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The Bose-Hubbard model is used to study bosons in optical lattices. In this thesis we will use an extended Bose-Hubbard model to study a type of completely localized solutions, called compactons. The compactons are a special case of the much studied solitons. The soliton is a familiar concept in non-linear physics. It is a stable, localized wave-solution, found in a range of different systems; from DNA-molecules to optical fibers. The compacton is a soliton that is completely localized, i.e. strictly zero outside a given area. The dynamics of the (extended) Bose-Hubbard model is based on the tunneling of particles between the lattice sites. The ordinary Bose-Hubbard model only accounts for one-particle tunneling processes. We will consider a model that also takes some two-particle tunneling processes into account, basically by considering long-range effects of the particle interaction. The aim of this thesis is to find and study the quantum analog of the compactons found in an extended Discrete Non-Linear Schrödinger equation. We will study analytical solutions and try to find if and under which conditions specific compactons exist. Numerical calculations are made to study the properties of the compactons and to study how compacton solutions arise in the classical limit.

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Hui Ka Ming = 非對稱Hubbard模型之量子蒙地卡羅與精準對角化研究 / 許嘉明. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2006. / Includes bibliographical references (leaves 85-88). / Text in English; abstracts in English and Chinese. / Hui Ka Ming = Fei dui cheng Hubbard mo xing zhi liang zi Mengdi Kaluo yu jing zhun dui jiao hua yan jiu / Xu Jiaming. / Chapter 1 --- Introduction --- p.1 / Chapter 2 --- The Asymmetric Hubbard Model And Its Physical Background / Chapter 2.1 --- Physical Motivation of the Study --- p.4 / Chapter 2.2 --- Symmetry Properties --- p.5 / Chapter 2.2.1 --- Particle-Hole Transformation --- p.5 / Chapter 2.2.2 --- U(l) Group --- p.5 / Chapter 2.2.3 --- SU(2) Group --- p.6 / Chapter 2.3 --- Relations with Other Models --- p.7 / Chapter 2.3.1 --- Falicov-Kimball Model --- p.7 / Chapter 2.3.2 --- Asymmetric t-J and Heisenberg Model --- p.7 / Chapter 2.3.3 --- Anderson and Kondo Model。 --- p.8 / Chapter 3 --- Exact Diagonalization and Density Matrix Renormalization Group --- p.11 / Chapter 3.1 --- Exact Diagonalization --- p.12 / Chapter 3.1.1 --- Second Quantization Representation --- p.13 / Chapter 3.1.2 --- Binary Representation Of Quantum States --- p.15 / Chapter 3.1.3 --- Two-Table Method --- p.15 / Chapter 3.1.4 --- Generation of Matrix Elements --- p.18 / Chapter 3.1.5 --- Lanczos Method --- p.20 / Chapter 3.1.6 --- Continued Fraction Dynamics At T=0 --- p.22 / Chapter 3.2 --- Density Matrix Renormalization Group --- p.23 / Chapter 3.2.1 --- Infinite system DMRG --- p.24 / Chapter 3.2.2 --- DMRG and Quantum Entanglement --- p.27 / Chapter 4 --- Determinant Quantum Monte Carlo And Finite Temperature Green Function --- p.29 / Chapter 4.1 --- Introduction --- p.29 / Chapter 4.2 --- General Scenario of Fermionic Monte Carol Method --- p.30 / Chapter 4.3 --- Review on Monte Carlo Method for Ising Model --- p.31 / Chapter 4.3.1 --- Ising Model --- p.31 / Chapter 4.3.2 --- Metropolis Algorithm --- p.31 / Chapter 4.3.3 --- Measurement --- p.32 / Chapter 4.3.4 --- Near The Critical Points --- p.33 / Chapter 4.4 --- Determinant Quantum Monte Carlo --- p.33 / Chapter 4.4.1 --- Suzuki-Trotter Decomposition --- p.34 / Chapter 4.4.2 --- Hubbard-St rant onovich Transformation --- p.34 / Chapter 4.5 --- Green Functions in DQMC And Wick's Theorem --- p.37 / Chapter 4.5.1 --- Equal-'Time' Green Functions --- p.37 / Chapter 4.5.2 --- Unequal-'Time' Green Functions --- p.38 / Chapter 4.5.3 --- Wick's Theorem --- p.38 / Chapter 4.6 --- Practical Consideration of DQMC --- p.39 / Chapter 4.6.1 --- Metropolis for DQMC --- p.39 / Chapter 4.6.2 --- Updating the Equal-time Green function --- p.41 / Chapter 4.6.3 --- Matrix Multiplication Stabilization --- p.42 / Chapter 4.6.4 --- A Survey In Negative Sign Problem --- p.43 / Chapter 4.6.5 --- Insight from Feynman --- p.44 / Chapter 4.6.6 --- Sign Problem As A NP-hard problem --- p.45 / Chapter 4.6.7 --- Personal Account on the Problem --- p.46 / Chapter 5 --- Dynamical Mean Field Theory --- p.47 / Chapter 5.1 --- Classical Mean Field Theory . --- p.49 / Chapter 5.2 --- DMFT as an Impurity Problem --- p.52 / Chapter 5.3 --- Scaling of Hopping Integral and Self-consistent Condition --- p.53 / Chapter 5.4 --- A QMC Impurity Solver --- p.55 / Chapter 5.5 --- Further development of DMFT --- p.56 / Chapter 6 --- Results and Discussion --- p.57 / Chapter 6.1 --- Physical Observable --- p.57 / Chapter 6.2 --- Methodology of the Studies --- p.58 / Chapter 6.3 --- Results --- p.59 / Chapter 6.4 --- Discussion and Suggestion --- p.83 / Chapter A --- DMRG of tight-binding model --- p.89 / Chapter B --- Scaling and Density of State of D-dimensional Hubbard Model --- p.94

Deng Shusa = 费米系統中的量子纠缠 : 在一维哈伯德模型中的研究 / 邓蜀萨. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2005. / Includes bibliographical references (leaves 85-90). / Text in English; abstracts in English and Chinese. / Deng Shusa = Feimi xi tong zhong de liang zi jiu chan : zai yi wei Habode mo xing zhong de yan jiu / Deng Shusa. / Abstract --- p.i / Acknowledgement --- p.iii / Chapter 1 --- Introduction --- p.1 / Chapter 1.1 --- Motivation --- p.1 / Chapter 1.2 --- Introduction to our study on quantum entanglement --- p.2 / Chapter 1.3 --- Introduction to Quantum Entanglement --- p.3 / Chapter 1.4 --- Introduction to Quantum Phase Transition --- p.7 / Chapter 1.5 --- Introduction to Extended Hubbard Model --- p.9 / Chapter 1.6 --- Arrangement of thesis writing --- p.14 / Chapter 2 --- Measurements of Entanglement --- p.15 / Chapter 2.1 --- Von neumann entropy --- p.16 / Chapter 2.2 --- Concurrence --- p.20 / Chapter 2.3 --- Negativity --- p.22 / Chapter 2.4 --- Other measurements --- p.24 / Chapter 3 --- Fermionic concurrence --- p.26 / Chapter 3.1 --- The model and formulism --- p.27 / Chapter 3.2 --- Extended Hubbard dimer with two electrons --- p.31 / Chapter 3.3 --- Dimer under a nonuniform field --- p.38 / Chapter 3.4 --- Large system for site=6 --- p.41 / Chapter 3.5 --- Negativity --- p.44 / Chapter 4 --- Block Entanglement --- p.48 / Chapter 4.1 --- The model and formulism --- p.50 / Chapter 4.2 --- Three-dimensional Phase diagram --- p.55 / Chapter 4.3 --- Entanglement change with block size and parameter --- p.62 / Chapter 4.4 --- Entanglement change with size and parameter --- p.66 / Chapter 4.5 --- Scaling behavior for block block entanglement --- p.70 / Chapter 4.6 --- Further discussion --- p.73 / Chapter 5 --- Conclusion --- p.82 / Bibliography --- p.85

Lu, Wenlong = 一维光格子中费米型原子混合物基态行为的玻色化研究 / 魯文龙. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2009. / Includes bibliographical references (p. 70-72). / Abstract also in Chinese. / Lu, Wenlong = Yi wei guang ge zi zhong Feimi xing yuan zi hun he wu ji tai xing wei de Bose hua yan jiu / Lu Wenlong. / Chapter 1 --- Introduction --- p.1 / Chapter 1.1 --- Cold-atom systems --- p.1 / Chapter 1.1.1 --- Optical lattices --- p.2 / Chapter 1.1.2 --- Feshbach resonance --- p.3 / Chapter 1.2 --- Outline of the thesis --- p.6 / Chapter 2 --- Bosonization method --- p.8 / Chapter 2.1 --- Special property of one-dimensional Fermion system --- p.9 / Chapter 2.2 --- Bosonization techniques --- p.13 / Chapter 2.2.1 --- Density operators as bosonic fields --- p.14 / Chapter 2.2.2 --- Bosonization Identities --- p.17 / Chapter 2.3 --- Renormalization analysis for Sine-Gordon field --- p.19 / Chapter 2.4 --- Summary --- p.25 / Chapter 3 --- Mass imbalance in the spin polarized fermion system --- p.26 / Chapter 3.1 --- Kinetic term --- p.29 / Chapter 3.2 --- Interaction term --- p.32 / Chapter 3.3 --- Phase separation --- p.38 / Chapter 3.4 --- Dominant order and pairing behavior --- p.47 / Chapter 3.5 --- Summary --- p.49 / Chapter 4 --- Mass imbalance in the strong repulsive interaction region --- p.50 / Chapter 4.1 --- Effective Hamiltonian at large U limit --- p.50 / Chapter 4.2 --- Bosonization of t-J-Jz model --- p.54 / Chapter 4.3 --- Phase separation --- p.60 / Chapter 4.4 --- Summary --- p.67 / Chapter 5 --- Conclusions --- p.68 / Bibliography --- p.70 / Chapter A --- Proofs of Bosonization --- p.73 / Chapter A.1 --- Anti-commutation relations between two branches of fermionic field operators --- p.73 / Chapter A.2 --- Bosonization-identities checking --- p.74 / Chapter B --- Diagonalization of Quadratic Hamiltonian with Two Bosonic Fields --- p.77 / Chapter C --- Correlation functions --- p.82

Wei, Bobo = 基於Bethe Ansatz解的兩個一維多體模型的研究 / 魏勃勃. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2008. / Includes bibliographical references (leaves 62-68). / Abstracts in English and Chinese. / Wei, Bobo = Ji yu Bethe Ansatz jie de liang ge yi wei duo ti mo xing de yan jiu / Chapter 1 --- Introduction --- p.1 / Chapter 1.1 --- Cold atoms systems --- p.1 / Chapter 1.1.1 --- Optical lattice --- p.2 / Chapter 1.1.2 --- Feshbach resonance --- p.4 / Chapter 1.2 --- Outline of this work --- p.6 / Chapter 2 --- Review of Bethe ansatz method --- p.8 / Chapter 2.1 --- Introduction --- p.8 / Chapter 2.2 --- Coordinate Bethe ansatz: One-dimensional Bose gas --- p.10 / Chapter 2.2.1 --- N = 2 bosons case --- p.11 / Chapter 2.2.2 --- N = 3 bosons case --- p.13 / Chapter 2.2.3 --- Arbitrary N bosons case --- p.15 / Chapter 3 --- Persistent currents in the one-dimensional mesoscopic Hubbard ring --- p.18 / Chapter 3.1 --- Introduction --- p.18 / Chapter 3.2 --- The model and its Bethe ansatz soluiton --- p.20 / Chapter 3.3 --- The charge persistent current --- p.23 / Chapter 3.3.1 --- The charge persistent current and the on-site interaction U --- p.24 / Chapter 3.3.2 --- The charge persistent current and the system size L --- p.28 / Chapter 3.4 --- The spin persistent current --- p.30 / Chapter 3.4.1 --- The spin persistent current and the on-site interaction U --- p.30 / Chapter 3.4.2 --- The spin persistent current and the system size L --- p.32 / Chapter 3.5 --- Conclusions --- p.33 / Chapter 4 --- Exact results of two-component ultra-cold Fermi gas in a hard wall trap --- p.36 / Chapter 4.1 --- Introduction --- p.36 / Chapter 4.2 --- The model and its exact solution --- p.37 / Chapter 4.3 --- The Theoretical Background --- p.41 / Chapter 4.4 --- N = 2 --- p.44 / Chapter 4.4.1 --- Single-particle reduced density matrix and Position density distributions --- p.44 / Chapter 4.4.2 --- Momentum density distributions --- p.45 / Chapter 4.5 --- N = 3 --- p.46 / Chapter 4.5.1 --- Single-particle reduced density matrix --- p.46 / Chapter 4.5.2 --- Natural orbitals and their populations --- p.48 / Chapter 4.5.3 --- Momentum density distribution --- p.51 / Chapter 4.5.4 --- Two-particle density distributions --- p.53 / Chapter 4.6 --- Conclusions --- p.53 / Chapter 5 --- Summary and prospects --- p.54 / Chapter 5.1 --- Summary --- p.54 / Chapter 5.2 --- Prospects for further study --- p.55 / Chapter 5.2.1 --- Recent experimental advancements on realization of quantum gas --- p.55 / Chapter 5.2.2 --- Some recent work on FTG gas --- p.57 / Bibliography --- p.62 / Chapter A --- Explicit form of Bethe ansatz wave function for N = 2 fermions --- p.69 / Chapter B --- "Simplified form of Bethe ansatz wave function for N = 3, M=1 fermions" --- p.73 / Chapter C --- Explicit form of Single-particle reduced density matrix for free fermions --- p.79

In this dissertation, I have studied four theoretical problems in quantum tunneling, quantum computing, and high-temperature superconductivity. I have developed a generally-useful numerical tool for analyzing impurity-induced resonant-state images observed with scanning tunneling microscope (STM) in high temperature superconductors. The integrated tunneling intensities on all predominant sites have been estimated. The results can be used to test the predictions of any tight-binding model calculation. I have numerically simulated two-dimensional time-dependent tunneling of a Gaussian wave packet through a barrier, which contains charged ions. We have found that a negative ion in the barrier directly below the tunneling tip can deflect the tunneling electrons and drastically reduce the probability for them to reach the point in the target plane directly below the tunneling tip. I have studied an infinite family of sure-success quantum algorithms, which are introduced by C.-R. Hu [Phys. Rev. A {\bf 66}, 042301 (2002)], for solving a generalized Grover search problem. Rigorous proofs are found for several conjectures made by Hu and explicit equations are obtained for finding the values of two phase parameters which make the algorithms sure success. Using self-consistent Hartree-Fock theory, I have studied an extended Hubbard model which includes quasi-long-range Coulomb interaction between the holes (characterized by parameter V). I have found that for sufficiently large V/t, doubly-charged-antiphase-island do become energetically favored localized objects in this system for moderate values of U/t, thus supporting a recent conjecture by C.-R. Hu [Int. J. Mod. Phys. B {\bf 17}, 3284 (2003)].

Thesis (Ph. D.)--University of Oregon, 2000. / Typescript. Includes vita and abstract. Includes bibliographical references (leaves 199-204). Also available for download via the World Wide Web; free to UO users.