本文關(guān)鍵詞：二維量子多體系統(tǒng)的張量網(wǎng)絡(luò)態(tài)算法　出處：《中國科學(xué)技術(shù)大學(xué)》2017年博士論文　論文類型：學(xué)位論文

  更多相關(guān)文章： 強(qiáng)關(guān)聯(lián) 量子多體問題 張量網(wǎng)絡(luò)態(tài) 副本交換分子動(dòng)力學(xué)方法 PEPS 梯度優(yōu)化 蒙特卡洛采樣

【摘要】：發(fā)展求解強(qiáng)關(guān)聯(lián)系統(tǒng)的高效的數(shù)值方法是現(xiàn)代物理最為核心的任務(wù)之一。在強(qiáng)關(guān)聯(lián)的相互作用體系中,由于傳統(tǒng)微擾論不再適用,研究強(qiáng)關(guān)聯(lián)系統(tǒng)的物理性質(zhì)主要依靠數(shù)值求解辦法,包括嚴(yán)格對角化方法,量子蒙特卡洛方法和密度矩陣重整化群方法。這些數(shù)值方法已被廣泛應(yīng)用于研究強(qiáng)關(guān)聯(lián)系統(tǒng),并且取得了巨大的成功。然而,上述方法都有其局限性:嚴(yán)格對角化方法會(huì)遇到所謂的"指數(shù)墻"問題;量子蒙特卡洛方法在處理費(fèi)米子問題和阻挫磁性問題時(shí)會(huì)遇到符號問題;而密度矩陣重整化方法主要用于處理一維或準(zhǔn)一維系統(tǒng),難以處理更高維系統(tǒng)。因此,發(fā)展新的高效的數(shù)值方法仍然是解決強(qiáng)關(guān)聯(lián)問題的當(dāng)務(wù)之急。近些年來,人們開始以量子信息理論的視角看待問題,通過對量子糾纏的深入理解,一種基于量子糾纏的張量網(wǎng)絡(luò)態(tài)(TNS)理論,包括矩陣乘積態(tài)(MPS)理論和投影糾纏對態(tài)(PEPS)理論逐漸建立起來。MPS和PEPS分別描述一維和二維系統(tǒng)時(shí)都滿足糾纏熵的面積定律和尺寸一致性,已經(jīng)被證明是研究強(qiáng)關(guān)聯(lián)系統(tǒng)的強(qiáng)有力的工具�；贛PS表示,人們建立起了描述一維量子多體系統(tǒng)的完善的理論。對于二維系統(tǒng),基于PEPS的相關(guān)算法還處于非常初級的階段。由于PEPS本身的復(fù)雜性和計(jì)算能力的限制,其在實(shí)際應(yīng)用中受到了很大限制。我們希望能夠發(fā)展一種高效地算法,使得PEPS可以能夠真正解決一些長期以來難以求解的問題�；诙S量子多體系統(tǒng)基態(tài)的張量網(wǎng)絡(luò)態(tài)表示,本論文講述了我們發(fā)展的求解二維量子多體系統(tǒng)基態(tài)的方法,主要內(nèi)容包括兩部分:第一部分著重講述了如何用副本交換的分子動(dòng)力學(xué)方法來解決用TNS求解多體問題時(shí)遇到的局域極小值問題。用TNS做為變分波函數(shù)求解多體系統(tǒng)的基態(tài)的很關(guān)鍵一步是如何有效優(yōu)化這個(gè)變分波函數(shù),使得其盡量避免陷入局域極小值。我們發(fā)展了一種可以大規(guī)模并行的高效地副本交換分子動(dòng)力學(xué)方法,用來解決這個(gè)問題。通過將TNS的元素看做廣義坐標(biāo),我們把這個(gè)優(yōu)化問題映射到一個(gè)經(jīng)典力學(xué)問題。在優(yōu)化時(shí),我們首先設(shè)定一系列不同的溫度,然后從隨機(jī)態(tài)出發(fā),根據(jù)這個(gè)經(jīng)典系統(tǒng)的勢能函數(shù),采用分子動(dòng)力學(xué)的方法對系統(tǒng)進(jìn)行演化,最終會(huì)得到不同溫度下的解,零溫下的解就是這個(gè)優(yōu)化問題的解。為了避免在分子動(dòng)力學(xué)的演化過程中陷入局域極小值,可以采用副本交換的方法將不同溫度下的構(gòu)型進(jìn)行充分交換來幫助其跳出局域極小值。第二部分著重講述了如何用PEPS的變分波函數(shù)來有效地求解二維量子自旋系統(tǒng)。PEPS可以很好地描述二維系統(tǒng)的基態(tài),但是由于其計(jì)算復(fù)雜度很高,在用它來模擬二維體系時(shí)受到很大限制。我們提出了用梯度優(yōu)化結(jié)合蒙特卡洛采樣的方法來優(yōu)化PEPS變分波函數(shù)。首先我們采用一種虛實(shí)演化的SU(simple update)方法來得到一個(gè)粗糙的PEPS波函數(shù)做為出發(fā)點(diǎn),然后通過梯度優(yōu)化來進(jìn)一步精確地優(yōu)化這個(gè)波函數(shù)來得到基態(tài)。在計(jì)算梯度和能量時(shí)我們采用了蒙特卡洛采樣的方法。與人們常用的方法相比,這種方法不僅大大地降低了計(jì)算復(fù)雜度,而且采用的梯度優(yōu)化算法可以更加精確地優(yōu)化變分波函數(shù),使得用PEPS解決一些長期以來難以求解的多體系統(tǒng)成為可能。
[Abstract]:For the development of efficient numerical methods for solving strongly correlated systems is one of the core tasks of modern physics. The interaction system in the strong correlation, because the traditional perturbation theory is no longer applicable, the physical properties of strongly correlated systems rely mainly on numerical solution, including the exact diagonalization method, quantum Monte Carlo method and the density matrix renormalization group method. These numerical methods have been widely used in the study of strongly correlated systems, and achieved great success. However, these methods have their limitations: the strict diagonalization method will encounter the so-called "refers to the number of wall" problem; quantum Monte Carlo method will encounter problems in dealing with problems and symbolic fermion frustrated magnetic the problem; and the density matrix renormalization method is mainly used for processing one dimensional system, difficult to deal with higher dimensional systems. Therefore, the development of efficient numerical methods is still new To solve the problem of strong association a pressing matter of the moment. In recent years, people began to look at the problem in quantum information theory, through in-depth understanding of quantum entanglement, a tensor network state based on quantum entanglement (TNS) theory, including the matrix product state (MPS) theory and projection of entanglement state (PEPS) theory has been gradually established.MPS and PEPS respectively describe the one-dimensional and two-dimensional system satisfies the entanglement entropy area law and uniform size, has proven to be a powerful tool to study the strong correlation systems. Based on MPS, people established a description of a dimension quantum many body system theory. For the two-dimensional system, correlation algorithm based on PEPS at a very early stage. Due to the complexity of PEPS and calculation of capacity constraints, it is limited in practical application. We hope to develop an efficient algorithm to make PEPS Can really solve some long-standing problems. It is difficult to solve that tensor network states in two-dimensional quantum system based on ground state, this paper describes the method for solving the two-dimensional quantum we develop multibody system ground state. The main contents include two parts: the first part describes how to use a replica exchange molecular dynamics method to solve local minima encountered by TNS to solve the many body problem when the value problem. Using TNS wave function for the ground state of multi-body system as the variable is the key step is how to effectively optimize the variational wave function, making it as far as possible to avoid falling into local minimum. We developed an efficient parallel can copy the exchange of molecular dynamics method is used to solve this problem. The TNS elements as generalized coordinates, we mapped the optimization problem to a problem in classical mechanics. When we first set a series of different temperature, and then starting from the random state, according to the potential energy function of this classic system, by the method of molecular dynamics of the evolution of the system, can be obtained under different temperature solution is the optimization solution of zero temperature. In order to avoid falling into local minimum evolution in the process of molecular dynamics, the replica exchange method can be used under different temperature and the configuration of full exchange to help them escape from the local minima. The second part focuses on how to use PEPS variational wave function to effectively solve the two-dimensional quantum spin system.PEPS can well describe the ground state of two dimensional systems, but due to its high computational complexity, it is limited it is used to simulate the two-dimensional system. We put forward the optimization method combining Monte Carlo sampling to optimize PEPS gradient Wave function. First we use a virtual evolution (simple update) SU method to obtain a rough PEPS wave function as the starting point, and then through the gradient optimization to further optimize the exact wave function to get the ground. In the calculation of gradient and energy when we use Monte Carlo sampling method. Compared with the one commonly used method, this method not only greatly reduces the computational complexity, and the use of the gradient optimization algorithm can accurately optimize the variational wave function, makes the use of PEPS to solve some long-standing difficulty in solving multi body system becomes possible.

【學(xué)位授予單位】：中國科學(xué)技術(shù)大學(xué)
【學(xué)位級別】：博士
【學(xué)位授予年份】：2017
【分類號】：O413.3

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