W\subset\mathbb{B}_{2}^{k+5}\cap W.$

The lemma then follows from taking $Q=\frac{1}{30}\tilde{Q}$. ∎

###  6.6 Perturbation Analysis and Proof of the Lower Bound

In this subsection, we study the number of edges of the intersection polygon
$Q\cap W$ after perturbation and prove our main theorem (Theorem 47). To show
that our construction has many edges even after perturbation, we require the
following two statements:

######  Lemma 55.

Let $\bar{a}_{1},\dots,\bar{a}_{n}\in\mathbb{R}^{d}$ be any points with
$r\mathbb{B}^{d}_{1}\subset\operatorname{conv}(\bar{a}_{1},\dots,\bar{a}_{n})$
for some $r>0$. If $\varepsilon\leq r/2$ and points
$a_{1},\dots,a_{n}\in\mathbb{R}^{d}$ satisfy
$\|a_{i}-\bar{a}_{i}\|_{1}\leq\varepsilon$ for all $i\in[n]$ then it follows
that

$(1-\frac{2\varepsilon}{r})\operatorname{conv}(\bar{a}_{1},\dots,\bar{a}_{n})\subset\operatorname{conv}(a_{1},\dots,a_{n})\subset(1+\frac{\varepsilon}{r})\operatorname{conv}(\bar{a}_{1},\dots,\bar{a}_{n}).$

###### Proof.

Write $Q=\operatorname{conv}(a_{1},\dots,a_{n})$ and
$\bar{Q}=\operatorname{conv}(\bar{a}_{1},\dots,\bar{a}_{n})$. The second
inclusion follows by
$Q\subset\bar{Q}+\varepsilon\mathbb{B}_{1}^{d}\subset\bar{Q}+\frac{\varepsilon}{r}\bar{Q}$.
For the first inequality, we observe that
$r\mathbb{B}_{1}^{d}\subset\bar{Q}\subset
Q+\varepsilon\mathbb{B}_{1}^{d}\subset Q+\frac{r}{2}\mathbb{B}_{1}^{d}$. This
implies that $\frac{r}{2}\mathbb{B}_{1}^{d}\subset Q$, for if there were to
exist $x\in\frac{r}{2}\mathbb{B}^{d}_{1}$ such that $x\notin Q$ then, since
$Q$ is closed and convex, we could find $y\in\mathbb{R}^{d}$ such that
$y^{\top}x>y^{\top}z$ for all $z\in Q$. Writing $f(S)=\max_{z\in S}y^{\top}z$
for $S\subset\mathbb{R}^{d}$, this would give

$f(r\mathbb{B}_{1}^{d})\geq
f(x+\frac{r}{2}\mathbb{B}_{1}^{d})=y^{\top}x+f(\frac{r}{2}\mathbb{B}_{1}^{d})>f(Q)+f(\frac{r}{2}\mathbb{B}_{1}^{d})\geq
f(Q+\frac{r}{2}\mathbb{B}_{1}^{d})\geq f(r\mathbb{B}_{1}^{d}).$

By contradiction it follows that $\frac{r}{2}\mathbb{B}_{1}^{d}\subset Q$.

Now the desired result follows by $\bar{Q}\subset
Q+\varepsilon\mathbb{B}_{1}^{d}\subset Q+\frac{\varepsilon}{r/2}Q$ and the
fact that $(1+x)^{-1}>1-x$ for every $x>-1$. ∎

######  Lemma 56.

If a polygon $T\subset\mathbb{R}^{2}$ satisfies
$\alpha\cdot\mathbb{B}_{2}^{2}\subset T\subset\beta\cdot\mathbb{B}_{2}^{2}$
for some $\alpha,\beta>0$ then $T$ has at least
$\Omega(\sqrt{\alpha/(\beta-\alpha)})$ edges.

###### Proof.

Without loss of generality, re-scale $T$ so that $\mathbb{B}_{2}^{2}\subset
T\subset(1+\varepsilon)\cdot\mathbb{B}_{2}^{2}$, where
$\varepsilon=\beta/\alpha-1>0$.

Consider any edge $[q_{1},q_{2}]\subset T$ and let $p\in[q_{1},q_{2}]$ denote
the minimum-norm point in this edge. Then we have
$\|q_{1}-p\|^{2}=\|q_{1}\|^{2}+\|p\|^{2}-2\langle q_{1},p\rangle$. Since $p$
is the minimum-norm point, we have $\langle p,q_{1}\rangle\geq\|p\|^{2}$, and
hence
$\|q_{1}-p\|^{2}\leq\|q_{1}\|^{2}-\|p\|^{2}\leq(1+\varepsilon)^{2}-\|p\|^{2}$.
Since $p$ lies on the boundary of $T$ we have $\|p\|\geq 1$, which implies
that $\|q_{1}-p\|^{2}\leq(1+\varepsilon)^{2}-1=2\varepsilon+\varepsilon^{2}$.
The analogous argument for $\|q_{2}-p\|$ and the triangle inequality tell us
that $\|q_{1}-q_{2}\|\leq 2\sqrt{2\varepsilon+\varepsilon^{2}}\leq
4\sqrt{\varepsilon}$. The choice of the edge $[q_{1},q_{2}]$ was arbitrary,
hence every edge of $T$ has length as most
$2\sqrt{2\varepsilon+\varepsilon^{2}}$.

But $T$ has perimeter at least $2\pi$. Since the perimeter of a polygon is
equal to the sum of the lengths of its edges, this implies that $T$ has at
least $\frac{2\pi}{4\sqrt{\varepsilon}}$ edges. ∎

Now, we can prove our generic lower bound Theorem 48 on the shadow size under
adversarial $\ell_{1}$-perturbations.

###### Proof of Theorem 48.

Fix any $d\geq 5$, let $k=d-5$ and observe that $n=4k+5$. Let
$\bar{a}_{1},\cdots,\bar{a}_{n}$ be as constructed in Lemma 54. Then we have

$\frac{1}{30(1+4^{-k})}\cdot\mathbb{B}_{2}^{k+5}\cap
W\subset\operatorname{conv}(\bar{a}_{1},\cdots,\bar{a}_{n})\cap
W\subset\frac{1}{30}\cdot\mathbb{B}_{2}^{k+5}\cap W$

and

$\displaystyle\frac{1}{45}\cdot\mathbb{B}_{1}^{k+5}\subset\operatorname{conv}(\bar{a}_{1},\cdots,\bar{a}_{n})\subset\mathbb{B}_{1}^{k+5}$

For any set of points $a_{1},\cdots,a_{n}$ such that
$\|a_{i}-\bar{a}_{i}\|_{1}\leq\varepsilon$ for each $i\in[n]$, by Lemma 55 we
have

$\frac{1}{30(1+4^{-k})}\cdot\mathbb{B}_{2}^{k+5}\cap
W\subset\operatorname{conv}(\bar{a}_{1},\cdots,\bar{a}_{n})\cap
W\subset(1+90\varepsilon)\operatorname{conv}(a_{1},\cdots,a_{n})\cap W$

and

$(1-90\varepsilon)\operatorname{conv}(a_{1},\cdots,a_{n})\cap
W\subset\operatorname{conv}(\bar{a}_{1},\cdots,\bar{a}_{n})\cap
W\subset\frac{1}{30}\cdot\mathbb{B}_{2}^{k+5}\cap W.$

Therefore, we can bound the inner and outer radius of
$\operatorname{conv}(a_{1},\cdots,a_{n})\cap W$ by

$\displaystyle\frac{1}{30\cdot(1+4^{-k})\cdot(1+90\varepsilon)}\cdot\mathbb{B}_{2}^{2}\subseteq\operatorname{conv}(a_{1},\cdots,a_{n})\cap
W\subseteq\cdot\frac{1}{30\cdot(1-90\varepsilon)}\cdot\mathbb{B}_{2}^{2}$

It follows from Lemma 56 that the polygon
$\operatorname{conv}(a_{1},\cdots,a_{n})\cap W$ has at least
$\Omega(\frac{1}{\sqrt{\varepsilon+4^{-k}}})$ edges. ∎

Finally, we can prove our main result using Gaussian tail bound:

###### Proof of Theorem 47.

Using concentration of Gaussian distribution in 13, we find that if
$\sigma\leq\frac{1}{360d\sqrt{\log n}}$, then with probability at least
$1-\binom{n}{d}^{-1}$, we have $\max_{i\in[n]}\|a_{i}-\bar{a}_{i}\|_{2}\leq
4\sigma\sqrt{d\log n}\leq\frac{1}{90\sqrt{d}}$. The result follows from
Theorem 48 and the fact that $\|x\|_{1}\leq\sqrt{d}\|x\|_{2}$ for every
$x\in\mathbb{R}^{d}$. ∎

###  6.7 Experimental Results

To measure whether analysis in Theorem 47 is tight or not, we ran numerical
experiments. Using Python and Gurobi 9.5.2, we constructed a matrix $A$ such
that

$P-(\bar{x},\bar{y},\bar{p}_{0},\bar{t},\bar{s})=\\{x\in\mathbb{R}^{k+5}:Ax\leq\mathbf{1}_{4k+5}\\},$

as described earlier in this section. Writing $R$ as the maximum Euclidean
norm among the row vectors of $A$, we sampled $\hat{A}$ with independent
Gaussian distributed entries with standard deviation $\sigma R$ and
$\mathbb{E}[\hat{A}]=A$. To approximate the shadow size, we optimized the
objective vectors
$\cos(\frac{(i+0.3)\pi}{2^{k+4}})x+\sin(\frac{(i+0.3)\pi}{2^{k+4}})y$, with
$i=0,\dots,2^{k+5}-1$, over the polyhedron
$\\{x\in\mathbb{R}^{k+5}:\hat{A}x\leq 1\\}$ and counted the number of distinct
values $(x,y)$ found among the solutions. When $\sigma=0$, our code found
$2^{k+1}$ such points. For $\sigma>0$, Theorem 47 shows that we expect to find
at least $\Omega\Big{(}\min\Big{(}\frac{1}{\sqrt{d\sigma\sqrt{\log
d}}},2^{k}\Big{)}\Big{)}$ distinct pairs $(x,y)$.

For $k=10,15,20$, we measured the shadow size for $20$ different values of
$\sigma$ ranging from $0.01$ to $0.0001/2^{k}$, The resulting data is depicted
in Figure 5 along with a graph of the function $\sigma\mapsto\sigma^{-3/4}$.
We observe that for each $k$, the measured shadow size appears to follow the
graphed function up to a point, plateauing slightly above $2^{k+1}$ when
$\sigma$ is small. The fact that some measurements come out higher than
$2^{k+1}$, the shadow size for $\sigma=0$, is not unexpected: the polytope $P$
is highly degenerate, whereas the perturbed polytope is simple and can thus
have many more vertices.

The measured shadow sizes appear to grow much faster than $1/\sqrt{\sigma}$ as
$\sigma$ gets small. These results suggest that the behaviour of the shadow
size is substantially different in $d=2$, where we have an upper bound of
$O\left(\frac{\sqrt[4]{\log(n)}}{\sqrt{\sigma}}+\sqrt{\log n}\right)$, and
$d>3$, where one might expect a lower bound with a higher dependence on
$\sigma$.

Figure 5: Measured shadow sizes for sampled perturbations of our construction,
for different values of $k$ and $\sigma$.

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# Spin Hall effect emerging from a chiral magnetic lattice without spin-orbit
coupling

Yang Zhang  Max Planck Institute for Chemical Physics of Solids, 01187
Dresden, Germany  Leibniz Institute for Solid State and Materials Research,
IFW Dresden, 01069 Dresden, Germany  Jakub Železný  Max Planck Institute for
Chemical Physics of Solids, 01187 Dresden, Germany  Yan Sun  Max Planck
Institute for Chemical Physics of Solids, 01187 Dresden, Germany  Jeroen van
den Brink  Leibniz Institute for Solid State and Materials Research, IFW
Dresden, 01069 Dresden, Germany  Binghai Yan<EMAIL_ADDRESS>Department of Condensed Matter Physics, Weizmann Institute of Science, 7610001
Rehovot, Israel

###### Abstract

The spin Hall effect (SHE), which converts a charge current into a transverse
spin current, has long been believed to be a phenomenon induced by the
spin–orbit coupling. Here, we propose an alternative mechanism to realize the
intrinsic SHE through a chiral magnetic structure that breaks the spin
rotation symmetry. No spin–orbit coupling is needed even when the scalar spin
chirality vanishes, different from the case of the topological Hall effect. In
known chiral antiferromagnetic compounds Mn${}_{3}X$ ($X=$ Ga, Ge, and Sn),
for example, we indeed obtain large spin Hall conductivities based on ab
initio calculations. Apart further developing the conceptual understanding of
the SHE, our work suggests an alternative strategy to design spin Hall
materials without involving heavy elements, which may be advantageous for
technological applications.

Introduction – The spin Hall effect (SHE) Sinova  _et al._ (2015) is one of
the most important ways to create and detect spin currents in the field of
spintronics, which aims to realize low-power-consumption and high-speed
devices Wolf  _et al._ (2001). It converts electric currents into transverse
spin currents and vice versa. The SHE in materials is generally believed to
rely on spin–orbit coupling (SOC) (Sinova  _et al._ , 2015; Maekawa  _et al._
, 2012; Hoffmann, 2013; Gradhand  _et al._ , 2012, for review). In typical SHE
devices, the generated spin current is injected into a ferromagnet (FM) and
consequently switches its magnetization via the spin-transfer torque Miron
_et al._ (2011); Liu  _et al._ (2012) or drives an efficient motion of
magnetic domain walls Parkin and Yang (2015); Yang  _et al._ (2015).

The SHE is conceptually similar to the well established anomalous Hall effect
(AHE). In recent decades, the understanding of the intrinsic AHE Nagaosa  _et
al._ (2010) and intrinsic SHE Murakami  _et al._ (2003); Sinova  _et al._
(2004) reached great success based on the concept of the Berry phase Xiao  _et
al._ (2010), which originates directly from the electronic band structure.
Although the AHE requires the existence of SOC in a FM, it also appears in a
non-coplanar magnetic lattice without SOC, where an electron acquires a Berry
phase by hopping through sites with a chiral magnetic structure (nonzero
scalar spin chirality) Ohgushi  _et al._ (2000); Taguchi  _et al._ (2001),
later referred as the topological Hall effect (THE) Bruno  _et al._ (2004).
Thus, in experiment the THE-induced Hall signal is treated as a signature of
the skyrmion phase with chiral spin texture (for instance Refs. Neubauer  _et
al._ , 2009; Kanazawa  _et al._ , 2011). Provoked by the THE, recent numerical
simulations of the spin scattering by a single skyrmion indicated a nonzero
SHE without considering SOC Yin  _et al._ (2015); Buhl  _et al._ (2017).
However, fundamental questions are still open. Is the skyrmion-like nonzero
spin chirality always necessary? What is the generic condition for a SHE
without SOC? Furthermore, heavy elements are found to enhance the Gilbert
damping Tserkovnyak  _et al._ (2002); Liu  _et al._ (2012) that hinders the
spin-current-switching in an adjacent thin-film magnet, although the search
for SHE compounds focuses on materials with strong SOC Guo  _et al._ (2008);
Tanaka  _et al._ (2008). From a technological point of view, SHE materials
with negligible or weak SOC will bring an advantage to constrain the Gilbert
damping.

The crucial role of SOC is to break the spin rotational symmetry (SRS) in SHE.
Alternatively, noncollinear magnetic textures can also violate SRS. Therefore,
we propose a mechanism to realize the SHE with the chiral magnetic structure
but without SOC. Different from the THE in symmetry, such an SHE appears
generally for the chiral magnetic lattice, even with zero scalar spin
chirality. Here, we first prove the principle from the symmetry analysis in a
simple lattice model. Then, we demonstrate the existence of a strong SHE in
several known materials Mn${}_{3}X$ ($X=$Ga, Ge, and Sn)Nayak  _et al._
(2016); Nakatsuji  _et al._ (2015) by ab initio calculations, both with and
without SOC.

Figure 1:  Existence of AHE and SHE in collinear FM, collinear AFM, coplanar,
and noncoplanar chiral magnetic lattices with and without SOC. We can deduce
whether AHE and SHE are allowed (A) or forbidden (F) by considering the
symmetry. Regardless of SOC, the AHE is forbidden in a collinear AFM caused by
the time-reversal symmetry combined with a half-lattice translation or an
inversion. In addition, a nonmagnetic system that respects time-reversal
symmetry is equivalent to a collinear AFM for both SHE and AHE.

Double-exchange model and symmetry analysis – We first consider a simple model
to illustrate the symmetry of SHE without SOC. We employed a double-exchange
model ($s$-$d$ model) Zener (1951); Anderson and Hasegawa (1955); de Gennes
(1960) which describes itinerant $s$ electrons interacting with local $d$
magnetic moments. Here we assume that magnetic moments are only contributed by
the spin degrees of freedom.

$\displaystyle
H=t\sum_{<ij>\alpha}c^{{\dagger}}_{i\alpha}c_{j_{\alpha}}-J\sum_{i\alpha\beta}(\bm{\sigma}\cdot\bm{n}_{i})_{\alpha\beta}c^{{\dagger}}_{i\alpha}c_{i\beta}.$
(1)

Here, $\alpha$ and $\beta$ stand for spin up and spin down, respectively. The
first term is the nearest neighbor hopping term with $<ij>$ denoting the
nearest neighbor lattice sites. In the second term, $J$ is the Hund’s coupling
strength between the conduction electron and the on-site spin moment,
$\bm{\sigma}$ is the vector of Pauli matrices, and $\bm{n}_{i}$ is the spin
magnetic moment on site $i$. The magnetic texture is defined by the pattern of
$\bm{n}_{i}$.

The intrinsic AHE and intrinsic SHE are well characterized via the Berry
curvature formalism Xiao  _et al._ (2010); Nagaosa  _et al._ (2010); Sinova
_et al._ (2015); Gradhand  _et al._ (2012). The anomalous Hall
conductivity(AHC) $\sigma_{\alpha\beta}$ can be evaluated by the integral of
the Berry curvature $\Omega^{n}_{\alpha\beta}(\mathbf{k})$ over the Brillouin
zone (BZ) for all the occupied bands, where $n$ is the band index. This method
can also be applied to the THE Ohgushi  _et al._ (2000), even if it is
commonly interpreted using the real space spin texture. Here,
$\sigma_{\alpha\beta}$ corresponds to a $3\times 3$ matrix and indicates a
transverse Hall current $j_{\alpha}$ generated by a longitudinal electric
field $\mathbf{E}$, which satisfies
$J_{\alpha}=\sigma_{\alpha\beta}E_{\beta}$. Within a linear response, Berry
curvature can be expressed as

$\displaystyle\Omega_{n,\alpha\beta}(\vec{k})=2i\hbar^{2}\sum_{m\neq
n}\dfrac{\bra{\psi_{n\mathbf{k}}}\hat{v}_{\alpha}\ket{\psi_{m\mathbf{k}}}\bra{\psi_{m\mathbf{k}}}\hat{v}_{\beta}\ket{\psi_{n\mathbf{k}}}}{(E_{n}(\vec{k})-E_{m}(\vec{k}))^{2}},$
(2)

where $n$ and $m$ are band indices, and $\psi_{n\mathbf{k}}$ and
$E_{n\mathbf{k}}$ denote the Bloch wave functions and eigenvalues,
respectively, and $\hat{\mathbf{v}}$ is the velocity operator. Replacing the
velocity operator with the spin current operator
$\hat{J}_{\alpha}^{\gamma}=\dfrac{1}{2}\\{\hat{v}_{\alpha},\hat{s}_{\gamma}\\}$,
where $\hat{s}_{\gamma}$ is the spin operator, we obtain the spin Berry
curvature and corresponding spin Hall conductivity (SHC),

$\displaystyle\sigma_{\alpha\beta}^{\gamma}$
$\displaystyle=\dfrac{e}{\hbar}\sum_{n}\int_{BZ}\dfrac{d^{3}\mathbf{k}}{(2\pi)^{3}}f_{n}(\mathbf{k})\Omega_{n,{\alpha\beta}}^{\gamma}(\mathbf{k}),$
(3) $\displaystyle\Omega_{n,\alpha\beta}^{\gamma}(\mathbf{k})$
$\displaystyle=2i\hbar^{2}\sum_{m\neq
n}\dfrac{\bra{\psi_{n\mathbf{k}}}\hat{J}_{\alpha}^{\gamma}\ket{\psi_{m\mathbf{k}}}\bra{\psi_{m\mathbf{k}}}\hat{v}_{\beta}\ket{\psi_{n\mathbf{k}}}}{(E_{n\mathbf{k}}-E_{m\mathbf{k}})^{2}}.$

The SHC ($\sigma_{\alpha\beta}^{\gamma}$; $\alpha,\beta,\gamma=x,y,z$) is a
third-order tensor ($3\times 3\times 3$) and represents the spin current
${J}_{s,\alpha}^{\gamma}$ generated by an electric field $\mathbf{E}$ via
${J}_{s,\alpha}^{\gamma}=\sigma_{\alpha\beta}^{\gamma}{E}_{\beta}$, where
${J}_{s,\alpha}^{\gamma}$ is a spin current flowing along the
$\alpha$-direction with the spin-polarization along the $\gamma$-direction.

We know that AHE vanishes while SHE remains if the time-reversal symmetry
(operator $T$) exists in the system. In Eq. 2, $T$ reverses the velocities
$\hat{v}_{\alpha,\beta}$ and brings an additional “$-$” sign by the complex
conjugate. Thus, $\sigma_{\alpha\beta}=0$ owing to
$\Omega_{n,\alpha\beta}(\vec{k})=-\Omega_{n,\alpha\beta}(\vec{-k})$. In
contrast, In Eq. 3, $T$ generates one more “$-$” sign by reversing the spin in
${J}_{\alpha}^{\gamma}$. Then, $\sigma_{\alpha\beta}^{\gamma}$ can be nonzero
since $\Omega_{n,\alpha\beta}^{\gamma}(\vec{k})$ is even in $k$-space. When
SOC is absent or negligible, a magnetic system usually exhibits the spin
rotation symmetry (operator $S$), which rotates only spins (or magnetic
moments) simultaneously but not the lattice. Although $T$ is broken here, some
systems may still preserve a combined symmetry by $T$ and a spin-rotation of
$180^{\circ}$ degree, noted as $TS$. For example, a coplanar magnetic system
shows a $TS$ symmetry, in which $S$ rotates all spins by $180^{\circ}$ around
the axis perpendicular to the plane. Since $S$ does not act on
$\hat{v}_{\alpha,\beta}$, $TS$ causes vanishing $\sigma_{\alpha\beta}$ just as
$T$ alone. In a general noncoplanar magnetic lattice, the TS symmetry is
naturally broken, because one cannot find a common axis about which all spins
can be rotated $180^{\circ}$ at the same time, and thus the AHE can exist
without SOC.

The situation is different for the SHE since ${J}_{\alpha}^{\gamma}$ in Eq. 3
contains an additional spin operator. As a consequence, (assuming that $S$ is
a rotation around the $z$ axis) $TS$ forces
$\Omega_{n,\alpha\beta}^{x/y}(\vec{k})$ to be odd where spin $\hat{s}_{x}$ or
$\hat{s}_{y}$ is reversed by $TS$, but $\Omega_{n,\alpha\beta}^{z}(\vec{k})$
to be even where $\hat{s}_{z}$ is unchanged by $TS$. Then, one can obtain zero
$\sigma_{\alpha\beta}^{x/y}$ but nonzero $\sigma_{\alpha\beta}^{z}$. In a
collinear magnetic lattice there exists more than one spin rotation $S$ such
that $TS$ is a symmetry of the system and thus all of the
$\sigma_{\alpha\beta}^{\gamma}$ components have to vanish. Therefore, we can
argue that SHE can exist without SOC in general noncollinear magnetic
lattices, regardless of FM, AFM, or the scalar spin chirality (coplanar or
noncoplanar). In contrast, the AHC is zero for a coplanar magnetic lattice
(zero scalar spin chirality), since $TS$ acts as $T$ alone in Eq. 2.

When SOC is included, SHE is allowed by symmetry in any crystalSeemann  _et
al._ (2015), while the AHE on the other hand can be present in magnetic
systems that are not symmetric under time reversal combined with a translation
or inversion (for example, a conventional collinear AFM). We summarize the
necessary conditions for the existence of AHE and SHE in systems with and
without SOC in Fig. 1.

To demonstrate that the SHE can indeed be nonzero without SOC, we consider the
$s$–$d$ Hamiltonian (1) projected on a kagome lattice with the so-called $q=0$
magnetic order, shown in Fig. 2(a). Such a coplanar AFM order is well studied
in theory Ohgushi  _et al._ (2000); Chen  _et al._ (2014) and appears in many
realistic materials even at room temperature, for example Mn${}_{3}X$ ($X=$
Ir, Ga, Ge, and Sn) Zhang  _et al._ (2013); Kübler and Felser (2014);
Nakatsuji  _et al._ (2015); Nayak  _et al._ (2016); Zhang  _et al._ (2016,
2017) as we discuss in the following. For comparison, the SOC effect is also
considered by adding to $H$ in Eq. 1,

$\displaystyle
H_{\text{so}}=it_{2}\sum_{<ij>\alpha\beta}v_{ij}(\bm{\sigma}\cdot\bm{n}_{ij})_{\alpha\beta}c^{{\dagger}}_{i\alpha}c_{j\beta},$
(4)

where $v_{ij}$ and $n_{ij}$ are defined as in Ref. Chen  _et al._ , 2014 and
$t_{2}$ is the SOC strength.

We first analyze the symmetry of the SHC tensor for the $q=0$ magnetic order.
Note that we use the Cartesian coordinate systems defined in Fig. 2. As
discussed above, the existence of the $TS$ symmetry leaves only
$\sigma_{\alpha\beta}^{z}$ terms in the absence of SOC. Further, the combined
symmetry $TM_{x}$, in which $M_{x}$ is the mirror reflection along $x$ and
flips $\hat{s}_{z}$ and $\hat{v}_{x}$ in Eq. 3, leads to
$\sigma_{xx}^{z}=\sigma_{yy}^{z}=0$. We further obtain only two nonzero SHC
tensor element $\sigma^{z}_{xy}=-\sigma^{z}_{yx}$ by considering the three-
fold rotation around $z$. The magnetic order shown in Fig. 2(b) will also be
relevant for the discussion in the following. This magnetic configuration
differs from the $q=0$ case only by a two-fold spin rotation around the
$y$-axis and thus, without SOC its symmetry is exactly the same as that of the
$q=0$ case.

Setting the Hund coupling constant $J=1.7t$ and SOC strength $t_{2}=0$, we
calculate the spin Berry curvature via Eq. 3. As expected, we find nonzero SHC
$\sigma^{z}_{xy}$ fully in agreement with the symmetry considerations. Figures
3a and 3b show the band structures with $t_{2}=0$ and $t_{2}=0.2t$,
respectively. One can see that SOC modifies slightly the band structure by
gaping some band crossing points such as the BZ corners ($K$). Without SOC, we
already observe nonzero $\sigma^{z}_{xy}$, while adding SOC reduces
$\sigma^{z}_{xy}$ slightly at the Fermi energy that is set between the first
and second bands at about –2.7 eV. We plot corresponding spin Berry curvature
$\Omega^{z}_{xy}$ in the hexagonal BZ In Figs. 3d and 3e. Large
$\Omega^{z}_{xy}$ appears in the BZ without SOC, leading to net
$\sigma^{z}_{xy}$. The SOC simply brings an extra contribution to
$\sigma^{z}_{xy}$ at the band anti-crossing region around the $K$ point.

Figure 2: Noncollinear order in kagome lattice and the magnetic structure of
Mn3Ir and Mn3X (X = Ga, Ge, and Sn). (a) The $q=0$ order in the kagome
lattice, with magnetic moments located as 2D Mn plane in Mn3Ir, (b) Two-fold
spin rotation around $y$-axis of configuration (a), corresponding to Mn planes
in Mn3X (X = Ga, Ge, Sn), (c) The face-centered cubic crystal structure of
Mn${}_{3}Ir$, (d) The hexagonal crystal structure of Mn3X (X = Ga, Ge, and
Sn).  Figure 3: Electronic band structures $q=0$ order in kagome lattice, (a)
without SOC, (b) with SOC. (c) Energy-dependent SHC $\sigma^{z}_{xy}$. (d)
Spin Berry curvature $\Omega^{z}_{xy}$ distribution of first BZ at Fermi
energy $-2.7$ eV (horizontal line) without SOC and (e) with SOC. Figure 4:
Electronic band structure for (a) Mn3Ga with SOC, (b) Mn3Ga without SOC, (d)
Mn3Ge with SOC, (e) Mn3Ge without SOC, (g) Mn3Sn with SOC, (h) Mn3Sn without
SOC, (j) Mn3Ir with SOC, (k) Mn3Ir without SOC. Energy-dependent SHC tensor
elements of $\sigma_{xy}^{z}$ with and without SOC for (c) Mn3Ga, (f) Mn3Ge,
(i) Mn3Sn, and (l) Mn3Ir. The Fermi energy is indicated by the dashed
horizontal line.

Realistic materials – After proving the principle, we now identify materials
that show strong SHE with negligible contribution from SOC. We naturally
consider Mn3X (X=Ga, Ge, Sn, and Ir) compounds, since they exhibit non-
collinear AFM order at room temperature (the AFM Néel temperature is over 365
K). Our recent ab initio calculations showed a sizable intrinsic SHE by
including SOC Zhang  _et al._ (2017). Here, we further point out that SHE
still presents without SOC and SOC actually plays a negligible effect for SHE
in these materials.

The primitive unit cell of Mn3Ga, Mn3Ge and Mn3Sn (space group $P6_{3}/mmc$,
No. 194) includes two Mn${}_{3}X$ planes that are stacked along the $c$-axis
according to a “–AB–AB–” sequence. Inside each plane, Mn atoms form a kagome-
type lattice with Ga, Ge, or Sn lying at the center of the hexagon formed by
Mn. Both the ab initio calculation Zhang  _et al._ (2013) and neutron
diffraction measurements Krén and Kádár (1970); Kádár and Krén (1971); Ohoyama
_et al._ (1961) show that the Mn magnetic moments exhibit noncollinear AFM
order, where the neighboring moments are aligned at an angle of 120∘, as in
Fig. 2(b). Large AHE in room temperature has recently been reported in Mn3Ge
and Mn3Sn Nakatsuji  _et al._ (2015); Nayak  _et al._ (2016). These materials
also exhibit other exotic phenomena including the Weyl semimetal phase Yang
_et al._ (2016), magneto-optical Kerr effect Feng  _et al._ (2015), anomalous
Nernst effect Li  _et al._ (2016), and topological defects Liu and Balents
(2017). Distinct from hexagonal Mn${}_{3}X$ compounds, the Mn3Ir (space group
$Pm\bar{3}m$, No. 221) crystallizes in a face-centered cubic structure with Mn
atoms in the [111] planes forming a kagome lattice with the $q=0$ magnetic
order.

The symmetry of the SHE without SOC in these materials can be understood using
a similar approach as we used for the 2D kagome lattice. The hexagonal Ga, Ge,
and Sn materials can be viewed as stacking versions of the kagome lattice and
thus we find that the symmetry of SHE is the same as the 2D kagome lattice,
i.e. only $\sigma_{xy}^{z}=-\sigma_{yx}^{z}$ is nonzero. However, we find that
SHE must vanish in Mn3Ir without SOC, which is imposed by the higher symmetry
of the cubic magnetic lattice. For completeness, we list the tensor matrices
without and with SOC for all these compounds in the appendix.

Since the SHC tensor shape imposed by the symmetry has been systematically
investigated for these materials in Ref. Zhang  _et al._ (2017), we only
discuss one of the largest SHC tensor elements $\sigma_{xy}^{z}$ based on the
ab initio calculations Kresse and Furthmüller (1996) of the SHC. For
comparison, we show the band structure and SHC without and together with SOC
in Fig. 4. In the absence of SOC, Ga, Ge, and Sn compounds indeed exhibit
nonzero SHC $\sigma_{xy}^{z}=-613,115,90~{}(\hbar/e)(\Omega\cdot cm)^{-1}$,
respectively, at the Fermi energy. One can see that SOC induces very few
changes in the band structure and thereafter modifies the SHC weakly,
especially at the Fermi energy for Ga, Ge, and Sn compounds. It is intuitive
to observe comparable $\sigma_{xy}^{z}$ values for Ge and Sn compounds,
despite the fact that Sn exhibits much larger SOC than Ge. These facts further
verifies that the noncollinear magnetic structure, rather than SOC, is
dominant for the SHE. The Ga compound shows an opposite sign in SHC compared
to the Ge/Sn compound, since Ga has one valence electron fewer than Ge/Sn and
the Fermi energy is lower in Mn3Ga than in Mn3Ge/Mn3Sn.

Summary – We proposed that the SHE can be realized by a chiral magnetic
structure without involving the SOC. The noncollinearity of the chiral
magnetic lattice can break the spin rotation symmetry and consequently allow
the existence of SHE. By ab initio calculations, we further predicted that
such an SHE without SOC can be observed in noncollinear AFM compounds
Mn${}_{3}X$ ($X=$ Ga, Ge, and Sn). Although only the intrinsic SHE is
considered here, we expect that an extrinsic SHE can also appear commonly in
the chiral magnetic lattice without SOC because of similar symmetry
considerations. By enabling a new understanding of the SHE, our work paves the
way to search for SHE materials among chiral magnetic systems, which are not
limited to the heavy element materials. In addition, the close relation
between SHE and the magnetic order suggests that SHE may be used as a probe
for different magnetic structures.

###### Acknowledgements.

We thank Prof. Claudia Felser at MPI-CPfS and Prof. Carsten Timm at TU-Dresden
for helpful discussions. Y.Z. and J.v.d.B. acknowledge the German Research
Foundation (DFG) SFB 1143. B.Y. thanks the support of the Ruth and Herman
Albert Scholars Program for New Scientists in Weizmann Institute of Science,
Israel.

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##  I Appendix

Table 1:  Shape of the AHC and SHC tensors obtained from symmetry analysis and numerical calculations for Mn${}_{3}X$ ($X$=Ga, Ge, and Sn). The calculated SHC tensor elements are set to zero when they are less than 5 $(\hbar/e)(\Omega\cdot cm)^{-1}$ . The coordinates used here are $x$ along [100], $y$ along [120], and $z$ along [001], as presented in Fig. 1(a, b). The SHC are given in units of $(\hbar/e)(\Omega\cdot cm)^{-1}$.  | AHC | SHC
---|---|---
| $\sigma$ | $\underline{\sigma}^{x}$ | $\underline{\sigma}^{y}$ | $\underline{\sigma}^{z}$
| tensor shape
---
with soc
$\left(\begin{array}[]{ccc}0&0&-\sigma_{zx}\\\ 0&0&0\\\ \sigma_{zx}&0&0\end{array}\right)$ | $\left(\begin{array}[]{ccc}0&0&0\\\ 0&0&\sigma_{yz}^{x}\\\ 0&\sigma_{zy}^{x}&0\end{array}\right)$ | $\left(\begin{array}[]{ccc}0&0&\sigma_{xz}^{y}\\\ 0&0&0\\\ \sigma_{zx}^{y}&0&0\end{array}\right)$ | $\left(\begin{array}[]{ccc}0&\sigma_{xy}^{z}&0\\\ \sigma_{yx}^{z}&0&0\\\ 0&0&0\end{array}\right)$
| Mn3Ga
---
$\left(\begin{array}[]{ccc}0&0&-81\\\ 0&0&0\\\ 81&0&0\end{array}\right)$ | $\left(\begin{array}[]{ccc}0&0&0\\\ 0&0&-15\\\ 0&12&0\end{array}\right)$ | $\left(\begin{array}[]{ccc}0&0&15\\\ 0&0&0\\\ -7&0&0\end{array}\right)$ | $\left(\begin{array}[]{ccc}0&-597&0\\\ 625&0&0\\\ 0&0&0\end{array}\right)$
| Mn3Ge
---
$\left(\begin{array}[]{ccc}0&0&330\\\ 0&0&0\\\ -330&0&0\end{array}\right)$ | $\left(\begin{array}[]{ccc}0&0&0\\\ 0&0&-21\\\ 0&18&0\end{array}\right)$ | $\left(\begin{array}[]{ccc}0&0&21\\\ 0&0&0\\\ -18&0&0\end{array}\right)$ | $\left(\begin{array}[]{ccc}0&112&0\\\ -115&0&0\\\ 0&0&0\end{array}\right)$
| Mn3Sn
---
$\left(\begin{array}[]{ccc}0&0&133\\\ 0&0&0\\\ -133&0&0\end{array}\right)$ | $\left(\begin{array}[]{ccc}0&0&0\\\ 0&0&-36\\\ 0&96&0\end{array}\right)$ | $\left(\begin{array}[]{ccc}0&0&36\\\ 0&0&0\\\ -96&0&0\end{array}\right)$ | $\left(\begin{array}[]{ccc}0&64&0\\\ -68&0&0\\\ 0&0&0\end{array}\right)$
| tensor shape
---
without soc
$\left(\begin{array}[]{ccc}0&0&0\\\ 0&0&0\\\ 0&0&0\end{array}\right)$ | $\left(\begin{array}[]{ccc}0&0&0\\\ 0&0&0\\\ 0&0&0\end{array}\right)$ | $\left(\begin{array}[]{ccc}0&0&0\\\ 0&0&0\\\ 0&0&0\end{array}\right)$ | $\left(\begin{array}[]{ccc}0&\sigma_{xy}^{z}&0\\\ -\sigma_{xy}^{z}&0&0\\\ 0&0&0\end{array}\right)$
| Mn3Ga
---
$\left(\begin{array}[]{ccc}0&0&0\\\ 0&0&0\\\ 0&0&0\end{array}\right)$ | $\left(\begin{array}[]{ccc}0&0&0\\\ 0&0&0\\\ 0&0&0\end{array}\right)$ | $\left(\begin{array}[]{ccc}0&0&0\\\ 0&0&0\\\ 0&0&0\end{array}\right)$ | $\left(\begin{array}[]{ccc}0&-613&0\\\ 631&0&0\\\ 0&0&0\end{array}\right)$
| Mn3Ge
---
$\left(\begin{array}[]{ccc}0&0&0\\\ 0&0&0\\\ 0&0&0\end{array}\right)$ | $\left(\begin{array}[]{ccc}0&0&0\\\ 0&0&0\\\ 0&0&0\end{array}\right)$ | $\left(\begin{array}[]{ccc}0&0&0\\\ 0&0&0\\\ 0&0&0\end{array}\right)$ | $\left(\begin{array}[]{ccc}0&115&0\\\ -115&0&0\\\ 0&0&0\end{array}\right)$
| Mn3Sn
---
$\left(\begin{array}[]{ccc}0&0&0\\\ 0&0&0\\\ 0&0&0\end{array}\right)$ | $\left(\begin{array}[]{ccc}0&0&0\\\ 0&0&0\\\ 0&0&0\end{array}\right)$ | $\left(\begin{array}[]{ccc}0&0&0\\\ 0&0&0\\\ 0&0&0\end{array}\right)$ | $\left(\begin{array}[]{ccc}0&90&0\\\ -90&0&0\\\ 0&0&0\end{array}\right)$
Table 2:  Intrinsic SHC tensors obtained from symmetry analysis and numerical calculations for Mn3Ir. The calculated SHC tensor elements are set to zero when they are less than 2 $(\hbar/e)(\Omega\cdot cm)^{-1}$. The coordinate axes correspond to $z$ oriented in the [111] direction, $x$ along $[1\overline{1}0]$ and $y$ along $[11\overline{2}]$. The SHC are in units of $(\hbar/e)(\Omega\cdot cm)^{-1}$.  | AHC | SHC
---|---|---
| $\sigma$ | $\underline{\sigma}^{x}$ | $\underline{\sigma}^{y}$ | $\underline{\sigma}^{z}$
| tensor shape
---
with soc
$\left(\begin{array}[]{ccc}0&\sigma_{xy}&0\\\ -\sigma_{xy}&0&0\\\ 0&0&0\end{array}\right)$ | $\left(\begin{array}[]{ccc}\sigma_{xx}^{x}&0&0\\\ 0&-\sigma_{xx}^{x}&-\sigma_{yz}^{x}\\\ 0&\sigma_{zy}^{x}&0\end{array}\right)$ | $\left(\begin{array}[]{ccc}0&-\sigma_{xx}^{x}&-\sigma_{yz}^{x}\\\ -\sigma_{xx}^{x}&0&0\\\ -\sigma_{zy}^{x}&0&0\end{array}\right)$ | $\left(\begin{array}[]{ccc}0&\sigma_{xy}^{z}&0\\\ -\sigma_{xy}^{z}&0&0\\\ 0&0&0\end{array}\right)$
| Mn3Ir
---
$\left(\begin{array}[]{ccc}0&-312&0\\\ 312&0&0\\\ 0&0&0\end{array}\right)$ | $\left(\begin{array}[]{ccc}-210&0&0\\\ 0&210&299\\\ 0&-7&0\end{array}\right)$ | $\left(\begin{array}[]{ccc}0&210&-299\\\ 210&0&0\\\ 7&0&0\end{array}\right)$ | $\left(\begin{array}[]{ccc}0&163&0\\\ -163&0&0\\\ 0&0&0\end{array}\right)$
| tensor shape
---
without soc
$\left(\begin{array}[]{ccc}0&0&0\\\ 0&0&0\\\ 0&0&0\end{array}\right)$ | $\left(\begin{array}[]{ccc}0&0&0\\\ 0&0&0\\\ 0&0&0\end{array}\right)$ | $\left(\begin{array}[]{ccc}0&0&0\\\ 0&0&0\\\ 0&0&0\end{array}\right)$ | $\left(\begin{array}[]{ccc}0&0&0\\\ 0&0&0\\\ 0&0&0\end{array}\right)$
| Mn3Ir
---
$\left(\begin{array}[]{ccc}0&0&0\\\ 0&0&0\\\ 0&0&0\end{array}\right)$ | $\left(\begin{array}[]{ccc}0&0&0\\\ 0&0&0\\\ 0&0&0\end{array}\right)$ | $\left(\begin{array}[]{ccc}0&0&0\\\ 0&0&0\\\ 0&0&0\end{array}\right)$ | $\left(\begin{array}[]{ccc}0&0&0\\\ 0&0&0\\\ 0&0&0\end{array}\right)$