Asymmetric Splitting of an Antiferromagnetic Resonance via Quartic Exchange Interactions in Multiferroic Hexagonal ${\mathrm{HoMnO}}_3$

The symmetric splitting of two spin-wave branches in an antiferromagnetic resonance (AFR) experiment has been an essential measurement of antiferromagnets for over half a century. In this work, circularly polarized time-domain THz spectroscopy experiments performed on the low symmetry multiferroic hexagonal ${\mathrm{HoMnO}}_3$ reveal an AFR of the Mn sublattice to split asymmetrically in an applied magnetic field, with an $\approx 50\%$ difference in $g$ factors between the high and low energy branches of this excitation. The temperature dependence of the $g$ factors, including a drastic renormalization at the Ho spin ordering temperature, reveals this asymmetry to unambiguously stem from Ho-Mn interactions. Theoretical calculations demonstrate that the AFR asymmetry is not explained by conventional Ho-Mn exchange mechanisms alone and is only reproduced if quartic spin interactions are also included in the spin Hamiltonian. Our results provide a paradigm for the optical study of such novel interactions in hexagonal manganites and low symmetry antiferromagnets in general.

Figure 1

Crystal structure of HMO in the ferroelectric phase ($T<T_c=875\mathrm{K}$) with views along the (left) $c$ axis and (right) $a$ axis, respectively. In this phase, the ${\mathrm{Ho}}^{+3}$ ions (green and blue spheres) lie in symmetrically distinct positions of the lattice resulting in a finite ferroelectric moment along the $c$ axis.

Figure 2

Image plots of the imaginary part of the index of refraction as a function of temperature and frequency for the orientations (a) ${\overrightarrow{\mathrm{h}}}_{\mathrm{ac}}\parallel c$ and (b) ${\overrightarrow{\mathrm{h}}}_{\mathrm{ac}}\perp c$, respectively. Horizontal dashed lines denote the three zero field transition temperatures while vertical dashed lines label the more prominent Ho crystal field (CF) excitations identified at temperatures $T\ge T_{\mathrm{Ho}}$. The excitation labeled $M$ is the AFR of the Mn sublattice whose resonant frequency is marked by white squares.

Figure 3

Field dependence of the imaginary part of the index of refraction of HMO at 20 K for (a) right-hand and (b) left-hand circular polarizations with $H\parallel c$. The Mn AFR is the linearly varying excitation at $\approx 1.3\mathrm{THz}$, which can be seen to naturally partition into low and high energy branches in the circular basis. White triangles mark the extracted resonant frequencies of the AFR.

Figure 4

(a) Resonant frequency of the AFR for both left-hand (blue, circles) and right-hand (red, squares) circular polarizations as a function of the magnetic field at $T=20\mathrm{K}$. The low energy branch of the AFR possesses a significantly larger $g$ factor than that of the high energy branch, regardless of polarization and field direction. (b) Temperature dependence of the $g$ factors which reveals a significant renormalization at $T_{\mathrm{Ho}}$. (c) Asymmetry ratio of the $g$ factors plotted with $\mathrm{\Delta }\chi$, the $H\parallel c$ magnetic susceptibility after the paramagnetic contribution has been subtracted.