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

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$_3X$ ($X=$ Ga, Ge, and Sn), for example, we indeed obtain large spin Hall conductivities based on \textit{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.


I. INTRODUCTION
The spin Hall effect (SHE) 1 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 highspeed devices. 2 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) 1, [3][4][5] . In typical SHE devices, the generated spin current is injected into a ferromagnet (FM) and consequently switches its magnetization via the spin-transfer torque 6,7 or drives an efficient motion of magnetic domain walls. 8,9 The SHE is conceptually similar to the well established anomalous Hall effect (AHE).
In recent decades, the understanding of the intrinsic AHE 10 and intrinsic SHE 11,12 was significantly advanced based on the concept of the Berry phase, 13 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 noncoplanar magnetic structure (nonzero scalar spin chirality) 14,15 , later referred as the topological Hall effect (THE) 16 .
Thus, in experiment the THE-induced Hall signal is considered as a signature of the skyrmion phase with chiral spin texture 17,18 . Provoked by the THE, recent numerical simulations of the spin scattering by a single skyrmion indicated the presence of a finite SHE even without SOC [19][20][21] , which is termed as a topological SHE. Thus, the topological SHE has been presumed to stem from the Berry phase due to the nonzero spin chirality of the skyrmion.
However, the origin of the spin current is illusive in the topological SHE, for it is hard to separate it from the spin-polarized charge current of the THE. Here, we pose new questions one step further. Is the skyrmion-like spin texture (nonzero scalar spin chirality) always necessary to generate a SHE without SOC? What is the generic condition for a SHE without SOC?
In this article, we propose a mechanism to realize the SHE with the noncollinear magnetic structure but without SOC. The crucial role of SOC is to break the spin rotational symmetry (SRS) in SHE. Alternatively, it is known that common noncollinear magnetic textures can also violate the SRS, thus resulting in the SHE. Different from the THE and topological SHE in symmetry, such an SHE appears universally for the noncollinear magnetic lattice, regardless of the scalar spin chirality. For example, it can even emerge in a coplanar magnetic structure where the scalar spin chirality is zero. 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) 22

Double-exchange model and symmetry analysis -
The existence of the SHE and AHE in metals is determined by symmetry (in insulators apart from symmetry also the topology of the electronic structure is important). The symmetry of magnetic systems is normally described in terms of magnetic space groups, which contain, apart from the spatial symmetry operations, also the time-reversal symmetry operation. In absence of SOC (or other terms in the Hamiltonian that couple the magnetic moments to the lattice such as the shape anisotropy), however, the symmetry of the magnetic systems is higher than that contained in the magnetic space groups since the spins can be rotated independently of the lattice.
This can be illustrated by considering the following minimal Hamiltonian We focus here only on the intrinsic contribution to the AHE and SHE, however, the other (extrinsic) contributions have the same symmetry and thus the symmetry discussion in the following is general. The intrinsic AHE and intrinsic SHE are well characterized via the Berry curvature formalism. 1,5,10,13 The anomalous Hall conductivity(AHC) σ αβ can be evaluated by the integral of the Berry curvature Ω n αβ (k) over the Brillouin zone (BZ) for all the occupied bands, where n is the band index. It should be noted that this method can also be applied to the THE, although it is commonly interpreted using the real space spin texture. Here, σ αβ corresponds to a 3 × 3 matrix and indicates a transverse Hall current j α generated by a longitudinal electric field E, which satisfies J α = σ αβ E β . Within a linear response, Berry curvature can be expressed as where n and m are band indices, and ψ nk and E nk denote the Bloch wave functions and eigenvalues, respectively, andv is the velocity operator. Replacing the velocity operator with the spin current operatorĴ γ α = 1 2 {v α ,ŝ γ }, whereŝ γ is the spin operator, we obtain the spin Berry curvature and corresponding spin Hall conductivity (SHC), The SHC (σ γ αβ ; α, β, γ = x, y, z) is a third-order tensor (3 × 3 × 3) and represents the spin current J γ s,α generated by an electric field E via J γ s,α = σ γ αβ E β , where J γ s,α is a spin current flowing along the α-direction with the spin-polarization along the γ-direction, and f n (k) is the temperature dependent Fermi-Diract distribution.
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 velocitiesv α,β and brings an additional "−" sign by the complex conjugate. Thus, σ αβ = 0 owing to Ω n,αβ ( k) = −Ω n,αβ ( −k). In contrast, In Eq. 3, T generates one more "−" sign by reversing the spin in J γ α . Then, σ γ αβ can be nonzero since Ω γ n,αβ ( k) is even in k-space. In a magnetic system without SOC, T is broken, but a combination of T and a spin rotation (operator S) can still be a symmetry.
For example, a coplanar magnetic system shows a T S symmetry, in which S rotates all spins by 180 • around the axis perpendicular to the plane. Since S does not act onv α,β , T S causes vanishing σ αβ just as T alone. In a general noncoplanar magnetic lattice, the T S symmetry is naturally broken, because one cannot find a common axis about which all spins can be rotated 180 • at the same time, and thus the AHE can exist without SOC.
The situation is different for the SHE since J γ α in Eq. 3 contains an additional spin operator. As a consequence, (assuming that S is a rotation around the z axis) T S forces Ω x/y n,αβ ( k) to be odd where spinŝ x orŝ y is reversed by T S, but Ω z n,αβ ( k) to be even whereŝ z is unchanged by T S. Then, one can obtain zero σ x/y αβ but nonzero σ z αβ . In a collinear magnetic lattice there exists more than one spin rotation S such that T S is a symmetry of the system and thus all of the σ γ αβ 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 T S acts as T alone in Eq. 2.
When SOC is included, SHE is allowed by symmetry in any crystal, 29 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 14,30 and appears in many realistic materials even at room temperature, for example Mn 3 X (X = Ir, Ga, Ge, and Sn) 22,23,31-34 as we discuss in the following. For comparison, the SOC effect is also considered by adding to H in Eq. 1, where v ij is the antisymmetric Levi-Civita symbol and n ij are a set of coplanar vectors anticlockwise perpendicular to the lattice vector R ij , as defined in Ref. 30 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 T S symmetry leaves only σ z αβ terms in the absence of SOC. Further, the combined symmetry T M x , in which M x is the mirror reflection along x and flipsŝ z andv x in Eq. 3, leads to σ z xx = σ z yy = 0. We further obtain only two nonzero SHC tensor element σ z xy = −σ 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 σ 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 σ z xy , while adding SOC reduces σ 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 Ω z xy in the hexagonal BZ In Figs. 3d and 3e. Large Ω z xy appears in the BZ without SOC, leading to net σ z xy . The SOC simply brings an extra contribution to σ z xy at the band anti-crossing region around the K point.    at the center of the hexagon formed by Mn. Both the ab initio calculation 31 and neutron diffraction measurements [35][36][37] 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 Mn 3 Ge and Mn 3 Sn. 22,23 These materials also exhibit other exotic phenomena including the Weyl semimetal phase, 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 σ z xy = −σ z yx is nonzero.
However, we find that SHE must vanish in Mn 3 Ir 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. 34, we only discuss one of the largest SHC tensor elements σ z xy based on the ab initio calculations 42 of the SHC. For comparison, we show the SHC without and together with SOC in Fig. 4. In the absence of SOC, Ga, Ge, and Sn compounds indeed exhibit nonzero SHC σ z xy = −613, 115, and 90 ( /e)(Ω · 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 σ z xy 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 Mn 3 Ga than in Mn 3 Ge/Mn 3 Sn.

III. DISCUSSION
Understanding the role SOC plays in the SHE is important for the fundamental understanding of the SHE, but also for practical reasons. It can help with the search for materials with large SHE since in non-magnetic or collinear magnetic materials, SOC is necessary for SHE and thus the presence of heavy elements is generally required for large SHE. The SHE without SOC proposed in this work suggests a new strategy to design SHE materials without necessarily involving heavy elements. In noncollinear systems, the Rashba effect can also appear without SOC 43 . The spin texture in the band structure may depend sensitively on the real space spin texture. For example, we found that band structure spin texture is different between the Kagome lattice and the triangular lattice.
We propose the general, necessary symmetry-breaking requirements (Fig. 1) for SHE without SOC. It is worth noting that SHE can become zero without SOC in some noncollinear magnetic lattice where additional symmetries forces the SHE to vanish. For example, we have shown that in Mn 3 Ir the SHE vanishes in absence of SOC even though it has a noncollinear magnetic structure. This is a consequence of its high-symmetrical cubic structure. Similar situation could happen for AHE without SOC in a noncoplanar magnetic lattice, such as the AFM skyrmion system 21 .
In conclusion, we have shown that the SHE can be realized by a non-chiral coplanar magnetic structure without involving the SOC. The noncollinearity of the 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). From our symmetry considerations, an extrinsic SHE can appear when breaking the spin rotational symmetry.
Thus, we expect the extrinsic effect to also exist in our systems. Its amplitude will depend on the details of the scattering, and cannot be estimated without microscopic calculations, though in general for SHE the intrinsic contribution tends to be the dominant contribution.
By providing a general theoretical, symmetry based understanding of the SHE, our work motivates a comprehensive search for SHE materials among noncollinear magnetic systems, that not necessarily involve heavy elements. In addition, the close relation between the SHE and the magnetic order suggests that the SHE may be used vice versa, as a probe to establish and symmetry restrict the ground state magnetic structures of long-range ordered antiferromagnets.
Regarding the strong correlated system, we would discuss the spin liquid material as an example. RuCl3 has a rich magnetic phase diagram, with a complex zig-zag type AFM longrange order at low temperatures, and even a quantum spin liquid phase in applied magnetic field. According to our work, non-magnetic and collinearly ordered phases have vanishing SHE without SOC. For the quantum spin liquid phase, it remains unclear whether SHE appears without SOC. If yes, SHE would be a promising probe to the spin liquid phase. We will study this very interesting question in the future.

IV. METHOD
To calculate the SHE in these compounds we obtain the DFT Bloch wave functions from Vienna ab − initio Simulation Package (vasp) 42 within the generalized gradient approximation (GGA). 44 By projecting the Bloch wave functions onto maximally localized Wannier functions (MLWFs), 45 we get a tight-binding Hamiltonian which we use for efficient evalua-