$G^{<}_{lk}(t,t^{\prime})\equiv-i\left\langle\hat{b}_{k}^{\dagger}(t^{\prime})\hat{\beta}_{l}(t)\right\rangle$
and
$G^{<}_{kl}(t,t^{\prime})\equiv-i\left\langle\hat{\beta}^{\dagger}_{l}(t^{\prime})\hat{b}_{k}(t)\right\rangle$.
The time-ordered Green functions

$\displaystyle
G^{t}_{lk}(t,t^{\prime})=-i\theta(t-t^{\prime})\langle\hat{\beta}_{l}(t)\hat{b}_{k}^{\dagger}(t^{\prime})\rangle-i\theta(t^{\prime}-t)\langle\hat{b}_{k}^{\dagger}(t^{\prime})\hat{\beta}_{l}(t)\rangle$
(347)

evolve according to

$\Big{(}-i\frac{\partial}{\partial
t^{\prime}}-\omega_{k}\Big{)}G^{t}_{lk}(t,t^{\prime})=\sum_{l^{\prime}}g_{k,l^{\prime}}{G}_{ll^{\prime}}^{t}(t,t^{\prime}),$
(348)

where the time-ordered Green function for the magnons reads

$\displaystyle{G}^{t}_{ll^{\prime}}(t,t^{\prime})=-i\theta(t-t^{\prime})\langle\hat{\beta}_{l}(t)\hat{\beta}_{l^{\prime}}^{\dagger}(t^{\prime})\rangle-i\theta(t^{\prime}-t)\langle\hat{\beta}_{l^{\prime}}^{\dagger}(t^{\prime})\hat{\beta}_{l}(t)\rangle.$
(349)

In terms of the non-interacting phonon Green function $-i{\partial}/{\partial
t^{\prime}}-\omega_{k}\equiv[{\cal G}^{t}_{k}(t^{\prime})]^{-1}$,

$G_{lk}^{t}(t,t^{\prime})=\sum_{l^{\prime}}\int
dt_{1}g_{k,l^{\prime}}{G}_{ll^{\prime}}^{t}(t,t_{1}){\cal
G}_{k}^{t}(t_{1},t^{\prime}).$ (350)

It is convenient to introduce retarded $G^{r}$ and advanced $G^{a}$ Green
functions (as explained in textbooks [430]), in terms of which the “lesser"
Green function in frequency space reads

$G_{lk}^{<}(\omega)=\sum_{l^{\prime}}g_{k,l^{\prime}}\big{[}{G}_{ll^{\prime}}^{r}(\omega){\cal
G}^{<}_{k}(\omega)+{G}_{ll^{\prime}}^{<}(\omega){\cal
G}_{k}^{a}(\omega)\big{]}.$ (351)

and the phonons pumping rate becomes

$\displaystyle{\cal
P}_{R(L)}=-2\sum_{k>0(<0)}\sum_{ll^{\prime}}\int\frac{d\omega}{2\pi}{\rm
Re}\left({g_{k,l}^{*}G^{r}_{ll^{\prime}}(\omega)}g_{k,l^{\prime}}{\cal
G}^{<}_{k}(\omega)+g_{k,l}^{*}G^{<}_{ll^{\prime}}(\omega)g_{k,l^{\prime}}{\cal
G}^{a}_{k}(\omega)\right).$ (352)

Full chirality means that either of ${\cal P}_{R}$ and ${\cal P}_{L}$ should
vanish, in which case the pumped phonon flows unidirectionally into half of
the elastic substrate, making an analogy to the magnon and photons as
reviewed.

The “lesser” and advanced non-interacting phonon Green functions read ${\cal
G}_{k}^{<}(\omega)=2\pi if(\omega)\delta(\omega-\omega_{k})$, where
$f(\omega)=1/(e^{\hbar\omega/(k_{B}T)}-1)$ is the Planck distribution at
temperature $T$, and ${\cal
G}_{k}^{a}(\omega)={1}/({\omega-\omega_{k}-i\eta_{k}})$, respectively.
Equation (341) is the retarded magnon Green function $G_{N}^{r}(\omega)$ and
the spectral function ${\mathcal{A}}(\omega)=2i\textrm{Im}G^{r}_{N}(\omega)$.
According to the fluctuation-dissipation theorem [430]

$\displaystyle
G_{ll^{\prime}}^{<}(\omega)=iF(\omega){\mathcal{A}}_{ll^{\prime}}(\omega),$
(353)

where $F(\omega)$ is the non-equilibrium magnon distribution. The function
${\cal
I}(\omega)\equiv\sum_{ll^{\prime}}g_{k,l}^{*}G_{ll^{\prime}}^{<}(\omega)g_{k,l^{\prime}}$
obeys ${\cal I}^{*}(\omega)=-{\mathcal{I}}(\omega)$ and

$\displaystyle{\mathcal{P}}_{R(L)}$
$\displaystyle=\sum_{k>0(<0)}\sum_{ll^{\prime}}{\rm
Re}\left(ig_{k,l}^{*}G_{ll^{\prime}}^{r}(\omega_{k})g_{k,l^{\prime}}\right)\left(F(\omega_{k})-f(\omega_{k})\right)-\sum_{k>0(<0)}\sum_{ll^{\prime}}{\rm
Re}\left(ig_{k,l}^{*}G_{ll^{\prime}}^{r}(\omega_{k})g_{k,l^{\prime}}\right)f(\omega_{k})$
$\displaystyle-\sum_{k>0(<0)}\sum_{ll^{\prime}}{\rm
Re}\left(ig_{k,l}^{*}G_{ll^{\prime}}^{r*}(\omega_{k})g_{k,l^{\prime}}\right)F(\omega_{k}).$
(354)

At thermal equilibrium, $F(\omega_{k})=f(\omega_{k})$ and
${\mathcal{P}}_{R(L)}$ vanishes. We recognize the passive phonon transmission
in Eq. (340b) with
$\sum_{ll^{\prime}}ig_{k,l}^{*}G_{ll^{\prime}}^{r}(\omega_{k})g_{k,l^{\prime}}=c_{r}(1-S_{21}(\omega_{k}))$.

We are interested in the scaling with the number of identical nanowires
excited by a monochromatic microwave of frequency $\omega_{\rm K}$. With
$F(\omega_{k})\rightarrow f(\omega_{k})+\delta f\delta(\omega_{k}-\omega_{\rm
K})$ Eq. (354) reduces to

$\displaystyle{\mathcal{P}}_{R(L)}=({1}/{c_{r}}){\mathcal{T}}_{R(L)}\delta f,$
(355)

where

$\displaystyle{\mathcal{T}}_{R}$
$\displaystyle=\frac{1}{2\pi}\sum_{ll^{\prime}}{\rm
Re}\left(ig_{k_{*},l}^{*}(G_{ll^{\prime}}^{r}(\omega_{\rm
K})-G_{ll^{\prime}}^{r*}(\omega_{\rm K}))g_{k_{*},l^{\prime}}\right),$ (356a)
$\displaystyle{\mathcal{T}}_{L}$
$\displaystyle=\frac{1}{2\pi}\sum_{ll^{\prime}}{\rm
Re}\left(ig_{-k_{*},l}^{*}(G_{ll^{\prime}}^{r}(\omega_{\rm
K})-G_{ll^{\prime}}^{r*}(\omega_{\rm K}))g_{-k_{*},l^{\prime}}\right).$ (356b)

For Ni nanowires on top of a GGG substrate and a magnetic field tuned to the
critical angle at which ${\mathcal{T}}_{L}=0$, we arrive at an approximately
linear scaling of ${\cal T}_{R}$ with the number of wires, which implies that
a nanowire array of high magnetic quality [76] can be a high-power and
unidirectional source of SAWs.

Numerical simulation.—Yamamoto et al. numerically solved the coupled
magnetization and surface acoustic wave dynamics formulated in [82] for a
magnetic wire with arbitrary thickness [75, 76] in Fig. 53(a). In Fig. 53(b)
we show results for a 50 nm thick YIG wire on a $1~{}{\mu}$m thick GGG slab,
while wire magnetization at an angle $\phi$ relative to the phonon propagation
direction $\hat{\bf x}$. We conclude that i) when the wire magnetization is
normal to the wire ($\phi=0$), the SAWs on both sides of the YIG film have the
same amplitude (upper panel) as expected from the analytical results in Ref.
[75]; ii), The symmetry is increasingly broken with $\phi$ until at
$\phi=0.36\pi$ the excitation becomes fully unidirectional (lower panel), as
also predicted [76].

Via the numerical simulation Cai et al. found that the non-reciprocity can
persist in the higher harmonic generation of SAWs by the magnetization
dynamics [458].

Figure 53: Numerical simulations of the magnon-phonon hybridization in a
magnetic film (YIG, 50 nm thick) on a dielectric substrate (GGG, 1 $\mu$m
thick). (a) illustrates the configuration, in which the magnetization
precession excites Rayleigh SAWs exclusively into one direction. (b) shows
snapshots of the spin waves and SAW excited by a magnetic field pulse at $t=1$
ns for the equilibrium magnetization directions $\phi=0$ (upper panel) and
$0.36\pi$ (lower panel) indicated by black arrows. The figures are taken from
Ref. [82].

##  5 Near-field spintronics: transverse spins as spin sources

The efficient transfer of spin information over large distances and between
different entities is a key objective in spintronics. As summarized in Tables
4 and 5, the near fields of microwaves, spin waves, surface acoustic waves,
and surface plasmon polaritons carry (transverse) angular moment normal to the
wave vector. When they act as spin carriers in the medium, these transverse
spins of evanescent waves can be unidirectionally excited because they are
locked to the momentum or surface, as reviewed in above Sec. 4. On the other
hand, they can act as “spin" sources that pump the spin current of other
carriers such as electrons. These effective transverse “spins” can be
transferred to electrons in spintronic devices with considerable efficiency.
This holds the promise of “near-field spintronics" with a number of novel
functionalities. In this section, we review the theoretical foundations and
address the experimental progress.

###  5.1 Non-contact spin pumping by evanescent microwaves

As reviewed in Sec. 3.5.2, in the vacuum, an electromagnetic field with
electric and magnetic field components E and H at frequency $\omega$ carries a
spin angular momentum density [21, 206, 40, 41]

${\boldsymbol{\mathcal{D}}}(\textbf{r})=\frac{1}{4\omega}\operatorname{Im}\left(\varepsilon_{0}\mathbf{E}^{\ast}\times\mathbf{E}+\mu_{0}\mathbf{H}^{\ast}\times\mathbf{H}\right),$
(357)

where $\mu_{0}$ $(\varepsilon_{0})$ is the vacuum permeability (permittivity).
In a dispersive medium, the transverse spin is generally given by Eq. (166)
[21, 40, 41], to be addressed later. In the microwave regime, the magnetic
field component dominates. In evanescent fields, ${\boldsymbol{\mathcal{D}}}$
may be finite close to an interface or boundary even when it vanishes in the
bulk [206, 17]. A photon transverse angular momentum is generated by excited
ferromagnets, and at the special lines of microwave cavities or waveguides, to
name a few. In analogy with the spin pumping by magnetization into metal
contacts, we refer to the spin transfer from these near-fields to electrons as
“near-field spin pumping”. This process is contactless and avoids possible
artifacts by magnetic proximity. The reciprocal progress of near-field spin
pumping is near-field spin transfer, that is, the transfer of spin angular
momentum from electrons to other degrees of freedom via contactless
interactions.

####  5.1.1 Near-field spin pumping by dipolar fields

Spin pumping refers to the generation of a spin current at the interface
between an excited ferromagnet and an electric conductor, which is mediated by
the exchange interaction between magnetic moments at the interface and the
conduction electron spins [290, 388]. This process is efficient when the
magnet and conductor form a good electric contact, but is strongly suppressed
by, e.g., surface contamination, and Schottky barriers. A good contact between
the magnet and the conductor also favors proximity effects that may impede a
straightforward interpretation of experimental results. However, spins can
also be pumped with high efficiency without an exchange interaction, but by
the dipolar stray fields emitted by an excited ferromagnet [85]. These fields
decay only slowly as a function of distance and may pump spins remotely, e.g.,
through tunnel junctions.

We illustrate the process for the configuration in Fig. 54(a). Here we
consider a magnetic nanowire of width $w$, thickness $d$, and equilibrium
magnetization $\mathbf{M}_{s}$ pointing along the wire into the $\hat{\bf
y}$-direction, deposited on top of a thin metal film with a thickness $s$.
According to Eqs. (180) and (181), the stray field emitted by the Kittel mode
at the FMR frequency $\omega_{\mathrm{K}}$ [47, 57, 58], is circularly
polarized in the $\hat{\bf x}$-$\hat{\bf z}$-plane (or spin along $\hat{\bf
y}$) near the edge of the magnetic wire. Figure 54(b) shows the degree of
transverse spin of a magnetic wire with $d=w=60$ nm, as reviewed in Sec.
3.5.4. In the following, we derive the spin current generated by this drive in
a nonmagnetic electron gas [85].

Figure 54: Near-field pumping of an electron spin current in a thin metal film
by the stray field of precessing magnetization in a nearby wire. (a) shows the
configuration with geometric parameters. (b) is the calculated DC component of
the transverse spin generated by the stray field that pumps a constant spin
current into both directions as in (c). The exact result is calculated by Eq.
(371). The red dashed line labeled by “estimated" assumes $k_{x}=\pi/(2w)$ in
the spin susceptibility [Eq. (372)]. (d) shows the calculated spin injection
rates into an electron gas with weak (SC) and strong (TI) spin-orbit
couplings. The figures are taken from Ref. [85].

The theory in terms of the Green function is addressed in the effective one-
dimensional case. The Zeeman interaction of an electron spin with the dynamic
magnetic field $\mathbf{H}(x,t)$

$\displaystyle\hat{H}_{\mathrm{Z}}\left(t\right)=\mu_{0}\gamma_{e}\int\hat{\mathbf{s}}(x,t)\cdot\mathbf{H}(x,t)dx$
(358)

perturbs the system Hamiltonian $\hat{H}_{0}=\hat{p}^{2}/(2m)$ of two-
dimensional free electron gas and modifies the Heisenberg equation of motion
for the electron spin density $\hat{\mathbf{s}}$ operator

$\frac{\partial\hat{\mathbf{s}}(x,t)}{\partial
t}=\frac{i}{\hbar}\left[\hat{H}_{0}+\hat{H}_{\mathrm{Z}},\hat{\mathbf{s}}(x,t)\right]=\frac{i}{\hbar}\left[\hat{H}_{0},\hat{\mathbf{s}}(x,t)\right]-\mu_{0}\gamma_{e}\hat{\mathbf{s}}(x,t)\times\mathbf{H}(x,t),$
(359)

where we use the commutator
$[\hat{s}_{\beta}(x^{\prime},t),\hat{s}_{\alpha}(x,t)]=i\hbar\varepsilon_{\beta\alpha\delta}\hat{s}_{\delta}(x)\delta(x-x^{\prime})$.
Here, $\gamma_{e}=-g_{e}\mu_{B}/\hbar$, while $\mu_{B}$ and $g_{e}$ are the
Bohr magneton and (effective) electron $g$-factor, respectively. We chose the
Heisenberg representation in which the wave function does not depend on time,

$\left\langle\frac{\partial\hat{\mathbf{s}}(x,t)}{\partial
t}\right\rangle=\frac{\partial\left\langle\hat{\mathbf{s}}(x,t)\right\rangle}{\partial
t},$ (360)

and the interaction representation for the perturbation [258, 293] with
operators $\hat{A}_{I}(t)=e^{iH_{0}t/\hbar}\hat{A}_{S}e^{-iH_{0}t/\hbar}$
evolving in time by $\hat{H}_{0}$. The spin-density rate of change then
contains two terms

$\displaystyle\left\langle\frac{\partial\hat{\mathbf{s}}(x,t)}{\partial
t}\right\rangle$ $\displaystyle=\frac{\partial\left\langle
U^{\dagger}(t)\hat{\mathbf{s}}_{I}(x,t)U(t)\right\rangle}{\partial t}$
$\displaystyle=\frac{i}{\hbar}\left\langle
U^{\dagger}(t)\left[\hat{H}_{\mathrm{Z}}(t),\hat{\mathbf{s}}_{I}(x,t)\right]U(t)\right\rangle+\left\langle
U^{\dagger}(t)\frac{\partial\hat{\mathbf{s}}_{I}(x,t)}{\partial
t}U(t)\right\rangle,$ (361)

where

$U(t)=T_{t}\exp\left(-\int_{-\infty}^{t}dt^{\prime}\hat{H}_{\mathrm{Z}}(t^{\prime})\right)$
(362)

is the evolution operator governed by the perturbation, where $T_{t}$ is the
time-ordering operator.

In the interaction representation the pump magnetic field does not appear in
the operators, so for the free electron gas
$\hat{H}_{0}^{I}=\hat{p}_{I}^{2}/(2m)$ and in the absence of spin decay
processes we arrive at the spin-conservation that reads

$\frac{\partial\hat{\mathbf{s}}_{I}(x,t)}{\partial
t}=\frac{i}{\hbar}[\hat{H}_{0}^{I},\hat{\mathbf{s}}_{I}(x,t)]=-\frac{\partial}{\partial
x}\hat{\boldsymbol{\mathcal{J}}}_{I}(x,t),$ (363)

where by the Fermion operator $\hat{f}_{\alpha}^{I}(x)$ (with “hat" for not
confusing with the distribution function) with spin $\alpha$

$\hat{\boldsymbol{\mathcal{J}}}_{I}=\frac{\hbar^{2}}{4im}\hat{f}_{\alpha}^{I\dagger}(x)\boldsymbol{\sigma}_{\alpha\alpha^{\prime}}\partial_{x}\hat{f}_{\alpha^{\prime}}^{I}(x)+\mathrm{H.c}.$
(364)

is the spin current density operator. The formal exact expectation value for
the spin current is therefore

$-\frac{\partial}{\partial
x}{\boldsymbol{\mathcal{J}}}(x,t)=\frac{\partial\left\langle
U^{\dagger}(t)\hat{\mathbf{s}}_{I}(x,t)U(t)\right\rangle}{\partial
t}-\frac{i}{\hbar}\left\langle
U^{\dagger}(t)\left[\hat{H}_{\mathrm{Z}}(t),\hat{\mathbf{s}}_{I}(x,t)\right]U(t)\right\rangle.$
(365)

Expanding the divergence of the spin current to first order in the microwave
field $\mathbf{H}$:

$\displaystyle\left\langle
U^{\dagger}(t)\hat{s}_{\alpha}^{I}(x,t)U(t)\right\rangle$
$\displaystyle\rightarrow\left\langle\hat{s}_{\alpha}(x,t)\right\rangle_{l}=\frac{i}{\hbar}\int_{0}^{t}dt^{\prime}\left\langle\left[\hat{H}_{\mathrm{Z}}^{I}(t^{\prime}),\hat{s}_{\alpha}^{I}(x,t)\right]\right\rangle$
$\displaystyle=-\mu_{0}\gamma_{e}\int
dx^{\prime}dt^{\prime}\chi_{\alpha\beta}(x-x^{\prime},t-t^{\prime})H_{\beta}(x^{\prime},t^{\prime})+\mathcal{O}(\chi^{\left(2\right)}H^{2}),$
(366)

The first term is the spin susceptibility [258, 293]

$\chi_{\alpha\beta}(x-x^{\prime},t-t^{\prime})=i\Theta(t-t^{\prime})\left\langle\left[\hat{s}_{\alpha}^{I}(x,t),\hat{s}_{\beta}^{I}(x^{\prime},t^{\prime})\right]\right\rangle$
(367)

and we disregard the quadratic response function $\chi^{\left(2\right)}$. The
second term in Eq. (365)

$\displaystyle-\frac{i}{\hbar}\left\langle
U^{\dagger}(t)\left[\hat{H}_{\mathrm{Z}}(t),\hat{s}_{\alpha}^{I}(x,t)\right]U(t)\right\rangle$
$\displaystyle\rightarrow\frac{\mu_{0}\gamma_{e}}{\hbar}i\varepsilon_{\beta\alpha\delta}H_{\beta}(x,t)\int_{0}^{t}dt^{\prime}\left\langle\left[\hat{H}_{\mathrm{Z}}(t^{\prime}),\hat{s}_{\delta}(x,t)\right]\right\rangle=\mu_{0}\gamma_{e}\varepsilon_{\beta\alpha\delta}H_{\beta}(x,t)\hat{s}_{\delta}(x,t)|_{l}$
(368)

scales like $\sim\chi H^{2}$ because the excited spin density ${\bf s}_{l}$ is
proportional to the field as well [Eq. (366)]. The spin current conservation
law then reads

$-\nabla\cdot{\boldsymbol{\mathcal{J}}}(x,t)=\frac{\partial\left\langle\mathbf{s}\right\rangle_{l}}{\partial
t}+\mu_{0}\gamma_{e}\left\langle\mathbf{s}\right\rangle_{l}\times\mathbf{H},$
(369)

with two source terms on the right-hand side, i.e., a direct spin injection
and its precession in the applied field, since the dissipative component
$\left\langle\mathbf{s}\right\rangle_{l}\nshortparallel\mathbf{H}$ feels an
instantaneous torque. The DC spin current generation is only contributed by
the second nonlinear terms of ${\bf H}$.

The spin-orbit coupling also introduces torques that can be included in the
definition of the spin current [459] in order to enforce global spin
conservation with spin injection rate
${\boldsymbol{\mathcal{R}}}(t)=\left\langle
U^{\dagger}(t)\partial_{t}\hat{\mathbf{s}}_{I}(\boldsymbol{\rho},t)U(t)\right\rangle$
as the only source term. The spin-injection rate beneath an evanescent field
is a source term for the kinetic spin Bloch equations [460, 461], from which
we can calculate the spin transport with spin relaxation. We consider here two
limiting cases of weak and strong spin-orbit coupling that correspond to spin
current injection into a free two-dimensional electron gas and into the
surface states of the topological insulator. The calculated results in Fig.
54(c) and (d) reveal the high efficiency or the dipolar pumping process. The
spin current flows with the same polarization and equal magnitude into
opposite directions from the local source, as in the spin pumping by the
exchange interaction. This example proves that quite generally chiral
excitation by dipolar radiation is not caused by a hidden symmetry but
requires poles in the spin susceptibility generated by degenerate electron-
hole pairs or the plasmon, magnon, and phonon excitations of a rigid ground
state.

####  5.1.2 Conservation of photon transverse spin and electron spin

Here we address the conservation of the total transverse spin of the stray
field and electron system, which is not obvious from inspection of the
Hamiltonian Eq. (358) since the near-field spin pumping is intrinsically a
nonlinear process in terms of quadratic ${\bf H}$. The free electron gas is a
convenient system to address this conservation law. Its (Pauli) spin
susceptibility is [258, 293],

$\chi(k,\omega)=\frac{\hbar^{2}}{2}\sum_{\mathbf{q}}\frac{f(\xi_{\mathbf{q}})-f(\xi_{\mathbf{k}+\mathbf{q}})}{\hbar\omega+i0_{+}+\xi_{\mathbf{q}}-\xi_{\mathbf{k}+\mathbf{q}}},$
(370)

where $\xi_{\mathbf{k}}=\hbar^{2}k^{2}/(2m^{\ast})-\mu$ is the electron energy
with effective mass $m^{\ast},$ relative to the chemical potential $\mu$, and
$f(\xi_{\mathbf{k}})=\\{\exp[\xi_{\mathbf{k}}/(k_{B}T)]+1\\}^{-1}$ is the
Fermi-Dirac distribution at temperature $T$. In the microwave regime and not
too narrow magnetic wires, $\left|k_{x}\right|<2k_{F},$ where $k_{F}$ is the
2DEG Fermi vector [in semiconductors $k_{F}=O\left(\mathrm{nm}^{-1}\right)$]
and $\mathrm{Re}\chi(|k_{x}|,\omega\rightarrow
0)=m^{\ast}/\left(\pi\hbar^{2}\right)$. The DC spin current derived above then
reads [85]

$\displaystyle{\boldsymbol{\cal J}}_{x}^{\mathrm{DC}}(x)$
$\displaystyle=(\mu_{0}\gamma_{e})^{2}\int_{0}^{x}dx\sum_{k_{x}}e^{ik_{x}x}\left.\partial_{\omega}\mathrm{\operatorname{Im}}\chi(|k_{x}|,\omega)\right|_{\omega=0}\left\langle\dot{\mathbf{H}}(k_{x},t)\times\mathbf{H}(x,t)\right\rangle_{\mathrm{DC}},$
(371)

where we used the symmetry relations ${\boldsymbol{\cal
J}}_{x}^{\mathrm{D}C}(k_{x})=-{\boldsymbol{\cal J}}_{x}^{\mathrm{DC}}(-k_{x})$
in terms of the stray field Eq. (180) of the Kittel mode of a magnetic
nanowire and ${\boldsymbol{\cal J}}_{x}^{\mathrm{DC}}(x=0)=0$. In the low
frequency limit $\chi(|k_{x}|,\omega)\approx\chi(k_{\mathrm{ave}},\omega)$
with $k_{\mathrm{ave}}\sim\pi/(2w)$

$\displaystyle{\boldsymbol{\cal J}_{x}^{\mathrm{DC}}(x)}$
$\displaystyle\approx\left.-2(\mu_{0}\gamma_{e})^{2}\omega_{\mathrm{K}}\partial_{\omega}\mathrm{\operatorname{Im}}\chi\left(|k_{x}|\rightarrow
k_{\mathrm{ave}},\omega\right)\right|_{\omega=0}\int_{0}^{x}dx^{\prime}\mathrm{\operatorname{Im}}\left[\tilde{\mathbf{H}}^{\ast}(x^{\prime})\times\tilde{\mathbf{H}}(x^{\prime})\right].$
(372)

Hence, the DC spin-current below the nanowire is (approximately) proportional
to the transverse spin of the magnetic field, implying the transfer of the
photon spin angular momentum to the electron spin with an efficiency governed
by
$\left.\partial_{\omega}\mathrm{\operatorname{Im}}\chi\left(|k_{x}|\rightarrow
k_{\mathrm{ave}},\omega\right)\right|_{\omega=0}$. The spin current is
polarized in the $-\hat{\bf y}$-direction, i.e., opposite to the magnetization
direction of the nanowire. This implies the transfer of transverse spin of
near fields to the electron spins, i.e., a conservation law.

The experimental demonstration of the transfer of the transverse spin of the
evanescent magnetic field to the electron spin needs a measurement of the
transverse spin current of electrons. The future theoretical study should
address parameter dependence such as the dependence of spin current flows on
the magnetization direction. The numerical calculation predicted a transverse
spin current density $10^{-13}~{}{\rm J/m}$ pumped into an InAs 2DEG [85],
which is almost 2 orders of magnitude larger than what has been observed for
the spin pumping by a Py slab into the graphene [462]. We note that its signal
should indeed be much smaller with a small $g$ factor. We therefore cannot
exclude that the observations are caused by dipolar fields at the edge of Py
and not exchange interactions at the interface.

####  5.1.3 Chiral spin pumping into carbon nanotube and Luttinger liquid

The near-field spin pumping turns out to be sensitive to the electron
dispersion [85]. The spin pumping current is chiral when the spin
susceptibility displays singularities that indicate collective states (e.g.,
magnons in Sec. 4.2.2). However, 1D systems with linear dispersion at the
Fermi energy, such as metallic carbon nanotubes, are an exception since spin
pumping is chiral even without interactions.

Carbon nanotube.— Three-dimensional conventional conductors behave like non-
interaction Fermi liquids, but when they are spatially confined, the electron-
electron interaction can become important. Here we illustrate the chiral spin
pumping [50] into a correlated system, viz., a one-dimensional electron gas in
a metallic nanotube with linear dispersion (along the $\hat{\bf x}$-direction)
perpendicular to a close-by magnetic nanowire (along the $\hat{\bf
y}$-direction). We showed above that the spin density pumped by the Kittel
magnon with frequency $\omega_{\mathrm{K}}$ can be written

$\langle\hat{s}_{\alpha}(x,t)\rangle_{l}=-\mu_{0}\gamma_{e}\sum_{k}e^{ikx-i\omega_{\mathrm{K}}t}\chi(k,\omega_{\mathrm{K}})H_{\alpha}(k,\omega_{\mathrm{K}})+\mathrm{H.c.},$
(373)

where $H_{\alpha}(\left|k\right|,\omega_{\mathrm{K}})=0$ and
$H_{\alpha}(-\left|k\right|,\omega_{\mathrm{K}})\neq 0$ [Eq. (180)] are chiral
in momentum space [50, 48]. The spin susceptibility Eq. (370) in one dimension
is

$\chi_{F}(k,\omega)=\frac{\hbar^{2}}{2}\sum_{\tau}\sum_{q}\frac{n_{F}(\xi_{q,\tau})-n_{F}(\xi_{k+q,\tau})}{\hbar\omega+i0_{+}+\xi_{q,\tau}-\xi_{k+q,\tau}},$
(374)

where $\tau$ labels the valley degree of freedom. For $n$-doped metallic
carbon nanotube the dispersion close to the neutrality poin
$\xi_{q,\tau}=\hbar v_{F}\left|q\right|-\mu$ is linear, where $v_{F}$ is the
Fermi velocity. We are interested in the long wavelength response with
$\left|k\right|\ll\mu/(\hbar v_{F})=k_{F}$, so $k+q$ is on the same branch as
$q$ and

$\xi_{k+q,\tau}=\hbar v_{F}\left(\left|q\right|+k\mathrm{sgn}(q)\right)-\mu.$
(375)

A low temperatures $n_{F}(\xi_{q})=1-\Theta(\xi_{q})$ and

$\displaystyle\chi_{F}(k,\omega)$
$\displaystyle=\frac{\hbar^{2}}{2}\sum_{\tau}\left(\frac{\sum_{q>0}\left(\Theta(\xi_{k+q})-\Theta(\xi_{q})\right)}{\hbar\omega+i0_{+}-\hbar
v_{F}k}+\frac{\sum_{q<0}\left(\Theta(\xi_{k+q})-\Theta(\xi_{q})\right)}{\hbar\omega+i0_{+}+\hbar
v_{F}k}\right)$ $\displaystyle=\frac{\hbar
kL}{2\pi}\left(\frac{1}{\omega+i0_{+}-v_{F}k}+\frac{1}{\omega+i0_{+}+v_{F}k}\right),$
(376)

where the valley degree of freedom contributes a factor of 2. The response
becomes singular because the linear dispersion renders the excited electron
hole pairs degenerate. For positive frequency $\omega$, the two poles
$k_{\pm}=\pm(\omega/(v_{F})+i0_{+})$ are in the upper and lower complex plane,
respectively. When $x>0$ we close the contour in the upper complex plane,
which leads to a vanishing spin density, while for $x<0$

$\langle\hat{s}_{\alpha}(x,t)\rangle_{l}=i\frac{\mu_{0}\gamma_{e}\omega_{\mathrm{K}}}{v_{F}^{2}}H_{\alpha}(k_{-},\omega_{\mathrm{K}})e^{ik_{-}x-i\omega_{\mathrm{K}}t}+\mathrm{H.c.},~{}~{}~{}~{}x<0.$
(377)

Therefore, in contrast to the results for conventional conductors, the excited
spin density lives in only half of the nanowire! Consequently, the DC spin
current

$\displaystyle{\boldsymbol{\mathcal{J}}}_{x}^{\mathrm{DC}}(x)$
$\displaystyle={\boldsymbol{\mathcal{J}}}_{x}^{\mathrm{DC}}\left(x=\frac{w}{2}\right)+\left\\{\begin{array}[c]{c}-\mu_{0}\gamma_{e}\int_{\frac{w}{2}}^{x}dx^{\prime}\left.\mathbf{s}_{l}(x^{\prime},t)\times\mathbf{H}(x^{\prime},t)\right|_{\mathrm{DC}}\\\
0\end{array}~{}~{}\text{for }\begin{array}[c]{c}x<0\\\ x>0\end{array}\right.$
(382)
$\displaystyle\rightarrow\left\\{\begin{array}[c]{c}-\mu_{0}\gamma_{e}\int_{0}^{x}dx^{\prime}\left.\mathbf{s}_{l}(x^{\prime},t)\times\mathbf{H}(x^{\prime},t)\right|_{\mathrm{DC}}\\\
0\end{array}~{}~{}\text{for }\begin{array}[c]{c}x<0\\\ x>0\end{array}\right.$
(387)

flows only in the same half-space because
$\boldsymbol{\mathcal{J}}_{x}^{\mathrm{DC}}\left(x=\frac{w}{2}\right)\rightarrow
0$ as we show now. The Fourier components of the spin current are

${\boldsymbol{\mathcal{J}}}_{x}(k)=-\frac{\mu_{0}\gamma_{e}}{ik}\sum_{k^{\prime}}\mathbf{s}_{l}(k^{\prime},t)\times\mathbf{H}(k-k^{\prime},t).$
(388)

Since the poles of $s_{l}(k,t)$ lie in the lower complex plane and for
$x=w/2>0$,

$\displaystyle{\boldsymbol{\mathcal{J}}}_{x}\left(x=\frac{w}{2},t\right)$
$\displaystyle=\sum_{k}e^{ikx}{\boldsymbol{\mathcal{J}}}_{x}\left(k,t\right)=i\mu_{0}\gamma_{e}\sum_{k}\sum_{k^{\prime}}e^{i(k+k^{\prime})x}\frac{1}{k+i0_{+}+k^{\prime}}\mathbf{s}_{l}(k^{\prime},t)\times\mathbf{H}(k,t)=0$
(389)

by closing the integral path of $k^{\prime}$ in the upper complex plane. The
mathematical origin for the chiral behavior are the singularities in the
susceptibilities Eq. (376) at low frequencies that are absent in higher
dimensions.

Luttinger liquid.—Similar singularities in the response functions render the
instability of the ground state of the 1D electron gas when the interaction is
switched on in favor of the Tomonaga-Luttinger liquid, but the results above
still hold. The Hamiltonian in the presence of interactions becomes [258, 293]

$\hat{H}_{0}=\hbar
v_{F}\sum_{k\sigma}\left|k\right|\hat{f}_{k\sigma}^{\dagger}\hat{f}_{k\sigma}+\frac{1}{2L}\sum_{k}V_{k}\hat{\rho}(k)\hat{\rho}(-k),$
(390)

where $V_{k}$ is the Fourier component of Coulomb potential in (quasi) one
dimension, and the Fourier component of the density operator reads

$\hat{\rho}(k)=\sum_{q\sigma}f_{q-k/2,\sigma}^{\dagger}\hat{f}_{q+k/2,\sigma}.$
(391)

This model supports two kinds of Bosonic excitations [258], i.e., charge and
spin density waves with different group velocities. The spin density
excitations have a dispersion $\omega_{k}=\hbar v_{F}\left|k\right|$ and
susceptibility (cf. Chap. 4 of Ref. [258])

$\chi(k,\omega)=\frac{\hbar\left|k\right|L}{4\pi}\left(\frac{1}{\omega-\omega_{k}+i0_{+}}-\frac{1}{\omega+\omega_{k}+i0_{+}}\right).$
(392)

Only the first term in Eq. (392) contributes to the resonant response to the
external microwaves with a positive $\omega,$ leading to exactly the same
response as the non-interacting case Eq. (376) except for a factor of 2 for
the valley degree of freedom. The Coulomb interaction dramatically modifies
the ground state, but the spin-excitation channel is not affected by the spin
pumping, so the chirality remains intact.

We conclude that chirality requires a singular susceptibility, which in
general requires a collective excitation that is indicative of rigidity in the
ground state. The one-dimensional system illustrates that this is not the same
as long-range order. The long-range correlation in the Tomonaga-Luttinger
liquid caused by the linear (or linearized) dispersion at the Fermi energy in
1D systems is sufficient to cause chiral spin pumping.

###  5.2 Surface plasmon polariton and electron spin

Metallic nanostructures are efficient antennas for high-frequency
electromagnetic fields by virtue of their surface plasmons. The plasmon
polaritons confine light far beyond the wave length and thereby strongly
enhance field amplitudes. Magneto-plasmonics addresses, for example, amplified
magneto-optical effects [438, 463], surface plasmon resonance in ferromagnetic
metals [464], and the integration of plasmonics and spintronics [87, 86, 89,
88, 91]. Before addressing the spin-current generation by surface plasmon
polariton, we address here the inverse Faraday effect, i.e., the dynamics
generation of magnetization by light.

####  5.2.1 Inverse Faraday effect in metals

In magneto-optics, the Faraday (Kerr) effect refers to the rotation of a
linearly-polarized light plane by a magnetic field or moment in transmission
(reflection) [208]. Pitaevskii predicted [465] an inverse effect, viz., the
generation of a magnetic moment by circularly polarized light, by the transfer
of photon angular momentum to that of the electrons by a Raman scattering
process. Here we follow the derivation by Hertel [466] which sketches a simple
physical picture, based on the charge conservation in an electron gas with
density $n({\bf r},t)$ and velocity field ${\bf v}({\bf r},t)$

$\displaystyle\partial n/\partial t+\nabla(n{\bf v})=0,$ (393)

and Ohm’s Law

$\displaystyle{\bf j}\equiv en{\bf v}=\sigma{\bf E},$ (394)

where $\sigma=ine^{2}/(m\omega)$ is the reactive dynamical conductivity of the
free electron gas. The electron gas responds to an electric field ${\bf
E}({\bf r},t)=\boldsymbol{\cal E}e^{i({\bf k}\cdot{\bf r}-\omega t)}$ dc and
ac components:

$\displaystyle n({\bf r},t)$ $\displaystyle\rightarrow\langle n({\bf
r},t)\rangle+\delta n({\bf r},t),$ (395a) $\displaystyle{\bf v}({\bf r},t)$
$\displaystyle\rightarrow\langle{\bf v}({\bf r},t)\rangle+\delta{\bf v}({\bf
r},t).$ (395b)

The dc response is a non-linear down-conversion process that follows from
substituting Eqs. (395a) and (395b) into Eqs. (393) and (394), i.e.,

$\displaystyle\delta n$ $\displaystyle=-\frac{i}{\omega e}\nabla(\sigma{\bf
E}),$ (396a) $\displaystyle\delta{\bf v}$ $\displaystyle=\frac{\sigma}{\langle
n\rangle e}{\bf E},$ (396b)

leading to

$\displaystyle{\bf j}({\bf r},t)=\langle\delta n({\bf r},t)\delta{\bf v}({\bf
r},t)\rangle\rightarrow-\frac{i}{4e\langle
n\rangle\omega}\nabla\times(\sigma^{*}{\bf E}^{*}\times\sigma{\bf E}).$ (397)

According to Maxwell’s equation the current density ${\bf j}=\nabla\times{\bf
M}$, which implies an induced orbital magnetization

$\displaystyle{\bf M}=-\frac{|e|\varepsilon_{0}\omega_{\rm
P}^{2}}{4m\omega^{3}}{\rm Im}({\bf E}^{*}\times{\bf E}).$ (398)

This result can be derived from the angular momentum of electromagnetic waves
[467] [see Eq. (166) and below]. It can be generalized to include dissipative
media [468]. In magnetic semiconductors, off-resonant polarized light can
shift the band edges by the spin-selective ac Stark effect, which is
equivalent to a strong effective magnetic field [469]. The orbital
magnetization induced by the electric components of circularly polarized light
is a non-linear and reactive effect that is believed to be a mechanism for
ultrafast all-optical magnetization dynamics [470, 471, 472]. In the full
theoretical treatment of light-induced magnetization reversal by the inverse
Faraday effect, the spin-orbit interaction including spin-relaxation should be
included.

####  5.2.2 Spin current generation by surface plasmon polariton

We now address the close connection of the inverse Faraday effect and the
transverse spin of electromagnetic waves. The surface plasmon polariton, much
like the surface acoustic wave, is an evanescent mode that can carry angular
momentum (refer to Sec. 3.5). Oue and Matsuo [88, 89] predicted angular
momentum transfer from optical transverse spins captured by the surface
plasmon polaritons to the orbital motion of conduction electrons, leading to a
steady-state magnetization and electron spin current. When the magnetic field
is linearly polarized, only the electric field contributes to the transverse
spin angular momentum (SI unit)

$\displaystyle{\bf S}$ $\displaystyle=\frac{1}{4\omega}{\rm
Im}\left(\varepsilon_{0}\frac{d(\omega\varepsilon_{r})}{d\omega}{\bf
E}^{*}\times{\bf E}+\mu_{0}\frac{d(\omega\mu_{r})}{d\omega}{\bf
H}^{*}\times{\bf H}\right)$
$\displaystyle=\left(\frac{\varepsilon_{0}\varepsilon_{r}}{4\omega}+\frac{\varepsilon_{0}}{4}\frac{d\varepsilon_{r}}{d\omega}\right){\rm
Im}\,{\bf E}^{*}\times{\bf E}\equiv{\bf S}_{\rm EM}+{\bf S}_{m}.$ (399)

The first term in the second line is purely electromagnetic, while the second
term invokes the conduction electron dynamics that affects the dielectric
constant. With gyromagnetic ratio $-|e|/(2m)$ and
$\varepsilon_{r}=1-\omega_{\rm P}^{2}/\omega^{2}$,

$\displaystyle{\bf M}=-|e|/(2m){\bf
S}_{m}=-\frac{|e|\varepsilon_{0}\omega_{\rm P}^{2}}{4m\omega^{3}}{\rm Im}({\bf
E}^{*}\times{\bf E}),$ (400)

which agrees with Eq. (398). When the surface plasmon polariton propagates
along the $\hat{\bf x}$-direction as in Fig. 55(a), the rotating electric
field lies in the $x$-$z$ plane and generates an in-plane magnetization along
$\hat{\bf y}$ that decays from the surface on the length scale $\lambda_{\rm
SP}$, as also addressed in Sec. 3.3.

Figure 55: Generation of spin currents by surface plasmons. (a) A surface
plasmon induces a constant magnetization profile whose gradient drives a spin
current in the metal. (b)The edge plasmon of a graphene ribbon is the one-
dimensional equivalent of the surface plasmon. It propagates along the edges
and excites an out-of-plane spin polarization and current between the edges.
(c) An experimental setup for a Pt|YIG bilayer with embedded Au nanoparticles.
The spin pumping is found to be enhanced at the surface plasmon resonance in
(d), providing evidence for the spin-current generation by surface plasmon
resonance. The figures are taken from Refs. [88] [(a)], [90] [(b)], and [86]
[(c) and (d)].

The magnetization is generated by a steady state orbital motion of the
electrons and its stray field equals that of their Oersted field. Its gradient
may drive an angular momentum current $-(\hbar\sigma_{0}/m)\nabla M_{y}$ in a
diffuse metal [88, 89], with conductivity $\sigma_{0}$. Oue et al. [88, 89]
treated the gradient as a source term in the spin diffusion equation. In terms
of the spin accumulation $\delta\mu$ [473]

$\displaystyle\nabla^{2}\delta\mu=\frac{\delta\mu}{\lambda_{s}^{2}}+\frac{\hbar
e}{m}\nabla^{2}M_{y},$ (401)

where $\lambda_{s}$ is the spin diffusion length. Here the connection between
the magnetization generated by the rotating electric field couples to the
electron spin by spin-flip relaxation that equilibrates the system in the
presence of the Zeeman field in favor of a net spin polarization [88, 89].

However, the gradient of a magnetic field should make a different effect from
that of an electric field, with the latter drifting the system that is
balanced by the momentum scattering. It is not really possible to bias the
spin system by the gradient of magnetic field in the steady state since the
spin flip quickly balances the spin-polarization gradient. Future studies
should also address how the electron spins couple to the orbital magnetization
that is induced optically.

The diffusive DC spin current driven by this spin accumulation is then

$\displaystyle{\bf
J}_{s}=\frac{\sigma_{0}}{e}\nabla\delta\mu=-\frac{2(2\lambda_{s}/\lambda_{\rm
SP})^{2}}{(2\lambda_{s}/\lambda_{\rm
SP})^{2}-1}\frac{\hbar\sigma_{0}}{m}\nabla M_{y}.$ (402)

The estimated magnitude of the spin current may achieve $10^{5}~{}{\rm
A/m^{2}}$ [88].

In Fig. 55(a) the excited transverse magnetization and spin polarization lie
in the plane of the surface, while the spin current flows perpendicular to it,
which renders it difficult to detect. Ukhtary et al. [90] proposed to employ
the one-dimensional plasmons that propagate along the edges of graphene
ribbons [474], that generate an out-of-plane spin polarization and current
that flows between the edges as shown in Fig. 55(b). The induced magnetization
$\mu_{0}M_{z}(x)/d$, where $d$ is the graphene thickness, polarizes the spin
of the electron in the direction of the magnetization. The spin accumulation
is proportional to the difference between spin-up $n_{\uparrow}$ and spin-down
$n_{\downarrow}$ electron densities [473]

$\displaystyle\delta\mu_{\rm
sou}(x)=\frac{2}{N_{0}}\left(n_{\uparrow}(x)-n_{\downarrow}(x)\right)=\frac{2\mu_{0}\mu_{B}}{d}M_{z}(x),$
(403)

where $N_{0}$ is the density of states at the Fermi energy. The out-of-plane
magnetization drives a spin current between the edges.

Experiments.—References [86, 91] reported experimental observation of spin
current generation by surface plasmon resonance. The experiment [86] employs
Au nanoparticles embedded in films composed by the paramagnetic Pt and
ferrimagnetic BiY2Fe5O12 layers, as illustrated in Fig. 55(c), illuminated by
unpolarized monochromatic light with wavelength $400$-$770$ nm. Surface
plasmon resonance happens at light wavelength $690$ nm. The voltage at the two
ends of the Pt layer comes from the inverse spin Hall effect and is measured
in device Fig. 55(c). The voltage is found to be enhanced when illuminated by
light at wavelength $690$ nm, as shown by the peak of the blue solid curve in
Fig. 55(d). This enhancement of spin pumping was attributed to the excitation
of magnons in BiY2Fe5O12 by the near field of surface plasmon polaritons [86].
The spin generation may be also explained by the spin Seebeck effect by
enhanced heating at the resonance of surface plasmon polariton. However,
recent measurements in Ref. [91] reported the direct spin-current generation
by surface plasmon resonance in the absence of the magnons.

###  5.3 Ac spin current generation by surface acoustic waves

We derived in Sec. 3.4.3 that the volume elements in surface acoustic waves
(SAWs) carry out a circular motion. This transverse phonon spin is locked to
its momentum and can drive electronic spin dynamics [475, 476, 477, 478], as
briefly reviewed in this subsection.

The close relation between electron angular momentum and mechanical angular
momentum was revealed by Einstein and de Haas in 1915 by measuring the
mechanical torque generated when reversing the magnetization of an iron
cylinder. The reciprocal effect, i.e., the induced magnetization by mechanical
rotation, was discovered by Barnett [479, 480]. A rigid rotation with angular
velocity $\boldsymbol{\omega}$ that in a rotating frame corresponds to an
effective Barnett field $\boldsymbol{\cal B}=\boldsymbol{\omega}/\gamma$ that
affects the magnetization by the Zeeman interaction $-{\bf
M}\cdot\boldsymbol{\cal B}$. The Barnett field is thus not a real field, but a
gauge field that arises when transforming to the rotation frame. It has no
effect unless there is relaxation to the ground state. The Barnett effect can
be employed to manipulate the magnetization of thin films and nanostructures
[481].

Matsuo et al. studied the mechanical generation of ac spin currents in non-
magnets by mechanical rotation and elastic deformations [482, 483, 484, 485,
486] starting from the relativistic Dirac equation. The electron spin ${\bf
s}$ couples with the Barnett-field density $\boldsymbol{\cal B}({\bf r},t)$
via the spin-rotation coupling [475, 476, 477, 478]

$\displaystyle\hat{H}_{\rm SR}=-\int{\bf s}({\bf r})\cdot\boldsymbol{\cal
B}({\bf r},t)d{\bf r}.$ (404)

A time-dependent effective magnetic field may thus parametrically pump an
electron spin current. In the coordinate system Fig. 56(a), the displacement
field of an SAW eigenmodes Eq. (147) reads

$\displaystyle u_{x}$
$\displaystyle=i\frac{s}{|k|}u_{0}\left(\frac{2k^{2}}{k^{2}+s^{2}}e^{-qy}-e^{-sy}\right)e^{ikx-i\omega_{k}t},$
(405a) $\displaystyle u_{y}$ $\displaystyle=-{\rm
sgn}(k)u_{0}\left(\frac{2qs}{k^{2}+s^{2}}e^{-qy}-e^{-sy}\right)e^{ikx-i\omega_{k}t},$
(405b)

where $u_{0}$ is the mode amplitude. The vorticity of fluid induces the spin-
vorticiy coupling in metallic fluids such as mercury [487]. The rotational
motion corresponds to an effective Barnett gauge field [482, 483, 484, 485,
486]

$\displaystyle\boldsymbol{\cal B}({\bf r},t)=\frac{1}{2}\nabla\times\dot{\bf
u}=\frac{1}{2}\omega_{k}\eta^{2}|k|u_{0}e^{-sy}e^{ikx-i\omega_{k}t}\hat{\bf
z}=\frac{1}{2c_{t}}\omega^{2}_{k}u_{0}e^{-sy}e^{ikx-i\omega_{k}t}\hat{\bf z},$
(406)

which is linearly polarized along the $\hat{\bf z}$-direction and not locked
to the propagation direction. In linear response, the induced spin
polarization ${\bf s}_{l}({\bf r},t)$ of a free electron gas is parallel to
the Barnett field which does not provide a torque on ${\bf s}_{l}({\bf r},t)$
or pump a dc spin current. On the other hand, the rate of change $\partial{\bf
s}_{l}({\bf r},t)/\partial t$ is a source for an ac spin current. SAW phonon
therefore cannot transfer their spins to conduction electrons, however,
different from electromagnetic waves.

Matsuo et al. solved the spin diffusion equation for the spin accumulation
$\delta\mu$ driven by the time-dependent Barnett gauge field Eq. (406) [482,
483, 473]

$\displaystyle\left(\frac{\partial}{\partial t}-D\nabla^{2}+\frac{1}{\tau_{\rm
sf}}\right)\delta\mu=\hbar\frac{\partial{\cal B}}{\partial t},$ (407)

where $D$ is the spin diffusion constant and $\tau_{\rm sf}$ is the spin-flip
time. According to Ohm’s Law, the ac spin current $J_{s}$ is polarized along
the $\hat{\bf z}$-direction and proportional to the conductivity $\sigma_{0}$.
In the strong scattering regime $\omega_{k}\tau_{\rm sf}\ll 1$

$\displaystyle
J_{s}=({\sigma_{0}}/{e})\nabla\delta\mu\approx\frac{\hbar\sigma_{0}}{2e}\tau_{\rm
sf}\omega_{k}^{4}\frac{u_{0}}{c_{t}^{2}}\frac{\sqrt{1-\eta^{2}}}{\eta}e^{-sy}e^{ikx-i\omega_{k}t}.$
(408)

It follows the spatiotemporal SAW profile, being a propagating wave in the
plane and evanescent normal to it. The ac spin current is proportional to the
spin-flip time, which is usually short for metals that have a large spin Hall
angle. This hinders its observation by the inverse spin Hall effect that
transforms it into an easily detectable transverse emf [488, 489, 490], which
can also detect an alternating spin current [491, 492, 493].

Experiments.—Instead, Kobayashi et al. detected the ac spin current via the
spin transfer torque effect on a ferromagnetic contact [494], as shown in Fig.
56. Here a NiFe|Cu bilayer is fabricated on a LiNbO3 substrate in which SAWs
are sent from IDT 1 to IDT 2. Matching the FMR and SAWs frequencies opens a
resonant dissipation channel between the phonons and the magnetic order that
damps the SAW transmission amplitude. Figure 56(c) shows the deduced
absorption power of the ferromagnet as a function of frequency $f$, for a
magnetic field parallel to the SAW propagation direction. The observed
resonant absorption is suppressed when rotating the field or inserting an
insulating SiO2 film, providing additional evidence that the Barnett effect
can generate a spin current.

However, a direct magneto-mechanical effect cannot yet be excluded with
certainty, since very similar effects might exists without the Cu spacer
layer, as summarized in Table 7. These experiments discovered non-reciprocal
propagation of SAWs rendered by magnetic cappings, which can be explained by
magnetostriction.

Figure 56: Generation of alternating spin currents in metal films by surface
acoustic waves (SAWs). (a) sketches the concept that the Barnett spin-rotation
coupling generates an alternating spin current that can be measured in the
configuration in panel (b) with a NiFe|Cu bilayer on top of a LiNbO3
substrate. Here the IDTs excite and detect the SAWs. (c) shows the absorption
power of the ferromagnet as a function of an applied magnetic field and the
SAW frequency. The figures are taken from Refs. [482] [(a)], and [494] [(b)
and (c)].

##  6 Summary and outlook

To summarize, we have reviewed the universal physical mechanisms of the
locking of the transverse spins of evanescent waves (or quasiparticles in
their quanta) in spintronic nanostructures to their momentum and surface
normal of their propagation plane. We have addressed the consequences of this
generalized spin-orbit interaction that have been theoretically and
experimentally explored in spintronics, magnetism, and magnonics.

The electron’s spin is an intrinsic degree of freedom originating from the
relativistic effect. The transverse spin, on the other hand, is a
generalization of the relativistic spin. As summarized in Fig. 57, many of
these transverse spins are even approximately conserved when transferring to
the electron spin via the non-contact spin pumping and spin transfer
processes, which thus may be useful family members of the “spin" carriers in
spintronics that may overcome the shortcomings of electron spin in the
generation, transport, and detection. For example, magnons, phonons, and
photons are important information carriers with very low dissipation in many
materials and devices.

The relativistic effect also induces the spin-orbit interaction of electrons.
It favors many chiral magnetic textures in magnetism and renders the static
chiral interaction between nanomagnets possible, which was successfully
exploited in the experiments for realizing logic operations. We have reviewed
that by the evanescence of many waves, rather than the relativistic effect,
the generalized transverse spin can be locked to its momentum (and surface
normal). Thus the chirality acts as the generalized spin-orbit interaction.
When these waves or quasiparticles interact with another object holding a
definite transverse “spin" due to the time-reversal symmetry breaking
(typically, magnons as reviewed, or circularly polarized electric dipoles),
the object favors to interact with the waves with the same spin, thus the
pumped waves propagating in one direction via the generalized spin-orbit
interaction. Figure 57 summarizes various chiral couplings, as highlighted by
“chiral" in blue, between these quasiparticles that render the interaction
non-reciprocal and even unidirectional. Many progresses in demonstrating the
chiral interaction and novel spin states and functionalities in this aspect
have been reviewed in this article.

Figure 57: Excitations with generalized transverse “spins" and chiral objects
in spintronic nanostructures. Examples contain the itinerant magnons in the
magnetic films, the confined magnons in the magnetic nanowires or spheres, the
evanescent microwaves from the stripline, waveguide or cavity, the near
optical fields of electric dipoles, the surface phonons, and the surface
plasmon polaritons. The spins of electrons can be along any direction when
itinerant, but can be fixed chirally in the chiral magnetic textures when they
are local. The mutual interaction between these quasiparticles allows the
transfer of (transverse) spin between each other. In particular, for the
quasiparticles holding transverse spins, their interaction can be chiral, as
highlighted by “chiral" in blue, which facilitates the unidirectional, non-
reciprocal, and sensitive control of spin transport and operation.

This chiral interaction remains to be exploited for realizing functional
spintronics devices (Sec. 6.1). The present theoretical and experimental
studies are performed favorably in one-dimensional structure and largely in
the linear regime, but the inevitable additional dimension and nonlinearity in
the practical devices can enrich the spintronics functionality as well (Sec.
6.2). The generalization of the spin concept is not limited to the transverse
case, but may also allow to include the mixed longitudinal spin, and, in a
broader view, the orbital angular momentum of wave beams (Sec. 6.2). It has
important implications for other materials such as low-dimensional,
antiferromagnetic, synthetic, and even meta-materials (Sec. 6.3). The
presented theory in this article is formulated in terms of quantum description
via Hamiltonian and operators, which becomes classical when replacing
operators with amplitudes, thus friendly to include the quantum effect such as
quantum fluctuation, quantum noise, and quantum qubit when going to the low
temperature, which we envision that chirality may bring unique functionalities
in quantum sensing, communication, and metrology based on spintronics in the
future (Sec. 6.4). We provide a detailed outlook for both static and dynamic
chiralities below.

###  6.1 Functional spintronic devices

####  6.1.1 Wave isolators

Optical, microwave, acoustic, spin-wave, and electric-polarization isolators
are those devices that block the propagating waves in one direction but allow
them to pass in the opposite direction. As reviewed in this article, the non-
reciprocal transmission of spin waves, microwaves, and acoustic waves etc. are
widely proposed when placing magnets in proximity to the magnetic films,
microwave waveguides, and dielectric substrate via the chiral interaction
mechanism [50, 48, 51, 49, 52, 53, 47, 56, 54, 55, 57, 58, 60, 61, 63, 68, 70,
66, 67, 59, 62, 64, 65, 71, 72, 73, 74, 75, 77, 78, 81, 80, 76, 79, 82, 83,
84, 86, 87, 88, 89, 91]. In particular, via this mechanism the non-reciprocal
transmission of microwaves [68] and surface acoustic waves [440, 443, 81, 80,
79, 78, 83, 84] have been experimentally observed. However, a simple non-
reciprocal transmission of waves does not imply the realization of an isolator
because an excellent isolator requires that the transmission in one direction
is completely blocked. As a wave phenomenon, the phenomenology of an isolator
is well captured by the scattering matrix in such as a two-terminal device in
Fig. 58, where $a_{1,2}$ is the input amplitude and $b_{1,2}$ is the output
amplitude of waves.

To date, chiral interaction via the generalized spin-orbit interaction
promises two schemes of wave isolators. Figure 58(a) shows that due to the
chiral interaction, the propagating waves with amplitude $a_{1}$ can interact
with the object while $a_{2}$ does not, via which the dissipation of the
object dampens $a_{1}$ with zero output $b_{2}$, but does not affect $a_{2}$.
The associated scattering matrix in Fig. 58(a) implies the unidirectional
transmission $e^{ikL}$, where $k$ is the wave vector and $L$ is the length of
the object. In this mechanism, the object itself suffers from the (chiral)
radiative damping as well when in proximity to the propagating waves. An
excellent isolator is thereby achieved only when balancing the radiative
damping and the wave absorption [76]. One can also use the gating effect to
achieve an isolator as shown in Fig. 58(b) that the gate only shifts the
dispersion of waves of one direction and thus block the waves from this
direction with a total reflection, such as the input $a_{1}$, but allows the
propagation via input $a_{2}$, which also leads to the broadband
unidirectional functionality without much complicated design [243, 436, 437].
This mechanism was proposed in gating the magnon by metal with interfacial DMI
[436, 437] without chiral interaction and superconductor with chiral
interaction [243], but may be generalized in other devices as well in the
future.

Figure 58: Wave isolators by chiral interactions with definitions in terms of
the different scattering matrices. $a_{1,2}$ is the input amplitude and
$b_{1,2}$ is the output amplitude of waves. In (a), the chiral interaction
renders that only the waves propagating in one direction with input $a_{1}$
interact with the object (the green scattering region) that dampens the
propagation unidirectionally, which is promising for the realization of an
isolator. In (b), the gate (the gray region) only shifts the dispersion of
waves propagating in one direction and thus blocks the waves with input
$a_{1}$ from this direction with a total reflection but allows the propagation
via input $a_{2}$, which also leads to the broadband unidirectional
functionality.

####  6.1.2 Diodes

A diode is a two-terminal (electronic) element that conducts momentum,
particle, or energy current primarily in one direction with asymmetric
conductance. That is, it has low resistance in one direction, but high
resistance in the other. In electronics, this is achieved via the nonlinear
modulation of the device. Obviously, this functionality is not defined via the
scattering matrix like the isolators (Sec. 6.1.1), but the conductance for the
current. This device does not require unidirectionality but achieves a high
on-off ratio may have this request.

Non-reciprocal wave transmission is a coherent process, but the waves in terms
of wave packets behave like a particle as well, so they carry current with the
group velocity. In this sense, the chiral waves reviewed in this article are
candidates for the realization of various diodes. Indeed, the thermal
incoherent transport in terms of these quasiparticles conducts thermal current
that is governed by thermal fluctuation. An important difference to the
electronics diodes occurs as well since the non-reciprocity holds in the
linear response, seemingly acting like a Maxwell demon. The behavior of
thermal noise itself under the chiral interaction, even at the equilibrium, is
of fundamental interest, e.g., the accumulation of spin noise at one edge,
which as well as holds the possible application such as the enhanced sensing
efficiency with (quantum) noise [495].

The control of thermal current in many spintronic nanostructures may have
straightforward applications in thermal control in miniaturized devices since
when the devices become smaller it is essential to steer thermal current or
noise away from the hot spots towards sinks. As reviewed, the thermal
transport in this respect is only studied in the context of the chiral spin
Seebeck effect theoretically [50] and experimentally [304]. Thereby this
largely unexplored area with transport in terms of various chiral
quasiparticles and their interplay awaits further theoretical explorations and
experimental observations in the future.

####  6.1.3 Logic device

The static chiral coupling between nanomagnets turns out to be useful for
logic devices via the domain wall [7, 8]. Figure 59 shows the complete set of
domain-wall logic elements, i.e., NOT gate, NAND gate, NOR gate, cross-over,
switch, fan-out, and diode in terms of the chirally coupled nanomagnets [8].
The domain-wall logic scheme can be easily integrated with magnetic
“racetrack" memories [104, 496, 105], allowing a logic-in-memory architecture,
which saves time and power. However, several technical challenges prevent as
yet industrial applications, such as intrinsic/extrinsic domain-wall pinning,
efficient domain-wall writing, and non-destructive reading [497]. Moreover,
the demands on a domain-wall logic device such as tunable magnetic anisotropy
and sizable DMI can be met only by several materials and requires additional
fabrication steps.

Figure 59: Complete set of domain-wall logic elements. i.e., NOT gate, NAND
gate, NOR gate, cross-over, switch, fan-out, diode. Red and blue shaded
regions indicate out-of-plane and in-plane anisotropy, respectively. The
direction of the domain wall flow is indicated by black arrows. The indicated
dimensions are those of experimental devices. The figures are reproduced with
permission from [8].

The spin-wave logic, or its extension to the devices with other waves, is an
important application in magnonics [498]. The proposed isolators or diodes
above may act as important elements in the spin-wave logic device.
Importantly, the chiral interaction promises that the pumped waves propagate
unidirectionally in only half space [50, 48, 51, 49, 52, 53, 47, 60, 61, 66,
67, 59, 64, 65, 71, 72, 73, 74, 75, 77, 78, 81, 80, 76, 79, 82, 83, 84, 86,
87, 88, 89, 91], which is, therefore, a switch that is governed by the
magnetization direction. The magnon gate by proximity superconductor becomes a
perfect switch for the spin waves that are controlled by the temperature
[243]. The future study may benefit from the dynamical chirality with the
generalized spin-orbit interaction to realize various spin-wave gates similar
to those by the static chiral interaction in Fig. 59.

###  6.2 Other degrees of freedom

####  6.2.1 High spatial dimension

Most chiral physics in this review is emphasized by the effective one-
dimensional models since the chirality index $Z=\hat{\bf
k}\cdot(\hat{\boldsymbol{{\bf n}}}\times\hat{\boldsymbol{\sigma}})$ is a
maximal integer $+1$ or $-1$, but in the other propagation directions, the
waves are affected by the (non-integer) chirality index as well. Examples may
vary from the Damon-Eshbach spin waves, magnetic fields of spin waves and
short striplines to the electric fields of short dipoles wires and electric
polarizations, where the waves’ spins become longitudinal with $Z=0$ only when
the waves are propagating along the magnetization or the wire directions.
Interestingly, the non-reciprocal coupling between magnons of nanowires and
films strongly depends on the propagation direction of magnons in the film, in
which the chirality completely vanishes at particular angles [50, 48, 51, 49,
52]. The chirality in the magnetoelastic coupling between proximity magnet and
surface acoustic waves is strongly modulated by the magnetization direction
[75, 76, 82]. These provide flexible tunability of chirality of the coherent
waves that can be switched on and off by changing the propagation direction of
waves. The associated dependence of chirality on the spatial dimension via the
generalized spin-orbit interaction thereby may be distinguished from the other
non-reciprocal mechanisms in various experiments in the future.

In incoherent thermal transport, the quasiparticles of all propagation
directions play a role, but the chirality survives for a large fraction of
them, thus easily manifesting in the transport measurement and providing
steering on the generation and control of thermal current. These thermal
dynamics are important design parameters in future spintronic nanostructures.

####  6.2.2 Nonlinearity

To date, the dynamical chirality in spintronics is focused more on the linear
regime. Despite its great success in demonstrating the key mechanisms and easy
observation, sometimes beyond the haven of linearity brings new functionality.
The magnetization switching by spin current relies on a large spin flow. Also,
the nonlinearity appears to be unavoidable in the miniaturized devices where
the relative driving force becomes strong and causes a large current flow.
Chiral interaction promises the unidirectional generation of the “spin"
current carried by various quasiparticles, which may modulate the ground state
by their interaction with local order parameters, such as the Doppler shift of
spin-wave dispersion under chiral pumping in thin magnetic films [59]. Another
source of nonlinearity brought by the chirality is that the energy tends to
accumulate at one edge when placing many magnets that are mediated by the
chiral waves. It favors the nonlinearity even with a small excitation power
[60, 61, 393]. However, from the experimental viewpoint, the nonlinear spin
dynamics in the presence of generalized spin-orbit interaction has not yet
been explored much in the literature.

The near electric or magnetic fields, by virtue of their strong spatial
localization at the sample surface, have enhanced strength that may serve as
the sources for driving nonlinear dynamics in the spin nanostructures. Surface
plasmon polaritons are successful examples that enhance many optical
interactions [254, 255, 256, 257] such as the magneto-optic processes [438].
The interaction between the Damon-Eshbach modes and optical whispering gallery
modes is strongly enhanced with their surface effects [212, 229, 226, 227,
228]. Future studies may benefit from this surface enhancement with other
evanescent chiral waves.

####  6.2.3 Chiral fluctuation beyond the quasiparticle scenario

The quasiparticle concept for the low energy excitations of ordered phases are
suitable theoretical instruments describing the dynamics of quantum materials
when the temperature is sufficiently lower than the critical ones. This review
focuses on the chirality of such low-energy excitations and their
interactions, such as the magnons [14, 13, 50, 231, 33, 251], microwave
photons [64, 65, 67, 60, 61, 63, 68, 69], optical photons [18, 21, 17],
phonons [16, 15, 77, 81, 80, 76, 75, 79, 83, 84, 82, 78, 75, 82, 76, 458],
surface plasmons [257, 19, 20, 86, 87, 88, 89, 91], and electron spins [85]
(refer to Sec. 3). Their chirality is intrinsic by evanescence and
characterized by the chirality index $Z$. The non-reciprocity of their
excitation, transport, and detection in the linear response regime emerges
from this chirality (refer to Sec. 4). The chirality affects transport in
nonlinear regimes that can be formulated by a mean-field theory that accounts
for the low-order interactions that arise between the quasiparticles [458, 59,
499, 500].

However, at temperatures close to the critical ones, the fluctuations
amplitudes become comparable to the equilibrium order parameter. They become
so short-lived that the quasiparticle description becomes inappropriate. In
such a beyond-quasiparticle regime, the role of the chirality, as initially
defined for the ground state and non-interacting quasiparticles with well-
defined wave vectors, raises theoretical challenges. The recently discovered
enhancement of the thermal Hall effect around the ordering temperature in two-
dimensional ferromagnets with perpendicular magnetic anisotropy emphasizes the
important role of chiral fluctuation as governed by a spin chirality with
three-spin index ${\bf S}_{i}\cdot({{\bf S}_{j}\times{\bf S}_{k}})$ [501,
502]. The numerical simulation by Monte Carlo and stochastic spin dynamics
[503] is beyond the scope of the present review.

####  6.2.4 Orbital angular momentum

The exploitation of the spin degree of freedom of electrons has led to the
fast development of spintronics, which focuses on the generation, detection,
and manipulation of spin information. Recently, attention is attracted to the
orbital degree of freedom of electrons as well, which is believed to be an
important alternative information “carrier" to spin. The associated branch is
termed as “orbitronics" [504, 505, 506]. The orbital degree of freedom is
often regarded as “frozen” due to the orbital quenching by the crystal fields.
As a surprise, recent studies accumulated evidence that the dynamics and
transport of the orbital information can be released under external stimuli
such as an electric field in many materials. Similar to those in spintronics,
orbital current can carry the orbital angular moment originating from orbital
Hall effect [507, 508, 509] and orbital Rashba effect [510, 511, 512, 513,
514], and exert torques on the magnetization. Hence, the orbitronics may
provide another route to manipulate and detect magnetization.

This review focuses on the transverse spin of numerous evanescent waves as a
generalization of the electron spin. Their locking to the momentum or surface
normal of the waves’ propagation plane is achieved by the integer chirality
index. A close analogy of orbital angular momentum of waves to the one of
electrons also exists, which has been well studied for decades in optics
[266]. Figure 60 shows that the optical wave beams also carry the orbital
angular momentum density that is indexed by an integer $l$, independent of the
beam’s polarization [515]. In this article, we have addressed the conservation
of transverse spin transferring to the electron one in the context of near-
field spin pumping in terms of microwaves [85]. The optical spin of surface
plasmon polaritons is also a source for the electron orbital magnetization by
the inverse Faraday effect [86, 87, 88, 89, 91]. The counterpart, i.e., the
mutual transfer between the orbital angular momentum of electromagnetic fields
and electron’s orbital one may be envisioned. The methodology and overview of
this article may inspire similar prospects in the future.

Figure 60: Orbital angular momentum of a light beam with wavefront (left
panel) and projection (right panel). The beams are labeled by the orbital
angular-momentum quantum number $l$. The associated beam’s orbital angular
momentum is $L=l\hbar$. The figure is reproduced with permission from [515].

####  6.2.5 Electric polarization

The close analogy between magnetic dipoles in magnetism and electric dipoles
in ferroelectricity originates from their duality, i.e. the associated
electric ${\bf E}$ and magnetic ${\bf H}$ fields have identical forms, which
have cross-fertilization of both fields for many years. Recently, their
excitations are found to have many common features [516, 517, 518]. The
electric polarization can transport momentum and energy in ferroelectrics,
similar to that in terms of magnons in a ferromagnet. Zhou et al. predicted
that coherent polarization waves or “ferrons" can be localized at the surface
of ferroelectric insulators by interacting with their own electric stray
fields [519]. Nevertheless, their properties turn out to be strikingly
different from the magnetic counterpart, i.e. the surface “Damon-Eshbach"
magnons [14, 13] in ferromagnets, although their generated stray electric
fields are chiral. We envision the dynamical chirality with generalized spin-
orbit interaction reviewed in this article should stimulate similar activities
in ferronics.

###  6.3 Antiferromagnetic, two-dimensional, and synthetic materials

In this article, both the static and dynamical chiralities are mostly reviewed
in the context of ferromagnets, dielectrics, conductors, and their
hybridizations for the simplicity of illustrating the concepts. Rich
functionalities of various spin textures and different alignments of
nanomagnets are governed by the static chiral interaction. Universal
viewpoints in terms of generalized spin-orbit interaction are addressed for
the motion of quasiparticles in these materials. The saturation magnetization
fixes the direction of the transverse spin of the magnon, so when being
evanescent they propagate unidirectionally at the surface of a magnet. For the
other evanescent waves, by the time-reversal symmetry, their transverse spins
are allowed to be reversed such that they are allowed to propagate along
opposite directions at one surface. The chiral interaction may rely on the
breaking of time reversal symmetry that chooses the preferred direction. These
implications should be broad that can be applied to other materials. For
example, treating the chirality as spins also bring about the tunability of
nematicity by the in-plane magnetic field in thin chiral superconductors
[520].

####  6.3.1 Antiferromagnetic spintronics

It is nowadays well appreciated that the antiferromagnet and ferrimagnet can
host magnons of both left-handed and right-handed circular polarizations or
“chiralities" [521, 522, 523] that are carried by the so-called $\alpha$ and
$\beta$ modes, which provides an additional degree of freedom and tunability
than that of ferromagnet. In particular, the magnon chirality can be
manipulated via the modulation of the ratio of the sublattice magnetization
[524, 525]. The manipulation of right-handed and left-handed chiralities in
antiferromagnets and ferrimagnets enables the design of chirality-based
magnonic devices such as field-effect transistor [526] and chirality-based
logic devices [528, 527].

Chirality already plays an important role in the magnonic excitations of the
antiferromagnet. The Damon-Eshbach surface modes exist as well in the
antiferromagnetic films [214, 215, 216, 217, 218, 219]. The dipolar
interaction, although being canceled out at the equilibrium, turns out to act
as a Rashba-like spin-orbit coupling between the $\alpha$ and $\beta$ magnon
modes [220], rendering an analogous spin dynamics to the electron. Thanks to
the time-reversal symmetry, spin-momentum locking of these modes are
envisioned. The dipolar interaction between two magnetic layers, i.e.,
synthetic antiferromagnet, is chiral [54, 55, 320, 319]. Exploiting the
accumulated experience of the chirality may bring inspiring functionalities in
the control of the magnon dynamics of an antiferromagnet.

####  6.3.2 Two-dimensional magnets

Previous works demonstrating the static and dynamic chiral interaction have
mainly focused on magnetic nanostructures including nanomagnets, thin films,
and nanowires, which are all very simple for clarifying the underlying
mechanisms. For example, the Damon-Eshbach modes vanish in the ultrathin
magnetic films, which after being ruled out can highlight the chiral dipolar
interaction between nanowires and films. However, the continuum model appears
to break down when the magnet is reduced to atomic thickness. On the other
hand, reducing the thickness of a magnet not only downscales the size of
spintronics devices but also brings physics to the new regime. The magnon
conductivity in high-quality subnanometer YIG films turns out to reach record
values because of the onset of two-dimensional (2D) diffusive magnon transport
[355].

But strictly speaking, these magnetic films are not “true" 2D magnets as there
are still many dangling bonds and many subbands for magnons. The inevitable
surface roughness at the interface plays an important role in the magnetic
properties and quality. Recently, along with intensive studies of 2D
nonmagnetic materials such as graphene [529] and monolayer transition-metal
dichalcogenide [530], isolated monolayer 2D magnets without dangling bonds,
e.g., Cr2Ge2Te6 [531], CrI3 [532], Fe3GeTe2 [533], and CrCl3 [534]) have been
successfully prepared by mechanical exfoliation with Scotch tape, liquid
exfoliation, and molecular-beam epitaxy (MBE). These materials provide an
ideal platform to explore magnetic excitations in 2D limit [535, 536, 537,
538]. Due to the atomic thickness, 2D magnets are susceptible to external
stimuli. Hence the chiral dynamic phenomena in 2D magnet-based systems can be
effectively modulated by the means of the electric field, photon irradiation,
pressure, chemical modification, and strain.

Similar to other 2D materials, 2D magnets can be stacked into high-quality
heterostructures via the van der Waals force. The observation of strong DMI in
2D magnet stacks promises its potential in ultra-compact next-generation
chiral spintronics [539, 540]. In particular, the relative twist in the
bilayer heterostructure creates moiré patterns [541, 542] and potentially
offers a versatile platform to study chirality-related physics as the twisting
operation has the helicity.

####  6.3.3 Synthetic and meta-materials

Chiral waves of very different types, such as electromagnetic waves, spin
waves, and acoustic waves, exist in many materials of very different
properties. Synthetic and meta-materials enable to bring various
functionalities together, such as the routing of various waves in one device.

Also, the static chirality is usually determined either by the structural
chirality of the system or by DMI polarity, which is an intrinsic property of
the materials. It usually keeps unchanged once constructed. On the other hand,
the capability of reversibly and locally modifying the chirality may offer
another dimension to manipulate the chiral magnetism and enable the
realization of multi-functional and reprogrammable devices. While the
structural chirality with respect to the shape of the structure is difficult
to change, it has been reported that the polarity of DMI can be reversibly
tuned by modulating electronic density with electric gate [543], by switching
the polarization of multiferroics [544] and by chemisorption/desorption of
functional atoms at the interface [545].

###  6.4 Quantum regime

Quantum information technology, harnessing the basic principles of quantum
mechanics such as superpositions and entanglement of quantum states, is of
great interest for precise measurement, information processing, and
communication tasks [546, 547, 548, 549, 550]. In analogous to the classical
bit that has the value of 0 and 1, the quantum information is encoded in the
so-called qubit (quantum bit). Taking advantage of the unique quantum physics
of superposition and entanglement, the simultaneous and coherent manipulation
of qubits enables the accomplishment of unprecedented computation tasks beyond
the capability of a classical supercomputer.

Although both spintronics and quantum information technology implement spins,
these two fields developed relatively independently at the early stage. In
recent years, the situation starts to change and their interdisciplinary has
opened up a new avenue for both two fields [551, 552, 553, 554, 555, 556,
557]. The sensing of magnon via superconducting qubit in the microwave cavity
achieves quantum resolution, i.e., a single quanta [556, 557]. The quantum
control of and communication with the quanta of surface acoustic waves has
been achieved experimentally [558, 559]. Microwave photon was intensively
studied in the context of quantum electrodynamics [560]. These ingredients
with chiral interaction may combine in the quantum regime.

The merit of chiral interaction between two objects is the lack of backaction
when they are mediated by the chiral waves, thus rendering novel manipulation
at the quantum level. The manipulation of microwave photons by the qubit is
the key functionality in circuit quantum electrodynamics, in which the chiral
interaction stabilizes the entanglement and quantum manipulation without
backaction [18]. In the classical regime, the chiral interaction between an
array of magnets favors the accumulation and greatly enhances the sensitivity
of detection of microwaves [60, 61, 393].

This review intensively formulates the chiral coupling of various
quasiparticles in terms of the quantum formalism, which may form a starting
point to study quantum effects in the spintronic nanostructures, such as spin-
pumping-induced magnetic quantum noise [495] at low temperatures, quantum
squeezing, one-way quantum steering [561], and entanglement of the
quasiparticles with microwaves via the magnet [562, 563]. The chiral
interaction may improve the performance and sensitivity of quantum
communication and quantum sensing with magnon, microwave photon, and surface
phonon in several aspects.

## Declaration of competing interest

The authors declare no competing financial interests that could have appeared
to influence the work reported in this paper.

## Acknowledgments

This work is financially supported by the National Natural Science Foundation
of China (Grant No. 0214012051), the startup grant of Huazhong University of
Science and Technology (Grants No. 3004012185 and No. 3004012198), the startup
grant of Peking University, as well as JSPS KAKENHI Grant No. 19H00645. T.Y.
started his research on chirality in spintronics with G.B. at the Kavli
Institute of Nanoscience of the TU Delft five years ago, acknowledging many
useful discussions with his colleagues Sanchar Sharma, Xiang Zhang, Yuguang
Chen, Yaroslav M. Blanter, and Toeno van der Sar. He continued these studies
at the Max Planck Institute for the Structure and Dynamics of Matter, thanking
Michael A. Sentef, Dante M. Kennes, and Angel Rubio for their support. We
thank Yu-Xiang Zhang, Weichao Yu, Kei Yamamoto, Bimu Yao, Jinwei Rao, Bowen
Zeng, Jin Lan, Haiming Yu, Mehrdad Elyasi, Ji Zou, Xiangyang Wei, Akashdeep
Kamra, Canming Hu, Ping Tang, Alejandro O. Leon, and Sergio Rezende for
numerous inspiring discussions.

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