Chern number in Ising models with spatially modulated real and complex fields

We study a one-dimensional quantum XY model with periodically modulated real and complex transverse fields. It is shown that both models can be mapped onto a pseudospin system in the k space with the aid of an extended Bogoliubov transformation. This allows us to introduce the geometric quantity, the Chern number, to identify the nature of quantum phases. Based on the exact solution, we find that the spatially modulated real and complex fields rearrange the phase boundaries from that of the ordinary Ising model, which can be characterized by the Chern numbers defined in the context of Dirac and biorthonormal inner products, respectively.