Dynamical bifurcation as a semiclassical counterpart of a quantum phase transition

TL;DR

This paper demonstrates how dynamical bifurcations in semiclassical models correspond to quantum phase transitions in the Bose-Hubbard system, providing analytical and numerical insights into critical behavior and finite-size effects.

Contribution

It establishes a connection between semiclassical bifurcations and quantum phase transitions, deriving critical exponents through a large-population expansion approach.

Findings

Broken gaussianity at the phase transition

Exact critical exponents derived

Numerical evidence supports the scaling hypothesis

Abstract

We illustrate how dynamical 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 level, respectively. After showing that the gaussianity of the quantum ground states is broken at the phase transition, we evaluate finite populations effects introducing a suitable scaling hypothesis; we work out the exact value of the critical exponents and we provide numerical evidences 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.