Toroid, Altermagnetic and Noncentrosymmetric ordering in metals
V. P. Mineev
TL;DR
The work analyzes how toroidal, altermagnetic, and noncentrosymmetric symmetries shape normal and superconducting properties in metals. It develops a symmetry-guided framework for electron spectra, Kramers degeneracy, and transport, deriving equilibrium currents, magnetoelectric couplings, and Berry-curvature–driven responses; it also treats superconductivity via mixed singlet–triplet pairing for toroid metals and interband pairing in altermagnets, highlighting potential gapless states and gradient terms. The findings show that toroid metals host equilibrium currents and current-induced magnetization, noncentrosymmetric metals exhibit magnetoelectric effects and anomalous Hall responses, and altermagnets present unique interband pairing tendencies and piezomagnetic Hall effects, with superconductivity in altermagnets remaining a delicate, possibly suppressed, possibility. Overall, the paper provides a unified, symmetry-based description of exotic transport and superconducting phenomena in metals with toroidal, altermagnetic, and noncentrosymmetric order, guiding material discovery and interpretation.
Abstract
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.