Parity-symmetry-adapted coherent states and entanglement in quantum phase transitions of vibron models

We propose coherent (`Schr\"odinger catlike') states adapted to the parity symmetry providing a remarkable variational description of the ground and first excited states of vibron models for finite-($N$)-size molecules. Vibron models undergo a quantum shape phase transition (from linear to bent) at a critical value $\xi_c$ of a control parameter. These trial cat states reveal a sudden increase of vibration-rotation entanglement linear ($L$) and von Neumann ($S$) entropies from zero to $L^{(N)}_{\rm cat}(\xi)\simeq 1-{2}/{\sqrt{\pi N}}$ [to be compared with $L^{(N)}_{\rm max.}(\xi)=1-{1}/{(N+1)}$] and $S^{(N)}_{\rm cat}(\xi)\simeq \frac{1}{2} \log_2(N+1)$, respectively, above the critical point, $\xi>\xi_c$, in agreement with exact numerical calculations. We also compute inverse participation ratios, for which these cat states capture a sudden delocalization of the ground state wave packet across the critical point. Analytic expressions for entanglement entropies and inverse participation ratios of variational states, as functions of $N$ and $\xi$, are given in terms of hypergeometric functions.