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Entanglement entropy (EE) is a quantitative measure of the effective degreesof freedom and the correlation between the sub-systems of a physical system.Using the replica trick, we can obtain the EE by evaluating the entanglementRenyi 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 isapparently easy to calculate the EE because the classical action represents thepartition function by the saddle point approximation and we do not need toperform the path integral for the evaluation of the partition function. Inprevious studies, it has been assumed that only the minimal-valued saddle pointcontributes to the EE. In this paper, we propose that all the saddle pointscontribute equally to the EE by dealing carefully with the semi-classical limitand then the $q \to 1$ limit. For example, we numerically evaluate the ERE oftwo disjoint intervals for the large $c$ Liouville field theory with $q \sim1$. We exploit the BPZ equation with the four twist operators, whose solutionis given by the Heun function. We determine the ERE by tuning the behavior ofthe Heun function such that it becomes consistent with the geometry of thereplica 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 tothe ERE. Based on this work, it shall be of interest to reconsider EE in othersemi-classical physical systems with multiple saddle points.

Entanglement Renyi entropy of two disjoint intervals for large $c$ Liouville field theory