$\displaystyle=\frac{1}{\sqrt{3}}\left(\widetilde{a}^{\phantom{{\dagger}}}_{\bf
q}+\widetilde{a}^{\phantom{{\dagger}}}_{{\bf q}+{\bf
Q}}+\widetilde{a}^{\phantom{{\dagger}}}_{{\bf q}-{\bf Q}}\right),$
$\displaystyle b^{\phantom{{\dagger}}}_{\bf q}$
$\displaystyle=\frac{1}{\sqrt{3}}\left(\widetilde{a}^{\phantom{{\dagger}}}_{\bf
q}+e^{i\theta}\,\widetilde{a}^{\phantom{{\dagger}}}_{{\bf q}+{\bf
Q}}+e^{-i\theta}\,\widetilde{a}^{\phantom{{\dagger}}}_{{\bf q}-{\bf
Q}}\right),$ (80) $\displaystyle c^{\phantom{{\dagger}}}_{\bf q}$
$\displaystyle=\frac{1}{\sqrt{3}}\left(\widetilde{a}^{\phantom{{\dagger}}}_{\bf
q}+e^{-i\theta}\,\widetilde{a}^{\phantom{{\dagger}}}_{{\bf q}+{\bf
Q}}+e^{i\theta}\,\widetilde{a}^{\phantom{{\dagger}}}_{{\bf q}-{\bf
Q}}\right),$

with the same $\theta\\!=\\!4\pi/3$. These expressions are general and apply
to the other cases where both single- and three-sublattice approaches are
possible, such as the 120° case discussed next.

###  B.2 120° state

For the 120° state, see Fig. 4(a) for a sketch, the angles
$\widetilde{\alpha}$ that define spin configuration and transformation to the
local reference frame in Eq. (12) can be written as

$\widetilde{\alpha}=\widetilde{\alpha}_{A}-{\bf Q}\cdot{\bf R}_{i}\,,$ (81)

where, according to the choice in Fig. 4,
$\widetilde{\alpha}_{A}\\!=\\!\beta\\!=\\!\pi$, ${\bf R}_{i}\\!=\\!{\bf
R}_{\ell}\\!+\\!\bm{\rho}_{\alpha}$ with $\bm{\rho}_{A}\\!=\\!0$,
$\bm{\rho}_{B}\\!=\\!-\bm{\delta}_{2}$, and
$\bm{\rho}_{C}\\!=\\!\bm{\delta}_{3}$, see Fig. 3, and ${\bf
Q}\\!=\\!(4\pi/3,0)$ as before. As a result, all mutual angles are the same up
to a sign, $\widetilde{\alpha}_{ij}\\!=\\!\pm 2\pi/3$, making the LSWT
Hamiltonian terms in (18) the same on each bond and rendering the division of
the lattice in three sublattices unnecessary. The Fourier transform (67) leads
to a standard LSWT Hamiltonian [62, 63, 64, 65, 66]

$\displaystyle\hat{\cal H}^{(2)}=\sum_{{\bf q}}\Big{[}A_{\bf
q}\widetilde{a}^{\dagger}_{{\bf q}}\widetilde{a}^{\phantom{{\dagger}}}_{{\bf
q}}-\frac{B_{\bf q}}{2}\Big{(}\widetilde{a}^{\dagger}_{{\bf
q}}\widetilde{a}^{\dagger}_{-{\bf q}}+{\rm H.c.}\Big{)}\Big{]},$ (82)

where

$\displaystyle A_{\bf q}=3J_{1}S\Big{(}1+\overline{\gamma}_{{\bf
q}}/2-b\big{(}1-4\overline{\gamma}_{{\bf
q}}\big{)}/2-2j_{2}\big{(}1-\gamma_{{\bf q}}^{(2)}\big{)}\Big{)},$
$\displaystyle B_{\bf q}=3J_{1}S\Big{(}3\big{(}\overline{\gamma}_{{\bf
q}}+b\big{)}/2\Big{)},$ (83)

with $\overline{\gamma}_{{\bf q}}$ from (69) and magnon energy with the
parameters of the Bogolyubov transformation given by

$\displaystyle\omega_{\bf q}\\!=\\!\sqrt{A_{\bf q}^{2}-B_{\bf q}^{2}},$
$\displaystyle U^{2}_{\bf q}+V^{2}_{\bf q}=\frac{A_{\bf q}}{\omega_{\bf q}},\
\ \ 2U_{\bf q}V_{\bf q}=\frac{B_{\bf q}}{\omega_{\bf q}}.$ (84)

Similarly to the FM phase considered above, the relation of the single-
sublattice branch $\omega_{\bf q}$ to the three-sublattice energies is
straightforward, with the three branches given by the “original” $\omega_{1\bf
q}\\!=\\!\omega_{\bf q}$ and the two “shifted” ones, $\omega_{2(3)\bf
q}\\!=\\!\omega_{{\bf q}\pm{\bf Q}}$, see Fig. 6(a).

The relationship of the three-flavor operators
$a^{\phantom{{\dagger}}}_{\alpha\bf q}$ ($a^{{\dagger}}_{\alpha\bf q}$) to the
single-sublattice operators $\widetilde{a}^{\phantom{{\dagger}}}_{\bf q}$
($\widetilde{a}^{{\dagger}}_{\bf q}$) is the same as in Eq. (80) as it is
simply a relation between one- and three-sublattice Fourier transforms in (21)
and (67). The eigenvectors of the generalized Bogolyubov transformation in the
three-sublattice case (36) are related to the single-sublattice ones in (84)
via a combination of shifts by $\pm{\bf Q}$ and the phase factors as in (80)
in the latter.

The electron-magnon Hamiltonian is obtained from the Kondo coupling (46) using
spin-rotation transformation (12) with the angles (81), subsequent spin-
operator bosonization, and Fourier (67) and Bogolyubov (84) transforms.
Because of the non-collinear structure, it contains both spin-flip and non-
spin-flip terms that have the same structure as the general form of electron-
magnon coupling in (49), which we will not rewrite here.

Following the steps outlined in Appendices C.2 and C.3, we obtain the
relaxation rate in the form of Eq. (51) with the 1D integral in (52) given by

$\displaystyle I_{k_{F}}(T,H)=$
$\displaystyle\,\int_{0}^{1}\frac{z^{2}\,dz}{\sqrt{1-z^{2}}}$ (85)
$\displaystyle\,\phantom{\int_{0}^{1}}\times\sum_{\gamma}\widetilde{\Phi}_{\gamma,{\bf
q}}\,{\rm n}^{0}_{\gamma,{\bf q}}({\rm n}^{0}_{\gamma,{\bf
q}}+1)\,\frac{\omega_{\gamma,{\bf q}}}{T}\,,$

with ${\bf q}\\!=\\!2k_{F}(z^{2},z\sqrt{1-z^{2}})$ and Bose function ${\rm
n}^{0}_{\gamma,{\bf q}}$ as before, and with $\gamma$ now numerating ${\bf q}$
and ${{\bf q}\pm{\bf Q}}$ branches.

The advantage of the form (85) over the general expression (52) is in an
explicit analytical form of the matrix elements contribution
$\widetilde{\Phi}_{\gamma,{\bf q}}$, which are given by

$\widetilde{\Phi}_{1,{\bf q}}=2\big{(}U_{\bf q}-V_{\bf q}\big{)}^{2},\ \ \
\widetilde{\Phi}_{2(3),{\bf q}}=\big{(}U_{{\bf q}\pm{\bf Q}}+V_{{\bf q}\pm{\bf
Q}}\big{)}^{2}.$ (86)

Using them, an analytic insight into the contributions of ${\bf
q}\\!\rightarrow\\!0$ or ${\bf q}\\!\rightarrow\\!{\bf Q}$ regions of
integration into the relaxation rate becomes possible.

###  B.3 Plateau state

The up-up-down (UUD), or magnetization plateau state is special as it does not
break rotational $U(1)$ symmetry about the field direction, with the spins
parallel (up) or antiparallel (down) to the field. For our choice in Fig. 4,
sublattice $A$ spins are down, $\widetilde{\alpha}_{A}\\!=\\!\beta\\!=\\!\pi$,
while $B$ and $C$ spins are up, $\widetilde{\alpha}_{B(C)}\\!=\\!0$. With
these angles, six out of twelve independent matrix elements in the matrices
$\hat{\bf A}^{\phantom{\dagger}}_{\bf q}$ and $\hat{\bf
B}^{\phantom{\dagger}}_{\bf q}$ (32) of the Hamiltonian matrix (25) vanish and
the remaining six are complementary, suggesting a rearrangement in the vector-
operator of the Holstein-Primakoff bosons that allows to reduce the rank of
the Hamiltonian matrix to $3\times 3$,

$\displaystyle\hat{\cal H}^{(2)}=\frac{3J_{1}S}{2}\sum_{\bf q}\hat{\bf y}_{\bf
q}^{\dagger}\hat{\bf H}_{\bf q}\hat{\bf y}_{\bf q}^{\phantom{\dagger}}\,,$
(87)

where $\hat{\bf y}^{\dagger}_{\bf q}\\!=\\!\left(a^{\phantom{{\dagger}}}_{-\bf
q},b^{\dagger}_{\bf q},c^{\dagger}_{\bf q}\right)$ is the rearranged vector-
operator with the Hamiltonian matrix

$\displaystyle\hat{\bf H}_{\bf q}=\left(\begin{array}[]{ccc}\bar{A}_{\bf
q}&\bar{P}_{\bf q}&\bar{P}^{*}_{\bf q}\\\ \bar{P}^{*}_{\bf q}&\bar{C}_{\bf
q}&\bar{H}_{\bf q}\\\ \bar{P}_{\bf q}&\bar{H}^{*}_{\bf q}&\bar{C}_{\bf
q}\end{array}\right),$ (91)

and matrix elements given by

$\displaystyle\bar{A}_{\bf q}=$
$\displaystyle\,2-h+4b-2j_{2}\big{(}1-\gamma^{(2)}_{\bf q}\big{)},$
$\displaystyle\bar{C}_{\bf q}=$
$\displaystyle\,h+4b-2j_{2}\big{(}1-\gamma^{(2)}_{\bf q}\big{)},$
$\displaystyle\bar{P}_{\bf q}=$ $\displaystyle\,-(1+2b)\gamma_{\bf q},$ (92)
$\displaystyle\bar{H}_{\bf q}=$ $\displaystyle\,(1-2b)\gamma_{\bf q},$

with all constants as before and $\gamma_{\bf q}$ and $\gamma^{(2)}_{\bf q}$
from (35).

This approach, first employed in Ref. [20], has been used in a number of more
recent works [38, 39]. Physically, such a rearrangement takes advantage of the
conservation of magnetization in the UUD state due to remaining $U(1)$
symmetry for the Heisenberg spins, which makes creation (annihilation) of spin
flips in the down-sublattice equivalent to the annihilation (creation) of spin
flips in the up-sublattices.

The standard diagonalization procedure of (91) now concerns $\hat{\bar{{\bf
g}}}\hat{{\bf H}}_{\bf q}$, where $\hat{\bar{\bf g}}$ is a diagonal matrix
$[-1,1,1]$. The characteristic equation $||\hat{\bar{{\bf g}}}\hat{{\bf
H}}_{{\bf q}}-\lambda\hat{{\bf I}}||\\!=\\!0$ yields

$\displaystyle\lambda^{3}+b\lambda^{2}+c\lambda+d=0\,,$ (93)

which is a cubic equation on $\lambda$ with real coefficients,

$\displaystyle b=$ $\displaystyle\,\bar{A}_{\bf q}-2\bar{C}_{\bf q},$
$\displaystyle c=$ $\displaystyle\,\bar{C}_{\bf q}^{2}-|\bar{H}_{\bf
q}|^{2}+2|\bar{P}_{\bf q}|^{2}-2\bar{A}_{\bf q}\bar{C}_{\bf q},$ (94)
$\displaystyle d=$ $\displaystyle\,\bar{A}_{\bf q}\big{(}\bar{C}_{\bf
q}^{2}-|\bar{H}_{\bf q}|^{2}\big{)}-2\bar{C}_{\bf q}|\bar{P}_{\bf
q}|^{2}+2\Re\big{(}\bar{P}_{\bf q}^{2}\bar{H}_{\bf q}\big{)},$

that can be solved in a number of standard ways [41], with the three roots
$\\{\lambda_{1},\lambda_{2},\lambda_{3}\\}$ giving magnon eigenenergies
$\\{-\omega_{1-{\bf q}},\omega_{2{\bf q}},\omega_{3{\bf q}}\\}$ in units of
$3J_{1}S$. At the $\Gamma$ point, analytic solution of (93) simplifies
considerably, making explicit the linear field-dependence of the magnon
energies throughout the UUD phase. Note that the numeration of the magnon
branches in Fig. 6(c) is from the lowest to highest in energy, not necessarily
according to the numbering of the solutions above.

The transformation from Holstein-Primakoff bosons in (87) to the
quasiparticles is similar to the generalized Bogolyubov transformation in (36)

$\displaystyle\bar{a}^{\phantom{{\dagger}}}_{\alpha,{\bf
q}}=\sum_{\gamma}U_{\alpha,{\bf
q}}^{({\gamma})}\,\bar{A}^{\phantom{{\dagger}}}_{\gamma,{\bf q}},$ (95)

but without the “anomalous” $V_{\alpha}^{({\gamma})}$ terms and with the
“mixed” original $\bar{a}^{\phantom{{\dagger}}}_{\alpha,{\bf
q}}\\!=\\!\big{\\{}a^{{\dagger}}_{-\bf q},b^{\phantom{{\dagger}}}_{\bf
q},c^{\phantom{{\dagger}}}_{\bf q}\big{\\}}$ and quasiparticle operators
$\bar{A}^{\phantom{{\dagger}}}_{\gamma,{\bf
q}}\\!=\\!\big{\\{}A^{{\dagger}}_{-\bf q},B^{\phantom{{\dagger}}}_{\bf
q},C^{\phantom{{\dagger}}}_{\bf q}\big{\\}}$ and normalization that respects
the metric $\hat{\bar{{\bf g}}}$

$\displaystyle\sum_{\gamma}\sigma_{\gamma}\big{|}U_{\alpha,{\bf
q}}^{({\gamma})}\big{|}^{2}=\sigma_{\alpha}\,,$ (96)

where
$\sigma_{\gamma}\\!\equiv\\!\hat{\bar{g}}_{\gamma\gamma}\\!=\\!\\{-1,1,1\\}$.

The transformation (95) from $\hat{\bf y}_{\bf
q}\\!=\\!\big{[}a^{{\dagger}}_{-\bf q},b^{\phantom{{\dagger}}}_{\bf
q},c^{\phantom{{\dagger}}}_{\bf q}\big{]}^{T}$ to $\hat{\bf z}_{\bf
q}\\!=\\!\big{[}A^{{\dagger}}_{-\bf q},B^{\phantom{{\dagger}}}_{\bf
q},C^{\phantom{{\dagger}}}_{\bf q}\big{]}^{T}$ vectors, written in a matrix
form is

$\displaystyle\hat{\bf y}_{\bf q}=\hat{\bf U}_{\bf q}\cdot\hat{\bf z}_{\bf
q},\ \ \ $ (97)

where the transformation matrix $\hat{\bf U}_{\bf q}$ is given by the
normalized eigenvectors of $\hat{\bar{{\bf g}}}\hat{{\bf H}}_{\bf q}$ that can
be obtained via a somewhat tedious, but straightforward diagonalization
procedure using explicit form of $\hat{{\bf H}}_{\bf q}$ in (91), yielding

$\displaystyle{\bf U}_{{\bf q}}^{({\gamma})}=\frac{1}{r_{\gamma,{\bf
q}}}\left(\begin{array}[]{c}\widetilde{R}_{\gamma,{\bf q}}\\\ R_{\gamma,{\bf
q}}\\\ R^{*}_{\gamma,{\bf q}}\end{array}\right),$ (101)

with

$\displaystyle\widetilde{R}_{\gamma,{\bf q}}=$
$\displaystyle\,\big{(}\lambda_{\gamma,{\bf q}}-\bar{C}_{\bf
q}\big{)}^{2}-|\bar{H}_{\bf q}|^{2},$ $\displaystyle R_{\gamma,{\bf q}}=$
$\displaystyle\,\bar{P}_{\bf q}\bar{H}_{\bf q}+\bar{P}_{\bf
q}^{*}\big{(}\lambda_{\gamma,{\bf q}}-\bar{C}_{\bf q}\big{)},$ (102)
$\displaystyle r_{\gamma,{\bf q}}^{2}=$
$\displaystyle\,\sigma_{\gamma}\big{(}2|R_{\gamma,{\bf
q}}|^{2}-\widetilde{R}_{\gamma,{\bf q}}^{2}\big{)},$

where $\lambda_{\gamma,{\bf q}}$ are the eigenvalues obtained from (93) and
(94). While somewhat cumbersome, the outlined formalism allows to perform
calculations without relying on numerical matrix diagonalization procedures.

The derivation of electron-magnon coupling and relaxation rate in the UUD
phase bears a lot of similarity to the polarized FM case in its three-
sublattice formulation, see Sec. B.1. Because of spin collinearity, only spin-
flip scattering is present, with either emission or absorption

${\cal H}_{int}^{+-}=\frac{2\widetilde{J}_{K}}{\sqrt{3N}}\sum_{{\bf k},{\bf
q},\gamma}\Big{[}\widetilde{M}_{\gamma,{\bf q}}f^{\dagger}_{{\bf k}-{\bf
q}\uparrow}f^{\phantom{{\dagger}}}_{{\bf
k}\downarrow}\bar{A}^{{\dagger}}_{\gamma,{\bf q}}\ +{\rm H.c.}\Big{]},$ (103)

where $\bar{A}^{{\dagger}}_{\gamma,{\bf
q}}\\!=\\!\big{\\{}A^{\phantom{{\dagger}}}_{-\bf q},B^{{\dagger}}_{\bf
q},C^{{\dagger}}_{\bf q}\big{\\}}$ as in (95) and matrix elements
$\widetilde{M}_{\gamma,{\bf q}}\\!=\\!\sum_{\alpha}U_{\alpha,-{\bf
q}}^{({\gamma})}$, with vectors ${\bf U}^{({\gamma})}_{\bf q}$ from (101).

As in the polarized FM case, the coupling to a mode $\gamma$ in (103) is given
by the sum of the components of a corresponding vector, ${\bf
U}^{({\gamma})}_{\bf q}$, but without further simplifications that follow in
the FM case.

Lastly, the 1D integral in the relaxation rate (51) is identical to Eq. (52)

$\displaystyle I_{k_{F}}(T,H)=$
$\displaystyle\,\int_{0}^{1}\frac{z^{2}\,dz}{\sqrt{1-z^{2}}}$ (104)
$\displaystyle\,\phantom{\int_{0}^{1}}\times\frac{1}{3}\sum_{\gamma}\widetilde{\Phi}_{\gamma,{\bf
q}}\,{\rm n}^{0}_{\gamma,{\bf q}}({\rm n}^{0}_{\gamma,{\bf
q}}+1)\,\frac{\omega_{\gamma,{\bf q}}}{T}\,,$

but with a simplified $\widetilde{\Phi}_{\gamma,{\bf
q}}\\!=\\!\big{|}\widetilde{M}_{\gamma,{\bf q}}\big{|}^{2}$.

##  Appendix C Transport formalism, $1/\tau$ approximation

Transport theory, including its Boltzmann version, is an essential chapter in
most of the advanced condensed matter textbooks [51, 52]. Discussion of the
electron-phonon scattering, which controls resistivity of metals in a wide
range of temperatures, is also a necessary part of the story. Still, the
technical aspects of its theory are usually avoided in this context, opting
for some physically justified but not rigorous considerations. While some of
the required steps seem surprisingly complex and formidable enough to justify
this approach, in the following narrative we would like to dispel this
accepted aura from the subject. In the process of doing so, we also provide a
solution of the problem at hand, that is, make available a compact expression
for the electron-magnon scattering rate in a 2D setting relevant to EuC6.

###  C.1 Basics and conventions

For an electric current, the key quantity to find from the Boltzmann formalism
is $\delta{\rm f}_{{\bf k},\sigma}\\!=\\!{\rm f}_{{\bf k},\sigma}\\!-{\rm
f}^{0}_{{\bf k},\sigma}$, a deviation of the non-equilibrium distribution
function from the equilibrium one, where ${\rm f}^{0}_{{\bf k},\sigma}$ is a
Fermi-distribution function. Then, the current density is

$\displaystyle{\bf j}=\sum_{\sigma}\sum_{\bf k}e{\bf v}_{{\bf k}}\delta{\rm
f}_{{\bf k},\sigma},$ (105)

where ${\bf v}_{{\bf k}}\\!=\\!\partial\varepsilon_{{\bf k}}/\partial{\bf k}$
is the electron velocity.

The linearized Boltzmann equation (LBE)

$\displaystyle e\big{(}{\bf E}\cdot{\bf v}_{\bf k}\big{)}\,\frac{\partial{\rm
f}^{0}_{\bf k}}{\partial\varepsilon_{\bf k}}={\rm St}_{\bf k}[{\rm f}_{\bf
k}],$ (106)

allows to determine $\delta{\rm f}_{\bf k}$ in the linear-response
approximation, here ${\bf E}$ is an electric field, ${\rm St}_{\bf k}[{\rm
f}_{\bf k}]$ is a collision integral, and we ignore electron spin for a
moment.

The crucial step is the relaxation-time approximation,

$\displaystyle{\rm St}_{\bf k}[{\rm f}_{\bf k}]\approx-\frac{\delta{\rm
f}_{\bf k}}{\tau_{\bf k}},$ (107)

which yields the expected linear relation
$j^{\alpha}\\!=\\!\sigma^{\alpha\beta}E^{\beta}$, with $\sigma^{\alpha\beta}$
being conductivity tensor. Making further approximation of a continuum renders
conductivity tensor diagonal, Fermi surface spherical (circular in 2D), and
naturally suggests the same scattering rate for all $|{\bf k}|\\!=\\!k_{F}$.
Together with a realization that the left-hand side of (106) is singular at
$T\\!\ll\\!E_{F}$ as $-\partial{\rm f}^{0}_{\bf k}/\partial\varepsilon_{\bf
k}\\!\approx\\!\delta(\varepsilon_{\bf k}-E_{F})$ and using the relation of
the density of states at $E_{F}$ to the electronic density $n$, leads to the
standard

$\displaystyle\rho=\sigma^{-1}=\frac{m}{e^{2}n}\cdot\frac{1}{\tau_{F}},$ (108)

where $1/\tau_{F}\\!=\\!1/\tau_{k_{F}}$ is responsible for all temperature-
and field-dependence of the resistivity in a metal.

While the achieved progress is not entirely hollow, it is clear that the
problem of finding resistivity is now converted to the problem of finding
electron transport relaxation rate. The latter depends on the type of
scattering and on the details of microscopic interaction that determine the
functional form of the collision integral in (106). However, the technical
problem at hand is more involved, as one needs to rigorously prove that the
collision integral for a given type of scattering indeed yields the
relaxation-rate approximation (107) and to derive a microscopic expression for
$1/\tau_{F}$ from it at the same time.

###  C.2 Phonon-like spin-conserving scattering

Figure 13: (a) Schematics of different terms in the collision integral. (b)
Momenta of electrons and a magnon and their mutual angles.

Let us first consider the problem of the non-spin-flip part of electron-magnon
scattering in (49). Since scattering channels for different magnon branches
are additive and electrons of opposite spin do not mix, interaction
Hamiltonian can be written in a more general form that is identical to that of
electron-phonon coupling

$\displaystyle{\cal H}_{int}=\frac{1}{\sqrt{N}}\sum_{{\bf k},{\bf
q}}\Big{(}V_{\bf q}\,f^{\dagger}_{{\bf k}-{\bf
q}}f^{\phantom{{\dagger}}}_{{\bf k}}a^{{\dagger}}_{\bf q}+{\rm H.c.}\Big{)},$
(109)

in which the coupling $V_{\bf q}$ depends only on the bosonic momentum because
of the adiabatic principle, $E_{F}\\!\gg\\!\Theta_{D}$, where $\Theta_{D}$ is
a phonon (magnon) Debye energy.

Collision integral for the scattering (109), schematically represented in Fig.
13(a), has a general form

$\displaystyle{\rm St}_{\bf k}$ $\displaystyle[{\rm f}_{\bf
k}]=\frac{2\pi}{\hbar}\sum_{\bf q}\big{|}V_{\bf q}\big{|}^{2}$ (110)
$\displaystyle\times$ $\displaystyle\,\Big{\\{}\Big{(}{\rm f}_{\bf
k^{\prime}}(1-{\rm f}_{\bf k})\,{\rm n}_{\bf q}-{\rm f}_{\bf k}(1-{\rm f}_{\bf
k^{\prime}})({\rm n}_{\bf q}+1)\Big{)}\,\delta_{\varepsilon_{\bf
k^{\prime}},\varepsilon_{\bf k}-\omega_{\bf q}}$ $\displaystyle+\\!\Big{(}{\rm
f}_{\bf k^{\prime}}(1-{\rm f}_{\bf k})({\rm n}_{\bf q}+1)-{\rm f}_{\bf
k}(1-{\rm f}_{\bf k^{\prime}})\,{\rm n}_{\bf
q}\Big{)}\,\delta_{\varepsilon_{\bf k^{\prime}},\varepsilon_{\bf
k}+\omega_{\bf q}}\Big{\\}},$

where ${\bf k^{\prime}}\\!=\\!{\bf k}-{\bf q}$ and
$\delta_{\varepsilon^{\prime},\varepsilon\pm\omega}\\!=\\!\delta(\varepsilon^{\prime}-\varepsilon\mp\omega)$.
Since bosons are in equilibrium, ${\rm n}_{\bf q}\\!=\\!{\rm n}^{0}_{\bf q}$,
linearization of (110) gives

$\displaystyle{\rm St}_{\bf k}$ $\displaystyle[{\rm f}_{\bf
k}]=\frac{2\pi}{\hbar}\sum_{\bf q}\big{|}V_{\bf q}\big{|}^{2}$ (111)
$\displaystyle\times$ $\displaystyle\,\Big{\\{}\Big{(}\delta{\rm f}_{\bf
k^{\prime}}\big{(}{\rm n}^{0}_{\bf q}+{\rm f}^{0}_{\bf k}\big{)}-\delta{\rm
f}_{\bf k}\big{(}{\rm n}^{0}_{\bf q}-{\rm f}^{0}_{\bf
k^{\prime}}+1\big{)}\Big{)}\,\delta_{\varepsilon_{\bf
k^{\prime}},\varepsilon_{\bf k}-\omega_{\bf q}}$
$\displaystyle+\\!\Big{(}\delta{\rm f}_{\bf k^{\prime}}\big{(}{\rm n}^{0}_{\bf
q}-{\rm f}^{0}_{\bf k}+1\big{)}-\delta{\rm f}_{\bf k}\big{(}{\rm n}^{0}_{\bf
q}+{\rm f}^{0}_{\bf k^{\prime}}\big{)}\Big{)}\,\delta_{\varepsilon_{\bf
k^{\prime}},\varepsilon_{\bf k}+\omega_{\bf q}}\Big{\\}}.$

The physically justified shortcut from (111) to the relaxation-time form (107)
capitalizes on the adiabatic approximation. It advocates an impurity-like,
quasi-elastic picture of the scattering of the “fast” electrons on the “slow”
bosons [51], which should adhere to the relaxation-time form by construction.
In this case, bosonic thermal population plays the role of the thermally-
excited “impurity” concentration. The resultant transport relaxation rate
yields qualitatively correct answers for the resistivity of a metal due to
phonon scattering in both low- and high-temperature limits, the Bloch-
Grüneisen and Ohm’s laws, respectively.

Technically, this approach amounts to neglecting $\omega_{\bf q}$ in the
energy conservation in (111), which is indeed well-justified, and neglecting
everything but ${\rm n}^{0}_{\bf q}$ in the innermost brackets of the
collision integral in Eq. (111). It is the last step that is substantially
harder to defend. While advanced monographs such as Ref. [53] offer a
significantly more refined path that we partially follow, there are still
additional constructs in it that seem unnecessary.

####  C.2.1 Ansatz and solution

A more intelligent way of dealing with the collision integral in (111) is to
reflect on the ultimate form of $\delta{\rm f}_{\bf k}$ that follows from the
LBE (106) and from the definition of the current in (105). This suggests an
ansatz

$\displaystyle\delta{\rm f}_{\bf k}=\frac{e}{m}\,\big{(}{\bf E}\cdot{\bf
k}\big{)}\bigg{(}-\frac{\partial{\rm f}^{0}_{\bf k}}{\partial\varepsilon_{\bf
k}}\bigg{)}\,\widetilde{\chi}_{\bf k}\,,$ (112)

which adheres to the linear-response level of consideration, $\delta{\rm
f}_{\bf k}\\!\sim\\!{\bf E}$, and respects the reflection anti-symmetry of
$\delta{\rm f}_{-\bf k}\\!=\\!-\delta{\rm f}_{\bf k}$ that follows from (105).
It also explicitly separates the part of the LBE solution (106) that is
strongly peaked near the Fermi energy, $\partial{\rm f}^{0}_{\bf
k}/\partial\varepsilon_{\bf k}$, from the slowly varying
$\widetilde{\chi}_{\bf k}$. By construction, the last quantity,
$\widetilde{\chi}_{\bf k}$, has the meaning of the relaxation time $\tau_{\bf
k}$ and is an even function of ${\bf k}$. Last but not the least, in (112) we
also use continuum-like approximations for electrons, which implies spherical
(circular) Fermi surface and allows us to replace ${\bf v}_{\bf k}$ with ${\bf
k}/m$, a move that makes the subsequent steps more straightforward.

We also note that since the dependence of the equilibrium distribution
functions on the momentum is only via the energy, ${\rm f}^{0}_{\bf
k}\\!\equiv\\!{\rm f}^{0}(\varepsilon_{\bf k})$ and ${\rm f}^{0}_{\bf
k^{\prime}}\\!\equiv\\!{\rm f}^{0}(\varepsilon_{\bf k}\pm\omega_{\bf q})$,
there is an effective separation of variables in the ansatz (112). This is
because the singular component of the ansatz, $\partial{\rm f}^{0}_{\bf
k}/\partial\varepsilon_{\bf k}$, depends sensitively on the differences
$\pm\omega_{\bf q}$ due to scattering near the Fermi energy, but not on the
direction of the momentum ${\bf k}$. At the same time, the $({\bf E}\cdot{\bf
k})$ factor in (112) is solely responsible for the directional dependence of
${\bf k}$, but is largely insensitive to the variations in energy due to
scattering, as is argued next.

A significant simplification follows from the hierarchy of energy scales,
$\varepsilon_{\bf k}\\!\gg\\!\omega_{\bf k}$. Since conducting electrons are
confined to a proximity of the Fermi energy, the energy conservation with an
emission or absorption of a boson imply a quasi-elastic nature of the
scattering process, $\varepsilon_{{\bf
k}^{\prime}}\\!\approx\\!\varepsilon_{\bf k}$, in a full accord with the
qualitative logic outlined above. The implication of this result is $|{\bf
k}^{\prime}|\\!\approx\\!|{\bf k}|$, valid with the accuracy
$O(\Theta_{D}/E_{F})\\!\ll\\!1$. Thus, while one should keep small differences
$\pm\omega_{\bf q}$ in the singular component of $\delta{\rm f}_{{\bf k}({\bf
k}^{\prime})}$, it is only the direction of the momentum that can be changing
significantly. This makes geometry of the scattering in Fig. 13(b)
particularly simple and allows to rewrite $\delta$-functions in (111) as

$\displaystyle\delta\left(\varepsilon_{\bf k^{\prime}}-\varepsilon_{\bf
k}\mp\omega_{\bf
q}\right)\approx\left(\frac{m}{\hbar^{2}\,kq}\right)\delta\left(\cos\varphi-\frac{q}{2k}\right),$
(113)

where $\varphi$ is the angle of ${\bf k}$ with ${\bf q}$, see Fig. 13(b).

The second important implication of this consideration is for the relation of
$\delta{\rm f}_{\bf k^{\prime}}$ to $\delta{\rm f}_{\bf k}$. Using (112), we
write

$\displaystyle\delta{\rm f}_{\bf k^{\prime}}=\frac{e}{m}\,\big{(}{\bf
E}\cdot{\bf k}^{\prime}\big{)}\bigg{(}-\frac{\partial{\rm f}^{0}_{\bf
k^{\prime}}}{\partial\varepsilon_{\bf
k^{\prime}}}\bigg{)}\,\widetilde{\chi}_{\bf k^{\prime}}\,.$ (114)

As is discussed already, ${\rm f}^{0}_{\bf k^{\prime}}\\!\equiv\\!{\rm
f}^{0}(\varepsilon_{\bf k}\pm\omega_{\bf q})$ and $\widetilde{\chi}_{\bf
k^{\prime}}$ is a slowly varying even function of ${\bf k^{\prime}}$. Since
$|{\bf k}^{\prime}|\\!\approx\\!|{\bf k}|$, it is natural to assume
$\widetilde{\chi}_{\bf k^{\prime}}\\!\approx\\!\widetilde{\chi}_{\bf k}$.
Consider Fig. 13(b). One can break ${\bf k}^{\prime}$ into components along
and perpendicular to ${\bf k}$

$\displaystyle{\bf k^{\prime}}=\cos\theta\,{\bf k}+\sin\theta\,{\bf
k}_{\perp}\,,$ (115)

with the latter component being odd with respect to the mirror reflection,
$\theta\\!\rightarrow\\!-\theta$. Making a reasonable assumption that
$\big{|}V_{\bf q}\big{|}^{2}$ and magnon energies $\omega_{\bf q}$ are only
weakly dependent on the direction of ${\bf q}$, it is easy to see that the
rest of the integrand in (111) is even under the mirror reflection.
Physically, the ${\bf k}_{\perp}$ component of the contribution of a scattered
${\bf k}^{\prime}$ state to the collision integral is identically canceled by
its mirror-pair.

Thus, up to a factor $\cos\theta$ and a shift of energy in ${\rm f}^{0}$ from
$\varepsilon_{\bf k}$ to $\varepsilon_{\bf k^{\prime}}$, the non-vanishing
contribution of $\delta{\rm f}_{\bf k^{\prime}}$ to (111) can be written as

$\displaystyle\delta{\rm f}_{\bf
k^{\prime}}\Rightarrow\bigg{(}1-\frac{q^{2}}{2k^{2}}\bigg{)}\,\delta{\rm
f}_{\bf k}(\varepsilon_{\bf k}\pm\omega_{\bf q})\,,$ (116)

where we have converted $\cos\theta$ to $(1-q^{2}/2k^{2})$ using the
trigonometry of Fig. 13(b). Combining (112), (113), and (116), transforms
collision integral in (111) to

$\displaystyle{\rm St}_{\bf k}[{\rm f}_{\bf k}]=\frac{e}{m}\,\big{(}{\bf
E}\cdot{\bf k}\big{)}\,\widetilde{\chi}_{\bf
k}\,\,\frac{2\pi}{\hbar^{3}}\sum_{\bf q}$ $\displaystyle\big{|}V_{\bf
q}\big{|}^{2}\left(\frac{m}{kq}\right)$ (117) $\displaystyle\times$
$\displaystyle\,\delta\left(\cos\varphi-\frac{q}{2k}\right)\Big{\\{}*\Big{\\}},$

with

$\displaystyle\Big{\\{}*\Big{\\}}=$
$\displaystyle\,\frac{1}{T}\,\bigg{\\{}\bigg{(}1-\frac{q^{2}}{2k^{2}}\bigg{)}\,{\rm
f}^{0}_{\varepsilon-\omega}\big{(}1-{\rm
f}^{0}_{\varepsilon-\omega}\big{)}\big{(}{\rm n}^{0}_{\omega}+{\rm
f}^{0}_{\varepsilon}\big{)}$
$\displaystyle\phantom{\,\frac{1}{T}\,\bigg{\\{}}-{\rm
f}^{0}_{\varepsilon}\big{(}1-{\rm f}^{0}_{\varepsilon}\big{)}\big{(}{\rm
n}^{0}_{\omega}+1-{\rm f}^{0}_{\varepsilon-\omega}\big{)}$ (118)
$\displaystyle\phantom{\,\frac{1}{T}\,}+\bigg{(}1-\frac{q^{2}}{2k^{2}}\bigg{)}\,{\rm
f}^{0}_{\varepsilon+\omega}\big{(}1-{\rm
f}^{0}_{\varepsilon+\omega}\big{)}\big{(}{\rm n}^{0}_{\omega}+1-{\rm
f}^{0}_{\varepsilon}\big{)}$
$\displaystyle\phantom{\,\frac{1}{T}\,\bigg{\\{}}-{\rm
f}^{0}_{\varepsilon}\big{(}1-{\rm f}^{0}_{\varepsilon}\big{)}\big{(}{\rm
n}^{0}_{\omega}+{\rm f}^{0}_{\varepsilon+\omega}\big{)}\bigg{\\}},$

where we used shorthand notations ${\rm f}^{0}_{\bf k}\\!=\\!{\rm
f}^{0}_{\varepsilon}$ and ${\rm f}^{0}_{\bf k^{\prime}}\\!=\\!{\rm
f}^{0}_{\varepsilon\pm\omega}$ with $\varepsilon_{\bf k}\\!=\\!\varepsilon$
and $\omega_{\bf q}\\!=\\!\omega$ and an identity

$\displaystyle-\frac{\partial{\rm
f}^{0}_{\varepsilon}}{\partial\varepsilon}=\frac{1}{T}\,{\rm
f}^{0}_{\varepsilon}\big{(}1-{\rm f}^{0}_{\varepsilon}\big{)}\,.$ (119)

For the sake of the subsequent discussion of the spin-flip scattering that
involves fewer terms in the collision integral, we note that the first (last)
two lines in (118) correspond to the first (second) lines in the bracket in
(111) and Fig. 13(a).

While Eq. (118) seems cumbersome and not intuitive, we will demonstrate that
it is, in fact, equivalent to a compact and substantially more physical
expression (124), which, upon an appropriate linearization, will yield both
the proof of the $1/\tau$ approximation for the collision integral and the
sought-after transport relaxation rate (127).

We note that the advocated approach of using ansatz of (112) and further
manipulations with it are not unfamiliar to the experts, see Refs. [53, 67,
68]. However, they typically resort to an additional integration of both sides
of the LBE (106) over electronic energy in order to eliminate the singular
component that is strongly peaked near the Fermi energy. Below we demonstrate
that such an ad hoc step is unnecessary.

First reorganization in (118) is an “extraction” of the bosonic distribution
function that makes explicit the dependence of the scattering on the boson
population

$\displaystyle{\rm f}^{0}_{\varepsilon-\omega}\,\big{(}{\rm
n}^{0}_{\omega}+{\rm f}^{0}_{\varepsilon}\big{)}=\,e^{\omega/T}\,{\rm
f}^{0}_{\varepsilon}\,{\rm n}^{0}_{\omega}\,,$ $\displaystyle{\rm
f}^{0}_{\varepsilon+\omega}\,\big{(}{\rm n}^{0}_{\omega}+1-{\rm
f}^{0}_{\varepsilon}\big{)}=\,{\rm f}^{0}_{\varepsilon}\,{\rm
n}^{0}_{\omega}\,,$ (120)

which, together with a “trade-off of $\omega$,”

$\displaystyle e^{\omega/T}\,{\rm f}^{0}_{\varepsilon}\,\big{(}1-{\rm
f}^{0}_{\varepsilon-\omega}\big{)}=\,{\rm
f}^{0}_{\varepsilon-\omega}\,\big{(}1-{\rm f}^{0}_{\varepsilon}\big{)}\,,$
(121)

reduces (118) to

$\Big{\\{}*\Big{\\}}\\!=\\!-\bigg{(}\frac{q^{2}}{2k^{2}}\bigg{)}\,\frac{{\rm
n}^{0}_{\omega}}{T}\,\Big{\\{}{\rm f}^{0}_{\varepsilon-\omega}\big{(}1-{\rm
f}^{0}_{\varepsilon}\big{)}+{\rm f}^{0}_{\varepsilon}\big{(}1-{\rm
f}^{0}_{\varepsilon+\omega}\big{)}\Big{\\}}.$ (122)

Last is a “bosonization” of the double-fermionic terms

$\displaystyle{\rm f}^{0}_{\varepsilon-\omega}\,\big{(}1-{\rm
f}^{0}_{\varepsilon}\big{)}=\,\big{(}{\rm
n}^{0}_{\omega}+1\big{)}\,\big{(}{\rm f}^{0}_{\varepsilon-\omega}-{\rm
f}^{0}_{\varepsilon}\big{)}\,,$ (123)

that leads to the final form

$\displaystyle\Big{\\{}*\Big{\\}}\equiv$
$\displaystyle\,\frac{1}{T}\,\bigg{(}\frac{q^{2}}{2k^{2}}\bigg{)}\,{\rm
n}^{0}_{\omega}\big{(}{\rm n}^{0}_{\omega}+1\big{)}\Big{\\{}{\rm
f}^{0}_{\varepsilon+\omega}-{\rm f}^{0}_{\varepsilon-\omega}\Big{\\}}\,.$
(124)

where the naturally-occurring factor $q^{2}/2k^{2}\\!=\\!(1-\cos\theta)$
differentiates transport scattering rate from the conventional one. The
bracket in (124) can now be expanded

$\displaystyle\Big{\\{}{\rm f}^{0}_{\varepsilon+\omega}-{\rm
f}^{0}_{\varepsilon-\omega}\Big{\\}}\approx
2\,\omega\bigg{(}\frac{\partial{\rm
f}^{0}_{\varepsilon}}{\partial\varepsilon}\bigg{)}\,,$ (125)

to yield the singular part of $\delta{\rm f}_{\bf k}$. Combining (124) and
(125), we rewrite collision integral (117)

$\displaystyle{\rm St}_{\bf k}$ $\displaystyle[{\rm f}_{\bf
k}]=-\bigg{[}\frac{e}{m}\,\big{(}{\bf E}\cdot{\bf
k}\big{)}\,\bigg{(}-\frac{\partial{\rm f}^{0}_{\bf
k}}{\partial\varepsilon_{\bf k}}\bigg{)}\,\widetilde{\chi}_{\bf
k}\bigg{]}\,\,$ (126) $\displaystyle\times$
$\displaystyle\,\frac{2\pi}{\hbar^{3}}\sum_{\bf q}\big{|}V_{\bf
q}\big{|}^{2}\bigg{(}\frac{mq}{k^{3}}\bigg{)}\,\delta\left(\cos\varphi-\frac{q}{2k}\right)\,\frac{\omega_{\bf
q}}{T}\,{\rm n}^{0}_{\bf q}\,\big{(}{\rm n}^{0}_{\bf q}+1\big{)},$

in the long-pursued relaxation-time form (107), which naturally replicates
$\delta{\rm f}_{\bf k}$ ansatz (112) highlighted with the square bracket, also
yielding transport relaxation rate

$\displaystyle\frac{\hbar}{\tau_{\bf k}}=\frac{\pi}{\varepsilon_{\bf
k}}\sum_{\bf q}\big{|}V_{\bf
q}\big{|}^{2}\bigg{(}\frac{q}{k}\bigg{)}\delta\Big{(}\cos\varphi\,-$
$\displaystyle\,\frac{q}{2k}\Big{)}$ (127) $\displaystyle\times$
$\displaystyle\,\bigg{(}\frac{\omega_{\bf q}}{T}\bigg{)}\,{\rm n}^{0}_{\bf
q}\,\big{(}{\rm n}^{0}_{\bf q}+1\big{)},$

where we have used $\varepsilon_{\bf k}\\!=\\!\hbar^{2}k^{2}/2m$. We also
point out that a comparison of the LBE (106) with the collision integral in
(126) makes explicit the equivalence of the auxiliary function
$\widetilde{\chi}_{\bf k}$ from the ansatz (112) with the transport relaxation
time $\tau_{\bf k}$ in (127).

One can easily check by power-counting that for the scattering on acoustic
phonons ($V_{\bf q}\\!\propto\\!\sqrt{q}$, $\omega_{\bf q}\\!\propto\\!q$),
the relaxation rate (127) yields Ohm’s law, $\rho\\!\sim\\!T$, for
$T\\!\gg\\!\Theta_{D}$ and Bloch-Grüneisen’s law, $\rho\\!\sim\\!T^{5}$ in 3D,
for $T\\!\ll\\!\Theta_{D}$.

We also note that while the provided consideration is for a spherical
(circular) Fermi surface, there is no question in our mind that the results
concerning validity of the $1/\tau$-approximation remain correct in general.
The main modification in the relaxation rate (127) should be expected in the
constraint on the surface of integration (contour in 2D), dictated by the
actual shape of the Fermi surface. Thus, one can expect the results obtained
using this approximation to be quantitatively correct.

####  C.2.2 2D case

In the present study, magnons are 2D and the relaxation rate (127) is given by
an integral along a circular 1D contour in Fig. 13(b) of the radius $k_{F}$
with the magnon momentum varying from 0 to $2k_{F}$. Assuming that $2k_{F}$ is
less than the size of the full BZ, one can make further progress and simplify
(127) by an appropriate parametrization. Let us first rewrite the summation
over the BZ in (127) as an integral in polar coordinates

$\frac{1}{V_{BZ}}\int
q\,dq\int_{0}^{2\pi}d\varphi\bigg{(}\frac{q}{k_{F}}\bigg{)}\,\delta\left(\cos\varphi-\frac{q}{2k_{F}}\right)\widetilde{F}_{\bf
q},$ (128)

where we have abbreviated part of the integrand as

$\widetilde{F}_{\bf q}=\big{|}V_{\bf q}\big{|}^{2}\bigg{(}\frac{\omega_{\bf
q}}{T}\bigg{)}\,{\rm n}^{0}_{\bf q}\,\big{(}{\rm n}^{0}_{\bf q}+1\big{)},$
(129)

and $V_{BZ}\\!=\\!8\pi^{2}/a^{2}\sqrt{3}$ is the 2D volume of the triangular-
lattice BZ in Fig. 5.

Introducing new variables $z\\!=\\!\cos\varphi$ and $y\\!=\\!q/2k_{F}$ and
using the mirror symmetry of $\widetilde{F}_{\bf q}$ with respect to
$\varphi\\!\rightarrow\\!-\varphi$ simplifies (128) to

$\frac{\sqrt{3}\,(k_{F}a)^{2}}{\pi^{2}}\int
y^{2}\,dy\int_{0}^{1}\frac{dz}{\sqrt{1-z^{2}}}\,\delta\left(z-y\right)\,2\widetilde{F}_{\bf
q},$ (130)

with the momentum ${\bf q}$ belonging to the 1D contour given by the
parametrization ${\bf q}\\!=\\!2k_{F}(z^{2},z\sqrt{1-z^{2}})$. Finally, Eq.
(127) transforms to

$\displaystyle\frac{\hbar}{\tau_{F}}=\frac{\sqrt{3}\,(k_{F}a)^{2}}{\pi\,E_{F}}\int_{0}^{1}\frac{z^{2}\,dz}{\sqrt{1-z^{2}}}\,2\,\widetilde{F}_{\bf
q},$ (131)

with $\widetilde{F}_{\bf q}$ from (129), $E_{F}\\!=\\!\hbar^{2}k_{F}^{2}/2m$,
and we have explicitly separated a factor of two for the sake of combining
this relaxation rate result with the one due to the spin-flip scattering
processes, discussed next.

Figure 14: Schematics of the collision integrals in (133).

###  C.3 Spin-flip scattering

The spin-flip component of electron-magnon scattering in (49) generally
contains two parts, one with emission by spin-down and one by spin-up
electrons, yielding scattering rates that have the same structure and are
additive. Therefore, we consider only one of them and write the interaction
Hamiltonian in a general form that is identical to that of electron-magnon
coupling in a ferromagnet

$\displaystyle{\cal H}_{int}=\frac{1}{\sqrt{N}}\sum_{{\bf k},{\bf
q}}\Big{(}V^{\pm}_{\bf q}f^{\dagger}_{{\bf k}-{\bf
q}\uparrow}f^{\phantom{{\dagger}}}_{{\bf k}\downarrow}a^{{\dagger}}_{\bf
q}+{\rm H.c.}\Big{)}.$ (132)

Here, similarly to Eq. (109), the coupling $V^{\pm}_{\bf q}$ depends only on
${\bf q}$ because of its local (Kondo) nature.

Owing to its relevance to the electron-magnon scattering mechanism of the
resistivity in ferromagnets, the kinetic theory for the model (132) has been
the subject of a number of works [67, 68, 69, 70], which, while yielding
correct results, have also used unnecessary ad hoc integrations.

The principal difference of the problem from the spin-conserving scattering is
in having two Boltzmann equations, one per spin projection, with their
collision integrals each containing only half of the terms of (110),

$\displaystyle{\rm St}_{\bf k}$ $\displaystyle[{\rm f}_{{\bf
k}\downarrow}]=\frac{2\pi}{\hbar}\sum_{\bf q}\big{|}V^{\pm}_{\bf
q}\big{|}^{2}\delta\left(\varepsilon_{{\bf
k^{\prime}}\uparrow}-\varepsilon_{{\bf k}\downarrow}+\omega_{\bf q}\right)$
$\displaystyle\times$ $\displaystyle\,\Big{\\{}\Big{(}{\rm f}_{{\bf
k^{\prime}}\uparrow}(1-{\rm f}_{{\bf k}\downarrow})\,{\rm n}_{\bf q}-{\rm
f}_{{\bf k}\downarrow}(1-{\rm f}_{{\bf k^{\prime}}\uparrow})({\rm n}_{\bf
q}+1)\Big{)}\Big{\\}},$ $\displaystyle{\rm St}_{\bf k}$ $\displaystyle[{\rm
f}_{{\bf k}\uparrow}]=\frac{2\pi}{\hbar}\sum_{\bf q}\big{|}V^{\pm}_{\bf
q}\big{|}^{2}\delta\left(\varepsilon_{{\bf
k^{\prime}}\downarrow}-\varepsilon_{{\bf k}\uparrow}-\omega_{\bf q}\right)$
(133) $\displaystyle\times$ $\displaystyle\,\Big{\\{}\Big{(}{\rm f}_{{\bf
k^{\prime}}\downarrow}(1-{\rm f}_{{\bf k}\uparrow})({\rm n}_{\bf q}+1)-{\rm
f}_{{\bf k}\uparrow}(1-{\rm f}_{{\bf k^{\prime}}\downarrow})\Big{)}\,{\rm
n}_{\bf q}\Big{\\}},$

with the energy $\delta$-functions as in the first and second rows of (110),
respectively, see also Fig. 14, ${\bf k^{\prime}}\\!=\\!{\bf k}-{\bf q}$ as
before. Linearization in (133) gives

$\displaystyle{\rm St}_{\bf k}$ $\displaystyle[{\rm f}_{{\bf
k}\downarrow}]=\frac{2\pi}{\hbar}\sum_{\bf q}\big{|}V^{\pm}_{\bf
q}\big{|}^{2}\delta\left(\varepsilon_{{\bf
k^{\prime}}\uparrow}-\varepsilon_{{\bf k}\downarrow}+\omega_{\bf q}\right)$
$\displaystyle\times$ $\displaystyle\,\Big{\\{}\Big{(}\delta{\rm f}_{{\bf
k^{\prime}}\uparrow}({\rm n}^{0}_{\bf q}+{\rm f}^{0}_{{\bf
k}\downarrow})-\delta{\rm f}_{{\bf k}\downarrow}({\rm n}^{0}_{\bf q}-{\rm
f}^{0}_{{\bf k^{\prime}}\uparrow}+1)\Big{\\}},$ $\displaystyle{\rm St}_{\bf
k}$ $\displaystyle[{\rm f}_{{\bf k}\uparrow}]=\frac{2\pi}{\hbar}\sum_{\bf
q}\big{|}V^{\pm}_{\bf q}\big{|}^{2}\delta\left(\varepsilon_{{\bf
k^{\prime}}\downarrow}-\varepsilon_{{\bf k}\uparrow}-\omega_{\bf q}\right)$
(134) $\displaystyle\times$ $\displaystyle\,\Big{\\{}\Big{(}\delta{\rm
f}_{{\bf k^{\prime}}\downarrow}({\rm n}^{0}_{\bf q}-{\rm f}^{0}_{{\bf
k}\uparrow}+1)-\delta{\rm f}_{{\bf k}\uparrow}({\rm n}^{0}_{\bf q}+{\rm
f}^{0}_{{\bf k^{\prime}}\downarrow})\Big{)}\Big{\\}}.$

We follow the narrative of the non-spin-flip consideration, with the ansatzes
for $\delta{\rm f}_{{\bf k}\sigma}$ as in (112)

$\displaystyle\delta{\rm f}_{{\bf k}\sigma}=\frac{e}{m}\,\big{(}{\bf
E}\cdot{\bf k}\big{)}\bigg{(}-\frac{\partial{\rm f}^{0}_{{\bf
k}\sigma}}{\partial\varepsilon_{{\bf
k}\sigma}}\bigg{)}\,\widetilde{\chi}_{{\bf k}\sigma}\,,$ (135)

and with the same continuum-like approximation for electrons. In the present
case, it is also important to recall that the equilibrium distribution
functions depend on the momentum and spin only via energy, ${\rm f}^{0}_{{\bf
k}\sigma}\\!\equiv\\!{\rm f}^{0}(\varepsilon_{{\bf k}\sigma})$, so that ${\rm
f}^{0}(\varepsilon_{{\bf k}^{\prime}\uparrow})\\!\equiv\\!{\rm
f}^{0}(\varepsilon_{{\bf k}\downarrow}-\omega_{\bf q})$ and ${\rm
f}^{0}(\varepsilon_{{\bf k}^{\prime}\downarrow})\\!\equiv\\!{\rm
f}^{0}(\varepsilon_{{\bf k}\uparrow}+\omega_{\bf q})$.

In the considered case of the field-induced effects in an otherwise spin-
compensated system, the only source of the difference in the Fermi momenta for
the electrons of different spin is Zeeman energy. Because
$g\mu_{B}H\\!\ll\\!E_{F}$, using the same consideration of the quasielastic
nature of electron-magnon scattering that is given before Eq. (113), leads to
$\varepsilon_{{\bf k}^{\prime}}\\!\approx\\!\varepsilon_{\bf k}$ and $|{\bf
k}^{\prime}|\\!\approx\\!|{\bf k}|$ in all scattering processes regardless of
the spin, keeping geometry of the scattering the same as in Fig. 13(b) and
allowing us to rewrite $\delta$-functions in (134) in the form of Eq. (113).

In principle, the same consideration implies
$k_{F\uparrow}\\!\approx\\!k_{F\downarrow}$ with the same accuracy of
$O(g\mu_{B}H/E_{F})\\!\ll\\!1$ and suggests that $\widetilde{\chi}_{{\bf
k}\uparrow}\\!\approx\\!\widetilde{\chi}_{{\bf k}\downarrow}$, but we will
arrive to the same conclusion via a different path.

Using the logic leading to (116), we can rewrite the non-vanishing
contributions of $\delta{\rm f}_{{\bf k}^{\prime}\sigma}$ to (134) as

$\displaystyle\delta{\rm f}_{{\bf k}^{\prime}\uparrow}\Rightarrow$
$\displaystyle\,\bigg{(}1-\frac{q^{2}}{2k^{2}}\bigg{)}\,\delta{\rm f}_{{\bf
k}\uparrow}\big{(}\varepsilon_{{\bf k}\downarrow}-\omega_{\bf q}\big{)}\,,$
(136) $\displaystyle\delta{\rm f}_{{\bf k}^{\prime}\downarrow}\Rightarrow$
$\displaystyle\,\bigg{(}1-\frac{q^{2}}{2k^{2}}\bigg{)}\,\delta{\rm f}_{{\bf
k}\downarrow}\big{(}\varepsilon_{{\bf k}\uparrow}+\omega_{\bf q}\big{)}\,.$

Combining (135), (113), and (136), transforms collision integrals in (134) to

$\displaystyle{\rm St}_{\bf k}[{\rm f}_{{\bf
k}\sigma}]=\frac{e}{m}\,\big{(}{\bf E}\cdot{\bf
k}\big{)}\,\,\frac{2\pi}{\hbar^{3}}\sum_{\bf q}$
$\displaystyle\big{|}V^{\pm}_{\bf q}\big{|}^{2}\left(\frac{m}{kq}\right)$
(137) $\displaystyle\times$
$\displaystyle\,\delta\left(\cos\varphi-\frac{q}{2k}\right)\Big{\\{}\sigma\Big{\\}},$

with

$\displaystyle\Big{\\{}\downarrow\Big{\\}}=$
$\displaystyle\frac{1}{T}\bigg{\\{}\widetilde{\chi}_{{\bf
k}\uparrow}\bigg{(}1-\frac{q^{2}}{2k^{2}}\bigg{)}{\rm
f}^{0}_{\varepsilon-\omega}\big{(}1-{\rm
f}^{0}_{\varepsilon-\omega}\big{)}\big{(}{\rm n}^{0}_{\omega}+{\rm
f}^{0}_{\varepsilon}\big{)}$
$\displaystyle\phantom{\frac{1}{T}\bigg{\\{}}-\widetilde{\chi}_{{\bf
k}\downarrow}{\rm f}^{0}_{\varepsilon}\big{(}1-{\rm
f}^{0}_{\varepsilon}\big{)}\big{(}{\rm n}^{0}_{\omega}+1-{\rm
f}^{0}_{\varepsilon-\omega}\big{)}\bigg{\\}},\ $ (138)
$\displaystyle\Big{\\{}\uparrow\Big{\\}}=$
$\displaystyle\frac{1}{T}\bigg{\\{}\widetilde{\chi}_{{\bf
k}\downarrow}\bigg{(}1-\frac{q^{2}}{2k^{2}}\bigg{)}{\rm
f}^{0}_{\varepsilon+\omega}\big{(}1-{\rm
f}^{0}_{\varepsilon+\omega}\big{)}\big{(}{\rm n}^{0}_{\omega}+1-{\rm
f}^{0}_{\varepsilon}\big{)}$
$\displaystyle\phantom{\frac{1}{T}\bigg{\\{}}-\widetilde{\chi}_{{\bf
k}\uparrow}{\rm f}^{0}_{\varepsilon}\big{(}1-{\rm
f}^{0}_{\varepsilon}\big{)}\big{(}{\rm n}^{0}_{\omega}+{\rm
f}^{0}_{\varepsilon+\omega}\big{)}\bigg{\\}},$

where $\varepsilon\\!=\\!\varepsilon_{{\bf k}\sigma}$ and
$\omega\\!=\\!\omega_{\bf q}$.

The same sequence of manipulations as in (120), (121), and (123) reduce Eq.
(138) identically to

$\displaystyle\Big{\\{}\downarrow\Big{\\}}\equiv\frac{1}{T}\,{\rm
n}^{0}_{\omega}\big{(}{\rm
n}^{0}_{\omega}+1\big{)}\bigg{[}\widetilde{\chi}_{{\bf k}\downarrow}-$
$\displaystyle\,\widetilde{\chi}_{{\bf
k}\uparrow}\bigg{(}1-\frac{q^{2}}{2k^{2}}\bigg{)}\bigg{]}$
$\displaystyle\times\Big{\\{}{\rm f}^{0}_{\varepsilon}-{\rm
f}^{0}_{\varepsilon-\omega}\Big{\\}},$ (139)
$\displaystyle\Big{\\{}\uparrow\Big{\\}}\equiv\frac{1}{T}\,{\rm
n}^{0}_{\omega}\big{(}{\rm
n}^{0}_{\omega}+1\big{)}\bigg{[}\widetilde{\chi}_{{\bf k}\uparrow}-$
$\displaystyle\,\widetilde{\chi}_{{\bf
k}\downarrow}\bigg{(}1-\frac{q^{2}}{2k^{2}}\bigg{)}\bigg{]}$
$\displaystyle\times\Big{\\{}{\rm f}^{0}_{\varepsilon+\omega}-{\rm
f}^{0}_{\varepsilon}\Big{\\}},$

which, in a retrospect, shows that the two groups of terms in the non-spin-
flip consideration (118) and Fig. 13, associated with two different energy
$\delta$-functions, can be reduced to a compact form (124) individually.

Linearization in the last brackets in (139) yields

$\displaystyle\Big{\\{}\dots\Big{\\}}\approx\,\omega\bigg{(}\frac{\partial{\rm
f}^{0}_{\varepsilon}}{\partial\varepsilon}\bigg{)}\,,$ (140)

where we note the factor of two difference with the result of an equivalent
step in (125). Bringing together (139) and (140) transforms collision
integrals in (137) to

${\rm St}_{\bf k}[{\rm f}_{{\bf
k}\sigma}]\\!=\\!\bigg{[}\frac{e}{m}\big{(}{\bf E}\cdot{\bf
k}\big{)}\bigg{(}\frac{\partial{\rm f}^{0}_{{\bf
k}\sigma}}{\partial\varepsilon_{{\bf k}\sigma}}\bigg{)}\bigg{]}\Big{(}A_{\bf
k}\widetilde{\chi}_{{\bf k}\sigma}-B_{\bf k}\widetilde{\chi}_{{\bf
k}\bar{\sigma}}\Big{)},$ (141)

where $\bar{\sigma}\\!=\\!-\sigma$ and $A_{\bf k}$ and $B_{\bf k}$ are
auxiliary functions. Given the symmetry of (141) under
$\\{\uparrow\leftrightarrow\downarrow\\}$, one can anticipate the following
result. The easiest way to proceed is to realize that the content of the
square bracket in the collision integrals in (141) is exactly the left-hand
side of their corresponding LBEs. Substitution of (141) into LBEs with a
cancellation in both sides reduces them to the two identical equations

$\displaystyle 1=$ $\displaystyle\,A_{\bf k}\widetilde{\chi}_{{\bf
k}\sigma}-B_{\bf k}\widetilde{\chi}_{{\bf k}\bar{\sigma}}\,,$ (142)

which finally gives $\widetilde{\chi}_{{\bf
k}\uparrow}\\!=\\!\widetilde{\chi}_{{\bf k}\downarrow}$ and provides the
relaxation rate for both spin projections in the form

$\displaystyle\frac{\hbar}{\tau_{{\bf
k}\sigma}}=\,\frac{\pi}{2\varepsilon_{\bf k}}\sum_{\bf q}\big{|}V^{\pm}_{\bf
q}\big{|}^{2}\bigg{(}\frac{q}{k}\bigg{)}\delta\Big{(}\cos\varphi\,-$
$\displaystyle\,\frac{q}{2k}\Big{)}$ (143) $\displaystyle\times$
$\displaystyle\,\bigg{(}\frac{\omega_{\bf q}}{T}\bigg{)}\,{\rm n}^{0}_{\bf
q}\,\big{(}{\rm n}^{0}_{\bf q}+1\big{)},$

where we note an extra factor $1/2$ compared to (127).

Further adaptation to the 2D case with spins on a triangular lattice
considered in the preceding section gives the same result as in (131)

$\displaystyle\frac{\hbar}{\tau_{F\sigma}}=\frac{\sqrt{3}\,(k_{F}a)^{2}}{\pi\,E_{F}}\int_{0}^{1}\frac{z^{2}\,dz}{\sqrt{1-z^{2}}}\,\widetilde{F}^{\pm}_{\bf
q},$ (144)

up to a factor of two that can be traced to the expansion in (125) and (140),
and, ultimately, to the fact that the spin-flip scattering in (132) involves
only half the terms of the non-spin-flip one in (109) for each spin species.
Here $\widetilde{F}^{\pm}_{\bf q}$ is given by

$\widetilde{F}^{\pm}_{\bf q}=\big{|}V^{\pm}_{\bf
q}\big{|}^{2}\bigg{(}\frac{\omega_{\bf q}}{T}\bigg{)}\,{\rm n}^{0}_{\bf
q}\,\big{(}{\rm n}^{0}_{\bf q}+1\big{)}.$ (145)

For the full interaction given in (49), collecting all contributions to the
scattering rate yields the final formula in (51)-(52), which takes into
account three different branches of magnons and all channels of scattering,
spin-flip and not.

###  C.4 Quasiparticle $1/\tau_{\rm qp}$, angular dependence

Our consideration of the transport relaxation rate due to magnon scattering
would not be complete without a brief note on the “regular,” or quasiparticle,
relaxation rate and an equally brief remark of the relationship between the
two. In turn, this analysis also allows us to discuss the angular dependence
of the quasiparticle relaxation rate, originating from the discrete lattice
symmetry that is encoded in the energies of spin excitations and matrix
elements of the coupling to them, while leaving a substantially more involved
study of their effect on its transport counterpart to a future work.

####  C.4.1 Quasiparticle vs transport $1/\tau$

The quasiparticle relaxation rate characterizes a probability of the
scattering of an electron from the momentum-${\bf k}$ state. It is given by

$\displaystyle\frac{\hbar}{\tau_{\rm qp,{\bf k}}}=-2\,{\rm
Im}\big{[}\Sigma_{\bf k}(\varepsilon_{\bf k})\big{]},$ (146)

where $\Sigma_{\bf k}(\varepsilon_{\bf k})$ is the on-shell electron self-
energy.

The phonon-like non-spin-flip scattering of the form (109) yields a well-known
answer for ${\rm Im}\big{[}\Sigma_{\bf k}(\varepsilon_{\bf k})\big{]}$ via a
standard second-order diagrammatic treatment [71]

$\displaystyle\frac{\hbar}{\tau_{\rm qp,{\bf k}}}=2\pi\sum_{\bf
q}\big{|}V_{\bf q}\big{|}^{2}$ $\displaystyle\,\Big{\\{}\big{(}{\rm
n}^{0}_{\bf q}-{\rm f}^{0}_{\bf k^{\prime}}+1\big{)}\,\delta_{\varepsilon_{\bf
k^{\prime}},\varepsilon_{\bf k}-\omega_{\bf q}}$ (147)
$\displaystyle+\\!\big{(}{\rm n}^{0}_{\bf q}+{\rm f}^{0}_{\bf
k^{\prime}}\big{)}\,\delta_{\varepsilon_{\bf k^{\prime}},\varepsilon_{\bf
k}+\omega_{\bf q}}\Big{\\}},$

where ${\bf k^{\prime}}\\!=\\!{\bf k}-{\bf q}$ and
$\delta_{\varepsilon^{\prime},\varepsilon\pm\omega}\\!=\\!\delta(\varepsilon^{\prime}-\varepsilon\mp\omega)$
as before.

While looking deceitfully different from its transport counterpart in (117),
simple manipulations using (121) and (123) bring the combinations of the
distribution functions in the first and second terms of (147) to

$\displaystyle\mp\,{\rm n}^{0}_{\omega}\big{(}{\rm
n}^{0}_{\omega}+1\big{)}\,\frac{\big{(}{\rm f}^{0}_{\varepsilon}-{\rm
f}^{0}_{\varepsilon\mp\omega}\big{)}}{{\rm f}^{0}_{\varepsilon}\big{(}1-{\rm
f}^{0}_{\varepsilon}\big{)}}\,,$ (148)

respectively. Recognizing the denominator in (148) for $(-T)(\partial{\rm
f}^{0}_{\varepsilon}/\partial\varepsilon)$ and expanding the numerators in
$\omega$ gives

$\displaystyle\frac{\hbar}{\tau_{\rm qp,{\bf k}}}\approx 4\pi\sum_{\bf
q}\big{|}V_{\bf q}\big{|}^{2}\,\frac{\omega_{\bf q}}{T}\,{\rm n}^{0}_{\bf
q}\,\big{(}{\rm n}^{0}_{\bf q}+1\big{)}\,\delta\big{(}{\varepsilon_{{\bf
k}-{\bf q}}-\varepsilon_{\bf k}}\big{)},\ \ \ \ \ \ \ $ (149)

where we have also neglected $\omega_{\bf q}$ in the delta-function. Further
simplification of the latter for the spherical (cylindrical) Fermi surface
(113) and parametrization of the electron mass the with energy
$\varepsilon_{\bf k}\\!=\\!\hbar^{2}k^{2}/2m$ used above, brings the
quasiparticle relaxation rate in (149) close to an expression of its transport
version (127)

$\displaystyle\frac{\hbar}{\tau_{{\rm qp},{\bf
k}}}=\frac{\pi}{\varepsilon_{\bf k}}\sum_{\bf q}\big{|}V_{\bf
q}\big{|}^{2}\bigg{(}\frac{2k}{q}\bigg{)}\delta\Big{(}\cos\varphi\,-$
$\displaystyle\,\frac{q}{2k}\Big{)}$ (150) $\displaystyle\times$
$\displaystyle\,\bigg{(}\frac{\omega_{\bf q}}{T}\bigg{)}\,{\rm n}^{0}_{\bf
q}\,\big{(}{\rm n}^{0}_{\bf q}+1\big{)}.$

As is expected, the only difference between the two rates is an extra factor
$q^{2}/2k^{2}\\!\equiv\\!(1-\cos\theta)\\!=\\!2\cos^{2}\varphi$ in the
integrand of the transport relaxation rate that is responsible for the
suppression of the small-angle scattering in it; as before, $\theta$ and
$\varphi$ are the angles between ${\bf k}$ and ${\bf k}^{\prime}$ and ${\bf
k}$ and ${\bf q}$, respectively, see Fig. 13(b).

Using the same approximations as in Appendix C.2 and considering the case of
2D, the quasiparticle relaxation rate (150) at $k\\!=\\!k_{F}$ reduces to

$\displaystyle\frac{\hbar}{\tau_{{\rm
qp},F}}=\frac{\sqrt{3}\,(k_{F}a)^{2}}{\pi\,E_{F}}\int_{0}^{1}\frac{dz}{\sqrt{1-z^{2}}}\,\widetilde{F}_{\bf
q},$ (151)

with $\widetilde{F}_{\bf q}$ from (129), ${\bf
q}\\!=\\!2k_{F}(z^{2},z\sqrt{1-z^{2}})$, and the difference from the transport
rate in (131) given by an extra factor $2z^{2}$ in the latter. Needless to
say, for the spin-flip scattering (132), an equivalent of the self-energy in
(147) contains only half the terms, resulting in an extra factor 1/2 in (150)
and (151), identically to the transport case (143) and (144) considered in
Appendix C.3.

While the technical tricks of converting standard expression for the
quasiparticle relaxation rate (147) into the form that is akin to the
transport one (150) may not be well-known, the resultant similarity between
them is in a full accord with the textbook expectations. Thus, it is prudent
on our side to investigate the resulting difference between the two rates for
our problem with the actual electron-magnon couplings and energies using Eqs.
(51)-(53) and their quasiparticle analogues.

Figure 15: Transport and quasiparticle relaxation rates from Eqs. (51)-(53)
and their quasiparticle analogues vs field for representative temperatures,
$b\\!=\\!0.06$, and $k_{F}\\!=\\!\pi/3$; other parameters are from Table 1.
Transport rates are from Fig. 10(a).

The result of such a comparison is presented in Fig. 15 for a representative
value of $b\\!=\\!0.06$ and $k_{F}\\!=\\!\pi/3$. The transport rates reproduce
the results shown in Fig. 10(a). Although the overall field-dependence of the
rates is quite similar, the quasiparticle $1/\tau_{\rm qp}$ exhibits more
pronounced variations from phase to phase, with an exception of the polarized
FM region, which shows only a modest overall offset from the transport one.

The evolution of $1/\tau_{\rm qp}$ with the biquadratic exchange reveals more
dramatic differences. Upon approaching the first-order transition,
$b\\!\rightarrow\\!b_{c}$, the Y-UUD and V-FM phase boundaries in $1/\tau_{\rm
qp}$ become sharper, turning into discontinuities at $b\\!=\\!b_{c}$. Further
increase to $b\\!>\\!b_{c}$ suggest truly divergent relaxation rates to occur
at the (metastable) boundaries discussed in Appendix A. This is in contrast
with the evolution of the transport relaxation rate, which remains continuous
at these phase boundaries up to $b\\!=\\!b_{c}$ and develops finite
discontinuities for $b\\!>\\!b_{c}$, as is demonstrated in Figs. 10(a) and
(b), respectively.

Since the difference of the rates is in the suppression of the small-angle
scattering in the transport case, the small-momentum contribution to the
quasiparticle rate is an obvious culprit for such a dichotomy. Detailed
analysis reveals the key role of the non-spin-flip scattering, present in the
noncollinear Y and V phases, in precipitating singular behavior of
$1/\tau_{\rm qp}$.

In the proximity of the transitions, $h\\!\rightarrow\\!h_{c1(s)}$, where the
Goldstone mode softens to $\omega_{\bf q}\\!\propto\\!q^{2}$, see Fig. 6,
these processes provide a dangerous $\propto\\!1/\omega_{\bf q}$ term to the
small-momentum equivalent of the kernel (54), resulting in the leading
contribution to $1/\tau_{\rm qp}$ that scales as
$\sim\\!\sin^{2}\\!\alpha_{1}/\sqrt{h_{c}-h}$. From Appendix A,
$\sin^{2}\\!\alpha_{1}\\!\propto\\!(h_{c}-h)$ for $b\\!<\\!b_{c}$, still
giving a continuous behavior of $1/\tau_{\rm qp}$ vs $h$, as is shown in Fig.
15. However, for $b\\!\rightarrow\\!b_{c}$, the field-dependence of the angle
switches to $\sin^{2}\\!\alpha_{1}\\!\propto\\!\sqrt{h_{c}-h}$, resulting in a
finite contribution to $1/\tau_{\rm qp}$ at $h_{c}$ and culminating in a
discontinuity. Further increase to $b\\!>\\!b_{c}$ leaves the angle finite at
the transition field $h_{c}$, see Fig. 12(b), producing a true singularity,
$1/\tau_{\rm qp}\\!\propto\\!1/\sqrt{h_{c}-h}$.

This discussion opens up an interesting possibility of the studies of the
field-tuned singular behavior in the scattering rate due to small-angle
scattering processes that can lead to a number of observables including
catastrophic violation of the Wiedemann-Franz law.

####  C.4.2 Angular dependence of $1/\tau_{\rm qp}$

Another aspect of the quasiparticle relaxation rate makes its consideration
worthwhile. The diagrammatics-derived expression for it in Eq. (147) is exact
to the second order in the coupling. The approximations leading to (149)
involves only a highly-justified energy hierarchy, $\omega_{\bf
q}\\!\ll\\!E_{F}$. The next step, resulting in (150), is a fairly reasonable
assumption of the spherical (cylindrical) Fermi surface. However, this last
assumption does not automatically make $1/\tau_{\rm qp}$ independent of the
direction of the momentum ${\bf k}$ on the Fermi surface. It is an additional
step of assuming an even mirror-symmetry of the entire kernel
$\widetilde{F}_{\bf q}$ with respect to a reflection of the momentum ${\bf q}$
about the direction of the momentum ${\bf k}$ which does that, leading to the
final expression (151), see the line preceding Eq. (130).

Although this assumption is valid for ${\bf k}$ along the high-symmetry
directions, dictated by the $C_{6}$ symmetry of the triangular lattice in our
case, the angular-dependence of the quasiparticle relaxation rate, encoded in
the energies of spin excitations and matrix elements of the coupling to them,
should remain.

A proper consideration of this problem for the transport relaxation rate has
an additional complicating factor because it also requires a consistency in
the assumption of the angle-dependence in relating scattered-state
distribution function $\delta{\rm f}_{\bf k^{\prime}}$ to $\delta{\rm f}_{\bf
k}$, thus modifying Eq. (116). However, the simplicity of this problem for the
quasiparticle relaxation rate makes it particularly appealing to solve. Thus,
we return to the relaxation rate in Eq. (150) and advance it one step further
to include such an angular-dependence properly, leaving its study in the
transport counterpart to a future work.

Although the direction of the electron momentum ${\bf k}$ is not explicitly
present in (150), it is implicitly tied to that of the magnon momentum ${\bf
q}$ via the mutual angle $\varphi$, see the scattering diagram in the right
inset of Fig. 16. This Figure shows that in the reference frame formed by
$({\bf\hat{k}},{\bf\hat{k}}_{\perp})$, where ${\bf\hat{k}}_{\perp}$ is the
axis perpendicular to the unit vector ${\bf\hat{k}}$, the magnon momentum
${\bf q}$ is parametrized as ${\bf q}\\!=\\!q(\cos\varphi,\sin\varphi)$.
However, this frame itself is rotated by the angle $\alpha_{0}$ with respect
to the laboratory reference frame, represented here by $(x,y)$, with the $x$
axis tied to one of the bond-directions of the triangular lattice. Then,
straightforwardly, momentum ${\bf q}$ parametrization in the $(x,y)$ reference
frame is ${\bf
q}\\!=\\!q\big{(}\\!\cos(\varphi\\!+\\!\alpha_{0}),\sin(\varphi\\!+\\!\alpha_{0})\big{)}$.

Figure 16: $1/\tau_{\rm qp}$ vs $\alpha_{0}$ for $H\\!=\\!H_{s}$ normalized to
its value at $\alpha_{0}\\!=\\!\pi/6$ for the parameters in Table 1 and
several values of $k_{F}$. Upper inset: $\tau_{\rm qp}(\pi/6)/\tau_{\rm
qp}(0)$ for the FM (solid line) and 120$\degree$ state (dashed line) vs
$k_{F}$. Right inset: scattering diagram, angles, and laboratory axes, cf.
Fig. 13(b).

Since the absolute value of the momentum ${\bf q}$ is tied to the angle
$\varphi$ via $q\\!=\\!2k_{F}\cos\varphi$, the integral in (150) can still be
reduced to a 1D form

$\displaystyle\frac{\hbar}{\tau_{{\rm
qp},F}}=\frac{\sqrt{3}\,(k_{F}a)^{2}}{\pi\,E_{F}}\int_{-\pi/2}^{\pi/2}d\varphi\,\frac{1}{2}\,\widetilde{F}_{\alpha_{0}}(\varphi),$
(152)

where $\widetilde{F}_{\alpha_{0}}(\varphi)\\!\equiv\\!\widetilde{F}_{\bf q}$
with

$\displaystyle\widetilde{F}_{\bf q}=\big{|}V_{\bf
q}\big{|}^{2}\Big{(}\frac{\omega_{\bf q}}{T}\Big{)}\,{\rm n}^{0}_{\bf
q}\,\big{(}{\rm n}^{0}_{\bf q}+1\big{)},\ \ \ {\rm and}$ (153)
$\displaystyle{\bf
q}=2k_{F}\cos\varphi\,\big{(}\\!\cos(\varphi\\!+\\!\alpha_{0}),\sin(\varphi\\!+\\!\alpha_{0})\big{)}\,.$

We note that Eq. (152) does not have any additional approximations beyond the
ones already present at the step (150). Ignoring $\alpha_{0}$, using symmetry
of $\widetilde{F}_{\bf q}$ to $\varphi\\!\rightarrow\\!-\varphi$, and
introducing $z\\!=\\!\cos\varphi$ brings (152) back to (151). Obviously, the
same considerations can be used to obtain the spin-flip equivalent of (152).

With this result (152), we can study the effect of the angular dependence of
the quasiparticle relaxation rate. The findings for the polarized FM phase at
the saturation field $H\\!=\\!H_{s}$ and for the 120$\degree$ $H\\!=\\!0$
state as representative points of our analysis are summarized in Fig. 16. In
Fig. 16 we used the same model parameters as in the rest of the paper when
applied to EuC6, see Table 1, and investigated the angle-dependence of
$1/\tau_{\rm qp}$ as a function of $k_{F}$, with very similar results in both
cases.

As is discussed in Sec. V for the transport case, the overall rates grow with
$k_{F}$. Since we are interested in the angular dependence, but not the
absolute values, the main panel in Fig. 16 shows the quasiparticle relaxation
rate from (152) as a function of $\alpha_{0}$ normalized to its minimum value,
which is in the middle of the two principal (bond) directions of the
triangular lattice, denoted as $\pi/6$. The results are shown for the FM phase
and for several values of $k_{F}$. The upper inset shows the ratio of the
rates at the maximum ($\alpha_{0}\\!=\\!0$) and minimum
($\alpha_{0}\\!=\\!\pi/6$) vs $k_{F}$ for both FM and 120$\degree$ states.

One can see that the effect of the angle-dependence in $1/\tau_{\rm qp}$ is
really negligible up to $k_{F}$ about $\pi/2$ and is still very modest up to
$k_{F}\\!\gtrsim\\!0.6\pi$, justifying our initial approximation that
neglected it and confirming the correctness of our results presented
throughout the main body of the paper. In a sense, the effect is reminiscent
of the accuracy of the sextupole field as nearly circular away from the close
vicinity of the poles.

However, upon approaching the value of $k_{F}\\!=\\!2\pi/3$, the rates for the
principal directions diverge while the rates at $\alpha_{0}\\!=\\!\pi/6$ stay
finite. As is discussed in Sec. V, this effect is due to a singularity
associated with the scattering by the gapless Goldstone magnons with ${\bf
Q}\\!=\\!\pm(4\pi/3,0)$ and equivalent ordering vectors matching $2k_{F}$ for
the momenta ${\bf k}$ in the principal directions. In that sense, the
$\alpha_{0}\\!=\\!0$ and $\alpha_{0}\\!=\\!\pi/6$ directions of the electron
momentum ${\bf k}$ for $k_{F}\\!\rightarrow\\!2\pi/3$ become close analogues
of the “hot” and “cold” spots for the scattering, familiar from the cuprate
superconductors [54]. As is mentioned above, while the same behavior is
expected to occur for the transport scattering rates, the derivation of it
becomes more complicated as one needs to account for the angular dependence in
relating non-equilibrium distribution functions for different directions self-
consistently.

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