量子信息中的度量空间方法在准周期系统中的应用

Application of metric space method in quantum information in quasi-periodic systems

得益于量子信息理论的发展,保真度、纠缠熵等概念被引入到量子相变的研究中,不仅能用来标识新奇的物质相,还能用来探测量子相变的临界点以及描绘其临界行为.从度量空间的角度来看,这些物理量都可以被理解为度量空间中两个函数的距离.本文利用波函数和实空间中密度分布函数的距离,研究了以广义Aubry-André-Harper模型为代表的准周期系统,发现该方法不仅能标识拓展相、临界相和局域相,还能找到准确的相变点并计算出临界指数.此外,不仅将度量空间方法推广到波包扩散动力学研究,还提出了一种新的量,即态密度分布函数的距离,发现上述定义的两种物理量都能标识不同的物质相及相变点.通过定义某个函数在不同参数下的距离,不仅为标识已知系统相变点提供了研究工具,还为探测未知系统的不同物质相、相变点及其临界行为提供了一种直观的方法.

Due to the rapid advancement of quantum information theory,some concepts such as fidelity and entanglement entropy have been introduced into the study of quantum phase transitions,which can be used not only to identify novel matter phases but also to detect the critical point and describe the critical behavior of the quantum phase transitions.From the point of view of the metric space,these physical quantities can be understood as the distance between the two functions in the metric space.In this work,we study a class of quasi-periodic system represented by the generalized Aubry-André-Harper(AAH)model,by using the distance between various wavefunctions or density distribution functions in real space.The generalized AAH model,an ideal platform to understand Anderson localization and other novel quantum phenomena,provides rich phase diagrams including extended,localized,even critical(multifractal)phases and can be realized in a variety of experimental platforms.In the standard AAH model,we find that the extended and localized phases can be identified.In addition,there exists a one-to-one correspondence between two distinct distances.We are able to precisely identify the critical point and compute the critical exponent by fitting the numerical results of different system sizes.In the off-diagonal AAH model,a complete phase diagram including extended phase,localized phase,and critical phase is obtained and the distance of critical phases is intermediate between the localized phase and extended phase.Meanwhile,we apply the metric space method to the wave packet propagation and discover that depending on the phase,the distance between wave functions or density functions exhibits varying dynamical evolution behavior,which is characterized by the exponent of the powerlaw relationship varying with time.Finally,the distance between the state density distribution functions is proposed,and it effectively identifies distinct matter phases and critical points.The critical phase which displays a multifractal structure,when compared with the other two phases,has the large state density distribution function distance.In a word,by defining the distances of a function under different parameters,we provide not only a physical quantity to identify familiar phase transitions but also an intuitive way to identify different matter phases of unknown systems,phase transition points,and their critical behaviors.