Microscopic origin of $p$-wave magnetism

Abstract

$P$-, $f$-, or $h$-wave antialtermagnets yield large non-relativistic spin splitting with out-of-plane spin polarization in momentum space perpendicular to the coplanar non-collinear local magnetic moments. We provide a microscopic explanation of this unconventional spin polarization by linking it to a previously overlooked site-compensated spin density that orders antiparallel when projected onto opposite momenta. We verify this result both by model derivation of the out-of-plane momentum-space spin polarization being proportional to the direct-space cross product of the coplanar non-collinear spin order, as well as by ab initio calculations in the material candidate CeNiAsO. By providing a general classification and analytic expression for the spin polarization of all spinful two-site tight-binding Hamiltonians, we reveal the momentum-resolved spin polarization as a probe of the Bloch-state geometry arising from spin-site coupling. Furthermore, our approach allows for geometric distinction between ferro-, alter-, and antialtermagnets. Our results provide a quantitative guidance for quantized out-of-plane momentum-space spin polarization and large spin splitting, and construction principles for antialtermagnets.

Figures (4)

• Figure 1: Schematics of a coplanar magnetic order (a) that yields non-relativistic band splitting with out-of-plane $p$-wave spin polarization (b). The band-projected spin polarization perpendicular to the in-plane magnetic moments is proportional to their cross product felt by forward and backward propagating electrons (c), arising from Bloch-state geometry. The hidden antiparallel real-space spin density becomes visible after directional selection of the electrons (d).
• Figure 2: (a) Momentum-dependence of the bands $E_1$ and $E_2$. The spin polarization $\langle \hat{S}_z\rangle_n$ of the lower (solid) and upper (dashed) band is $\mathcal{T}$-even and nearly quantized to $\pm\hbar/2$ near $\Gamma=(0,0)$. (b) The spin splitting $\Delta$ yields a sinusoidal angle dependence, maximal at orthogonal in-plane magnetic moments. In contrast, the spin polarization shows a weak angular dependence that increases with the amplitude $J$. (c) The splitting increases quadratically with the amplitude of the in-plane magnetic moments, while the quantization of the spin polarization is gradually lifted, as shown for different relative angles.
• Figure 3: (a) The real-space crystal structure of $\text{CeNiAsO}$ with coplanar non-collinear spin texture. (b) The band structure near the Fermi level with $\Gamma = (0,0,0)$ and $X=(\pi,0,0)$. (c) The angle dependence of spin splitting and spin polarization of the bands 1 and 2 near the Fermi level at two momenta near $\Gamma$, as indicated in (b); the physical angle is marked in red.
• Figure 4: The spin density of band 3 integrated over the left (a) and right (b) half of the Brillouin zone.