Antiferroaxial altermagnetism

Abstract

The antiferroaxial state is emerging as an important ferroic order in condensed matter systems. Here, we establish antiferroaxial altermagnetism as a broadly prevalent, generic, and microscopically grounded multiferroic mechanism, in which antiferroaxial counter-rotating distortions both induce altermagnetism and enable its deterministic and reversible switching. Within a unified Landau-theory and symmetry framework, we identify a symmetry-allowed trilinear invariant coupling the antiferroaxial order, the Néel vector, and the altermagnetic order, and derive general symmetry criteria for its occurrence. This coupling locks the induced altermagnetism to the antiferroaxial order, so that reversing the latter reverses the spin splitting and associated time-reversal-odd responses, such as anomalous Hall conductivity. We provide a practical spin group dictionary mapping Néel-vector representations to the resulting $d$-, $g$-, and $i$-wave antiferroaxial altermagnetism, validate the mechanism with ligand-rotation tight-binding models and first-principles calculations, and identify many candidate materials by screening the MAGNDATA and C2DB databases. Our results elevate antiferroaxiality to a universal ferroic control knob for structurally programmable altermagnetic spintronics.

Figures (4)

• Figure 1: Antiferroaxial altermagnetism and multiferroic control of the spin splitting. (a) Altermagnet with a spin-splitting isoenergy surface with the red and blue denoting magnetic sublattices; both inversion-time ($\mathcal{IT}$) and translation-time-reversal ($\mathcal{T}\boldsymbol\tau$) symmetries are broken. (b) Antiferroaxiality, characterized by counter-rotating sublattices, is defined as $\mathbf{G} = \mathbf{A}_a - \mathbf{A}_b$ with $\mathbf{A}_{a/b}$ the ferroaxial order on the sublattice $a/b$. Antiferroaxial order breaks $\mathcal{I}$ and $\boldsymbol\tau$ that swap the two sublattices, but preserves those acting within each sublattice. (c) In antiferroaxial altermagnets, the coexistence of Néel and antiferroaxial orders breaks $\mathcal{IT}$ and $\mathcal{T}\boldsymbol\tau$ while preserving mirror-time ($M\mathcal{T}$) and rotation-time ($R_{\parallel}\mathcal{T}$) symmetries, where $\parallel$ denotes the axis parallel to the antiferroaxial order. Antiferroaxial altermagnetism realizes a broadly prevalent and generic multiferroic mechanism enabling antiferroaxiality-controlled altermagnetic spin splitting.
• Figure 2: Antiferroaxial altermagnetism with distinct spin splitting types. Red and blue spheres denote spin-up and spin-down atoms, gray spheres represent ligands, and green arrows indicate rotational distortions. (a)-(c) Illustration of the mechanism (top), band structures (middle), and spin splittings (bottom). (a) $d$-wave altermagnetism from parent group $P4/mmm$: $N\sim M_1^+$ couples to $G_z\sim M_3^+$ via a $\mathbf{O}^{(2)}$ magnetic multipole in the $\Gamma_3^+(B_{2g})$ channel (Table S1 in SM SM). Upon symmetry lowering to $P4/mbm$, the Néel vector transforms as $\Gamma_4^+(B_{2g})$, giving a $d_{xy}$-wave spin splitting in point group $4/mmm$ (Table \ref{['table:TB1']}). (b) $g$-wave altermagnetism from parent group $P4/mmm$: $N\sim M_1^+$ couples to $G_z\sim M_2^+$ via the $\Gamma_2^+(A_{2g})$ magnetic multipole channel. In $P4/mbm$, the altermagnetic Néel vector is $\Gamma_2^+(A_{2g})$, corresponding to a planar $g$-wave spin splitting in $4/mmm$ (Table \ref{['table:TB1']}). (c) $i$-wave altermagnetism from parent group $P6/mmm$: $N\sim\Gamma_3^-$ couples to $G_z\sim \Gamma_4^-$ via the $\Gamma_2^+(A_{2g})$ magnetic multipole channel. The symmetry lowering to $P\bar{6}m2$ gives an altermagnetic Néel vector $\Gamma_2(A_2^\prime)$, corresponding to a planar $i$-wave spin splitting in $\bar{6}m2$ (Table \ref{['table:TB1']}). The common model parameters are $J=t$ and $t_M=0.5t$, with specific values: (a) $\theta=20^\circ$, $V_1=1.69t$, $V_2=0.37t$; (b) $\theta=20^\circ$, $V_0=1.2t$, $t_1=0.7t$, $t_2=0.15t$; (c) $\theta=30^\circ$, $V_0=1.2t$, $V_1=0.875t$, $V_2=0.5t$, $t_1=0.6t$, $t_2=0.15t$.
• Figure 3: Antiferroaxial altermagnets. The blue/red spheres and green arrows denote opposite spins and antiferroaxial orders, respectively. (a, c) Monolayer MnS$_2$ (parent group $P\bar{4}m2$), where Néel ($M_1$) and antiferroaxial ($M_2$) orders couple with a $\mathbf{O}^{(4)}$ magnetic multipole in the $\Gamma_2(A_2)$ channel (Table S1 in SM SM). The symmetry lowers to space group $P\bar{4}2_1m$ with Néel vector $\Gamma_2(A_2)$, corresponding to planar $g$-wave spin splitting (Table \ref{['table:TB1']}). (b, d) Bulk La$_2$NiO$_4$ (parent group $I4/mmm$), where Néel ($X_1^+$) and antiferroaxial ($X_3^+$) orders couple with the $\Gamma_5^+(E_{g})$ magnetic multipole channel. The symmetry lowers to space group $Pccn$ with Néel vector $\Gamma_4^+(B_{3g})$, corresponding to $d_{yz}$-wave splitting (Table \ref{['table:TB1']}). The insets in the band structures (c, d) illustrate the spin splitting within the Brillouin zone.
• Figure 4: Multiferroicity with antiferroaxial switchable spin splitting and anomalous Hall conductivity in altermagnets. (a) Top and side views of the crystal structure of FeF$_3$. Red and blue spheres denote Fe atoms with opposite magnetic moments, while gray spheres represent F atoms. The green curved arrows indicate the antiferroaxial rotation of the Fe-F$_6$ octahedra, parameterized by the angle $\theta$. (b) Band structure for the lowest-energy configuration at $\theta = 12^\circ$. The inset shows the 3D Brillouin zone with the bulk $g$-wave spin splitting. (c) Total energy as a function of the rotation angle $\theta$, revealing a double-well potential with ground-state minima at $\theta \approx \pm 12^\circ$. The insets illustrate that reversing the structural rotation direction (top) switches the sign of the spin splitting (bottom). The effective Landau free energy is well fit to the model in Eq. \ref{['Eq:eff_Landau']}, yielding $\mathcal{F}_\mathrm{eff}(\theta) = 32.1 - 0.446\theta^2 + 1.45 \times 10^{-3} \theta^4$, where $\theta$ parameterizes the structural distortion $G$ (in degrees) and $\mathcal{F}_\mathrm{eff}$ is in meV per unit cell. (d) Anomalous Hall conductivity, calculated as a function of chemical potential at the equilibrium angles $\theta = \pm 12^\circ$, is switchable by antiferroaxial order.