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 (ELC) phases, multipole expansions, and altermagnetism (AM). At the heart of these phenomena lies a shared foundation: spin-momentum locking in the nonrelativistic regime. Originally proposed in the context of ELC phases, nonrelativistic spin-momentum locking (NRSML) 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 AMs, which host localized magnetic moments with anisotropic magnetization densities distributed over sublattices. In the second part, we examine superconducting (SC) phenomena associated with AM from three complementary perspectives. First, we investigate SC associated with NRSML Fermi surfaces (FSs), the unifying theme of the first part, highlighting a rich variety of unconventional SC states. These include finite-momentum pairing,-wave and spin-triplet SC, and topological Bogoliubov FSs, among others. We then review SC emerging from either static AM order or AM fluctuations. Finally, we discuss the possible competition and intertwining between AM order and SC, 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 AM and a conceptual guide for researchers aiming to harness the ensuing unconventional SC in the development of future quantum technologies.