Unconventional Mixed-Parity Magnetism in Rare-Earth Tetraborides

Technical Summary: Unconventional Mixed-Parity Magnetism in Rare-Earth Tetraborides

Problem Statement
Altermagnetism (AM) has recently expanded the understanding of compensated magnets by revealing non-relativistic spin splitting characterized by even-parity spin textures (e.g., $d$-wave). Conversely, recent developments have identified odd-parity compensated magnets where non-collinear, coplanar orders drive spin splitting satisfying $\varepsilon(s, k) = \varepsilon(-s, -k)$, enabling bulk non-relativistic Edelstein effects. However, even- and odd-parity spin textures have historically been treated as distinct symmetry classes. A significant gap remains in identifying a natural bulk material where these textures coexist within symmetry-distinct spin channels, thereby producing unique, experimentally accessible transport fingerprints that combine magnetoelectric and spin-transport responses associated with separate parity classes.

Methodology
The authors investigate the rare-earth tetraborides family ($RB_4$), specifically focusing on Gd$B_4$, Tm$B_4$, and Tb$B_4$, as candidate platforms for this phenomenon. The study employs a multi-faceted theoretical approach:

1. First-Principles Calculations: Performed using the Vienna Ab initio Simulation Package (VASP) with projector augmented-wave (PAW) pseudo-potentials. Strong electron correlations in the rare-earth $f$-electrons are treated using the GGA + $U$ scheme ($U=6$ eV, $J=0.75$ eV).
2. Modeling: Tight-binding and $k \cdot p$ models are constructed to analyze the electronic structure and spin textures.
3. Symmetry Analysis: A rigorous spin-group symmetry analysis is conducted to classify emergent spin textures and determine allowed transport tensors. This includes examining the effects of spin-orbit coupling (SOC) by transitioning to the magnetic space group framework.

Key Contributions and Results

• Comparative Analysis of $RB_4$ Compounds:

  • Gd$B_4$: Exhibits a non-collinear, coplanar antiferromagnetic (AFM) order ($m\Gamma^-_1$) that preserves combined $PT$ symmetry. Consequently, it shows no spin splitting, zero non-relativistic Edelstein effect (NREE), and zero net Berry curvature.
  • Tm$B_4$: Features a collinear AFM order ($m\Gamma^+_2$) along the $c$-axis. It preserves inversion symmetry ($P$) but breaks time-reversal ($T$) and $PT$. It acts as a prototypical $d$-wave altermagnet with even-parity spin splitting but enforces zero NREE and zero Berry curvature dipole (BCD) due to spatial inversion.
  • Tb$B_4$: The central focus of the study. Its magnetic structure is a linear superposition of the Gd$B_4$ and Tm$B_4$ orders ($m\Gamma^-_1 \oplus m\Gamma^+_2$), resulting in a non-coplanar AFM state.

• Discovery of Component-Resolved Mixed-Parity Spin Splitting:
  The authors demonstrate that Tb$B_4$ does not merely exhibit a linear combination of the parent phases' properties. Instead, the non-coplanar ground state enforces a unique momentum-space spin texture where parity is resolved by spin component:

  • Out-of-plane component ($s_z$): Retains even-parity ($d$-wave) altermagnetic character, satisfying $\varepsilon(s_z, k) = \varepsilon(s_z, -k)$.
  • In-plane components ($s_x, s_y$): Exhibit odd-parity textures, satisfying $\varepsilon(s_{\parallel}, k) = \varepsilon(-s_{\parallel}, -k)$. These textures are characterized by a coexistence of $p$-wave-like and $f$-wave-like patterns.

• Mechanism of Odd-Parity Textures:
  Crucially, the paper establishes that the coexistence of in-plane $p$- and $f$-wave-like textures in Tb$B_4$ is not driven by relativistic spin-orbit coupling. Instead, it arises from a staggered Berry phase induced by the inherent scalar spin chirality of the non-coplanar magnetic order. This mechanism is distinct from previously reported $p$-wave magnetism.

• Transport Fingerprints:
  The mixed-parity structure dictates distinct transport responses:

  • Non-Relativistic Edelstein Effect (NREE): The odd-parity in-plane textures generate a finite bulk NREE. Symmetry analysis restricts the Edelstein tensor $\chi_{ij}$ to non-zero diagonal components $\chi_{xx}$ and $\chi_{yy}$ (with $\chi_{xx} = -\chi_{yy}$ for intra-band contributions), while $\chi_{zz}$ is forced to zero by specific symmetries.
  • Non-Relativistic Spin Hall Effect (NR SHE): The even-parity out-of-plane texture supports an intrinsic spin Hall response. While the odd-parity contributions cancel upon Brillouin zone integration, the even-parity $s_z$ texture yields a finite macroscopic spin Hall conductivity $\sigma^z_{xy}$.
  • Berry Curvature Dipole (BCD): Upon including SOC, the symmetry constraints are modified. While $PT$ symmetry in Gd$B_4$ and inversion in Tm$B_4$ forbid BCD, Tb$B_4$ hosts a finite macroscopic BCD. The magnetic rotational symmetries restrict the BCD to a diagonal form ($D_{xx} = -D_{yy}$, $D_{zz}=0$), and the specific momentum dependence of the Berry curvature prevents cancellation, resulting in a robust BCD.

Significance and Claims
The paper claims to establish rare-earth tetraborides, and specifically Tb$B_4$, as a robust, natural material platform for engineering complex spin-charge conversion phenomena. The primary significance lies in the demonstration of a "fundamentally different regime" of magnetism: a fully three-dimensional compensated magnet hosting component-resolved mixed-parity spin splitting.

Unlike previous theoretical proposals for mixed-parity states that rely on relativistic spin-orbit coupling or external driving fields (e.g., circularly polarized light), the mixed-parity state in Tb$B_4$ emerges intrinsically from its non-coplanar magnetic ground state and non-symmorphic spin-group constraints. This work provides a definitive route to stabilizing odd-parity spin textures independent of SOC and establishes Tb$B_4$ as a unique system where distinct parity classes coexist to produce complementary transport signatures (NREE and NR SHE) within a single magnetic phase.