Non-relativistic spin splitting: Features and Functionalities

TL;DR

This work surveys non-relativistic spin splitting (NRSS) in compensated antiferromagnets, emphasizing symmetry-based principles that enable spin splitting without net magnetization and, in many cases, without spin–orbit coupling. It classifies NRSS into three symmetry-breaking scenarios of $\mathcal{I}$ and $\mathcal{T}$, presenting distinct band-structure signatures for collinear and noncollinear spin configurations, including planar, bulk, and nodal patterns in $\Delta \epsilon(\vec{k})$. The review compiles candidate materials from theory and highlights experimental detection approaches such as ARPES and magnetic Compton scattering, while outlining key functionalities that NRSS enables—transverse spin currents without SOC, longitudinal spin polarization with zero net magnetization, anomalous Hall effects, and junction phenomena with superconductors. It also discusses richer phenomena such as hidden magnetic order, magnon splitting, piezomagnetism, kinetomagnetism, and magnetoelectric multiferroicity, and points to promising future directions including 2D NRSS, electric-field control, and topological contexts with practical implications for AFM spintronics and superconducting devices.

Abstract

Recently, spin splitting of non-relativistic origin in compensated antiferromagnets has drawn growing attention in condensed matter research. Although many materials, now known to exhibit such spin splitting, have been studied for decades, their manifestation along non-high-symmetry momentum directions initially hindered their recognition. In recent years, significant progress has been made in uncovering the symmetry principles that allow non-relativistic spin splitting in the absence of net magnetization, revealing the unconventional physics arising from their coexistence. In this review, we provide a concise overview of non-relativistic spin splitting in compensated antiferromagnets with various spin configurations, including collinear, coplanar, and non-coplanar spin arrangements. We summarize practical identification guidelines, highlight characteristic features in electronic band structures, and discuss the emerging functionalities, with an emphasis on promising directions for future exploration.

Figures (7)

• Figure 1: Schematic illustration of (a) spin splitting in ferromagnets due to exchange splitting, (b) spin-degenerate bands in conventional $\mathcal{IT}$-symmetric AFMs, (c) relativistic Rashba-like spin splitting, and (d) NRSS arising from broken $\mathcal{IT}$ symmetry in the absence of magnetization. The lower panel shows the corresponding energy cross-sections on the $k_x$–$k_y$ plane, as indicated by the shaded plane in the upper panel.
• Figure 2: Schematic illustration of the effect of time-reversal $\cal T$ symmetry on (a) ferromagnets, (b) AFMs with equivalent surrounding nonmagnetic environments around the two magnetic sublattices, and (c) AFMs with inequivalent surrounding nonmagnetic environments around the two magnetic sublattices. While all ferromagnets break $\cal T$ symmetry due to the reversal of the spin moment direction, as shown in (a), AFMs may or may not preserve the global $\cal T$ symmetry, i.e., time-reversal plus translation $t$, as shown in (b) and (c), respectively.
• Figure 5: Detection of NRSS. Measured ARPES data along the $D$–$U$ line at (a) $T > T_N$ and (b) $T < T_N$ for $\alpha$-MnTe, as reported in Ref. Lee2024. Schematic illustration of (c) magnetic Compton scattering with circularly polarized photons, a photon-in–photon-out process, and (d) the theoretically predicted magnetic Compton profile $J_{\rm mag}(\vec{p})$ in $d$-wave NRSS AFMs, depicting the reversal of the profile as the momentum direction changes from $[110]$ to $[1\bar{1}0]$. Panel (a) and (b) are reprinted with permission from Reference Lee2024 Copyright 2024 by the American Physical Society.
• Figure 6: Schematic illustration of emerging concepts (inner circle) and functionalities (outer ring) associated with NRSS.
• Figure 7: Spin currents from relativistic and NRSS bands. (a) Schematic showing reciprocal-space spin-momentum locking in the relativistic Rashba system at equilibrium (dashed circle) and non-equilibrium (solid circle) conditions. At non-equilibrium, the displacement of the Fermi surface, denoted by the black arrow, leads to the generation of a spin accumulation at the surface. (b) Schematics of transverse spin current $J_s \equiv J_{xz}^y$ generated by the applied charge current $J_c$ due to the spin Hall effect (SHE) driven by SOC. (c) Illustration of a typical NRSS, generating transverse spin current without SOC upon application of an electric field along $\hat{x}$. (d) The spin polarization, $\sigma$, of the generated $J_s$ in NRSS materials is collinear with the Néel vector, $\cal N$, and depends on the crystal axis, offering a re-orientable spin current, $J_{xz}^\sigma$, unlike the SHE. (e) NRSS producing longitudinal spin-polarised current, $J_{xx}^\sigma$, upon application of an electric field along $[110]$ direction. (f) Spin to charge conversion from the NRSS bands in the inverse spin-splitter effect. (g) NRSS bands together with SOC results in avoided crossing and there by AHE. (h) Schematic experimental set-up for probing AHE and ANE from NRSS effects by applying electric current ($J_C$) and thermal gradient ($\vec{\nabla} T$), respectively. The color map in (a,c,e,g) schematically represents the orientation of the spins in the $k$ space with the up (down) spin moment denoted by the red (blue) color.