Toroid, Altermagnetic and Noncentrosymmetric ordering in metals

This article is dedicated to the 60-th anniversary of the Landau Institute for Theoretical Physics and presents a review of normal and superconducting properties of toroidal, altermagnetic, and noncentrosymmetric metals. Metals with toroidal order are compounds not possessing symmetry in respect of space and time inversion but are symmetric in respect of the product of these operations. An electric current propagating through samples of such a material causes its magnetisation. Superconducting states in toroidal metals are a mixture of singlet and triplet states. Superconductivity is gapless even in ideal crystals without impurities. Altermagnets are antiferromagnetic metals that have a specific spin splitting of electron bands determined by time inversion in combinations with rotations and reflections of a crystal lattice. Similar splitting takes place in metals whose symmetry does not have a spatial inversion operation. Both of these types of materials have an anomalous Hall effect. A current propagating through a noncentrosymmetric metal causes magnetization, but this is not the case in altermagnets. On the other hand, in altermagnets, there is a specific piezomagnetic Hall effect. Superconducting pairing in non-centrosymmetric metals occurs between electrons occupying states in one zone, whereas, in altermagnets, we are dealing with interband pairing, which is unfavorable for the formation of a superconducting state.