Spin-orbital exchange as a route to intertwined dipole-quadrupole orbital order in MnV 2 O 4 under strong trigonal crystal field

The gist The system stabilizes a two-in/two-out magnetic configuration featuring spin canting and intertwined dipole-quadrupole orbital order in MnV2O4 under strong trigonal crystal field

How it works

The research investigates the interplay between spin and orbital degrees of freedom in MnV2O4 by combining first-principles calculations with an effective spin-orbital model to address the nature of the orbital state The analysis begins by constructing a multiorbital Hamiltonian for the t2g manifold, which includes terms for on-site Coulomb interaction (HC), trigonal crystal field (HtriCEF), atomic spin-orbit coupling (HSOC), and electron hopping (Hkin) The energy hierarchy derived from Density Functional Theory calculations reveals that the trigonal crystal field splitting is significantly larger than the SOC strength, leading to a hierarchy where the trigonal crystal field plays the dominant role in determining the low-energy orbital manifold even in the cubic phase

The effective spin-orbital Hamiltonian derived using degenerate perturbation theory consists of two main contributions: Hex, which describes intersite spin-orbital exchange interactions generated by virtual hopping processes, and Hλ, an on-site term induced by SOC The exchange term is given by a complex expression (Equation 3) that includes bilinear orbital pseudospin interactions and terms involving spin correlations and orbital quadrupoles

Key Model Parameters and Hopping Effects

The hopping matrix between neighboring V ions is constrained by the crystal symmetry of the spinel structure, with specific hopping integrals like t1, t2, t3, and t4 defined in terms of the global t2g orbital basis The analysis shows that including subdominant hopping processes significantly modify the exchange interactions and play a key role in determining the orbital order and magnetic structure Specifically, when all hoppings are included, the system stabilizes a two-in/two-out (2I2O) magnetic configuration with spin canting and intertwined dipole-quadrupole orbital order

The hopping matrix in the effective angular momentum basis reveals that ta is the dominant hopping integral in the lZeff basis The ratio of subdominant to dominant hopping processes, |td/ta|, is shown to be strongly suppressed compared to the dominant-hopping limit where |ta| = |td| This contrast leads to a strong enhancement of the Kzz exchange parameter, which drives the system away from the compass-model limit

Magnetic and Orbital Structures

The mean-field analysis identifies three distinct ordered phases in the (x, ζ) parameter space, where x is the hopping tuning parameter and ζ is the SOC constant Phase I corresponds to an all-in/all-out (AIAO) magnetic structure together with orbital dipole order associated with τZ, with no quadrupolar components Phases II and III exhibit a two-in/two-out (2I2O) magnetic configuration with spin canting and intertwined dipole-quadrupole orbital order

In Phase III, corresponding to MnV2O4 at (x, ζ) = (1, 27 meV), the system exhibits a finite Kzz interaction, which couples orbital dipoles through a spin-dependent prefactor This term is responsible for stabilizing the 2I2O configuration over the AIAO state Furthermore, the κ term, which induces orbital quadrupole moments via spin correlations, further induces the ferroquadrupolar orbital order This results in the space group I41/amd and an occupation pattern where "⟨nxy⟩ > ⟨nyz⟩ = ⟨nzx⟩, indicating an instability toward tetragonal symmetry"

Microscopic Mechanisms and Conclusion

The magnetic structures are understood in terms of longitudinal and transverse components of the order parameters, where the longitudinal components determine the AIAO or 2I2O configurations The transverse spin components lead to spin canting and noncoplanar structures The resulting spin exchange constants split into J∥ and J⊥, where J∥ is enhanced by the Kzz term, leading to a noncoplanar magnetic structure observed in Phase III The orbital quadrupole moments couple to lattice degrees of freedom, driving a structural distortion that is consistent with the experimentally observed tetragonal compression This work demonstrates that going beyond the compass-model limit by incorporating subdominant hoppings and all symmetryallowed exchange interactions is essential for capturing the competing interactions that stabilize complex spin-orbital orders The effective model successfully stabilizes the 2I2O configuration of spin and orbital dipolar moments accompanied by spin canting and ferroquadrupolar orbital order The results clarify how competing and cooperating exchange interactions govern the intertwined spin-orbital state and the associated instability toward tetragonal distortion.

1. Focus on developing a predictive model for spin-orbital phase transitions by incorporating subdominant hopping processes, as subdominant hopping processes strongly modify the spin-orbital exchange interactions. The improved AI system can predict whether a material will stabilize an all-in/all-out (AIAO) magnetic structure or a two-in/two-out (2I2O) magnetic configuration, moving beyond models based on dominant hopping.

2. Implement a mechanism for determining the resulting crystal symmetry based on the effective Hamiltonian analysis, as the study shows that including subdominant hoppings leads to an order consistent with the experimentally observed tetragonal distortion and identifies a space group of I41/amd. The improved system can predict phase transitions in orbital ordering based on input material parameters.

3. Develop an AI capable of identifying the resulting orbital occupation patterns from spin configurations, specifically by using the relationship where the direction of the orbital pseudospin in the τX-τY plane determines the occupations of the three t2g orbitals. The improved system can predict whether "unequal orbital occupations ⟨nxy⟩ > ⟨nyz» = ⟨nzx⟩" will occur based on spin configuration parameters.

4. Create a framework for correlating microscopic exchange interactions with structural distortion by analyzing how the orbital quadrupole moments become finite in the 2I2O state, which modifies the occupation of the t2g orbitals and can drive a structural distortion. The improved system can predict lattice instabilities (like tetragonal compression) arising from specific spin-orbital order parameters.

5. Establish a method for quantifying the influence of competing energy scales, such as the trigonal crystal field splitting ∆tri = 42 meV, on the low-energy manifold, as this is shown to play the dominant role in determining the low-energy orbital manifold even in the cubic phase. The improved system can prioritize physical degrees of freedom when constructing effective Hamiltonians for correlated materials.