Chiral Spin Hall Effect in Magnetic Systems

• Chiral Spin Hall effect is a transport phenomenon where lattice chirality and noncoplanar spin textures induce unique transverse spin and charge currents.
• It leverages Berry phase mechanisms and symmetry-breaking in magnetic lattices to produce quantized, geometry-sensitive Hall responses in systems like Mn₃Ga and ferromagnetic graphene.
• Applications in spintronics and quantum materials stem from its ability to manipulate spin currents through intrinsic magnetic order and engineered chiral textures.

The chiral spin Hall effect encompasses a set of transport phenomena in condensed matter and magnetic systems that arise from the interplay between electron spin, lattice chirality, and noncoplanar spin textures. Unlike the conventional spin Hall and anomalous Hall effects—primarily attributed to spin–orbit coupling and net magnetization—the chiral spin Hall effect reflects the presence of complex magnetic structures or intrinsic lattice chirality that generate transverse spin or charge currents. In such systems, Berry-phase physics, asymmetric scattering, and symmetry-breaking produce unique Hall-type responses, often exhibiting quantized, geometry-sensitive, or sign-reversed features relative to standard mechanisms.

1. Scalar Spin Chirality and Berry Phase Mechanism

A central concept underlying many chiral Hall phenomena is scalar spin chirality, typically defined for a triad of local spins $(\mathbf{S}_i, \mathbf{S}_j, \mathbf{S}_k)$ as: $$\chi_{ijk} = \mathbf{S}_i \cdot (\mathbf{S}_j \times \mathbf{S}_k)$$ This parameter encodes the solid angle subtended by three spins and acts as an emergent (topological) gauge field for itinerant electrons. When electrons traverse a noncoplanar spin arrangement, they acquire a real-space Berry phase, leading to a fictitious Lorentz force that deflects their trajectory transversely. For example, in Fe$_{1.3}$Sb, the formation of heptamer spin clusters and DM-induced canting breaks the cancellation of scalar chirality contributions, resulting in a net Berry curvature and observable chiral Hall resistivity component $\rho_{yx}^\chi$ that reverses sign relative to the conventional anomalous Hall effect (Shiomi et al., 2011).

In chiral spin liquids, such as in the extended kagome Heisenberg model, uniform chiral order $\langle \chi_i \rangle$ leads to quantized spin transport analogous to the Laughlin-type fractional quantum Hall effect, marked by a fractionally quantized Chern number and the possibility of edge spin pumping—the prototypical chiral spin Hall effect (Gong et al., 2013).

2. Chiral Magnetic Lattice Effects and Spin Rotation Symmetry Breaking

Chiral spin Hall effects can emerge solely from the magnetic texture or lattice symmetry, without requiring spin–orbit coupling. In noncollinear antiferromagnets (e.g., Mn$_3$X compounds), a 120$^\circ$ spin arrangement on a kagome lattice naturally breaks continuous spin rotational symmetry, enabling intrinsic spin Hall currents even when scalar chirality vanishes and in the absence of heavy elements (Zhang et al., 2017). The key formula for the intrinsic spin Hall conductivity (SHC) in such systems is: $$\sigma_{\alpha\beta}^\gamma = \frac{e}{\hbar} \sum_n \int_{\mathrm{BZ}} \frac{d^3k}{(2\pi)^3} f_n(\mathbf{k}) \Omega_{n,\alpha\beta}^\gamma(\mathbf{k}),$$ where $\Omega_{n,\alpha\beta}^\gamma(\mathbf{k})$ is the spin Berry curvature, calculated via the spin current and velocity operator matrix elements. The allowed SHC tensor components are highly anisotropic, dictated by the magnetic point group symmetry; large values (e.g., $\sim$600 $(\hbar/e)(\Omega\cdot\mathrm{cm})^{-1}$ in Mn$_3$Ga) are observed (Zhang et al., 2016).

Ab initio calculations confirm that this mechanism operates with or without SOC, providing new strategies for spintronic materials design focused on chiral magnetic order rather than heavy-element engineering.

3. Chiral Spin Hall Effects in Electronic Band Structures and Semimetals

In ferromagnetic graphene with a "spin chiral configuration"—where one spin species is electron-like and the other hole-like—the interplay of exchange splitting $h$ and spin–orbit coupling $\lambda$ leads to a charge Hall current (not a spin current), even in zero external magnetic field. The key semiclassical result for undoped ferromagnetic graphene is: $$j_c = \frac{e^3\tau^2\lambda C}{\pi} h (\hat{z} \times \mathbf{E}),$$ where both spin subbands contribute to a transverse charge current due to carrier type inversion, in contrast to the conventional spin Hall effect (Rameshti et al., 2013). In topological chiral semimetals, such as CoSi, the intrinsic SHC exhibits an odd dependence on Fermi energy, vanishing at the spin-1 node and displaying antisymmetric extrema around it; thus, precise tuning of the Fermi level can reversibly modulate the chiral spin Hall response (Tang et al., 2021, Hsieh et al., 2022). Furthermore, the presence of multiple independent SHC tensor elements correlates with crystal helicity, making spin Hall transport a sensitive probe of structural chirality.

4. Chiral Spin Hall Effect in Systems with Skyrmions and Chiral Spin Textures

Transport signatures of chiral spin textures—e.g., skyrmions, spin spirals—include topological Hall effects and their chiral spin Hall analogs. The microscopic theory separates the Hall resistivity into spin-independent (charge) and spin-dependent (Berry phase/chirality) contributions: $\rho_{yx}^{\mathrm{T}} = \rho_c + \rho_a,$ with spin polarization $P_s$ controlling conversion between spin and charge signals (Denisov et al., 2018). The adiabatic parameter $\lambda_a$ (incorporating exchange, texture size, and Fermi energy) determines the crossover between regimes. In the adiabatic limit, the chiral spin Hall effect becomes dominant; in the nonadiabatic regime, direct charge skew scattering prevails. Notably, magnetic skyrmions, characterized by $S_i \cdot (S_j \times S_k) \neq 0$, impart a distinct Berry curvature, enabling electrical detection of their helicity or type via chiral Hall conductivity contributions that vary for Bloch versus Néel skyrmions (Terasawa et al., 2023).

In epitaxial films of MnGe, thermally excited spin clusters yield a "giant" anomalous Hall effect dominated by spin-chirality skew scattering, producing large Hall angles and conductivities exceeding conventional intrinsic and extrinsic mechanisms (Fujishiro et al., 2020). This effect demonstrates the importance of short-range chiral correlations and their potential for high-efficiency spintronic devices.

5. Chiral Spin Hall Effect in Ultracold Atomic and Quantum Spin Systems

Quantum order-by-disorder in spinor Bose gases within momentum-space double-well bands can stabilize a chiral spin superfluid, where opposite spins condense at $\pm K$ momenta. This generates staggered real-space spin loop currents and a spontaneous chiral spin Hall effect. The Berry curvature at $\pm K$ drives spin-dependent transverse velocities under applied forces: $$\dot{\mathbf{r}}_s = \pm\, \mathbf{F} \times [\mathbf{\Omega}(K) - \mathbf{\Omega}(-K)],$$ enabling direct experimental access to chiral spin transport properties via time-of-flight imaging or Bloch oscillation protocols (Li et al., 2014).

In chiral spin liquids (e.g., on the kagome lattice), spontaneous symmetry breaking leads to fractional quantum Hall states for spin, with edge spin pumping and a half-integer topological Chern number. Experimental detection relies on thermal Hall measurements, entanglement spectroscopy, or flux insertion protocols, all indicating robust chiral edge modes and topological spin conductance (Gong et al., 2013).

6. Symmetry, Material Design, and Experimental Detection

The chiral spin Hall effect is highly sensitive to lattice symmetries, magnetic ordering, and the presence of inversion or time-reversal symmetry. In centrosymmetric systems, only second-derivative gradient corrections contribute to the Hall response, suppressing one-derivative (chiral Hall) terms as found in Rashba models (Terasawa et al., 2023). In contrast, noncentrosymmetric structures, or those with engineered chirality (such as self-assembled chiral molecules modulating the inverse spin Hall effect via the CISS effect), show pronounced asymmetry in spin-to-charge conversion, dependent on molecular length, orientation, and dephasing (Liu et al., 4 Sep 2025).

Experimental discrimination between chiral edge-mode-origin thermal Hall effects and conventional phonon/magnon-driven thermal Hall effects can be achieved by geometric constriction: only the chiral edge mode gives a geometry-dependent enhancement of thermal Hall resistance at low temperatures, providing a practical route to identify chiral spin liquid states (Halász, 6 May 2025).

Oscillographic and AC transport studies in metals with strong SOC (e.g., platinum) reveal chiral nonequilibrium spin Hall features—including double-frequency voltage signals and insensitivity to current or field direction—that can be leveraged for robust spintronic device platforms (Chiang et al., 2021).

7. Distinction from Related Effects and Future Directions

Chiral spin Hall effects differ fundamentally from conventional anomalous Hall effects (AHE), topological Hall effects (THE), and crystal Hall effects. Whereas AHE depends on net magnetization and SOC, and THE on net topological winding, chiral spin Hall effects result from either the handedness of the spin texture (vector chirality, winding of spiral states) or the structural chirality of the lattice. In many systems, chiral Hall contributions scale linearly with magnetization gradients or wavevector $q$, distinct from quadratic or topological scaling seen in THE or AHE (Kipp et al., 2021, Kipp et al., 2020).

The application potential for chiral spin Hall phenomena includes spin current generation and manipulation in antiferromagnetic spintronics, topological spintronics based on unconventional chiral fermions, molecular spintronic interfaces exploiting CISS, and precision quantum sensing in spin liquids or ultracold atomic gases. The sensitivity to crystal, magnetic, or molecular chirality provides new avenues for control and detection in spin transport, enabling functional devices that operate with reduced energy consumption or enhanced environmental robustness.

Continued research aims to clarify the interplay of Berry-phase physics, symmetry constraints, and quantum fluctuation effects—across magnetic materials, quantum gases, and topological semimetals—illuminating the full scope of the chiral spin Hall effect and its impact on emergent quantum technologies.