Altermagnetism and superconductivity: A short historical review

This review is organized into three parts. In the first part, we explore 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, nonrelativistic 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 host localized magnetic moments with anisotropic magnetization densities distributed over sublattices. In the second part, we examine superconducting phenomena associated with altermagnetism from three complementary perspectives. First, we investigate superconductivity associated with nonrelativistic spin-momentum locked Fermi surfaces, the unifying theme of the first part, highlighting a rich variety of unconventional superconducting states. These include finite-momentum pairing, $d$-wave and spin-triplet superconductivity, and topological Bogoliubov Fermi surfaces, among others. We then review superconductivity emerging from either static altermagnetic order or altermagnetic fluctuations. Finally, we discuss the possible competition and intertwining between altermagnetic order and superconductivity, illustrated using the repulsive Hubbard model. Additional related topics are addressed in the concluding part. 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.