Many-Body Invaraints for Electric Multipoles in Higher-order Topological Insulators

According to the modern theory of polarization, the electric polarization in solids is defined via Berry phase, which definition is fundamentally different from the classical definition of polarization involving charge distribution. In certain classes of crystalline topological insulators, the electric polarization is known to take quantized value due to underlying crystalline symmetries. Recently, the theory of multipole moments in crystalline insulators was developed where the quadrupole moment and higher-order multipole moments can be defined quantum mechanically. Although crytalline symmetries are useful they are not essential in understanding multiple moments, and so far the theory is only applicable to non-interacting crystalline band insulators. In this talk, I will introduce a many-body order parameter for quadrupole moment which provides a way to extend the quantum theory of multipole moments to the most general setting including interacting case. I will also discuss the bulk-boundary correspondence for quadrupole moment, relating seemingly unrelated quantities from the bulk and from the boundary.