張量網(wǎng)絡(luò)算法及其在多體動(dòng)力學(xué)中的研究
發(fā)布時(shí)間：2018-11-18 20:18
【摘要】：強(qiáng)關(guān)聯(lián)多體物理是凝聚態(tài)物理領(lǐng)域的核心課題,它與很多我們關(guān)心的物理體系,比如高溫超導(dǎo)體系,自旋液體體系密切相關(guān)。然而無(wú)論在解析上還是數(shù)值上,這些強(qiáng)關(guān)聯(lián)系統(tǒng)都很難研究,原因是其希爾伯特空間的維數(shù)隨著系統(tǒng)粒子數(shù)的增長(zhǎng)而指數(shù)增加。然而最近20年來(lái),藉由人們對(duì)于多體量子系統(tǒng)中最重要的性質(zhì)—糾纏和關(guān)聯(lián)的深入理解,及其帶來(lái)的密度矩陣重整化群(Density Matrix Renormalization Group),張量網(wǎng)絡(luò)態(tài)(Tensor Network States),含時(shí)變分原理(Time Dependent Variational Principle)等先進(jìn)的數(shù)值工具的快速發(fā)展,我們終于可以研究多體物理里最為人們關(guān)注的基態(tài),低能激發(fā)態(tài)以及動(dòng)力學(xué)演化等等問(wèn)題。本篇論文中,從發(fā)展張量網(wǎng)絡(luò)算法,到應(yīng)用張量網(wǎng)絡(luò)算法研究量子多體動(dòng)力學(xué),主要完成了以下三方面的工作:1.基于相對(duì)熵的幾何度量,運(yùn)用并行蒙特卡洛算法,發(fā)展了一套統(tǒng)一計(jì)算所有關(guān)聯(lián)的數(shù)值方法。為了研究各種不同的量子系統(tǒng),人們定義了多種不同形式的糾纏和關(guān)聯(lián)。盡管這些定義都在特定問(wèn)題中發(fā)揮著重要的作用,人們?nèi)匀幌Ｍ軌蛴媒y(tǒng)一的方式度量所有的關(guān)聯(lián),并對(duì)它們的大小進(jìn)行比較。從這個(gè)問(wèn)題入手,我們借助相對(duì)熵的幾何度量,將所有的關(guān)聯(lián)度量都轉(zhuǎn)化為特定希爾伯特空間內(nèi)的極值問(wèn)題,并基于并行蒙特卡洛算法,構(gòu)建了一套可以高效求解各種關(guān)聯(lián)的相對(duì)熵的數(shù)值方法,彌補(bǔ)了相對(duì)熵難以解析計(jì)算的缺點(diǎn),同時(shí)推進(jìn)了關(guān)聯(lián)定義的統(tǒng)一化。2.將尺寸一致性(Size Consistency)和面積定律(Area Law)進(jìn)行結(jié)合,共同構(gòu)建一套描述張量網(wǎng)絡(luò)性質(zhì)的方法。對(duì)于一個(gè)能隙不為零的量子系統(tǒng),取其基態(tài)的一部分為A,則A與剩余部分的糾纏大小由A邊界的測(cè)度決定—這條被稱(chēng)為面積定律(Area Law)的規(guī)則,被認(rèn)為描述了DMRG在一維成功的關(guān)鍵,并被作為構(gòu)建合理張量網(wǎng)絡(luò)的指導(dǎo)原則。但在實(shí)踐中我們意識(shí)到,面積定律不是描述張量網(wǎng)絡(luò)的唯一標(biāo)準(zhǔn),它只關(guān)心了張量網(wǎng)絡(luò)的糾纏性質(zhì),但忽略了其能量可加性。我們將尺寸一致性與面積定律相結(jié)合,構(gòu)建了一套更全面的描述張量網(wǎng)絡(luò)性質(zhì)的準(zhǔn)則。3.利用矩陣直積態(tài)(Matrix Product State)以及含時(shí)變分原理,研究長(zhǎng)程關(guān)聯(lián)下的Kibble-Zurek動(dòng)力學(xué)機(jī)制。近年來(lái),一方面離子阱實(shí)驗(yàn)技術(shù)的突飛猛進(jìn),使得在實(shí)驗(yàn)上對(duì)量子多體系統(tǒng)的的非平衡動(dòng)力學(xué)研究成為可能;另一方面,通過(guò)選取合適的張量網(wǎng)絡(luò),再運(yùn)用含時(shí)變分原理,我們可以精確的模擬一維粒子數(shù)為100的格點(diǎn)系統(tǒng)長(zhǎng)達(dá)500個(gè)單位時(shí)間的實(shí)時(shí)演化。以這兩者為基礎(chǔ),我們研究了量子多體系統(tǒng),在長(zhǎng)程關(guān)聯(lián)情況下的Kibble-Zurek動(dòng)力學(xué)機(jī)制,為量子區(qū)間的KZ機(jī)制提供了新的方向和觀點(diǎn)。
[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.
【學(xué)位授予單位】：中國(guó)科學(xué)技術(shù)大學(xué)
【學(xué)位級(jí)別】：博士
【學(xué)位授予年份】：2017
【分類(lèi)號(hào)】：O413
本文編號(hào)：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.
【學(xué)位授予單位】：中國(guó)科學(xué)技術(shù)大學(xué)
【學(xué)位級(jí)別】：博士
【學(xué)位授予年份】：2017
【分類(lèi)號(hào)】：O413
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