The Study Of Quantum Phase Transition In The Spin-Boson Model

In recent years,with the booming development of quantum information and quantum computing,the dynamics of quantum open systems has attracted much attention.Due to the existence of the environment,quantum systems inevitably interact with the environment.Quantum dissipation phenomena induced by the environment play a fundamental role in many fields.For example,it has become a technical bottleneck to improve the coherence time of qubit in quantum computing.In the photosynthetic reaction center system,In light-harvesting protein complexes,the excitation energy transport process is also modulated by the environment.To study the quantum dissipation phenomenon,many quantum dynamical methods have been developed.Among various quantum dynamics methods,the hierarchical equations of motion(HEOM)is a numerically exact method,which can describe the quantum system interacting with the environment without any approximation.It has become a mainstream method in open quantum system.The original HEOM is hard to deal with the bath at low temperature and with slow decaying time correlation function,e.g.power-law decaying.It limits the application of HEOM.In order to overcome these difficulties,we expand the time correlation function over a orthonormal complete basis,we reconstruct the dynamical complete basis including the reduced density matrix and other auxiliary density operators,so we can deal with arbitrary bath even at zero temperature theoretically.In this thesis,we focuses on the quantum phase transition problem in zero-temperature spin-boson model.Based on numerical exact calculation of extended HEOM,we obtain the phase diagram of coherent-incoherent dynamical transition and delocalized-localized phase transition.This paper includes two dynamical studies of spin-boson model at zero temperature.On the one hand,the dynamical scaling behavior of the mean magnetic moment is studied.On the other hand,the linear response function and linear absorption spectrum are calculated to study the different delocalized-localized phase transition between deep sub-Ohmic and Ohmic spin-boson model.The thesis is organized as follows:(1)In Chapter One,we introduce the background of quantum phase transition in the open quantum system and give the outline of this thesis.(2)In Chapter Two,the background of open quantum system is given.We introduce some important physical concepts including reduced density matrix,quantum master equation,spin-boson model and quantum phase transition.(3)In Chapter Three,we focus on the theory of hierarchical equations of motion.Firstly,we review the history of HEOM and its recent development.They we introduce the decomposition of bath time correction function.By computing the time derivative over the n-th order auxiliary density operator,we derive the complete form of the extended HEOM.(4)In Chapter Four,we study the dynamical scaling phenomenon in the zero temperature spin-boson model.A characteristic time is defined accordingly as the inverse of the zeroth-order moment of the rate kernel.For a given Kondo parameter in the incoherent regime,the time evolution of average magnetic moments gradually collapses onto a master curve after rescaling the time variable with the characteristic time.The rescaled spin dynamics is nearly independent of the cutoff frequency and the form of cutoff functions.For a given cutoff frequency,the inverse of the characteristic time is fitted excellently by a nonlinear function of the renormalized tunneling amplitude.Despite a significant difference in definition,our result is in good agreement with the characteristic time of the noninteracting blip approximation.(5)In Chapter Five,we calculate the linear response function through the extended HEOM.Furthermore,we obtain the linear absorption spectrum from the Fourier transformation.From a series of numerical calculation over different bath component and Kondo parameter,we obtain the phase diagram of the delocalized-localized phase transition and coherent-incoherent dynamical transition.We also provide a energy level picture to under the different phase transition behavior between sub-Omic and Ohmic bath.(6)In Chapter Six,we review the main conclusions and give the plan in the following study.