# Exact and variational investigations of Hubbard rings

## Electronic Theses and Dissertations

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- title
- Exact and variational investigations of Hubbard rings
- author
- Hodge, William B.
- abstract
- We study the one-dimensional Hubbard model for interacting electrons in a crystal lattice, using two methods: solution of the Lieb-Wu equations in the atomic limit of no intersite hopping, and variational techniques to obtain a lower bound for the ground state energy. From the Lieb-Wu equations we derive an exact expression for the ground state energy in the atomic limit. We obtain our result by exploiting the spin symmetry of the Hubbard Hamiltonian. Our energy is lower than that obtained by previous published results, and we are able to explain why our results differ. Our method also gives information about the magnetic ordering of the ground state as a function of the number of electrons. Our results are valid for arbitrary electron density values. The variational techniques we use are applied to the two-particle reduced-density matrix (2-RDM), because this quantity suffices to obtain expectation values of all one- and two-particle operators, and because it is less computationally intensive than obtaining the complete N -particle wave function. Because the 2-RDM must be positive semidefinite, we use an efficient programming algorithm known as the Semidefinite Programming Algorithm. A naive search through the space of two-particle matrices encounters a difficult problem, known as the " N -representability problem", which is to assure that the trial density matrices actually come from a wave function that has the required exchange symmetry. This problem is ameliorated by imposing constraints on the variational search that eliminate trial matrices with the incorrect exchange symmetry. These constraints express known physical properties of the ground state (for example its total spin and its translational symmetry) and the so-called D, Q , and G two-positivity conditions that express the required exchange symmetry. To calculate matrix elements, we use the two-particle eigenstates of the reduced Hubbard Hamiltonian; this choice of basis significantly decreases the number of matrix elements needed for energy calculations. After an optimal two-particle density matrix has been obtained, we calculate additional properties of the ground state, including spin correlation functions, chemical potentials, and the momentum distribution. We compare all of these properties to values obtained by exact numerical solution of the Schrödinger equation.
- subject
- Hubbard rings
- Hubbard model
- Strongly correlated electrons
- Semidefinite programming
- One-dimensional systems
- Bethe ansatz
- contributor
- Akbar Salam, Ph.D. (committee chair)
- William Kerr, Ph.D. (committee member)
- Keith Bonin, Ph.D. (committee member)
- Natalie Holzwarth, Ph.D. (committee member)
- Freddie Salsbury, Jr., Ph.D. (committee member)
- date
- 2011-07-29T14:44:16Z (accessioned)
- 2011-07-29T14:44:16Z (available)
- 2008 (issued)
- degree
- Physics (discipline)
- Wake Forest University (grantor)
- Ph.D. (level)
- identifier
- http://hdl.handle.net/10339/33588 (uri)
- language
- en_US (iso)
- type
- Thesis

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