An efficient formalism for inertial spin waves: Dzyaloshinskii-Moriya antiferromagnets as case studies

Technical Summary: An Efficient Formalism for Inertial Spin Waves: Dzyaloshinskii-Moriya Antiferromagnets as Case Studies

Problem Statement
While magnetic inertia has been experimentally confirmed and theoretically incorporated into the Landau-Lifshitz-Gilbert (LLG) equation as a second-order time derivative term, a systematic formalism for fully characterizing inertial spin waves (SWs) remains lacking. Specifically, there is no unified framework to define the intrinsic chirality and polarization of these waves, particularly in spatially nonuniform magnetic configurations. Existing studies have largely focused on uniform ferromagnetic or antiferromagnetic ground states, often restricted to the long-wavelength regime, leaving the behavior of inertial SWs in complex textures (such as canted and spiral states) and the precise relationship between eigenfrequency signs and chirality poorly explored.

Methodology
The authors develop a general, efficient formalism to calculate inertial SWs applicable to both uniform and nonuniform magnetic configurations. The approach involves:

1. Theoretical Framework: Starting from the discrete inertial Landau-Lifshitz (ILL) equation, the authors introduce a local coordinate frame attached to the spatially varying equilibrium magnetization.
2. Linearization and Reformulation: By expanding the magnetization fluctuations to second order and linearizing the equations of motion, the original second-order differential system is recast into a first-order system with doubled dimensionality. This is achieved by introducing auxiliary phase-space variables.
3. Eigenvalue Solution: Assuming plane-wave solutions, the system is transformed into a matrix eigenvalue problem. This allows for the direct calculation of eigenfrequencies and eigenvectors, from which amplitudes, phases, and dispersion relations are derived.
4. Unified Definition of Chirality and Polarization: The authors propose a general definition of SW chirality and polarization using a single parameter: the ellipticity angle ($\chi$). This angle encodes the frequency sign, the phase difference between dynamic components, and the ratio of the elliptical axes. It is defined as $\sin 2\chi_l = \text{sgn}(\omega) \frac{2A_{\theta,l}A_{\phi,l}}{A_{\theta,l}^2 + A_{\phi,l}^2} \sin \delta$, where $\delta$ is the phase difference. This definition distinguishes between linear, elliptical, and circular polarizations and assigns handedness (left or right) based on the sign of $\chi$.

Case Studies
The formalism is validated through a systematic investigation of uniaxial antiferromagnets (AFMs) with both staggered and homogeneous Dzyaloshinskii-Moriya interactions (DMIs). The study covers three equilibrium configurations determined by the DMI strength:

• Uniform Collinear AFM: Occurs when DMI is below a critical value.
• Canted Configuration: Occurs in staggered DMI systems above the critical value.
• Spiral Configuration: Occurs in homogeneous DMI systems above the critical value.

Key Results

1. Degeneracy and Symmetry:

   • Staggered DMI: Preserves the combined parity-time ($PT$) symmetry. Consequently, small staggered DMI preserves the degeneracy of both precessional and nutational SWs.
   • Homogeneous DMI: Breaks $PT$ symmetry. Small homogeneous DMI lifts the spin-wave degeneracy, shifting dispersion curves in opposite directions along the wave vector ($k$) axis.
   • Nonuniform Configurations: In canted and spiral structures, further breaking of space-time inversion symmetry fully removes spin-wave degeneracy across the entire Brillouin zone.

2. Dispersion and Group Velocity:

   • Backward Waves: Long-wavelength nutational SWs generally behave as backward waves, exhibiting a group velocity opposite to the wave vector.
   • Flat Bands: In canted and spiral configurations, a critical inertial relaxation time ($\eta_c$) exists. Near this value, the group velocity of one nutational branch vanishes, leading to the emergence of flat bands and sharp peaks in the density of states.
   • Band Folding: The dispersion spectra of the canted configuration can be derived from those of the spiral configuration via band folding. Specifically, the low-frequency branches of the canted state coincide with the spiral state's branches, while the high-frequency branches arise from folding the extended Brillouin zone.

3. Chirality and Polarization:

   • Uniform vs. Nonuniform: In uniform configurations, polarization is wavenumber-insensitive, and circularly polarized modes with opposite chiralities appear in pairs. In nonuniform (canted/spiral) configurations, polarization becomes strongly dispersive.
   • Mode Specificity: In canted and spiral configurations, nutational modes are consistently left-handed, while precessional modes are consistently right-handed.
   • Frequency Sign Independence: The paper demonstrates that flipping the sign of the eigenfrequency does not necessarily reverse chirality. Chirality depends on the phase difference between components; in canted, spiral, and staggered-DMI AFMs, the phase difference adjusts such that chirality remains invariant under frequency sign reversal. This contrasts with homogeneous-DMI AFMs where chirality does reverse with frequency sign.

4. Instability: The study identifies conditions where the precessional gap becomes complex (e.g., when DMI exceeds the critical value in certain configurations), indicating the instability of the equilibrium magnetic state.

Significance and Claims
The paper claims to advance the fundamental understanding of magnetic inertial dynamics by providing a robust, unified framework for characterizing inertial SWs. The proposed formalism offers a more direct route to determining chirality and polarization compared to quantum approaches (like Bogoliubov transformations), which may obscure these specific classical properties. By revealing the distinct behaviors of nutational and precessional modes in complex magnetic textures, the work provides theoretical insights necessary for the development of ultrafast magnonic devices, particularly those leveraging backward wave propagation and terahertz frequency operations. The authors emphasize that their method is general and can be extended to investigate inertial SWs in a broad range of magnetic configurations beyond the specific cases studied.