  We perform an analytical study of the energy and entanglement spectrum of
non-interacting fermionic bilayer honeycomb lattices in the presence of
trigonal warping in the energy spectrum, on-site energy difference and uniform
magnetic field. Employing single particle correlation functions, we present an
explicit form for layer-layer entanglement Hamiltonian whose spectrum is
entanglement spectrum. We demonstrate that in the absence of trigonal warping,
at zero on-site energy difference exact correspondence is established between
entanglement spectrum and energy spectrum of monolayer which means that the
entanglement spectrum perfectly reflects the edge state properties of the
bilayer. We also show that trigonal warping breaks down such a perfect
correspondence, however, in $\Gamma$-K direction in hexagonal Brillouin zone,
their behaviors are remarkably the same for particular relevances of hopping
parameters. In the presence of an on-site energy difference the symmetry of
entanglement spectrum is broken with opening an indirect entanglement gap. We
also study the effects of a perpendicular magnetic field on both energy and the
entanglement spectrum of the bilayer in the presence of trigonal warping and
on-site energy difference. We demonstrate that the entanglement spectrum versus
magnetic flux has a self similar fractal structure, known Hofstadter butterfly.
Our results also show that the on-site energy difference causes a transition
from the Hofstadter butterfly to a tree-like picture.
