Bulk-boundary correspondence from the intercellular Zak phase

Abstract

The Zak phase $\ensuremath{\gamma}$, the generalization of the Berry phase to Bloch wave functions in solids, is often used to characterize inversion-symmetric one-dimensional (1D) topological insulators. Due to its dependence on the real-space origin and unit cell, however, there is an ambiguity in its use in a bulk-boundary correspondence. Here, we extract an origin-independent part of $\ensuremath{\gamma}$, the so-called intercellular Zak phase ${\ensuremath{\gamma}}^{\mathrm{inter}}$, and show that it is a bulk quantity that unambiguously predicts the number of surface modes. Specifically, a neutral finite 1D tight-binding system has ${n}_{s}={\ensuremath{\gamma}}^{\mathrm{inter}}/\ensuremath{\pi}$ (mod 2) in-gap surface modes below the Fermi level if there exists a commensurate inversion-symmetric bulk unit cell. We demonstrate this in two steps: First, we verify that $\ifmmode\pm\else\textpm\fi{}e{\ensuremath{\gamma}}^{\mathrm{inter}}/2\ensuremath{\pi}$ (mod $e)$ equals the extra charge accumulation in the surface region in a terminated system of a translationally invariant 1D insulator, while the remnant part of $\ensuremath{\gamma}$, the intracellular Zak phase ${\ensuremath{\gamma}}^{\mathrm{intra}}$, corresponds to the electronic part of the bulk's unit-cell dipole moment. Second, we show that the extra charge accumulation is related to the number of surface modes when the unit cell is inversion symmetric. We study several tight-binding models to quantitatively check both the relation between the extra charge accumulation and the intercellular Zak phase, and the bulk-boundary correspondence using the intercellular Zak phase.