  We study the entanglement spectrum of a Chern insulator on a cylinder
geometry, with the cut separating the two partitions taken parallel to the
cylinder edge, at varying distances from the edge. In contrast to similar
studies on a torus, there is only one cut, and hence only one virtual edge mode
in the entanglement spectrum. The entanglement spectrum has a gap when the cut
is close enough to the physical edge of the cylinder such that the edge mode
spatially extends over the cut. This effect is suppressed for parameter choices
where the edge mode is sharply localized at the edge. In the extreme case of a
perfectly localized edge mode, the entanglement spectrum is gapless even if the
smaller partition consists of a single edge row. For the single-row cut, we
construct the corresponding entanglement Hamiltonian, which is a
one-dimensional tight-binding Hamiltonian with complex long-range hopping and
interesting properties. We also study and explain the effect of two different
schemes of flux insertion through a ring described by such an entanglement
Hamiltonian.
