Coupled Spin-Orbital $p$-Wave Magnetism via Structural and Magnetic Chirality

Abstract: Helical spin textures represent the minimal realization of $p$-wave magnetism which is characterized by momentum-odd spin polarization. Independently, structurally chiral crystals exhibit momentum-odd orbital polarization arising from broken inversion symmetry. Here, we demonstrate that spin-orbit coupling couples these two independent microscopic chirality degrees of freedom, allowing the orbital polarization of a chiral crystal to generate an additional contribution to the $p$-wave spin splitting. The resulting spin-orbital state is naturally classified by the relative chirality $η=χ{\mathrm c}χ{\mathrm m}$, giving rise to two symmetry-distinct $p$-wave phases corresponding to homochiral and heterochiral configurations which can be directly probed by the longitudinal conductivity. These phases exhibit distinct transport signatures, establishing a unified framework linking orbitronics and unconventional magnetism through coupled spin-orbital $p$-wave order.

• The paper presents a unified symmetry framework revealing how structural chirality induces momentum-odd orbital polarization and p-wave spin splitting via SOC.
• It employs a tight-binding helical chain model to show that the relative chirality parameter (η = χcχm) organizes electronic and spin responses into homochiral and heterochiral sectors.
• Experimental transport measurements, such as longitudinal conductivity and Edelstein responses, serve as effective fingerprints for identifying coupled spin-orbital p-wave phases.

Coupled Spin-Orbital $p$-Wave Magnetism Driven by Structural and Magnetic Chirality

Introduction

This work introduces a unified symmetry framework for $p$-wave magnetism, establishing structural and magnetic chirality as independent microscopic order parameters. Via spin-orbit coupling (SOC), the study demonstrates that structurally chiral crystals with momentum-odd orbital polarization can generate a distinctive $p$-wave spin splitting, extending the landscape of momentum-odd (odd-parity) ordered states. The resulting spin-orbital $p$-wave phases are classified by the relative chirality $\eta = \chi_\mathrm{c}\chi_\mathrm{m}$, organizing electronic and transport properties into symmetry-distinct homochiral and heterochiral sectors. These results provide a conceptual bridge between orbitronics and unconventional magnetism.

Background and Motivation

Magnetic order has evolved beyond simple exchange-driven $s$-wave ferromagnetism to include compensated, higher-multipole phases with complex momentum-space polarization textures. Altermagnetism has established that compensated systems host momentum-dependent spin splitting in the absence of SOC, expanding the space of unconventional phases [Smejkal et al., Phys. Rev. X 12, 031042 (2022)]. More recently, odd-parity (momentum-odd) $p$-wave magnetism—exhibited by helical spin textures—has been recognized as a symmetry-protected magnetic phase with reversed spin polarization under momentum inversion, but real-space cancellation [Hellenes et al., (Hellenes et al., 2023)].

Parallel advances in structurally chiral materials, such as carbon nanotubes and B20 helimagnets, have identified momentum-odd orbital polarization resulting from broken inversion symmetry, leading to orbital Edelstein effects and chiral orbital textures even in the absence of explicit atomic orbital angular momentum [Go et al., Phys. Rev. Lett. 121, 086602 (2018)].

This paper addresses whether these orbital and spin analogues of $p$-wave order can coexist and interact, and how their coupling manifests in distinct electronic and transport phenomena.

Model and Symmetry Classification

The analysis employs an analytically tractable tight-binding model: a 1D helical chain supports a single $s$ orbital per site, with structural chirality $\chi_\mathrm{c} = \pm 1$ encoded solely in the geometry of the hoppings. A helical (non-collinear) spin texture induces magnetic chirality $p$0. SOC is introduced via spin-dependent hopping amplitudes. In the absence of SOC, the chiral lattice produces a momentum-odd orbital polarization, while the spin spiral induces a momentum-odd spin polarization.

SOC mediates orbital-to-spin transfer, enabling both spin and orbital $p$1-wave order to coexist and couple. The key symmetry descriptor is the relative chirality $p$2, which collapses the four possible microscopic configurations—left/right-handed structure and spin helix—into two symmetry-distinct sectors:

• Homochiral phase: $p$3 (structural and magnetic chiralities have the same sign)
• Heterochiral phase: $p$4 (opposite chiralities)

Within a phase, inversion of either chirality reverses the corresponding spin or orbital polarization.

Spin-Orbital Coupling and Emergence of Distinct $p$5-Wave Phases

The momentum-dependent orbital polarization from structural chirality survives even for compensated magnetic order. When SOC is present, this orbital polarization induces a corresponding $p$6-wave spin splitting, even in collinear antiferromagnets, demonstrating that structural chirality constitutes an independent microscopic route to spin $p$7-wave order.

When both chiralities are present, the coupling via SOC produces electronic structures and spin textures whose qualitative features depend solely on $p$8. Distinctity between homochiral and heterochiral phases is evident in the band structure and in momentum-dependent spin textures, highlighting the joint effect of the two chirality channels.

Transport Signatures and Experimental Identification

An analysis of the longitudinal conductivity ($p$9), spin Edelstein response ($p$0), and orbital Edelstein response ($p$1) under applied electric fields reveals that:

• Reversing structural chirality ($p$2) reverses only the orbital Edelstein response.
• Reversing magnetic chirality ($p$3) reverses only the spin Edelstein response.
• The longitudinal conductivity is insensitive to the sign of individual chiralities; instead, it directly probes the electronic reconstruction arising from their coupling.

The magnitude and profile of longitudinal conductivity serve as experimental fingerprints for the homochiral and heterochiral coupled spin-orbital $p$4-wave phases. This enables direct access to the symmetry classification via standard transport measurements.

Implications, Material Realizations, and Outlook

The general symmetry framework established here has immediate ramifications for materials hosting both structural and magnetic chirality. Chiral B20 helimagnets, such as Mn$p$5Fe$p$6Ge and the $p$7-Mn Co–Zn–Mn family, are highlighted as promising candidates due to their established structural chirality and tunability of the helical magnetic texture (and thus $p$8) via chemical substitution. These systems provide experimental routes for toggling between homochiral and heterochiral phases by controlling relative chirality $p$9.

The theoretical findings identify relative chirality as a new symmetry degree of freedom controlling coupled spin-orbital $p$0-wave order. This links the orbitronics domain with unconventional (momentum-odd) magnetism and enables material-by-design strategies where electronic topology, spin textures, and orbital response can be engineered through control of crystal and magnetic structure.

Future research will likely explore multidimensional generalizations, interaction effects, and functional switching of coupled spin-orbital $p$1-wave phases for device applications, particularly where spin and orbital currents are both operationally relevant. The framework is extensible to higher multipolar orders and topologically nontrivial systems.

Conclusion

Coupling between structural and magnetic chirality under SOC yields two symmetry-distinct spin-orbital $p$2-wave phases, classified by the relative chirality $p$3. The transport response unambiguously identifies these phases, providing experimental access to a unified order parameter space bridging orbitronics and momentum-odd magnetic phenomena. These insights broaden the fundamental taxonomy of symmetry-protected magnetic phases and pave the way for novel chiral spintronic and orbitronic functionalities (2607.02378).