Entanglement entropy approach for examining quantum phase transition in the framework of semiclassical approximation: Testing its validity in Casten triangle

The entanglement entropy of s and d bosons in the framework of Interacting Boson Model -1 (IBM-1) has been obtained using consistent-Q formalism and semiclassical approximation. This has been possible by using Schmidt decomposition and expressing s and d bosons entanglement entropy in terms of Schmidt numbers. In this research, a simple method in the framework of IBM-1 has been presented for deriving the entanglement entropy in the Casten triangle. The results indicated that the entanglement entropy is sensitive to the shape-phase transition in the various regions of the Casten triangle. It was demonstrated that the entanglement entropy of s and d bosons in the semiclassical approximation depends only on the values of the deformation parameter (β) and is independent of the angular parameter (γ). Also, the entanglement entropy between s and d bosons reaches its maximum value in the $O\left(6\right)$ limit, while it decreases in the $SU\left(3\right)$ limit, and reaches zero in the $U\left(5\right)$ limit. Based on the results obtained via Schmidt decomposition, it is shown that the probability distribution functions of the number of s bosons in IBM-1 are the binomial distributions. For $N_B\gg 1$, it was proved that the distribution function in the $SU\left(3\right)$, $O\left(6\right)$ and $\overline{SU\left(3\right)}$ limits is the Gaussian, and in the $U\left(5\right)$ limit is the Poissonian.