[Paper Review] Magnetic octupoles as the order parameter for unconventional antiferromagnetism

Sayantika Bhowal, Nicola A. Spaldin|arXiv (Cornell University)|Dec 7, 2022

Magnetic properties of thin films4 citations

TL;DR

This paper proposes that magnetic octupoles serve as the fundamental order parameter for unconventional antiferromagnets with non-relativistic spin splitting (NRSS), particularly in centrosymmetric systems like MnF₂. By describing the ferroic ordering of these octupoles, the authors unify the description of time-reversal symmetry breaking and spin splitting, enabling control over spin-splitting and predicting novel phenomena such as non-zero magnetic Compton scattering for direct experimental detection.

ABSTRACT

We show that time-reversal symmetry broken, centrosymmetric antiferromagnets with nonrelativistic spin-splitting are conveniently described in terms of the ferroic ordering of magnetic octupoles. The magnetic octupoles are the lowest-order ferroically ordered magnetic quantity in this case, and so are the natural order parameter for the transition into the magnetically ordered state. They provide a unified description of the broken time-reversal symmetry and the non-relativistic spin splitting as well as a platform for manipulating the latter, and account for other phenomena, such as piezomagnetism, characteristic of this class of antiferromagnets. Unusually for antiferromagnets, we show that the magnetic octupoles cause a non-zero magnetic Compton scattering, providing a route for their direct experimental detection. We illustrate these concepts using density-functional and model calculations for the prototypical non-relativistic spin-split antiferromagnet, rutile-structure manganese difuoride, MnF2.

Motivation & Objective

• To identify a unified order parameter for unconventional antiferromagnets that break time-reversal symmetry but preserve inversion symmetry.
• To address the limitation of conventional antiferromagnetic order parameters in describing non-relativistic spin splitting and time-reversal symmetry breaking.
• To provide a ferroic description of magnetic octupoles as the lowest-order multipoles in this class of antiferromagnets.
• To enable manipulation of non-relativistic spin splitting through octupolar control, relevant for spintronic applications.
• To predict new experimental signatures, such as non-zero magnetic Compton scattering, for direct detection of octupolar order.

Proposed method

• Employing a multipole expansion of magnetic interaction energy, the authors identify magnetic octupoles as the lowest-order ferroic multipoles in time-reversal-symmetry-broken, centrosymmetric antiferromagnets.
• Using density-functional theory (DFT) with LDA+SOC+U and the ELK code, the electronic structure of MnF₂ is computed to analyze spin-split bands and multipoles.
• Applying the Nth-order muffin-tin orbital (NMTO) method to downfold DFT results into a tight-binding model with effective d-d hoppings and onsite energies.
• Calculating atomic-site multipoles via decomposition of the density matrix into tensor moments, focusing on parity-even contributions from d-d and p-p orbitals.
• Computing the magnetic Compton profile (MCP) from spin-polarized momentum densities to predict a non-zero signal for octupolar order.
• Validating results with alternative DFT calculations using the VASP code and analyzing strain-induced piezomagnetic effects via Hellmann-Feynman forces.

Experimental results

Research questions

• RQ1 What is the appropriate order parameter for unconventional antiferromagnets that break time-reversal symmetry but remain centrosymmetric?
• RQ2 How can magnetic octupoles unify the description of non-relativistic spin splitting and time-reversal symmetry breaking in these systems?
• RQ3 Can the ferroic ordering of magnetic octupoles enable control over non-relativistic spin splitting for device applications?
• RQ4 What measurable experimental signatures arise from magnetic octupolar order in antiferromagnets?
• RQ5 How does the magnetic Compton scattering signal differ in octupole-ordered antiferromagnets compared to conventional antiferromagnets?

Key findings

• Magnetic octupoles are identified as the lowest-order ferroic multipoles in time-reversal-symmetry-broken, centrosymmetric antiferromagnets, making them the natural order parameter.
• The ferroic ordering of magnetic octupoles unifies the description of non-relativistic spin splitting and time-reversal symmetry breaking in these systems.
• The model predicts a non-zero magnetic Compton scattering signal in octupole-ordered antiferromagnets, enabling direct experimental detection.
• The octupolar description accounts for the piezomagnetic effect and predicts an anti-piezomagnetic response in these materials.
• DFT and tight-binding calculations for MnF₂ confirm the presence of non-relativistic spin splitting and consistent octupolar multipoles at the Mn site.
• The band structure remains qualitatively unchanged under sublattice exchange in a modified MnF₂ structure, but the orbital character of bands reverses, confirming the role of sublattice-dependent octupolar order.