Static electric field in one-dimensional insulators without boundaries

Abstract

In this brief report, we show that in a one-dimensional insulating system with periodic boundary conditions, the coefficient of the θ term in the effective theory is not only determined by the topological index ∫i∑[superscript α]∈[superscript occ]〈u[superscript k]α|∂/∂k|u[superscript kα]〉dk. Specifically, the relative position between the electronic orbitals and the ions also alters the coefficient, as one would expect when one identifies −eθ/2π as the polarization. This resolves a paradox when we apply our previous result to the Su-Shreiffer-Heeger model, where the two ground states related by a lattice translation have θ differed by π. We also show that the static dielectric screening is the same with or without boundaries, in contrast to comments made in our previous paper.