  We systematically investigate the finite size scaling behavior of the R\'enyi
entanglement entropy (EE) of several representative 2d quantum many-body
systems between a subregion and its complement, with smooth boundaries as well
as boundaries with corners. In order to reveal the subleading correction, we
investigate the quantity ``subtracted EE" $S^s(l) = S(2l) - 2S(l)$ for each
model, which is designed to cancel out the leading perimeter law. We find that
$\mathbf{(1)}$ for a spin-1/2 model on a 2d square lattice whose ground state
is the Neel order, the coefficient of the logarithmic correction to the
perimeter law is consistent with the prediction based on the Goldstone modes;
$\mathbf{(2)}$ for the $(2+1)d$ O(3) Wilson-Fisher quantum critical point
(QCP), realized with the bilayer antiferromagnetic Heisenberg model, a
logarithmic subleading correction exists when there is sharp corner of the
subregion, but for subregion with a smooth boundary our data suggests the
absence of the logarithmic correction to the best of our efforts;
$\mathbf{(3)}$ for the $(2+1)d$ SU(2) J-Q$_2$ and J-Q$_3$ model for the
deconfined quantum critical point (DQCP), we find a logarithmic correction for
the EE even with smooth boundary.
