一維與準(zhǔn)一維量子多體系統(tǒng)的相變和臨界現(xiàn)象
發(fā)布時間：2018-09-10 17:34
【摘要】：本文使用了以張量網(wǎng)絡(luò)表述為背景的一維無限虛時間演化塊算法(infinite time evolving block decimation,iTEBD)和準(zhǔn)一維的沿梯度定向行走的梯子算法(projected entangled-pair states,PEPS對梯子系統(tǒng)的幾何適應(yīng)),對一維和準(zhǔn)一維量子自旋多體系統(tǒng)進(jìn)行了模擬研究。在熱力學(xué)極限下,通過近似逼近的辦法,獲得了很好的一維無限鏈和準(zhǔn)一維無限梯子的量子多體系統(tǒng)的基態(tài)波函數(shù)。然后由表示基態(tài)波函數(shù)的張量表示出了量子多體系統(tǒng)對應(yīng)的約化密度矩陣。進(jìn)而使用基態(tài)單位格點(diǎn)保真度理論和其它可觀測物理量的算符作用在約化密度矩陣上,從而得到關(guān)于量子相變、量子相變點(diǎn)以及量子多體系統(tǒng)所處的態(tài)的序參量等這些讓人感興趣的物理信息。本文的第一章是緒論,介紹了和本課題密切相關(guān)的一些背景知識。首先以熱力學(xué)相變?yōu)閰⒄战榻B了什么是量子相變的概念。其次介紹了基態(tài)能譜、保真度、自發(fā)對稱性破缺、局域序參量、重整化群論和標(biāo)度理論等基本概念。再次介紹了二分糾纏熵——馮諾伊曼熵,以及使用馮諾伊曼熵時必須滿足的面積定理。本文第二章第一部分介紹了準(zhǔn)一維沿梯度定向行走的梯子張量網(wǎng)絡(luò)算法更新基態(tài)波函數(shù)、得到近似基態(tài)波函數(shù)的過程。第二部分運(yùn)用梯度算法結(jié)合單位格點(diǎn)保真度理論,對“具有反鐵磁交叉交換相互作用的海森堡受挫自旋梯子”的相圖構(gòu)成進(jìn)行了模擬研究,研究的原因在于:該區(qū)域存在關(guān)于CD相是否存在的爭議。單位格點(diǎn)保真度相圖捕捉到了一個相變點(diǎn)和連續(xù)變化的過程,保真度之所以能夠探測到相變點(diǎn)和相變變化的過程是因?yàn)?波函數(shù)內(nèi)部結(jié)構(gòu)本質(zhì)上的不同,由于內(nèi)部結(jié)構(gòu)的不同導(dǎo)致了擠點(diǎn)(相變點(diǎn))的出現(xiàn);另一方面對于連續(xù)相變,當(dāng)參考態(tài)的選擇不是自身時,保真度的值總是沿著重整化群流的方向單調(diào)的減小(連續(xù)變化的原因)。同時,本文還首次將單位格點(diǎn)保真度探測簡并基態(tài)的功能引入到了受挫兩腿自旋梯子量子多體系統(tǒng)里面。結(jié)果表明:在反鐵磁區(qū)域,沒有屬于Landau-Ginzberg-Wilson范式的由自發(fā)對稱性破缺所引起的簡并基態(tài)。最后從超越Landau理論非局域長程序——弦序參量,對系統(tǒng)的兩個拓?fù)湫蜻M(jìn)行了刻畫,結(jié)果表明:只有Handane相和rung-singlet相的存在(CD相的局域序參量的值一直保持為零)。另外,本文還首次從哈密頓量對稱性對相圖的限制進(jìn)行了討論。幾個方面的結(jié)果一致贊成所研究區(qū)域的相圖的本質(zhì)是:該區(qū)域僅由Handane相和rung-singlet相構(gòu)成(沒有CD相的存在),兩相之間發(fā)生的是連續(xù)相變。本文第三章第一部分介紹了觀測量——單位格點(diǎn)幾何糾纏的實(shí)現(xiàn)過程。第二部分基于準(zhǔn)一維沿梯度定向行走的梯子張量網(wǎng)絡(luò)算法結(jié)合單位格點(diǎn)幾何糾纏和基態(tài)單位格點(diǎn)保真度理論,對“具有反鐵磁交叉交換相互作用的海森堡受挫自旋梯子模型”的相圖相變級數(shù)進(jìn)行了模擬研究。研究相變級數(shù)的原因在于:該模型在強(qiáng)弱耦合作用不同的條件下,相變級數(shù)存在著爭議。為了讓數(shù)據(jù)更具有說服力,我們從幾何糾纏、弦序參量和單位格點(diǎn)基態(tài)保真度分別進(jìn)行了研究。三個方面的結(jié)果一致支持:在弱耦合作用區(qū),系統(tǒng)發(fā)生的是連續(xù)相變;而在強(qiáng)耦合作用區(qū),系統(tǒng)發(fā)生的是非連續(xù)相變。目前整條相變線的數(shù)據(jù)還沒有計(jì)算完,根據(jù)現(xiàn)有階段性結(jié)果能判斷出由于耦合作用強(qiáng)弱的不同,相圖相變的級數(shù)會相應(yīng)的變化。完整的相圖會在下一步工作中去實(shí)現(xiàn)。本文第四章第一部分介紹了iTEBD算法。第二部分基于iTEBD算法結(jié)合基態(tài)波函數(shù)的約化密度矩陣導(dǎo)出的各種觀測量模擬研究了擴(kuò)展的量子羅盤模型(extended quantum compass model,EQCM)的相圖。首次從序參量的角度深入研究了這個模型的相圖。為了合適地描述相圖當(dāng)中相的序參量,我們套用了從保真度理論研究相變的三大步驟:第一,確定出相變點(diǎn)和各個相所在區(qū)域,用基態(tài)能量和導(dǎo)數(shù)確定了相變點(diǎn)和各個相所在區(qū)域。第二,刻畫指定相中選出作為代表態(tài)的局域序參量(如果有任何可能的局域序參量),用兩點(diǎn)關(guān)聯(lián)和局域磁化強(qiáng)度導(dǎo)出了局域序參量,并用局域序參量刻畫對應(yīng)相。第三,刻畫非局域長程序,用短程弦關(guān)聯(lián)函數(shù)的單調(diào)行為和振蕩行為定義了可能的弦序參量,然后做飽和驗(yàn)證,滿足飽和驗(yàn)證就導(dǎo)出弦序參量,并用弦序參量刻畫對應(yīng)相。同時,我們對序參量在臨界區(qū)域附近的臨界行為、馮諾伊曼熵在臨界點(diǎn)處的有限糾纏標(biāo)度行為做了標(biāo)度。臨界指數(shù)?=1/8、中心荷c?0.5表明:連續(xù)相變線的相變類型屬于Ising普適類。第五章介紹主要結(jié)論和后續(xù)研究工作展望。
[Abstract]:In this paper, the infinite time evolving block decimation (iTEBD) algorithm and the projected entangled-pair states (PEPS) algorithm for one-dimensional and quasi-one-dimensional quantum spin multibody systems are used. The ground state wave functions of quantum multibody systems with one-dimensional infinite chains and quasi-one-dimensional infinite ladders are obtained by approximate approximation under the thermodynamic limit. Then the reduced density matrices corresponding to the quantum multibody systems are shown by the tensors representing the ground state wave functions. Degree theory and operators of other observable physical quantities act on the reduced density matrix to obtain interesting physical information about quantum phase transitions, quantum phase transition points and the order parameters of the states in which quantum multibody systems are located. In this paper, the concept of quantum phase transition is introduced with reference to thermodynamic phase transition. Secondly, the basic concepts of ground state energy spectrum, fidelity, spontaneous symmetry breaking, local order parameter, renormalization group theory and scaling theory are introduced. In the second chapter, the process of updating the ground state wave function and obtaining the approximate ground state wave function by the quasi-one-dimensional gradient-oriented ladder tensor network algorithm is introduced. The simulation of the phase diagram is carried out because there is a controversy about whether the CD phase exists in the region. The unit lattice fidelity phase diagram captures a phase transition point and a continuous change process. The reason why the fidelity can detect the phase transition point and the phase transition process is that the internal structure of the wave function is essentially different. On the other hand, for the continuous phase transition, the fidelity decreases monotonously along the direction of the renormalization group flow (the cause of the continuous change). At the same time, the function of detecting degenerate ground state with the fidelity of the unit lattice is introduced for the first time. The results show that in the antiferromagnetic region, there is no degenerate ground state caused by spontaneous symmetry breaking in the Landau-Ginzberg-Wilson normal form. Finally, the two topological orders of the system are characterized by the nonlocal long program, the string order parameter, which transcends the Landau theory. It is shown that only Handane phase and rung-singlet phase exist (the value of local order parameter of CD phase remains zero all the time). In addition, the limitation of phase diagrams from Hamiltonian symmetry is discussed for the first time. The results from several aspects agree that the nature of phase diagrams in the studied region is that the region is composed only of Handane phase and rung-singlet phase. In the third chapter, we introduce the realization of the geometric entanglement of the unit lattice. In the second part, we combine the geometric entanglement of the unit lattice and the fidelity theory of the ground state unit lattice with the algorithm of the ladder tensor network based on the quasi-one-dimensional directional gradient walk. The phase transition series of Heisenberg frustrated spin ladder model with antiferromagnetic cross-exchange interaction is simulated. The reason for studying the phase transition series is that the phase transition series of the model is controversial under different coupling conditions. To make the data more convincing, we consider the geometric entanglement and the chord order parameter. In the weak coupling region, the continuous phase transition occurs; in the strong coupling region, the discontinuous phase transition occurs. In the fourth chapter, the iTEBD algorithm is introduced. In the second part, based on the iTEBD algorithm and the reduced density matrix of the ground state wave function, the extended quantum compass model is studied. The phase diagram of the extended quantum compass model (EQCM) is studied from the point of view of order parameters for the first time. In order to describe the order parameters of the phase diagram appropriately, three steps are applied to study the phase transition from the fidelity theory. Firstly, the phase transition point and the region of each phase are determined, and the ground state energy and derivative are used to determine the phase transition. Secondly, the local order parameter (if any possible local order parameter) selected from the specified phase is characterized, and the local order parameter is derived by two-point correlation and local magnetization, and the corresponding phase is characterized by the local order parameter. Thirdly, the non-local long program is described, and the short-range chord correlation function is used. The possible chord order parameters are defined for the monotonic and oscillatory behavior of numbers, and then saturation tests are performed to derive the chord order parameters and characterize the corresponding phase by chord order parameters. The exponent?=1/8 and the central charge c?0.5 indicate that the phase transition types of the continuous phase change lines belong to the Ising universal class.
【學(xué)位授予單位】：重慶大學(xué)
【學(xué)位級別】：博士
【學(xué)位授予年份】：2016
【分類號】：O469
[Abstract]:In this paper, the infinite time evolving block decimation (iTEBD) algorithm and the projected entangled-pair states (PEPS) algorithm for one-dimensional and quasi-one-dimensional quantum spin multibody systems are used. The ground state wave functions of quantum multibody systems with one-dimensional infinite chains and quasi-one-dimensional infinite ladders are obtained by approximate approximation under the thermodynamic limit. Then the reduced density matrices corresponding to the quantum multibody systems are shown by the tensors representing the ground state wave functions. Degree theory and operators of other observable physical quantities act on the reduced density matrix to obtain interesting physical information about quantum phase transitions, quantum phase transition points and the order parameters of the states in which quantum multibody systems are located. In this paper, the concept of quantum phase transition is introduced with reference to thermodynamic phase transition. Secondly, the basic concepts of ground state energy spectrum, fidelity, spontaneous symmetry breaking, local order parameter, renormalization group theory and scaling theory are introduced. In the second chapter, the process of updating the ground state wave function and obtaining the approximate ground state wave function by the quasi-one-dimensional gradient-oriented ladder tensor network algorithm is introduced. The simulation of the phase diagram is carried out because there is a controversy about whether the CD phase exists in the region. The unit lattice fidelity phase diagram captures a phase transition point and a continuous change process. The reason why the fidelity can detect the phase transition point and the phase transition process is that the internal structure of the wave function is essentially different. On the other hand, for the continuous phase transition, the fidelity decreases monotonously along the direction of the renormalization group flow (the cause of the continuous change). At the same time, the function of detecting degenerate ground state with the fidelity of the unit lattice is introduced for the first time. The results show that in the antiferromagnetic region, there is no degenerate ground state caused by spontaneous symmetry breaking in the Landau-Ginzberg-Wilson normal form. Finally, the two topological orders of the system are characterized by the nonlocal long program, the string order parameter, which transcends the Landau theory. It is shown that only Handane phase and rung-singlet phase exist (the value of local order parameter of CD phase remains zero all the time). In addition, the limitation of phase diagrams from Hamiltonian symmetry is discussed for the first time. The results from several aspects agree that the nature of phase diagrams in the studied region is that the region is composed only of Handane phase and rung-singlet phase. In the third chapter, we introduce the realization of the geometric entanglement of the unit lattice. In the second part, we combine the geometric entanglement of the unit lattice and the fidelity theory of the ground state unit lattice with the algorithm of the ladder tensor network based on the quasi-one-dimensional directional gradient walk. The phase transition series of Heisenberg frustrated spin ladder model with antiferromagnetic cross-exchange interaction is simulated. The reason for studying the phase transition series is that the phase transition series of the model is controversial under different coupling conditions. To make the data more convincing, we consider the geometric entanglement and the chord order parameter. In the weak coupling region, the continuous phase transition occurs; in the strong coupling region, the discontinuous phase transition occurs. In the fourth chapter, the iTEBD algorithm is introduced. In the second part, based on the iTEBD algorithm and the reduced density matrix of the ground state wave function, the extended quantum compass model is studied. The phase diagram of the extended quantum compass model (EQCM) is studied from the point of view of order parameters for the first time. In order to describe the order parameters of the phase diagram appropriately, three steps are applied to study the phase transition from the fidelity theory. Firstly, the phase transition point and the region of each phase are determined, and the ground state energy and derivative are used to determine the phase transition. Secondly, the local order parameter (if any possible local order parameter) selected from the specified phase is characterized, and the local order parameter is derived by two-point correlation and local magnetization, and the corresponding phase is characterized by the local order parameter. Thirdly, the non-local long program is described, and the short-range chord correlation function is used. The possible chord order parameters are defined for the monotonic and oscillatory behavior of numbers, and then saturation tests are performed to derive the chord order parameters and characterize the corresponding phase by chord order parameters. The exponent?=1/8 and the central charge c?0.5 indicate that the phase transition types of the continuous phase change lines belong to the Ising universal class.
【學(xué)位授予單位】：重慶大學(xué)
【學(xué)位級別】：博士
【學(xué)位授予年份】：2016
【分類號】：O469
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