Topological properties of a one-dimensional excitonic model combining local excitation and charge transfer

We have computed the Zak phase for a one-dimensional excitonic model, which takes into account dimerisation, local and charge-transfer excited states. There are four hopping parameters, which can be varied to give rise to a rich spectrum of physics. By turning on more than one parameters, we can find (i) the topological phase could be $\pi$ even for a uniform chain, which is related to topological order, (ii) there exist topologically nontrivial flat bands, suggesting an interesting correlation between flat bands and topology, (iii) exotic fractional phases, which are due to quantum interference and relevant to anyon and fractional statistics, and (iv) a phase transition related to second-order hopping event - excitonic hopping. We have also developed the concept of effective chiral states (linear combination of excitonic states) to interpret our calculations. Our model is sufficiently general to describe excitonic topological properties for one-dimensional chain structures formed by physical unit such as atom, molecule, semiconductor dopant, and quantum dot.