Polarization in inhomogeneous crystals and its relationship to quadrupole moments

The formulae for polarization and orbital magnetization in crystals are expressed in terms of the Bloch wavefunctions and they are closely related to Berry curvature. As a result, their properties as bulk quantities are well established. In contrast, definitions of higher-order electric/magnetic multipole moments in crystals as bulk quantities are still elusive. We revisit this problem by considering the polarization in inhomogeneous crystals. While it has been calculated by semiclassical theory in a previous work (Y. Zhang, Y. Gao, D. Xiao, Phys. Rev. B 104, 144203 (2021)), in the present work, we calculate it by linear response theory and compare it with previous works. Since such polarization is expected to be equal to a gradient of the quadrupole moment, we define the electric quadrupole moment from this result, and discuss its properties, including its relationship to polarizability. Orbital magnetic quadrupole moments are also discussed along the same line.