  Using the infinite density matrix renormalization group method on an infinite
cylinder geometry, we characterize the $1/3$ fractional Chern insulator state
in the Haldane honeycomb lattice model at $\nu=1/3$ filling of the lowest band
and check its stability. We investigate the chiral and topological properties
of this state through (i) its Hall conductivity, (ii) the topological
entanglement entropy, (iii) the $U(1)$ charge spectral flow of the many body
entanglement spectrum, and (iv) the charge of the anyons. In contrast to
numerical methods restricted to small finite sizes, the infinite cylinder
geometry allows us to access and characterize directly the metal to fractional
Chern insulator transition. We find indications it is first order and no
evidence of other competing phases. Since our approach does not rely on any
band or subspace projection, we are able to prove the stability of the
fractional state in the presence of interactions exceeding the band gap, as has
been suggested in the literature. As a by-product we discuss the signatures of
Chern insulators within this technique.
