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

We propose coherent ('Schrodinger cat-like') 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 in vibration-rotation entanglement linear (L) and von Neumann (S) entropies from zero to L-cat((N))(xi) similar or equal to 1-2/root pi N (to be compared with L-max((N))(xi) = 1-1/(N + 1)) and S-cat((N))(xi) similar or equal to 12 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.