# Spin pumping effect in non-Fermi liquid metals

Xiao-Tian Zhang  Xu-Ping Yao  Kavli Institute for Theoretical Sciences and
HKU-UCAS Joint Institute for Theoretical and Computational Physics, University
of Chinese Academy of Sciences, Beijing 100190, China  Yuya Ominato  Waseda
Institute for Advanced Study, Waseda University, Shinjuku, Tokyo 169-8050,
Japan.  Long Zhang  Kavli Institute for Theoretical Sciences and HKU-UCAS
Joint Institute for Theoretical and Computational Physics, University of
Chinese Academy of Sciences, Beijing 100190, China  CAS Center for Excellence
in Topological Quantum Computation, University of Chinese Academy of Sciences,
Beijing 100190, China  Mamoru Matsuo  Kavli Institute for Theoretical Sciences
and HKU-UCAS Joint Institute for Theoretical and Computational Physics,
University of Chinese Academy of Sciences, Beijing 100190, China  RIKEN Center
for Emergent Matter Science (CEMS), Wako, Saitama 351-0198, Japan  Advanced
Science Research Center, Japan Atomic Energy Agency, Tokai, 319-1195, Japan

###### Abstract

Spin pumping effect is a sensitive and well-established experimental method in
two-dimensional (2D) magnetic materials. We propose that spin pumping effect
can be a valuable probe for non-Fermi liquid (NFL) behaviors at the 2D
interface of magnetic heterostructures. We show that the modulations of
ferromagnetic resonance (FMR) exhibiting power-law scalings in frequency and
temperature for NFL metals induced near a quantum critical point (QCP). The
exponents are universal parameters inherited from the QCP and reflect the non-
quasiparticle nature of spin relaxation in NFLs. We demonstrate the divergent
FMR modulations with a particular focus on the Ising nematic QCP at both zero
and finite temperatures, which contradicts the conventional Gilbert damping
mechanism in normal Fermi liquids. Our theoretical proposal could potentially
catalyze a surge in research harnessing insights from spin-dependent non-
equilibrium physics at nanoscale, primarily developed for room-temperature
functioning spin devices in the rapidly progressing field of spintronics, to
access magnetic properties in strongly correlated electron systems.

INTRODUCTION
Dimensionality plays a vital role in many-body physics, particularly, the
universal critical phenomena near a quantum critical point (QCP) [1]. Two-
dimensional (2D) correlated systems with strong quantum fluctuation have been
a fertile land for various unconventional quantum phases of matter and
associated QCPs, including the cuprate oxide layers in high-$T_{c}$
superconductors [2, 3], non-Fermi liquid metals [4] and exotic quantum magnets
[5]. The atomic-level 2D nature facilitates the formation of the proximity
effect, thus, offering a practical platform for co-integrating distinct
physical ingredients by artificial design of heterostructures [6]. The
stacking and twisting of 2D materials have expanded the boundary of condensed
matter physics and give birth to the van der Waals heterostructure [5] and
“twistronics” [7]. The lower dimensionality enables otherwise unattainable
fabrication, manipulation, and measurement as their bulk counterparts [8]. The
strain [9], gating [10], light [11] and electric field [10] can couple with
various internal degrees of freedom — charge, spin, orbit, lattice, etc. This
offers unprecedent tunability towards magnetic QCPs. In the mediated quantum
critical region, wild quantum fluctuations can lead to Fermi liquid
instability for the conduction electrons [12]. The breakdown of coherent
Fermi-liquid quasiparticles [13] is the most dramatic manifestation of the
many-body correlation, which is known as non-Fermi liquid (NFL) behavior.

Despite the success in fabricating 2D magnetic thin layers, the experimental
detection of the critical spin fluctuation poses significant challenges. The
conventional magnetic probes, such as neutron scattering, superconducting
quantum interference device magnetometry, are designed for bulk magnets,
therefore, are insensitive provided the weak signals from 2D magnetic
materials. The effectiveness and efficiency of optical probes, such as Raman
spectroscopy or magneto-circular dichroism, are currently under investigation
[14]. It is therefore greatly desired that a clever implementation of the
magnetic heterostructure probes the many-body correlation in directly in the
spin channel. Many-body correlations have been probed using the pure spin
current in the spintronics community [15]. The interface at the
heterostructure of various 2D magnetic materials is particularly beneficial
for harnessing spintronic functionalities [16]. The spin current can be
mediated by various quasiparticles given the rich combination of
heterstructures.

The spin pumping effect [17, 18], originally designed at the interface of
magnetic heterostructure to generate pure spin current, develops potential as
a sensitive probe in measuring the magnetic phase transitions in 2D thin
layers [19]. The heterostructure comprises a magnetic insulator subjected to
dynamical driven field and a 2D quantum magnet as the spin current receiver.
Conventionally, one consider a ferromagnetic insulator (FI) with a
precessional magnetization at its resonance states, the spin current is
injected into the adjacent material via interfacial spin exchange interaction
[20, 21, 22]. The spin injection has a backaction on the FI in modulating the
frequency of feromagnetic resonance (FMR) and the Gilbert damping. The
modulated FMR signal carries information about the dynamical spin
susceptibility of the 2D magnetic thin layer, making it a effective probe for
studying its spin characteristics of 2D magnetic materials. It is therefore
appealing to probe the spin correlation, particularly the critical spin
fluctuation near the QCP, in 2D magnetic heterostructure using the spin
pumping.

Figure 1: Schematic plot of the NFL/FI bilayer structure for the FMR-driven
spin pumping experiment. The pink arrow in the FI indicates the spin
${\bm{S}}$ precessing in the external ac magnetic field ${\bm{h}}_{\rm ac}$.
The blue arrows in the NFL metal indicate the itinerant electrons exchanging
spin angular momentum at the interface with magnons in the FI layer. The
gradually fainted green balls illustrate the incoherent quasiparticles in the
NFL metal. Right panel demonstrates the FMR signal modulation due to the
interlayer coupling.

In this article, we consider the spin pumping effect in the magnetic
heterostructure composed of a 2D NFL metal and a ferromagnetic insulator (FI)
thin film schematically plotted in Fig. 1. We demonstrate that the spin
pumping is an effective method to probe the critical spin fluctuation for 2D
magnetic heterostructure in the dynamic regime near the QCP. The QCP is
modeled by gapless Fermi surface coupled with critical bosons. The critical
boson can be either the order parameter near a magnetic phase transition or
the collective mode of electrons in the Pomeranchuk instability. We start with
generic analysis based on universal scaling rule near the magnetic QCP; Then,
we take the 2D Ising nematic QCP induced NFL as a concrete example which is
relevant to underdoped cuprates [3] and Fe-based superconductors [23]. The FI
is at the resonant frequency under the microwave radiation, the magnon
excitations are damped by the phenomenlogical Gilbert mechanism. Pure spin
current is injected into the adjacent NFL where the spin angular momentum is
carried by “non-quasiparticles”. The backaction of the spin injection can be
read off from the magnon self-energy correction. The perturbation is carried
out up to second order in interfacial exchange coupling in the Keldysh
representation which gives rise to a direct relation between the magnon self-
energy and the dynamical spin susceptibility in the NFL metal.

To be concrete, we evaluate the dynamical spin susceptibility $\chi_{\rm
uni}(\omega)$ near the Ising nematic QCP at both zero and finite temperatures.
We show the simple power counting within the Migdal-Eliashberg framework and
systematical inclusion of the vertex corrections yield consistent power-law
scaling for $\chi_{\rm uni}(\omega)$ in zero temperature limit. The scaling
exponent is inferred from the university class of the underlying QCP. The FMR
signals, namely the shift of resonance frequency and enhanced Gilbert damping
coefficient, are significantly modulated: The enhanced Gilbert damping
coefficient exhibits a power-law divergence in the low-energy limit indicating
that the conventional Gilbert mechanism becomes invalid. At finite
temperatures, we adopt revised Eliashberg equations for fermions and bosons
where bosonic correlation length and electronic scattering rate acquire
characteristic temperature dependence. The FMR modulations show distinct
power-law scaling with both frequency and temperature compared with the zero
temperature results. We point out that, at both zero and finite temperatures,
the shift of resonance frequency captures characteristics of quasi-particle
disappearance.

RESULTS
Magnetic Heterostructure. Let’s consider the magnetic heterostructure composed
of a NFL metal and a FI thin film shown in Fig. 1. The FI is driven by an
external _ac_ magnetic field in resonance with the precession of local spins.
The spin current is injected into the NFL metal via the spin exchange coupling
at the interface. The full Hamiltonian of the NFL/FI heterostructure comprises
three parts,

${\cal H}(t)={\cal H}_{\rm FI}(t)+{\cal H}_{\rm ex}+{\cal H}_{\rm NFL}.$ (1)

The first term describes the FI with the ferromagnetic Heisenberg model
subjected to an oscillating magnetic field,

${\cal H}_{\rm FI}(t)=-J\sum_{\langle
i,j\rangle}{\bm{S}}_{i}\cdot{\bm{S}}_{j}+\gamma_{g}H\sum_{i}S_{i}^{z}\\\
-\gamma h_{\rm ac}\sum_{i}\big{(}S_{i}^{x}\cos\omega t-S_{i}^{y}\sin\omega
t\big{)},$ (2)

in which $J>0$ is the ferromagnetic exchange coupling constant. ${\bm{S}}_{i}$
stands for the local spin at site $i$ in the FI. $H$ is the magnitude of the
Zeeman field. $\gamma_{g}$ ($<0$) is the gyromagnetic ratio. $h_{\rm ac}$ and
$\omega$ are the amplitude and the frequency of the circularly oscillating
external magnetic field, respectively. The second term in Eq. (1) is the
exchange coupling at the interface between local spins in the FI and itinerant
electrons in the NFL metal,

$\displaystyle{\cal H}_{\rm ex}$ $\displaystyle=\sum_{i}\int d^{2}{\bm{r}}\
J({\bm{r}},{\bm{r}}_{i}){\bm{S}}_{i}\cdot{\bm{s}}({\bm{r}})$
$\displaystyle=\sum_{{\bm{k}},{\bm{q}}}\big{(}J_{{\bm{k}},{\bm{q}}}s_{\bm{q}}^{+}S_{\bm{k}}^{-}+{\rm
H.c.}\big{)}+\cdots,$ (3)

where
${\bm{s}}({\bm{r}})=\frac{1}{2}c^{\dagger}_{\alpha}({\bm{r}}){\bm{\sigma}}_{\alpha\beta}c_{\beta}({\bm{r}})$
is the itinerant electron spin operator. The exchange coupling function in the
reciprocal space is approximately given by [24]

$|J_{{\bm{k}},{\bm{q}}}|^{2}=\frac{J_{1}^{2}}{N}\delta_{{\bm{k}},{\bm{q}}}+\frac{J_{2}^{2}l^{2}}{AN},$
(4)

where $A$ is the area of the NFL at the 2D interface, and $N$ is the number of
sites in FI. The first and second terms describe averaged uniform and
spatially uncorrelated roughness respectively. $l$ is introduced as an atomic
scale length for the continuous model. $J_{1}$ and $J_{2}$ are the mean value
and the variance of the exchange coupling.

The third term in Eq. (1) is a fermion-boson coupled model describing the NFL
induced by critical boson modes. The mechanism and classification of different
types of NFL has a long and distinguished history [25, 26, 27, 4] with ever-
increasing new developments [28, 29, 30, 31, 32]. The critical boson induced
NFL can be described by a generic form of Hamiltonian,

$\mathcal{H}_{\rm
NFL}=\int\mathrm{d}^{2}{\bm{r}}\Big{[}c^{\dagger}_{\alpha}({\bm{r}})\epsilon(-i\partial_{{\bm{r}}})c_{\alpha}({\bm{r}})-\lambda
O({\bm{r}})\phi({\bm{r}})\\\
+\frac{1}{2}\big{(}\partial_{\bm{r}}\phi\big{)}^{2}+\frac{r}{2}\phi^{2}+\cdots\Big{]},$
(5)

Here $c^{\dagger}_{\alpha}$($c_{\alpha}$) is the electron creation
(annihilation) operator for spin the spin up/down $\alpha=\pm$.
$\epsilon({\bm{k}})$ is the spin degenerated, bare electronic dispersion near
the Fermi surface. $\phi$ is a critical fluctuating bosonic field admitting a
Ginzburg-Landau expansion near a QCP tuned by $r\rightarrow 0$. $\lambda$ is
the coupling constant of electrons and the bosonic order parameter field.
$O({\bm{r}})$ is a fermion bilinear operator, which transforms inversely as
$\phi({\bm{r}})$ under symmetry actions guaranteeing the invariance of the
coupling term. We note that the boson field $\phi({\bm{r}})$ is _not_ directly
coupled to the FI via magnetic exchange interactions. It’s reasonable to
assume that the boson field is either regarded as non-magnetic for the Ising
nematic case, or being ineffective in coupling to the local magnetic moment
${\bm{S}}_{i}$ in FI. Invoking the direct coupling, e.g. in the itinerant
quantum magnets where the boson $\phi$-field represents the magnetic ordering
fluctuation, would introduce additional complexity [33], which is left for
future studies.

Spin pumping and FMR modulations. The spin pumping, at a microscopic level, is
initiated by the magnetic exchange interaction at the heterostructure
interface between the magnetization in the FI and the electron spin of the
NFL. This interaction promotes spin injection and generates a self-energy
correction for the magnons in the FI as a backaction, modulating the frequency
of FMR and Gilbert damping [20, 21, 22]. The modulated FMR signal carries
information about the dynamical spin susceptibility and is shown to be related
to the magnon self-energy. To be specific, the retarded magnon Green’s
function takes a form,

$G^{-1}({\bm{k}},\omega)=\omega-\omega_{\bm{k}}+i\alpha\omega-\Sigma({\bm{k}},\omega),$
(6)

in which the bare magnon dispersion reads
$\omega_{\bm{k}}=Dk^{2}-\gamma_{g}H$, and $\Sigma({\bm{k}},\omega)$ is the
magnon self-energy due to the backaction of the spin pumping. In a generic
bosonic system, the self-energy has an imaginary part proportional to $\omega$
[34], and the coefficient $\alpha$ is called the Gilbert damping constant
[17]. The FMR modulation is determined by the uniform component (${\bm{k}}=0$)
of the magnon Green’s function, in which the pole dictates the resonance
condition, $\omega+\gamma_{g}H-{\rm Re}\Sigma_{{\bm{k}}=0}(\omega)=0$, thus
the resonance frequency is shifted by $\delta H=\gamma_{g}^{-1}{\rm
Re}\Sigma_{{\bm{k}}=0}(\omega)$. The imaginary part of the self-energy leads
to an enhanced Gilbert damping coefficient,
$\delta\alpha=-\omega^{-1}\mathrm{Im}\Sigma_{{\bm{k}}=0}(\omega)$.

The magnon self-energy can be calculated perturbatively up to the second order
in terms of the external oscillating magnetic field $h_{\rm ac}$ and the
exchange coupling $|J_{{\bm{k}},{\bm{q}}}|$ [20, 22],

$\Sigma({\bm{k}},\omega)=-\sum_{{\bm{q}}}|J_{{\bm{k}},{\bm{q}}}|^{2}\chi({\bm{q}},\omega),$
(7)

with $\chi({\bm{q}},\omega)\equiv i\int
dte^{i(\omega+i0^{+})t}\Theta(t)\langle[s_{\bm{q}}^{+}(t),s_{-{\bm{q}}}^{-}(0)]\rangle$
being the retarded spin susceptibility for NFL metals. Inserting the exchange
coupling function in Eq. (4), the magnon self-energy is given by [24]

$\Sigma_{{\bm{k}}=0}(\omega)=-\frac{J_{1}^{2}}{N}\chi_{\rm
uni}(\omega)-\frac{J_{2}^{2}l^{2}}{AN}\chi_{\rm loc}(\omega).$ (8)

where two terms corresponds to uniform and uncorrelated roughness contribution
of the interfacial exchange interaction. $J_{1},J_{2}$ are the mean value and
variance, singling out the uniform and the local components of the dynamical
spin susceptibility. They are expressed as $\chi_{\rm
uni}(\omega)\equiv\chi({\bm{q}}=0,\omega)$ and $\chi_{\rm
loc}(\omega)\equiv\sum_{\bm{q}}\chi({\bm{q}},\omega)$.

Figure 2: Feynman diagrams for the perturbative calculation. (a) Dyson
equation for the magnon Green’s function. The thin and the thick wavy lines
are the bare and the renormalized magnon Green’s functions, respectively. The
circle stands for the magnon self-energy $\Sigma$. (b) The magnon self-energy
equals to the dynamical spin susceptibility that can be expressed in terms of
charge polarization bubble under Migdal approximation. The thick straight line
is the renormalized electron propagator.

In the vicinity of the QCP at $r=r_{c}$, the dynamical spin susceptibility
takes the following universal scaling form at $T=0$,
$\chi({\bm{q}},\omega,r-r_{c})=\xi^{d_{\chi}}{\chi}({\bm{q}}\xi,\omega\xi^{z})$,
in which $\xi$ is the spatial correlation length, $d_{\chi}$ is the scaling
dimension of the spin susceptibility, and $z$ is the dynamical exponent. At
the QCP, the correlation length diverges, leading to the reduced scaling form,
$\chi({\bm{q}},\omega,T=0)=\omega^{-d_{\chi}/z}\tilde{\chi}({\bm{q}}/\omega^{1/z})$.
The magnon self-energy correction has different power-law scaling forms
inherited from the uniform and local components in respective limits,

$\Sigma_{{\bm{k}}=0}(\omega)=\begin{cases}-\frac{J_{1}^{2}}{N}\chi_{\rm
uni}(\omega)\simeq\omega^{-d_{\chi}/z},&\eta\gg 1\\\
-\frac{J_{2}^{2}l^{2}}{AN}\chi_{\rm
loc}(\omega)\simeq\omega^{(d-d_{\chi})/z},&\eta\ll 1.\end{cases}$ (9)

Here $d_{\chi}$ and $z$ are two independent critical exponents which can be
uniquely determined by tuning the interfacial roughness
$\eta=J_{1}\sqrt{A}/(J_{2}l)$. And, $d_{\chi}$ is often related to the
physical dimension $d$ where deviation are encountered when hyperscaling is
violated approaching certain QCPs [35, 36]. The power-law scalings are in
sharp contrast to the conventional linear-in-frequency Gilbert damping term,
thus reflects the NFL nature associated with the QCP.

The quantum critical regime extends to nonzero temperatures above the QCP,
where a different set of critical exponents $d_{\chi}^{\prime}$ and
$z^{\prime}$ emerge
$\chi({\bm{q}},\omega,r-r_{c},T)=L_{\tau}^{d_{\chi}^{\prime}/z^{\prime}}\tilde{\chi}\big{(}{\bm{q}}L_{\tau}^{1/z^{\prime}},\omega
L_{\tau},\xi(T)/L_{\tau}^{1/z^{\prime}}\big{)}$. $L_{\tau}=1/(k_{B}T)$ is the
characteristic scale in the imaginary time. Note that the correlation length
(or the boson mass) is originally a quantum parameter, now, acquires a
$T$-dependence endowed by higher order interactions. The uniform and local
components are expressed in terms of correlation length as

$\displaystyle\chi_{\rm uni}(\omega,T)={}$
$\displaystyle\xi(T)^{d_{\chi}^{\prime}}\tilde{\chi}\big{(}\omega/T,\xi(T)T^{1/z^{\prime}}\big{)},$
(10a) $\displaystyle\chi_{\rm loc}(\omega,T)={}$
$\displaystyle\xi(T)^{(d_{\chi}^{\prime}-d)}\tilde{\chi}\big{(}\omega/T,\xi(T)T^{1/z^{\prime}}\big{)}.$
(10b)

Henceforth, we consider the smooth interface $\eta\gg 1$ and focus on the
uniform component of the spin susceptibility $\chi_{\rm uni}(\omega)$, namely
the dynamical spin susceptibility. The frequency dependence in the dynamic
extreme with ${\bm{q}}=0$ and finite $\omega$ is rarely studied theoretically.
The main reasons are two-fold: on one hand, the dynamic exponent is usually
$z>1$ in the vicinity of QCP which amounts to the static limit $\omega\sim
q^{z}\ll v_{F}q$; on the other hand, the uniform component susceptibility
should vanish for any finite frequency, indicating the conservation of
respective physical variables such as charges and spins. However, the
dynamical spin susceptibility contains an important piece of information of
the correlated electron system that is non-vanishing and can be readily probed
by the spin pumping experiment. In the magnetic heterostructure plotted in
Fig. 1, only the total spin, that includes both the itinerant electronic spins
in NFL metal and the local magnetic moments in FI, is a conserved quantity.
The desired dynamic spin susceptibility $\chi_{\rm uni}(\omega)$ of the
itinerant electrons is generically non-zero due to the spin-fermion model [see
Eq.(4)] at the interface.

Figure 3:  Schematic illustration of the patch decomposition. for 2D circular
Fermi surface (dashed grey circle). (a) The critical Fermi circle in 2D is
approximated by the orange polygon. where each red line segment stands for
decoupled patches with the width $2\Lambda$. Orange fainted rectangle is the
low-energy region for a single decoupled patch labelled by index
$s\in[1,2N_{p}]$. (b) The small momentum variation $\delta{\bm{k}}$ in the
$s$’th patch centered at ${\bm{k}}_{F,s}$ is decomposed into radical component
$\delta k_{x}$ and tangential component $\delta k_{y}$.

Divergent FMR modulation at Ising nematic QCP. The Pomeranchuk instability
near the Ising nematic QCP ($d=2,z=3$) is described by critical bosonic modes
of the collectively distorting Fermi surface. The four-fold lattice rotational
symmetry is spontaneously broken down to two-fold in the Ising nematic phase.
The order parameter field $\phi({\bm{r}})$ is coupled to the $d$-wave
component of the fermion density $O({\bm{r}})$ in Eq. (5)

$O({\bm{r}})=\frac{1}{A}\sum_{\bm{q}}\sum_{{\bm{k}},\alpha}d_{\bm{k}}c^{\dagger}_{{\bm{k}}-{\bm{q}}/2,\alpha}c_{{\bm{k}}+{\bm{q}}/2,\alpha}e^{i{\bm{q}}\cdot{\bm{r}}},$
(11)

where the $d$-wave form factor can take a form $d_{\bm{k}}=\cos k_{x}-\cos
k_{y}$. The dynamic charge and spin responses are related to the boson
susceptibility. The boson susceptibility at ${\bm{q}}=0$ can have non-trivial
dynamics since the order parameter is conjugated to a non-conserved physical
quantity which is not strictly constrained by the conservation law [37]. At
$T=0$, the bosonic self-energy has a generic scaling in the dynamic limit
$|\omega|\gg v_{F}q$ as: $\Pi({\bm{q}}=0,\omega)=c_{0}+c_{1}{\rm
sgn}(\omega)|\omega|^{-d_{\chi}(z)/z}$. The universal frequency scaling
exponent $d_{\chi}(z)$ can be probed by Raman scattering [38], optical
conductivity [39] and evaluated by Monte Carlo simulations [40].

Theoretical investigation of the dynamic susceptibility relies on various
perturbative expansions and non-perturbative methods which has led to
controversial results. (i) One way to tackle with the strongly correlated
fermion-boson problem is the large fermion flavor limit
$N_{f}\rightarrow\infty$. Large-$N_{f}$ limit corresponds to the random phase
approximation and leads to the celebrated expression
$d_{\chi}(z)=z-4\xrightarrow{z=3}-1$. Subsequently, the large-$N_{f}$
expansion is accused of being uncontrolled in 2D [41]. A more careful
evaluation yields vanishing leading order contribution $c_{1}=0$ for a generic
convex Fermi surface [42] and subleading term scales as $\sim\omega^{5/3}$.
Importantly, the leading order term is found to be finite ($c_{1}\neq 0$) for
a concave Fermi surface [43, 44]. (ii) Another way is to apply the Migdal
approximation neglecting all vertex corrections, which leads to the Eliashberg
coupled equations for the fermions and bosons [37, 45, 46]. The self-
consistent solution yields $-d_{\chi}/z=1/3$ with $c_{1}\neq 0$ [47]
contradicting the result from large-$N_{f}$ expansion. Since the Migdal
approximation is not generically acknowledged, theorists bypass this ambiguity
by introducing randomness into interactions, which is under the name of SYK. A
generalization of SYK, Yukawa-SYK model [48, 49, 50], has bee shown to
generate the same set of Eliashberg equations [42]. Unfortunately, the
microscopic origin of SYK model is not clear and the validity of the entire
Migdal-Eliashberg framework is questioned [51]. (iii) Non-perturbative method
anomaly has been particularly useful in dictating dynamical response in the
presence of Fermi surface. Quantum anomaly predicts a trivial frequency
scaling with $c_{0}\neq 0,c_{1}=0$ [52], yet, it is not applicable to the case
of 2D Ising nematic QCP given that the boson-fermion interaction in $d$-wave
channel is irrelevant. Despite the tortuous exploration process, results based
on distinct theoretical frameworks remain contradicting and controversial.
Proposing a measurable phenomenon based on an established experimental
framework can greatly facilitate the resolution of theoretical disputes.

Anticipating a non-vanishing dynamic response in the spin pumping setup, we
evaluate the spin susceptibility using the Eliashberg theory described above.
For starters, we adopt the Migdal approximation, retaining the self-energy
corrections while neglecting all vertex corrections. Subsequently, we go
beyond this simple approximation and take into account of the ladder diagrams
of vertex corrections properly. In this study, we omit the contributions from
Aslamazov-Larkin (AL) diagrams since they are one order higher than the
corresponding self-energy and vertex diagrams in terms of $o(J_{1})$. AL
diagrams are crucial in restoring the spin conservation[53] which is lack for
the itinerant electron spins at the magnetic heterostructure interface. In a
similar fashion, this has been verified for itinerant quantum magnets where
interaction between itinerant electron spins and local magnetic moments is
described by the spin-fermion model[33]. Moreover, the AL diagrams in the
$d$-wave channel mediated by critical boson $\phi$ only yield quantitative
modifications[47].

We first proceed by assuming the Migdal approximation so that the spin
susceptibility is given by the renormalized charge polarization bubble in Fig.
2(b). Explicitly, it is expressed as
$\chi_{\mathrm{uni}}(\omega)=3i\sum_{{\bm{k}}}\int\frac{d\omega^{\prime}}{2\pi}g_{R}({\bm{k}},\omega^{\prime})g_{R}({\bm{k}},\omega^{\prime}+\omega)$.
The spin index $\alpha$ is omitted due to the spin degeneracy and the spin
Pauli matrices at the external vertex are traced out ${\rm
tr}(\sigma^{+}\sigma^{-})=3$. Here, $g_{R}({\bm{k}},\omega)=-i\int d\tau
e^{i(\omega+i0^{+})\tau}\langle{\cal
T}c_{{\bm{k}},\alpha}(\tau)c^{\dagger}_{{\bm{k}},\alpha}(0)\rangle$ is the
renormalized retarded fermion Green function. We employ the patch theory to
describe the NFL [4] and calculate the polarization bubble which involves only
a single patch. As illustrated in Fig. 3(a), the Fermi surface can be divided
into $2N_{p}$ number of patches labelled by $s$. As a result, the dynamic spin
susceptibility is expressed as a summation over single patch polarization
bubbles $\Pi_{s}$ [54],

$\displaystyle\chi_{\mathrm{uni}}(\omega)={}$ $\displaystyle
3A\sum_{s=1}^{2N_{p}}\Pi_{s}({\bm{q}}=0,\omega),$ (12a)
$\displaystyle\Pi_{s}({\bm{q}}=0,\omega)={}$ $\displaystyle
i\int\frac{d^{2}\delta{\bm{k}}}{(2\pi)^{2}}\frac{d\Omega}{2\pi}g_{s,R}(\delta{\bm{k}},\omega)g_{s,R}(\delta{\bm{k}},\omega+\Omega),$
(12b) $\displaystyle g_{s,R}(\delta{\bm{k}},\omega)={}$
$\displaystyle\big{(}ic_{F}d^{2}_{{\bm{k}}_{F,s}}|\omega|^{2/3}-v_{F}\delta
k_{x}-\frac{\kappa}{2}\delta k_{y}^{2}\big{)}^{-1}.$ (12c)

Note that the polarization bubble $\Pi_{s}$ is different from the exact boson
self-energy $\Pi$ by a pair of $d$-wave form factors at the two external
vertex. Here, $g_{s,R}(\delta{\bm{k}},\omega)$ is the fully renormalized
retarded fermion Green function for the $s$’th patch centered at the Fermi
momentum ${\bm{k}}_{F,s}$. As depicted in Fig. 3(b), the momentum nearby is
expanded as ${\bm{k}}={\bm{k}}_{F,s}+\delta{\bm{k}}$. The small variation
$\delta{\bm{k}}$ can be further decomposed into radial and tangential
components $\delta k_{x},\delta k_{y}$ with respect to the directional vector
$\hat{k}={\bm{k}}_{F,s}/k_{F,s}$. The bare dynamic term $i\omega$ is
overwritten by the fermionic self-energy $\Sigma_{F}({\bm{k}}_{F,s},\omega)$
in the low-energy regime $|\omega|<\omega_{c}=c_{F}^{3}$. The real part of the
self-energy is omitted in Eq.(12) which can be derived from the imaginary part
using Kramer-Kronig relation in the low-energy regime

$\displaystyle{\rm Re}\Sigma_{F}({\bm{k}}_{F,s},\omega)$
$\displaystyle=\frac{1}{\pi}\int_{-\omega_{c}}^{+\omega_{c}}d\omega^{\prime}\frac{{\rm
Im}\Sigma_{F}({\bm{k}}_{F,s},\omega^{\prime})}{\omega^{\prime}-\omega}$
$\displaystyle\simeq-\sqrt{3}d^{2}_{{\bm{k}}_{F,s}}c_{F}{\rm
sgn}(\omega)|\omega|^{2/3}.$ (13)

The NFL nature is captured by the vanishing quasiparticle weight at
$k_{F,x}\neq\pm k_{F,y}$ which is given by
$Z_{{\bm{k}}_{F,s}}(\omega)\equiv\big{[}1-\partial_{\omega}{\rm
Re}\Sigma_{F}({\bm{k}}_{F,s},\omega)\big{]}^{-1}\simeq
d_{{\bm{k}}_{F,s}}^{-2}\omega^{1/3}\xrightarrow{\omega\rightarrow 0}0$. The
dynamical exponent $z=3$ holds up to three-loop perturbations [46].
Substituting the expression of fermion Green function into the polarization
bubble leads to

$\displaystyle\Pi_{s}({\bm{q}}=0,\omega)\simeq
d_{{\bm{k}}_{F,s}}^{-2}|\omega|^{-d_{\chi}(z)/z},$ (14a) $\displaystyle
d_{\chi}(z)/z=(d-z)/z=-1/3.$ (14b)

The expression in Eq. (14) is different from the well-known Landau damping
term that is obtained in the opposite static limit $|\omega|\ll v_{F}q$.

Figure 4: One-loop Feynman diagrams of (a) the polarization bubble (boson
self-energy) and (b) the electron self-energy in the NFL metal. The straight
and wavy lines represent the electron and the boson propagators, respectively.
(c) Fully renormalized polarization bubble beyond the Migdal-Eliashberg
framework. Gray cicle $\Gamma$ labels the fully renormalized vertex. (d) The
Dyson equation for the vertex function $\Gamma(k;q)$. The thin and thick lines
are for the bare and renormalized propagators, respectively. The panels on the
left/right hand side of the equation are the fully renormalized/bare vertex
function.

Furthermore, we go beyond the Migdal-Eliashberg framework by additionally
including the vertex ladder diagrams in Fig. 4(c). The polarization bubble in
the dynamic limit is constrained by the charge/spin conservation; With a
constant form factor $d_{\bm{k}}=d_{0}$, the Ward identity dictates a
vanishing result for $\Pi_{B}({\bm{q}}=0,\omega)$ [37]. We show that with the
presence of the $d$-wave form factor at the internal vertex,
$\Pi_{s}({\bm{q}}=0,\omega)$ can be nonzero for a non-circular Fermi surface.
The fully renormalized polarization bubble at $q=({\bm{q}}=0,q_{0})$ is
written as,

$\Pi_{s}(q)=-\int\frac{d^{3}k}{(2\pi)^{3}}\Gamma(k;q)g_{s}(k)g_{s}(k+q),$ (15)

where the $k=(\delta{\bm{k}},k_{0})$ represents the three-momentum with
$k_{0},q_{0}$ being the fermionic and bosonic Matsubara frequency,
respectively. The vertex function
$\Gamma(k;q)\simeq\Gamma(\hat{k};k_{0},q_{0})$ satisfies the Dyson equation
depicted in Fig. 4(d) which reads

$\Gamma(\hat{k};k_{0},q_{0})=-\lambda^{2}\int\frac{d^{3}k^{\prime}}{(2\pi)^{3}}\Gamma(\hat{k^{\prime}};k_{0}^{\prime},q_{0})g_{s}(k^{\prime})g_{s}(k^{\prime}+q)\\\
\times
D(k-k^{\prime})d^{2}\left(\frac{\delta{\bm{k}}+\delta{\bm{k}}^{\prime}}{2}\right).$
(16)

The ansatz takes a modified form of the Ward identity

$\Gamma(\hat{k};k_{0},q_{0})=\frac{iq_{0}-\big{[}\Sigma(k+q)-\Sigma(k)\big{]}}{iq_{0}\big{[}1-\delta\Gamma^{(1)}(\hat{k};k_{0},q_{0})\big{]}}.$
(17)

Here the small correction $\delta\Gamma^{(1)}$ represents violation of the
conventional Ward identity originated from the $d$-wave form factor and the
non-circular nature on the Fermi surface. Substituting the ansatz into Eq.
(16), we obtain the solution self-consistently as [see Method section for
details]

$\delta\Gamma^{(1)}(\hat{k};k_{0},q_{0})\simeq\lambda^{2}|{q_{0}}|^{1/3}F_{1}(\hat{k},k_{0}/q_{0}).$
(18)

where $F_{1}(\hat{k},x)$ is a dimensionless function. Inserting this
expression into Eq. (15) we obtain

$\Pi_{s}({\bm{q}}=0,q_{0})\simeq-\frac{\lambda^{2}}{2\pi
v_{F}}|q_{0}|^{1/3}.\\\ $ (19)

Comparing Eq. (14) and Eq. (19), we arrive at a consistent scaling form of the
polarization bubble obtained within and beyond the Migdal-Eliashberg
framework. The dynamic spin susceptibility is readily obtained and the scaling
exponent can be experimentally probed by the FMR modulations. To see this, we
write down the retarded magnon self-energy

$\Sigma_{{\bm{k}}=0}(\omega)=\frac{4\pi^{2}J_{1}^{2}\rho_{F}}{N}\Big{|}\frac{\omega}{\omega_{c}}\Big{|}^{1/3}\Big{[}\sqrt{3}-i{\rm
sgn}(\omega)\Big{]},$ (20)

in which $\rho_{F}=Ak_{F}/(\pi v_{F})$ is the density of states at the Fermi
energy. The FMR modulations, namely the resonance frequency shift and the
enhanced Gilbert damping, acquire the following scaling forms

$\delta\alpha\sim|\omega|^{-2/3},\quad\delta
H\sim\gamma_{g}^{-1}|\omega|^{1/3}.$ (21)

which are schematically plotted in Fig. 5. In the low energy and zero
temperature limit: $T<|\omega|<\omega_{c}$, the diverging coefficient
$\delta\alpha$ indicates a novel spin relaxation mechanism, which is in sharp
contrast to the conventional linear-in-$\omega$ Gilbert damping. The exponents
reflect the universal scaling behavior near the Ising nematic QCP in the
dynamic limit which is currently under heated debt. Moreover, by comparing
with the quasiparticle weights, the FMR modulations capture the disappearance
of the quasiparticles in the low-energy limit.

Figure 5:  Schematic illustrations of the FMR modulations scaling forms. The
frequency and temperature are rescaled by a characteristic temperature $T_{0}$
($\gg T$). (a) Temperature dependence of the resonance frequency shift $\delta
H(T)$. The finite temperature($T>\omega$) $\delta H(T)$ shows a rapid increase
approaching low-$T$ owing to the divergent correlation length. (b) Frequency
dependence of the enhanced Gilbert damping coefficient
$|\delta\alpha(\omega)|$. The magnitude of $\delta\alpha$ displays different
scaling forms in the low-frequency($\omega<T$, left) and high-
frequency($\omega>T$, right) regions.

FMR modulations at finite temperatures. The finite-temperature properties of
NFLs in the quantum critical regime mediated by the Ising nematic QCP cannot
be simply inferred from the zero-temperature results. The quantum fluctuation
of the bosons obeys the $\omega/T$ scaling rule and leads to $T^{\frac{2}{3}}$
dependence for fermion self-energy; while, the contribution from thermal
fluctuation is drastically different and dominates at low temperatures [45].
When both thermal and quantum fluctuations are included, the vertex
corrections are shown to be subdominant and the Migdal approximation becomes
valid at finite temperatures [55]. The finite temperature version of
Eliashberg equations are solved leading to the solution [55, 54]

$\displaystyle g^{-1}_{R}({\bm{k}},\omega)$
$\displaystyle=\omega+i0^{+}-\epsilon_{\bm{k}}+i\gamma_{{\bm{k}}_{F}}(T),$
(22a) $\displaystyle D_{R}^{-1}({\bm{q}},\Omega)$
$\displaystyle=\xi^{-2}(T)+a|{\bm{q}}|^{2}-ib\frac{\Omega}{\gamma(T)}.$ (22b)

where $a$ and $b$ are nonuniversal constants. The correlation length and
electron scattering rate scale as $\xi(T)^{-1}\sim\sqrt{T\ln(\epsilon_{F}/T)}$
and $\gamma(T)\sim\sqrt{T/\ln(\epsilon_{F}/T)}$, respectively. The imaginary
part of the retarded self-energy at $\omega=0$ is given by
$\gamma_{{\bm{k}}_{F}}(T)=\lambda^{2}d^{2}_{{\bm{k}}_{F}}T\xi(T)/(4v_{F}\sqrt{a})$
for $T\ll T_{0}$. This expression can not be derived from the $T=0$ fermion
self-energy by assuming the $\omega/T$ scaling rule since the
thermal/classical fluctuation of bosons plays a dominant role in mediating
interaction between fermions. The upper bound
$T_{0}=\epsilon_{F}e^{-\lambda^{2}/v_{F}^{2}}$ is specified by the condition
$|\gamma_{{\bm{k}}_{F}}(T)|/v_{F}\ll\xi^{-1}(T)/\sqrt{a}$. As a result, the
finite temperature regime is defined as $|\omega|<T\ll T_{0}(<\omega_{c})$,
while, the zero-temperature limit is in the opposite regime $T<|\omega|$.

The dynamic polarization bubble $\Pi(q)=-3(\beta A)^{-1}\sum_{q}g(q)g(k+q)$ is
calculated at finite temperatures $|\omega|<T\ll T_{0}$ more conveniently in
real frequencies,

${\rm
Im}{\Pi}_{R}({\bm{q}},\omega)=3\int\frac{d^{2}{\bm{k}}}{(2\pi)^{2}}\int\frac{d\omega^{\prime}}{\pi}\big{[}n_{\rm
F}(\omega)-n_{\rm F}(\omega^{\prime}+\omega)\big{]}\\\ \times{\rm
Im}g_{R}({\bm{q}}+{\bm{k}},\omega^{\prime}+\omega){\rm
Im}g_{R}({\bm{k}},\omega^{\prime}).$ (23)

Here we consider a circular Fermi surface
$\epsilon_{\bm{k}}=\frac{{\bm{k}}^{2}}{2m_{F}}-\epsilon_{F}$ which is simply
described by a continuous phase variable instead of the patch decomposition in
previous section. We constrain the external boson momentums to the dynamic
limit $v_{F}|{\bm{q}}|\ll|\omega|\ll\epsilon_{F}$, whereas, the
${\bm{q}}\rightarrow 0$ limit of Eq. (23). Therefore, the momentum dependence
can be neglected in the fermion Green functions, which leads to

${\rm
Im}g_{R}({\bm{k}},\omega)=\frac{-\gamma_{\bm{k}}}{(\omega-\epsilon_{\bm{k}})^{2}+\gamma_{\bm{k}}^{2}}\simeq-\pi\delta(\omega-\epsilon_{\bm{k}}).$
(24)

where the Lorenzian is approximated by a Dirac delta function assuming
$|\gamma_{\bm{k}}|\ll\epsilon_{F}$. In addition, the low temperature limit
$|\omega|/T\ll 1$ simplifies the distribution functions as $n_{\rm
F}(\omega^{\prime})-n_{\rm F}(\omega^{\prime}+\omega)\simeq\omega/4T$.
Finally, we arrive at

$\displaystyle{\rm Im}{\Pi}_{R}(0,\omega)$ $\displaystyle\simeq
3\omega\int_{-T}^{+T}d\omega^{\prime}\int\frac{d^{2}{\bm{k}}}{(2\pi)^{2}}\frac{\delta(\omega^{\prime}-\epsilon_{\bm{k}})\gamma_{{\bm{k}}}}{(\omega+\omega^{\prime}-\epsilon_{\bm{k}})^{2}+\gamma_{{\bm{k}}}^{2}}$
$\displaystyle=\frac{3k_{F}/v_{F}}{4\pi}\frac{\omega}{\gamma(T)}{\cal
F}\big{[}|\omega|/\gamma(T)\big{]}$
$\displaystyle\simeq\frac{3\pi\rho_{F}}{2\sqrt{2}A}{\rm
sgn}(\omega)\big{(}|\omega|/\gamma(T)\big{)}^{1/2},$ (25)

where the function ${\cal F}(x)$ contains the polar angle $\theta$-integral
and is approximated as

${\cal
F}(x)\equiv\int_{0}^{2\pi}\frac{d\theta}{2\pi}\frac{\cos^{2}(2\theta)}{x^{2}+\cos^{4}(2\theta)}\xrightarrow{|x|\rightarrow
0}\frac{\sqrt{2}\pi}{\sqrt{|x|}}.$ (26)

Correspondingly, the real part is obtained applying the Kramer-Kronig relation

$\displaystyle{\rm Re}{\Pi}_{R}(0,\omega)$
$\displaystyle\equiv\frac{1}{\pi}\int_{-\omega_{c}}^{\omega_{c}}d\omega^{\prime}\frac{{\rm
Im}{\Pi}_{R}(0,\omega^{\prime})}{\omega^{\prime}-\omega}$
$\displaystyle\simeq\frac{3\rho_{F}}{2\sqrt{2\gamma(T)}A}\int_{-T}^{T}d\omega^{\prime}\
{\rm
sgn}(\omega^{\prime})\frac{|\omega^{\prime}|^{1/2}}{\omega^{\prime}-\omega}$
$\displaystyle\simeq\frac{3\sqrt{2}\rho_{F}}{\pi
A}\sqrt{\frac{T}{\gamma(T)}}.$ (27)

Collecting terms from Eqs. (Spin pumping effect in non-Fermi liquid metals)
and (Spin pumping effect in non-Fermi liquid metals), we arrive at the
dynamical spin susceptibility, then, the magnon self-energy at finite
temperature, which reads

$\Sigma_{{\bm{k}}=0}(\omega)=-\frac{3\pi
J_{1}^{2}\rho_{F}}{2\sqrt{2}N}\left[\frac{4}{\pi}{\xi^{-\frac{1}{2}}(T)}+i{\rm
sgn}(\omega)\sqrt{\frac{|\omega|}{\gamma(T)}}\right],$ (28)

where we have used the relation $\xi(T)=\gamma(T)/T$. This expression is
consistent with generic scaling form in Eq. (10) with $d^{\prime}_{\chi}=-1/2$
at finite temperatures (note that $d_{\chi}=-1$ at $T=0$). Accordingly, the
FMR frequency shift and the enhanced Gilbert damping coefficient have the
following scaling form

$\delta\alpha\sim\left[|\omega|\gamma(T)\right]^{-\frac{1}{2}},\quad\delta
H\sim-\gamma_{g}^{-1}\xi^{-\frac{1}{2}}(T),$ (29)

which are schematically plotted in Fig. 5. Particularly, the information of
the correlation length can be extracted from the FMR frequency shift which, at
the same time, reflects the non-quasiparticle nature in the quantum critical
regime. To see this, we evaluate the real part of the fermionic self-energy
$\mathrm{Re}\Sigma_{F}({\bm{k}}_{F},\omega)\sim-
d_{{\bm{k}}_{F}}\omega\xi(T)^{-1}$ for $|\omega|<T<\omega_{c}$ which leads to
a vanishing quasiparticle weight due to the divergent correlation length [see
Method section for derivations]

$Z({\bm{k}}_{F},\omega;T)\simeq d_{{\bm{k}}_{F}}^{-2}\xi(T)^{-1},\quad
k_{F,x}\neq\pm k_{F,y}.$ (30)

SUMMARY AND DISCUSSION
We have presented a practical implementation of the magnetic heterostructure
that probes the many-body spin/charge correlations. We study the FMR-driven
spin pumping effect at the interface of NFL/FI heterostructure in the absence
of quasiparticles. The dynamic spin susceptibility of itinerant electrons is
not constrainted of the spin conservation law due to the local moments at the
interface of magnetic heterostructure, thus, acquires universal power-law
scalings in frequency and temperature. The experimental measurable FMR
modulations, namely the resonance frequency shift and the enhanced Gilbert
damping, conceive valuable informations characterizing the NFL behaviors and
the disappearance of the quasiparticles. We conclude that the spintronics
experiments, particularly spin pumping, can take full advantage of the
magnetic heterostructure, meanwhile, shed light on the non-quasiparticle
feature of spin relaxation in NFLs. Our proposal is also helpful in
reconciling the current theoretical debates in the scaling forms of dynamic
correlation functions near Ising nematic QCP.

The magnetic heterostructure is far from simple stacking of two different
materials. The heterostructure by design breaks inversion symmetry, thus,
allowing the existence of anti-symmetric spin exchange interaction, which is
also known as the Dzyaloshinski–Moriya interaction (DMI). On one hand, the DMI
can drive the magnetic insulators into forming more complex magnetic orders,
such as the skyrmion lattice structures where local magnetic moments swirl
spatially in a non-collinear way while forming a periodical lattice. The
formation of more complex magnetic ordered states inevitably makes the
phenomena of the spin pumping effect more diverse and interesting. It has
already been reported that the magnetic fluctuation of the skyrmion lattices
can mediate exotic type of electron interaction in normal metals. Owing to the
non-collinear nature of the magnetic ground state, topological
superconductivity can be induced directly [56]. The competition between
topological superconductivity and novel type of NFLs is an interesting topic
on its own; while, probing the spin pumping signals at the interface of a
skyrmion lattice and a topological superconductor/NFL appealing to the
spintronics society. On the other hand, the DMI is an essential ingredient
when the magnetic insulator approaches its criticality. Infinite many critical
bosons can simultaneously reach criticality due to the presence of DMI in
three-dimensional chiral magnet [30] as well as 2D magnetic heterostructure
interface [31]. The infinite many critical boson can induce a novel type of
NFL state for the itinerant magnets nearby [29, 57, 58, 30, 31].

Finally, we remark on the non-equilibrium theoretical framework adopted in the
present study. The FMR modulations are derived perturbatively in terms of
interfacial exchange coupling strength $J_{1}$ in the Keldysh representation.
On the contrary, the self-energy correction of the NFL is independent of
$J_{1}$, namely, the stability of the NFL state is assumed as a prior. From a
theoretical point of view, it is greatly desired that the coupled two sides of
the magnetic heterostructure are solved self-consistently; yet, it poses great
difficult. Thanks to the development of Sachdev-Ye-Kitaev (SYK) model [59],
the magnetic exchange interaction can be regarded as a random variable in
fictitious flavor space. The replica trick used in solving SYK model can lead
to a solvable equation set for the entire magnetic heterostructure. The mutual
interactions between the itinerant magnets and driven magnetic insulator can
be determined self-consistently via numerical calculations. This can greatly
improve the current theoretical treatments in analyzing the spin pumping
effect, particularly, the non-steady non-equilibrium spin dynamics, which we
study in a future work [60].

METHODS
Evaluation of the polarization bubble. In the literatures, the static limit
$|q_{0}|\ll v_{F}q$ of various susceptibility functions is extensively
investigated since the dynamic exponent $z>1$ approaching the quantum
criticality. While, the opposite dynamic limit $v_{F}q\ll|q_{0}|$ is desired
for the purpose of spin pumping as dictated in Eq. (9). Moreover, the dynamic
polarization bubble $\Pi({\bm{q}}=0,q_{0})$ is proven to be non-vanishing for
the Ising nematic QCP of $d$-wave symmetry [47]. The peculiar frequency-
dependence persists in the non-Fermi liquid regime and can be probed by the
Raman scattering experiment [47, 61].

The charge and spin susceptibility are essentially same when the spin are
degenerated; yet, they are both different from the bosonic self-energy by the
$d$-wave factors at external vertex. The fully renormalized, single-patch
polarization bubble are evaluated using the renormalized fermion Green
function and a renormalized vertex function, which reads

$\Pi_{s}(q)=-\int\frac{d^{3}k}{(2\pi)^{3}}\Gamma(k;q)g_{s}(k)g_{s}(k+q).$ (31)

The vertex function in the charge/spin density channel $\Gamma(k;q)$ is
approximated as,

$\Gamma(k;q)\simeq 1+\delta\Gamma(k;q)$ (32)

and the ladder equation for $\delta\Gamma(k;q)$ is obtained as

$\delta\Gamma(k;q)=-\lambda^{2}\int\frac{d^{3}k^{\prime}}{(2\pi)^{3}}\big{[}1+\delta\Gamma(k^{\prime};q)\big{]}\\\
\times
g(k^{\prime})g(k^{\prime}+q)D(k-k^{\prime})d^{2}\Big{(}\frac{{\bm{k}}+{\bm{k}}^{\prime}}{2}\Big{)}.$
(33)

The main contribution of the integral comes from small momentum exchange
${\bm{k}}-{\bm{k}}^{\prime}$. We make a change of variable
${\bm{k}}^{\prime}\rightarrow{\bm{k}}+{\bm{p}}$ where ${\bm{k}}$ is fixed at a
given patch on the Fermi surface and the ${\bm{p}}$ is the small boson
momentum. The vertex correction function is simplified at
$q=({\bm{q}}=0,q_{0})$ as $\Gamma(k;q)=\Gamma(\hat{k};k_{0},q_{0})$. The boson
momentum can be decomposed as,

${\bm{p}}={\bm{p}}_{\parallel}+{\bm{p}}_{\perp},\ \
{\bm{p}}_{\perp}=(\hat{k}\cdot{\bm{p}})\hat{k},\ \
{\bm{p}}_{\parallel}=p_{\parallel}\ \hat{z}\times\hat{k},$ (34)

where $\hat{k}={\bm{k}}_{F}/k_{F}$ is the directional vector on the Fermi
surface and ${\bm{p}}_{\perp}$, ${\bm{p}}_{\parallel}$ are the radical and
tangential components respectively. The $d$-wave form factor in Eq. (36) is
transformed and expanded near the Fermi surface,

$d({\bm{k}}+{\bm{p}}/2)\simeq
d({\bm{k}}+{\bm{p}}_{\parallel}/2)+\frac{p_{\perp}}{2}\hat{k}\cdot\partial_{\bm{k}}d({\bm{k}}+{\bm{p}}_{\parallel}/2).$
(35)

The first term represents a circular Fermi surface with
$d({\bm{k}})=d(\hat{k})$ whose coordinate is labbeled by the tangential
component $\bm{p}_{\parallel}$. And, the second term stands for the radical
distortion of the Fermi surface. In terms of extracting leading order
frequency dependence for the vertex corrections, we can further ignore the
${\bm{p}}_{\parallel}$ dependence in Eq.(35). As a result, the integral
equation for the vertex correction is rewritten as,

$\delta\Gamma({\hat{k}};k_{0},q_{0})=-\frac{\lambda^{2}}{(2\pi)^{3}}\int
dp_{0}dp_{\parallel}dp_{\perp}\frac{|p_{\parallel}|}{|p_{\parallel}|^{3}+c_{B}|p_{0}|}\\\
\times\frac{\big{[}1+\delta\Gamma(\hat{k}^{\prime};p_{0}+k_{0},q_{0})\big{]}\big{[}d^{2}({\bm{k}})+p_{\perp}d({\bm{k}})d^{\prime}_{k}({\bm{k}})\big{]}}{\big{[}i\tilde{\Sigma}(k+p+q)-v_{F}p_{\perp}\big{]}\big{[}i\tilde{\Sigma}(k+p)-v_{F}p_{\perp}\big{]}},$
(36)

where we have used a modified self-energy for later convenience,

$\displaystyle i\tilde{\Sigma}(k)$ $\displaystyle\equiv ik_{0}-\Sigma(k),$
(37a) $\displaystyle\Sigma(k)$ $\displaystyle=-ic_{F}d^{2}({\bm{k}}){\rm
sgn}(k_{0})|k_{0}|^{2/3}.$ (37b)

We first carry out the $p_{\perp}$-integral which yields

$\displaystyle\int\frac{dp_{\perp}}{2\pi}\frac{d^{2}({\bm{k}})+p_{\perp}d({\bm{k}})d^{\prime}_{k}({\bm{k}})}{\big{[}i\tilde{\Sigma}(k+p+q)-v_{F}p_{\perp}\big{]}\big{[}i\tilde{\Sigma}(k+p)-v_{F}p_{\perp}\big{]}}$
$\displaystyle={}$
$\displaystyle\frac{1}{v_{F}}\big{[}\Theta(k_{0}+q_{0}+p_{0})-\Theta(k_{0}+p_{0})\big{]}$
$\displaystyle\times\frac{d^{2}({\bm{k}})+iv_{F}^{-1}d({\bm{k}})d^{\prime}_{k}({\bm{k}})\tilde{\Sigma}(k+p+q)}{\tilde{\Sigma}(k+p+q)-\tilde{\Sigma}(k+p)},$
(38)

where we have used the relation $\Theta\big{[}{\rm
Im}\tilde{\Sigma}(k)\big{]}=\Theta(k_{0})$. We split the vertex correction
into two terms taking account the contribution from the circular Fermi surface
and radical distortion in Eq.(35), which reads

$\delta\Gamma({\hat{k}};k_{0},q_{0})\simeq\delta\Gamma^{(0)}({\hat{k}};k_{0},q_{0})+\delta\Gamma^{(1)}({\hat{k}};k_{0},q_{0}).$
(39)

Accordingly, the ladder equation is solved term by term. We firstly assume a
circular Fermi surface in Eq. (36) and obtain

$\delta\Gamma^{(0)}({\hat{k}};k_{0},q_{0})=-\frac{\lambda^{2}d^{2}({\bm{k}})}{(2\pi)^{2}v_{F}}\int
dp_{\parallel}\frac{|p_{\parallel}|}{|p_{\parallel}|^{3}+c_{B}|p_{0}|}\\\
\times\int
dp_{0}\frac{1+\delta\Gamma^{(0)}(\hat{k}^{\prime};p_{0}+k_{0},q_{0})}{\tilde{\Sigma}(k+p+q)-\tilde{\Sigma}(k+p)}.$
(40)

$\delta\Gamma^{(0)}({\hat{k}};k_{0},q_{0})$ satisfies the Ward identity
underlying the charge/spin conservation

$1+\delta\Gamma^{(0)}(k;q)=\frac{\tilde{\Sigma}(k+q)-\tilde{\Sigma}(k)}{q_{0}}.$
(41)

This can be readily checked by deriving the self-energy difference

$\displaystyle\Delta\Sigma\equiv{}$
$\displaystyle\Sigma(k+q)-\Sigma(k)=\frac{\lambda^{2}}{(2\pi)^{3}}$
$\displaystyle\times\int
d^{3}p\big{[}g(k+p+q)-g(k+p)\big{]}D(p)d^{2}\Big{(}{\bm{k}}+\frac{\bm{p}}{2}\Big{)}$
$\displaystyle\simeq{}$
$\displaystyle\frac{i\lambda^{2}d^{2}({\bm{k}})}{(2\pi)^{2}v_{F}}\int
dp_{\parallel}\frac{|p_{\parallel}|}{|p_{\parallel}|^{3}+c_{B}|p_{0}|}$
$\displaystyle\times\int dp_{0}\big{[}{\rm sgn}(k_{0}+p_{0}+q_{0})-{\rm
sgn}(k_{0}+p_{0})\big{]}$ $\displaystyle={}$
$\displaystyle\frac{i\lambda^{2}d^{2}({\bm{k}})}{(2\pi)^{2}v_{F}}\int_{-q_{0}}^{0}dp_{0}\int
dp_{\parallel}\frac{|p_{\parallel}|}{|p_{\parallel}|^{3}+c_{B}|p_{0}-k_{0}|}.$
(42)

The validity of the solution in Eq. (41) is verified self-consistently by
substituting into Eq. (40) and comparing with Eq. (Spin pumping effect in non-
Fermi liquid metals).

Next, we include the radical distortion terms in Eq. (35) and invoke an
additional vertex correction term $\delta\Gamma^{(1)}$ listed in Eq. (39)

$\displaystyle\delta\Gamma^{(0)}({\hat{k}};k_{0},q_{0})+\delta\Gamma^{(1)}({\hat{k}};k_{0},q_{0})$
$\displaystyle={}$
$\displaystyle\frac{-\lambda^{2}}{(2\pi)^{2}v_{F}}\int_{-q_{0}}^{0}dp_{0}dp_{\parallel}\frac{|p_{\parallel}|}{|p_{\parallel}|^{3}+c_{B}|p_{0}|}$
$\displaystyle\times\frac{1+\delta\Gamma^{(0)}(\hat{k}^{\prime};p_{0}+k_{0},q_{0})}{\tilde{\Sigma}(k+p+q)-\tilde{\Sigma}(k+p)}$
$\displaystyle\times\big{[}d^{2}({\bm{k}})+iv_{F}^{-1}d({\bm{k}})d^{\prime}_{k}({\bm{k}})\tilde{\Sigma}(k+p+q)\big{]}.$
(43)

This expression is a refined version of vertex corrections compared to
Eq.(40). The additional term $\delta\Gamma^{(1)}$ modifies both sides of
Eq.(41), yet, leaving the Ward identity unchanged,

$\displaystyle\Gamma(k;q)$
$\displaystyle=1+\delta\Gamma^{(0)}(k;q)+\delta\Gamma^{(1)}(k;q)$
$\displaystyle=\frac{\tilde{\Sigma}(k+q)-\tilde{\Sigma}(k)}{q_{0}\big{[}1-\delta\Gamma^{(1)}(k;q)\big{]}}.$
(44)

Explicitly, we have

$\displaystyle\delta\Gamma_{0}({\hat{k}};k_{0},q_{0})+\delta\Gamma_{1}({\hat{k}};k_{0},q_{0})$
$\displaystyle={}$ $\displaystyle\int_{-q_{0}}^{0}\frac{dp_{0}}{q_{0}}\int
dp_{\parallel}\frac{|p_{\parallel}|}{|p_{\parallel}|^{3}+c_{B}|p_{0}|}\frac{-\lambda^{2}}{(2\pi)^{2}v_{F}}$
$\displaystyle\times\big{[}d^{2}({\bm{k}})+iv_{F}^{-1}d({\bm{k}})d^{\prime}_{k}({\bm{k}})\tilde{\Sigma}(k+p+q)\big{]}$
(45)

The self-energy difference can be written in a similar form $\Delta\Sigma$ as
Eq.(Spin pumping effect in non-Fermi liquid metals) accompanied by an
additional term $\delta\Sigma$

$\displaystyle\Sigma(k+$ $\displaystyle
q)-\Sigma(k)=\Delta\Sigma+\delta\Sigma,$ (46a) $\displaystyle\Delta\Sigma={}$
$\displaystyle\frac{i\lambda^{2}}{(2\pi)^{2}v_{F}}\int_{-q_{0}}^{0}dp_{0}\int
dp_{\parallel}\frac{|p_{\parallel}|}{|p_{\parallel}|^{3}+c_{B}|p_{0}|}$
$\displaystyle\times\big{[}d^{2}({\bm{k}})+iv_{F}^{-1}d({\bm{k}})d^{\prime}_{k}({\bm{k}})\tilde{\Sigma}(k+p+q)\big{]},$
(46b) $\displaystyle\delta\Sigma={}$
$\displaystyle-\frac{\lambda^{2}d({\bm{k}})d^{\prime}_{k}({\bm{k}})}{(2\pi)^{2}v_{F}^{2}}\int
dp_{0}\int
dp_{\parallel}\frac{|p_{\parallel}|}{|p_{\parallel}|^{3}+c_{B}|p_{0}|}$
$\displaystyle\times{\rm
sgn}(k_{0}+p_{0})\big{[}\tilde{\Sigma}(k+p+q)-\tilde{\Sigma}(k+p)\big{]}.$
(46c)

Comparing Eq. (Spin pumping effect in non-Fermi liquid metals) and Eq. (46),
we obtain

$\displaystyle\delta\Gamma_{1}({\hat{k}};k_{0},q_{0})=-i\delta\Sigma/q_{0}$
$\displaystyle={}$
$\displaystyle\frac{i\lambda^{2}d({\bm{k}})d^{\prime}_{k}({\bm{k}})}{(2\pi)^{2}v_{F}^{2}}\int_{-\infty}^{+\infty}\frac{dp_{0}}{q_{0}}\int
dp_{\parallel}\frac{|p_{\parallel}|}{|p_{\parallel}|^{3}+c_{B}|p_{0}|}$
$\displaystyle\times{\rm
sgn}(p_{0}+k_{0})\big{[}\tilde{\Sigma}(p+k+q)-\tilde{\Sigma}(p+k)\big{]}$
$\displaystyle\simeq{}$
$\displaystyle\frac{\lambda^{2}d^{3}({\bm{k}})d^{\prime}_{k}({\bm{k}})c_{F}}{(2\pi)^{2}v_{F}^{2}}\int
dp_{\parallel}\frac{|p_{\parallel}|}{|p_{\parallel}|^{3}+c_{B}|p_{0}-k_{0}|}\int_{-\infty}^{+\infty}\frac{dp_{0}}{q_{0}}$
$\displaystyle\times{\rm sgn}(p_{0})\big{[}{\rm
sgn}(p_{0}+q_{0})|p_{0}+q_{0}|^{2/3}-{\rm sgn}(p_{0})|p_{0}|^{2/3}\big{]}$
$\displaystyle={}$
$\displaystyle\frac{\lambda^{2}d^{3}({\bm{k}})d^{\prime}_{k}({\bm{k}})c_{F}}{(2\pi)^{2}v_{F}^{2}}\frac{4\pi}{3\sqrt{3}c_{B}^{1/3}}\int_{-\infty}^{+\infty}\frac{dp_{0}}{q_{0}}$
$\displaystyle\times\frac{{\rm sgn}(p_{0}){\rm
sgn}(p_{0}+q_{0})|p_{0}+q_{0}|^{2/3}-|p_{0}|^{2/3}}{|p_{0}-k_{0}|^{1/3}}$
$\displaystyle={}$
$\displaystyle\lambda^{2}\Big{|}\frac{q_{0}}{q_{0}^{c}}\Big{|}^{1/3}F_{1}(\hat{k},k_{0}/q_{0}).$
(47)

where $q_{0}^{c}=\big{(}3\sqrt{3}\pi v_{F}^{2}c_{B}^{1/3}/c_{F}\big{)}^{3}$
and the dimensionless function is defined as

$F_{1}(\hat{k},y)\equiv
d^{3}({\bm{k}})d^{\prime}_{k}({\bm{k}})\int_{0}^{+\infty}dx\Bigg{\\{}\frac{|1+x|^{2/3}-|x|^{2/3}}{|x-y|^{1/3}}\\\
-\frac{|1-x|^{2/3}-|x|^{2/3}}{|x+y|^{1/3}}\Bigg{\\}}.$ (48)

With the vertex function at hand, we evaluate polarization bubble in Eq. (31).
The main contributions are from low and high frequency set apart by a cutoff
$\omega_{u}$

$\Pi_{s}(0,q_{0})=\Pi^{\rm low}_{s}(0,q_{0})+\Pi^{\rm high}_{s}(0,q_{0}).$
(49)

It is shown that the high and low energy contributions from
$\delta\Gamma^{(0)}$ cancel each other leading to a vanishing result [37].
Here, we show that $\delta\Gamma^{(1)}$ endows the polarization bubble with a
characteristic frequency dependence. In the low frequency regime
$\tilde{\Sigma}\ll\omega_{u}$, we can simply set
$\omega_{u}\rightarrow+\infty$

$\displaystyle\Pi^{\rm low}_{s}({\bm{q}}=0,q_{0})$ $\displaystyle\simeq{}$
$\displaystyle-\frac{1}{2\pi
v_{F}}\int_{-q_{0}}^{0}dk_{0}\int\frac{d\phi}{2\pi}\frac{1+\delta\Gamma_{0}(\phi;k_{0},q_{0})}{\tilde{\Sigma}(p+q)-\tilde{\Sigma}(p)}$
$\displaystyle\times\big{[}1+\delta\Gamma_{1}(\phi;k_{0},q_{0})\big{]}$
$\displaystyle={}$ $\displaystyle-\frac{1}{2\pi
v_{F}}\int_{-q_{0}}^{0}\frac{dk_{0}}{q_{0}}\int\frac{d\phi}{2\pi}\big{[}1+\delta\Gamma_{1}(\phi;k_{0},q_{0})\big{]}$
$\displaystyle={}$ $\displaystyle-\frac{1}{2\pi v_{F}}-\frac{1}{2\pi
v_{F}}\int_{-q_{0}}^{0}\frac{dk_{0}}{q_{0}}\int\frac{d\phi}{2\pi}\delta\Gamma_{1}(\phi;k_{0},q_{0}),$
(50)

where the directional vector ${\hat{k}}$ is replaced by a polar angle $\phi$.
In the high frequency regime $\tilde{\Sigma}(q)\gg\omega_{u}$, we keep a
finite cutoff $\omega_{u}$. And, the self-energy and vertex function are
approximated as

$\tilde{\Sigma}(q)\sim q_{0},\ \ \Gamma(k;q)\sim 1.$ (51)

which yields

$\displaystyle\Pi^{\rm high}_{s}({\bm{q}}=0,q_{0})$ $\displaystyle={}$
$\displaystyle-\frac{1}{(2\pi)^{2}v_{F}}\int
dk_{0}\int\frac{d\phi}{2\pi}\frac{\Gamma(k;q)}{i\big{[}\tilde{\Sigma}(k+q)-\tilde{\Sigma}(k)\big{]}}$
$\displaystyle\times\Big{\\{}-2i\arctan\frac{\omega_{u}\big{[}\tilde{\Sigma}(k+q)-\tilde{\Sigma}(k)\big{]}}{\omega_{u}^{2}+\tilde{\Sigma}^{2}(k)}\Big{\\}}$
$\displaystyle\simeq{}$ $\displaystyle 2\frac{1}{(2\pi)^{2}v_{F}}\int
dk_{0}\int\frac{d\phi}{2\pi}\
\Gamma(k;q)\frac{\omega_{u}}{\omega_{u}^{2}+\tilde{\Sigma}^{2}(k)}$
$\displaystyle\simeq{}$ $\displaystyle 2\frac{\lambda^{2}}{2\pi
v_{F}}\int_{-\infty}^{+\infty}\frac{dk_{0}}{2\pi}\frac{\omega_{u}}{\omega_{u}^{2}+k_{0}^{2}}$
$\displaystyle={}$ $\displaystyle\frac{\lambda^{2}}{2\pi v_{F}}.$ (52)

Collecting terms from Eqs. (Spin pumping effect in non-Fermi liquid metals)
and (Spin pumping effect in non-Fermi liquid metals), we arrive at

$\displaystyle\Pi_{s}({\bm{q}}=0,q_{0})$ $\displaystyle=-\frac{1}{2\pi
v_{F}}\int_{-q_{0}}^{0}\frac{dk_{0}}{q_{0}}\int\frac{d\phi_{k}}{2\pi}\delta\Gamma_{1}(\phi_{k};p_{0},q_{0})$
$\displaystyle=-C^{\prime}\frac{\lambda^{2}}{2\pi
v_{F}}\Big{|}\frac{q_{0}}{q_{0}^{c}}\Big{|}^{1/3}$ (53)

where $\phi_{k}$ labels the polar coordinate of a circular Fermi surface. The
$d$-wave factor can take a form $d({\bm{k}})=\cos(2\phi_{k})$ and
$d^{\prime}_{k}({\bm{k}})$ dictates the deviation from the perfect circle
leading to a non-zero constant $C^{\prime}$,

$C^{\prime}\equiv\int_{-1}^{0}dx\int\frac{d\phi_{k}}{2\pi}F_{1}(\phi_{k},x)\propto\int_{0}^{\frac{\pi}{2}}\phi_{k}d^{3}(\phi_{k})d^{\prime}_{k}(\phi_{k})\neq
0.$ (54)

Evaluation of the quasiparticle weight. The electron quasiparticle weight is
evaluated by using the analytic expressions of the fermion self-energy at both
zero and finite temperatures. The imaginary part of the retarded self-energy
at zero and finite temperatures are written in Eq. (12) and Eq. (22),
respectively, which we re-write here as a step-wise function

${\rm
Im}\Sigma_{F}({\bm{k}},\omega)=\begin{cases}-\gamma_{{\bm{k}}_{F}}(T),&|\omega|<T;\\\
-c_{F}d^{2}_{{\bm{k}}_{F}}|\omega|^{2/3},&T<|\omega|<\omega_{c}.\end{cases}$
(55)

Then, we apply the Kramer-Kronig relation to obtain the real part

${\rm
Re}\Sigma_{F}({\bm{k}},\omega)=\frac{1}{\pi}\int_{-\omega_{c}}^{+\omega_{c}}d\omega^{\prime}\frac{{\rm
Im}\Sigma_{F}({\bm{k}},\omega^{\prime})}{\omega^{\prime}-\omega},$ (56)

where we set a UV-cutoff for the frequency integral $|\omega|\leq\omega_{c}$.
Explicitly, we have

$\displaystyle{\rm Re}\Sigma_{F}({\bm{k}}_{F},\omega)$ $\displaystyle={}$
$\displaystyle\frac{1}{\pi}\int_{-T}^{T}d\omega^{\prime}\frac{-\gamma_{{\bm{k}}_{F}}(T)}{\omega^{\prime}-\omega}-\frac{d^{2}_{{\bm{k}}_{F}}}{\pi}\Big{\\{}\int_{-\omega_{c}}^{-T}+\int_{T}^{\omega_{c}}\Big{\\}}d\omega^{\prime}\frac{|\omega^{\prime}|^{2/3}}{\omega^{\prime}-\omega}$
$\displaystyle={}$
$\displaystyle-\frac{2}{\pi}\omega\gamma_{{\bm{k}}_{F}}(T)\int_{0}^{T}d\omega^{\prime}\frac{1}{{\omega^{\prime}}^{2}-\omega^{2}}$
$\displaystyle-d^{2}_{{\bm{k}}_{F}}\frac{2}{\pi}\omega\int_{T}^{\omega_{c}}d\omega^{\prime}\frac{{\omega^{\prime}}^{2/3}}{{\omega^{\prime}}^{2}-\omega^{2}}$
$\displaystyle\simeq{}$
$\displaystyle\frac{2}{\pi}\omega\frac{\gamma_{{\bm{k}}_{F}}(T)}{T}-\frac{6}{\pi}d^{2}_{{\bm{k}}_{F}}\omega(T^{-1/3}-\omega_{c}^{-1/3})$
$\displaystyle={}$
$\displaystyle\frac{2}{\pi}\omega\frac{\gamma_{{\bm{k}}_{F}}(T)}{T}-\frac{6d^{2}_{{\bm{k}}_{F}}}{\pi}\omega
T^{-1/3}+\frac{6d^{2}_{{\bm{k}}_{F}}}{\pi\omega_{c}^{1/3}}\omega.$ (57)

where the principal value of the first term is calculated and approximated in
the finite temperature regime $|\omega|<T$

$\int_{0}^{T}\frac{d\Omega}{\Omega^{2}-\omega^{2}}=\frac{1}{|\omega|}\frac{1}{2}\ln\frac{1-|\omega|/T}{1+|\omega|/T}\simeq-\frac{1}{|\omega|}\frac{|\omega|}{T}=-\frac{1}{T}.$
(58)

And, the second and third terms are obtained from the zero temperature regime
$T<|\omega|<\omega_{c}$

$\displaystyle\int_{T}^{\omega_{c}}d\omega^{\prime}\frac{{\omega^{\prime}}^{2/3}}{x^{2}-{\omega^{\prime}}^{2}}$
$\displaystyle={}$
$\displaystyle\frac{1}{4|\omega|^{1/3}}\Bigg{\\{}2\sqrt{3}\tan^{-1}\frac{1+2(\omega^{\prime}/|\omega|)^{1/3}}{\sqrt{3}}$
$\displaystyle-2\sqrt{3}\tan^{-1}\frac{1-2(\omega^{\prime}/|\omega|)^{1/3}}{\sqrt{3}}$
$\displaystyle+2\ln\Big{|}\frac{1-{\omega^{\prime}}^{1/3}}{1+{\omega^{\prime}}^{1/3}}\Big{|}+\ln\Big{|}\frac{1-{\omega^{\prime}}^{1/3}+{\omega^{\prime}}^{2/3}}{1+{\omega^{\prime}}^{1/3}+{\omega^{\prime}}^{2/3}}\Big{|}\Bigg{\\}}_{T}^{\omega_{c}}$
$\displaystyle\simeq{}$
$\displaystyle\frac{1}{4|\omega|^{1/3}}\big{\\{}-12(|\omega|/\omega_{c})^{1/3}+12(|\omega|/T)^{1/3}\big{\\}}$
$\displaystyle={}$ $\displaystyle 3\big{(}T^{-1/3}-\omega_{c}^{-1/3}\big{)}.$
(59)

The leading order contribution to the real part of retarded self-energy is
from the first term in Eq. (Spin pumping effect in non-Fermi liquid metals)
for $T<T_{1}$

${\rm
Re}\Sigma_{F}({\bm{k}}_{F},\omega)\simeq\frac{2}{\pi}\omega\frac{\gamma_{{\bm{k}}_{F}}(T)}{T}=\frac{2d_{{\bm{k}}_{F}}^{2}}{\pi}\omega\xi(T)^{-1}.$
(60)

The first term dominates over the second term in Eq. (Spin pumping effect in
non-Fermi liquid metals) in a temperature regime $T<T_{1}$ set by the
condition $\ln(T/\epsilon_{F})<C(T/\epsilon_{F})^{-1/3}$ with
$C=\lambda^{2}/(12v_{F}\epsilon_{F}^{1/3}\sqrt{a})$. It is reasonable to
assume that the finite temperature regime $T<T_{0}$, in which the self-
consistent solutions are valid, fits into a hierarchy of temperature scales
$T<T_{0}<T_{1}$.

The electron/fermion quasiparticle weight at finite temperatures
$|\omega|<T<T_{0}<\omega_{c}$ can be readily obtained as

$\displaystyle Z_{{\bm{k}}_{F}}(\omega=0;T)$
$\displaystyle=\big{[}1-\partial_{\omega}{\rm
Re}\Sigma_{F}({\bm{k}}_{F},\omega;T)\big{]}^{-1}_{\omega\rightarrow 0}$
$\displaystyle\simeq d^{2}_{{\bm{k}}_{F}}\xi(T)^{-1}.$ (61)

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ACKNOWLEDGEMENTS
We thank Fu-Chun Zhang for valuable discussions and financial supports. This
work is supported by China’s Ministry of Science and Technology (Grant No.
2022YFA1403902) and National Natural Science Foundation of China(NSFC) (Grant
No. 11920101005). L.Z. is supported by the National Key R&D Program of China
(Grant No. 2018YFA0305800), the NSFC (Grant No. 12174387), the Strategic
Priority Research Program of CAS (Grant No. XDB28000000), and the CAS Youth
Innovation Promotion Association. M.M. is supported by the Priority Program of
Chinese Academy of Sciences under Grant No. XDB28000000, and by JSPS KAKENHI
for Grants (Nos. 21H01800, and 21H04565, and 23H01839) from MEXT, Japan.

AUTHOR CONTRIBUTIONS
X.T.Z. conceived this idea. X.T.Z., L.Z. and M.M. supervised the project.
X.T.Z. and Y.O. performed the theoretical calculation. X.T.Z., X.P.Y, L.Z. and
M.M. wrote the manuscript.

COMPETING INTERESTS
The authors declare no competing interests.