[Paper Review] Altermagnets and beyond: Nodal magnetically-ordered phases

T. Jungwirth, Rafael M. Fernandes+2 more|arXiv (Cornell University)|Sep 16, 2024

Magnetic and transport properties of perovskites and related materials5 citations

TL;DR

A comprehensive review of altermagnets with nodal even-parity-wave magnetic ordering, detailing symmetry-based identification, material realizations, electronic-structure signatures, relativistic/topological effects, and extensions to non-collinear spin densities.

ABSTRACT

The recent discovery of altermagnets has opened new perspectives in the field of ordered phases in condensed matter. In strongly-correlated superfluids, the nodal p-wave and d-wave ordered phases of $^{3}$He and cuprates play a prominent role in physics for their rich phenomenology of the symmetry-breaking order parameters. While the p-wave and d-wave superfluids have been extensively studied over the past half a century, material realizations of their magnetic counterparts have remained elusive for many decades. This is resolved in altermagnets, whose recent discovery was driven by research in the field of spintronics towards highly scalable information technologies. Altermagnets feature d, g or i-wave magnetic ordering, with a characteristic alternation of spin polarization and spin-degenerate nodes. Here we review how altermagnetism can be identified from symmetries of collinear spin densities in crystal lattices, and can be realized at normal conditions in a broad family of insulating and conducting materials. We highlight salient electronic-structure signatures of the altermagnetic ordering, discuss extraordinary relativistic and topological phenomena that emerge in their band structures, and comment on strong-correlation effects. We then extend the discussion to non-collinear spin densities in crystals, including the prediction of p-wave magnets, and conclude with a brief summary of the reviewed physical properties of the nodal magnetically-ordered phases.

Motivation & Objective

• Motivate the study of altermagnetism as a distinct symmetry class with nodal spin-polarized electronic structure and vanishing net magnetization.
• Explain how spin-symmetry groups classify collinear magnets into ferromagnets, antiferromagnets, and altermagnets.
• Summarize salient band-structure characteristics and the principles enabling material realizations of altermagnetic ordering.
• Discuss the impact of relativistic spin-orbit coupling, topology, and strong correlations on altermagnetic bands.
• Extend the discussion to non-collinear spin densities and the prediction of p-wave magnetic ordering.

Proposed method

• Describe the three-way spin-group symmetry classification for collinear magnets, highlighting the non-trivial spin Laue groups for altermagnets.
• Explain how even-parity l>0 magnetic order (d, g, i waves) arises from spin densities in crystals and is reflected in momentum-space band structures.
• Summarize how altermagnetism can be identified from symmetries of collinear spin densities and realized in insulating and conducting materials at normal conditions.
• Outline the impact of relativistic spin-orbit coupling on altermagnetic band structures and possible topological phenomena.
• Discuss computational (DFT) and spectroscopic approaches supporting material predictions and experimental verifications.

Figure 1: Altermagnets and ferromagnets. a, Nodal altermagnetic ordering in the crystal space (top) and in the momentum-space energy iso-surfaces/Fermi surfaces (bottom). The cartoons illustrate a d-wave ordering. Spin-degenerate nodes are highlighted by black lines in the 2D cartoons. Arrows and co

Experimental results

Research questions

• RQ1 What symmetry principles distinguish altermagnets from ferromagnets and antiferromagnets in collinear spin systems?
• RQ2 How can d-, g-, and i-wave altermagnetic order manifest in real materials and what are their band-structure signatures?
• RQ3 What roles do spin-density symmetries and crystal potentials play in enabling altermagnetic phases under ambient conditions?
• RQ4 How does spin-orbit coupling influence non-relativistic altermagnetic band structures and emergent topological phenomena?
• RQ5 Can the framework be extended to non-collinear spin densities to predict p-wave magnetic ordering?

Key findings

• Altermagnets form a distinct symmetry class with nodal spin-polarized band structures and zero net magnetization.
• Nodal altermagnetic ordering features even-parity-wave spin densities (d, g, i) with spin-degenerate nodes and opposite-spin iso-surfaces.
• Relativistic spin-orbit coupling can lift spin degeneracy on nodal surfaces and produce large spin-polarization magnitudes and Berry-curvature effects.
• Experimental signatures include anomalous Hall and Nernst effects, dichroism measurements, and spin-resolved ARPES confirming spin textures and nodal features.
• Material candidates hosting altermagnetic order have been identified by symmetry analyses (spin Laue groups) and DFT, with realizations in MnTe, Mn5Si3, CrSb, RuO2, MnTe, FeSe, and MnTe-based systems, among others.
• Extensions to non-collinear spin densities predict p-wave magnetism and broaden the landscape beyond strictly collinear altermagnetic order.

Figure 2: Nodal ordering in superfluids and magnets. Schematic comparison of the Cooper-pairing Fermi-liquid instabilities with p-wave and d-wave gap functions ( $\pm$ refers to the phase of the order parameter), their searched-for counterpart magnetic Pomeranchuk Fermi-liquid instabilities, and the