Exploring dynamical phase transitions of spin-boson models with Krylov complexity

We report numerical and theoretical analysis of Krylov complexity in the dynamics of Dicke model, a paradigmatic quantum system that describes a collective spin coupled to a bosonic mode. Recently the Krylov complexity has been shown to provide valuable insights about quantum dynamics. For example, it serves as a measure to characterize operator growth and to probe phenomena such as chaos and scrambling. We extend the Krylov analysis to the quench dynamics of the Dicke model, which is known to exhibit rich behaviors such as integrability, chaos, and saddle point scrambling across different parameter regimes. We systematically investigate Krylov complexity in different dynamical phases and demonstrate that it can be used to probe dynamical phase transitions. We further compare our results with integrable collective spin models studied previously to shed light on the role of the bosonic degree of freedom in the complexity measure.