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

The main work of this thesis involves the calculation, using the Bethe ansatz, of two of the signature quantities of the one-dimensional delta-function Bose gas. These are the density matrix and concomitantly its Fourier transform the occupation numbers, and the correlation function and concomitantly its Fourier transform the structure factor. The coefficient of the delta-function is called the coupling constant; these quantities are calculated in the finite-coupling regime, both expansions around zero coupling and infinite coupling are considered. Further to this, the density matrix in the infinite coupling limit, and its first order correction, is recast into Toeplitz determinant form. From this the occupation numbers are calculated up to 36 particles for the ground state and up to 26 particles for the first and second excited states. This data is used to fit the coefficients of an ansatz for the occupation numbers. The correlation function in the infinite coupling limit, and its first order correction, is recast into a form which is easy to calculate for any N, and is determined explicitly in the thermodynamic limit.