Thesis advisor: Ying Ran / Classification and simulation of quantum phases are one of main themes in condensed matter physics. Quantum phases can be distinguished by their symmetrical and topological properties. The interplay between symmetry and topology in condensed matter physics often leads to exotic quantum phases and rich phase diagrams. Famous examples include quantum Hall phases, spin liquids and topological insulators. In this thesis, I present our works toward a more systematically understanding of symmetric topological quantum phases in bosonic systems. In the absence of global symmetries, gapped quantum phases are characterized by topological orders. Topological orders in 2+1D are well studied, while a systematically understanding of topological orders in 3+1D is still lacking. By studying a family of exact solvable models, we find at least some topological orders in 3+1D can be distinguished by braiding phases of loop excitations. In the presence of both global symmetries and topological orders, the interplay between them leads to new phases termed as symmetry enriched topological (SET) phases. We develop a framework to classify a large class of SET phases using tensor networks. For each tensor class, we can write down generic variational wavefunctions. We apply our method to study gapped spin liquids on the kagome lattice, which can be viewed as SET phases of on-site symmetries as well as lattice symmetries. In the absence of topological order, symmetry could protect different topological phases, which are often referred to as symmetry protected topological (SPT) phases. We present systematic constructions of tensor network wavefunctions for bosonic symmetry protected topological (SPT) phases respecting both onsite and spatial symmetries.

當絶熱地改變哈密頓量的參數時，波函數會獲得一個幾何相位，既 Berry相。它可以表示為參數空間內一個局域的規範場，叫作 Berry曲率。Berry曲率在凝聚態物理的許多領域的研究中起著至關重要的作用，例如量子霍爾效應以及拓撲絶緣體。因此它已成為固體的最基本的性質之一。在量子霍爾效應中，霍爾電導可以表示為 Berry曲率在布里淵區上的積分。這個積分是一個量子化的 Chern數，並且反映了系統的拓撲結構。然而由於時間反演對稱性，拓撲絶緣體的霍爾電導等於零。因此對時間反演不變絶緣體的 Berry曲率的直接以及非破壞性的測量已經成為凝聚態物理中的重要問題。 / 在這篇論文中，我們提出標準的非線性光譜學可以用來探測時間反演不變絶緣體的性質，而且非線性光譜學的實驗比直流實驗更可控。通過計算，我們發現當遠紅外光和 THz光入射到樣品上時，系統的三階光學響應不為零，這與輸運實驗的結果相比形成了鮮明的對比。更重要的是響應函數正比於能帶的非阿貝爾 Berry曲率。這個結果提供了直接測量時間反演不變系統的 Berry曲率的可能性。 / 對具有(近似的 )空間旋轉對稱性的時間反演不變絶緣體，我們發現三階光學響應與等能球面的 Berry曲率通量直接相關。由於 Berry曲率通量給出了能帶簡併點處的奇異單子的拓撲電荷，因此人們可以利用這種方法直接測量能帶的拓撲結構。作為一個例子，這個方法被應用於 III-V族化合物半導體的八帶模型，並給出了一個拓撲電荷為 3的非線性響應。 / Berry phase, a geometric phase acquired by a wave function by adiabatically varying the parameters of the Hamiltonian, can be expressed in terms of a local gauge field in parameter space, called Berry curvature. The Berry curvature plays an essential role in many fields of condensed matter physics, such as the quantum Hall eect and in the study of Topological insulators (TI) and hence it has become one of the most fundamental properties of solids. In Quantum Hall eect, the Hall conductance can be expressed as an integral of the Berry curvature over the Brillouin zone, which is a quantized Chern number and reflects the topology of the system. However in TI, the Hall conductance is equal to zero as a result of the time-reversal (TR) symmetry. Thus, the direct and nondestructive measurement of the Berry curvature of a TR invariant insulator is an important issue in condensed matter physics. / In this thesis, we show that the standard nonlinear optical spectroscopy, being more experimentally controllable than DC experiments, can be used to detect the bulk properties of TR invariant insulators. Through a general calculation, we nd that, when optical and terahertz light fields are employed, the third order optical eect is nonzero compared with the transport method. And the susceptibility is exactly proportional to the non-Abelian Berry curva-ture of the energy band, which provides the possibility of determining Berry curvature directly. / For the TR invariant insulator with (approximate) rotational symmetry, the third order optical susceptibility is related to the the Berry curvature flux through the iso-energy sphere, which gives the topological charge of the monopole at the degeneracy point. Hence it enables one to measure the topo¬logical property of the energy band explicitly. As an example, the method is applied to the eight-band model of III-V compound semiconductors and gives a quantized susceptibility with topological charge equal to 3. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Yang, Fan = 時間反演不變絶緣體的Berry曲率的非線性光學測量 / 楊帆. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2012. / Includes bibliographical references (leaves 77-[80]). / Abstracts also in Chinese. / Yang, Fan = Shi jian fan yan bu bian jue yuan ti de Berry qu lu de fei xian xing guang xue ce liang / Yang Fan. / Chapter 1 --- Introduction --- p.1 / Chapter 1.1 --- Introduction of Berry phase --- p.1 / Chapter 1.1.1 --- Basic concepts of the Berry phase and Berry curvature --- p.2 / Chapter 1.1.2 --- Degeneracy and monopole --- p.5 / Chapter 1.1.3 --- Berry phase in Bloch bands --- p.7 / Chapter 1.1.4 --- Non-Abelian Berry curvature --- p.8 / Chapter 1.2 --- Quantum Hall effect and topological insulator --- p.10 / Chapter 1.2.1 --- Anomalous velocity and Quantum Hall effect --- p.11 / Chapter 1.2.2 --- Topological insulator --- p.14 / Chapter 1.3 --- Introduction of nonlinear optics --- p.16 / Chapter 1.3.1 --- Nonlinear optical susceptibilities --- p.16 / Chapter 1.3.2 --- Density matrix formalism --- p.19 / Chapter 1.3.3 --- Diagrammatic analysis of nonlinear optical processes --- p.21 / Chapter 1.4 --- Polarization operator of band electrons --- p.24 / Chapter 1.5 --- Outline of this thesis --- p.26 / Chapter 2 --- Third-order Optical Response of a General Insulator --- p.28 / Chapter 2.1 --- Introduction --- p.28 / Chapter 2.2 --- Microscopic mechanism --- p.30 / Chapter 2.3 --- Third-order nonlinear susceptibility --- p.31 / Chapter 2.3.1 --- A general model --- p.31 / Chapter 2.3.2 --- Perturbative calculation I --- p.35 / Chapter 2.3.3 --- Perturbative calculation II --- p.40 / Chapter 2.3.4 --- Total response --- p.43 / Chapter 2.4 --- Diagrammatic calculation of the third-order response --- p.45 / Chapter 2.5 --- Application to topological insulators --- p.56 / Chapter 2.6 --- Summary --- p.59 / Chapter 3 --- Nonlinear Optical Measurement of Topological Charge --- p.61 / Chapter 3.1 --- Introduction --- p.61 / Chapter 3.2 --- Third-order response with resonant interband transitions --- p.62 / Chapter 3.3 --- Third-order response and topological charge in a rotationally symmetric insulator --- p.66 / Chapter 3.4 --- Quantized susceptibility of III-V compound semiconductors --- p.70 / Chapter 3.5 --- Summary --- p.74 / Chapter 4 --- Summary and Conclusions --- p.75 / Bibliography --- p.77 / Chapter A --- Calculation of equation (2.32) --- p.81 / Chapter B --- Proof of formula (3.20) --- p.89 / Chapter C --- Third-order response with multiple conduction and valence bands --- p.92

Wong Hon Man = 量子力學中之幾何相位 / 黃漢文. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2005. / Includes bibliographical references (leaves 75-78). / Text in English; abstracts in English and Chinese. / Wong Hon Man = Liang zi li xue zhong zhi ji he xiang wei / Huang Hanwen. / Chapter 1 --- Introduction --- p.1 / Chapter 1.1 --- Geometric Phase in Orthogonal States --- p.2 / Chapter 1.2 --- Projective Phase --- p.3 / Chapter 1.3 --- Bose-Einstein Condensate --- p.3 / Chapter 1.4 --- Arrangement of the Thesis --- p.3 / Chapter 2 --- Review of Geometric Phase --- p.5 / Chapter 2.1 --- Introduction --- p.5 / Chapter 2.2 --- Berry's Phase --- p.6 / Chapter 2.3 --- Aharonov-Anadan Phase --- p.9 / Chapter 2.4 --- Pancharatnam Phase --- p.10 / Chapter 2.5 --- Example of a Two-State System --- p.13 / Chapter 2.6 --- Remark on Phase Factor --- p.15 / Chapter 3 --- Review of Geometric Phase near Orthogonal States --- p.16 / Chapter 3.1 --- π Phase Change --- p.17 / Chapter 3.2 --- Off-diagonal Geometric Phase --- p.18 / Chapter 4 --- Projective Phase --- p.21 / Chapter 4.1 --- Projective Measurement --- p.21 / Chapter 4.2 --- Definition in Differential Geometry --- p.22 / Chapter 4.3 --- Gauge Transformation --- p.24 / Chapter 4.4 --- Two-state System and Monopole --- p.26 / Chapter 5 --- π Phase Change --- p.30 / Chapter 5.1 --- Projective Phase near Orthogonal States --- p.30 / Chapter 5.2 --- Two-state system --- p.32 / Chapter 5.3 --- Three-state systems --- p.33 / Chapter 5.4 --- Topological restriction for Spin-1 System --- p.36 / Chapter 5.5 --- Remark on Continuity of Curves --- p.37 / Chapter 6 --- Topological Number Associated with Projective Phase --- p.38 / Chapter 6.1 --- Curves Separated by Zero Dot Products --- p.38 / Chapter 6.2 --- Definition of Phase without 2π arbitrary phase --- p.41 / Chapter 6.3 --- 1st Chern Number Associated with Closed Loops --- p.42 / Chapter 6.4 --- Example with a Spin-m System --- p.43 / Chapter 6.5 --- πphase change --- p.45 / Chapter 6.6 --- Variation of Projection States and Curves --- p.46 / Chapter 7 --- Off-diagonal geometric phase --- p.50 / Chapter 7.1 --- Experiment to Measure Projective Phase --- p.52 / Chapter 8 --- Summary on Geometric Phase near Orthogonal States --- p.53 / Chapter 9 --- Review of Bose-Einstein Condensate --- p.55 / Chapter 9.1 --- Gross-Pitaevskii Equation --- p.55 / Chapter 9.2 --- One-Dimensional GP equation --- p.57 / Chapter 9.3 --- Elementary Excitations --- p.57 / Chapter 10 --- Geometric Phase in Bose-Einstein Condensate --- p.59 / Chapter 10.1 --- Two-state Geometric Phase --- p.59 / Chapter 10.2 --- Geometric Phase in Perturabation --- p.60 / Chapter 10.3 --- Numerical Results --- p.63 / Chapter 10.4 --- Discussions --- p.64 / Chapter A --- Monopole --- p.69 / Chapter B --- Definition of Fibre Bundle --- p.71 / Chapter C --- Evaluation of Bargmann invariant by Pancharatnam Phase Formula --- p.73 / Bibliography --- p.75

published_or_final_version / abstract / Physics / Master / Master of Philosophy

Thesis advisor: Ying Ran / In this thesis, I investigated and implemented various numerical and simulation methods, including mean field theory, functional renormalization group method (fRG), density matrix renormalization group (DMRG) method etc., to find different quantum phases and quantum phase diagrams on models of correlated electronic systems. I found different phase diagrams with phases such as magnetism, superconductivity. By summarizing the strength and limitations of these methods, I investigated the projected entangled paired states (PEPS) with symmetry quantum number to sharply distinguish phases into crude classes and applied a variation of fast full update (FFU) prototype[58] to simulate different phases numerically. This method provides a promising, powerful and efficient way to simulate unconventional quantum phases and quantum phase diagrams in correlated electronic systems. / Thesis (PhD) — Boston College, 2016. / Submitted to: Boston College. Graduate School of Arts and Sciences. / Discipline: Physics.

Thesis (M. Phil.)--University of Hong Kong, 2007. / Title proper from title frame. Also available in printed format.

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published_or_final_version / abstract / toc / Physics / Master / Master of Philosophy

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