Odd-Parity Magnetism and Gate-Tunable Edelstein Response in van der Waals Heterostructures

TL;DR

The paper proposes van der Waals trilayer heterostructures composed of two stripe antiferromagnets separated by a metallic layer to realize electrically tunable $p$-wave (odd-parity) magnetism. A geometric cancellation of the bilinear RKKY exchange $J_{\rm eff}$, together with a dominant biquadratic term $\beta_2$, drives a filling-controlled transition from a collinear to an orthogonal spin configuration, enabling a non-relativistic, gate-tunable Edelstein response that survives sizable SOC. The authors derive a microscopic free-energy via Schrieffer-Wolff transformation and path-integral methods, show symmetry-protected band features including a Dirac nodal line at $\theta=\pi/2$, and map a phase diagram as a function of electronic filling. They propose GdTe$_3$-based vdW platforms as realizations and argue that this setup provides a practical route to electrically control odd-parity spin textures for non-relativistic spintronics with potential applications in spin-based information processing.

Abstract

Odd-parity magnetism has attracted significant interest for its unconventional spin splitting. However, a concrete microscopic route for its realization remains elusive. In this work, we propose van der Waals heterostructures of stripe antiferromagnets (sAFMs) as an ideal platform for electrically controllable $p$-wave magnetism. In the sAFM/metal/sAFM structure, the leading RKKY-type exchange interaction is canceled due to the symmetry of the stacking pattern. This exposes a higher-order biquadratic interaction as a dominant contribution that drives a filling-controlled transition from a collinear phase to an orthogonal $p$-wave configuration. The resulting $p$-wave phase exhibits a gate-tunable Edelstein response, which originates from magnetic symmetry breaking rather than conventional relativistic spin-momentum locking and remains robust even under substantial spin-orbit coupling. Finally, we propose material candidates for the realization of our theory. Our results establish van der Waals heterostructures as a practical platform for non-relativistic spintronics with electric control of odd-parity spin textures.

Figures (8)

• Figure 1: (a) Schematic of the sAFM/metal/sAFM heterostructure. $\mathbf{N}_{1}$ and $\mathbf{N}_{2}$ form a relative in-plane angle $\theta$. (b) Band structure for the collinear state ($\theta=0$), where $\mathcal{PT}$-symmetry enforces global spin degeneracy. (c) Band structure for the orthogonal state ($\theta=\pi/2$). The breaking of $\mathcal{PT}$-symmetry lifts the degeneracy, resulting in $p$-wave band splitting. (d) Relative free energy $\mathcal{F}(\theta)$ as a function of $\theta$ for different electronic fillings of the metallic layer. The color gradient indicates filling levels from $n=1/3$ (blue) to $n=1/2$ (red). The system undergoes a transition from an orthogonal $p$-wave phase ($\theta=\pm\pi/2$) at low filling to a collinear phase ($\theta=0, \pi$) at high filling.
• Figure 2: (a) Evolution of the Bloch bands and density of states (DOS) for different relative angles. The shaded DOS regions highlight the emergence of spectral splitting as $\theta$ approaches $\pi/2$. (b) Momentum-space spin polarization map $\langle S_{z}(\mathbf{k})\rangle$ at the 1/3-filling. The texture exhibits the characteristic odd-parity symmetry, $\langle S_{z}(\mathbf{k})\rangle=-\langle S_{z}(-\mathbf{k})\rangle$. (c) Spin-resolved band dispersion along the $Y\rightarrow -Y$ path [white solid line in (b)]. The crossing of opposite spin branches (red and blue) at $E_{F}$ (dashed line) demonstrates the symmetry-protected $p$-wave splitting without relativistic SOC.
• Figure 3: Microscopic origin of biquadratic exchange and magnetic phase diagram. (a–c) Momentum-resolved distribution of the $\beta_2$ integrand components : (a) Fermi surface contribution $(\propto n'_{F})$, dominating near the chemical potential; (b) Fermi sea contribution $(\propto n_{F})$, representing energy redistribution of all occupied states (c) Magnetic phase diagram in the filling $(n)$–temperature $(T/t)$ plane. The orthogonal $p$-wave phase $(\theta \approx \pi/2)$ is stabilized at low fillings by kinetic energy gain, whereas the collinear phase $(\theta = 0, \pi)$ is favored near half-filling $(n=1/2)$.
• Figure 4: (a) Edelstein kernel $\chi_{zy}$ as a function of electronic filling for different SOC strengths. The shaded regions indicate the $p$-wave phase ($\theta \approx \pi/2$) where the response is significant. (b) Filling dependence of representative tensor components $\chi_{ij}$ in the presence of Rashba SOC ($\lambda_{\rm SO}/t=0.1$), showing that transverse components (e.g., $\chi_{xy}$ and $\chi_{yx}$) remain subdominant to $\chi_{zy}$ in the $p$-wave regime. (c) Momentum-resolved kernel of the response, showing the odd-parity distribution over the Fermi surface (black lines). (d) Relative total energy $\mathcal{F}(\theta)$ for representative fillings of $1/2$ (collinear stable) and $1/3$ (orthogonal stable).
• Figure S1: Stripe antiferromagnetic order in top and bottom layers for visual clarification.