Electric polarization as a nonquantized topological response and boundary Luttinger theorem

TL;DR

This paper introduces a nonperturbative framework for understanding electric polarization in crystalline insulators, linking it to magnetic monopoles and boundary phenomena, applicable even to strongly interacting systems.

Contribution

It develops a novel approach to bulk polarization based on flux responses and monopole properties, extending to boundary theorems and strongly interacting systems.

Findings

Polarization related to magnetic monopoles and translation symmetries.

Boundary Luttinger theorem is modified by bulk polarization.

Applicable to strongly interacting insulators.

Abstract

We develop a nonperturbative approach to the bulk polarization of crystalline electric insulators in $d\geq1$ 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 condition, 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.