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Mutual interplay between the electronic degrees of freedom in solids, such ascharge, spin, orbital, sublattice, and bond degrees of freedom, is a source ofcross-correlated phenomena with unconventional electronic ordered states. Suchdegrees of freedom can be described by four types of multipoles (electric,magnetic, magnetic toroidal, and electric toroidal) in a unified way, whichenable us to tightly connect the microscopic degrees of freedom withmacroscopic physical responses in a transparent manner. We complete aclassification of the multipoles in all 122 magnetic point groups based on thegroup theory. The classification is useful to identify potentially activemultipoles not only in ordinary ferromagnetic and antiferromagnetic orderingsbut also in exotic orderings breaking time-reversal symmetry, e.g., aloop-current state. Moreover, the classification gives an insight into themicroscopic origin of the cross-correlated responses and quantum transports. Byanalyzing response functions up to the second order, we summarize theindispensable multipole moments for various responses, such as the linearmagnetoelectric, piezoelectric, and elastic responses, and the nonlinearconductivity and Nernst coefficient. Our results highly promote a furtherdiscovery of functional multiferroic materials, guided by the bottom-upmaterial design based on the symmetry-adapted multipoles.

Multipole classification in 122 magnetic point groups for unified understanding of multiferroic responses and transport phenomena