Inversion-symmetric topological insulators in cut-and-project binary chains
We investigate the electronic properties of binary tight-binding chains generated by the cut-and-project method for rational slopes $α=p/q$, leading to periodic and inversion symmetric chains with $n=p+q$ sites. The binary structure is encoded in two hopping amplitudes $t_a$ and $t_b$. For fixed $t_a \neq t_b$, the support of the energy spectrum as a function of $p/n$ gives rise to a "Cut-and-Project butterfly". We concentrate on insulators with $M$ filled bands among a total of $n$ bands and vary $t_a/t_b$. Inversion symmetry constrains the electric polarization $P$ to $0$ or $P_q/2$ modulo a polarization quantum $P_q = \gcd(M,n)/n$. A topological transition, between two insulators that differ by their quantized polarization, occurs if and only if $n/\gcd(M,n)$ is odd. When $n/\gcd(M,n)$ is even, the two insulating regimes have a vanishing polarization and no topological transition occurs, despite the gap closing at $t_a=t_b$. When $n$ is even and $M$ odd, we find an adiabatic path between $t_a>t_b$ and $t_a<t_b$ that maintains inversion symmetry and a gap.