Topological invariants to characterize universality of boundary charge in one-dimensional insulators beyond symmetry constraints

In the absence of any symmetry constraints we address universal properties of the boundary charge $Q_B$ for a wide class of nearest-neighbor tight-binding models in one dimension with one orbital per site but generic modulations of on-site potentials and hoppings. We provide a precise formulation of the bulk-boundary correspondence relating the boundary charge of a single band uniquely to the Zak phase evaluated in a particular gauge. We reveal the topological nature of $Q_B$ by proving the quantization of a topological index $eI=\mathrm{\Delta }{Q}_B-\overline{\rho }$, where $\mathrm{\Delta }{Q}_B$ is the change of $Q_B$ when shifting the lattice by one site towards a boundary and $\overline{\rho }$ is the average charge per site. For a single band we find this index to be given by the winding number of the fundamental phase difference of the Bloch wave function between the two lattice sites defining the boundary of a half-infinite system. For a given chemical potential we establish a central topological constraint $I\in \{-1,0\}$ related only to charge conservation of particles and holes. Our results are shown to be stable against disorder and we propose generalizations to multichannel and interacting systems.

Figure 1

Sketch of the model. $Z$ denotes the number of sites within a unit cell. The black bar indicates the left boundary such that the eigenstate of the infinite system has to vanish on the site left to the boundary. The index $m=1,2,\cdots$ labels the lattice sites starting from the boundary.

Figure 2

Illustration of the band structure and of the phase dependence of the edge states (blue color) connecting the bands for $Z=5$, $t=1$, $V=0.5$, $\delta t=0.1$, and $F_{\gamma }\left(\phi \right)$ defined via three random Fourier components for $\gamma =v,t$ (see Supplemental Material [47] for the concrete parameters). The chemical potentials ${\mu }_{\nu }$ in gap $\nu$ are indicated by dashed horizontal lines, for which $Q_B$ is calculated in Fig. 3. To the right we state the total numbers $M_{\pm }\left({\mu }_{\nu }\right)$ of edge states entering/leaving the system corresponding to the four chemical potentials ${\mu }_{\nu }$.

Figure 3

Boundary charge $Q_B$ and the invariant $I$ as a function of $\phi$ for a half-infinite system using the parameters of Fig. 2 for several ${\mu }_{\nu }$. We show $Q_B+\nu /2$ to offset the different curves. Up to a $2\pi /Z$-periodic function, $Q_B$ shows on average a linear slope with jumps at the positions where edge states move above/below ${\mu }_{\nu }$. As shown in the right inset, $Q_F$ or $Q_P$ alone do not show any linear behavior. The invariant is always quantized to $I\in \{-1,0\}$. The dashed line is the invariant with additional staggered on-site disorder drawn from a uniform distribution (0,0.05] for a very large finite system of $5\times {10}^5$ lattice sites. In the lowest panel we show the invariant $I_{\alpha }$ with $\alpha =2$ for a single band. It can only take the values $I_{\alpha }\in \{0,\pm 1\}$.