Unconventional Mixed-Parity Magnetism in Rare-Earth Tetraborides

Abstract: Altermagnetism has advanced the study of compensated magnets by revealing non-relativistic spin splitting, traditionally classified into strictly even- or odd-parity spin textures. Here, we unveil a fundamentally different regime: component-resolved mixed-parity spin splitting in a fully three-dimensional compensated magnet. Using first-principles calculations, tight-binding and $\mathbf{k} \cdot \mathbf{p}$ models, along with spin-group symmetry analysis, we demonstrate that the non-coplanar ground state of $\mathrm{TbB}_4$ enforces a unique momentum-space spin texture. The in-plane spin components exhibit odd-parity $p$- and $f$-wave-like textures, whereas the out-of-plane component retains an even-parity $d$-wave altermagnetic character. Crucially, the coexistence of the in-plane odd-parity textures is driven not by relativistic spin-orbit coupling, but by a staggered Berry phase arising from the inherent scalar spin chirality. This mixed-parity structure dictates distinct transport fingerprints, including bulk non-relativistic Edelstein and spin Hall responses, as well as a symmetry-allowed Berry curvature dipole. These results establish the rare-earth tetraborides as a robust platform for engineering complex spin-charge conversion phenomena.

• The paper demonstrates that TbB4 exhibits mixed-parity spin splitting by resolving in-plane odd-parity and out-of-plane even-parity components through first-principles calculations and symmetry analysis.
• It employs tight-binding and k·p modeling to uncover symmetry-enforced constraints that yield distinct non-relativistic Edelstein and spin Hall responses.
• The study further shows that including spin-orbit coupling enables a Berry curvature dipole, indicating promising avenues for symmetry-driven spintronic applications.

Unconventional Mixed-Parity Magnetism in Rare-Earth Tetraborides

Introduction

This work establishes the rare-earth tetraboride family ($R$B$_4$) as a model system for realizing and probing mixed-parity spin textures in compensated magnetic materials. Unlike conventional even- or odd-parity spin textures observed in altermagnets and odd-parity magnets, the study demonstrates that $\mathrm{TbB}_4$ exhibits a unique coexistence of even-parity and odd-parity spin splitting resolved in separate spin components. The analysis employs first-principles calculations, tight-binding and $k \cdot p$ modeling, and symmetry analysis, highlighting the symmetry-enforced constraints that produce this qualitatively new regime of magnetism. The paper further details the implications for non-relativistic spin-charge conversion phenomena, including the non-relativistic Edelstein effect, spin Hall response, and the emergence of a symmetry-allowed Berry curvature dipole under spin-orbit coupling (SOC).

Symmetry-Driven Magnetic Phenomena in $R$B$_4$ Compounds

The investigation systematically contrasts the electronic and spin structures of $\mathrm{GdB}_4$, $\mathrm{TmB}_4$, and $\mathrm{TbB}_4$. $\mathrm{GdB}_4$, with a coplanar non-collinear AFM order, preserves $_4$0 symmetry and exhibits no spin splitting or non-trivial spin-charge conversion, retaining only a nonzero non-relativistic spin Hall conductivity for the $_4$1 channel due to symmetry constraints. $_4$2, a collinear AFM along $_4$3, realizes a $_4$4-wave altermagnetic spin texture but, due to additional space group symmetries, all net spin-charge responses vanish.

By contrast, $_4$5 hosts a non-coplanar ground state that is a direct superposition of the irreducible representations relevant to both $_4$6 and $_4$7, but the resulting spin texture exhibits component-resolved mixed-parity: in-plane components ($_4$8, $_4$9) are odd-parity ($\mathrm{TbB}_4$0- and $\mathrm{TbB}_4$1-wave-like), while the out-of-plane component ($\mathrm{TbB}_4$2) preserves even-parity ($\mathrm{TbB}_4$3-wave) altermagnetic character.

Figure 1: Comparative summary illustrating symmetry-enforced phenomena in $\mathrm{TbB}_4$4, $\mathrm{TbB}_4$5, and $\mathrm{TbB}_4$6, including their magnetic structures and resulting transport properties.

Momentum-Space Spin Texture and Symmetry Constraints

First-principles calculations reveal that $\mathrm{TbB}_4$7's band structure supports pronounced spin splitting with mixed-parity textures strictly dictated by its non-symmorphic spin-group symmetries. The spin-projected band analysis demonstrates that the $\mathrm{TbB}_4$8 and $\mathrm{TbB}_4$9 components satisfy odd-parity relations, whereas $k \cdot p$0 remains even-parity. Effective $k \cdot p$1 modeling, further supported by tight-binding calculations, elucidates the coexistence of $k \cdot p$2-wave (linear-in-momentum) and $k \cdot p$3-wave (cubic-in-momentum) features in the in-plane channels, directly arising from a staggered Berry phase generated by the ground state's scalar spin chirality rather than SOC. The $k \cdot p$4 channel remains $k \cdot p$5-wave dominated and even-parity.

Figure 2: Component-wise spin splitting for $k \cdot p$6 across the Brillouin zone, highlighting the coexistence of odd-parity ($k \cdot p$7- and $k \cdot p$8-wave-like) in-plane spin components and even-parity ($k \cdot p$9-wave) out-of-plane component.

Symmetry analysis, incorporating generalized Seitz notation, demonstrates that the mixed-parity relations are rigorously enforced by operations such as $R$0 and $R$1. These yield, in band energies:

$R$2

where $R$3, explaining the component-resolved parity.

Transport Signatures: Edelstein and Spin Hall Effects

The odd-parity in-plane spin components in $R$4 generate a bulk non-relativistic Edelstein effect (NREE), characterized by tensor elements $R$5, which remain finite under all examined conditions, while all off-diagonal and $R$6 components are symmetry forbidden. Calculations decompose the Edelstein tensor into intra- and interband contributions, confirming the dictated symmetry relations.

Figure 3: Calculated non-relativistic Edelstein response tensors and spin Berry curvature in $R$7, confirming symmetry-allowed diagonal elements and zero out-of-plane response.

The even-parity $R$8 channel yields a non-relativistic spin Hall conductivity, with only $R$9 symmetry-allowed. Although local spin Berry curvature exists for in-plane spin components, the Brillouin zone integration enforces cancellation, yielding a net macroscopic spin Hall effect solely for the $_4$0 channel, consistent with symmetry predictions.

Spin-Orbit Coupling and Berry Curvature Dipole

Upon inclusion of SOC, which breaks certain spin-group symmetries, further exotic transport phenomena become symmetry-allowed. Notably, $_4$1 realizes a bulk Berry curvature dipole (BCD), $_4$2, while $_4$3. This effect is entirely absent in $_4$4 and $_4$5, where symmetry (either $_4$6 or $_4$7) forbids macroscopic Berry curvature. The remaining magnetic rotations and $_4$8 symmetry strictly dictate the allowed BCD components. The presence of a finite BCD in $_4$9 demonstrates the system's capacity for nonlinear Hall-like responses, representing a further symmetry-engineered transport channel.

Figure 4: Calculated momentum-space Berry curvature distribution in $\mathrm{GdB}_4$0 with SOC, demonstrating the non-cancelling behavior responsible for the symmetry-allowed Berry curvature dipole.

Implications and Prospects

These findings position $\mathrm{GdB}_4$1 and, by extension, rare-earth tetraborides as key platforms for engineering spin textures and responses via symmetry manipulation. The demonstration that non-coplanar compensated magnetic structures can stabilize mixed-parity spin textures opens paths for designing materials with tunable spin-charge interconversion characteristics without reliance on SOC. Practically, $\mathrm{GdB}_4$2B$\mathrm{GdB}_4$3 families provide candidate materials for non-relativistic, symmetry-driven spintronics, including Edelstein and spin Hall devices, and nonlinear Hall effect-based architectures.

Theoretically, the symmetry-enforced separation of parity channels in different spin components, with underlying origins in scalar spin chirality and Berry-phase physics, offers new ground for exploration in quantum materials and the manipulation of emergent electronic phases. Extensions involving external tuning, interfacial engineering, or pressure could further expand the phase space of accessible mixed-parity phenomena.

Conclusion

This work provides a comprehensive demonstration that rare-earth tetraborides, and specifically $\mathrm{GdB}_4$4, realize component-resolved mixed-parity spin splitting in a fully three-dimensional compensated magnet. This regime supports distinct, symmetry-enforced transport signatures—non-relativistic Edelstein and spin Hall effects, and, under SOC, a Berry curvature dipole—rooted in the material's non-coplanar magnetic ground state and non-symmorphic symmetry. Rare-earth tetraborides thus represent a promising class of compounds for advancing both fundamental understanding and application of symmetry-engineered spintronic phenomena.

(2607.02117)