Date of Award
1-1-2025
Document Type
Thesis
Degree Name
M.S. in Physics
First Advisor
Kevin S. D. Beach
Second Advisor
Gavin Davies
Third Advisor
Nicholas MacDonald
School
University of Mississippi
Relational Format
dissertation/thesis
Abstract
Spin chains are thoroughly studied models used throughout statistical physics. They make considerable simplifications to complex many-body systems that allow us to analyze complex behaviors such as phase transitions. With these large interacting systems, it is often impossible to solve them analytically. However, with the advent of digital computers, we are able to investigate these models numerically and to extrapolate to the thermodynamic limit. Our area of study concerns the Fredkin Spin Chain model. The Fredkin Spin Chain is a well studied model which has an exactly solvable ground state and an unusually large dynamical exponent, z. Much analysis has been done on the construction of the chain, its Hamiltonian, and the properties of its first excited state. However, exploration of the full excitation spectrum has yet to be done. In this paper, we analyze the Fredkin Spin Chain using Monte Carlo analysis with simple sampling techniques. We sweep over spin sectors 1–149 on spin chains ranging from sizes of 4 sites up to 300 sites. The main question that we wished to answer was if z has a single consistent value, or if it is dependent on the Hilbert space sector. There are three primary contributions to the body of work around the Fredkin Spin Chain in this paper. The first contribution is the creation of new Monte Carlo algorithms which allow one to generate arbitrary chain configurations. Prior work using Monte Carlo focused solely on configurations in which there were either zero, or one, canted bond. With our algorithm, one may generate spin chains with any (valid) number of canted, as well as mismatch, bonds in order to investigate behavior within every Hilbert subspace. Second, we were able to find closed form counting formulas, using novel generating function techniques, which count every possible configuration of the Fredkin Spin Chain. Finally, we investigate the behavior of the dynamical exponent, z, for an arbitrary number of canted bonds. We have found that, in the thermodynamic limit, z is a constant. For finite chain sizes, however, it becomes a function of the spin sector because of the geometric constraints of the system.
Recommended Citation
Irby, Coleman Kristopher, "Measuring the Dynamic Exponent Across the Excitation Spectrum of the Fredkin Spin Chain" (2025). Electronic Theses and Dissertations. 3298.
https://egrove.olemiss.edu/etd/3298