Fractional Chern insulators on finite cylinders and their bulk-edge correspondence*

It has been recently realized that strong interactions in topological Bloch bands give rise to the appearance of novel states of matter. Here we study these systems -- fractional Chern insulators -- via generalization of a gauge-fixed Wannier-Qi construction in the cylinder geometry. Our setup offers a number of important advantages compared to the earlier exact diagonalization studies on a torus. Most notably, it gives access to edge states and to a single-cut orbital entanglement spectrum, hence to the physics of bulk-edge correspondence. It is also readily implemented using density matrix renormalization group method which allows for numerical simulations of significantly larger systems. Previously [Z. Liu et al., Phys. Rev. B {\bf 88}, 081106(R) (2013)], this approach was applied to bosons on the ruby lattice model at filling $\nu=1/2$ and $\nu=1$, which show the signatures of (non)-Abelian phases, and we establish the correspondence between the physics of edge states and entanglement in the bulk. Here, we generalize this to other fillings such as $\nu=2/3$.