Symmetry Classification of Antiferromagnets with Four Types of Multipoles

A plethora of antiferromagnetic structures have been so far found in condensed matter physics, where the antiferromagnetic phase transition is characterized by symmetry lowering under the magnetic point group. Depending on the types of symmetry lowering, various cross-correlation phenomena, such as the anomalous Hall effect, magneto-electric effect, and magneto-piezoelectric effect, emerge below the critical temperature. We revisit a close relationship between the symmetry of the antiferromagnetic structures and cross-correlations based on the augmented multipoles consisting of electric, magnetic, magnetic toroidal, and electric toroidal multipoles with different spatial inversion and time-reversal parities. The symmetry classification will be useful for further exploration of functional antiferromagnetic materials.