# Magnonic thermal transport using the quantum Boltzmann equation

Kouki Nakata  Advanced Science Research Center, Japan Atomic Energy Agency,
Tokai, Ibaraki 319-1195, Japan  Yuichi Ohnuma  Research Center for Advanced
Science and Technology (RCAST), The University of Tokyo, Meguro-ku, Tokyo
153-8904, Japan

###### Abstract

We present a formula for thermal transport in the bulk of Bose systems based
on the quantum Boltzmann equation (QBE). First, starting from the quantum
kinetic equation and using the Born approximation for impurity scattering, we
derive the QBE of Bose systems and provide a formula for thermal transport
subjected to a temperature gradient. Next, we apply the formula to magnons.
Assuming a relaxation time approximation and focusing on the linear response
regime, we show that the longitudinal thermal conductivity of the QBE exhibits
the different behavior from the conventional. The thermal conductivity of the
QBE reduces to the Drude-type in the limit of the quasiparticle approximation,
while not in the absence of the approximation. Finally, applying the
quasiparticle approximation to the QBE, we find that the correction to the
conventional Boltzmann equation is integrated as the self-energy into the
spectral function of the QBE, and this enhances the thermal conductivity. Thus
we shed light on the thermal transport property of the QBE beyond the
conventional.

##  I Introduction

The last decade has seen a rapid development of magnon-based spintronics,
dubbed magnonics, aiming at utilizing the quantized spin waves, magnons, as a
carrier of information Chumak  _et al._ (2015). The main subject is the
realization of efficient transmission of information using spins in insulating
magnets. For this purpose, taking into account the fundamental difference of
the quantum-statistical properties between electrons and magnons, i.e.,
fermions and bosons, respectively, many magnonic analogues of electron
transport have been established both theoretically and experimentally Nakata
_et al._ (2017a), with a particular focus on thermal transport, e.g., the
thermal Hall effect Onose  _et al._ (2010); Katsura  _et al._ (2010);
Matsumoto and Murakami (2011a, b) and the Wiedemann-Franz law Nakata  _et al._
(2015, 2017b, 2017c, 2019a) for magnon transport.

The key ingredient in the study of thermal transport phenomena is the
Boltzmann equation Ashcroft and Mermin (1976). As well as electron transport,
the conventional Boltzmann equation has been playing a central role in the
study of magnon transport, e.g., see Refs. Rezende  _et al._ (2014a, b); Basso
_et al._ (2016a, b, c); Okuma  _et al._ (2017); Schmidt  _et al._ (2018);
Nakata  _et al._ (2019b); Liu  _et al._ (2019); Jiang  _et al._ (2013); the
temperature gradient ${\mathbf{\nabla}}T$ drives a magnonic system out of
equilibrium. The conventional Boltzmann equation describes the property of the
system at time $t$ as

$\displaystyle\Big{(}$ $\displaystyle\frac{\partial}{\partial
t}+{\mathbf{v}}_{\mathbf{k}}\cdot{\mathbf{\nabla}}T\frac{\partial}{\partial
T}\Big{)}f_{{\mathbf{k}},{\mathbf{r}},t}=I_{{\mathbf{k}},{\mathbf{r}},t},$ (1)

where
${\mathbf{v}}_{\mathbf{k}}:=\partial\omega_{\mathbf{k}}/(\partial{\mathbf{k}})$
is the magnon velocity for the energy dispersion relation
$\hbar\omega_{\mathbf{k}}$ in the wavenumber space ${\mathbf{k}}$, $\hbar$
represents the Planck constant, $f_{{\mathbf{k}},{\mathbf{r}},t}$ is the
nonequilibrium Bose distribution function of the absolute temperature $T$, and
$I_{{\mathbf{k}},{\mathbf{r}},t}$ is the collision integral for a position
${\mathbf{r}}$. Within the relaxation time approximation, the longitudinal
thermal conductivity of magnons in the bulk of magnets is proportional to the
relaxation time Nakata  _et al._ (2017c, 2019b); Jiang  _et al._ (2013). Under
some conditions, the relaxation time coincides with the lifetime of magnons,
which is proportional to the inverse of the Gilbert damping constant $\alpha$
See . Thus the magnonic thermal conductivity of the conventional Boltzmann
equation reduces to the Drude-type Dru  as a function of $\alpha$ in that it
is proportional to $1/\alpha$.

From the viewpoint of quantum field theory, the conventional Boltzmann
equation [Eq. (1)] is derived from the quantum kinetic equation 111 Ref. Haug
and Jauho (2007) refers to the quantum kinetic equation as the Kadanoff-Baym
equation Kadanoff and Baym (1962), or the Keldysh equation Keldysh (1965). For
the details, see also Ref. Danielewicz (1984).  by taking several
approximations Mahan (2000); Haug and Jauho (2007); Kita (2010); Danielewicz
(1984). Assuming that the variation of the center-of-mass coordinates is slow
compared with that of the relative coordinates, Kadanoff and Baym Kadanoff and
Baym (1962) applied an approximation, called the gradient expansion, to the
quantum kinetic equation for the nonequilibrium Green’s function Keldysh
(1965). The quantum kinetic equation of the lowest order gradient
approximation becomes the quantum Boltzmann equation (QBE). The QBE describes
the equation of motion for the lesser Green’s function. The spectral function
is assumed to be the Dirac delta function in the quasiparticle approximation
Mahan (2000); Haug and Jauho (2007); Kita (2010); Matsuo  _et al._ (2017);
Nakata  _et al._ (2019b). In the limit of the quasiparticle approximation, the
QBE reduces to the conventional Boltzmann equation [Eq. (1)] for the
nonequilibrium distribution function of three variables
$({\mathbf{k}},{\mathbf{r}},t)$.

This hierarchical structure of quantum field theory indicates that by relaxing
some of the approximations, quantum-mechanical corrections to the conventional
Boltzmann equation can be evaluated Haug and Jauho (2007). Such sound
development has been made successfully as to electrons Kita (2010). The
lifetime of electrons in metals subjected to a strong impurity potential
becomes substantial, and the quasiparticle approximation is not applicable. To
solve the issue, Prange and Kadanoff introduced an alternative approach Prange
and Kadanoff (1964), which was developed for the application to
superconductors and superfluids Eilenberger (1968); Serene and Rainer (1983);
Larkin and Ovchinnikov (1975); Kita (2001), e.g., the Eilenberger equation
Eilenberger (1968). However, those are for Fermi systems. Since the approach
is based on the assumption that Kita (2010) there is a Fermi surface and the
Fermi energy, the developed formula is not applicable to Bose systems, e.g.,
magnons. Thus, to the best of our knowledge Loss , as for magnons in the bulk
of magnets, the thermal transport property beyond the conventional Boltzmann
equation remains an open issue.

In this paper, we provide a solution to this fundamental challenge by starting
from the quantum kinetic equation and developing the QBE for Bose systems. The
purpose of any useful formalism is to provide a method for calculation of
measurable quantities. First, using the QBE we develop a formula for thermal
transport in the bulk of Bose systems, including the nonlinear response to the
temperature gradient. Next, as a platform, we apply it to magnons. In the
conventional spintronics study, the Landau-Lifshitz-Gilbert equation is
playing the central role Tserkovnyak  _et al._ (2005). To develop a relation
with it, using the Gilbert damping constant we describe the spectral function
of magnons, and study the longitudinal thermal conductivity of the QBE.
Finally, by applying the quasiparticle approximation to the thermal
conductivity of the QBE, we find the correction to the conventional Boltzmann
equation and discuss thermal transport properties beyond the conventional.

We remark that the conventional Boltzmann equation Rezende  _et al._ (2014a,
b); Basso  _et al._ (2016a, b, c); Okuma  _et al._ (2017); Schmidt  _et al._
(2018); Nakata  _et al._ (2019b); Liu  _et al._ (2019); Jiang  _et al._
(2013), i.e., the transport theory based on the quasiparticle approximation,
can not describe paramagnons in the bulk of paramagnets Mahan (2000). The
quasiparticle approximation assumes that the spectrum has the form of the
Dirac delta function. However, the spectrum of paramagnons is broad in
general, and has a peak with a sufficient width of a nonzero value associated
with the inverse of the finite lifetime Doniach (1967); Hertz (1976); Doubble
_et al._ (2010); the spectrum can not be approximated by the Dirac delta
function. Therefore, the conventional Boltzmann equation can not describe
paramagnons. In this paper, we will also shed light on this issue.

This paper is organized as follows. In Sec. II starting from the quantum
kinetic equation of the lowest order gradient approximation and using the Born
approximation for impurity scattering, we derive the QBE for Bose systems. In
Sec. III, first, using the QBE and assuming a steady state in terms of time,
we provide a formula for thermal transport in the bulk of Bose systems
subjected to a temperature gradient, including the nonlinear response. Next,
we apply the formula to magnons in Sec. III.1. To develop a relation with the
conventional spintronics study, we describe the spectral function of magnons
in terms of the Gilbert damping constant. Then, assuming a relaxation time
approximation and focusing on the linear response regime, we evaluate the
longitudinal thermal conductivity of magnons in the bulk of magnets based on
the QBE. Finally, in Sec. III.2 applying the quasiparticle approximation to
the magnonic thermal conductivity of the QBE, we discuss the difference from
the one of the conventional Boltzmann equation. Comparing also with the linear
response theory, we comment on our formula in Sec. IV. We remark on open
issues in Sec. V and give some conclusions in Sec. VI. Technical details are
deferred to the Appendices.

##  II Quantum Boltzmann equation for Bose system

We consider a Bose system where the center-of-mass coordinates, the position
and time in center-of-mass (${\mathbf{r}},t$), respectively, vary slowly
compared to the relative coordinates. Up to the lowest order of the gradient
expansion, the quantum kinetic equation Mahan (2000); Haug and Jauho (2007);
Kita (2010); Danielewicz (1984) for the system reduces to

$\displaystyle-i\Big{(}$
$\displaystyle\frac{\partial{\mathscr{H}}_{{\mathbf{k}},\omega}}{\partial
t}\frac{\partial}{\partial\omega}-\frac{\partial{\mathscr{H}}_{{\mathbf{k}},\omega}}{\partial\omega}\frac{\partial}{\partial
t}-\frac{\partial{\mathscr{H}}_{{\mathbf{k}},\omega}}{\partial{\mathbf{r}}}\frac{\partial}{\partial{\mathbf{k}}}+\frac{\partial{\mathscr{H}}_{{\mathbf{k}},\omega}}{\partial{\mathbf{k}}}\frac{\partial}{\partial{\mathbf{r}}}\Big{)}G^{<}_{{\mathbf{k}},\omega,{\mathbf{r}},t}$
$\displaystyle=$
$\displaystyle(G^{<}_{{\mathbf{k}},\omega,{\mathbf{r}},t}\Sigma^{>}_{{\mathbf{k}},\omega,{\mathbf{r}},t}-G^{>}_{{\mathbf{k}},\omega,{\mathbf{r}},t}\Sigma^{<}_{{\mathbf{k}},\omega,{\mathbf{r}},t}),$
(2)

where
${\mathscr{H}}_{{\mathbf{k}},\omega}:=\hbar\omega-\hbar\omega_{\mathbf{k}}$
for a frequency $\omega$ and the functions,
$G^{<(>)}_{{\mathbf{k}},\omega,{\mathbf{r}},t}$ and
$\Sigma^{<(>)}_{{\mathbf{k}},\omega,{\mathbf{r}},t}$, are the lesser (greater)
component of the bosonic nonequilibrium Green’s function and that of the self-
energy, respectively; the variables $({\mathbf{k}},\omega)$ arise from the
Fourier transform of the relative coordinates. Following Ref. Mahan (2000), we
refer to Eq. (2) as the QBE. The QBE is the equation of motion for the lesser
Green’s function $G^{<}_{{\mathbf{k}},\omega,{\mathbf{r}},t}$ and consists of
four variables $({\mathbf{k}},\omega,{\mathbf{r}},t)$, while the conventional
Boltzmann equation is for the nonequilibrium Bose distribution function
$f_{{\mathbf{k}},{\mathbf{r}},t}$ and of three variables
$({\mathbf{k}},{\mathbf{r}},t)$. The QBE in the limit of the quasiparticle
approximation reduces to the conventional Boltzmann equation. In this paper we
study thermal transport of the QBE for Bose systems, and find the properties
beyond the conventional Boltzmann equation. Then, applying the quasiparticle
approximation to the thermal conductivity of the QBE, we discuss the
difference from the conventional.

Within the Born approximation, the self-energy due to impurity scattering of
the impurity potential $V_{{\mathbf{k}},{\mathbf{k}}^{\prime}}$ is given as
$\Sigma_{{\mathbf{k}},\omega,{\mathbf{r}},t}=\sum_{{\mathbf{k}}^{\prime}}G_{{\mathbf{k}}^{\prime},\omega,{\mathbf{r}},t}\mid
V_{{\mathbf{k}},{\mathbf{k}}^{\prime}}\mid^{2}$. Assuming that the function
${\mathscr{H}}_{{\mathbf{k}},\omega}$ is time-independent and spatially
uniform, the QBE becomes

$\displaystyle($
$\displaystyle\partial_{t}+{\mathbf{v}}_{\mathbf{k}}\cdot\partial_{\mathbf{r}})G^{<}_{{\mathbf{k}},\omega,{\mathbf{r}},t}$
(3) $\displaystyle=$
$\displaystyle\frac{1}{i\hbar}\sum_{{\mathbf{k}}^{\prime}}\mid
V_{{\mathbf{k}},{\mathbf{k}}^{\prime}}\mid^{2}(G^{<}_{{\mathbf{k}},\omega,{\mathbf{r}},t}G^{>}_{{\mathbf{k}}^{\prime},\omega,{\mathbf{r}},t}-G^{>}_{{\mathbf{k}},\omega,{\mathbf{r}},t}G^{<}_{{\mathbf{k}}^{\prime},\omega,{\mathbf{r}},t}).$

The QBE consists of the lesser (greater) Green’s functions
$G^{<(>)}_{{\mathbf{k}},\omega,{\mathbf{r}},t}$. The Kadanoff-Baym ansatz
ensures that Mahan (2000); Haug and Jauho (2007); Kita (2010) the Green’s
functions are associated with the spectral function
${\cal{A}}_{{\mathbf{k}},\omega,{\mathbf{r}},t}$ and the nonequilibrium
distribution function $\phi_{{\mathbf{k}},\omega,{\mathbf{r}},t}$ as
$G^{<}_{{\mathbf{k}},\omega,{\mathbf{r}},t}=-i{\cal{A}}_{{\mathbf{k}},\omega,{\mathbf{r}},t}\phi_{{\mathbf{k}},\omega,{\mathbf{r}},t}$
and
$G^{>}_{{\mathbf{k}},\omega,{\mathbf{r}},t}=-i{\cal{A}}_{{\mathbf{k}},\omega,{\mathbf{r}},t}(1+\phi_{{\mathbf{k}},\omega,{\mathbf{r}},t})$
for bosons, while
$G^{<}_{{\mathbf{k}},\omega,{\mathbf{r}},t}=i{\cal{A}}_{{\mathbf{k}},\omega,{\mathbf{r}},t}\phi_{{\mathbf{k}},\omega,{\mathbf{r}},t}$
and
$G^{>}_{{\mathbf{k}},\omega,{\mathbf{r}},t}=-i{\cal{A}}_{{\mathbf{k}},\omega,{\mathbf{r}},t}(1-\phi_{{\mathbf{k}},\omega,{\mathbf{r}},t})$
for fermions. The nonequilibrium Bose distribution function in the wavenumber
space is given as $f_{{\mathbf{k}},{\mathbf{r}},t}^{\textrm{QBE}}:=\int[\hbar
d\omega/(2\pi)]{\cal{A}}_{{\mathbf{k}},\omega,{\mathbf{r}},t}\phi_{{\mathbf{k}},\omega,{\mathbf{r}},t}$.
Using the Kadanoff-Baym ansatz for bosons, finally, we obtain the QBE of the
functions ${\cal{A}}_{{\mathbf{k}},\omega,{\mathbf{r}},t}$ and
$\phi_{{\mathbf{k}},\omega,{\mathbf{r}},t}$ as

$\displaystyle($
$\displaystyle\partial_{t}+{\mathbf{v}}_{\mathbf{k}}\cdot\partial_{\mathbf{r}})({\cal{A}}_{{\mathbf{k}},\omega,{\mathbf{r}},t}\phi_{{\mathbf{k}},\omega,{\mathbf{r}},t})$
(4) $\displaystyle=-$
$\displaystyle\frac{1}{\hbar}\sum_{{\mathbf{k}}^{\prime}}\mid
V_{{\mathbf{k}},{\mathbf{k}}^{\prime}}\mid^{2}{\cal{A}}_{{\mathbf{k}},\omega,{\mathbf{r}},t}{\cal{A}}_{{\mathbf{k}}^{\prime},\omega,{\mathbf{r}},t}(\phi_{{\mathbf{k}},\omega,{\mathbf{r}},t}-\phi_{{\mathbf{k}}^{\prime},\omega,{\mathbf{r}},t}).$

The QBE is useful to a wide range of Bose system subjected to impurity
scattering. For convenience, we define the collision integral as
${\cal{I}}_{{\mathbf{k}},\omega,{\mathbf{r}},t}:=-\sum_{{\mathbf{k}}^{\prime}}\mid
V_{{\mathbf{k}},{\mathbf{k}}^{\prime}}\mid^{2}{\cal{A}}_{{\mathbf{k}},\omega,{\mathbf{r}},t}{\cal{A}}_{{\mathbf{k}}^{\prime},\omega,{\mathbf{r}},t}(\phi_{{\mathbf{k}},\omega,{\mathbf{r}},t}-\phi_{{\mathbf{k}}^{\prime},\omega,{\mathbf{r}},t})/\hbar$.
Hereafter, for simplicity, we drop the indices (${\mathbf{r}},t$) when those
are not important.

##  III Thermal transport in
Bose system

The temperature gradient drives the Bose system out of equilibrium and
generate a heat current. The QBE [Eq. (4)] describes the transport property of
a steady state in terms of time as

$\displaystyle{\mathbf{v}}_{\mathbf{k}}\cdot{\mathbf{\nabla}}T\frac{\partial}{\partial
T}({\cal{A}}_{{\mathbf{k}},\omega}\phi_{{\mathbf{k}},\omega})={\cal{I}}_{{\mathbf{k}},\omega},$
(5)

where we assume that the temperature gradient is spatially uniform
${\mathbf{\nabla}}T=({\textrm{const.}})$. In this section, first, using the
functions ${\cal{A}}_{{\mathbf{k}},\omega}$ and $\phi_{{\mathbf{k}},\omega}$
of the QBE [Eq. (5)], we provide a formula for the heat current in the bulk of
two-dimensional Bose systems. Next, as a platform, in Sec. III.1 we apply the
formula to magnons in two-dimensional insulating magnets. Assuming a
relaxation time approximation for the function
${\cal{A}}_{{\mathbf{k}},\omega}\phi_{{\mathbf{k}},\omega}$ and focusing on
the linear response regime, we evaluate the thermal conductivity in the bulk
of the magnet. Finally, in Sec. III.2 applying the quasiparticle approximation
to the magnonic thermal conductivity of the QBE, we discuss the difference
from the one of the conventional Boltzmann equation.

The applied temperature gradient drives the system out of equilibrium and the
Bose distribution function $\phi_{{\mathbf{k}},\omega}$ deviates from the one
$\phi_{0}=({\text{e}}^{\beta\hbar\omega}-1)^{-1}$ in equilibrium, where
$\beta:=1/(k_{\textrm{B}}T)$ is the inverse temperature and $k_{\textrm{B}}$
represents the Boltzmann constant. The deviation is characterized as the
function
$\delta\phi_{{\mathbf{k}},\omega}:=\phi_{{\mathbf{k}},\omega}-\phi_{0}$. Since
the self-energy arises from impurity scattering, we assume that the spectral
function ${\cal{A}}_{{\mathbf{k}},\omega}$ is little influenced by temperature
and neglect the temperature dependence. Therefore the nonequilibrium Bose
distribution function in the wavenumber space is given as
$f_{\mathbf{k}}^{\textrm{QBE}}=f_{0}^{\textrm{QBE}}+\delta
f_{\mathbf{k}}^{\textrm{QBE}}$ with $f_{0}^{\textrm{QBE}}:=\int[\hbar
d\omega/(2\pi)]{\cal{A}}_{{\mathbf{k}},\omega}\phi_{0}$ and $\delta
f_{\mathbf{k}}^{\textrm{QBE}}:=\int[\hbar
d\omega/(2\pi)]{\cal{A}}_{{\mathbf{k}},\omega}\delta\phi_{{\mathbf{k}},\omega}$.
The function consists of two parts; the equilibrium component
$f_{0}^{\textrm{QBE}}$ and the nonequilibrium one $\delta
f_{\mathbf{k}}^{\textrm{QBE}}$. Since each mode $\omega$ subjected to a
chemical potential $\mu$ carries the energy $\hbar\omega$, the heat current
density in the bulk of two-dimensional Bose systems,
${\mathbf{j}}_{Q}=(j_{Q_{x}},j_{Q_{y}})$, is given as

$\displaystyle{\mathbf{j}}_{Q}=\int\frac{d^{2}{\mathbf{k}}}{(2\pi)^{2}}{\mathbf{v}}_{\mathbf{k}}\int\frac{\hbar
d\omega}{2\pi}(\hbar\omega-\mu){\cal{A}}_{{\mathbf{k}},\omega}\delta\phi_{{\mathbf{k}},\omega}.$
(6)

This is the formula for the heat current density of the QBE, including the
nonlinear response to the temperature gradient. The formula [Eq. (6)] is
useful to Bose systems (e.g., insulators and metals) with the spectral
function of arbitrary shape.

###  III.1 Magnonic thermal conductivity

As a platform, we apply the formula for the heat current of the QBE [Eq. (6)]
to magnons in a two-dimensional insulating magnet where time-reversal symmetry
is broken, e.g., due to an external magnetic field. At sufficiently low
temperatures, the effect of magnon-magnon interactions and that of phonons are
negligibly small, and impurity scattering makes a major contribution to the
self-energy. Therefore, we work under the assumption that the spectral
function ${\cal{A}}_{{\mathbf{k}},\omega}$ is little influenced by
temperature, and neglect the temperature dependence.

First, we comment on the chemical potential of magnons subjected to the
temperature gradient Nakata  _et al._ (2015, 2017b, 2017c). The applied
temperature gradient induces magnon transport, which leads to an accumulation
of magnons at the boundaries and builds up a nonuniform magnetization in the
sample. This magnetization gradient plays a role of an effective magnetic
field gradient and works as the gradient of a nonequilibrium spin chemical
potential Basso  _et al._ (2016b); Cornelissen  _et al._ (2016); Du  _et al._
(2017); Demokritov  _et al._ (2006) for magnons. This generates a counter-
current of magnons and thus the nonequilibrium spin chemical potential
contributes to the thermal conductivity. See Ref. Nakata  _et al._ (2017c) for
details. In this paper, for simplicity, we consider a sufficiently large
system and work under the assumption that the effect of the boundaries is
negligibly small. Consequently, the nonequilibrium magnon accumulation becomes
negligible and the nonequilibrium spin chemical potential of magnons vanishes.
In the magnonic system, the heat current is identified with the energy current
Matsumoto and Murakami (2011a); Nakata  _et al._ (2017b). Note that Basso  _et
al._ (2016b); Cornelissen  _et al._ (2016); Du  _et al._ (2017); Demokritov
_et al._ (2006) the spin chemical potential of magnons is peculiar to the
system out of equilibrium 222 Ref. Basso  _et al._ (2016b) indicates that the
nonequilibrium spin chemical potential can be regarded as a Johnson-Silsbee
potential Johnson and Silsbee (1987)..

Then, we consider thermal transport carried by magnons with the energy
dispersion relation $\hbar\omega_{\mathbf{k}}=Dk^{2}+\Delta$, where $D$
represents the spin stiffness constant, $k:=|{\mathbf{k}}|$ denotes the
magnitude of the wavenumber, and $\Delta$ is the magnon energy gap, e.g., due
to an external magnetic field and a spin anisotropy, etc. Assuming a
relaxation time approximation for the function
${\cal{A}}_{{\mathbf{k}},\omega}\phi_{{\mathbf{k}},\omega}$ of

$\displaystyle{\cal{I}}_{{\mathbf{k}},\omega}=-\frac{{\cal{A}}_{{\mathbf{k}},\omega}\phi_{{\mathbf{k}},\omega}-{\cal{A}}_{{\mathbf{k}},\omega}\phi_{0}}{\tau_{{\mathbf{k}},\omega}^{\textrm{R}}}$
(7)

and focusing on the linear response regime, from Eq. (5) the nonequilibrium
component $\delta\phi_{{\mathbf{k}},\omega}$ is given as

$\displaystyle\delta\phi_{{\mathbf{k}},\omega}=-\tau_{{\mathbf{k}},\omega}^{\textrm{R}}{\mathbf{v}}_{\mathbf{k}}\cdot{\mathbf{\nabla}}T\frac{\partial\phi_{0}}{\partial
T}+O(({\mathbf{\nabla}}T)^{2}),$ (8)

where $\tau_{{\mathbf{k}},\omega}^{\textrm{R}}$ is the relaxation time for the
magnonic system. Under the assumption that impurity scattering is elastic and
that the relaxation time depends solely on the magnitude of the wavenumber, it
is evaluated as
${1}/{\tau_{{\mathbf{k}},\omega}^{\textrm{R}}}=\sum_{{\mathbf{k}}^{\prime}}\mid
V_{{\mathbf{k}},{\mathbf{k}}^{\prime}}\mid^{2}{\cal{A}}_{{\mathbf{k}}^{\prime},\omega}(1-{{\mathbf{v}}_{\mathbf{k}}\cdot{\mathbf{v}}_{{\mathbf{k}^{\prime}}}}/{\mid{\mathbf{v}}_{\mathbf{k}}\mid^{2}})/\hbar$.
We remark that the relaxation time $\tau_{{\mathbf{k}},\omega}^{\textrm{R}}$
is different from the magnon lifetime
$\tau_{{\mathbf{k}},\omega}^{\textrm{L}}$ in general. Those are distinct
quantities. However, when the impurity potential is localized in space, the
Fourier component $V_{{\mathbf{k}},{\mathbf{k}}^{\prime}}$ becomes independent
of the wavenumber, and the relaxation time coincides with the magnon lifetime,
which takes a wavenumber-independent value as
$\tau_{\omega}^{\textrm{R}}=\tau_{\omega}^{\textrm{L}}$. From the Landau-
Lifshitz-Gilbert equation Tserkovnyak  _et al._ (2005), the magnon lifetime is
associated with the inverse of the Gilbert damping constant $\alpha$ and it is
described as Adachi  _et al._ (2011); Ohnuma  _et al._ (2014); Abrikosov  _et
al._ (1975); Tatara (2015a); Kovalev and Tserkovnyak (2012); Jiang  _et al._
(2013) $\hbar/(2\tau_{\omega}^{\textrm{L}})=\alpha\hbar\omega$. Therefore,
under the assumption that the real part of the self-energy is negligibly small
compared with the magnon energy gap, the spectral function is given as
${\cal{A}}_{{\mathbf{k}},\omega}=2\alpha\hbar\omega/[({\mathscr{H}}_{{\mathbf{k}},\omega})^{2}+(\alpha\hbar\omega)^{2}]$.
See Appendices for details See .

Finally, from Eqs. (6) and (8), we obtain the longitudinal thermal
conductivity of magnons in the bulk of the two-dimensional insulating magnet,
$\kappa_{xx}:=-j_{Q_{x}}/(\partial_{x}T)$, as

$\displaystyle\kappa_{xx}=$
$\displaystyle\frac{1}{2}\Big{(}\frac{D}{\pi}\Big{)}^{2}\frac{1}{k_{\textrm{B}}T^{2}}\int_{0}^{\infty}dkk^{3}$
$\displaystyle\cdot$ $\displaystyle\int
d\omega\frac{(\hbar\omega)^{2}}{({\mathscr{H}}_{{\mathbf{k}},\omega})^{2}+(\alpha\hbar\omega)^{2}}\frac{{\textrm{e}}^{\beta\hbar\omega}}{({\textrm{e}}^{\beta\hbar\omega}-1)^{2}},$
(9)

where we assume that the temperature gradient is applied along the $x$ axis.
In contrast to the conventional Boltzmann equation (cf., Sec. I), the thermal
conductivity of the QBE in the absence of the quasiparticle approximation does
not reduce to the Drude-type Dru , as a function of the Gilbert damping
constant $\alpha$, in that it is not proportional to $1/\alpha$. The factor
$1/\alpha$ arises from the relaxation time $\tau_{\omega}^{\textrm{R}}$ in
$\delta\phi_{{\mathbf{k}},\omega}$ [Eq. (8)] as
$\tau_{\omega}^{\textrm{R}}=\tau_{\omega}^{\textrm{L}}={1}/(2\alpha\omega)$.
However, it cancels out by the factor $2\alpha\hbar\omega$ of the spectral
function
${\cal{A}}_{{\mathbf{k}},\omega}=2\alpha\hbar\omega/[({\mathscr{H}}_{{\mathbf{k}},\omega})^{2}+(\alpha\hbar\omega)^{2}]$
[Eq. (6)]. Therefore the integrand remains the Lorentz-type 333 In this paper
we refer to the function of $\alpha$, $F(\alpha):=C_{1}/({\alpha}^{2}+C_{2})$
with $C_{1}>0$ and $C_{2}>0$, as a Lorentz-type to distinguish from the Drude-
type Dru . , and the thermal conductivity of the QBE does not reduce to the
Drude-type in the absence of the quasiparticle approximation.

###  III.2 Comparison: Conventional Boltzmann equation

The QBE in the limit of the quasiparticle approximation reduces to the
conventional Boltzmann equation, which provides the thermal conductivity of
the Drude-type Dru  in terms of the Gilbert damping constant $\alpha$ (cf.,
Sec. I). This agrees with our formula of the QBE [Eq. (6)]; when we employ the
quasiparticle (qp) approximation
${\cal{A}}_{{\mathbf{k}},\omega}\approx{\cal{A}}_{{\mathbf{k}},\omega}^{\textrm{qp}}:=2\pi\delta({\mathscr{H}}_{{\mathbf{k}},\omega})$
for Eq. (6), the thermal conductivity of the QBE reduces to
$\kappa_{xx}\approx\kappa_{xx}^{\textrm{qp}}$ as

$\displaystyle\kappa_{xx}^{\textrm{qp}}=\frac{1}{2}\Big{(}\frac{D}{\pi}\Big{)}^{2}\frac{1}{k_{\textrm{B}}T^{2}}\frac{\pi}{\alpha}\int_{0}^{\infty}dkk^{3}\omega_{\mathbf{k}}\frac{{\textrm{e}}^{\beta\hbar\omega_{\mathbf{k}}}}{({\textrm{e}}^{\beta\hbar\omega_{\mathbf{k}}}-1)^{2}}.$
(10)

This is consistent with Eq. (9) in the limit of $\alpha\rightarrow 0$. Thus we
find in the limit of the quasiparticle approximation that the thermal
conductivity of the QBE becomes the Drude-type, as a function of $\alpha$, in
that it is proportional to $1/\alpha$ as
$\kappa_{xx}\approx\kappa_{xx}^{\textrm{qp}}\propto{1}/{\alpha}$. The factor
$1/\alpha$ arises from the relaxation time
$\tau_{\omega}^{\textrm{R}}=\tau_{\omega}^{\textrm{L}}={1}/(2\alpha\omega)$ in
$\delta\phi_{{\mathbf{k}},\omega}$ [Eq. (8)].

Figure 1:  (Left) Plots of the rescaled thermal conductivity of magnons,
${\tilde{\kappa}}_{xx}:=[2\pi^{2}\hbar/(k_{\textrm{B}}^{2}T)]{\kappa}_{xx}$,
as a function of the Gilbert damping constant $\alpha$ Heinrich  _et al._
(2011); Tserkovnyak  _et al._ (2002) obtained by numerically solving Eqs. (9)
and (10) for $\Delta=k_{\textrm{B}}T$ with a fixed temperature. Red line: Eq.
(9) of the QBE. Black line: Eq. (10) for
$\kappa_{xx}\approx\kappa_{xx}^{\textrm{qp}}$ in the limit of the
quasiparticle approximation, i.e., the conventional Boltzmann equation. The
correction to the conventional Boltzmann equation enhances the thermal
conductivity. Inset: The enhancement still works and remains significant even
for small values of the parameter $\alpha$. (Right) The plot of the ratio,
$\kappa_{xx}/\kappa_{xx}^{\textrm{qp}}=1+(\kappa_{xx}-\kappa_{xx}^{\textrm{qp}})/\kappa_{xx}^{\textrm{qp}}$,
as a function of the Gilbert damping constant $\alpha$, i.e., the ratio of the
magnonic thermal conductivity $\kappa_{xx}$ of the QBE to the one
$\kappa_{xx}^{\textrm{qp}}$ of the conventional Boltzmann equation (i.e., the
QBE in the limit of the quasiparticle approximation). In units of
$\kappa_{xx}^{\textrm{qp}}$, the correction to the conventional one and the
resulting enhancement of the magnonic thermal conductivity increase as the
value of $\alpha$ becomes large.

In conclusion, using the QBE we have developed the formula for thermal
transport in the bulk of Bose systems, with a particular focus on magnons. As
a function of the Gilbert damping constant, the thermal conductivity of the
QBE reduces to the Drude-type Dru  in the limit of the quasiparticle
approximation, while not in the absence of the approximation. This is the main
difference in the thermal conductivity between the QBE and the conventional
one.

We remark that our formula based on the QBE reduces to the Drude-type Dru  in
the limit of the quasiparticle approximation. This means that, by relaxing the
quasiparticle approximation, the correction to the conventional Boltzmann
equation is integrated as the self-energy into the spectral function
${\cal{A}}_{{\mathbf{k}},\omega,{\mathbf{r}},t}$ of the QBE; Fig. 1 shows that
the correction to the conventional Boltzmann equation enhances the thermal
conductivity of the QBE. Thus our thermal transport theory of the QBE with
four variables $({\mathbf{k}},\omega,{\mathbf{r}},t)$ is identified as an
appropriate extension of the one of the conventional Boltzmann equation with
three variables $({\mathbf{k}},{\mathbf{r}},t)$.

##  IV Discussion

To conclude a few comments on our approach are in order. First, the
conventional Boltzmann equation Rezende  _et al._ (2014a, b); Basso  _et al._
(2016a, b, c); Okuma  _et al._ (2017); Schmidt  _et al._ (2018); Nakata  _et
al._ (2019b); Liu  _et al._ (2019); Jiang  _et al._ (2013) [Eq. (1)], i.e.,
the transport theory based on the quasiparticle approximation, can not
describe paramagnons Mahan (2000). The quasiparticle approximation assumes
that the spectrum has the form of the Dirac delta function. However, the
spectrum of paramagnons is broad in general, and has a peak with a sufficient
width of a nonzero value associated with the inverse of the finite lifetime
Doniach (1967); Hertz (1976); Doubble  _et al._ (2010); the spectrum of
paramagnons can not be approximated by the Dirac delta function. Therefore,
the conventional Boltzmann equation can not describe paramagnons. On the other
hand, our formula based on the QBE is applicable to magnons with the spectrum
of arbitrary shape [Eq. (6)]. In that sense, it is expected that our thermal
transport theory is useful also to paramagnons in the bulk of paramagnets 444
As for spin transport in paramagnets, see Refs. Shiomi and Saitoh (2014); Wu
_et al._ (2015); Oyanagi  _et al._ (2019); Okamoto (2016). .

We remark that as well as insulators, our formula [Eq. (6)] is applicable, in
principle, also to metals; the spectral function is described in terms of the
Gilbert damping constant See , and the value for metals (e.g., transition
metal ferromagnets) Bhagat and Lubitz (1974) $\alpha=O(10^{-2})$ is large
compared with that for insulators $\alpha=O(10^{-3})$ in general Tserkovnyak
_et al._ (2005); Heinrich  _et al._ (2011); Tserkovnyak  _et al._ (2002). Fig.
1 shows the behavior of the magnonic thermal conductivity in the region
$O(10^{-3})\leq\alpha\leq O(10^{-2})$. As seen in the right of Fig. 1,
$\kappa_{xx}/\kappa_{xx}^{\textrm{qp}}=1+(\kappa_{xx}-\kappa_{xx}^{\textrm{qp}})/\kappa_{xx}^{\textrm{qp}}$,
in units of $\kappa_{xx}^{\textrm{qp}}$ the correction to the conventional
Boltzmann equation and the resulting enhancement of the magnonic thermal
conductivity increase as the value of $\alpha$ becomes large. The damping
constant is associated with the inverse of the magnon lifetime See .

Next, our formula based on the QBE has an advantage over the linear response
theory that it includes the nonlinear response; Eq. (6) is the formula for the
heat current density including the nonlinear response to the temperature
gradient. Here, using the QBE we develop an analysis on the nonlinear
response. We apply the relaxation time approximation [Eq. (7)] for the
function ${\cal{A}}_{{\mathbf{k}},\omega}\phi_{{\mathbf{k}},\omega}$ to the
QBE [Eq. (5)]. Since the relaxation is induced by impurity scattering, we
assume that the relaxation time is little influenced by temperature, and
neglect the temperature dependence. Using the method of successive
substitution, the nonequilibrium component of the Bose distribution function
$\delta\phi_{{\mathbf{k}},\omega}$ [Eq. (6)] is evaluated, beyond the liner
response regime, as
$\delta\phi_{{\mathbf{k}},\omega}=\sum_{n=1}^{\infty}(-\tau_{{\mathbf{k}},\omega}^{\textrm{R}}{\mathbf{v}}_{\mathbf{k}}\cdot{\mathbf{\nabla}}T)^{n}[{\partial^{n}\phi_{0}}/({\partial
T^{n}})]$, where the nonequilibrium component
$\delta\phi_{{\mathbf{k}},\omega}$ is arranged in terms of the function
$({\mathbf{\nabla}}T)^{n}$. Lastly, combining this equation with Eq. (6), we
can obtain each coefficient of the nonlinear response to the temperature
gradient. The formula is not restricted to magnonic systems, and it is useful
to a wide range of Bose systems.

Finally, we add a comment to the linear response theory. According to Ref.
Tatara (2015a), which focuses on the region $\alpha\ll 1$, the linear response
theory (i.e., Kubo formula) results in the magnonic thermal conductivity of
the Drude-type Dru , the same as the conventional Boltzmann equation, in that
it is proportional to the inverse of the Gilbert damping constant. Note that
the temperature gradient is not mechanical force but thermodynamic force; the
temperature gradient is not described by a microscopic Hamiltonian. Therefore
it is not straightforward to integrate the temperature gradient into the
linear response theory. Ref. Tatara (2015a) described the effect of the
temperature gradient with the help of the thermal vector potential Tatara
(2015b) associated with the Luttinger’s approach Luttinger (1964). For the
details, see Ref. Tatara (2015a). We remark that the Boltzmann equation has no
difficulty in integrating the temperature gradient into the formalism, e.g.,
see Eq. (1).

##  V Outlook

We give a few perspectives on further research. As a platform, in Sec. III.1
and Sec. III.2 focusing on insulating magnets at sufficiently low temperature,
we have studied thermal transport of magnons under the assumption that the
effect of magnon-magnon interactions and that of phonons are negligibly small,
and that elastic impurity scattering makes a major contribution to the self-
energy. It is of interest to study those effects on magnonic thermal transport
of the QBE, with considering also the case that impurity scattering is
inelastic 555It is expected from Ref. Arakawa and Ohe (2018) that the weak
localization of magnons is induced in a disordered magnonic system where
impurities are randomly distributed and time-reversal symmetry holds
effectively..

In this paper relaxing the quasiparticle approximation and using the QBE for
Bose systems, we have found that the correction to the conventional Boltzmann
equation enhances the thermal conductivity. Therefore, based on the QBE it is
intriguing to study the effect of the correction on the magnonic Wiedemann-
Franz law Nakata  _et al._ (2015, 2017b, 2017c, 2019a); at low temperatures
magnon transport obeys a magnonic analogue of the Wiedemann-Franz law Franz
and Wiedemann (1853), a universal law, in that the ratio of heat to spin
conductivity is linear in temperature and does not depend on material
parameters except the $g$-factor. Note that the magnonic Wiedemann-Franz law
for the bulk of magnets has been proposed in Ref. Nakata  _et al._ (2017c)
based on the conventional Boltzmann equation. For the magnonic Wiedemann-Franz
law of the QBE, the spin conductivity and the off-diagonal elements of the
Onsager coefficient Nakata  _et al._ (2017c) remain to be obtained. We believe
it can be evaluated by following the study Mahan (2000) on the electrical
conductivity of the QBE and developing it into the Bose system. Using the QBE,
it will be intriguing also to study the magnonic Hall coefficients in
topologically nontrivial magnonic systems Katsura  _et al._ (2010); Matsumoto
and Murakami (2011a, b); Nakata  _et al._ (2017b, c, 2019a).

##  VI Conclusion

Developing the quantum Boltzmann equation, we have provided the formula for
thermal transport in the bulk of Bose systems, including the nonlinear
response to the temperature gradient. We have then applied the formula to
magnons and have shown that thermal transport of the quantum Boltzmann
equation exhibits the different behavior from the conventional. The
longitudinal thermal conductivity of the quantum Boltzmann equation reduces to
the Drude-type in the limit of the quasiparticle approximation, while not in
the absence of the approximation. Relaxing the quasiparticle approximation, we
have found that the correction to the conventional Boltzmann equation is
integrated as the self-energy into the spectral function of the quantum
Boltzmann equation, and this enhances the thermal conductivity. Our formula is
useful to Bose systems, including metals as well as insulators, with the
spectral function of arbitrary shape. Using the quantum Boltzmann equation we
have found the thermal transport property beyond the conventional.

###### Acknowledgements.

We would like to thank D. Loss for the collaborative work on the related study
and for turning our attention to this subject; this work is motivated by the
discussion at Basel in 2015 (K.N.). We are grateful also to T. Kita for
educating the author (K.N.) on the theoretical method of this study and
creating an opportunity to work on this subject, and to Y. Araki and H. Chudo
for helpful feedback. We acknowledge support by JSPS KAKENHI Grant Number
JP20K14420 (K. N.), by Leading Initiative for Excellent Young Researchers,
MEXT, Japan (K. N.), and by JST ERATO Grant No. JPMJER1601(Y. O.).

##  Appendix A Relaxation time and magnon lifetime

In this Appendix, we show that the relaxation time coincides with the magnon
lifetime and takes a wavenumber-independent value under the assumption that;
the relaxation time depends solely on the magnitude of the wavenumber,
impurity scattering is elastic, and the impurity potential is localized in
space. We remark that at sufficiently low temperatures, the effect of magnon-
magnon interactions and that of phonons are negligibly small, and impurity
scattering makes a major contribution to the self-energy. Therefore, we assume
that the spectral function is little influenced by temperature and neglect the
temperature dependence.

We consider magnons with the energy dispersion relation of
$\hbar\omega_{\mathbf{k}}=Dk^{2}+\Delta$, where $|{\mathbf{k}}|=:k$ denotes
the magnitude of the wavenumber. First, assuming the steady state in terms of
time and applying the relaxation time approximation for the function
${\cal{A}}_{{\mathbf{k}},\omega}\phi_{{\mathbf{k}},\omega}$ to the QBE (see
the main text), within the linear response regime we obtain the nonequilibrium
component $\delta\phi_{{\mathbf{k}},\omega}$ as
$\delta\phi_{{\mathbf{k}},\omega}=-\tau_{{\mathbf{k}},\omega}^{\textrm{R}}{\mathbf{v}}_{\mathbf{k}}\cdot{\mathbf{\nabla}}T[\partial\phi_{0}/(\partial
T)]+O(({\mathbf{\nabla}}T)^{2})$. Next, using the relaxation time
approximation for the collision integral
${\cal{I}}_{{\mathbf{k}},\omega}=-\sum_{{\mathbf{k}}^{\prime}}\mid
V_{{\mathbf{k}},{\mathbf{k}}^{\prime}}\mid^{2}{\cal{A}}_{{\mathbf{k}},\omega}{\cal{A}}_{{\mathbf{k}}^{\prime},\omega}(\phi_{{\mathbf{k}},\omega}-\phi_{{\mathbf{k}}^{\prime},\omega})/\hbar$,
we reach
$\delta\phi_{{\mathbf{k}},\omega}/\tau_{{\mathbf{k}},\omega}^{\textrm{R}}=\sum_{{\mathbf{k}}^{\prime}}\mid
V_{{\mathbf{k}},{\mathbf{k}}^{\prime}}\mid^{2}{\cal{A}}_{{\mathbf{k}}^{\prime},\omega}(\delta\phi_{{\mathbf{k}},\omega}-\delta\phi_{{\mathbf{k}}^{\prime},\omega})/\hbar$.
Finally, combining the equations under the assumption that the relaxation time
depends solely on the magnitude of the wavenumber and that impurity scattering
is elastic, we obtain the relaxation time as
${1}/{\tau_{{\mathbf{k}},\omega}^{\textrm{R}}}=\sum_{{\mathbf{k}}^{\prime}}\mid
V_{{\mathbf{k}},{\mathbf{k}}^{\prime}}\mid^{2}{\cal{A}}_{{\mathbf{k}}^{\prime},\omega}(1-{{\mathbf{v}}_{\mathbf{k}}\cdot{\mathbf{v}}_{{\mathbf{k}^{\prime}}}}/{\mid{\mathbf{v}}_{\mathbf{k}}\mid^{2}})/\hbar$.

Since we assume that impurities are dilute, the effect can be taken into
account within the Born approximation, see the main text for details. When the
impurity potential is localized in space, the Fourier component becomes
independent of the wavenumber and it is described as
$|V_{{\mathbf{k}},{\mathbf{k}}^{\prime}}|^{2}=:u^{2}n_{\textrm{imp}}$, where
$n_{\textrm{imp}}$ is the impurity concentration Haug and Jauho (2007). Then,
the self-energy becomes independent of the wavenumber ${\mathbf{k}}$, and it
is given as
$\Sigma_{{\mathbf{k}},\omega}=\Sigma_{\omega}:=u^{2}n_{\textrm{imp}}\sum_{{\mathbf{k}}^{\prime}}G_{{\mathbf{k}}^{\prime},\omega}$.
Therefore, the spectral function ${\cal{A}}_{{\mathbf{k}},\omega}$ depends
solely on the magnitude of the wavenumber $|{\mathbf{k}}|=:k$ and it is
denoted as ${\cal{A}}_{{\mathbf{k}},\omega}={\cal{A}}_{k,\omega}$. We remark
that the spectral function ${\cal{A}}_{{\mathbf{k}},\omega}$ consists of the
self-energy $\Sigma_{{\mathbf{k}},\omega}=\Sigma_{\omega}$ and the function
${\mathscr{H}}_{{\mathbf{k}},\omega}={\mathscr{H}}_{k,\omega}$; since we
assume that the energy dispersion relation of magnons takes the form of
$\hbar\omega_{\mathbf{k}}=Dk^{2}+\Delta$, the function
${\mathscr{H}}_{{\mathbf{k}},\omega}:=\hbar\omega-\hbar\omega_{\mathbf{k}}$
depends only on the magnitude of the wavenumber and it is represented as
${\mathscr{H}}_{{\mathbf{k}},\omega}={\mathscr{H}}_{k,\omega}$. Thus, the
spectral function becomes dependent only on the magnitude of the wavenumber as
${\cal{A}}_{{\mathbf{k}},\omega}={\cal{A}}_{k,\omega}$. Using this result with
the relation $\int_{0}^{2\pi}d\theta{\textrm{cos}}{\theta}=0$, finally, we
obtain the relaxation time as
${1}/{\tau_{{\mathbf{k}},\omega}^{\textrm{R}}}={1}/{\tau_{\omega}^{\textrm{R}}}:=u^{2}n_{\textrm{imp}}\sum_{{\mathbf{k}}^{\prime}}{\cal{A}}_{{\mathbf{k}}^{\prime},\omega}/\hbar$,
and find that it is independent of the wavenumber ${\mathbf{k}}$.

The relaxation time coincides with the lifetime for the impurity potential of
$|V_{{\mathbf{k}},{\mathbf{k}}^{\prime}}|^{2}=u^{2}n_{\textrm{imp}}$. The
lifetime ${\tau_{{\mathbf{k}},\omega}^{\textrm{L}}}$ is associated with the
imaginary part of the self-energy and it is described as
$\hbar/(2{\tau_{{\mathbf{k}},\omega}^{\textrm{L}}}):=-{\textrm{Im}}\Sigma_{{\mathbf{k}},\omega}^{\textrm{r}}$
in general, where $\Sigma_{{\mathbf{k}},\omega}^{\textrm{r}}$ represents the
retarded component of the self-energy and it is given as
$\Sigma_{{\mathbf{k}}\omega}^{\textrm{r}}=\Sigma_{\omega}^{\textrm{r}}:=u^{2}n_{\textrm{imp}}\sum_{{\mathbf{k}}^{\prime}}G_{{\mathbf{k}}^{\prime},\omega}^{\textrm{r}}$
for the impurity potential of
$|V_{{\mathbf{k}},{\mathbf{k}}^{\prime}}|^{2}=u^{2}n_{\textrm{imp}}$. Since
the imaginary part of the retarded Green’s function is associated with the
spectral function as
${\textrm{Im}}G_{{\mathbf{k}}^{\prime},\omega}^{\textrm{r}}=-{\cal{A}}_{{\mathbf{k}}^{\prime},\omega}/2$,
the lifetime becomes
${1}/{\tau_{{\mathbf{k}},\omega}^{\textrm{L}}}={1}/{\tau_{\omega}^{\textrm{L}}}:=u^{2}n_{\textrm{imp}}\sum_{{\mathbf{k}}^{\prime}}{\cal{A}}_{{\mathbf{k}}^{\prime},\omega}/\hbar$,
and takes the wavenumber-independent value. Thus the lifetime coincides with
the relaxation time
${1}/{\tau_{\omega}^{\textrm{L}}}={1}/{\tau_{\omega}^{\textrm{R}}}$.

We stress that the relaxation time is different from the lifetime in general.
Those are distinct quantities. However, under the assumption that the
relaxation time depends solely on the magnitude of the wavenumber, impurity
scattering is elastic, and the impurity potential is localized in space, the
relaxation time coincides with the lifetime and takes the wavenumber-
independent value.

##  Appendix B Magnon spectral function and
Gilbert damping constant

In this Appendix, we describe the spectral function of magnons in terms of the
Gilbert damping constant $\alpha$. The Landau-Lifshitz-Gilbert equation is
playing the central role in the conventional spintronics study Tserkovnyak
_et al._ (2005). Therefore to develop a relation with it, it is useful to
describe the spectrum as a function of the Gilbert damping constant. First, as
seen above, the spectral function
${\cal{A}}_{{\mathbf{k}},\omega}={\cal{A}}_{k,\omega}$ consists of the
function ${\mathscr{H}}_{{\mathbf{k}},\omega}={\mathscr{H}}_{k,\omega}$ and
the self-energy $\Sigma_{{\mathbf{k}},\omega}=\Sigma_{\omega}$, and it is
described as Mahan (2000); Haug and Jauho (2007); Kita (2010)
${\cal{A}}_{k,\omega}=-2{\textrm{Im}}\Sigma_{\omega}^{\textrm{r}}/[({\mathscr{H}}_{k,\omega})^{2}+({\textrm{Im}}\Sigma_{\omega}^{\textrm{r}})^{2}]$,
where we assume that the real part of the self-energy is negligibly small
compared with the magnon energy gap. Next, the lifetime is defined as the
imaginary part of the self-energy in general as
$\hbar/(2{\tau_{\omega}^{\textrm{L}}}):=-{\textrm{Im}}\Sigma_{\omega}^{\textrm{r}}$.
Since the lifetime of magnons is associated with the inverse of the Gilbert
damping constant as Adachi  _et al._ (2011); Ohnuma  _et al._ (2014);
Abrikosov  _et al._ (1975); Tatara (2015a); Kovalev and Tserkovnyak (2012);
Jiang  _et al._ (2013)
$\hbar/(2{\tau_{\omega}^{\textrm{L}}})=\alpha\hbar\omega$, the imaginary part
of the self-energy is characterized in terms of the Gilbert damping constant.
Finally, the spectral function is described as a function of the Gilbert
damping constant as
${\cal{A}}_{k,\omega}=2\alpha\hbar\omega/[({\mathscr{H}}_{k,\omega})^{2}+(\alpha\hbar\omega)^{2}]$.

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