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Why are there so few non-altermagnetic antiferromagnets?

Nicola A. Spaldin, Sang-Wook Cheong, Sinead Griffin

TL;DR

The work addresses why non-altermagnetic antiferromagnets are scarce by arguing that non-relativistic spin splitting is the default when magnetic order breaks time-reversal symmetry. It develops a framework that starts with Standard-Model Altermagnets (SMALM) and extends to Beyond SMALMs (BSMALM) by relaxing symmetry constraints or including spin-orbit coupling and non-collinearity, thereby broadening the class of compensated AFMs that can exhibit NRSS. It identifies two robust routes to preserving spin degeneracy: (i) PT symmetry in antiferromagnets like Cr$_2$O$_3$, and (ii) translation-based global time-reversal symmetry in NiO/MnO, which require specific inversion and multipolar ordering. The findings have practical implications for materials discovery and design, clarifying why conventional antiferromagnets are common and under what symmetry conditions NRSS can be controlled or suppressed, with multipolar order playing a central role.

Abstract

We review the conditions that cause or prohibit non-relativistic spin splitting of the energy bands in antiferromagnets. We propose that the existence of spin splitting in magnetically ordered systems is the default scenario and outline the criteria that must be met to avoid it. We discuss some of the properties of those special antiferromagnets that succeed in preserving their spin degeneracy.

Why are there so few non-altermagnetic antiferromagnets?

TL;DR

The work addresses why non-altermagnetic antiferromagnets are scarce by arguing that non-relativistic spin splitting is the default when magnetic order breaks time-reversal symmetry. It develops a framework that starts with Standard-Model Altermagnets (SMALM) and extends to Beyond SMALMs (BSMALM) by relaxing symmetry constraints or including spin-orbit coupling and non-collinearity, thereby broadening the class of compensated AFMs that can exhibit NRSS. It identifies two robust routes to preserving spin degeneracy: (i) PT symmetry in antiferromagnets like CrO, and (ii) translation-based global time-reversal symmetry in NiO/MnO, which require specific inversion and multipolar ordering. The findings have practical implications for materials discovery and design, clarifying why conventional antiferromagnets are common and under what symmetry conditions NRSS can be controlled or suppressed, with multipolar order playing a central role.

Abstract

We review the conditions that cause or prohibit non-relativistic spin splitting of the energy bands in antiferromagnets. We propose that the existence of spin splitting in magnetically ordered systems is the default scenario and outline the criteria that must be met to avoid it. We discuss some of the properties of those special antiferromagnets that succeed in preserving their spin degeneracy.
Paper Structure (11 sections, 9 equations, 3 figures)

This paper contains 11 sections, 9 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Standard-Model Altermagnets
  3. Beyond Standard-Model Altermagnets
  4. Removing the constraint of the symmetry operation connecting opposite spins
  5. Allowing spin-orbit coupling and / or magnetic non-collinearity
  6. So what is needed in order to be non-altermagnetic?
  7. First possibility: Not breaking time-reversal symmetry
  8. Second possibility: Breaking time-reversal symmetry but having an additional operation restore the Kramers degeneracy
  9. Antiferromagnets with $\cal{P T}$ symmetry
  10. Antiferromagnets in which a lattice translation restores global time-reversal symmetry
  11. Summary

Figures (3)

  • Figure 1: Upper panel: The crystal field from the ligand (red circles) polyhedra surrounding the metal ions (gray circles) splits the energies of the metal orbitals oriented in the $x$ and $y$ directions. Lower panel: When the pattern of antiferromagnetic spin ordering matches that of the structural motif, the split energy bands are decorated by the spin orientation. Note that the orientation of the opposite spins relative to the lattice, shown here as up and down, is arbitrary.
  • Figure 2: Left: Unit cell of cartoon LaMnO$_3$ consisting of two five-atom unit cells with oppositely spin polarized Mn ions and a small structural distortion introduced into one of the units. La ions are shown in green, Mn in purple and O in red. Right: The calculated band structure with the two spin channels shown in solid orange and dashed blue lines. Note in particular the spin splitting at the $\Gamma$ point.
  • Figure 3: Crystal structure of ferroelectric BiFeO$_3$. The ground state $R3c$ crystal structure (right) is reached from the ideal cubic perovskite structure (left) via an antiferrodistortive rotation of the light blue FeO$_6$ octahedra (black arrows) around, and a displacement of the orange Bi ions along, the pseudocubic [111] direction.