Electric polarization as a nonquantized topological response and boundary Luttinger theorem

We develop a nonperturbative approach to the bulk polarization of crystalline electric insulators in d >= 1 dimensions. Formally, we define polarization via the response to background fluxes of both charge and lattice translation symmetries. In this approach, the bulk polarization is related to properties of magnetic monopoles under translation symmetries. Specifically, in 2D, the monopole is a source of 2 pi flux, and the polarization is determined by the crystal momentum of the 2 pi flux. In 3D, the polarization is determined by the projective representation of translation symmetries on Dirac monopoles. Our approach also leads to a concrete scheme to calculate polarization in 2D, which in principle can be applied even to strongly interacting systems. For open boundary conditions, the bulk polarization leads to an altered boundary Luttinger theorem (constraining the Fermi surface of surface states) and also to modified Lieb-Schultz-Mattis theorems on the boundary, which we derive.

Automated summary

A nonperturbative approach to bulk polarization of crystalline electric insulators in different dimensions is developed, defining polarization through response to background fluxes of charge and lattice translation symmetries. The relationship between bulk polarization and magnetic monopoles is explored, leading to a concrete scheme for calculating 2D polarization. The study also yields insights into the altered boundary Luttinger theorem for open boundary conditions and modified Lieb-Schultz-Mattis theorems on the boundary.