张量网络算法及其在多体动力学中的研究
发布时间：2018-11-18 20:18
【摘要】：强关联多体物理是凝聚态物理领域的核心课题,它与很多我们关心的物理体系,比如高温超导体系,自旋液体体系密切相关。然而无论在解析上还是数值上,这些强关联系统都很难研究,原因是其希尔伯特空间的维数随着系统粒子数的增长而指数增加。然而最近20年来,藉由人们对于多体量子系统中最重要的性质—纠缠和关联的深入理解,及其带来的密度矩阵重整化群(Density Matrix Renormalization Group),张量网络态(Tensor Network States),含时变分原理(Time Dependent Variational Principle)等先进的数值工具的快速发展,我们终于可以研究多体物理里最为人们关注的基态,低能激发态以及动力学演化等等问题。本篇论文中,从发展张量网络算法,到应用张量网络算法研究量子多体动力学,主要完成了以下三方面的工作:1.基于相对熵的几何度量,运用并行蒙特卡洛算法,发展了一套统一计算所有关联的数值方法。为了研究各种不同的量子系统,人们定义了多种不同形式的纠缠和关联。尽管这些定义都在特定问题中发挥着重要的作用,人们仍然希望能够用统一的方式度量所有的关联,并对它们的大小进行比较。从这个问题入手,我们借助相对熵的几何度量,将所有的关联度量都转化为特定希尔伯特空间内的极值问题,并基于并行蒙特卡洛算法,构建了一套可以高效求解各种关联的相对熵的数值方法,弥补了相对熵难以解析计算的缺点,同时推进了关联定义的统一化。2.将尺寸一致性(Size Consistency)和面积定律(Area Law)进行结合,共同构建一套描述张量网络性质的方法。对于一个能隙不为零的量子系统,取其基态的一部分为A,则A与剩余部分的纠缠大小由A边界的测度决定—这条被称为面积定律(Area Law)的规则,被认为描述了DMRG在一维成功的关键,并被作为构建合理张量网络的指导原则。但在实践中我们意识到,面积定律不是描述张量网络的唯一标准,它只关心了张量网络的纠缠性质,但忽略了其能量可加性。我们将尺寸一致性与面积定律相结合,构建了一套更全面的描述张量网络性质的准则。3.利用矩阵直积态(Matrix Product State)以及含时变分原理,研究长程关联下的Kibble-Zurek动力学机制。近年来,一方面离子阱实验技术的突飞猛进,使得在实验上对量子多体系统的的非平衡动力学研究成为可能;另一方面,通过选取合适的张量网络,再运用含时变分原理,我们可以精确的模拟一维粒子数为100的格点系统长达500个单位时间的实时演化。以这两者为基础,我们研究了量子多体系统,在长程关联情况下的Kibble-Zurek动力学机制,为量子区间的KZ机制提供了新的方向和观点。
[Abstract]:Strong correlation multibody physics is a core subject in condensed matter physics. It is closely related to many physical systems that we are concerned about, such as high temperature superconducting systems and spin liquid systems. However these strongly correlated systems are difficult to study both analytically and numerically because the dimension of Hilbert space increases exponentially with the increase of the number of particles in the system. However, in the last 20 years, through the deep understanding of the most important properties in the multi-volume subsystem, entanglement and correlation, and the density matrix renormalization group (Density Matrix Renormalization Group), Zhang Liang network state (Tensor Network States), With the rapid development of advanced numerical tools such as time-varying fractional principle (Time Dependent Variational Principle), we can finally study the ground state, low-energy excited state and dynamic evolution of multi-body physics. In this thesis, from the development of Zhang Liang network algorithm to the study of quantum multi-body dynamics by the algorithm of Zhang Liang network, the following three aspects of work have been accomplished: 1. Based on the geometric metric of relative entropy and using the parallel Monte Carlo algorithm, a unified numerical method for computing the correlation is developed. In order to study various quantum systems, many different forms of entanglement and correlation have been defined. Although these definitions play an important role in specific problems, people still hope to be able to measure all associations in a unified way and compare their size. From this point of view, we use the geometric metric of relative entropy to transform all the correlation metrics into extremum problems in a particular Hilbert space, and based on the parallel Monte Carlo algorithm, A set of numerical methods which can efficiently solve the relative entropy of various correlations are constructed, which make up for the disadvantage of the relative entropy which is difficult to calculate analytically, and promote the unification of the definition of correlation. 2. A method of describing Zhang Liang network properties is constructed by combining the size consistent (Size Consistency) with the area law (Area Law). For a quantum system whose energy gap is not zero, if a part of the ground state is A, the entanglement between A and the rest is determined by the measure of the boundary of A, which is called the rule of the law of area (Area Law). It is considered to describe the key to the success of DMRG in one-dimensional and as a guiding principle for the construction of a reasonable Zhang Liang network. However, in practice, we realize that the area law is not the only standard for describing Zhang Liang's network. It only concerns about the entanglement property of Zhang Liang network, but neglects its energy additivity. We combine the size consistency with the area law, and construct a more comprehensive description of Zhang Liang network properties. 3. The dynamic mechanism of Kibble-Zurek with long range correlation is studied by using matrix direct product state (Matrix Product State) and time-varying fractional principle. In recent years, with the rapid development of ion trap experimental technology, it is possible to study the non-equilibrium dynamics of quantum multi-body system experimentally. On the other hand, by selecting the appropriate Zhang Liang network and applying the time-varying principle, we can accurately simulate the real-time evolution of one-dimensional lattice system with 100 particles per unit time up to 500 units. Based on these two methods, we study the Kibble-Zurek dynamical mechanism of quantum multibody systems under the condition of long range correlation, which provides a new direction and viewpoint for the quantum interval KZ mechanism.
【学位授予单位】：中国科学技术大学
【学位级别】：博士
【学位授予年份】：2017
【分类号】：O413
本文编号：2341097
[Abstract]:Strong correlation multibody physics is a core subject in condensed matter physics. It is closely related to many physical systems that we are concerned about, such as high temperature superconducting systems and spin liquid systems. However these strongly correlated systems are difficult to study both analytically and numerically because the dimension of Hilbert space increases exponentially with the increase of the number of particles in the system. However, in the last 20 years, through the deep understanding of the most important properties in the multi-volume subsystem, entanglement and correlation, and the density matrix renormalization group (Density Matrix Renormalization Group), Zhang Liang network state (Tensor Network States), With the rapid development of advanced numerical tools such as time-varying fractional principle (Time Dependent Variational Principle), we can finally study the ground state, low-energy excited state and dynamic evolution of multi-body physics. In this thesis, from the development of Zhang Liang network algorithm to the study of quantum multi-body dynamics by the algorithm of Zhang Liang network, the following three aspects of work have been accomplished: 1. Based on the geometric metric of relative entropy and using the parallel Monte Carlo algorithm, a unified numerical method for computing the correlation is developed. In order to study various quantum systems, many different forms of entanglement and correlation have been defined. Although these definitions play an important role in specific problems, people still hope to be able to measure all associations in a unified way and compare their size. From this point of view, we use the geometric metric of relative entropy to transform all the correlation metrics into extremum problems in a particular Hilbert space, and based on the parallel Monte Carlo algorithm, A set of numerical methods which can efficiently solve the relative entropy of various correlations are constructed, which make up for the disadvantage of the relative entropy which is difficult to calculate analytically, and promote the unification of the definition of correlation. 2. A method of describing Zhang Liang network properties is constructed by combining the size consistent (Size Consistency) with the area law (Area Law). For a quantum system whose energy gap is not zero, if a part of the ground state is A, the entanglement between A and the rest is determined by the measure of the boundary of A, which is called the rule of the law of area (Area Law). It is considered to describe the key to the success of DMRG in one-dimensional and as a guiding principle for the construction of a reasonable Zhang Liang network. However, in practice, we realize that the area law is not the only standard for describing Zhang Liang's network. It only concerns about the entanglement property of Zhang Liang network, but neglects its energy additivity. We combine the size consistency with the area law, and construct a more comprehensive description of Zhang Liang network properties. 3. The dynamic mechanism of Kibble-Zurek with long range correlation is studied by using matrix direct product state (Matrix Product State) and time-varying fractional principle. In recent years, with the rapid development of ion trap experimental technology, it is possible to study the non-equilibrium dynamics of quantum multi-body system experimentally. On the other hand, by selecting the appropriate Zhang Liang network and applying the time-varying principle, we can accurately simulate the real-time evolution of one-dimensional lattice system with 100 particles per unit time up to 500 units. Based on these two methods, we study the Kibble-Zurek dynamical mechanism of quantum multibody systems under the condition of long range correlation, which provides a new direction and viewpoint for the quantum interval KZ mechanism.
【学位授予单位】：中国科学技术大学
【学位级别】：博士
【学位授予年份】：2017
【分类号】：O413
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