Geometrical description of fractional Chern insulators based on static structure factor calculations

We study the static structure factor of the fractional Chern insulator Laughlin-like state and provide analytical forms for this quantity in the long-distance limit. In the course of this we identify the averaged over the Brillouin zone Fubini-Study metric as the relevant metric in the long-distance limit. We discuss under which conditions the static structure factor will assume the usual behavior of a Laughlin-like fractional quantum Hall system, i.e., the scenario of S. M. Girvin, A. H. MacDonald, and P. M. Platzman [Phys. Rev. B 33, 2481 (1986)]. We study the influence of the departure of the averaged over the Brillouin zone Fubini-Study metric from its fractional quantum Hall value which appears in the long-distance analysis as an effective change of the filling factor. According to our exact-diagonalization results on the Haldane model and analytical considerations we find persistence of the fractional Chern insulator state even in this region of the parameter space.