量子相变点附近的动力学行为及其半经典研究
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摘要
为刻画系统在其量子相变点附近的行为,除传统的方法之外,人们发现还可以借用量子信息领域中的相关概念,例如fidelity, Loschmidt echo (LE),纠缠以及量子失协等。具体一些而言,在许多模型的临界点附近,人们发现这些物理量有特殊的行为与性质,例如出现快速衰减、奇异性等,从而可以标志量子相变的发生。无论是fidelity还是Loschmidt echo,从解析的角度对其性质进行分析,都需涉及临界点附近的基态以及相关激发态之间的关系。从一般的角度对此进行分析,并非易事。例如,可以利用微扰论等方法来进行,也可以利用Kibble-Zurek机制来研究缓慢通过相变点的过程；这些方法都是针对特定类型的问题而适用。
在本篇论文的研究中,我们为量子相变点附近动力学行为的研究,提供另一种有一定普适性的、新的研究框架。为此,我们研究其临界点基态拥有无穷重简并的量子相变系统。我们的工作表明,在适当的条件下,半经典理论可适用于该种量子相变系统之临界点附近行为的研究。作为应用,在不同系统的临界点附近,利用半经典理论,我们给出了Loschmidt echo衰减行为的解析预言。具体而言,依据经典对应系统维度的不同,LE有两种不同的行为：若经典对应系统为一维规则系统,则LE有短时高斯衰减跟着长时的幂次衰减(1/t)；若经典对应系统为多维规则系统,则LE有指数衰减。
在研究量子相变的性质时,人们总会关心可能出现的标度行为,即表征系统特性的物理量在临界点附近是如何依赖于控制参数的。例如,在临界点附近,人们发现系统特征能量会表现为Ec～|λ—λc|Φ,其中λ。是临界点,Φ是临界指数。在通常的研究中,所关心的物理量常常只与一个控制参数有关,然而,对于fidelity以及Loschmidt echo而言,它们由两个控制参数λ和λ'所给出。在这种双控制参数情况下,临界点附近的标度行为会有何特点?2011年,Rams与Damski给出了Ising模型中fidelity的标度行为的表示式。我们研究了与Ising模型属于不同普适类的另一类模型,即有单玻色零模的量子相变系统。通过解析分析,我们发现该类系统中的fidelity具有一类新的参数标度行为。具体而言,这种标度行为不独立依赖于两个控制参数,而仅仅依赖于这两个控制参数的比值η,其中η=(λ—λc)/(λ'-λc)。利用半经典理论分析,对该类量子相变系统的时间做一个重标度后,我们在Loschmidt echo中也发现了该种标度行为。
在单模Dicke模型,横场1维Ising模型中,通过具体的解析与数值计算,我们验证了上述解析预言,即在临界点附近半经典理论的适用性、以及新发现的参数标度行为。除此之外,作为对临界点附近半经典理论研究的深化,我们还讨论了半经典理论预言在横场1维Ising模型量子临界点附近较深量子区域内的适用情况,并研究了半经典理论预言从适用到不适用的过程。在量子混沌模型如sawtooth模型中,我们也讨论了该预言的适用情况。在这些研究中,我们发现半经典理论有可能适用于较深量子区域的研究,尤其是在量子混沌模型的所谓FGR区域内。
Presently, there are many concepts and quantities in the field of the quantum in-formation, such as, fidelity, Loschmidt echo(LE), entanglement and quantum discord, which are used to analyze properties of the critical points in the quantum phase tran-sitions (QPT). In many models, researchers found that these concepts and quantities have special properties such as fast decay or singularity, which can characterize the ap-pearance of the quantum phase transition. However, it's hard to get the nature of these quantities from the view of the analytical analysis, because the analytical analysis needs the whole information of the ground state and excitation states with different parameter-s. Although there are some studies which use the perturbation theory or Kibble-Zurek mechanism to analyse some properties of fidelity or Loschmidt echo, they are limit-ed in one model or one class of models, and can't give the whole properties of these quantities.
In our research, we give an analytical method to study the dynamical behaviors in the vicinity of the critical points. In one class of QPT which has infinite degenerate of the ground states at the critical points, we prove the semi-classical theory can be used to analyse the dynamic property of the system in the nearby of the critical points, and discuss the condition for validity of this semi-classical theory near the critical points. These are rare in the previous researches. As an application, the prediction of the behav-iors of LE in different systems are given, too. Using the Van Vleck propagator which is obtained from the path integral semi-classical theory, one can easily obtain such LE behaviors. In the vicinity of the critical points, depending on the different dimension of the classic corresponding system, LE has two kinds of behaviors. Namely, in a one-dimensional classic corresponding regular system, LE has a Gaussian decay for initial times and a l/t decay for long time. When the dimension of the classic corresponding regular system is sufficiently large, the LE has the exponential decay.
At present, there are many studies focus on the scaling behaviors of the quantities near the critical points, and try to figure out how these quantities depend on their con-trol parameters. In the previous studies, the characteristic energy scale usually takes the form Ec～|λ-λc|φ in the vicinity of the critical points, with Ac indicating the critical point and φ a critical exponent. However, these quantities studied are only the function of one parameter. There are two control parameters A and (?) for fidelity and LE. In this situation, how these quantities depend on their double control parameters? In2011, Rams and Damski give the scaling behaviors of fidelity in the Ising model, which depending on their double control parameters independently. Through the ana-lytical analysis, in a single bosonic zero mode QPT, we found a special scaling behavior for fidelity. This new scaling behavior only depends on the ratio η of the parameters (?) and A', with η=(?)/(?). This scaling property is valid also for the time-dependent quantities such as the Loschmidt echo, provided time is measured in units of the inverse frequency of the critical mode.
Next, in the vicinity of the critical point of the single mode Dicke model and Ising model, we validate the predictions of the semi-classical theory for LE and the new found scaling behavior. These predictions fit well with the numerical calculation. In addition to this, we also discuss the validity of the semi-classical prediction in the deep quantum region of the Ising model and give the change from validity to breakdown. In one quantum chaotic model such as the sawtooth model, we discuss this validity too. In our study, we found the semi-classical predictions for the LE work well even in the deep quantum region, especially in the FGR regime of the quantum chaotic systems.
在本篇论文的研究中,我们为量子相变点附近动力学行为的研究,提供另一种有一定普适性的、新的研究框架。为此,我们研究其临界点基态拥有无穷重简并的量子相变系统。我们的工作表明,在适当的条件下,半经典理论可适用于该种量子相变系统之临界点附近行为的研究。作为应用,在不同系统的临界点附近,利用半经典理论,我们给出了Loschmidt echo衰减行为的解析预言。具体而言,依据经典对应系统维度的不同,LE有两种不同的行为：若经典对应系统为一维规则系统,则LE有短时高斯衰减跟着长时的幂次衰减(1/t)；若经典对应系统为多维规则系统,则LE有指数衰减。
在研究量子相变的性质时,人们总会关心可能出现的标度行为,即表征系统特性的物理量在临界点附近是如何依赖于控制参数的。例如,在临界点附近,人们发现系统特征能量会表现为Ec～|λ—λc|Φ,其中λ。是临界点,Φ是临界指数。在通常的研究中,所关心的物理量常常只与一个控制参数有关,然而,对于fidelity以及Loschmidt echo而言,它们由两个控制参数λ和λ'所给出。在这种双控制参数情况下,临界点附近的标度行为会有何特点?2011年,Rams与Damski给出了Ising模型中fidelity的标度行为的表示式。我们研究了与Ising模型属于不同普适类的另一类模型,即有单玻色零模的量子相变系统。通过解析分析,我们发现该类系统中的fidelity具有一类新的参数标度行为。具体而言,这种标度行为不独立依赖于两个控制参数,而仅仅依赖于这两个控制参数的比值η,其中η=(λ—λc)/(λ'-λc)。利用半经典理论分析,对该类量子相变系统的时间做一个重标度后,我们在Loschmidt echo中也发现了该种标度行为。
在单模Dicke模型,横场1维Ising模型中,通过具体的解析与数值计算,我们验证了上述解析预言,即在临界点附近半经典理论的适用性、以及新发现的参数标度行为。除此之外,作为对临界点附近半经典理论研究的深化,我们还讨论了半经典理论预言在横场1维Ising模型量子临界点附近较深量子区域内的适用情况,并研究了半经典理论预言从适用到不适用的过程。在量子混沌模型如sawtooth模型中,我们也讨论了该预言的适用情况。在这些研究中,我们发现半经典理论有可能适用于较深量子区域的研究,尤其是在量子混沌模型的所谓FGR区域内。
Presently, there are many concepts and quantities in the field of the quantum in-formation, such as, fidelity, Loschmidt echo(LE), entanglement and quantum discord, which are used to analyze properties of the critical points in the quantum phase tran-sitions (QPT). In many models, researchers found that these concepts and quantities have special properties such as fast decay or singularity, which can characterize the ap-pearance of the quantum phase transition. However, it's hard to get the nature of these quantities from the view of the analytical analysis, because the analytical analysis needs the whole information of the ground state and excitation states with different parameter-s. Although there are some studies which use the perturbation theory or Kibble-Zurek mechanism to analyse some properties of fidelity or Loschmidt echo, they are limit-ed in one model or one class of models, and can't give the whole properties of these quantities.
In our research, we give an analytical method to study the dynamical behaviors in the vicinity of the critical points. In one class of QPT which has infinite degenerate of the ground states at the critical points, we prove the semi-classical theory can be used to analyse the dynamic property of the system in the nearby of the critical points, and discuss the condition for validity of this semi-classical theory near the critical points. These are rare in the previous researches. As an application, the prediction of the behav-iors of LE in different systems are given, too. Using the Van Vleck propagator which is obtained from the path integral semi-classical theory, one can easily obtain such LE behaviors. In the vicinity of the critical points, depending on the different dimension of the classic corresponding system, LE has two kinds of behaviors. Namely, in a one-dimensional classic corresponding regular system, LE has a Gaussian decay for initial times and a l/t decay for long time. When the dimension of the classic corresponding regular system is sufficiently large, the LE has the exponential decay.
At present, there are many studies focus on the scaling behaviors of the quantities near the critical points, and try to figure out how these quantities depend on their con-trol parameters. In the previous studies, the characteristic energy scale usually takes the form Ec～|λ-λc|φ in the vicinity of the critical points, with Ac indicating the critical point and φ a critical exponent. However, these quantities studied are only the function of one parameter. There are two control parameters A and (?) for fidelity and LE. In this situation, how these quantities depend on their double control parameters? In2011, Rams and Damski give the scaling behaviors of fidelity in the Ising model, which depending on their double control parameters independently. Through the ana-lytical analysis, in a single bosonic zero mode QPT, we found a special scaling behavior for fidelity. This new scaling behavior only depends on the ratio η of the parameters (?) and A', with η=(?)/(?). This scaling property is valid also for the time-dependent quantities such as the Loschmidt echo, provided time is measured in units of the inverse frequency of the critical mode.
Next, in the vicinity of the critical point of the single mode Dicke model and Ising model, we validate the predictions of the semi-classical theory for LE and the new found scaling behavior. These predictions fit well with the numerical calculation. In addition to this, we also discuss the validity of the semi-classical prediction in the deep quantum region of the Ising model and give the change from validity to breakdown. In one quantum chaotic model such as the sawtooth model, we discuss this validity too. In our study, we found the semi-classical predictions for the LE work well even in the deep quantum region, especially in the FGR regime of the quantum chaotic systems.
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