# Flavors of Magnetic Noise in Quantum Materials

Shu Zhang<EMAIL_ADDRESS>Department of Physics and Astronomy,
University of California, Los Angeles, California 90095, USA  Yaroslav
Tserkovnyak  Department of Physics and Astronomy, University of California,
Los Angeles, California 90095, USA

###### Abstract

The complexity of electronic band structures in quantum materials offers new
charge-neutral degrees of freedom stable for transport, a promising example
being the valley (axial) degree of freedom in Weyl semimetals (WSMs). A
noninvasive probe of their transport properties is possible by exploiting the
frequency dependence of the magnetic noise generated in the vicinity of the
material. In this work, we investigate the magnetic noise generically
associated with diffusive transport using a systematic Langevin approach.
Taking a minimal model of magnetic WSMs for demonstration, we show that
thermal fluctuations of the charge current, the valley current, and the
magnetic order can give rise to magnetic noise with distinctively different
spectral characters, which provide a theoretical guidance to separate their
contributions. Our approach is extendable to the study of magnetic noise and
its spectral features arising from other transport degrees of freedom in
quantum materials.

Introduction.—Many recently discovered novel quantum materials are featured by
the complexity of their electronic band structures, as a result of spin-orbit
coupling Hasan and Kane (2010), magnetic order Watanabe  _et al._ (2018),
twist engineering Zou  _et al._ (2018), etc. These structures may exhibit new
degrees of freedom stable for transport, in addition to charge and spin, due
to the protection either by symmetry or topology, which can be explored for
the next-generation information devices. One prominent example is the valley
degree of freedom present in some hexagonal two-dimensional semiconductors or
Weyl semimetals (WSMs), the possibility of manipulating which gives rise to
the field of valleytronics Schaibley  _et al._ (2016).

Transport is generally noisy. As the Johnson-Nyquist noise (thermally excited
electric currents) is important in electronic devices, understanding the
generation of noise from these new degrees of freedom is of practical
relevance in spintronic or valleytronic devices. At the same time, the
electromagnetic noise emitted by a material into its environment encodes rich
information about its intrinsic excitation dynamics and transport properties.
For example, the magnetic noise in the vicinity of a conductor is directly
related to its impedance Johnson (1928); Varpula and Poutanen (1984). The
recent development of magnetic noise spectroscopy using single qubits,
especially the nitrogen-vacancy (NV) centers in diamond Taylor  _et al._
(2008), has provided a nanoscale probe to access such information
noninvasively, and with high frequency resolution Kolkowitz  _et al._ (2015);
Agarwal  _et al._ (2017); Ariyaratne  _et al._ (2018). NVs have also turned
out to be useful in the study of magnetic insulators Van der Sar  _et al._
(2015); Du  _et al._ (2017); Casola  _et al._ (2018); Chatterjee  _et al._
(2019) by probing the magnetic noise generated by spin excitations. While
thermodynamic valley fluctuations have been accessed by optical methods Goryca
_et al._ (2019), the noise associated with valley transport in low-frequency
regimes is so far rarely explored.

In this work, we offer a qualitative perspective to the study of the magnetic
noise in quantum materials, which can be flavorful due to the presence of
various transport degrees of freedom, focusing on their generic diffusive
aspect. To this end, we take an example of a magnetic WSM, which naturally
involves three sources of noise, namely, charge, spin, and valley.
Intriguingly, each flavor can contribute a distinct spectral character, as
shown in Fig. 1. Our perspective can be extended to consider other pseudospin
degrees of freedom in general and may inspire future experimental work in the
NV probe of quantum materials in light of its advantage in frequency
resolution in the GHz regime.

Figure 1: An illustration of the three flavors of magnetic noise in a magnetic
WSM and their different spectral characters.

WSMs are a family of topological quantum materials promising for valleytronic
applications, because the band crossing at Weyl points is topologically
protected, and the valley relaxation time can be very long in a clean system
Hosur and Qi (2013). Magnetic WSMs, such as those in magnetic Heusler
compounds Wang  _et al._ (2016); Kübler and Felser (2016); Manna  _et al._
(2018), allow the existence of a single pair of energy-degenerate Weyl valleys
Wang  _et al._ (2019), and are thus ideal for the study of valley transport.
The interplay between charge, valley, and spin Zhang  _et al._ (2019);
Tserkovnyak (2021) also makes them attractive for spintronics. The detection
of valley transport often relies on the conversion from valley excitations to
optic or electric signals, for instance, with the help of the chiral anomaly
effect in a nonlocal geometry Parameswaran  _et al._ (2014). In magnetic WSMs,
however, this is not totally unambiguous due to the presence of spin
excitations, which usually have a long diffusion length as well. The magnetic
noise spectroscopy can serve as a direct probe of the intrinsic transport
properties in the absence of external perturbations and electric contacts. As
we will show, it is possible to distinguish the valley and spin contributions
to the magnetic noise based on their spectral characteristics.

Main concepts.—We first briefly summarize our conceptual understanding of why
the three flavors contribute differently, even all under a diffusive
treatment. Focusing on the scenario with a nonvanishing carrier density, where
the electric charge density fluctuations are screened by Coulomb interactions,
transverse fluctuations of the charge current dominate the charge channel. In
contrast, longitudinal fluctuations of the valley current are important due to
its charge neutrality and hence the absence of screening of the axial charge
density. In our model (see below), the Weyl nodes are induced by the broken
time-reversal symmetry associated with the magnetic order, which dictates the
response of valley currents to magnetic fields and thus the generation of
magnetic noise by valley fluctuations. Consequently, only the longitudinal
component of the valley current generates magnetic fields in the environment:
The valley current $\mathbf{j}^{v}$ behaves as a magnetization, and determines
a “magnetic charge” distribution in the bulk
$\rho_{M}\propto-\bm{\nabla}\cdot\mathbf{j}^{v}$, while that on the surface
vanishes due to the boundary condition
$\sigma_{M}\propto\mathbf{n}\cdot\mathbf{j}^{v}=0$. In the spin channel,
however, the spin density (rather than the spin current) plays such a role.
See Fig. 1 for comparison.

Figure 2: Quantum-impurity relaxometry of a magnetic Weyl semimetal. A three-
dimensional sample of a magnetic WSM with a band structure given by Eq. (1)
has a magnetic order $\mathbf{M}$ in the $z$ direction and a surface lying in
the $xy$ plane. An NV center is placed at distance $d$ from the surface, with
a tunable resonance frequency $\omega$.

Model.—The following minimal model is considered for a magnetic WSM with four
bands Burkov  _et al._ (2011):

$\displaystyle
H=v\tau_{z}\otimes(\bm{\sigma}\cdot\hbar\mathbf{k})+\Delta\tau_{x}+J\bm{\sigma}\cdot\mathbf{M},$
(1)

where $\bm{\sigma}$ and $\bm{\tau}$ are vectors of the Pauli matrices in the
spin and valley spaces, respectively, $v$ is the Fermi velocity, $\Delta$ is a
Dirac mass, and $J$ is the magnetic exchange between the itinerant electrons
and the magnetic order $\mathbf{M}$. When $|J|M>|\Delta|$, this model realizes
a single pair of Weyl nodes with opposite chiralities separated in the
momentum space. Here after, $|J|M\gg|\Delta|$ is assumed to approximately
conserve the valley index Tserkovnyak (2021). We also assume a small but
finite Fermi surface, and thus a finite carrier density at the valleys, i.e.,
a Weyl metal.

Defining $\mathbf{j}^{+(-)}$ as the number flux density operator of electrons
belonging to the valley with positive (negative) chirality, the valley current
becomes

$\displaystyle\mathbf{j}^{v}=\mathbf{j}^{+}-\mathbf{j}^{-}=\tau_{z}\partial_{\mathbf{k}}H/\hbar=v\bm{\sigma},$
(2)

at a single-particle level. The assumptions about the spin and parity
symmetries Tserkovnyak (2021) in our model (1), therefore, establish a
proportionality between the valley current and the itinerant spin density at
the Fermi surface. Since a Zeeman term $\bm{\sigma}\cdot\mathbf{B}$ is
allowed, the valley current can directly couple to magnetic fields, with its
fluctuations generating magnetic noise.

We employ a simple setup of magnetic noise measurement using an NV center
placed at a nanoscale distance $d$ from a flat surface of the three-
dimensional material, as shown in Fig. 2. We choose the magnetic order
$\mathbf{M}\parallel\hat{\mathbf{z}}$ and the surface plane to be the $xy$
plane. With this geometry, we focus on the contributions from bulk transport,
separated from the Fermi arcs. The central object of our study is the magnetic
noise tensor in the frequency domain at the position of the NV center
$\mathbf{r}_{\text{NV}}$, expressed in the symmetrized correlation functions
of the magnetic field operators,

$\displaystyle\mathcal{B}_{ii^{\prime}}(\omega)=\frac{1}{2}\int dt\,e^{i\omega
t}\langle\\{B_{i}(\mathbf{r}_{\text{NV}},t),B_{i^{\prime}}(\mathbf{r}_{\text{NV}},0)\\}\rangle.$
(3)

Its components determine the NV relaxation rate depending on the NV
orientation Van der Sar  _et al._ (2015); Flebus and Tserkovnyak (2018).
Keeping the typical GHz frequency of NV centers in mind, we focus on the
magnetostatic limit, where the wavelength $\lambda$ of the fluctuating
electromagnetic field and the skin depth $\lambda_{s}$ of the material are
both much larger than $d$. This frequency also puts us in the classical limit
($\hbar\omega\ll k_{B}T$), with the exception only for very low temperatures.
Accordingly, we develop a simplified but systematic treatment of the magnetic-
noise generation from charge, valley, and spin degrees of freedom, which we
now turn to.

Charge.—To introduce our approach, we first reproduce the established results
for the Johnson-Nyquist noise. The Langevin dynamics Chaikin and Lubensky
(1995); Sinitsyn and Pershin (2016) of the electron current (the number flux
density) $\mathbf{j}^{c}=-\sigma\bm{\nabla}\mu^{c}+\bm{\epsilon}^{c}$ is
considered, where $\sigma$ is the conductivity (neglecting the Hall effect
111A small Hall conductivity would not affect our result since magnetic noise
is governed by dissipative properties. In the presence of a large Hall angle,
the theory can be generalized using a full conductivity tensor.), $\mu^{c}$ is
the electrochemical potential, and $\bm{\epsilon}^{c}$ is a Gaussian white
noise with $\langle\epsilon^{c}_{i}(\mathbf{r},t)\rangle=0$ and
$\langle\epsilon^{c}_{i}(\mathbf{r},t)\epsilon^{c}_{i^{\prime}}(\mathbf{r}^{\prime},t^{\prime})\rangle=2\sigma
k_{B}T\delta_{ii^{\prime}}\delta(\mathbf{r}-\mathbf{r}^{\prime})\delta(t-t^{\prime})$.
The coefficient in the correlation follows from the equipartition theorem
Nyquist (1928), and is consistent with the fluctuation dissipation theorem
Kubo (1966). This treatment assumes the diffusive regime, focusing on dynamics
on the length scale much larger than the mean free path of the electrons, and
at frequencies much lower than the momentum relaxation rate. Focusing on the
transverse components of the charge current,
$\bm{\nabla}\cdot\mathbf{j}^{c}=0$, we have the following differential
equation for the electrochemical potential $\mu^{c}$:

$\displaystyle\sigma\bm{\nabla}^{2}\mu^{c}=\bm{\nabla}\cdot\bm{\epsilon}^{c}.$
(4)

The charge current does not flow across the surface
$j^{c}_{z}(\mathbf{r},t)\Big{|}_{z=0}=0$, giving the Neumann boundary
condition
$\sigma\partial_{z}\mu^{c}(\mathbf{r},t)\Big{|}_{z=0}=\epsilon^{c}_{z}(\mathbf{r},t)\Big{|}_{z=0}$.
Solving the differential equation yields

$\displaystyle\mathbf{j}^{c}(\mathbf{r},t)=\bm{\epsilon}^{c}(\mathbf{r},t)+\int_{\Omega}d^{3}\mathbf{r}^{\prime}\,\bm{\epsilon}^{c}(\mathbf{r}^{\prime},t)\cdot\bm{\nabla}_{\mathbf{r}^{\prime}}\bm{\nabla}_{\mathbf{r}}\mathcal{G}_{L}(\mathbf{r},\mathbf{r}^{\prime}),$
(5)

where $\mathcal{G}_{L}(\mathbf{r},\mathbf{r}^{\prime})$ is the Green function
for the Laplacian compatible with the Neumann boundary condition sup .Bass and
Pratt Jr. (2007)

The magnetic field generated by the charge current is given by the Biot-Savart
law,

$\displaystyle\mathbf{B}^{c}(\mathbf{r},t)$
$\displaystyle=-\frac{e}{c}\int_{\Omega}d^{3}\mathbf{r}^{\prime}\,\frac{\mathbf{j}^{c}(\mathbf{r}^{\prime},t)\times(\mathbf{r}-\mathbf{r}^{\prime})}{|\mathbf{r}-\mathbf{r}^{\prime}|^{3}},$
(6)

where $c$ is the speed of light, and we have taken the charge carriers to have
charge $-e$. Inserting the magnetic field (6) into the definition (3), we
obtain the magnetic noise due to charge fluctuations,

$\displaystyle\mathcal{B}^{c}_{ii^{\prime}}(\omega)$ $\displaystyle=\frac{\pi
e^{2}k_{B}T\sigma}{c^{2}d}\Lambda_{ii^{\prime}},$ (7)

where $\Lambda=\text{diag}(1/2,1/2,1)$. This reproduces exactly the result
from the formulation in terms of transmission and reflection of
electromagnetic fields at a metal surface Ford and Weber (1984); Henkel  _et
al._ (1999); Henkel (2005), in the magnetostatic limit.

Valley.—We next apply this approach to the valley degree of freedom. Different
from the charge current, the valley fluctuations are neutral and thus
compressible, which, as we have pointed out, turns out to be crucial for
generating magnetic noise. The weak intervalley scattering allows us to
consider electron conservation in each valley separately: For the valley $p$
($p=\pm$),

$\displaystyle\partial_{t}\rho^{p}+\bm{\nabla}\cdot\mathbf{j}^{p}=0,$ (8)

where $\rho^{p}=(\nu/2)\mu^{p}$ and
$\mathbf{j}^{p}=-(\sigma/2)\bm{\nabla}\mu^{p}+\bm{\epsilon}^{p}$. The total
density of states $\nu$ at the Fermi surface includes both valleys, consistent
with the convention of the total charge density $\rho^{c}=\rho^{+}+\rho^{-}$
and the average electrochemical potential $\mu^{c}=(\mu^{+}+\mu^{-})/2$. The
Langevin noise here obeys
$\langle\epsilon^{p}_{i}(\mathbf{r},t)\epsilon^{p^{\prime}}_{i^{\prime}}(\mathbf{r}^{\prime},t^{\prime})\rangle=\sigma
k_{B}T\delta_{pp^{\prime}}\delta_{ii^{\prime}}\delta(\mathbf{r}-\mathbf{r}^{\prime})\delta(t-t^{\prime})$,
supposing fluctuations in the two valleys are uncorrelated. We therefore
arrive at the stochastic diffusion equation

$\displaystyle\partial_{t}\mu^{p}-D\bm{\nabla}^{2}\mu^{p}=-\frac{2}{\nu}\bm{\nabla}\cdot\bm{\epsilon}^{p},$
(9)

where $D=\sigma/\nu$ is the diffusion coefficient, as given by the Einstein
relation. Under the boundary condition
$j_{z}^{p}(\mathbf{r},t)\Big{|}_{z=0}=0$, we solve for $\mu^{p}$ to obtain the
currents

$\displaystyle\mathbf{j}^{p}(\mathbf{r},t)$
$\displaystyle=\bm{\epsilon}^{p}(\mathbf{r},t)-D\int_{\Omega}d^{3}\mathbf{r}^{\prime}\int_{-\infty}^{t}dt^{\prime}\,$
(10)
$\displaystyle\qquad\qquad\quad\bm{\epsilon}^{p}(\mathbf{r}^{\prime},t^{\prime})\cdot\bm{\nabla}_{\mathbf{r}^{\prime}}\bm{\nabla}_{\mathbf{r}}\mathcal{G}_{D}(\mathbf{r},\mathbf{r}^{\prime};t,t^{\prime}),$

where $\mathcal{G}_{D}(\mathbf{r},\mathbf{r}^{\prime};t,t^{\prime})$ is the
Green function for the diffusion equation satisfying the homogeneous boundary
condition sup .

Our model dictates that the valley current (2) generates magnetic fields
equivalently to a local magnetization, via the demagnetization kernel,

$\displaystyle\mathbf{B}^{v}(\mathbf{r},t)=\frac{g_{v}\mu_{B}}{v}\int_{\Omega}d^{3}\mathbf{r}^{\prime}\left(-\bm{\nabla}_{\mathbf{r}}\bm{\nabla}_{\mathbf{r}^{\prime}}\frac{1}{|\mathbf{r}-\mathbf{r}^{\prime}|}\right)\cdot\mathbf{j}^{v},$
(11)

where $g_{v}$ is the effective g factor characterizing the coupling of the
valley degree of freedom to an external magnetic field, and $\mu_{B}$ is the
Bohr magneton. The valley contribution to the magnetic noise can then be
calculated, yielding

$\displaystyle\mathcal{B}^{v}_{ii^{\prime}}(\omega)$
$\displaystyle=\left(\frac{g_{v}\mu_{B}}{v}\right)^{2}\frac{4\pi
k_{B}T\sigma}{d^{3}}\Lambda_{ii^{\prime}}\int
d\xi\,\xi^{2}e^{-2\xi}I^{v}(\xi,\zeta),$ (12)

with dimensionless quantities $\xi=Kd$, $\zeta=\omega d^{2}/D$, and
$a=\sqrt{-i\omega/DK^{2}+1}=\sqrt{-i\zeta/\xi^{2}+1}$,

$\displaystyle
I^{v}(\xi,\zeta)=1+\frac{1+aa^{*}}{(a+a^{*})aa^{*}}-\frac{1}{a^{*}}-\frac{1}{a}.$
(13)

The integration is essentially taken over the magnitude of the wavevector
$K=|\mathbf{K}|$ in the $xy$ plane, with $\xi^{2}e^{-2\xi}$ as a form factor.
The anisotropy tensor $\Lambda$ is the same as in $\mathcal{B}^{c}$ (7). The
frequency dependence of the magnetic noise is contained in $I^{v}$, which
scales as $\omega^{2}$ for $\omega\rightarrow 0$ and approaches $1$ for
$\omega\rightarrow\infty$, $1-I^{v}\sim\omega^{-1/2}$. The magnetic noise
vanishes at zero frequency, consistent with our understanding that transverse
components of the valley current do not contribute, in contrast to the charge
current.

We estimate the magnitude of the valley contribution relative to the charge
contribution,

$\displaystyle\frac{\mathcal{B}^{v}_{zz}}{\mathcal{B}^{c}_{zz}}\sim
g_{v}^{2}\left(\frac{c\alpha}{v}\right)^{2}\left(\frac{a_{0}}{d}\right)^{2}\int
d\xi\,\xi^{2}e^{-2\xi}I^{v}(\xi,\zeta),$ (14)

where we have reduced the result to physical constants, namely the fine-
structure constant $\alpha$ and the Bohr radius $a_{0}$. The effective g
factor $g_{v}$ can be greatly enhanced by the strong spin-orbit coupling,
especially in topological semimetals Cohen and Blount (1960); Singh  _et al._
(1982); Fuseya  _et al._ (2015). Taking $g_{v}\sim 100$, the Fermi velocity
$v\sim 10^{5}$ m/s Grassano  _et al._ (2018), the NV distance $d\sim 100$ nm,
and evaluating the integral numerically at $\zeta=5$, we obtain
$\mathcal{B}^{v}_{zz}/\mathcal{B}^{c}_{zz}\sim 0.2$. Recalling that
$\mathcal{B}^{c}$ (7) is spectrally flat, in our treatment, $\mathcal{B}^{v}$
can be easily recognized, for example, by taking a frequency derivative of the
total measured noise $\partial_{\omega}\mathcal{B}$, exploiting the high
frequency resolution Casola  _et al._ (2018) of NV probes. Moreover, this will
not be spoiled by the presence of spin contribution, which will be seen to
have a distinct frequency behavior.

Figure 3: The frequency-dependent magnetic noise generated by valley (green)
and spin (red) fluctuations in a linear (top) and a log-log (bottom) plot. The
total magnetic noise (black) exhibits a “dip” feature at frequency
$\omega_{\text{dip}}$. Two NV distances $d=100$ nm (solid lines) and $d=150$
nm (dashed lines) are plotted. Inset: $\omega_{\text{dip}}$ as a function of
$d$. The vertical axis of the bottom plot spans three orders of magnitude (in
arbitrary units).

Spin.—Fluctuations of the magnetic order, in the form of thermal magnon
excitations, constitute a third ingredient of the magnetic noise in magnetic
WSMs. Considering the magnon gap in magnetic WSMs can be several meV Itoh  _et
al._ (2016); Liu  _et al._ (2021), we focus on the subgap magnetic noise
generated by the longitudinal (relative to the order parameter $\mathbf{M}$)
spin fluctuations Flebus and Tserkovnyak (2018); McCullian  _et al._ (2020).
The Langevin dynamics of the longitudinal spin density $s_{z}$ can be
described by a stochastic diffusion equation, sometimes referred to as the
Cahn-Hilliard model Cahn and Hilliard (1958); Chaikin and Lubensky (1995):

$\displaystyle\partial_{t}s_{z}-D^{s}\bm{\nabla}^{2}s_{z}=\epsilon^{s},$ (15)

where $D^{s}$ is the spin diffusion constant. Here, the Gaussian white noise
$\epsilon^{s}$ has the correlation function
$\langle\epsilon^{s}(\mathbf{r},t)\epsilon^{s}(\mathbf{r}^{\prime},t^{\prime})\rangle=-2\sigma^{s}k_{B}T\bm{\nabla}^{2}\delta(\mathbf{r}-\mathbf{r}^{\prime})\delta(t-t^{\prime})$,
where the spin conductivity $\sigma^{s}$ is related to the spin diffusion
constant $D^{s}=\sigma^{s}/\chi_{0}$ by the static longitudinal spin
susceptibility $\chi_{0}$. The corresponding boundary condition
$j^{s}_{z}(\mathbf{r},t)\Big{|}_{z=0}=0$ is specified in terms of the spin
current $\mathbf{j}^{s}=-D^{s}\bm{\nabla}s_{z}$. The solution is

$\displaystyle
s_{z}(\mathbf{r},t)=\int_{\Omega}d^{3}\mathbf{r}^{\prime}\int_{-\infty}^{t}dt^{\prime}\mathcal{G}_{D}^{s}(\mathbf{r},\mathbf{r}^{\prime};t,t^{\prime})\,\epsilon^{s}(\mathbf{r}^{\prime},t^{\prime}),$
(16)

where $\mathcal{G}_{D}^{s}(\mathbf{r},\mathbf{r}^{\prime};t,t^{\prime})$ is
simply the Green function of Eq. (10) with $D$ replaced by $D^{s}$.

The spin density generates demagnetization fields, and their the correlation
function gives the magnetic noise

$\displaystyle\mathcal{B}^{s}_{ii^{\prime}}(\omega)=\left(g_{s}\mu_{B}\right)^{2}\frac{2\pi
k_{B}T\chi_{0}}{\omega d^{3}}\Lambda_{ii^{\prime}}\int
d\xi\,\xi^{2}e^{-2\xi}I^{s}(\xi,\eta),$ (17)

where $g_{s}$ is the g factor of localized spins, and

$\displaystyle
I^{s}(\xi,\eta)=-i\left[\frac{1}{b(b+1)}-\frac{1}{b^{*}(b^{*}+1)}\right]$ (18)

where $b=\sqrt{-i\eta/\xi^{2}+1}$, $\xi=Kd$ and $\eta=\omega d^{2}/D^{s}$. As
$\omega\rightarrow 0$, $I^{s}$ scales as $\omega$ and $\mathcal{B}^{s}$
converges to a finite value. For $\omega\rightarrow\infty$, $I^{s}\sim
1/\omega$ and $\mathcal{B}^{s}$ vanishes asymptotically as $1/\omega^{2}$.

According to recent experiments around room temperature, the resultant NV
transition rate from charge fluctuations in a good conductor Kolkowitz  _et
al._ (2015) and that from longitudinal spin fluctuations in a magnetic
insulator Wang  _et al._ (2022) are both of ms-1 order. In magnetic WSMs with
relatively low electrical conductivity, spin fluctuations may dominate the
magnetic noise at low frequencies. However, the spin contribution quickly
drops a few orders of magnitude with increasing frequency. Consequently, the
total magnetic noise coming from valley and spin exhibits a “dip” feature in
its frequency dependence, as shown in Fig. 3, on top of the flat charge
contribution. The frequency $\omega_{\text{dip}}$ at the bottom of the dip
shifts lower at a larger distance $d$. For the plot, we have assumed similar
charge and spin conductivities $\sigma\sim\sigma^{s}$ 222The electric
conductivity in a ferromagnetic WSM and the spin conductivity in a magnetic
insulator can both be around $10^{5}-10^{6}$ S/m according to literature so
far (see, for example, Cornelissen  _et al._ (2015); Wang  _et al._ (2022);
Hirschberger  _et al._ (2016); Liu  _et al._ (2018))., and taken both electron
and magnon mean free paths $\sim 30$ nm, Fermi velocity $\sim 10^{5}$ m/s and
magnon velocity $\sim 10^{3}$ m/s, yielding $D\sim 10^{-3}$ m2/s and
$D^{s}\sim 10^{-5}$ m2/s.

Discussion.—It should be remarked that the valley noise derived here is based
on a specific model (1), as a demonstration of our formalism. The resulted
prediction is not universal for all WSMs. Different symmetry assumptions about
spin and parity in the band model can lead to a different coupling scheme of
the valley degree of freedom to magnetic fields. The frequency and distance
dependence of the magnetic noise can therefore also reflect the applicability
of the effective theories. For specific materials, one may need to construct
the coupling scheme from first-principles calculations.

We have considered the magnetic noise primarily driven by the diffusive
transport, neglecting the valley and spin relaxation effects. This applies to
NV distances much smaller than the corresponding diffusion lengths (while
larger than the mean free paths), which is often the case in a clean material.
A finite valley relaxation time modifies the valley diffusion equation, and in
our Langevin approach, is accompanied by an intervalley fluctuation of the
axial charge density. Likewise, spin density relaxation and fluctuation can be
added to the Cahn-Hilliard model. We refer to the discussion in the
supplemental material sup . Furthermore, our treatment focuses on the low-
frequency regime, where dynamic effects of the demagnetization fields, such as
the Faraday induction, are negligible.

The Fermi arc states are universal in WSMs and are generally dissipative
Kourtis  _et al._ (2016); Gorbar  _et al._ (2016). The magnetic noise they
generate, which is not detected in our choice of geometry, may also be
interesting to look into, by placing the NV near material surfaces parallel to
the magnetic order $\mathbf{M}$. The Fermi-arc contribution may be recognized
by its distance dependence Burrello  _et al._ (2019), which is different from
the bulk one. For the spin channel, the change in the surface orientation
relative to the order parameter would acquire a geometric factor Guslienko and
Slavin (2011). It would also be straightforward to account for an anisotropic
electric conductivity under our framework for practical application.

Going beyond the diffusive regime, a nanoscale probe such as NV can also probe
the ballistic and perhaps the hydrodynamic transport of charge or spin, when
the NV to material distance is smaller than or comparable to the electron or
magnon mean free paths Kolkowitz  _et al._ (2015); Fang  _et al._ (2022). The
spectral analysis of magnetic noise would also be particularly useful in the
nonequilibrium regime, for example, in the presence of a steady spin or valley
current, which may provide a cleaner detection of the corresponding nonlinear
responses.

###### Acknowledgements.

This work is supported by NSF under Grant No. DMR-2049979. We thank Chunhui
Rita Du for inspiring discussions.

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Supplemental Material for
“Flavors of Magnetic Noise in Quantum Materials”

Shu Zhang and Yaroslav Tserkovnyak

Department of Physics and Astronomy, University of California, Los Angeles,
California 90095, USA

In this document, we present the details on the generation of the magnetic
noise by the fluctuations of the charge, valley, and spin degrees of freedom
in a magnetic Weyl semimetal, via different generating kernels. Specifically,
we look into the magnetic field $\mathbf{B}(\mathbf{r}_{\text{NV}},t)$ at the
position of the NV center $\mathbf{r}_{\text{NV}}=(0,0,d)$ generated by the
material occupying the half space $\Omega$ with $z\leq 0$, assuming the
material dimensions are much larger than $d$.

The following identity and its variants will be frequently used, which is
convenient for the broken translational invariance in the $z$ direction,

$\displaystyle\frac{1}{|\mathbf{r}-\mathbf{r}^{\prime}|}=\int\frac{d^{2}\mathbf{K}}{(2\pi)^{2}}\,\frac{2\pi}{K}e^{i\mathbf{K}\cdot(\mathbf{R}-\mathbf{R}^{\prime})-K|z-z^{\prime}|},$
(S.1)

where $\mathbf{R}=(x,y,0)$, and $\mathbf{K}=(k_{x},k_{y},0)$ denotes the
wavevector in the $x$ and $y$ directions, $K=|\mathbf{K}|$.

##  Appendix A I. Transport

###  A.1 A. Charge

We take a stochastic approach and study the response of the charge current to
a Langevin noise, and verify that our method reproduces the known result for
the Johnson-Nyquist noise from the thermal fluctuations of electric charge
current in a metal. The charge transport obeys the continuity equation

$\displaystyle\partial_{t}\rho^{c}+\bm{\nabla}\cdot{\mathbf{j}^{c}}=0.$ (S.2)

In the low frequency limit, the charge density fluctuation is efficiently
screened by Coulomb interactions. We thus consider the transverse current
fluctuations only, $\bm{\nabla}\cdot{\mathbf{j}^{c}}=0$, where
${\mathbf{j}^{c}}=-\sigma\bm{\nabla}\mu^{c}+\bm{\epsilon}^{c}$. Here $\sigma$
is the conductivity, $\mu^{c}$ is the chemical potential, and
$\bm{\epsilon}^{c}$ is a Gaussian white noise satisfying
$\langle\epsilon^{c}_{i}(\mathbf{r},t)\rangle=0$ and
$\langle\epsilon^{c}_{i}(\mathbf{r},t)\epsilon^{c}_{i^{\prime}}(\mathbf{r}^{\prime},t^{\prime})\rangle=2\sigma
k_{B}T\delta_{ii^{\prime}}\delta(\mathbf{r}-\mathbf{r}^{\prime})\delta(t-t^{\prime})$,
as a result of the fluctuation dissipation theorem at high temperatures. We
find the response of the charge current ${\mathbf{j}^{c}}$ to the thermal
noise $\bm{\epsilon}^{c}$ by solving the following differential equation

$\displaystyle\sigma\bm{\nabla}^{2}\mu^{c}=\bm{\nabla}\cdot\bm{\epsilon}^{c}.$
(S.3)

In a free space, it is easy to show by the Fourier transform that
$\sigma\mathbf{k}^{2}\mu^{c}=\mathbf{k}\cdot\bm{\epsilon}^{c}$ and
${\mathbf{j}^{c}}=\bm{\epsilon}^{c}-\mathbf{k}\mathbf{k}\cdot\bm{\epsilon}^{c}/k^{2}$,
which gives the familiar transverse current correlation functions $\langle
j^{c}_{i}(\mathbf{r},t)j^{c}_{i^{\prime}}(\mathbf{r}^{\prime},t^{\prime})\rangle=\left(\delta_{ii^{\prime}}-k_{i}k_{i^{\prime}}/k^{2}\right)2\sigma
k_{B}T\delta(\mathbf{r}-\mathbf{r}^{\prime})\delta(t-t^{\prime})$.

Here, since the current does not flow out of the material, we solve Eq. (S.3)
in the half space $\Omega$ with the Neumann boundary condition
$\sigma\partial_{z}\mu^{c}(\mathbf{r},t)\Big{|}_{z=0}=\epsilon^{c}_{z}(\mathbf{r},t)\Big{|}_{z=0}$.
The Green function for the Laplacian with the homogeneous Neumann boundary
condition is [the time dependent factor is simply $\delta(t-t^{\prime})$]

$\displaystyle\mathcal{G}_{L}(\mathbf{r},\mathbf{r}^{\prime})=-\frac{1}{4\pi}\left[\frac{1}{\sqrt{(x-x^{\prime})^{2}+(y-y^{\prime})^{2}+(z-z^{\prime})^{2}}}+\frac{1}{\sqrt{(x-x^{\prime})^{2}+(y-y^{\prime})^{2}+(z+z^{\prime})^{2}}}\right].$
(S.4)

Therefore,

$\displaystyle\mu^{c}(\mathbf{r},t)$
$\displaystyle=\int_{\Omega}d^{3}\mathbf{r}^{\prime}\,\mathcal{G}_{L}(\mathbf{r},\mathbf{r}^{\prime})\frac{\bm{\nabla}_{\mathbf{r}^{\prime}}\cdot\bm{\epsilon}^{c}(\mathbf{r}^{\prime},t)}{\sigma}-\int_{\partial\Omega}d^{2}\mathbf{r}^{\prime}\,\mathcal{G}_{L}(\mathbf{r},\mathbf{r}^{\prime})\frac{\epsilon^{c}_{z}(\mathbf{r}^{\prime},t)}{\sigma},$
(S.5)

and integrating by parts

$\displaystyle{\mathbf{j}^{c}}(\mathbf{r},t)$
$\displaystyle=\bm{\epsilon}^{c}(\mathbf{r},t)+\int_{\Omega}d^{3}\mathbf{r}^{\prime}\,\bm{\epsilon}^{c}(\mathbf{r}^{\prime},t)\cdot\bm{\nabla}_{\mathbf{r}^{\prime}}\bm{\nabla}_{\mathbf{r}}\mathcal{G}_{L}(\mathbf{r},\mathbf{r}^{\prime}).$
(S.6)

Using the Fourier transform of the Green function (S.4) in the $x$ and $y$
directions with $\mathbf{R}=(x,y,0)$,

$\displaystyle\mathcal{G}_{L}(\mathbf{K},z,z^{\prime})=-\frac{1}{2K}\left(e^{-K|z-z^{\prime}|}+e^{-K|z+z^{\prime}|}\right),$
(S.7)

we take the Fourier transfrom of ${\mathbf{j}^{c}}(\mathbf{r},t)$ in the
following form,

$\displaystyle{\mathbf{j}^{c}}(\mathbf{K},t)$
$\displaystyle=\int_{-\infty}^{0}dz\,e^{Kz}\int
d^{2}\mathbf{R}\,e^{-i\mathbf{K}\cdot\mathbf{R}}{\mathbf{j}^{c}}(\mathbf{r},t)$
(S.8) $\displaystyle=\int_{-\infty}^{0}dz\,e^{Kz}\int
d^{2}\mathbf{R}\,e^{-i\mathbf{K}\cdot\mathbf{R}}\,\left[\bm{\epsilon}^{c}(\mathbf{r},t)-\epsilon_{z}^{c}(\mathbf{r},t)\hat{\mathbf{z}}\right]$
$\displaystyle\qquad\qquad-\frac{1}{2K}\int_{-\infty}^{0}dz^{\prime}\,e^{Kz^{\prime}}\int
d^{2}\mathbf{R}^{\prime}\,e^{-i\mathbf{K}\cdot\mathbf{R}^{\prime}}\bm{\epsilon}^{c}(\mathbf{r}^{\prime},t)\cdot\left(\mathbf{K}\bm{\kappa}^{*}\frac{1}{K}-\bm{\kappa}\bm{\kappa}z^{\prime}\right),$

where $\bm{\kappa}=\mathbf{K}+iK\hat{\mathbf{z}}$. The second term in the
square brackets comes from the Dirac $\delta$-function in
$\bm{\nabla}_{\mathbf{r}^{\prime}}\bm{\nabla}_{\mathbf{r}}G^{c}(\mathbf{r},\mathbf{r}^{\prime})$.

In the magnetostatic limit, where the wavelength of the electromagnetic field
and the skin depth are both much larger than the NV distance $d$, the Biot-
Savart law gives the magnetic field generated by the electric charge current
distribution,

$\displaystyle\mathbf{B}^{c}(\mathbf{r}_{\text{NV}},t)$
$\displaystyle=-\frac{e}{c}\int_{\Omega}d^{3}\mathbf{r}\,{\mathbf{j}^{c}}(\mathbf{r},t)\times\bm{\nabla}_{\mathbf{r}}\frac{1}{|\mathbf{r}_{\text{NV}}-\mathbf{r}|}$
(S.9)
$\displaystyle=-\frac{e}{c}\int\frac{d^{2}\mathbf{K}}{(2\pi)^{2}}\,\frac{2\pi}{K}e^{-Kd}(i\bm{\kappa})\times{\mathbf{j}^{c}}(\mathbf{K},t)$
$\displaystyle=-\frac{e}{c}\int\frac{d^{2}\mathbf{K}}{(2\pi)^{2}}\,\frac{2\pi}{K}e^{-Kd}\left[i\bm{\kappa}\times\bm{\epsilon}^{c}(\mathbf{K},t)+\frac{1}{K}(\hat{\mathbf{z}}\times\mathbf{K})\bm{\epsilon}^{c}(\mathbf{K},t)\cdot\bm{\kappa}\right],$

where $\bm{\epsilon}^{c}(\mathbf{K},t)=\int_{-\infty}^{0}dz\,e^{Kz}\int
d^{2}\mathbf{R}\,e^{-i\mathbf{K}\cdot\mathbf{R}}\,\bm{\epsilon}^{c}(\mathbf{r},t)$.
We have used $\bm{\kappa}\times\bm{\kappa}^{*}=i2K\mathbf{z}\times\mathbf{K}$
and $\bm{\kappa}\times\bm{\kappa}=0$. This result reproduces that in Ref.
Henkel (2005) derived from Maxwell equations with proper boundary conditions,
and is consistent with the approach of finding the Fresnel coefficients Ford
and Weber (1984); Henkel  _et al._ (1999). Finally, we obtain the magnetic
noise tensor contributed from charge fluctuations, according to Eq. (3),

$\displaystyle\mathcal{B}^{c}_{ii^{\prime}}(\omega)$
$\displaystyle=\left(\frac{e}{c}\right)^{2}\int d^{2}\mathbf{K}\frac{2\sigma
k_{B}T}{2K^{3}}e^{-2Kd}\left(\epsilon^{ijk}\epsilon^{i^{\prime}j^{\prime}k}\kappa_{j}\kappa_{j^{\prime}}^{*}+\frac{i}{K}\epsilon^{ijk}\epsilon^{i^{\prime}zk^{\prime}}\kappa_{j}K_{k^{\prime}}\kappa_{k}^{*}-\frac{i}{K}\epsilon^{i^{\prime}j^{\prime}k^{\prime}}\epsilon^{izk}\kappa^{*}_{j^{\prime}}K_{k}\kappa_{k^{\prime}}+2\epsilon^{izk}\epsilon^{i^{\prime}zk^{\prime}}K_{k}K_{k^{\prime}}\right)$
(S.10) $\displaystyle=\left(\frac{e}{c}\right)^{2}\int dKe^{-2Kd}2\pi\sigma
k_{B}T\Delta_{ii^{\prime}}$ $\displaystyle=\frac{\pi
e^{2}k_{B}T\sigma}{c^{2}d}\Delta_{ii^{\prime}},$

where the tensor $\Delta=\text{diag}(1/2,1/2,1)$ and Einstein summation is
taken over repeated indices.

###  A.2 B. Valley

Following the same approach, we next look into the contribution from the
valley current. We first focus on the particle conservation in a single valley
$p$, assuming the interaction and scattering between the two Weyl nodes are
weak. We do not consider the chiral-anomaly related perturbations here.

$\displaystyle\partial_{t}\rho^{p}+\bm{\nabla}\cdot\mathbf{j}^{p}=0,$ (S.11)

where $\rho^{p}=(\nu/2)\mu^{p}$ and
$\mathbf{j}^{p}=-(\sigma/2)\bm{\nabla}\mu^{p}+\bm{\epsilon}^{p}$. Here $\nu$
refers to the total density of states at the Fermi surface, including both
valleys. Assuming no correlation between $\bm{\epsilon}^{+}$ and
$\bm{\epsilon}^{-}$,
$\langle\epsilon^{p}_{i}(\mathbf{r},t)\epsilon^{p^{\prime}}_{i^{\prime}}(\mathbf{r}^{\prime},t^{\prime})\rangle=\sigma
k_{B}T\delta_{pp^{\prime}}\delta_{ii^{\prime}}\delta(\mathbf{r}-\mathbf{r}^{\prime})\delta(t-t^{\prime})$.
The differential equation to be solved is then

$\displaystyle\partial_{t}\mu^{p}-D\bm{\nabla}^{2}\mu^{p}=-\frac{2}{\nu}\bm{\nabla}\cdot\bm{\epsilon}^{p}$
(S.12)

where the diffusion coefficient $D=\sigma/\nu$, as is consistent with the
Einstein relation and the Drude law. The boundary condition
$j_{z}^{p}(\mathbf{r},t)\Big{|}_{z=0}=0$ is similar to the case with charge.
We use the Green function for the diffusion equation satisfying the
homogeneous Neumann boundary condition, and its Fourier transform

$\displaystyle\mathcal{G}_{D}(\mathbf{r},\mathbf{r}^{\prime};t,t^{\prime})=\frac{\Theta(t-t^{\prime})}{[4\pi
D(t-t^{\prime})]^{3/2}}e^{-[(x-x^{\prime})^{2}+(y-y^{\prime})^{2}]/4D(t-t^{\prime})}\left(e^{-(z-z^{\prime})^{2}/4D(t-t^{\prime})}+e^{-(z+z^{\prime})^{2}/4D(t-t^{\prime})}\right),$
(S.13)
$\displaystyle\mathcal{G}_{D}(\mathbf{K},z,z^{\prime};\omega)=\frac{1}{2DaK}\left(e^{-aK|z-z^{\prime}|}+e^{-aK|z+z^{\prime}|}\right),$

where $\Theta(t-t^{\prime})$ is the Heaviside step function. Here, the
dimensionless parameter $a=\sqrt{-i\omega/DK^{2}+1}$ carries the frequency
dependence in the following context, and is evaluated in the branch with
$\text{Re}\,a>0$. The current distribution is therefore

$\displaystyle\mathbf{j}^{p}(\mathbf{r},t)$
$\displaystyle=\bm{\epsilon}^{p}(\mathbf{r},t)-D\int_{\Omega}d^{3}\mathbf{r}^{\prime}\int_{-\infty}^{t}dt^{\prime}\,\bm{\epsilon}^{p}(\mathbf{r}^{\prime},t^{\prime})\cdot\bm{\nabla}_{\mathbf{r}^{\prime}}\bm{\nabla}_{\mathbf{r}}\mathcal{G}_{D}(\mathbf{r},\mathbf{r}^{\prime};t,t^{\prime}),$
(S.14)

and

$\displaystyle\mathbf{j}^{p}(\mathbf{K},\omega)$
$\displaystyle=\int_{-\infty}^{0}dz\,e^{Kz}\int d^{2}\mathbf{R}\int
dt\,e^{i\omega t-i\mathbf{K}\cdot\mathbf{R}}\mathbf{j}^{p}(\mathbf{r},t)$
(S.15) $\displaystyle=\int_{-\infty}^{0}dz\,e^{Kz}\int d^{2}\mathbf{R}\int
dt\,e^{i\omega
t-i\mathbf{K}\cdot\mathbf{R}}\,\left[\bm{\epsilon}^{p}(\mathbf{r},t)-{\epsilon}_{z}^{p}(\mathbf{r},t)\hat{\mathbf{z}}\right]$
$\displaystyle-\frac{1}{2aK^{2}}\int_{-\infty}^{0}dz^{\prime}\int
d^{2}\mathbf{R}^{\prime}\int dt\,e^{i\omega
t^{\prime}-i\mathbf{K}\cdot\mathbf{R}^{\prime}}\,\bm{\epsilon}^{p}(\mathbf{r}^{\prime},t^{\prime})\cdot\left[{\textbf{\textkappa}}^{\star}{\textbf{\textkappa}}^{\star}\frac{e^{Kz^{\prime}}}{a+1}+\textbf{\textkappa}\textbf{\textkappa}\frac{e^{Kz^{\prime}}-e^{aKz^{\prime}}}{a-1}+\textbf{\textkappa}{\textbf{\textkappa}}^{\star}\frac{e^{aKz^{\prime}}}{a+1}\right],$

where $\textbf{\textkappa}=\mathbf{K}+iaK\hat{\mathbf{z}}$, and
$\textbf{\textkappa}^{\star}=\mathbf{K}-iaK\hat{\mathbf{z}}$.

The valley current density behaves like a local magnetization in generating
the magnetic field:

$\displaystyle\mathbf{B}^{v}(\mathbf{r}_{\text{NV}},\omega)$
$\displaystyle=\frac{g_{v}\mu_{B}}{v}\int_{\Omega}d^{3}\mathbf{r}\left(-\bm{\nabla}_{\mathbf{r}_{\text{NV}}}\bm{\nabla}_{\mathbf{r}}\frac{1}{|\mathbf{r}_{\text{NV}}-\mathbf{r}|}\right)\cdot{\mathbf{j}^{v}}(\mathbf{r},\omega)$
(S.16)
$\displaystyle=-\frac{g_{v}\mu_{B}}{v}\int\frac{d^{2}\mathbf{K}}{(2\pi)^{2}}\,\frac{2\pi}{K}e^{-Kd}\bm{\kappa}\bm{\kappa}\cdot\left[\mathbf{j}^{+}(\mathbf{K},\omega)-\mathbf{j}^{-}(\mathbf{K},\omega)\right]$
$\displaystyle=-\frac{g_{v}\mu_{B}}{v}\int\frac{d^{2}\mathbf{K}}{(2\pi)^{2}}\,\frac{2\pi}{K}e^{-Kd}\int_{-\infty}^{0}dz\int
d^{2}\mathbf{R}\int dt\,e^{i\omega
t-i\mathbf{K}\cdot\mathbf{R}}\,\bm{\kappa}\left(e^{Kz}\bm{\kappa}-\frac{e^{aKz}}{a}\textbf{\textkappa}\right)\cdot\left[\bm{\epsilon}^{+}(\mathbf{r},t)-\bm{\epsilon}^{-}(\mathbf{r},t)\right],$

where $g_{v}$ is the effective g factor describing the coupling of the valley
degrees of freedom to an external magnetic field and $v$ is the Fermi
velocity. We have used $\bm{\kappa}\cdot\textbf{\textkappa}=-K^{2}(a-1)$ and
$\bm{\kappa}\cdot\textbf{\textkappa}^{\star}=K^{2}(a+1)$. With the help of

$\displaystyle\Big{\langle}\int_{-\infty}^{0}dz\,e^{c_{1}z}\int
d^{2}\mathbf{R}\int dt\,e^{-i\omega
t+i\mathbf{K}\cdot\mathbf{R}}\,\epsilon^{p}_{i}(\mathbf{r},t)\int_{-\infty}^{0}dz^{\prime}\,e^{{c_{2}}^{*}z^{\prime}}\int
d^{2}\mathbf{R}^{\prime}\int
dt^{\prime}\,e^{i\omega^{\prime}t^{\prime}-i\mathbf{K}^{\prime}\cdot\mathbf{R}^{\prime}}\epsilon^{p^{\prime}}_{i^{\prime}}(\mathbf{r}^{\prime},t^{\prime})\Big{\rangle}$
(S.17) $\displaystyle=\frac{\sigma
k_{B}T}{c_{1}+{c_{2}}^{*}}(2\pi)^{3}\delta_{pp^{\prime}}\delta_{ii^{\prime}}\delta(\mathbf{K}-\mathbf{K}^{\prime})\delta(\omega-\omega^{\prime}),$

we obtain the magnetic noise tensor generated by valley fluctuations

$\displaystyle\mathcal{B}^{v}_{ii^{\prime}}(\omega)$
$\displaystyle=\frac{1}{2\pi}\langle
B_{i}(\mathbf{r}_{\text{NV}},\omega)B_{i^{\prime}}^{*}(\mathbf{r}_{\text{NV}},\omega)\rangle$
(S.18)
$\displaystyle=\left(\frac{g_{v}\mu_{B}}{v}\right)^{2}\int\frac{d^{2}\mathbf{K}}{(2\pi)^{2}}\frac{(2\pi)^{2}}{K}e^{-2Kd}2\sigma
k_{B}T\kappa_{i}{\kappa}_{i^{\prime}}^{*}\left[1+\frac{1+aa^{*}}{(a+a^{*})aa^{*}}-\frac{1}{a^{*}}-\frac{1}{a}\right]$
$\displaystyle=\left(\frac{g_{v}\mu_{B}}{v}\right)^{2}\frac{4\pi
k_{B}T\sigma}{d^{3}}\Lambda_{ii^{\prime}}\int
d\xi\,\xi^{2}e^{-2\xi}I^{v}(\xi,\zeta),$

where

$\displaystyle
I^{v}(\xi,\zeta)=1+\frac{1+aa^{*}}{(a+a^{*})aa^{*}}-\frac{1}{a^{*}}-\frac{1}{a},$
(S.19)

with dimensionless quantities $\xi=Kd$, $\zeta=\omega d^{2}/D$, and
$a=\sqrt{-i\zeta/\xi^{2}+1}$. The anisotropy tensor $\Delta$ is the same as in
$\mathcal{B}^{c}$.

###  A.3 C. Spin

We invoke the Cahn-Hilliard model Cahn and Hilliard (1958) to describe the
Langevin dynamics of the longitudinal spin density $s_{z}$

$\displaystyle\partial_{t}s_{z}-D^{s}\bm{\nabla}^{2}s_{z}=\epsilon^{s},$
(S.20)

where $D^{s}$ is the spin diffusion constant and $\epsilon^{s}$ is a scalar
Gaussian white noise with the correlation function
$\langle\epsilon^{s}(\mathbf{r},t)\epsilon^{s}(\mathbf{r}^{\prime},t^{\prime})\rangle=-2\sigma^{s}k_{B}T\bm{\nabla}^{2}\delta(\mathbf{r}-\mathbf{r}^{\prime})\delta(t-t^{\prime})$,
containing the Laplacian operator. The spin conductivity $\sigma^{s}$ is
related to the spin diffusion constant $D^{s}=\sigma^{s}/\chi_{0}$ by the
static longitudinal spin susceptibility $\chi_{0}$. The boundary condition
$j^{s}_{z}(\mathbf{r},t)\Big{|}_{z=0}=0$ is specified in terms of the spin
current $\mathbf{j}^{s}=-D^{s}\bm{\nabla}s_{z}$. The solution is

$\displaystyle
s_{z}(\mathbf{r},t)=\int_{\Omega}d^{3}\mathbf{r}^{\prime}\int_{-\infty}^{t}\mathcal{G}_{D}^{s}(\mathbf{r},\mathbf{r}^{\prime};t,t^{\prime})\,\epsilon^{s}(\mathbf{r}^{\prime},t^{\prime}).$
(S.21)

The Green function
$\mathcal{G}_{D}^{s}(\mathbf{r},\mathbf{r}^{\prime};t,t^{\prime})$ is simply
Eq. (S.13) with $D\rightarrow D^{s}$,

$\displaystyle\mathcal{G}_{D}^{s}(\mathbf{r},\mathbf{r}^{\prime};t,t^{\prime})=\frac{\Theta(t-t^{\prime})}{[4\pi
D^{s}(t-t^{\prime})]^{3/2}}e^{-[(x-x^{\prime})^{2}+(y-y^{\prime})^{2}]/4D^{s}(t-t^{\prime})}\left(e^{-(z-z^{\prime})^{2}/4D^{s}(t-t^{\prime})}+e^{-(z+z^{\prime})^{2}/4D^{s}(t-t^{\prime})}\right)$
(S.22)
$\displaystyle\mathcal{G}_{D}^{s}(\mathbf{K},z,z^{\prime};\omega)=\frac{1}{2DbK}\left(e^{-bK|z-z^{\prime}|}+e^{-bK|z+z^{\prime}|}\right),$

where $b=\sqrt{-i\omega/D^{s}K^{2}+1}$. Taking the Fourier transform,

$\displaystyle s_{z}(\mathbf{K},\omega)$
$\displaystyle=\int_{-\infty}^{0}dz\,e^{Kz}\int d^{2}\mathbf{R}\int
dt\,e^{i\omega t-i\mathbf{K}\cdot\mathbf{R}}s_{z}(\mathbf{r},t)$ (S.23)
$\displaystyle=\int_{\Omega}d^{3}\mathbf{r}^{\prime}\int_{-\infty}^{t}dt^{\prime}e^{i\omega
t^{\prime}-i\mathbf{K}\cdot\mathbf{R}^{\prime}}\,\frac{1}{D^{s}bK^{2}}\frac{be^{Kz^{\prime}}-e^{bKz^{\prime}}}{(b+1)(b-1)}\epsilon^{s}(\mathbf{r}^{\prime},t^{\prime})$

The spin density generates stray fields via the demagnetization kernel,

$\displaystyle\mathbf{B}^{s}(\mathbf{r}_{\text{NV}},\omega)$
$\displaystyle=g_{s}\mu_{B}\int_{\Omega}d^{3}\mathbf{r}\left(-\bm{\nabla}_{\mathbf{r}_{\text{NV}}}\bm{\nabla}_{\mathbf{r}}\frac{1}{|\mathbf{r}_{\text{NV}}-\mathbf{r}|}\right)\cdot
s_{z}(\mathbf{r},\omega)\hat{\mathbf{z}}$ (S.24)
$\displaystyle=g_{s}\mu_{B}\int\frac{d^{2}\mathbf{K}}{(2\pi)^{2}}\,\frac{2\pi}{K}e^{-Kd}\bm{\kappa}\int_{-\infty}^{0}dz\int
d^{2}\mathbf{R}\int_{-\infty}^{t}dte^{i\omega
t-i\mathbf{K}\cdot\mathbf{R}}\,\frac{1}{iD^{s}bK}\frac{be^{Kz}-e^{bKz}}{(b+1)(b-1)}\epsilon^{s}(\mathbf{r},t).$

Using the correlation relation of the Gaussian noise, we obtain the magnetic
noise tensor

$\displaystyle\mathcal{B}^{s}_{ii^{\prime}}(\omega)$
$\displaystyle=(g_{s}\mu_{B})^{2}\int\frac{d^{2}\mathbf{K}}{(2\pi)^{2}}\,\frac{(2\pi)^{2}}{K}e^{-2Kd}\kappa_{i}\kappa^{*}_{i^{\prime}}\frac{\sigma^{s}k_{B}T}{iD^{s}\omega}\left[\frac{1}{b(b+1)}-\frac{1}{b^{*}(b^{*}+1)}\right]$
(S.25) $\displaystyle=\left(g_{s}\mu_{B}\right)^{2}\frac{2\pi
k_{B}T\chi_{0}}{\omega d^{3}}\Lambda_{ii^{\prime}}\int
d\xi\,\xi^{2}e^{-2\xi}I^{s}(\xi,\eta),$

where $g_{s}$ is the g factor of localized spins, and

$\displaystyle
I^{s}(\xi,\eta)=-i\left[\frac{1}{b(b+1)}-\frac{1}{b^{*}(b^{*}+1)}\right]$
(S.26)

with $\xi=Kd$, $\eta=\omega d^{2}/D^{s}$ and $b=\sqrt{-i\eta/\xi^{2}+1}$.

##  Appendix B II. Relaxation

###  B.1 A. Spin

In the Langevin approach, the spin relaxation can be handled by replacing the
correlation functions of the Gaussian noise by
$\langle\widetilde{\epsilon}^{s}(\mathbf{r},t)\widetilde{\epsilon}^{s}(\mathbf{r}^{\prime},t^{\prime})\rangle=2k_{B}T\chi_{0}(1/\tau^{s}-D^{s}\bm{\nabla}^{2})\delta(\mathbf{r}-\mathbf{r}^{\prime})\delta(t-t^{\prime})$
in

$\displaystyle\left(\partial_{t}+\frac{1}{\tau^{s}}\right)s_{z}-D^{s}\bm{\nabla}^{2}s_{z}=\widetilde{\epsilon}^{s},$
(S.27)

where $\tau^{s}$ is the spin relaxation time. This is because the
consideration of the spin relaxation has to be accompanied by a spin density
fluctuation, as dictated by the fluctuation-dissipation theorem. Here, the
Green function and its Fourier transform are, respectively,

$\displaystyle\widetilde{\mathcal{G}}^{s}_{D}(\mathbf{r},\mathbf{r}^{\prime};t,t^{\prime})=\frac{\Theta(t-t^{\prime})}{[4\pi
D^{s}(t-t^{\prime})]^{3/2}}e^{-[(x-x^{\prime})^{2}+(y-y^{\prime})^{2}]/4D^{s}(t-t^{\prime})}\left(e^{-(z-z^{\prime})^{2}/4D^{s}(t-t^{\prime})}+e^{-(z+z^{\prime})^{2}/4D^{s}(t-t^{\prime})}\right)e^{-(t-t^{\prime})/\tau^{s}},$
(S.28)
$\displaystyle\widetilde{\mathcal{G}}^{s}_{D}(\mathbf{K},z,z^{\prime};\omega)=\frac{1}{2D^{s}\widetilde{b}K}\left(e^{-\widetilde{b}K|z-z^{\prime}|}+e^{-\widetilde{b}K|z+z^{\prime}|}\right),$

where $\widetilde{b}=\sqrt{(-i\omega+1/\tau^{s})/D^{s}K^{2}+1}$. The magnetic
noise thus becomes

$\displaystyle\widetilde{\mathcal{B}}^{s}_{ii^{\prime}}(\omega)$
$\displaystyle=(g_{s}\mu_{B})^{2}\int\frac{d^{2}\mathbf{K}}{(2\pi)^{2}}\,\frac{(2\pi)^{2}}{K^{3}}e^{-2Kd}\kappa_{i}\kappa^{*}_{i^{\prime}}\frac{2k_{B}T\chi_{0}}{D^{s}}\frac{(1+\widetilde{b}+\widetilde{b}^{*})(\widetilde{b}+1)(\widetilde{b}^{*}+1)+[2(\widetilde{b}+\widetilde{b}^{*}+1)^{2}+\widetilde{b}\widetilde{b}^{*}(\widetilde{b}+\widetilde{b}^{*})]/2K^{2}D^{s}\tau^{s}}{\widetilde{b}\widetilde{b}^{*}(\widetilde{b}+1)^{2}(\widetilde{b}^{*}+1)^{2}(\widetilde{b}+\widetilde{b}^{*})}$
(S.29) $\displaystyle=(g_{s}\mu_{B})^{2}\frac{\pi
k_{B}T\chi_{0}}{D^{s}}\Delta_{ii^{\prime}}\int
dKe^{-2Kd}\,\left[\frac{2+\widetilde{b}}{\widetilde{b}(\widetilde{b}+1)^{2}}+\frac{2+\widetilde{b}^{*}}{\widetilde{b}^{*}(\widetilde{b}^{*}+1)^{2}}\right].$

In the limit $1/\tau^{s}\rightarrow 0$,
$\text{Re}\left[(2+\widetilde{b})/\widetilde{b}(\widetilde{b}+1)^{2}\right]\rightarrow\text{Im}\left[(2D^{s}K^{2}/\omega)/b(b+1)\right]$,
we arrive at the same result as in Eq. (S.25). In the opposite limit
$\tau^{s}\rightarrow 0$,
$(2+\widetilde{b})/\widetilde{b}(\widetilde{b}+1)^{2}\approx
D^{s}K^{2}\tau^{s}$,
$\mathcal{B}^{s}_{ii^{\prime}}(\omega)\rightarrow\left(g_{s}\mu_{B}\right)^{2}\pi
k_{B}T\chi_{0}\tau^{s}\Delta_{ii^{\prime}}/2d^{3}$ is independent of $D^{s}$,
as expected from the relaxation dominated equation of motion.

Alternatively, the noise can be computed from the dynamic spin susceptibility,
with the help of the fluctuation dissipation theorem. We derive dynamic spin
susceptibility from the diffusion equation for the longitudinal spin dynamics

$\displaystyle\partial_{t}s_{z}+\bm{\nabla}\cdot\mathbf{j}^{s}=-\frac{\chi_{0}}{\tau^{s}}\mu^{s},$
(S.30)

where $\mathbf{j}^{s}=-\sigma^{s}\bm{\nabla}\mu^{s}$ is the spin current,
$\mu^{s}=s_{z}/\chi_{0}-h$ is the spin chemical potential, and $h$ is a force
thermodynamically conjugate to $s_{z}$, given by the external magnetic field.
The spin distributionr relaxes towards $\chi_{0}h$ ($\mu^{s}=0$ in
equilibrium), characterized by the spin relaxation time $\tau^{s}$. The
differential equation

$\displaystyle\left(\partial_{t}+\frac{1}{\tau^{s}}\right)\mu^{s}-D^{s}\bm{\nabla}^{2}\mu^{s}=-\partial_{t}h.$
(S.31)

again has the boundary condition
$\partial_{z}\mu^{s}(\mathbf{r},t)\Big{|}_{z=0}=0$. From the solution of
$\mu^{s}(\mathbf{r},t)$, we have

$\displaystyle s_{z}(\mathbf{r},t)$
$\displaystyle=\chi_{0}\int_{\Omega}d^{3}\mathbf{r}^{\prime}\int_{-\infty}^{t}dt^{\prime}\,\left[\delta(\mathbf{r}-\mathbf{r}^{\prime})\delta(t-t^{\prime})-\widetilde{\mathcal{G}}_{D}^{s}(\mathbf{r},\mathbf{r}^{\prime};t,t^{\prime})\partial_{t^{\prime}}\right]h(\mathbf{r}^{\prime},t^{\prime}),$
(S.32)

which defines the longitudinal susceptibility
$\chi(\mathbf{r},\mathbf{r}^{\prime};t,t^{\prime})$. The Fourier transform of
the dynamic spin susceptibility is therefore

$\displaystyle\chi(z,z^{\prime},\mathbf{K},\omega)=\chi_{0}\left[\delta(z-z^{\prime})+\frac{i\omega}{2D^{s}\widetilde{b}K}\left(e^{-\widetilde{b}K|z-z^{\prime}|}+e^{-\widetilde{b}K|z+z^{\prime}|}\right)\right].$
(S.33)

The spin induced magnetic noise is then

$\displaystyle\widetilde{\mathcal{B}}^{s}_{ii^{\prime}}(\omega)$
$\displaystyle=\left(g\mu_{B}\right)^{2}\int\frac{d^{2}\mathbf{K}}{(2\pi)^{2}}\frac{(2\pi)^{2}}{K^{2}}e^{-2Kd}\kappa_{i}{\kappa}_{i^{\prime}}^{*}\kappa_{j}{\kappa}_{j^{\prime}}^{*}\int_{-\infty}^{0}dz\int_{-\infty}^{0}dz^{\prime}\,e^{K(z+z^{\prime})}\,\frac{2k_{B}T}{\omega}\text{Im}\chi(z,z^{\prime},\mathbf{K},\omega)$
(S.34) $\displaystyle=\left(g\mu_{B}\right)^{2}\frac{2\pi
k_{B}T\chi_{0}}{D^{s}}\Delta_{ii^{\prime}}\int
dKe^{-2Kd}\,\text{Re}\left[\frac{2+\widetilde{b}}{\widetilde{b}(\widetilde{b}+1)^{2}}\right],$

which reproduces the result (S.25) from the Langevin approach.

###  B.2 B. Valley

In the same spirit, we consider the following diffusion and relaxation of the
valley degree of freedom

$\displaystyle\left(\partial_{t}+\frac{1}{\tau^{v}}\right)\rho^{v}+\bm{\nabla}\cdot\mathbf{j}^{v}=\widetilde{\epsilon}^{v},$
(S.35)

where $\rho^{v}=(\nu/2)\mu^{v}$, $\nu$ is the total density of states at the
Fermi surface, $\mu^{v}=(\mu^{+}-\mu^{-}$)/2 is the valley chemical potential,
$\mathbf{j}^{v}=-(\sigma/2)\bm{\nabla}\mu^{v}+\bm{\epsilon}^{v}$, and
$\tau^{v}$ is the valley relaxation time. The intravalley noise has the same
correlation as before
$\langle\epsilon^{v}_{i}(\mathbf{r},t)\epsilon^{v}_{i^{\prime}}(\mathbf{r}^{\prime},t^{\prime})\rangle=2\sigma
k_{B}T\delta_{ii^{\prime}}\delta(\mathbf{r}-\mathbf{r}^{\prime})\delta(t-t^{\prime})$,
and we also account for an intervalley fluctuation of the axial charge density
$\langle\widetilde{\epsilon}^{v}(\mathbf{r},t)\rangle=0$ and
$\langle\widetilde{\epsilon}^{v}(\mathbf{r},t)\widetilde{\epsilon}^{v}(\mathbf{r}^{\prime},t^{\prime})\rangle=(\nu/\tau^{v})2k_{B}T\delta(\mathbf{r}-\mathbf{r}^{\prime})\delta(t-t^{\prime})$.
The differential equation to be solved is then

$\displaystyle\left(\partial_{t}+\frac{1}{\tau^{v}}\right)\mu^{v}-D\bm{\nabla}^{2}\mu^{v}=\frac{2}{\nu}(-\bm{\nabla}\cdot\bm{\epsilon}^{v}+\widetilde{\epsilon}^{v}),$
(S.36)

with the Neumann boundary condition. The Green function in this case is

$\displaystyle\widetilde{\mathcal{G}}_{D}(\mathbf{r},\mathbf{r}^{\prime};t,t^{\prime})=\frac{\Theta(t-t^{\prime})}{[4\pi
D(t-t^{\prime})]^{3/2}}e^{-[(x-x^{\prime})^{2}+(y-y^{\prime})^{2}]/4D(t-t^{\prime})}\left(e^{-(z-z^{\prime})^{2}/4D(t-t^{\prime})}+e^{-(z+z^{\prime})^{2}/4D(t-t^{\prime})}\right)e^{-(t-t^{\prime})/\tau^{s}},$
(S.37)
$\displaystyle\widetilde{\mathcal{G}}_{D}(\mathbf{K},z,z^{\prime};\omega)=\frac{1}{2D\widetilde{a}K}\left(e^{-\widetilde{a}K|z-z^{\prime}|}+e^{-\widetilde{a}K|z+z^{\prime}|}\right),$

where $\widetilde{a}=\sqrt{(-i\omega+1/\tau^{v})/DK^{2}+1}$. Following Eqs.
(S.14-S.18),

$\displaystyle\widetilde{\mathbf{j}}^{v}(\mathbf{r},t)$
$\displaystyle=\bm{\epsilon}^{v}(\mathbf{r},t)-D\int_{\Omega}d^{3}\mathbf{r}^{\prime}\int_{-\infty}^{t}dt^{\prime}\,\left[\,\widetilde{\epsilon}^{v}(\mathbf{r}^{\prime},t^{\prime})+\bm{\epsilon}^{v}(\mathbf{r}^{\prime},t^{\prime})\cdot\bm{\nabla}_{\mathbf{r}^{\prime}}\right]\bm{\nabla}_{\mathbf{r}}\widetilde{\mathcal{G}}_{D}(\mathbf{r},\mathbf{r}^{\prime};t,t^{\prime}),$
(S.38) $\displaystyle\widetilde{\mathbf{j}}^{v}(\mathbf{K},\omega)$
$\displaystyle=\int_{-\infty}^{0}dz\,e^{Kz}\int d^{2}\mathbf{R}\int
dt\,e^{i\omega
t-i\mathbf{K}\cdot\mathbf{R}}\,\left[\bm{\epsilon}^{v}(\mathbf{r},t)-{\epsilon}_{z}^{v}(\mathbf{r},t)\hat{\mathbf{z}}\right]$
(S.39)
$\displaystyle-\frac{1}{2\widetilde{a}K^{2}}\int_{-\infty}^{0}dz^{\prime}\int
d^{2}\mathbf{R}^{\prime}\int dt\,e^{i\omega
t^{\prime}-i\mathbf{K}\cdot\mathbf{R}^{\prime}}\,\bm{\epsilon}^{v}(\mathbf{r}^{\prime},t^{\prime})\cdot\left({\widetilde{\textbf{\textkappa}}}^{\star}{\widetilde{\textbf{\textkappa}}}^{\star}\frac{e^{Kz^{\prime}}}{\widetilde{a}+1}+\widetilde{\textbf{\textkappa}}\widetilde{\textbf{\textkappa}}\frac{e^{Kz^{\prime}}-e^{\widetilde{a}Kz^{\prime}}}{\widetilde{a}-1}+\widetilde{\textbf{\textkappa}}{\widetilde{\textbf{\textkappa}}}^{\star}\frac{e^{\widetilde{a}Kz^{\prime}}}{\widetilde{a}+1}\right)$
$\displaystyle-\frac{i}{2\widetilde{a}K^{2}}\int_{-\infty}^{0}dz^{\prime}\int
d^{2}\mathbf{R}^{\prime}\int dt\,e^{i\omega
t^{\prime}-i\mathbf{K}\cdot\mathbf{R}^{\prime}}\,\widetilde{\epsilon}^{v}(\mathbf{r}^{\prime},t^{\prime})\left(\widetilde{\textbf{\textkappa}}\frac{e^{Kz^{\prime}}-e^{\widetilde{a}Kz^{\prime}}}{\widetilde{a}-1}+\widetilde{\textbf{\textkappa}}^{\star}\frac{e^{Kz^{\prime}}+e^{\widetilde{a}Kz^{\prime}}}{\widetilde{a}+1}\right),$

where
$\widetilde{\textbf{\textkappa}}=\mathbf{K}+i\widetilde{a}K\hat{\mathbf{z}}$
and
$\widetilde{\textbf{\textkappa}}^{\star}=\mathbf{K}-i\widetilde{a}K\hat{\mathbf{z}}$.

$\displaystyle\widetilde{\mathbf{B}}^{v}(\mathbf{r}_{\text{NV}},\omega)$
$\displaystyle=-\frac{g_{v}\mu_{B}}{v}\int\frac{d^{2}\mathbf{K}}{(2\pi)^{2}}\,\frac{2\pi}{K}e^{-Kd}\bm{\kappa}\bm{\kappa}\cdot\widetilde{\mathbf{j}}^{v}(\mathbf{K},\omega)$
(S.40)
$\displaystyle=\frac{g_{v}\mu_{B}}{v}\int\frac{d^{2}\mathbf{K}}{(2\pi)^{2}}\,\frac{2\pi}{K}e^{-Kd}\int_{-\infty}^{0}dz\int
d^{2}\mathbf{R}\int dt\,e^{i\omega
t-i\mathbf{K}\cdot\mathbf{R}}\,\bm{\kappa}\left[\left(-e^{Kz}\bm{\kappa}+\frac{e^{\widetilde{a}Kz}}{\widetilde{a}}\widetilde{\textbf{\textkappa}}\right)\cdot\bm{\epsilon}^{v}(\mathbf{r},t)+\frac{ie^{\widetilde{a}Kz}}{\widetilde{a}}\widetilde{\epsilon}^{v}(\mathbf{r}^{\prime},t^{\prime})\right].$

The magnetic noise turns out to be

$\displaystyle\widetilde{\mathcal{B}}^{v}_{ii^{\prime}}(\omega)$
$\displaystyle=\left(\frac{g_{v}\mu_{B}}{v}\right)^{2}\int\frac{d^{2}\mathbf{K}}{(2\pi)^{2}}\frac{(2\pi)^{2}}{K}e^{-2Kd}(2\sigma
k_{B}T)\kappa_{i}{\kappa}_{i^{\prime}}^{*}\left[\frac{(\widetilde{a}-1)(\widetilde{a}^{*}-1)(1+\widetilde{a}+\widetilde{a}^{*})+1/D\tau^{v}K^{2}}{(\widetilde{a}+\widetilde{a}^{*})\widetilde{a}\widetilde{a}^{*}}\right]$
(S.41) $\displaystyle=\left(\frac{g_{v}\mu_{B}}{v}\right)^{2}(4\pi\sigma
k_{B}T)\Delta_{ii^{\prime}}\int
K^{2}dKe^{-2Kd}\left[1-\frac{\widetilde{a}+\widetilde{a}^{*}}{2\widetilde{a}\widetilde{a}^{*}}\right],$

which goes back to Eq. (S.18) in the limit $1/\tau^{v}\rightarrow 0$. In the
opposite limit $\tau^{s}\rightarrow 0$, the equation for the chemical
potential is dominated by the intervalley fluctuation and relaxation, and
essentially decouples from the intravalley fluctuations and the valley current
transport. The latter continues to contribute to the magnetic noise
$\widetilde{\mathcal{B}}^{v}_{ii^{\prime}}(\omega)\rightarrow(g_{v}\mu_{B}/v)(\pi\sigma
k_{B}T/d^{3})\Delta_{ii^{\prime}}.$

##  Appendix C III. Simple metal

For a paramagnetic metal, the spin fluctuations are associated with electrons
on the Fermi surface and can be compared with the charge fluctuations on an
equal footing. The spin diffusion equation remains a valid description in this
scenario. We follow the discussion below Eq. (S.29), replacing $D^{s}$ and by
the charge diffusion constant $D$. For a system with a long spin relaxation
time, the magnetic noise induced by spin transport yields

$\displaystyle\frac{B^{s}_{zz}}{B^{c}_{zz}}\sim\left(g_{s}\mu_{B}\right)^{2}\frac{2c^{2}\chi_{0}}{e^{2}\sigma\omega
d^{2}}\int d\xi\,\xi^{2}e^{-2\xi}I^{s}(\xi,\zeta).$ (S.42)

For an estimation, we take $\chi_{0}$ to be the Pauli paramagnetic
susceptibility $\chi_{p}\sim\nu$ the density of states at the Fermi surface,

$\displaystyle\frac{B^{s}_{zz}}{B^{c}_{zz}}\sim\frac{g_{s}^{2}}{2}\left(\frac{c\alpha}{v}\right)^{2}\left(\frac{3a_{0}}{\ell_{\text{mfp}}}\right)^{2}\int
d\xi\,\xi^{2}e^{-2\xi}I^{s}(\xi,\zeta)/\zeta,$ (S.43)

resulting in a ratio of $0.001$ with $g_{s}\sim 2$, $v\sim 10^{5}$ m/s,
$\ell_{\text{mfp}}\sim 30$ nm, $D\sim v\ell_{\text{mfp}}/3$ and $\zeta\sim
0.5$. The presence of strong spin-orbit interactions may lead to an extremely
short spin relaxation time, such as in Pt, and thus the spin decay length can
be even shorter than the electron mean free path Bass and Pratt Jr. (2007).
For $v\tau^{s}\sim\ell_{\text{mfp}}$, an estimation of

$\displaystyle\frac{B^{s}_{zz}}{B^{c}_{zz}}\sim(g_{s}\mu_{B})^{2}\frac{c^{2}\chi_{0}\tau^{s}}{2e^{2}\sigma
d^{2}}\;\overset{\chi_{0}\sim\nu}{\sim}\;\frac{3g_{s}^{2}}{8}\left(\frac{c\alpha}{v}\right)^{2}\left(\frac{v\tau^{s}}{\ell_{\text{mfp}}}\right)\left(\frac{a_{0}}{d}\right)^{2}$
(S.44)

is also typically much smaller than $1$. Therefore, the charge fluctuations
dominate the magnetic noise in most circumstances for simple metals. However,
the spin susceptibility $\chi_{0}$ can be greatly enhanced near the Stoner
instability, leading to a larger contribution of the spin fluctuations to the
magnetic noise.


reddening the cQSO composite with a simple PL gives the best fit to the rQSO
composite. The dashed grey lines correspond to the major emission lines.

Fig. 15 displays the comparison between our cQSO composite, and those from V01
and S16, all normalized to 1450 Å. Below $\sim 4000$ Å, the three composites
are fairly similar. Above 4000 Å, the V01 template flattens out compared to
both S16 and our composites; this is expected due to the strong host-galaxy
contamination from the low redshift QSOs included in the V01 template. The S16
composite appears to be redder relative to our cQSO composite. Comparing the
$g^{*}-i^{*}$ colours of the seven X-shooter QSOs included in the S16
composite to our red and control QSO colour selections, we find that six have
colours consistent with our cQSOs and one is slightly redder (although not red
enough to be selected as an rQSO); this could be driving the differences in
the composite. Equally, the low source statistics in both of the composites,
the difference in luminosities between the samples, or the level of host-
galaxy contamination could also be affecting the shape.

##  Appendix E Spectra, [O iii] and C iv profiles

Figure 17: Finalized spectra for rQSO sample. The name, sample, whether a
source is radio-loud (RL) or radio-quiet (RQ) and the radio morphology is
displayed at the top right of each spectrum. The dashed grey lines correspond
to the major emission lines, labelled in the top panel. Figure 18: Finalized
spectra for cQSO sample, the including C15 sources. The name, sample, whether
a source is radio-loud (RL) or radio-quiet (RQ) and the radio morphology is
displayed at the top right of each spectrum. The dashed grey lines correspond
to the major emission lines, labelled in the top panel. Figure 19: Finalized
spectra for the QSOs excluded from our analyses (see Section 2.1); 2229-0832
is the Fermi-detected source and 2109+0045 and 2239+1222 had an incorrect
level of Galactic extinction originally applied that moves them out of our
rQSO threshold. The name, sample, whether a source is radio-loud (RL) or
radio-quiet (RQ) and the radio morphology is displayed at the top right of
each spectrum. The dashed grey lines correspond to the major emission lines,
labelled in the top panel.

Figs. 17, 18, and 19 display the spectra for the rQSOs, cQSOs and three QSOs
not included in our final sample (see Section 2.1), respectively.

Figure 20: C iv profiles for the cQSOs. The y-axis is scaled to 1.4 times the
maximum flux. The red and green lines indicate the narrow and broad Gaussian
fits, respectively, and the blue line shows the overall fit profile. The
source name and rest wavelength of C iv is also displayed. Figure 21: H
$\upbeta$–[O iii] profiles for the cQSOs. The y-axis is scaled to 1.3 times
the maximum flux. The red and green lines indicate the narrow core and broad
wing fits, respectively, and the blue line shows the overall fit profile. The
source name and whether it is radio-quiet (RQ) or radio-loud (RL) is also
displayed. Significant [O iii] detections are indicated by a star next to the
source name. Figure 22: C iv profiles for the rQSOs. The y-axis is scaled to
1.4 times the maximum flux. The red and green lines indicate the narrow and
broad Gaussian fits, respectively, and the blue line shows the overall fit
profile. The source name and rest wavelength of C iv is also displayed. The
rQSO 1049+1157 shows a strong BAL feature blueward of the C iv peak which will
have an effect on the quality of the fit; after removing this source from our
sample, we still do not find any differences in the C iv blueshift between the
rQSOs and cQSOs (see Section 7.2). Figure 23: H $\upbeta$–[O iii] profiles for
the rQSOs. The y-axis is scaled to 1.3 times the maximum flux. The red and
green lines indicate the narrow core and broad wing fits, respectively, and
the blue line shows the overall fit profile. The source name and whether it is
radio-quiet (RQ) or radio-loud (RL) is also displayed. Significant [O iii]
detections are indicated by a star next to the source name.

Figs. 20 and 21 display the C iv and H $\upbeta$–[O iii] thumbnails for the
cQSOs, respectively, and Figs. 22 and 23 display the C iv and H $\upbeta$–[O
iii] thumbnails for the rQSOs, respectively. The red and green lines indicate
the narrow and broad Gaussian fits, respectively, and the blue line indicates
the overall fitted profile. There are a number of rQSOs and cQSOs that display
both broad [O iii] and extreme C iv blueshifts; however, there is are no
significant differences between the rQSOs and cQSOs. This suggests that,
although both rQSOs and cQSOs can host powerful outflows, it is not clear that
they are more prevalent in the rQSO population.