Altermagnetism and Hall of fame: sorting out ordinary, anomalous and topological Hall effects

Hall effect (HE) refers to a transverse electric current arising in time-reversal symmetry broken solids subjected to an applied electric field. In old, and even many new papers, an empirical classification is being used, whereby the Hall conductivity (HC) is represented as a sum of two contributions, one proportional to the external magnetic field, B, dubbed "ordinary" (OHC), and one proportional to the sample magnetization, M , "anomalous" (AHC). It has been realized quite some time ago, though, that this classification is generally incorrect, and that OHC is not always proportional to B (if the electronic structure itself changes with the field) and AHC is not necessarily (unless in multidomain ferromagnets) proportional to M and may appear even if M = 0, e.g. in noncollinear compensated magnets and altermagnets. This original misleading empirical classification was further shattered with the discovery of the so-called "topological" HC (THC), but led to another, albeit a more subtle misconception that the total HC can always be written as a sum of three terms, one, OHC, ∝B, another, AHC, ∝M , and the third, THC, independent of both. In this talk I will explain the internal difference in the physical mechanisms of OHC, AHC, and THC on a microscopic level, relating it to which symmetries are broken and why. We particularly emphasize the recent theoretical and experimental developments regarding the "altermagnetic" AHE, arising without an external magnetic field, without net magnetization, and, in some cases, even without exchange-driven spin-splitting of electronic bands.