  Entanglement entropy (EE) is a quantitative measure of the effective degrees
of freedom and the correlation between the sub-systems of a physical system.
Using the replica trick, we can obtain the EE by evaluating the entanglement
Renyi entropy (ERE). The ERE is a $q$-analogue of the EE and expressed by the
$q$ replicated partition function. In the semi-classical approximation, it is
apparently easy to calculate the EE because the classical action represents the
partition function by the saddle point approximation and we do not need to
perform the path integral for the evaluation of the partition function. In
previous studies, it has been assumed that only the minimal-valued saddle point
contributes to the EE. In this paper, we propose that all the saddle points
contribute equally to the EE by dealing carefully with the semi-classical limit
and then the $q \to 1$ limit. For example, we numerically evaluate the ERE of
two disjoint intervals for the large $c$ Liouville field theory with $q \sim
1$. We exploit the BPZ equation with the four twist operators, whose solution
is given by the Heun function. We determine the ERE by tuning the behavior of
the Heun function such that it becomes consistent with the geometry of the
replica manifold. We find the same two saddle points as previous studies for $q
\sim 1$ in the above system. Then, we provide the ERE for the large but finite
$c$ and the $q \sim 1$ in case that all the saddle points contribute equally to
the ERE. Based on this work, it shall be of interest to reconsider EE in other
semi-classical physical systems with multiple saddle points.
