Dynamical bifurcation as a semiclassical counterpart of a quantum phase transition

We illustrate howdynamical transitions in nonlinear semiclassical models can be recognized as phase transitions in the corresponding-inherently linear-quantum model, where, in a statistical-mechanics framework, the thermodynamic limit is realized by letting the particle population go to infinity at fixed size. We focus on lattice bosons described by the Bose-Hubbard (BH) model and discrete self-trapping (DST) equations at the quantum and semiclassical levels, respectively. After showing that the Gaussianity of the quantum ground states is broken at the phase transition, we evaluate finite-population effects by introducing a suitable scaling hypothesis; we work out the exact value of the critical exponents and provide numerical evidence confirming our hypothesis. Our analytical results rely on a general scheme obtained from a large-population expansion of the eigenvalue equation of the BH model. In this approach the DST equations resurface as solutions of the zeroth-order problem.