Berry curvature multipoles and topological properties of altermagnets

Altermagnets are magnetically ordered states that are invariant under a combination of time-reversal symmetry and rotation. These combined symmetries lead not only to unique spectroscopic features and responses, like a nodal spin-splitting of the electronic structure and piezomagnetism, but also to constraints on the topological properties of altermagnets embedded in their Berry curvature. In this talk, I will present the features that characterize the Berry curvature of altermagnets by employing both phenomenology and microscopic modeling. In the absence of spin-orbit coupling, the Berry curvature has a multipolar character (such as quadrupole and hexadecapole), thus averaging to zero around the Brillouin zone, which in turn implies a vanishing anomalous Hall effect. In the presence of spin-orbit coupling, a net Berry curvature monopole appears only for certain moment directions, while in the so-called "pure" altermagnets at least one of the Berry curvature components acquires a quadrupolar character. I will show that such a Berry curvature quadrupole is manifested in an experimentally accessible quantity dubbed the elasto-Hall conductivity, by which the combination of uniaxial strain and electric field induces a transverse current, resulting in an anomalous Hall conductivity (AHC). I will discuss the different contributions to the AHC within the microscopic Lieb lattice model for altermagnetism, whose band structure displays spin-polarized Dirac cones that are gapped by the spin-orbit coupling. Finally, I will present other manifestations of the non-trivial topology of altermagnets, such as the Hall viscosity.