Altermagnetism and Superconductivity: A Short Historical Review

Abstract

This article explores the deep interconnections among three seemingly unrelated concepts in condensed matter physics: electronic liquid crystal phases, multipole expansions, and altermagnetism. At the heart of these phenomena lies a shared foundation: spin-momentum locking in the nonrelativistic regime. Originally proposed in the context of electronic liquid crystal phases, spin-momentum locking was later elegantly incorporated into the formalism of multipole expansions. This framework can be further extended across multiple atomic sites, making it particularly effective for describing altermagnets, which feature localized magnetic moments distributed over at least two sublattices. In the second part of the article, we examine superconducting phenomena that stem from this shared mechanism, focusing on superconductivity in systems with spin-momentum locked Fermi surfaces. We highlight a rich variety of unconventional superconducting states, including finite-momentum pairing, $d$-wave and spin-triplet superconductivity, and topological Bogoliubov Fermi surfaces, among others. Additional related topics are addressed in the concluding section. Overall, this work offers both an accessible introduction to the newly identified magnetic order known as altermagnetism and a conceptual guide for researchers aiming to harness the ensuing unconventional superconductivity in the development of future quantum technologies.

Figures (8)

• Figure 1: A tale of three, wheresignificant progress is chronologically displayed and labelled by year. While the idea of electronic liquid crystal phase is yet to be materialized, the multipole basis analysis of a collinear antiferromagnetic order with sublattices Hayami2019 and the symmetry classification based on spin-group theory Smejkal2022-1 eventually lead to the discovery of altermagnetism.
• Figure 2: Fermi surfaces of different electronic liquid crystal phases. (a) charge nematic state, (b) itinerant FM state, (c) $l=1$ or $p$-wave $\alpha$ state, (d) $l=2$ or $d$-wave $\alpha$ state, which bears similarity to the $d$-wave AM, (e)-(h) various $\beta$ phases. The dashed line denotes the isotropic Fermi surfaces.
• Figure 3: Multipole basis up to $l=2$. The dashed line denotes the Fermi surfaces with vanishing multipoles. "NA" means not allowed by symmetry. It is useful to compare the various phases in this figure with the ones in Fig.\ref{['fig:Fig-ELC']}. In particular, the dispersion relation of the $\textbf{MT}$ quadrupole $T_{xy}$ should be contrasted with that of $F^{A}_{2}$$\alpha$-phase shown in Fig.\ref{['fig:Fig-ELC']}(d), both of which exhibit a similar feature as the $d$-wave AM.
• Figure 4: (a) Schematic representation of a collinear magnet, here the local magnetic moment $S_i$ is perpendicular to the paper plane ($xoy$ plane). (b) Spin-polarized band structure of collinear magnets. (c) Top: both $C^{(s)}_{2\perp}$ and $\mathcal{T}$ flips local magnetic moment $S_i$ in real space. Bottom: the combination of $C^{(s)}_{2\perp}$ and $\mathcal{T}$ functions as $\mathcal{P}$ for the energy spectrum. (d) $[C^{(s)}_{2\perp}||\mathcal{P}]$ works as $\mathcal{T}$ for the energy spectrum. (e) $[C^{(s)}_{2\perp}||\boldsymbol{\tau}]$ protects spin degeneracy.
• Figure 5: (a) A Ńeel state on an alternating anisotropic square lattice is a $d$-wave altermagnet. It is invariant under $[C^{(s)}_{2\perp}||C^+_{4z}]$ symmetry, where the red dot labels the rotation center. The shaded region is the paramagnetic unit cell. (b) Orbital interaction diagram for the two sublattices.