Chiral-Angle-Controlled Altermagnetic Spin Splitting in Nanotubes

Abstract: Altermagnets exhibit momentum-dependent spin splitting despite having zero net magnetization. Here, we show that rolling a two-dimensional (2D) $d$-wave altermagnet into a nanotube transforms this momentum-dependent spin splitting into chiral-angle-controlled one-dimensional (1D) spin splitting through dimensional projection. Using a minimal tight-binding model and first-principles calculations, we demonstrate that the nanotube spin splitting follows a characteristic $\cos(2θ)$ dependence, vanishing for nodal orientations and reaching extrema for antinodal orientations. The mechanism remains robust across a broad class of nanotubes derived from 2D altermagnets. Our results establish dimensional projection as a general route for transferring momentum-dependent altermagnetic spin splitting into 1D systems and provide a framework for engineering spin-split quantum states in low-dimensional magnetic materials.

• The paper establishes that the chiral angle in nanotubes controls 1D spin splitting following a cos(2θ) law derived from 2D d-wave altermagnets.
• Tight-binding and first-principles calculations confirm that spin splitting is maximized at antinodal orientations and suppressed at nodal orientations.
• The findings enable geometric engineering of spin states for spintronic applications without net magnetization, even amidst curvature-induced perturbations.

Chiral-Angle-Dependent Altermagnetic Spin Splitting in Nanotubes

Introduction

This work introduces and systematically investigates the dimensional projection of momentum-dependent altermagnetic spin splitting from two-dimensional (2D) d-wave altermagnets into one-dimensional (1D) spin-split states in nanotube geometries. The study establishes that rolling a 2D altermagnet into a nanotube transforms the anisotropic d-wave spin splitting—highly dependent on crystal momentum in the parent lattice—into a chiral-angle-dependent spin splitting in the 1D projected system. Both tight-binding modeling and first-principles calculations reveal that the orientation of the nanotube axis relative to the nodal and antinodal directions of the parent d-wave altermagnet critically determines the resultant spin splitting.

Theoretical Framework and Symmetry Analysis

The central analytical construct is a minimal tight-binding Hamiltonian for a 2D square-lattice d-wave altermagnet featuring two magnetic sublattices coupled via a staggered exchange field. The Hamiltonian includes a momentum-dependent form factor, $g_a(\mathbf{k}) = \cos k_x - \cos k_y$, which encodes the $d_{x^2-y^2}$ symmetry of the spin splitting. This anisotropic form factor changes sign under $\pi/2$ rotation, establishing symmetry-protected nodal lines where the splitting vanishes, and antinodal directions with maxima.

Projection onto a 1D nanotube geometry is implemented by replacing the momentum vectors with those aligned and perpendicular to the tube axis, with the rolling (chiral) angle $\theta$ as a control parameter. The analysis yields a key prediction: for a given subband $k_\perp=0$, the projected spin splitting is proportional to $\cos(2\theta)$, with exact suppression (degeneracy) at high-symmetry nodal orientations ($\theta=45^\circ$ for $d$-wave symmetry) and extrema at antinodal orientations ($\theta=0^\circ$, $90^\circ$), including an explicit sign reversal reflecting the parent symmetry.

Numerical Validation via First-Principles Calculations

First-principles DFT calculations using the PBE functional and norm-conserving pseudopotentials validate the predictions for both checkerboard V$d_{x^2-y^2}$0O-based and lower-symmetry Janus nanotubes (e.g., V$d_{x^2-y^2}$1OSe$d_{x^2-y^2}$2, V$d_{x^2-y^2}$3OSeTe, Fe$d_{x^2-y^2}$4SSe). The calculations focus on several high-symmetry rolling angles, comparing band structures of large rectangular supercells (to mimic momentum folding) with those of actual nanotubes.

For both the parent monolayer and resulting nanotube geometries, the electronic structure evolution corroborates the tight-binding model: pronounced spin splitting appears for antinodal rolling angles, while complete cancellation is observed for nodal orientations. The calculations reveal that even substantial curvature-induced or symmetry-breaking perturbations (leading to small net magnetization or inner-outer-side asymmetry) do not qualitatively disrupt the chiral-angle dependence or the nodal-antinodal selection rule. This robustness extends to a diverse class of nanotubes and persists for physically relevant tube radii.

Implications for Spintronics and Low-Dimensional Magnetism

The study delineates dimensional projection as a rigorous and generalizable route for engineering spin-split states in 1D nanostructures derived from higher-dimensional altermagnets. The results provide a symmetry-based prescription to realize, suppress, or invert spin splitting in the absence of net magnetization, solely via geometric parameters. This capability for precise geometric control of spin splitting affords new degrees of freedom for the design of spintronic devices, particularly for realizing tunable spin-polarized currents in nanostructures with vanishing macroscopic stray fields.

Moreover, the cos($d_{x^2-y^2}$5) dependence of the 1D spin splitting, rooted in inherent crystal symmetries, implies that quantum transport properties—such as spin Hall conductance or spin filtering—can be dynamically tuned by adjusting the nanotube rolling angle during fabrication. The persistence of the effect in the presence of secondary perturbations suggests immediate translatability to experimental platforms, such as transition metal dichalcogenide or van der Waals Janus nanotubes, many of which are now experimentally accessible or computationally predicted.

Future Directions

Several avenues for further research are evident. First, systematic experimental realization and transport studies of altermagnetic nanotubes with controlled chirality are needed to validate the predictions regarding spin splitting tunability. Second, the extension to other symmetry classes (e.g., $d_{x^2-y^2}$6-wave, $d_{x^2-y^2}$7-wave, noncollinear orders) may reveal more complex angular dependencies and richer physics. Third, integrating the chiral-angle control mechanism with external fields, strain, or proximity effects could allow for complex, multi-parameter spin manipulation in low-dimensional systems.

Developments in high-throughput computational materials discovery will be crucial for identifying candidate 2D altermagnets suitable for exfoliation and roll-up into robust nanotube spintronic architectures.

Conclusion

This paper establishes that the chiral angle in nanotubes derived from 2D d-wave altermagnets provides deterministic control over 1D spin splitting via dimensional projection, governed by a strict cos($d_{x^2-y^2}$8) law. Theoretical modeling and first-principles calculations confirm that the mechanism is symmetry-protected and remains robust against practical material perturbations. These findings lay a foundation for geometric engineering of quantum spin states and provide actionable insights for realizing spin-orbit-free spintronic devices based on low-dimensional altermagnets.