  We investigate the von Neumann entanglement entropy and Schmidt gap in the
ground state of the Kitaev model on the honeycomb lattice for square and
cylindrical subsystems. We find that, for both the subsystems, the
free-fermionic contribution to the entanglement entropy $S_E$ exhibits
signatures of the phase transitions between the gapless and gapped phases.
However, we find that $S_E$ does not show an expected monotonic behaviour as a
function of the coupling $J_z$ between the suitably defined one-dimensional
chains within the gapless phase for either geometry; moreover the system
generically reaches a point of minimum entanglement within the gapless phase
before the entanglement saturates or increases again until the gapped phase is
reached. This may be attributed to the competition between the nearest
neighbour correlation functions of the $x-x$ and $z-z$ bonds, and to the onset
of gapless $\vec{k}$-modes in the bulk spectrum. On the other hand $S_E$ always
monotonically varies with $J_z$ in the gapped phase independent of the
subregion size or shape. Finally, further confirming the Li-Haldane conjecture,
we find that the Schmidt gap $\Delta$ defined from the entanglement spectrum
also signals the topological transitions but only if there are corresponding
zero energy Majorana edge states that simultaneously appear or disappear across
the transitions. We analytically corroborate some of our results on
entanglement entropy, Schmidt gap and the bulk-edge correspondence using
perturbation theory.
