Symmetry-Enforced Nodal $f$-Wave Magnets

Abstract

Owing to their relevance for spintronics, electronic band splitting and spin-polarization textures in magnets are active areas of research. In non-collinear magnets, alternating spin textures can arise both for isolated bands and for intersecting band pairs with nodal splitting. This raises the question of whether $p,f,...$-wave magnets should be defined by their spin polarization or their band splitting. To resolve this ambiguity, we introduce spin-space symmetries that couple the spin polarization and splitting textures for all bands. Focusing on the nodal $f$-wave magnet, we construct a tight-binding model of itinerant electrons on a honeycomb bilayer coupled to a non-collinear magnetic texture. Analytic expressions for spin polarization and splitting reveal the dependence on hopping and exchange coupling. We predict a canting-induced spin conductivity arising from the nodal structure of the splitting. Furthermore, the $f$-wave magnet in the bulk can induce $p$-wave magnetism on the surface. This surface $p$-wave character leads to a bulk-forbidden Edelstein effect with $f$-wave anisotropy.

Figures (5)

• Figure 1: Spin splitting, spin polarization for an $f$-wave magnet. (a) Comparison $s,p,d,f,g$-wave spin splitting. (b) Minimal $f$-wave magnetic texture with symmetries $\widetilde{C}^s, \widetilde{\mathcal{T}},$ and $C_{3z}$. Top/bottom layer with dark-/light-colored spheres. (c) Electronic band structure exhibiting spin-degenerate lines on M-K-$\Gamma$. (d) Difference of band-resolved spin $s_z$ polarization $s_{z,n}(\mathbf{k})$ between the first and second bands. (e) Spin splitting $\Delta_{s,1}(\mathbf{k})$ between the first and second bands. $t_1 = 1$, $t_2 = 1/2$, $J = 3$.
• Figure 2: Origin of $f$-wave splitting. (a) Comparison of numerical fit (orange) up to $k^5$ and the analytical low-energy expansion (green). (b) $f$-wave splitting divided by size of parabolic part as function of spin polarization for the lower band pair. The black arrow highlights the effect of increasing hopping $t_1$. The green dashed line is the analytical result for $t_1 = t_2$. (c) An in-plane field $B_\parallel$ gaps the nodal points on the Fermi surfaces and creates a polarization $s_\parallel$. (d) The spin conductivity $\sigma^s_{xx}$ divided by electric conductivity $\sigma_{xx}$ and spin polarization $\vert s_{n,k} \vert$ depends qualitatively on the $f$-wave splitting $\Delta_s^{(1)}$.
• Figure 3: Anisotropic Edelstein effect on $f$-wave magnet ribbons. (a) Ribbon geometry in $x$ termination leads to $p$-wave splitting, whereas $y$ termination does not. (b) Schematic Edelstein effect for different splitting, where filled circles denote occupied states. (c) Band structure of an x-terminated ribbon, spin polarization shown in color. (d) Edelstein susceptibility $\chi_{zy}$ for the ribbon in (c); solid (dashed) lines denote the contribution from edge (bulk) sites at a filling $E_\text{F}$. (e) Edelstein effect over position $x$ on the ribbon at Fermi energies marked in (d). $t_1 = 1,\, t_2 = 0.5, \, J = 3$. Ribbon width: 40 unit cells
• Figure 4: Parameter dependence of $f$-wave splitting for the upper band pair in Fig. \ref{['Fig2']}(a). (a) Parabolic dispersion and (b) $f$-wave splitting as function of exchange coupling $J$. (c) Ratio of (a,b) over spin polarization. Black arrows qualitatively highlight the effect of increasing hopping $t_1$. The green dashed line is the analytical result for $t_1 = t_2$, which does not match the fitted parameters for small $J$ where the two-band approximation fails. Due to the band hybridization, the curves (a-c) are non-monotonous, unlike Fig. \ref{['Fig2']}(a). Vanishing $E_0^{(1)}$ leads to divergences in (c).
• Figure 5: Normalized linear spin conductivity, Eqs. \ref{['Eq_BoltzSigXX']} and \ref{['Eq_BoltzSpinSigXX']}, as a function of splitting $\Delta_s$ and Fermi energy $E_f$ for $B_\parallel = 0.001$ and $k_BT = 0.01$. The Hamiltonians are $k_x^2 + k_y^2 + \Delta_s(\mathbf{k}) \tilde{\sigma}_z$, where the splitting $\Delta_s(\mathbf{k})$ is defined in Fig. \ref{['Fig1']}(a) with $\Delta_s$ as additional prefactor.