  We study the universal scaling behavior of the entanglement entropy of
critical theories in $2+1$ dimensions. We specially consider two fermionic
scale-invariant models, free massless Dirac fermions and a model of fermions
with quadratic band touching, and numerically study the two-cylinder
entanglement entropy of the models on the torus. We find that in both cases the
entanglement entropy satisfies the area law and has the subleading term which
is a scaling function of the aspect ratios of the cylindrical regions. We test
the scaling of entanglement in both the free fermion models using three
possible scaling functions for the subleading term derived from a) the
quasi-one-dimensional conformal field theory, b) the bosonic quantum Lifshitz
model, and c) the holographic AdS/CFT correspondence. For the later case we
construct an analytic scaling function using holography, appropriate for
critical theories with a gravitational dual description. We find that the
subleading term in the fermionic models is well described, for a range of
aspect ratios, by the scaling form derived from the quantum Lifshitz model as
well as that derived using the AdS/CFT correspondence (in this case only for
the Dirac model). For the case where the fermionic models are placed on a
square torus we find the fit to the different scaling forms is in agreement to
surprisingly high precision.
