Keffer-like form of the symmetric Heisenberg exchange integral: Contribution to the Landau--Lifshitz--Gilbert equation and spin wave dispersion dependence

Technical Summary: Keffer-like form of the symmetric Heisenberg exchange integral

Problem Statement
The paper addresses the microscopic origins of magnetoelectric phenomena in multiferroic materials, specifically focusing on the mechanisms driving electric polarization. While the Dzyaloshinskii-Moriya (DM) interaction is well-established as a source of non-collinear spin order and electric polarization (via the spin-current model), the symmetric Heisenberg exchange interaction is typically treated as isotropic or possessing even anisotropy. The author proposes that the ligand shift (the displacement of non-magnetic ions from the line connecting two magnetic ions), which is the microscopic origin of the Keffer form of the DM vector constant, may also induce an "odd anisotropy" in the symmetric Heisenberg exchange integral. This suggests that the symmetric exchange constant in antiferromagnetic (AFM) or ferrimagnetic materials is not solely a function of the distance between magnetic ions but also depends on the projection of this distance onto the ligand shift vector.

Methodology
The study employs a quantum hydrodynamic method to derive macroscopic equations from a microscopic many-particle Hamiltonian.

1. Microscopic Model: The author constructs a modified Heisenberg Hamiltonian where the exchange integral $U_{ij}$ includes terms dependent on the ligand shift $\delta$. Specifically, two forms of odd anisotropy are proposed: one involving a single ligand shift parallel to the inter-ion distance, and another involving a cross-product of two ligand shifts.
2. Derivation of Macroscopic Equations: Using the quantum hydrodynamic formalism, the author derives:
   • The spin torque contribution to the Landau–Lifshitz–Gilbert (LLG) equation.
   • The force field in the Euler momentum balance equation.
   • The macroscopic energy density.
3. Spin-Current Model Application: The derived spin torques are utilized within the spin-current model to calculate the resulting electric polarization and microscopic electric dipole moments.
4. Dispersion Analysis: The paper analyzes the spin wave dispersion relations for antiferromagnetic materials under two equilibrium conditions: collinear order (easy-axis and easy-plane regimes) and cycloidal (non-collinear) order. The analysis incorporates the XYZ model for uniaxial crystals and compares the effects of the new interaction against standard DM interactions.

Key Contributions and Results

• Novel Exchange Integral Form: The paper introduces a "Keffer-like" form for the symmetric Heisenberg exchange integral, $U_{ij} = U_{0,ij} + U_{1,ij}(\mathbf{r}_{ij} \cdot \boldsymbol{\delta}_{3}) + U_{2,ij}(\mathbf{r}_{ij} \cdot [\boldsymbol{\delta}_1 \times \boldsymbol{\delta}_2])$. This form is shown to be symmetric ($U_{ij} = U_{ji}$) and valid specifically for antiferromagnetic materials.
• New Spin Torque and Energy Density: The odd anisotropy generates a spin torque term proportional to the first spatial derivative of the spin density, $\partial_t \mathbf{S} \propto [\mathbf{S} \times (\boldsymbol{\delta}_{eff} \cdot \nabla)\mathbf{S}]$. Consequently, the energy density acquires a term containing one spatial derivative of the spin density, distinct from the standard second-derivative exchange terms or the DM interaction terms.
• Spin Wave Dispersion:
  • In collinear AFM order, the new interaction contributes to the spin wave dispersion frequency squared ($\omega^2$) proportional to the square of the projection of the wave vector ($\mathbf{k} \cdot \boldsymbol{\delta}_{eff}$). This contrasts with the DM interaction, which shifts the minimum frequency linearly with the wave vector projection.
  • In cycloidal order, the interaction leads to an interference between the dynamics of the antiferromagnetic vector ($\mathbf{L}$) and the magnetization vector ($\mathbf{M}$), complicating the dielectric response compared to cases where these branches evolve independently.
• Electric Polarization: A novel mechanism for electric polarization is derived. The effective spin current associated with this interaction yields a polarization vector $\mathbf{P} \propto \mathbf{S}_A(\mathbf{S}_B \cdot \boldsymbol{\delta}_{eff}) - \mathbf{S}_B(\mathbf{S}_A \cdot \boldsymbol{\delta}_{eff})$. Unlike polarization mechanisms derived from the DM interaction (which often involve spatial derivatives of spin density), this new form depends on the local deviation of spin projections along the effective ligand direction without explicit spatial derivatives in the macroscopic expression.
• Microscopic Dipole Moment: A corresponding microscopic electric dipole moment operator is identified as $\hat{d}_{ij} \propto r_{ij}^2 l(r_{ij}) [\boldsymbol{\delta}_{eff} \times [\hat{\mathbf{S}}_i \times \hat{\mathbf{S}}_j]]$.

Significance and Claims
The paper claims to demonstrate that the ligand shift, traditionally associated solely with the antisymmetric DM interaction, also fundamentally alters the symmetric Heisenberg exchange in antiferromagnetic systems. This "odd anisotropy" of the symmetric exchange leads to:

1. Qualitatively new spin torques and modifications to the static and dynamic behavior of magnetically ordered systems.
2. A distinct contribution to the dielectric response in multiferroics, offering a new structural form for spin-origin electric polarization.
3. A more complex dielectric response in multiferroics with cycloidal order due to the coupling of $\delta L_z$ and $\delta M_z$ modes.

The author positions this work as a refinement of the spin-current model, suggesting that the physical mechanisms behind magnetoelectric effects in multiferroics are more diverse than previously modeled, particularly regarding the role of symmetric exchange anisotropy. The study remains purely theoretical, with no experimental data presented or specific experimental proposals made beyond the derivation of the theoretical framework.