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1. Wang, Chunguang. Topics in Bethe ansatz.

Degree: PhD, Physics (Arts and Sciences), 2017, University of Miami

URL: https://scholarlyrepository.miami.edu/oa_dissertations/1827

Integrable quantum spin chains have close connections to integrable quantum field
theories, modern condensed matter physics, string and Yang-Mills theories. Bethe ansatz is one of the most important approaches for solving quantum integrable spin chains. At the heart of the algebraic structure of integrable quantum spin chains is the quantum Yang-Baxter equation and the boundary Yang-Baxter equation. This thesis focuses on four topics in Bethe ansatz. The Bethe equations for the isotropic periodic spin-1/2 Heisenberg chain with N sites have solutions containing ±i/2 that are singular: both the corresponding energy and the algebraic Bethe ansatz vector are divergent. Such solutions must be carefully regularized. We consider a regularization involving a parameter that can be determined using a generalization of the Bethe equations. These generalized Bethe
equations provide a practical way of determining which singular solutions correspond to eigenvectors of the model. The Bethe equations for the periodic XXX and XXZ spin chains admit singular solutions, for which the corresponding eigenvalues and eigenvectors are ill-defined. We use a twist regularization to derive conditions for such singular solutions to be physical, in which case they correspond to genuine eigenvalues and eigenvectors of the Hamiltonian. We analyze the ground state of the open spin-1/2 isotropic quantum spin chain with a non-diagonal boundary term using a recently proposed Bethe ansatz solution. As the coefficient of the non-diagonal boundary term tends to zero, the Bethe roots split evenly into two sets: those that remain finite, and those that become infinite. We argue that the former satisfy conventional Bethe equations, while the latter satisfy a
generalization of the Richardson-Gaudin equations. We derive an expression for the
leading correction to the boundary energy in terms of the boundary parameters. We argue that the Hamiltonians for A(2) 2n open quantum spin chains corresponding to two choices of integrable boundary conditions have the symmetries Uq(Bn) and Uq(Cn), respectively. The deformation of Cn is novel, with a nonstandard coproduct. We find a formula for the Dynkin labels of the Bethe states (which determine the degeneracies of the corresponding eigenvalues) in terms of the numbers of Bethe roots of each type. With the help of this formula, we verify numerically (for a generic value of the anisotropy parameter) that the degeneracies and multiplicities of the spectra implied by the quantum group symmetries are completely described by the Bethe ansatz.
*Advisors/Committee Members: Rafael I. Nepomechie, Orlando Alvarez, Chaoming Song, Rodrigo A. Pimenta.*

Subjects/Keywords: Bethe ansatz; Integrable model; Spin chain; Singular; Boundary condition

…career and life.
Chunguang Wang
*University* *of* *Miami*
May 2017
iv
Table of Contents
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Record Details Similar Records

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^{th} Edition):

Wang, C. (2017). Topics in Bethe ansatz. (Doctoral Dissertation). University of Miami. Retrieved from https://scholarlyrepository.miami.edu/oa_dissertations/1827

Chicago Manual of Style (16^{th} Edition):

Wang, Chunguang. “Topics in Bethe ansatz.” 2017. Doctoral Dissertation, University of Miami. Accessed August 08, 2020. https://scholarlyrepository.miami.edu/oa_dissertations/1827.

MLA Handbook (7^{th} Edition):

Wang, Chunguang. “Topics in Bethe ansatz.” 2017. Web. 08 Aug 2020.

Vancouver:

Wang C. Topics in Bethe ansatz. [Internet] [Doctoral dissertation]. University of Miami; 2017. [cited 2020 Aug 08]. Available from: https://scholarlyrepository.miami.edu/oa_dissertations/1827.

Council of Science Editors:

Wang C. Topics in Bethe ansatz. [Doctoral Dissertation]. University of Miami; 2017. Available from: https://scholarlyrepository.miami.edu/oa_dissertations/1827

University of Miami

2. Murgan, Rajan. Bethe Ansatz and Open Spin-1/2 XXZ Quantum Spin Chain.

Degree: PhD, Physics (Arts and Sciences), 2008, University of Miami

URL: https://scholarlyrepository.miami.edu/oa_dissertations/69

The open spin-1/2 XXZ quantum spin chain with general integrable boundary terms is a fundamental integrable model. Finding a Bethe Ansatz solution for this model has been a subject of intensive research for many years. Such solutions for other simpler spin chain models have been shown to be essential for calculating various physical quantities, e.g., spectrum, scattering amplitudes, finite size corrections, anomalous dimensions of certain field operators in gauge field theories, etc. The first part of this dissertation focuses on Bethe Ansatz solutions for open spin chains with nondiagonal boundary terms. We present such solutions for some special cases where the Hamiltonians contain two free boundary parameters. The functional relation approach is utilized to solve the models at roots of unity, i.e., for bulk anisotropy values eta = i pi/(p+1) where p is a positive integer. This approach is then used to solve open spin chain with the most general integrable boundary terms with six boundary parameters, also at roots of unity, with no constraint among the boundary parameters. The second part of the dissertation is entirely on applications of the newly obtained Bethe Ansatz solutions. We first analyze the ground state and compute the boundary energy (order 1 correction) for all the cases mentioned above. We extend the analysis to study certain excited states for the two-parameter case. We investigate low-lying excited states with one hole and compute the corresponding Casimir energy (order 1/N correction) and conformal dimensions for these states. These results are later generalized to many-hole states. Finally, we compute the boundary S-matrix for one-hole excitations and show that the scattering amplitudes found correspond to the well known results of Ghoshal and Zamolodchikov for the boundary sine-Gordon model provided certain identifications between the lattice parameters (from the spin chain Hamiltonian) and infrared (IR) parameters (from the boundary sine-Gordon S-matrix) are made.
*Advisors/Committee Members: Rafael I. Nepomechie, James Nearing, Orlando Alvarez, Changrim Ahn.*

Subjects/Keywords: Integrable Models; Bethe Ansatz; Quantum Spin Chain

Record Details Similar Records

❌

APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^{th} Edition):

Murgan, R. (2008). Bethe Ansatz and Open Spin-1/2 XXZ Quantum Spin Chain. (Doctoral Dissertation). University of Miami. Retrieved from https://scholarlyrepository.miami.edu/oa_dissertations/69

Chicago Manual of Style (16^{th} Edition):

Murgan, Rajan. “Bethe Ansatz and Open Spin-1/2 XXZ Quantum Spin Chain.” 2008. Doctoral Dissertation, University of Miami. Accessed August 08, 2020. https://scholarlyrepository.miami.edu/oa_dissertations/69.

MLA Handbook (7^{th} Edition):

Murgan, Rajan. “Bethe Ansatz and Open Spin-1/2 XXZ Quantum Spin Chain.” 2008. Web. 08 Aug 2020.

Vancouver:

Murgan R. Bethe Ansatz and Open Spin-1/2 XXZ Quantum Spin Chain. [Internet] [Doctoral dissertation]. University of Miami; 2008. [cited 2020 Aug 08]. Available from: https://scholarlyrepository.miami.edu/oa_dissertations/69.

Council of Science Editors:

Murgan R. Bethe Ansatz and Open Spin-1/2 XXZ Quantum Spin Chain. [Doctoral Dissertation]. University of Miami; 2008. Available from: https://scholarlyrepository.miami.edu/oa_dissertations/69