Static Electric Field in a 1D Systems without Boundaries

In this brief report, we show that in a 1D system with unit-cell doubling, the coefficient of the $θ$-term is not only determined the topological index, $\int i\bra{u_k}\frac{\d}{\d k}\ket{u_k}{\rm d}k$. Specifically, the relative position between the electronic orbitals and the ions also alters the coefficient. 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, on the contrary to what we have commented in our previous paper.

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