# Topological Spin Hall Effect in Antiferromagnets Driven by Vector Néel
Chirality

Kazuki Nakazawa1, Koujiro Hoshi1, Jotaro J. Nakane2, Jun-ichiro Ohe3, Hiroshi
Kohno2 1Department of Applied Physics, University of Tokyo, Bunkyo, Tokyo
113-8656, Japan
2Department of Physics, Nagoya University, Nagoya 464-8602, Japan
3Department of Physics, Toho University, 2-2-1 Miyama, Funabashi, Chiba
274-8510, Japan

###### Abstract

Spin Hall effect of spin-texture origin is explored theoretically for
antiferromagnetic (AF) metals. It is found that a vector chirality formed by
the Néel vector gives rise to a topological spin Hall effect. This is
topological since it is proportional to the winding number counted by in-plane
vector chirality along the sample edge, which can be nonvanishing for AF
merons but not for AF skyrmions. The effect is enhanced when the Fermi level
lies near the AF gap, and, surprisingly, at weak coupling with small AF gap.
These features are confirmed numerically based on the Landauer-Büttiker
formula. Important roles played by nonadiabatic processes and spin dephasing
are pointed out.

Spin-charge interconversion has been extensively studied in spintronics with
the aim of application to next-generation devices. It is typically achieved by
the spin Hall effect (SHE) [1] originating from the relativistic spin-orbit
coupling (SOC), mostly in nonmagnetic materials [2, 3, 4, 5, 6]. Ferromagnets
(FMs) are another class of materials that enable spin-charge conversion, not
just as a simple spin source, but also by emergent electromagnetism due to
spatiotemporal magnetization dynamics. In particular, a magnetization texture
forming a finite scalar spin chirality simulates a magnetic field that affects
electrons’ orbital motion but in a spin-dependent way. The resulting Hall
effect, often called the topological Hall effect (THE), is thus the SHE in
essence [7].

Antiferromagnets (AF) are a material having both aspects, magnetic at the
microscopic scale but nonmagnetic at the (semi)macroscopic scale, and offers a
unique platform to generate pure spin currents. A large SHE was reported in
${\rm Ir_{20}Mn_{80}}$ [9], which originates from SOC. Recently, there are
some proposals of SHE that arise from the antiferromagnetic spin texture,
providing another means of pure spin-current generation without relying on
relativistic SOC.

In this Letter, we explore theoretically the SHE in AF originating from AF
spin textures. From the analogy with FMs, an AF with a textured Néel vector
${\bm{n}}$ is expected to generate a spin Hall current,

$\displaystyle\tilde{j}_{{\rm s},i}^{z}$ $\displaystyle=\tilde{{\sigma}}_{\rm
SH}\,{{\bm{n}}}\cdot({\partial}_{i}{{\bm{n}}}\times{\partial}_{j}{{\bm{n}}})eE_{j},$
(1)

under an applied electric field $E_{j}$ ($\tilde{{\sigma}}_{\rm SH}$ is a
coefficient, and $e>0$ is the elementary charge). Because of the scalar
chirality,
${{\bm{n}}}\cdot({\partial}_{i}{{\bm{n}}}\times{\partial}_{j}{{\bm{n}}})$,
this effect may be termed as a topological spin Hall (TSH) effect [10, 11,
12]. Such a texture has opposite spin chiralities on different sublattices and
deflects spin-up and spin-down electrons in mutually opposite directions; the
charge Hall currents then cancel out, but the spin Hall currents add up.
However, Eq. (1) is not a macroscopically observable quantity since its sign
depends on the definition of sublattice or ${\bm{n}}$; it changes sign under
${\bm{n}}\to-{\bm{n}}$. We define the physical spin current ${\bm{j}}_{{\rm
s},i}$ through $\tilde{j}_{{\rm s},i}^{z}={\bm{n}}\cdot{\bm{j}}_{{\rm s},i}$,
hence by

$\displaystyle j_{{\rm s},i}^{\alpha}$ $\displaystyle=\tilde{{\sigma}}_{\rm
SH}({\partial}_{i}{{\bm{n}}}\times{\partial}_{j}{{\bm{n}}})^{\alpha}eE_{j}.$
(2)

The factor $({\partial}_{i}{{\bm{n}}}\times{\partial}_{j}{{\bm{n}}})^{\alpha}$
may be identified as an emergent magnetic field in spin channel, and
interestingly, it can be expressed as
$(\partial_{i}a_{j}^{\alpha}-\partial_{j}a_{i}^{\alpha})/2$ with an emergent
vector potential,

$\displaystyle a_{i}^{\alpha}=({\bm{n}}\times\partial_{i}{\bm{n}})^{\alpha}.$
(3)

This is the vector chirality ($\sim{\bm{S}}_{1}\times{\bm{S}}_{2}$ for two
spins) formed by the Néel vector, and we call it “vector Néel chirality” [13].
Spatially-averaged spin current $\langle j_{{\rm s},i}^{\alpha}\rangle$ is
proportional to a winding number defined by the vector chirality, hence is
“topological.” To date, the vector spin chirality is known to induce charge
[14, 15] and (equilibrium) spin currents [16, 17, 18], but its AF counterpart
in terms of the Néel vector has been less focused on.

In the following, we derive Eqs. (1) and (2) and demonstrate the topological
character of Eq. (2). The effect is present in systems with AF merons [19] but
not with AF skyrmions [20, 21, 22, 23, 24, 25, 26], and is enhanced in the
weak-coupling regime. These results are confirmed numerically based on the
Landauer-Büttiker formula.

Figure 1:  (a) Static magnetic structure considered in this work, a
checkerboard type AF on a square lattice with a very slow spatial modulation.
The two sublattices (A or B) are indicated by color (red or blue). (b)
Electron dispersion in a uniform AF state. Each subband is spin degenerate.

We consider electrons on a square lattice and coupled to a static spin
texture. The Hamiltonian

$\displaystyle H$ $\displaystyle=-t\sum_{(i,j)}c_{i}^{\dagger}c_{j}-J_{\rm
sd}\sum_{i}{\bm{S}}_{i}\cdot(c_{i}^{\dagger}{\bm{\sigma}}c_{i})+u_{\rm
i}{\sum_{i}}^{\prime}c_{i}^{\dagger}c_{i},$ (4)

consists of nearest-neighbor hopping (first term), s-d exchange coupling to
localized spins ${\bm{S}}_{i}$ (second term), and on-site impurity potential
(third term), with electron operators
$c_{i}={}^{t}(c_{i\uparrow},c_{i\downarrow})$ at site $i$ and Pauli matrices
${\bm{\sigma}}=(\sigma^{x},\sigma^{y},\sigma^{z})$. We assume a slowly-varying
checkerboard type AF texture, ${\bm{S}}_{i}=S(-1)^{i}{\bm{n}}_{i}$, where
${\bm{n}}_{i}$ is the Néel vector varying slowly in space [Fig. 1 (a)].

With a unitary transformation, $c_{i}=U_{i}\tilde{c}_{i}$, which diagonalizes
the s-d coupling,
$U_{i}^{\dagger}({\bm{n}}_{i}\cdot{\bm{\sigma}})U_{i}=\sigma^{z}$, $H$ is
transformed into $H=-t\sum_{(i,j)}\tilde{c}_{i}^{\dagger}{\rm
e}^{iA_{ij}}\tilde{c}_{j}-J\sum_{i}(-)^{i}\tilde{c}_{i}^{\dagger}\sigma^{z}\tilde{c}_{i}+u_{\rm
i}{\sum_{i}}^{\prime}\tilde{c}_{i}^{\dagger}\tilde{c}_{i}$, where $J=J_{\rm
sd}S$, and $A_{ij}$ is the spin gauge field defined by
$U_{i}^{\dagger}U_{j}={{\rm e}}^{iA_{ij}}$ [27, 28]. Because of slow
variations of the texture, $A_{ij}$ is small and can be treated
perturbatively. The unperturbed state (with $A_{ij}=0$) is a uniform AF, and
the electron band splits into spin-degenerate upper and lower bands, $\pm
E_{\bm{k}}$, with an AF gap $2|J|$ in between [Fig. 1 (b)]. Here,
$E_{\bm{k}}\equiv\sqrt{\varepsilon_{\bm{k}}^{2}+J^{2}}$ with
$\varepsilon_{\bm{k}}=-2t(\cos k_{x}+\cos k_{y})$. Also, $A_{ij}$ can be
treated in the continuum approximation, $A_{ij}\to A_{\mu}$, where $\mu$
($=x,y$) specifies the bond direction of $(i,j)$, and expanded as

$\displaystyle
A_{\mu}=\frac{1}{2}A_{\mu}^{\alpha}\sigma^{\alpha}=\frac{1}{2}(A_{\mu}^{z}\sigma^{z}+{\bm{A}}_{\mu}^{\perp}\cdot{\bm{\sigma}}^{\perp}),$
(5)

where ${\alpha}=x,y,z$ and $\perp=x,y$. The spin-conserving component $A^{z}$
describes adiabatic processes, whereas the spin-flip component
${\bm{A}}^{\perp}$ induces nonadiabatic transitions. In FM, the latter can be
important only in the weak-coupling regime [3], but in AF, both are important
because of spin degeneracy of the AF bands. Both produce the same effective
field,
$(\nabla\times{\bm{A}}^{z})_{z}=({\bm{A}}_{x}^{\perp}\times{\bm{A}}_{y}^{\perp})^{z}={{\bm{n}}}\cdot({\partial}_{x}{{\bm{n}}}\times{\partial}_{y}{{\bm{n}}})$.

To calculate the spin Hall conductivity, $\sigma_{\rm
SH}\equiv\frac{1}{2}(\sigma_{xy}^{z}-\sigma_{yx}^{z})$, we assume a good AF
metal and focus on the Fermi-surface contribution [30],

$\displaystyle\sigma_{ij}^{z}({{\bm{Q}}})$
$\displaystyle=-\frac{e\hbar}{4\pi}{\rm Tr}\left\langle{\cal J}_{{\rm
s},i}^{z}G_{{{\bm{k}}}_{+},{{\bm{k}}}^{\prime}}^{\rm R}{\cal
J}_{j}G_{{{\bm{k}}}^{\prime},{{\bm{k}}}_{-}}^{\rm A}\right\rangle_{\rm i},$
(6)

where ${\cal J}_{{\rm s},i}^{z}$ and ${\cal J}_{j}$ are spin-current and
number-current vertices, $\rm Tr$ means the trace in spin, sublattice, and
${{\bm{k}}}$ spaces (${{\bm{k}}}$, ${{\bm{k}}}^{\prime}$-integrals), and
$\langle\cdots\rangle_{\rm i}$ represents impurity average. The Green’s
function $G_{{{\bm{k}}},{{\bm{k}}}^{\prime}}^{\rm R(A)}=(\mu-H\pm
i0)_{{{\bm{k}}},{{\bm{k}}}^{\prime}}^{-1}$ takes full account of impurities
and the gauge field, and ${{\bm{k}}}_{\pm}={{\bm{k}}}\pm{{\bm{Q}}}/2$. We
treat the impurity scattering in the Born approximation with ladder vertex
corrections (VC) [31]. The superscript $z$ on $\sigma_{ij}^{z}$ and ${\cal
J}_{{\rm s},i}^{z}$ indicates the spin component in the rotated frame, thus it
is the component projected to the local Néel vector ${\bm{n}}$.

After a standard procedure (see Supplemental Material [31]), we obtain Eq. (1)
with $\tilde{\sigma}_{\rm SH}=\tilde{\sigma}_{\rm
SH}^{(0)}+\tilde{\sigma}_{\rm SH}^{(1)}$,

$\displaystyle\tilde{\sigma}_{\rm SH}^{(0)}$
$\displaystyle=(J\tau)^{2}\frac{t^{2}\nu}{\mu}\left(1-\frac{J^{2}}{\mu^{2}}\right)C_{xy},$
(7) $\displaystyle\tilde{\sigma}_{\rm SH}^{(1)}$
$\displaystyle=(J\tau)^{2}\frac{t^{2}\nu}{\mu}\frac{8t^{2}}{\mu^{2}+J^{2}}\left(\frac{\tau^{-1}}{\tau_{\varphi}^{-1}+\tau_{\rm
s}^{-1}}\right)C_{xx}^{2},$ (8)

where $\tilde{\sigma}_{\rm SH}^{(0)}$ is the contribution without VC, which
comes from both adiabatic and nonadiabatic processes, and $\tilde{\sigma}_{\rm
SH}^{(1)}$ is the contribution with VC, coming only from nonadiabatic
processes. Here, $\nu=\nu(\mu)$ is the density of states (per spin) at
chemical potential $\mu$, $C_{ij}=\langle 1-\cos k_{i}\cos k_{j}\rangle_{\rm
FS}$ is the Fermi surface average [32],
$\tau=[{\gamma}_{0}+(J/\mu){\gamma}_{3}]^{-1}/2$ is the scattering time
(${\gamma}_{0}=\pi n_{\rm i}u_{\rm i}^{2}\nu$ and
${\gamma}_{3}=(J/\mu){\gamma}_{0}$ are the sublattice-independent and
dependent parts, respectively, of the damping, and $n_{\rm i}$ is the impurity
concentration), and

$\displaystyle\frac{1}{\tau_{\varphi}}=\frac{4J}{\mu}\frac{\mu^{2}+J^{2}}{\mu^{2}-J^{2}}{\gamma}_{3}=\frac{2J^{2}}{\mu^{2}-J^{2}}\frac{1}{\tau},$
(9)

is the “spin dephasing” rate [2]. We introduced a finite spin relaxation rate
$\tau_{\rm s}^{-1}$ by hand [34]; without $\tau_{\rm s}^{-1}$, we would have
an unphysical result that $\tilde{\sigma}_{\rm SH}^{(1)}$ does not vanish in
the limit $J\to 0$. Note that $\tau_{\varphi}^{-1}$ differs from $\tau_{\rm
s}^{-1}$ in that it does not require spin-dependent scattering, randomizes
only the transverse ($\perp{\bm{n}}$) components of the electron spin (see
below), and vanishes as $J\to 0$. The results (7) and (8) are obtained at the
leading order, i.e., second order in spatial gradient and second order in
$\tau$.

Figure 2: (a,b) Normalized topological spin Hall conductivity vs. chemical
potential $\mu$ for several choices of $J/t$. (a) $\tilde{\sigma}_{\rm
SH}^{(0)}\tilde{\gamma}^{2}$ and $\tilde{\sigma}_{\rm
SH}^{(1)}\tilde{\gamma}^{2}$, where $\tilde{\gamma}=\pi n_{\rm i}u_{\rm
i}^{2}/t^{2}$ is a dimensionless damping parameter. (b) $\tilde{\sigma}_{\rm
SH}=\tilde{\sigma}_{\rm SH}^{(0)}+\tilde{\sigma}_{\rm SH}^{(1)}$. In (b),
$\tilde{\sigma}_{\rm SH}$ with finite $q$ are also shown (dotted lines). These
are odd functions of $\mu$, hence plotted only for the lower AF band. The
parameters used are $\tilde{\gamma}=0.2$ and $\tau_{\rm s}^{-1}=10^{-4}t$. (c)
Characteristic parameter regions for the TSH conductivity. The red dashed
line, given by $J/|\mu|=ql/\sqrt{4+(ql)^{2}}$ in the diffusive regime, is a
crossover line separating the local and nonlocal field regions, and
$\varepsilon_{\rm m}=\sqrt{(4t)^{2}+J^{2}}$. The analytical results, Eqs. (7),
(8), and (13), apply to the blue shaded region, while the numerical results
(Fig. 4) apply to the green shaded region.

The coefficients $\tilde{\sigma}_{\rm SH}^{(0)}$ and $\tilde{\sigma}_{\rm
SH}^{(1)}$ are plotted in Fig. 2 (a). They are comparable in magnitude at
large $J$ ($\sim 1.5t$), but as $J$ is reduced, $\tilde{\sigma}_{\rm
SH}^{(1)}$ grows markedly whereas $\tilde{\sigma}_{\rm SH}^{(0)}$ decreases.
The sum $\tilde{\sigma}_{\rm SH}=\tilde{\sigma}_{\rm
SH}^{(0)}+\tilde{\sigma}_{\rm SH}^{(1)}$ is plotted in Fig. 2 (b) by solid
lines, which grows as $J$ is reduced, especially near the AF gap edge, but
finally vanishes at $J=0$. Since $\tilde{\sigma}_{\rm SH}^{(1)}$ comes solely
from nonadiabatic processes, these results show that the combined effect of
nonadiabaticity and the VC is important for the present SHE [35]. Physically,
a nonadiabatic process produces a transverse spin polarization, and the VC
describes its collective transport, which is however limited by spin dephasing
[2, 28, 36]. The origin of the enhancement at small $J$ can be traced to the
reduced dephasing at small $J$ [31]. As seen from Eq. (9), the spin dephasing
arises through ${\gamma}_{3}$, a sublattice asymmetry in (nonmagnetic)
scattering [28, 37], and its physical picture is illustrated in Fig. 3.

The obtained result, Eq. (1), needs to be interpreted with care. It arises
with the scalar chirality formed by the Néel vector ${\bm{n}}$, and changes
sign under ${\bm{n}}\to-{\bm{n}}$. This is not a pleasant situation since any
physical quantity measurable by (semi)macroscopic means should not depend on
the sign of ${\bm{n}}$, or on the definition of sublattice. This (apparent)
puzzle is resolved if we note that the spin component of the calculated spin
current $\tilde{j}_{x,{\rm s}}^{z}$ is the one projected to the Néel vector
$\bm{n}$. Therefore, we write $\tilde{j}_{\rm
s}^{z}={\bm{n}}\cdot{\bm{j}}_{\rm s}$ and identify ${\bm{j}}_{\rm s}$ as a
physical spin current. The physical spin Hall current is thus given by Eq.
(2).

It is in fact possible to obtain Eq. (2) directly. By assuming $J$ is small
and treating it perturbatively, we found the spin current arises at second
order in $J$ [31],

$\displaystyle j_{{\rm s},i}^{\alpha}$
$\displaystyle=(J\tau)^{2}\frac{t^{2}\nu}{\mu}C_{xy}(\partial_{i}{\bm{n}}\times\partial_{j}{\bm{n}})^{\alpha}eE_{j}.$
(10)

This contrasts with the THE in FM caused by scalar spin chirality, which
starts at third order ($\sim J^{3}$) [38], and demonstrates that the essential
quantity for the present SHE is the vector (not scalar) chirality. That Eq.
(10) is an even function of $J$ (or $J{\bm{n}}$) is consistent with the fact
that the spin current is even under time reversal.

Figure 3:  Physical picture of electron spin transport in a uniform
antiferromagnet. The blue sphere with an arrow represents an electron, the
green arrow a localized spin, and the red star a nonmagnetic impurity. (a) The
electron spin precesses around the local moment, alternating its sense from
site to site. (b) Interaction with impurities locally modifies the precession.
(c) A “collective” transverse spin density contributed from many electrons
decays and loses its original information through the impurity scattering.
This is because the degree of the modification, mentioned in (b), varies from
electron to electron. This is called “dephasing” and the characteristic length
is the “dephasing length” $\lambda_{\varphi}=\sqrt{D\tau_{\varphi}}$. The
orange stars represent averaged impurities.

The expression Eq. (2) holds locally in space (as far as the variation of
${\bm{n}}$ is sufficiently slow). As a spin current measured experimentally,
we consider a spatially-averaged one, $\langle{\bm{j}}_{\rm
s}^{\alpha}\rangle=\Omega^{-1}\int{\bm{j}}_{\rm s}^{\alpha}dxdy$ (in two
dimensions), where $\Omega$ is the sample area. It can be written as

$\displaystyle\langle{\bm{j}}_{\rm s}^{\alpha}\rangle$
$\displaystyle=\pi\tilde{{\sigma}}_{\rm
SH}\frac{m^{\alpha}}{\Omega}\left(e{\bm{E}}\times\hat{z}\right),$ (11)

where

$\displaystyle m^{\alpha}$
$\displaystyle=\frac{1}{2\pi}\int(\nabla\times{\bm{a}}^{\alpha})_{z}dxdy=\frac{1}{2\pi}\oint{\bm{a}}^{\alpha}\cdot
d{\bm{\ell}},$ (12)

and $a_{i}^{\alpha}=({\bm{n}}\times\partial_{i}{\bm{n}})^{\alpha}$ [Eq. (3)]
is an emergent vector potential in spin channel. The line integral is taken
along the sample perimeter. If the system has easy-plane magnetic anisotropy,
and the Néel vector on the sample edge is constrained to lie in-plane, e.g.,
$x$-$y$ plane, the line integral of the vector chirality defines a topological
winding number $m^{z}\in\mathbb{Z}$ in $\pi_{1}(S^{1})$. The spin Hall
conductivity is thus proportional to the topological number density
$m^{z}/\Omega$, and this fact resurrects the naming “topological” spin Hall
effect. We emphasize that it is characterized by the vector chirality of Néel
vector along the sample edge. Therefore, the present TSHE is absent for AF
skyrmions, in which the Néel vector at the edge is uniaxial. On the other
hand, it is finite for AF merons, which have finite in-plane winding of the
Néel vector along the edge.

Figure 4:  Topological spin Hall conductance $(G_{\rm SHC}^{z})$ based on the
Landauer-Büttiker formula for finite systems with $L\times L$ sites. (a) AF
skyrmion system. (b) AF meron system. (c) $L$-dependence of the peak value of
$G_{\rm SHC}^{z}$. The data are fitted with functions, $f(x)=286x-0.504$ and
$g(x)=0.237/x-2.22$. (d) AF meron system with $L=70$ for several choices of
$J/t$. We took $J/t=0.3$ [except in (d)] and meron/skyrmion radius $r=15$. The
data are symmetrized with respect to $J\to-J$, as explained in [31].

To verify these results, we have conducted numerical works based on the four-
terminal Landauer-Büttiker formula [39]. We consider ballistic systems with
$L\times L$ sites without disorder, and containing a single AF skyrmion or a
single AF meron. For both textures, the spin Hall conductance $G_{\rm
SHC}^{z}$ shows a strong peak just below the AF gap [Fig. 4 (a) and (b)],
which, however, behave oppositely as $L$ is increased (with the skyrmion/meron
size fixed); for the AF skyrmion the peak decreases with $L$ and seems to
vanish in the thermodynamic limit. In contrast, for the AF meron it increases
with $L$ [Fig. 4 (c)]. This is consistent with the analytical result, which is
valid for infinite-size systems. Plots for several $J/t$ values are shown in
Fig. 4 (d) for the AF meron system, showing that it is indeed enhanced at
small $J/t$. All these agree with the analytic results, except for the
detailed shape of $\mu$-dependence.

The discrepancy in shape ($\mu$-dependence) between the numerical [Fig. 4 (d)]
and analytic results [Fig. 2 (b)] may be understood as due to the nonlocality
effect in the former. To illustrate this, let us first consider the diffusive
regime. As the typical wave number $q$ of the Néel texture (i.e., inverse of
meron/skyrmion size) is increased, Eq. (8) is modified as

$\displaystyle(\tau_{\varphi}^{-1}+\tau_{\rm
s}^{-1})^{-1}\to(\tau_{\varphi}^{-1}+\tau_{\rm s}^{-1}+Dq^{2})^{-1},$ (13)

in the denominator, where $D$ is the diffusion constant. When electron spin
diffusion ($Dq^{2}$) occurs faster than spin dephasing
($\tau_{\varphi}^{-1}$), the effective field becomes “nonlocal”. Similar
feature has been noted for FMs, in which $Dq^{2}$ is compared to the exchange
splitting [3]. Here in AF, it is compared to the (much smaller) spin dephasing
rate, $\tau_{\varphi}^{-1}$, hence the present SHE enters the nonlocal regime
rather easily compared to the THE in FM. More explicitly, the nonlocality
appears if

$\displaystyle ql>\frac{2J}{\sqrt{\mu^{2}-J^{2}}},\ \ \ {\rm or}\ \ \
|\mu|>J\sqrt{1+(2/ql)^{2}},$ (14)

where $l$ is the mean free path. In Fig. 2 (b), the analytic results with
$q^{-2}=6000$ (with lattice constant taken unity) are plotted by dotted lines.
The suppression due to nonlocality is more significant at larger $|\mu|$ (away
from the AF gap), leaving a sharp peak in the vicinity of the AF gap edge.
Since cleaner systems enter the nonlocal regime more easily [see Eq. (14) and
a red dotted line in Fig. 2 (c)], this feature is expected to persist into the
ballistic regime with a wider nonlocality region. The shape of Fig. 4 (d) may
thus be understood as due to the nonlocality effect.

Thus, as in the case of THE in FM [3], the present TSHE in AF exhibits various
characteristic regimes [Fig. 2 (c)]. These are summarized as follows. First,
for a ballistic and local regime, the effect is truly topological. As $q$ is
increased and the nonlocal effects become important, the SHC deviates from the
topological expression. In the diffusive case, it is difficult to have the
topological expression because of dephasing (and nonlocality), but the effect
is enhanced for weak-coupling AF with small AF gap. An interesting possibility
may be found in mesoscopic systems, for which the effect can be topological
even if the system is in a diffusive regime.

The emergent vector potential ${\bm{a}}^{\alpha}$ in the spin channel,
identified here through TSHE, has more generality. In a study on THE in canted
AF [28], an emergent vector potential in charge channel was identified as
$l^{\alpha}{\bm{a}}^{\alpha}$, where $l^{\alpha}$ is the canting (uniform)
moment. Also, ${\bm{a}}^{\alpha}$ can be expressed as $a_{i}^{\alpha}=-({\cal
R}{\bm{A}}^{\perp}_{i})^{\alpha}$ [40], where ${\cal R}$ is an SO(3) matrix
that connects the rotated and the original frames (e.g., ${\bm{n}}={\cal
R}\hat{z}$), showing its conformity with the spin gauge field
${\bm{A}}^{\perp}_{i}$. These facts reinforce our interpretation of
${\bm{a}}^{\alpha}$ as an effective vector potential in spin channel.

To realize the present TSHE experimentally, a prime candidate texture is
${\bm{n}}$-meron. Such texture was found very recently in $\alpha$-Fe2O3 [19],
which is however an insulator; search for metallic systems is desired. Another
candidate is a canted AF; if the ferromagnetic moment ${\bm{l}}$ (due to
canting) forms a skyrmion (called “${\bm{l}}$-skyrmion” in [28]), topological
consideration shows that the Néel vector winds at least twice around the
skyrmion, i.e., $m^{z}=2$ per skyrmion [28, 41]. A recent experiment on thin
films of Ce-doped CaMnO3, a canted AF, observed skyrmion bubbles formed by the
(weak) ferromagnetic moment [42]. Therefore, this system can also be a
candidate for the present TSHE.

Finally, we discuss the relationship with previous theoretical studies. In
Ref. [11], the TSH conductivity was investigated in an AF skyrmion lattice
with a focus on the intrinsic (Berry curvature) contribution. It is a future
issue to investigate such intrinsic contribution in our framework. For
example, one may consider a “meron-antimeron lattice” which contains a vortex
with $m=\pm 2$ per unit cell. In Ref. [12], the Landauer-Büttiker method was
used to study the TSHE in AF skyrmion systems, and a finite TSHE was found for
finite-size systems, which does not contradict with our result because of
finite size. More importantly, an increase of the spin Hall conductivity was
pointed out for special impurity configurations, and it is also interesting to
investigate how the impurity configuration affect the spin dephasing and TSHE.

To summarize, we have studied a spin Hall effect due to magnetic textures in
AF metals. By analytic calculations, we found a topological contribution
proportional to the winding number defined by vector chirality. This is finite
for AF merons but not for AF skyrmions, and is enhanced at weak coupling.
These results may provide hints to enhance spin currents and give directions
for experimental investigations and device fabrications. The results are
confirmed by numerical calculations based on the Landauer-Büttiker formula.
Important roles played by nonadiabatic processes and spin dephasing are
pointed out.

We would like to thank A. Yamakage, J. Fujimoto, T. Yamaguchi, Y. Imai, A.
Matsui, and T. Nomoto for valuable discussions. This work was partly supported
by JSPS KAKENHI Grant Numbers JP15H05702, JP17H02929, JP19K03744 and No.
JP21H01799, and the Center of Spintronics Research Network of Japan. KN is
supported by JSTCREST (JP-MJCR18T2) and JSPS KAKENHI Grant Number JP21K13875.
JJN is supported by a Program for Leading Graduate Schools “Integrative
Graduate Education and Researchin Green Natural Sciences” and Grant-in-Aid for
JSPS Research Fellow Grant Number JP19J23587.

## References

  * [1] J. Sinova, S. O. Valenzuela, J. Wunderlich, C. H. Back, and T. Jungwirth, Rev. Mod. Phys. 87, 1213 (2015).
  * [2] M. I. Dyakonov and V. I. Perel, JETP, Lett. 13, 467 (1971); Phys. Lett. A35, 459 (1971).
  * [3] S. Murakami, N. Nagaosa, and S. -C. Zhang, Science 301, 1348 (2003).
  * [4] J. Sinova, D. Culcer, Q. Niu, N. A. Sinitsyn, T. Jungwirth, and A. H. MacDonald, Phys. Rev. Lett. 92, 126603 (2004).
  * [5] Y. K. Kato, R. C. Myers, A. C. Gossard, and D. D. Awschalom, Science 306, 1910 (2004).
  * [6] J. Wunderlich, B. Kastner, J. Sinova, and T. Jungwirth, Phys. Rev. Lett. 94, 047204 (2004).
  * [7] This applies to the strong $s$-$d$ coupling regime. In the weak-coupling regime, there is a proposal recently that the essence is the Hall effect in the charge channel [8].
  * [8] K. S. Denisov, I. V. Rozhansky, N. S. Averkiev, and E. Lähderanta, Phys. Rev. Lett. 117, 027202 (2016); Phys. Rev. B 98, 195439 (2018); I. V. Rozhansky, K. S. Denisov, M. B. Lifshits, N. S. Averkiev, and E. Lähderanta, Phys. Status Solidi (b) 1900033 (2019).
  * [9] J. B. S. Mendes, R. O. Cunha, O. Alves Santos, P. R. T. Ribeiro, F. L. A. Machado, R. L. Rodríguez-Suárez, A. Azevedo, and S. M. Rezende et al., Phys. Rev. B 89, 140406(R) (2014).
  * [10] P. M. Buhl, F. Freimuth, S. Blügel, and Y. Mokrousov, Phys. Status Solidi RRL 11, 1700007 (2017).
  * [11] B. Göbel, A. Mook, J. Henk, I. Mertig, Phys. Rev. B 96, 060406(R) (2017).
  * [12] C. A. Akosa, O. A. Tretiakov, G. Tatara, and A. Manchon, Phys. Rev. Lett. 121, 097204, (2018).
  * [13] One may also think of “Néel vector chirality”, which, however, has more flavor of scalar chirality formed by the Néel vector, i.e., ${{\bm{n}}}\cdot({\partial}_{i}{{\bm{n}}}\times{\partial}_{j}{{\bm{n}}})$.
  * [14] K. Taguchi and G. Tatara, Phys. Rev. B 79, 054423 (2009)
  * [15] H. Ishizuka and N. Nagaosa, New J. Phys. 20, 123027 (2018).
  * [16] G. Tatara and N. Garcia, Phys. Rev. Lett. 91, 076806 (2003).
  * [17] H. Katsura, N. Nagaosa, and A. V. Balatsky, Phys. Rev. Lett. 95, 057205 (2005).
  * [18] T. Kikuchi, T. Koretsune, R. Arita, and G. Tatara, Phys. Rev. Lett. 116, 247201 (2016).
  * [19] H. Jani, J. -C. Lin, J. Chen, J. Harrison, F. Maccherozzi, J. Schad, S. Prakash, C. -B. Eom, A. Ariando, T. Venkatesan and P. G. Radaelli, Nature 590, 74-79 (2021).
  * [20] C. Jin, C. Song, J. Wang, and Q. Liu, Appl. Phys. Lett. 109, 182404 (2016).
  * [21] X. Zhang, Y. Zhou, and M. Ezawa, Sci. Rep. 6, 24795 (2016).
  * [22] H. Velkov, O. Gomonay, M. Beens, G. Schwiete, A. Brataas, J. Sinova, and R. A. Duine, New J. Phys. 18, 075016 (2016).
  * [23] J. Barker and O. A. Tretiakov, Phys. Rev. Lett. 116, 147203 (2016).
  * [24] T. Dohi, S. DuttaGupta, S. Fukami, and H. Ohno, Nat. Commun. 10, 5153 (2019).
  * [25] W. Legrand, D. Maccariello, F. Ajejas, S. Collin, A. Vecchiola, K. Bouzehouane, N. Reyren, V. Cros, and A. Fert, Nat. Mater. 19, 34 (2020).
  * [26] R. Yambe and S. Hayami, Phys. Rev B 107, 014417 (2023).
  * [27] G. Tatara, H. Kohno, and J. Shibata, Phys. Rep. 468, 213-301 (2008).
  * [28] J. J. Nakane, K. Nakazawa, and H. Kohno, Phys. Rev. B 101, 174432 (2020).
  * [29] K. Nakazawa, M. Bibes, and H. Kohno, J. Phys. Soc. Jpn. 87, 033705 (2018); K. Nakazawa, and H. Kohno, Phys. Rev. B 99, 174425, (2019).
  * [30] P. Středa, J. Phys. C 15, L717 (1982).
  * [31] See Supplemental Material for calculational details, explicit formula of the adiabatic and nonadiabatic contributions, analysis on the enhancement at weak couling, perturbative treatment of $J$, and further plots including the spin Hall angle.
  * [32] Explicitly, they are evaluated as [28] $\displaystyle C_{xy}$ $\displaystyle=\frac{1}{4\pi^{2}}\frac{|\mu|}{t^{2}\nu}\left[\frac{2}{\tilde{\mu}}K(x)-\left(1+\frac{1}{\tilde{\mu}}\right)E(x)\right],$ $\displaystyle C_{xx}$ $\displaystyle=\frac{1}{4\pi^{2}}\frac{|\mu|}{t^{2}\nu}\left[\left(1+\frac{1}{\tilde{\mu}}\right)E(x)-2K(x)\right],$ where $K(x)$ and $E(x)$ are the complete elliptic integrals of the first and the second kind, respectively, and $\tilde{\mu}=\sqrt{\mu^{2}-J^{2}}/4t$ and $x=(1-\tilde{\mu})/(1+\tilde{\mu})$ are dimensionless parameters.
  * [33] A. Manchon, J. Phys. Condens. Matter 29, 104002 (2017). Their $\beta$ seems to be $\sin\chi_{k_{\rm F}}$ instead of $\cos\chi_{k_{\rm F}}$.
  * [34] A microscopic calculation of the spin relaxation rate in AF metal can be found in Ref. [37], in which magnetic impurities are considered as a spin sink.
  * [35] The nonadiabatic process is important at a more fundamental level. Both the adiabatic and nonadiabatic contributions to $\tilde{\sigma}_{\rm SH}^{(0)}$ contain terms of $O(J^{0})$, which do not vanish at $J=0$, but when added they cancel each other and the resulting $\tilde{\sigma}_{\rm SH}^{(0)}$ vanishes at $J=0$ [31].
  * [36] S. Hikami, A. I. Larkin, and Y. Nagaoka, Prog. Theor. Phys. 63, 707 (1980).
  * [37] J. J. Nakane and H. Kohno, Phys. Rev. B 103, L180405 (2021).
  * [38] G. Tatara and H. Kawamura, J. Phys. Soc. Jpn. 71, 2613 (2002).
  * [39] J. Ohe, T. Ohtsuki, and B. Kramer, Phys. Rev. B 75, 245313 (2017).
  * [40] H. Kohno and J. Shibata, J. Phys. Soc. Jpn. 76, 063710 (2007).
  * [41] O. Meshcheriakova, S. Chadov, A. K. Nayak, U. K. Rößler, J. Kübler, G. André, A. A. Tsirilin, J. Kiss, S. Hausdorf, A. Kalache, W. Schnelle, M. Nicklas, and C. Felser, Phys. Rev. Lett. 113, 087203 (2014).
  * [42] L. Vistoli, W. Wang, A. Sander, Q. Zhu, B. Casals, R. Cichelero, A. Barthélémy, S. Fusil, G. Herranz, S. Valencia, R. Abrudan, E. Weschke, K. Nakazawa, H. Kohno, J. Santamaria, W. Wu, V. Garcia, and M. Bibes, Nat. Phys. 15, 67 (2019).

Supplemental material
Topological Spin Hall Effect in Antiferromagnets Driven by Vector Néel
Chirality

##  I Preliminary

###  I.1 Green’s function and vertex corrections

Before presenting the calculation of the topological spin Hall conductivity
(TSHC), we define some building blocks of the calculation. The first one is
the Green’s function that does not contain the spin gauge field but with the
effects of impurities evaluated in the Born approximation [Fig. S1(a)].

$\displaystyle G_{{\bm{k}}}^{\rm R}(\varepsilon)=\mu^{\rm R}(1\otimes
1)+J^{\rm R}(\sigma^{z}\otimes\tau_{3})+T_{{\bm{k}}}^{\rm
R}(1\otimes\tau_{1}),$ (S1)

where $\mu^{\rm R}=(\varepsilon+\mu+i\gamma_{0})/D_{\bm{k}}^{\rm R}$, $J^{\rm
R}=(i\gamma_{3}-J)/D_{\bm{k}}^{\rm R}$, $T_{\bm{k}}^{\rm
R}=T_{{\bm{k}}}/D_{\bm{k}}^{\rm R}=-2t(\cos k_{x}+\cos k_{y})/D_{\bm{k}}^{\rm
R}$, with $D_{\bm{k}}^{\rm
R}(\varepsilon)=(\varepsilon+\mu)^{2}-E_{\bm{k}}^{2}+2i\left[(\varepsilon+\mu)\gamma_{0}+J\gamma_{3}\right]$,
and $E_{\bm{k}}=\sqrt{T_{\bm{k}}^{2}+J^{2}}$. The damping constants are given
by ${\gamma}_{0}=\pi n_{\rm i}u_{\rm i}^{2}\nu$ and
${\gamma}_{3}=(J/\mu){\gamma}_{0}$, where $n_{\rm i}$ is the impurity
concentration and $\nu=(1/N)\sum_{{\bm{k}}}\delta(|\mu|-E_{{\bm{k}}})$ is the
density of states per spin at the chemical potential $\mu$, and we define
elastic scattering (mean free) time $\tau$ by
$(2\tau)^{-1}={\gamma}_{0}+(J/\mu){\gamma}_{3}=(1+J^{2}/\mu^{2}){\gamma}_{0}$.
In Eq. (S1), the spin degree of freedom is described by the Pauli matrices
${\bm{{\sigma}}}=({\sigma}^{x},{\sigma}^{y},{\sigma}^{z})$, and the sublattice
degree of freedom by another Pauli matrices
${\bm{\tau}}=(\tau_{1},\tau_{2},\tau_{3})$. We occasionally suppress/simplify
the wave-vector dependence, such as $G=G_{{\bm{k}}}$,
$G_{\pm}=G_{{{\bm{k}}}\pm{{\bm{q}}}/2}$, $G^{\rm R(A)}\equiv G_{{\bm{k}}}^{\rm
R(A)}$, or $G_{\pm}^{\rm R(A)}\equiv G_{{{\bm{k}}}\pm{{\bm{q}}}/2}^{\rm
R(A)}$.

To be consistent with the self-energy considered above, we also consider the
ladder-type vertex corrections [Fig. S1(b)]. The particle-hole correlation
with opposite spins (i.e., transverse spin diffusion propagator) plays an
essential role in the present TSHE. Explicit form of the spin diffusion
propagator (red broken line in Fig. S3) is [1, 2]

$\displaystyle\Pi^{({\alpha}0)({\alpha}0)}({{\bm{q}}},\omega)$
$\displaystyle=\frac{4}{\pi\nu\tau^{2}}\frac{\mu^{2}}{\mu^{2}-J^{2}}\frac{1}{\tau_{\rm\varphi}^{-1}+\tau_{\rm
s}^{-1}+Dq^{2}-i\omega}\equiv\Pi_{\bar{\sigma}\sigma},$ (S2)
$\displaystyle\Pi^{({\alpha}3)({\alpha}3)}({{\bm{q}}},\omega)$
$\displaystyle=\frac{4}{\pi\nu\tau}\frac{\mu^{2}}{\mu^{2}+J^{2}},$ (S3)

where ${\alpha}=x,y$, $D=\frac{\mu^{2}-J^{2}}{\mu^{2}}\langle(2t\sin
k_{x})^{2}\rangle_{\rm FS}$ is the diffusion constant, $\tau_{\varphi}$ is the
spin dephasing time [Eq. (9) in the main text], $\tau_{\rm s}$ is the
(isotropic) spin relaxation time, and $\Pi^{({\alpha}0)({\alpha}0)}$
corresponds to Eq. (13) in the main text. These have been derived in Ref. [1].

Figure S1:  Feynman diagrams for the self energy (a) and the ladder-type
vertex correction (b). The blue solid line with arrow is a retarded or
advanced Green’s function. The blue broken line with a cross represents
(averaged) impurity scattering.

###  I.2 Current and density vertices

The vertices we use in the following are defined as follows,

$\displaystyle j_{i}$ $\displaystyle\equiv 2t\sin k_{i}\sigma^{0}\tau_{1},$
(S4) $\displaystyle j_{i}^{\alpha}$ $\displaystyle\equiv 2t\sin
k_{i}\sigma^{\alpha}\tau_{1},$ (S5) $\displaystyle\rho_{ij}^{\alpha}$
$\displaystyle\equiv 2t\cos k_{i}\sigma^{\alpha}\tau_{1}\delta_{ij},$ (S6)
$\displaystyle\rho_{ij}^{{\alpha}{\beta}}$ $\displaystyle\equiv 2t\cos
k_{i}\sigma^{\alpha}\sigma^{\beta}\tau_{1}\delta_{ij}+{\rm h.c.},$ (S7)

where $\alpha,\beta=x,y,z$. We write as $j_{i\pm}\equiv 2et\sin(k_{i}\pm
q_{i}/2)\sigma^{0}\tau_{1}$ and $j_{i\pm}^{\alpha}\equiv 2t\sin(k_{i}\pm
q_{i}/2)\sigma^{\alpha}\tau_{1}$. The spin-current and number-current vertices
are given by

$\displaystyle{\cal J}_{{\rm s},i}^{\alpha}$
$\displaystyle=j_{i}^{\alpha}+\rho_{ij}^{{\alpha}{\beta}}A_{j}^{\beta},$ (S8)
$\displaystyle{\cal J}_{i}$
$\displaystyle=j_{i}+\rho_{ij}^{\alpha}A_{j}^{\alpha}.$ (S9)

##  II Calculation of TSHC without VC

We first study the contribution without vertex corrections (VC). After a
perturbative treatment of the gauge field and taking impurity averaging, we
divide the perturbation terms into two parts; adiabatic part (which contains
$A^{z}$ only) and nonadiabatic part (which contains $A^{\perp}$ only). The
adiabatic contribution consists of three terms,

$\displaystyle\sigma_{xy}^{z,z}=-\frac{e}{4\pi}\cdot\frac{1}{2}A_{i}^{z}\sum_{\bm{k}}{\rm
tr}\left[j_{x}^{z}G_{+}^{\rm R}j_{i}^{z}G_{-}^{\rm R}j_{y-}G_{-}^{\rm
A}+j_{x}^{z}G_{+}^{\rm R}j_{y-}G_{+}^{\rm A}j_{i}^{z}G_{-}^{\rm
A}+j_{x}^{z}G_{+}^{\rm R}\rho_{iy}^{z}G_{-}^{\rm A}\right],$ (S10)

and the nonadiabatic contribution consists of eight terms,

$\displaystyle\sigma_{xy}^{z,\perp}=\sigma_{xy}^{z,1a}+\sigma_{xy}^{z,1b}+\sigma_{xy}^{z,1c}+\sigma_{xy}^{z,2a}+\sigma_{xy}^{z,2b}+\sigma_{xy}^{z,3a}+\sigma_{xy}^{z,3b}+\sigma_{xy}^{z,4},$
(S11)

where

$\displaystyle\sigma_{xy}^{z,1a}$
$\displaystyle=-\frac{e}{4\pi}\cdot\frac{1}{4}A_{i}^{\alpha}A_{j}^{\beta}\sum_{\bm{k}}{\rm
tr}\left[j_{x}^{z}G^{\rm R}j_{i}^{\alpha}G^{\rm R}j_{j}^{\beta}G^{\rm
R}j_{y}G^{\rm A}\right],$ (S12) $\displaystyle\sigma_{xy}^{z,1b}$
$\displaystyle=-\frac{e}{4\pi}\cdot\frac{1}{4}A_{i}^{\alpha}A_{j}^{\beta}\sum_{\bm{k}}{\rm
tr}\left[j_{x}^{z}G^{\rm R}j_{i}^{\alpha}G^{\rm R}j_{y}G^{\rm
A}j_{j}^{\beta}G^{\rm A}\right],$ (S13) $\displaystyle\sigma_{xy}^{z,1c}$
$\displaystyle=-\frac{e}{4\pi}\cdot\frac{1}{4}A_{i}^{\alpha}A_{j}^{\beta}\sum_{\bm{k}}{\rm
tr}\left[j_{x}^{z}G^{\rm R}j_{y}G^{\rm A}j_{i}^{\alpha}G^{\rm
A}j_{j}^{\beta}G^{\rm A}\right],$ (S14) $\displaystyle\sigma_{xy}^{z,2a}$
$\displaystyle=-\frac{e}{4\pi}\cdot\frac{1}{4}A_{i}^{\alpha}A_{j}^{\beta}\sum_{\bm{k}}{\rm
tr}\left[\rho_{ix}^{z\alpha}G^{\rm R}j_{j}^{\beta}G^{\rm R}j_{y}G^{\rm
A}\right],$ (S15) $\displaystyle\sigma_{xy}^{z,2b}$
$\displaystyle=-\frac{e}{4\pi}\cdot\frac{1}{4}A_{i}^{\alpha}A_{j}^{\beta}\sum_{\bm{k}}{\rm
tr}\left[\rho_{ix}^{z\alpha}G^{\rm R}j_{y}G^{\rm A}j_{j}^{\beta}G^{\rm
A}\right],$ (S16) $\displaystyle\sigma_{xy}^{z,3a}$
$\displaystyle=-\frac{e}{4\pi}\cdot\frac{1}{4}A_{i}^{\alpha}A_{j}^{\beta}\sum_{\bm{k}}{\rm
tr}\left[j_{x}^{z}G^{\rm R}j_{i}^{\alpha}G^{\rm R}\rho_{jy}^{\beta}G^{\rm
A}\right],$ (S17) $\displaystyle\sigma_{xy}^{z,3b}$
$\displaystyle=-\frac{e}{4\pi}\cdot\frac{1}{4}A_{i}^{\alpha}A_{j}^{\beta}\sum_{\bm{k}}{\rm
tr}\left[j_{x}^{z}G^{\rm R}\rho_{jy}^{\beta}G^{\rm A}j_{i}^{\alpha}G^{\rm
A}\right],$ (S18) $\displaystyle\sigma_{xy}^{z,4}$
$\displaystyle=-\frac{e}{4\pi}\cdot\frac{1}{4}A_{i}^{\alpha}A_{j}^{\beta}\sum_{\bm{k}}{\rm
tr}\left[\rho_{ix}^{z\alpha}G^{\rm R}\rho_{jy}^{\beta}G^{\rm A}\right].$ (S19)

The corresponding Feynman diagrams are shown in Fig. S2.

Figure S2:  Feynman diagrams for the TSH conductivity without vertex
corrections. The blue and red solid lines represent the Green’s functions with
mutually opposite spin states. The total wave vector
${\bm{Q}}={{\bm{q}}}+{{\bm{q}}}^{\prime}$ in the nonadiabatic terms
($\sigma_{xy}^{z,\perp}$) is provided by the gauge fields, which will be set
${{\bm{Q}}}\to{\bm{0}}$ later in the calculation. The diagrams in the green-
shaded region turned out to vanish because of symmetry.

Most of the nonadiabatic terms can be disregarded, however. First,
$\sigma_{xy}^{z,1a},\sigma_{xy}^{z,1b}$, and $\sigma_{xy}^{z,1c}$ do not
contain anti-symmetric components, so they are not studied here. Also, we find
$\sigma_{xy}^{z,2a}=\sigma_{xy}^{z,2b}=0$ and $\sigma_{xy}^{z,4}=0$ because of
the cancellation with the Hermitian conjugate part of $\rho_{ij}^{z{\alpha}}$.
Hence, it is sufficient to consider
$\sigma_{xy}^{z,\perp}=\sigma_{xy}^{z,3a}+\sigma_{xy}^{z,3b}$, or

$\displaystyle\sigma_{xy}^{z,\perp}=-\frac{e}{4\pi}\cdot\frac{1}{4}A_{i}^{\alpha}A_{j}^{\beta}\sum_{\bm{k}}{\rm
tr}\left[j_{x}^{z}G^{\rm R}j_{i}^{\alpha}G^{\rm R}\rho_{jy}^{\beta}G^{\rm
A}+j_{x}^{z}G^{\rm R}\rho_{jy}^{\beta}G^{\rm A}j_{i}^{\alpha}G^{\rm
A}\right].$ (S20)

Let us first calculate the adiabatic contribution, Eq. (S10), which is
expressed as.

$\displaystyle\sigma_{xy}^{z,z}$
$\displaystyle=-\frac{e}{4\pi}A_{i}^{z}\sum_{\bm{k}}{\rm
tr}\left[G_{{\bm{k}}-}^{\rm A}j_{x}^{z}G_{{\bm{k}}+}^{\rm
R}\left\\{j_{i}^{z}\left(G^{\rm R}j_{y}-G^{\rm
A}j_{y}\right)_{-}-\left(j_{y}G^{\rm R}-j_{y}G^{\rm
A}\right)_{+}j_{i}^{z}\right\\}\right].$ (S21)

Expanding it with respect to $q$, we write

$\displaystyle\sigma_{xy}^{z,z}=-\frac{e}{16\pi}q_{j}A_{i}^{z}\left(K_{ij}-K_{ji}\right)=-\frac{e}{16\pi}\left({{\bm{q}}}\times{\bm{A}}^{z}\right)_{z}K,$
(S22)

where

$\displaystyle K_{ij}$ $\displaystyle=\sum_{\bm{k}}{\rm tr}\left[G^{\rm
A}j_{x}^{z}G^{\rm R}j_{i}^{z}\left(G^{\rm R}-G^{\rm
A}\right)\rho_{jy}^{0}\right]=\delta_{ix}\delta_{jy}K,$ (S23) $\displaystyle
K$ $\displaystyle=(2t)^{3}\sum_{\bm{k}}\sin^{2}k_{x}\cos k_{y}\cdot{\rm
tr}\left[\sigma^{z}\tau_{1}G^{\rm R}\sigma^{z}\tau_{1}G^{\rm
R}\sigma^{0}\tau_{1}G^{\rm A}-\sigma^{z}\tau_{1}G^{\rm
R}\sigma^{z}\tau_{1}G^{\rm A}\sigma^{0}\tau_{1}G^{\rm A}\right]$
$\displaystyle=8i(2t)^{3}\sum_{\bm{k}}\sin^{2}k_{x}\cos k_{y}\cdot{\rm
Im}\left[T^{\rm A}\left\\{(\mu^{\rm R})^{2}-(J^{\rm R})^{2}+(T^{\rm
R})^{2}\right\\}+2T^{\rm R}\left(\mu^{\rm R}\mu^{\rm A}-J^{\rm R}J^{\rm
A}\right)\right].$ (S24)

Using

$\displaystyle\sum_{{\bm{k}}}\frac{A_{{\bm{k}}}}{(D^{\rm R})^{2}D^{\rm
A}}=-\frac{i\pi\tau^{2}}{2\mu|\mu|}\left[A_{{\bm{k}}}\right]-\frac{\pi\tau}{2|\mu|}\left[A_{{\bm{k}}}\right]^{\prime},$
(S25)

where

$\displaystyle\left[A_{{\bm{k}}}\right]$
$\displaystyle=\sum_{{\bm{k}}}A_{{\bm{k}}}\delta(\mu^{2}-E_{{\bm{k}}}^{2}),$
(S26) $\displaystyle\left[A_{{\bm{k}}}\right]^{\prime}$
$\displaystyle=\sum_{{\bm{k}}}A_{{\bm{k}}}\frac{\partial}{\partial\mu^{2}}\delta(\mu^{2}-E_{{\bm{k}}}^{2}),$
(S27)

with $(2\tau)^{-1}={\gamma}=(1+J^{2}/\mu^{2}){\gamma}_{0}$, we obtain

$\displaystyle\sigma_{xy}^{z,z}=-\frac{et^{2}\nu}{\mu}\left(\mu^{2}-\frac{3}{4}J^{2}\right)\tau^{2}\left(1-\frac{J^{2}}{\mu^{2}}\right)\left\langle
1-\cos k_{x}\cos k_{y}\right\rangle_{\rm
FS}\,{\bm{n}}\cdot(\partial_{x}{\bm{n}}\times\partial_{y}{\bm{n}}).$ (S28)

This contains a term independent to $J$ (which survives the limit $J\to 0$).

Next, the nonadiabatic terms [Eq. (S20)] are calculated as

$\displaystyle\sigma_{xy}^{z,\perp}$
$\displaystyle=\frac{e}{16\pi}\left({\bm{A}}_{x}^{\perp}\times{\bm{A}}_{y}^{\perp}\right)^{z}L$
(S29) $\displaystyle L$ $\displaystyle=8(2t)^{3}\sum_{\bm{k}}\sin^{2}k_{x}\cos
k_{y}\cdot{\rm Im}\left[T^{\rm A}\left\\{(\mu^{\rm R})^{2}+(J^{\rm
R})^{2}+(T^{\rm R})^{2}\right\\}+2T^{\rm R}\mu^{\rm R}\mu^{\rm A}\right].$
(S30)

Similar manipulation leads to

$\displaystyle\sigma_{xy}^{z,\perp}=\frac{et^{2}\nu}{\mu}\left(\mu^{2}+\frac{1}{4}J^{2}\right)\tau^{2}\left(1-\frac{J^{2}}{\mu^{2}}\right)\left\langle
1-\cos k_{x}\cos k_{y}\right\rangle_{\rm
FS}\,{\bm{n}}\cdot(\partial_{x}{\bm{n}}\times\partial_{y}{\bm{n}}).$ (S31)

Here we also find a $J$-independent term.

Summing the adiabatic and nonadiabatic terms, the VC-free contribution is
obtained as

$\displaystyle\sigma_{\rm SH}^{(0)}$
$\displaystyle=\sigma_{xy}^{z,z}+\sigma_{xy}^{z,\perp}$
$\displaystyle=\frac{et^{2}\nu}{\mu}(J\tau)^{2}\left(1-\frac{J^{2}}{\mu^{2}}\right)\left\langle
1-\cos k_{x}\cos k_{y}\right\rangle_{\rm
FS}\,{\bm{n}}\cdot(\partial_{x}{\bm{n}}\times\partial_{y}{\bm{n}}).$ (S32)

This is Eq. (7) in the main text. Note that the $J$-independent term is now
absent; those in $\sigma_{xy}^{z,z}$ and $\sigma_{xy}^{z,\perp}$ have been
canceled. This fact clearly shows the importance of nonadiabatic processes.

##  III Calculation of TSHC with VC

We next consider the effect of vertex corrections (VC). The terms with a
longitudinal spin diffusion propagator (green broken line in Fig. S3) with
same-spin electrons vanish for the uniform and DC component of spin Hall
current. This can be justified by taking the ${{\bm{Q}}}\to{\bm{0}}$ limit
before taking the $\omega\to 0$ limit, similar to the case of ferromagnets
[3].

Figure S3:  Feynman diagrams for the TSH conductivity which contain vertex
corrections. The red and green broken lines are the longitudinal and
transverse spin diffusion propagators, respectively. The primes on
$\rho_{jy}^{{}^{\prime}{\alpha}}$, $j_{i}^{{}^{\prime\prime}{\alpha}}$, etc.
are used to distinguish loop momenta. The upside-down diagrams are also
considered in the calculation. The diagrams in the green-shaded regions are
disregarded.

Thus, we consider the nonadiabatic contribution with transverse-spin diffusion
propagator, Eq. (S2). Assuming a very slow spatial variation of the texture,
we set $q=0$ in Eq. (S2). As in the preceding section, the diagrams that
contain $\rho_{ij}^{{\alpha}{\beta}}$ vanish, and it suffices to consider

$\displaystyle\sigma_{xy}^{z,(1)}=-\frac{e}{4\pi}\cdot\frac{1}{4}A_{i}^{\alpha}A_{j}^{\beta}\cdot\frac{1}{16}\left(P+\bar{P}\right)_{ix}^{\alpha,a}\Pi^{ab}\left[(R+\bar{R})-(S+\bar{S})\right]_{jy}^{\beta,b},$
(S33)

where ${\alpha},{\beta}=x,y$, and

$\displaystyle P_{ix}^{{\alpha},a}=\sum_{{\bm{k}}}{\rm tr}\left[G^{\rm
A}j_{x}^{z}G^{\rm R}j_{i}^{\alpha}G^{\rm R}{\lambda}^{a}\right],$ (S34)
$\displaystyle\bar{P}_{ix}^{{\alpha},a}=\sum_{{\bm{k}}}{\rm tr}\left[G^{\rm
A}j_{i}^{\alpha}G^{\rm A}j_{x}^{z}G^{\rm R}{\lambda}^{a}\right],$ (S35)
$\displaystyle R_{jy}^{{\beta},b}=\sum_{{\bm{k}}}{\rm tr}\left[G^{\rm
R}j_{j}^{\beta}G^{\rm R}j_{y}G^{\rm A}{\lambda}^{b}\right],$ (S36)
$\displaystyle\bar{R}_{jy}^{{\beta},b}=\sum_{{\bm{k}}}{\rm tr}\left[G^{\rm
R}j_{y}G^{\rm A}j_{j}^{\beta}G^{\rm A}{\lambda}^{b}\right],$ (S37)
$\displaystyle S_{jy}^{{\beta},b}=\sum_{{\bm{k}}}{\rm tr}\left[G^{\rm
R}j_{y}G^{\rm R}j_{j}^{\beta}G^{\rm A}{\lambda}^{b}\right],$ (S38)
$\displaystyle\bar{S}_{jy}^{{\beta},b}=\sum_{{\bm{k}}}{\rm tr}\left[G^{\rm
R}j_{i}^{\beta}G^{\rm A}j_{y}G^{\rm A}{\lambda}^{b}\right].$ (S39)

Here, $\Pi^{ab}$ are matrix elements of the vertex correction, with indices
$a,b$ taking $({\gamma}\delta)$, where ${\gamma}=x,y$ and $\delta=0,1,2,3$,
which specify the channel
${\lambda}^{({\gamma}\delta)}\equiv\sigma^{\gamma}\otimes\tau_{\delta}$.
Taking the trace in $\left(P+\bar{P}\right)_{ix}^{\alpha,a}$,
$\left(R+\bar{R}\right)_{jy}^{\beta,b}$ and
$\left(S+\bar{S}\right)_{jy}^{\beta,b}$, we have

$\displaystyle\left(P+\bar{P}\right)_{ix}^{\alpha,({\gamma}0)}$
$\displaystyle=-8(2t)^{2}\delta_{ix}\varepsilon^{{\alpha}{\gamma}}P_{0},$
(S40) $\displaystyle\left(R+\bar{R}\right)_{jy}^{\beta,({\gamma}0)}$
$\displaystyle=8(2t)^{2}\delta_{jy}\delta^{{\gamma}{\beta}}R_{0},$ (S41)
$\displaystyle\left(S+\bar{S}\right)_{jy}^{\beta,({\gamma}0)}$
$\displaystyle=8(2t)^{2}\delta_{jy}\delta^{{\gamma}{\beta}}S_{0},$ (S42)
$\displaystyle\left(P+\bar{P}\right)_{ix}^{\alpha,({\gamma}3)}$
$\displaystyle=8(2t)^{2}\delta_{ix}\delta^{{\alpha}{\gamma}}P_{3},$ (S43)
$\displaystyle\left(R+\bar{R}\right)_{jy}^{\beta,({\gamma}3)}$
$\displaystyle=8(2t)^{2}\delta_{jy}\varepsilon^{{\gamma}{\beta}}R_{3},$ (S44)
$\displaystyle\left(S+\bar{S}\right)_{jy}^{\beta,({\gamma}3)}$
$\displaystyle=-8(2t)^{2}\delta_{jy}\varepsilon^{{\gamma}{\beta}}S_{3},$ (S45)

with

$\displaystyle P_{0}$ $\displaystyle=\sum_{{\bm{k}}}\sin^{2}k_{x}\,{\rm
Im}\left[\mu^{\rm A}\left\\{(\mu^{\rm R})^{2}+(J^{\rm R})^{2}+(T^{\rm
R})^{2}\right\\}-2J^{\rm R}J^{\rm A}\mu^{\rm R}+2T^{\rm R}T^{\rm A}\mu^{\rm
R}\right],$ (S46) $\displaystyle R_{0}$
$\displaystyle=\sum_{{\bm{k}}}\sin^{2}k_{y}\,{\rm Re}\left[\mu^{\rm
A}\left\\{(\mu^{\rm R})^{2}+(J^{\rm R})^{2}+(T^{\rm R})^{2}\right\\}-2J^{\rm
R}J^{\rm A}\mu^{\rm R}+2T^{\rm R}T^{\rm A}\mu^{\rm R}\right],$ (S47)
$\displaystyle S_{0}$ $\displaystyle=\sum_{{\bm{k}}}\sin^{2}k_{y}\,{\rm
Re}\left[\mu^{\rm A}\left\\{(\mu^{\rm R})^{2}-(J^{\rm R})^{2}+(T^{\rm
R})^{2}\right\\}+2T^{\rm R}T^{\rm A}\mu^{\rm R}\right],$ (S48) $\displaystyle
P_{3}$ $\displaystyle=\sum_{{\bm{k}}}\sin^{2}k_{x}\,{\rm Re}\left[J^{\rm
A}\left\\{(\mu^{\rm R})^{2}+(J^{\rm R})^{2}+(T^{\rm R})^{2}\right\\}-2\mu^{\rm
R}\mu^{\rm A}J^{\rm R}\right],$ (S49) $\displaystyle R_{3}$
$\displaystyle=\sum_{{\bm{k}}}\sin^{2}k_{y}\,{\rm Im}\left[J^{\rm
A}\left\\{(\mu^{\rm R})^{2}+(J^{\rm R})^{2}+(T^{\rm R})^{2}\right\\}-2\mu^{\rm
R}\mu^{\rm A}J^{\rm R}\right],$ (S50) $\displaystyle S_{3}$
$\displaystyle=\sum_{{\bm{k}}}\sin^{2}k_{y}\,{\rm Im}\left[J^{\rm
A}\left\\{(\mu^{\rm R})^{2}-(J^{\rm R})^{2}+(T^{\rm R})^{2}\right\\}-2T^{\rm
R}T^{\rm A}J^{\rm R}\right].$ (S51)

Using Eq. (S25), we obtain

$\displaystyle P_{0}$
$\displaystyle=-\pi\nu\tau^{2}\frac{\mu^{2}-J^{2}}{\mu^{2}}\left\langle\sin^{2}k_{x}\right\rangle_{\rm
FS},$ (S52) $\displaystyle R_{0}-S_{0}$
$\displaystyle=-\frac{\pi\nu\tau}{2\mu}\frac{J^{2}}{\mu^{2}+J^{2}}\left\langle\sin^{2}k_{y}\right\rangle_{\rm
FS}.$ (S53)

Therefore, Eq. (S33) becomes

$\displaystyle\sigma_{\rm SH}^{(1)}$
$\displaystyle=\frac{4et^{4}}{\pi}({\bm{A}}_{x}\times{\bm{A}}_{y})^{z}\cdot
P_{0}\,\Pi_{\bar{\sigma}\sigma}(R_{0}-S_{0})$ (S54)
$\displaystyle=\frac{2et^{2}\nu}{\mu}(J\tau)^{2}\frac{(2t)^{2}}{\mu^{2}+J^{2}}\frac{\tau^{-1}}{\tau_{\varphi}^{-1}+\tau_{\rm
s}^{-1}}\left\langle\sin^{2}k_{x}\right\rangle_{\rm
FS}^{2}\,{\bm{n}}\cdot(\partial_{x}{\bm{n}}\times\partial_{y}{\bm{n}}).$ (S55)

This is Eq. (8) in the main text. We note that the contribution with
$\Pi^{({\gamma}3)({\gamma}3)}$ is smaller than the one with
$\Pi^{({\gamma}0)({\gamma}0)}$ by a factor of $(J/\mu^{2}\tau)^{2}$, and is
zeroth order in $\tau$; thus it can be disregarded for good metals considered
here.

Figure S4:  Comparison of $J$-dependence of the peak value of $\sigma_{\rm
SH}^{(1)}$ (red lines) with (a) $\Pi_{\bar{\sigma}\sigma}$ and
$\tau_{\varphi}$, (b) $|P_{0}|$, and (c) $|R_{0}-S_{0}|$ (blue lines). See
Eqs. (S2) and (S46)-(S48) for the definition of $\Pi_{\bar{\sigma}\sigma}$,
$P_{0}$, $R_{0}$, and $S_{0}$, and Eq. (S54) for the relation among them. In
the plots, we set $e$, $t$, and the lattice constant unity, and
$\tau_{\varphi}$ is divided by 40.

##  IV Enhancement at weak coupling

As discussed in the main text, one of the most spectacular aspect of the
present TSHE is the enhancement at weak coupling (small $J$). More
specifically, the peak value of the spin Hall conductivity near the AF gap
edge increases as $J$ is decreased. To see the origin of this enhancement, we
compare in Fig. S4 the $J$-dependence of the peak value of $\sigma_{\rm
SH}^{(1)}$ with that of $\Pi_{\bar{{\sigma}}{\sigma}}$ (vertex correction; see
Eq. (S2)), $\tau_{\varphi}$ (spin dephasing time), $P_{0}$ (left loop
integral), and $R_{0}-S_{0}$ (right loop integral) [see Eq. (S54)] in the weak
coupling regime, $J/t<1$. The value of $\mu$ is taken at which $\sigma_{\rm
SH}^{(1)}$ is peaked. As seen, the overall $J$-dependence of $\sigma_{\rm
SH}^{(1)}$ follows that of $\Pi_{\bar{\sigma}\sigma}$ and $\tau_{\varphi}$,
rather than $P_{0}$ and $R_{0}-S_{0}$, implying that the spin dephasing time
is the key parameter for the enhancement of TSHE. At very small $J/t$
($<0.2$), $|R_{0}-S_{0}|$ shows a remarkable increase, which may also be
important. Overall, it is very important to consider $\tau_{\varphi}$ in the
weak coupling regime.

##  V Perturbative treatment of $J$

Figure S5:  Feynman diagrams for the TSH conductivity at second order in $J$.

In this section, we calculate the TSH conductivity by treating the exchange
coupling $J$ perturbatively. Relevant diagrams are shown in Fig. S5. The
Green’s functions in this section are those at $J=0$. Extracting ${{\bm{q}}}$
to the lowest (second) order, we write the antisymmetric part as

$\displaystyle\sigma_{xy}^{{\alpha},{\rm A}}$
$\displaystyle=-\frac{e}{8\pi}J^{2}q_{x}q^{\prime}_{y}n_{{\bm{q}}}^{\beta}n_{{{\bm{q}}}^{\prime}}^{\gamma}(-2t)^{3}\sum_{{\bm{k}}}\cos
k_{x}\sin^{2}k_{y}{\rm
tr}\left[{\sigma}^{\alpha}{\sigma}^{\beta}{\sigma}^{\gamma}\otimes\tau_{1}G^{\rm
R}\tau_{3}G^{\rm R}\tau_{1}G^{\rm R}\ \tau_{3}G^{\rm R}\tau_{1}G^{\rm
A}\right],$ (S56) $\displaystyle\sigma_{xy}^{{\alpha},{\rm B}}$
$\displaystyle=-\frac{e}{8\pi}J^{2}q^{\prime}_{x}q_{y}n_{{\bm{q}}}^{\beta}n_{{{\bm{q}}}^{\prime}}^{\gamma}(-2t)^{3}\sum_{{\bm{k}}}\cos
k_{x}\sin^{2}k_{y}{\rm
tr}\left[{\sigma}^{\alpha}{\sigma}^{\beta}{\sigma}^{\gamma}\otimes\tau_{1}G^{\rm
R}\tau_{1}G^{\rm A}\tau_{3}G^{\rm A}\tau_{1}G^{\rm A}\tau_{3}G^{\rm
A}\right],$ (S57) $\displaystyle\sigma_{xy}^{{\alpha},{\rm C1}}$
$\displaystyle=\frac{e}{8\pi}J^{2}q^{\prime}_{x}q_{y}n_{{\bm{q}}}^{\beta}n_{{{\bm{q}}}^{\prime}}^{\gamma}(-2t)^{3}\sum_{{\bm{k}}}\cos
k_{x}\sin^{2}k_{y}{\rm
tr}\left[{\sigma}^{\alpha}{\sigma}^{\beta}{\sigma}^{\gamma}\otimes\tau_{1}G^{\rm
R}\tau_{1}G^{\rm R}\tau_{3}G^{\rm R}\tau_{1}G^{\rm A}\tau_{3}G^{\rm
A}\right],$ (S58) $\displaystyle\sigma_{xy}^{{\alpha},{\rm C2}}$
$\displaystyle=\frac{e}{8\pi}J^{2}q_{x}q^{\prime}_{y}n_{{\bm{q}}}^{\beta}n_{{{\bm{q}}}^{\prime}}^{\gamma}(-2t)^{3}\sum_{{\bm{k}}}\cos
k_{x}\sin^{2}k_{y}{\rm
tr}\left[{\sigma}^{\alpha}{\sigma}^{\beta}{\sigma}^{\gamma}\otimes\tau_{1}G^{\rm
R}\tau_{3}G^{\rm R}\tau_{1}G^{\rm A}\tau_{3}G^{\rm A}\tau_{1}G^{\rm
A}\right],$ (S59)

where $\sigma_{xy}^{{\alpha},{\rm C1}}+\sigma_{xy}^{{\alpha},{\rm
C2}}=\sigma_{xy}^{{\alpha},{\rm C}}$. Noting $\left(\mu^{\rm
R(A)}\right)^{2}-\left(T^{\rm R(A)}\right)^{2}=1/D^{\rm R(A)}$ and Eq. (S25),
we proceed as

$\displaystyle\sigma_{xy}^{{\alpha},{\rm A}}+\sigma_{xy}^{{\alpha},{\rm B}}$
$\displaystyle=\frac{ie}{2\pi}J^{2}\left(\partial_{x}{\bm{n}}\times\partial_{y}{\bm{n}}\right)^{\alpha}(-2t)^{3}2i{\rm
Im}\sum_{{\bm{k}}}\cos k_{x}\sin^{2}k_{y}\left[(\mu^{\rm R})^{2}-(T^{\rm
R})^{2}\right]^{2}T^{\rm A}$ (S60)
$\displaystyle=-\frac{e}{\pi}J^{2}\left(\partial_{x}{\bm{n}}\times\partial_{y}{\bm{n}}\right)^{\alpha}(-2t)^{3}{\rm
Im}\sum_{{\bm{k}}}\frac{\cos k_{x}\sin^{2}k_{y}T_{{\bm{k}}}}{\left(D^{\rm
R}\right)^{2}D^{\rm A}}$ (S61)
$\displaystyle=\frac{1}{2}\frac{et^{2}\nu}{\mu}(J\tau)^{2}\left\langle 1-\cos
k_{x}\cos k_{y}\right\rangle_{\rm
FS}\left(\partial_{x}{\bm{n}}\times\partial_{y}{\bm{n}}\right)^{\alpha},$
(S62) $\displaystyle\sigma_{xy}^{{\alpha},{\rm C1}}+\sigma_{xy}^{{\alpha},{\rm
C2}}$
$\displaystyle=\frac{ie}{2\pi}J^{2}\left(\partial_{x}{\bm{n}}\times\partial_{y}{\bm{n}}\right)^{\alpha}(-2t)^{3}2i{\rm
Im}\sum_{{\bm{k}}}\cos k_{x}\sin^{2}k_{y}\left[(\mu^{\rm R})^{2}-(T^{\rm
R})^{2}\right]\left[(\mu^{\rm A})^{2}-(T^{\rm A})^{2}\right]T^{\rm R}$ (S63)
$\displaystyle=-\frac{e}{\pi}J^{2}\left(\partial_{x}{\bm{n}}\times\partial_{y}{\bm{n}}\right)^{\alpha}(-2t)^{3}{\rm
Im}\sum_{{\bm{k}}}\frac{\cos k_{x}\sin^{2}k_{y}T_{{\bm{k}}}}{\left(D^{\rm
R}\right)^{2}D^{\rm A}}$ (S64)
$\displaystyle=\frac{1}{2}\frac{et^{2}\nu}{\mu}(J\tau)^{2}\left\langle 1-\cos
k_{x}\cos k_{y}\right\rangle_{\rm
FS}\left(\partial_{x}{\bm{n}}\times\partial_{y}{\bm{n}}\right)^{\alpha}.$
(S65)

Therefore, we obtain

$\displaystyle j_{{\rm
s},x}^{\alpha}=\frac{et^{2}\nu}{\mu}(J\tau)^{2}\left\langle 1-\cos k_{x}\cos
k_{y}\right\rangle_{\rm
FS}\left(\partial_{x}{\bm{n}}\times\partial_{y}{\bm{n}}\right)^{\alpha}E_{y}.$
(S66)

This is Eq. (10) in the main text, and agrees with Eq. (S32) at order $J^{2}$.

##  VI Spin Hall angle

For a comparison to experiments, we plot a normalized spin Hall angle
$\tilde{\theta}_{\rm SH}\equiv\tilde{\sigma}_{\rm SH}/\tilde{\sigma}_{\rm c}$,
where $\tilde{\sigma}_{\rm c}=\sigma_{\rm c}\tilde{\gamma}$ ($\sigma_{\rm
c}=2e^{2}D\nu$) is the normalized longitudinal conductivity [Fig. S6(a)].

Figure S6:  (a) and (b) show the chemical potential dependence of the
normalized longitudinal conductivity (a) and the spin Hall angle (b) for
several choices of $J/t$. (c) and (d) are color plots in the plane of $n$
(average electron number per site) and $J/t$ of the normalized topological
spin Hall conductivity (c) and the normalized spin Hall angle (d). ($n=1$
corresponds to completely filled lower and empty upper AF bands.)

The spin Hall angle $\tilde{\theta}_{\rm SH}$ is shown in Fig. S6(b). As $J$
is increased from $J=0.1t$, the peak value decreases first quickly, then
moderately and becomes stationary. This is because the longitudinal
conductivity also decreases but rather constantly; see Fig. S6(a).

The normalized TSH conductivity $\tilde{\sigma}_{\rm SH}\tilde{\gamma}^{2}$
and the normalized TSH angle $\tilde{\theta}_{\rm SH}$ are plotted in Fig. S6
(c) and (d), respectively, in the plane of $J$ and electron filling $n$. As
for the former [Fig. S6(c)], we see a very sharp peak in a very narrow region
at weak coupling ($J\lesssim 0.5t$) and $n\gtrsim 0.4$. This is discussed in
the main text. The spin Hall angle $\tilde{\theta}_{\rm SH}$ [Fig. S6(d)] has
a similar structure, but also has a broad tail (or plateau) extended to larger
$J$.

##  VII nonlocality

As the wave-vector ${\bm{q}}$ of spin texture (or spin gauge field) is
increased, the effective field becomes nonlocal with respect to the spin
texture. This is induced through spin diffusion. To study this, we retain the
${\bm{q}}$-dependence of the texture, and write the TSH conductivity as

$\displaystyle\sigma_{\rm
SH}({{\bm{Q}}})\propto\Pi({{\bm{q}}}^{\prime})[{\bm{A}}_{x}^{\perp}({{\bm{q}}})\times{\bm{A}}_{y}^{\perp}({{\bm{q}}}^{\prime})]^{z}+({{\bm{q}}}\leftrightarrow{{\bm{q}}}^{\prime})-(x\leftrightarrow
y).$ (S67)

In real space, it reads

$\displaystyle\sigma_{\rm SH}$ $\displaystyle\propto\int
d{{\bm{r}}}^{\prime}\Pi({{\bm{r}}}-{{\bm{r}}}^{\prime})[{\bm{A}}_{x}^{\perp}({{\bm{r}}})\times{\bm{A}}_{y}^{\perp}({{\bm{r}}}^{\prime})]^{z}-(x\leftrightarrow
y)$ $\displaystyle={\bm{n}}\cdot(\partial_{x}{\bm{n}}\times{\cal
R}(\hat{z}\times\tilde{\bm{A}}_{y}^{\perp}))-(x\leftrightarrow y)$
$\displaystyle={\bm{n}}\cdot(\partial_{x}{\bm{n}}\times\tilde{\bm{d}}_{y})-(x\leftrightarrow
y),$ (S68)

with

$\displaystyle\tilde{\bm{A}}_{i}^{\perp}$ $\displaystyle=\int
d{{\bm{r}}}^{\prime}\Pi({{\bm{r}}}-{{\bm{r}}}^{\prime}){\bm{A}}_{i}^{\perp}({{\bm{r}}}^{\prime}),$
(S69) $\displaystyle\tilde{\bm{d}}_{i}$ $\displaystyle=\int
d{{\bm{r}}}^{\prime}\Pi({{\bm{r}}}-{{\bm{r}}}^{\prime}){\cal
R}({{\bm{r}}}){\cal
R}^{-1}({{\bm{r}}}^{\prime})\partial_{i}{\bm{n}}({{\bm{r}}}^{\prime}),$ (S70)

and

$\displaystyle\Pi({\bm{R}})=\sum_{{\bm{q}}}\Pi_{\bar{\sigma}\sigma}({{\bm{q}}})\,{{\rm
e}}^{i{{\bm{q}}}\cdot{\bm{R}}}=\frac{2}{\pi^{2}\nu\tau}\frac{\mu^{2}}{\mu^{2}-J^{2}}\frac{1}{D\tau}K_{0}(R/\ell_{{\varphi}{\rm
s}}),$ (S71)

where $\ell_{{\varphi}{\rm s}}=\sqrt{D\tau_{{\varphi}{\rm s}}}$ with
$\tau_{{\varphi}{\rm s}}^{-1}=\tau_{{\varphi}}^{-1}+\tau_{\rm s}^{-1}$,
${\bm{R}}={{\bm{r}}}-{{\bm{r}}}^{\prime}$, and $K_{0}(x)$ is the modified
Bessel function of the second kind. We identify the physical spin Hall current
as

$\displaystyle{\bm{j}}_{{\rm
s},i}\propto(\partial_{i}{\bm{n}}\times\tilde{\bm{d}}_{j}-\partial_{j}{\bm{n}}\times\tilde{\bm{d}}_{i})E_{j}.$
(S72)

##  VIII Numerics with Landauer-Büttiker formula

Figure S7:  Chemical potential dependence of the spin Hall conductance before
the “symmetrization” for systems with AF skyrmion (upper panels) and AF meron
(lower panels), and for $J=0.3t$ (left panels) and $J=-0.3t$ (right panels).

In the calculation based on the Landauer-Büttiker formula, we employ the four-
terminal geometry with nonmagnetic leads and obtained numerically the spin
Hall conductance from the transmission coefficients [4]. The magnetic textures
employed are as follows. For AF skyrmion, we take

$\displaystyle{\bm{n}}({{\bm{r}}})=\left(\cos\varphi\sin\theta,\
\sin\varphi\sin\theta,\ \cos\theta\right),$ (S73)

where $\varphi={\rm Arg}(x+iy)+\pi/2$ and $\theta=\theta_{+}+\theta_{-}$ with
$\sin\theta_{\pm}=\tanh\frac{2(|{{\bm{r}}}|\pm R_{\rm sk})}{R_{\rm sk}}$ [5].
For AF meron, we take

$\displaystyle{\bm{n}}({{\bm{r}}})$
$\displaystyle=\left(\cos\varphi\sin\theta,\ \sin\varphi\sin\theta,\
\cos\theta\right)\quad(\cos\theta>0)$ (S74)
$\displaystyle{\bm{n}}({{\bm{r}}})$ $\displaystyle=\left(\cos\varphi,\
\sin\varphi,\ 0\right)\quad({\rm otherwise}),$ (S75)

with same $\varphi$ and $\theta$.

The results are shown in Fig. S7 for $J=0.3t$ and $J=-0.3t$. While the
analytical result indicates that contributions from vector chirality are even
in $J$ (see the main text), a substantial $J$-odd component is seen in Fig. S7
for AF skyrmion whereas it is almost absent for AF meron. We have thus
extracted the $J$-even contribution by symmetrizing as

$\displaystyle G_{\rm SHC}^{z}\equiv\left[G_{\rm SHC}^{z}(J)+G_{\rm
SHC}^{z}(-J)\right]/2.$ (S76)

Figure 3 in the main text shows this “symmetrized” data. Since the
(disregarded) $J$-odd components do not depend much on the system size, they
are likely to come from the boundary between the sample and the lead.

## References

  * [1] J. J. Nakane, K. Nakazawa, and H. Kohno, Phys. Rev. B 101, 174432 (2020).
  * [2] A. Manchon, J. Phys. Condens. Matter 29, 104002 (2017).
  * [3] K. Nakazawa, M. Bibes, and H. Kohno, J. Phys. Soc. Jpn. 87, 033705 (2018); K. Nakazawa and H. Kohno, Phys. Rev. B 99, 033705 (2018).
  * [4] J. Ohe, T. Ohtsuki, B. Kramer, Phys. Rev. B 75, 245313 (2007).
  * [5] N. Romming, A. Kubetzka, C. Hanneken, K. von Bergmann, and R. Wiesendanger, Phys. Rev. Lett. 114, 177203 (2015).