  The multipole moment is an established concept of electrons in solids.
Entanglement of spin, orbital, and sublattice degrees of freedom is described
by the multipole moment, and spontaneous multipole order is a ubiquitous
phenomenon in strongly correlated electron systems. In this paper, we present
group-theoretical classification theory of multipole order in solids.
Intriguing duality between the real space and momentum space properties is
revealed for odd-parity multipole order which spontaneously breaks inversion
symmetry. Electromagnetic responses in odd-parity multipole states are
clarified on the basis of the classification theory. A direct relation between
the multipole moment and the magnetoelectric effect, Edelstein effect,
magnetopiezoelectric effect, and dichromatic electron transport is
demonstrated. More than 110 odd-parity magnetic multipole materials are
identified by the group-theoretical analysis. Combining the list of materials
with the classification tables of multipole order, we predict emergent
responses of the candidate materials.
