The Husimi distribution is proposed for a phase space analysis of quantum phase transitions in the Dicke model of spin-boson interactions. We show that the inverse participation ratio and Wehrl entropy of the Husimi distribution give sharp signatures of the quantum phase transition. The analysis has been done using two frameworks: a numerical treatment and an analytical variational approximation. Additionally we have proposed a new characterization of the Dicke model quantum phase transition by means of the zeros of the Husimi distribution in the variational approach.

The problems arising when quantizing systems with periodic boundary conditions are analysed, in an algebraic (group-) quantization scheme, and the "failure" of the Ehrenfest theorem is clarified in terms of the already defined notion of good (and bad) operators. The analysis of "constrained" Heisenberg-Weyl groups according to this quantization scheme reveals the possibility for new quantum (fractional) numbers extending those allowed for Chern classes in traditional Geometric Quantization. This study is illustrated with the examples of the free particle on the circumference and the charged particle in a homogeneous magnetic field on the torus, both examples featuring "anomalous" operators, non-equivalent quantization and the latter, fractional quantum numbers. These provide the rationale behind flux quantization in superconducting rings and Fractional Quantum Hall Effect, respectively.

We propose coherent ('Schrödinger 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 ξc 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)max.(ξ) = 1−1/(N + 1)] and S (N) cat (ξ) ≃ 1 2 log 2 (N +1), respectively, above the critical point, ξ > ξc, 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 ξ, are given in terms of hypergeometric functions.

The Husimi distribution is proposed for a phase space analysis of quantum phase transitions in the two-dimensional U (3) vibron model for N -size molecules. We show that the inverse participation ratio and Wehrl's entropy of the Husimi distribution give sharp signatures of the quantum (shape) phase transition from linear to bent. Numerical results are complemented with a variational approach using parity-symmetry-adapted U (3) coherent states, which reach the minimum Wehrl entropy, in the rigidly linear phase, according to a generalized Wehrl-Lieb conjecture. We also propose a characterization of the vibron-model quantum phase transition by means of the zeros of the Husimi distribution.

We present a phase-space study of first-, second-and third-order quantum phase transitions in the Lipkin-Meshkov-Glick model by means of the Husimi function. By analyzing the distribution of zeros of the ground state Husimi function we have characterized each phase and each type of quantum phase transition in this model. We show that Rényi-Wehrl entropies of the ground state Husimi function give a good description of quantum phase transitions. The study has been done using a numerical treatment and a variational approximation in terms of coherent states. Additionally, we have analyzed quantum phase transitions using the fidelity and fidelity susceptibility concepts.