Study On Localized Transition In The Spin-boson Model By Variational Calculation

Spin-boson model(or the dissipative two-state system) is an important toy model in quantum dissipative systems. One important issue in this model is the phonon-induced localization, which is considered as some kind of boundary phase transition in quantum phase transition. The decoherence mechanism of the two state system due to the boson bath has important basic applications in quantum information theory and has attracted a lot of interest i n recent years. In this thesis, we present variational calculations based on both the displaced-oscillator state(DOS) and the displaced-squeezed-state(DSS) to study the phonon-induced localization. The self-consistent equations are derived and the evolution of the solutions near the critical point is analyzed by using Landau theory. With the help of Landau theory, the physics picture of the phonon-induced localization is constructed and the correct way to determined the true critical point for a discontinuous transition in sub-Ohmic case is demonstrated. We also present a mathematical analysis on the self-consistent equations obtained from both kinds of variational scheme. The geometric version of the evolution of the solutions near the critical point is demonstrated, from which the boundary of phonon-induced transition is found. The dependence of the transition behavior on the parameter s defined in the spectrum function of the bath is also elucidated. We also present the variational calculation based on displaced-squeezed state with double parameters in the appendix.