Anomalous-Hall Neel textures in altermagnetic materials

Recently, the altermagnets, a new kind of collinear antiferromagnet with nearly zero net magnetization and momentum-dependent spin-splitting of bands, have sparked great interest. Despite simple magnetic structures, these altermagnets exhibit intriguing and intricate dependence of anomalous Hall effect (AHE) on the N\'eel vector, in contrast to the conventional perpendicular configuration of Hall current with magnetization in ferromagnets. However, the fundamental relationship between the AHE and the N\'eel vector remains largely elusive. Here, we reveal all the unconventional anomalous Hall textures in the N\'eel vector space, dubbed anomalous-Hall N\'eel textures (AHNTs) for altermagnets. Specifically, we identify 10 types across four categories of AHNTs for all altermagnets. Notably, we find that AHNTs resemble the known spin textures in momentum space, and further reveal their symmetry origin. Meanwhile, we examine our key discoveries in prototypical altermagnets. Our work offers a thorough understanding of AHE in altermagnets and a complete and pictorial classification of altermagnets based on the geometry of response functions.