparrow\downarrow}\,,$
(165b)

which we classify as an optical theorem for the transport of spins transverse
to the magnetization direction. The optical theorem (165) relates sums of
reflection and transmission probabilities to the real part of the reflection
and transmission amplitudes.

We also evaluate

$\displaystyle\sum_{nm}W_{\downarrow\uparrow LL}^{nm}W_{\downarrow\uparrow
LL}^{mn}$
$\displaystyle=-\left(\frac{1}{4\pi}\right)^{2}2\left(A_{\downarrow\uparrow}-C_{\downarrow\uparrow}\right)\,,$
(166)

where

$A_{\downarrow\uparrow}=A_{\uparrow\downarrow}^{*}\,,$ (167)

$A_{\uparrow\downarrow}$ was defined in Eq. (155a), and

$C_{\downarrow\uparrow}=\frac{1}{2}\text{Tr}\left[1-\left(r_{\downarrow}^{\dagger}r_{\uparrow}+t_{\downarrow}^{\dagger}t_{\uparrow}\right)^{2}\right]\,.$
(168)

We note that in general, $C_{\downarrow\uparrow}\neq B_{\downarrow\uparrow}$.
Furthermore, we find

$\sum_{nm}W_{\downarrow\uparrow LR}^{nm}W_{\downarrow\uparrow
RL}^{mn}=-\left(\frac{1}{4\pi}\right)^{2}2D_{\downarrow\uparrow}\,,$ (169)

where

$D_{\downarrow\uparrow}=\text{Tr}\left[\left(r_{\downarrow}^{\dagger}\tilde{t}_{\uparrow}+t_{\uparrow}^{\dagger}\tilde{r}_{\uparrow}\right)\left(\tilde{t}_{\downarrow}^{\dagger}r_{\uparrow}+\tilde{r}_{\downarrow}^{\dagger}t_{\uparrow}\right)\right]\,.$
(170)

We also find

$\sum_{nm}W_{\downarrow\uparrow RL}^{nm}W_{\downarrow\uparrow
LR}^{mn}=-\left(\frac{1}{4\pi}\right)^{2}2D_{\downarrow\uparrow}\,,$ (171)

and

$\sum_{nm}W_{\downarrow\uparrow RR}^{nm}W_{\downarrow\uparrow
RR}^{mn}=-\left(\frac{1}{4\pi}\right)^{2}2\left(\tilde{A}_{\downarrow\uparrow}-\tilde{C}_{\downarrow\uparrow}\right)\,.$
(172)

We use these results to compute the transverse spin accumulation noise matrix
elements as defined in Eq. (92):

$\displaystyle\tilde{\sigma}=\frac{1}{8\pi
S}\left(\begin{array}[]{cc}A_{\downarrow\uparrow}-C_{\downarrow\uparrow}&D_{\downarrow\uparrow}\\\
D_{\downarrow\uparrow}&\tilde{A}_{\downarrow\uparrow}-\tilde{C}_{\downarrow\uparrow}\end{array}\right)\,,$
(175)

which can be expressed as in Eq. (27).

##  Appendix B Transport Coefficients

The transport coefficients can be computed by ab initio band structure
calculations. This has been carried out for the spin transfer and spin pumping
efficiencies reflected in the mixing conductances in Eq. (12) in Ref.
Zwierzycki  _et al._ , 2005.

Here, we consider some scenarios for electron transport.

###  B.1 Magnetic Insulator

In the case of a magnetic insulator, $t=0$, and $\tilde{t}=0$; thus, that we
find from Eq. (155) that the spin transfer and spin pumping parameters are
determined by the reflection coefficients:

$\displaystyle A_{\uparrow\downarrow}$ $\displaystyle=g_{r}\,,$ (176a)
$\displaystyle\tilde{A}_{\uparrow\downarrow}$ $\displaystyle=\tilde{g}_{r}\,.$
(176b)

The spin transfer/spin pumping efficiencies are determined by the mixing
conductance associated with the reflection of the incoming waveguide modes.

We also find that

$\displaystyle C_{\downarrow\uparrow}$
$\displaystyle=\frac{1}{2}\text{Tr}\left[1-\left(r_{\uparrow}^{\dagger}r_{\uparrow}\right)^{2}\right]\,,$
(177) $\displaystyle\tilde{C}_{\downarrow\uparrow}$
$\displaystyle=\frac{1}{2}\text{Tr}\left[1-\left(\tilde{r}_{\uparrow}^{\dagger}\tilde{r}_{\uparrow}\right)^{2}\right]\,,$
(178) $\displaystyle D_{\downarrow\uparrow}$ $\displaystyle=0\,,$ (179)
$\displaystyle\tilde{D}_{\downarrow\uparrow}$ $\displaystyle=0\,.$ (180)

In the simplest case of complete dephasing, $g_{r}=N=\tilde{g}_{r}$, where $N$
is the number of transverse waveguide modes. Furthermore, from Eq. (160), we
find that

$\displaystyle B_{\uparrow\downarrow}$ $\displaystyle=0\,,$ (181a)
$\displaystyle\tilde{B}_{\uparrow\downarrow}$ $\displaystyle=0\,,$ (181b)

and $C_{\downarrow\uparrow}=N/2=C_{\uparrow\downarrow}$. The latter implies
that there is noise associated with transverse spin accumulation transport.
This is due to the discrete angular momentum carried by each electron that may
be flipped inside the magnet.

In contrast, there are no shot noise contributions arising from
$B_{\uparrow\downarrow}$ due to the fluctuations of magnetic insulators. This
is because $B_{\uparrow\downarrow}$ governs the transport of longitudinal
spins that is absent in magnetic insulators.

###  B.2 Clean Junctions

In a simple model of a clean junction, the reflection and transmission
coefficients are diagonal in the transverse mode basis. To illustrate the main
physics, let us furthermore assume that only the phases of the reflection and
transmission coefficients differ. Then, for each mode, the spin-dependent
scattering coefficients are

$\displaystyle r_{s}$ $\displaystyle=\sqrt{1-T}\exp{i\phi_{rs}}\,,$ (182a)
$\displaystyle t_{s}$ $\displaystyle=\sqrt{T}\exp{i\phi_{rt}}\,,$ (182b)
$\displaystyle\tilde{r}_{s}$
$\displaystyle=\sqrt{1-T}\exp{i\phi_{\tilde{r}s}}\,,$ (182c)
$\displaystyle\tilde{t}_{s}$
$\displaystyle=\sqrt{T}\exp{i\phi_{\tilde{t}s}}\,.$ (182d)

Then, we find for a single mode

$\displaystyle A_{\uparrow\downarrow}(1)$ $\displaystyle=\frac{1}{4\pi
S}\left(1-(1-T)e^{i(\phi_{r\uparrow}-\phi_{r\downarrow})}-T^{i(\phi_{t\uparrow}-\phi_{t\downarrow})}\right)\,,$
(183a) $\displaystyle\tilde{A}_{\uparrow\downarrow}(1)$
$\displaystyle=\frac{1}{4\pi
S}\left(1-(1-T)e^{i(\phi_{r\uparrow}-\phi_{r\downarrow})}-T^{i(\phi_{t\uparrow}-\phi_{t\downarrow})}\right)\,,$
(183b) $\displaystyle B_{\uparrow\downarrow}(1)$
$\displaystyle=T\left(1-T\right)\left[1-e^{-i(\phi_{r\uparrow}-\phi_{r\downarrow})}e^{-i(\phi_{t\uparrow}-\phi_{t\downarrow})}\right]\,,$
$\displaystyle\tilde{B}_{\uparrow\downarrow}(1)$
$\displaystyle=T\left(1-T\right)\left[1-e^{-i(\phi_{\tilde{r}\uparrow}-\phi_{\tilde{r}\downarrow})}e^{-i(\phi_{\tilde{t}\uparrow}-\phi_{\tilde{t}\downarrow})}\right]\,,$
(184a)
$C_{\downarrow\uparrow}=\frac{1}{2}\left[1-Re^{-i(\phi_{r\uparrow}-\phi_{r\downarrow})}+Te^{-i(\phi_{t\uparrow}-\phi_{t\downarrow})}\right]\,,$
(185) $D_{\downarrow\uparrow}=0\,.$ (186)

As is common in the shot noise characteristics for fermions, the shot noise
parameter $B_{\uparrow\downarrow}$ is proportional to the factor $T(1-T)$ due
to the Fermi exclusion principle ensuring that only one particle can propagate
at the same time. Furthermore, when we have many channels and average over all
of them with different scattering phases, dephasing results in

$\displaystyle A_{\uparrow\downarrow}(N)$ $\displaystyle=N\,,$ (187a)
$\displaystyle\tilde{A}_{\uparrow\downarrow}(N)$ $\displaystyle=N\,,$ (187b)
$\displaystyle B_{\uparrow\downarrow}(N)$
$\displaystyle=NT\left(1-T\right)\,,$
$\displaystyle\tilde{B}_{\uparrow\downarrow}(N)$
$\displaystyle=NT\left(1-T\right)\,,$ (188a) $C_{\downarrow\uparrow}=N/2\,,$
(189)

and

$D_{\downarrow\uparrow}=0\,,$ (190)

where $N$ is the number of transverse waveguide modes. Eq. (187) is the well-
known result that the mixing conductance roughly equals the Sharvin
conductance.

###  B.3 Disordered Junctions

Let us consider a random potential scatter in the magnet and normal metals.
Furthermore, we assume that the random impurity scattering differs for spin-up
and spin-down electrons and that these scatterings are uncorrelated. The
statistical averages of the spin-up and spin-down electrons are then
independent. When the number of transverse modes $N$ is large, we use the
averages

$\displaystyle\langle\lvert t_{nm}\rvert^{2}\rangle$
$\displaystyle=g/N^{2}\,,$ (191) $\displaystyle\langle\lvert
r_{nm}\rvert^{2}\rangle$ $\displaystyle=\frac{1-g/N}{N}\,,$ (192)

where $g$ is the dimensionless conductance. We then find that
$B_{\uparrow\downarrow}=\frac{g_{\uparrow}+g_{\downarrow}}{N^{2}}\rightarrow^{N\rightarrow\infty}0$,
so

$\displaystyle A_{\uparrow\downarrow}^{dis}$ $\displaystyle=N\,,$ (193a)
$\displaystyle\tilde{A}_{\uparrow\downarrow}^{dis}$ $\displaystyle=N\,,$
(193b) $\displaystyle B_{\uparrow\downarrow}^{dis}$ $\displaystyle=0\,,$
$\displaystyle\tilde{B}_{\uparrow\downarrow}^{dis}$ $\displaystyle=0\,,$
(194a) $\displaystyle C_{\downarrow\uparrow}$ $\displaystyle=N/2\,,$
$\displaystyle\tilde{C}_{\downarrow\uparrow}$ $\displaystyle=N/2\,,$ (195)
$\displaystyle D_{\downarrow\uparrow}$ $\displaystyle=0\,,$
$\displaystyle\tilde{D}_{\downarrow\uparrow}$ $\displaystyle=0\,.$ (196)

##  Appendix C Fourier Transform

We define the Fourier transform of the Green functions as

$\displaystyle G_{0}(\omega)$
$\displaystyle=\int_{-\infty}^{\infty}d(t-\tau)G(t-\tau)e^{i\omega(t-\tau)}\,,$
(197a) $\displaystyle G_{0}(t-\tau)$
$\displaystyle=\frac{1}{2\pi}\int_{-\infty}^{\infty}d\omega
G_{0}(\omega)e^{-i\omega(t-\tau)}\,.$ (197b)

###### Acknowledgements.

This work was supported by the Research Council of Norway through its Centres
of Excellence funding scheme, project number 262633, ”QuSpin”.

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