Magnetic dipole is a source of magnetism, and an electric charge and quadrupole are responsible for charge and orbital orderings, respectively. Such fundamental degrees of freedoms, described by electronic multipoles in general, play important roles in phase transitions and emergent responses under their orderings with broken symmetry. So far, multipoles are mainly discussed within single species of orbital at single site, in which only even-parity multipoles can be activated [1].
Recently, we generalize a concept of electronic multipoles in two ways: (i) a cluster multipole, which is defined in sublattice systems [2]; a particular alignment of magnetic/electric dipoles in a cluster can be regarded as an inter-atomic multipole, and (ii) a hybrid multipole, which is defined in multi-orbital systems. These extensions allow us to consider odd-parity multipoles, and electric/magnetic toroidal type of multipoles in addition to ordinary electric/magnetic multipoles, which are essential for cross-correlated phenomena such as magneto-electric, magneto-elastic responses, current-induced magnetization (Edelstein effect) and so on [3].
I will talk about the concept of augmented multipole, and peculiar electromagnetic responses under their spontaneous orderings [4].

A series of works has been done in collaboration with S. Hayami, Y. Yanagi, Y. Motome, M.-T. Suzuki, R. Arita, T. Nomoto, M. Yatsushiro, M. Naka, and H. Seo. These works were supported by JSPJ KAKENHI Grants Numbers 15K05176, 15H05885 (J-Physics), and 16H06590.

References:
[1] Y. Kuramoto, H. Kusunose, and A. Kiss, J. Phys. Soc. Jpn. 78, 072001 (2009), and references therein.
[2] S. Hayami, H. Kusunose, and Y. Motome, J. Phys.: Condens. Matter 28, 395601 (2016), and references therein.
[3] S. Hayami and H. Kusunose, J. Phys. Soc. Jpn. 87, 033709 (2018).
[4] S. Hayami, M. Yatsushiro, Y. Yanagi, and H. Kusunose, Phys. Rev. B (in press), and references therein.