Ferroaxial magnets: time-reversal-even mirror symmetry violation from spin order

Abstract

We investigate ferroaxial magnets, a new class of spin-order-driven multiferroic magnets in which magnetic ordering induces mirror-symmetry breaking while preserving both time-reversal and spatial-inversion symmetries. These systems exhibit a ferromagnet-like axial anisotropy that allows optical control of the ferroaxial polarization, while their macroscopic time-reversal symmetry makes them attractive for antiferromagnetic spintronics. Using spin crystallographic group analysis, we identify the candidate materials and the nonrelativistic ferroaxial nature stemming from the strong exchange splitting of magnets. Furthermore, a symmetry-based identification shows magnetic materials that host ferroaxial order and metallic conductivity, realizing the ferroaxial metal state that undergoes a ferroaxial phase transition while remaining metallic. As a direct probe for the ferroaxial metal, we propose a third-order nonlinear Hall effect originating from the transverse coupling between the electric field and Berry curvature dipole mediated by the ferroaxial anisotropy. Our results establish ferroaxial magnets as a platform for nonrelativistic multiferroicity and spintronic applications.

Figures (4)

• Figure 1: Sketch of axial-magnetic multiferroicity given by the combination of (antiferro-)magnetic and ferroaxial order (upper panel). In the absence of relativistic SOC, the multiferroic state can be classified by the orbital-active ($\bm{A}_\text{orb}$) and spin-active ($\bm{A}_\text{sp}$) ferroaxial anisotropy, which are defined by the $\theta$-even axial vector in the real space ($x,y,z$) and spin space ($s_x,s_y,s_z$), respectively. $\bm{A}_\text{orb}$ can be active both in collinear and noncollinear magnets, while $\bm{A}_\text{sp}$ in noncollinear magnets.
• Figure 2: (a) Second-order nonlinear Hall effect originated from the Berry-curvature-dipole (BCD) vector $\bm{d}$ defined by the cross product of the momentum asymmetry (${\bm{k}}$) and Berry curvature $\bm{\Omega}$. The current (colored in orange) is rectified along $\bm{d}$, and thus $\bm{d} \perp \bm{E}$ is required for the Hall response. (b) Third-order nonlinear current response in an isotropic system. Field-induced BCD $\partial_{\bm{E}}\bm{d}$ is parallel to the electric field $\bm{E}$ [$d_0$ in Eq. \ref{['ferroaxial_bcd']}], leading to no Hall response. (c) Third-order nonlinear current response in the presence of ferroaxial order ($\bm{A}$). Owing to the lateral correlation between $\bm{E}$ and $\partial_{\bm{E}} \bm{d}$ by $\bm{A}$, the tilting component $d_\text{ax} \bm{A} \times \bm{E}$ allows for the Hall response as in the panel (a).
• Figure 3: (a) Crystal and spin structures for the model study (right panel). The purple balls are magnetic sites surrounded by gray-colored ligands, which define the staggered chirality of magnetic sites (left panel). The spin order doubling the unit cell to the $z$ axis allows for the toroidal moment $\bm{t}$ in each original unit cell due to the coupling of the spin magnetic moment and chirality. (b) Fermi surface ($k_z=0,~\mu=-1.2$) without the nonsymmorphic symmetry [i.e. local chirality depicted in (a)]. (c,d) Spin-driven ferroaxial states illustrated by yellow-colored ferroaxial vectors (left panel), fermi surface ($k_z=0,~\mu=-1.2$) tilted from that in (b) due to the ferroaxial symmetry (center panel), and the Berry connection polarizability $G_{xx} ({\bm{k}})$ summed over the bands below $\mu=0$ (right panel).
• Figure 4: Third-order nonlinear conductivity $\sigma_{y;xxx}$ and its chemical potential ($\mu$) depedence. Two curves are for the spin-order-driven ferroaxial states with opposite polarizations ($A_\pm$). Note that a negligible response is obtained in the insulator regime ($\mu \gtrsim -0.6$). We used $q=1$, temperature $T=10^{-2}$, and the Brillouin zone meshing $N=100^3$ for the summation over ${\bm{k}}$.