Doctor of Philosophy, The Ohio State University, 2024, Physics

Phase transitions occur in many classical and quantum systems, and are the subject of many an open problem in physics. In the past decade, the conformal bootstrap has provided new perspectives for looking at the critical point of a transition. With this formalism, it is possible to exploit the conformal symmetry intrinsically present at the critical point, and derive general results about classes of transitions that obey the same symmetries. This thesis presents the application of this method to two problems of note in classical and quantum phase transitions. The first is a classical model of O(N) spins in the presence of a boundary. We use the numerical conformal bootstrap to prove rigorously the existence of a new boundary phase in three-dimensional Heisenberg (O(3)) and O(4) magnets, deemed the extraordinary-log universality class. The results agree well with a parallel numerical study but are more rigorous due to the bounded nature of the error. The second case is the inherently quantum problem of Anderson transitions between metals and insulators. It has been discovered that at criticality, the wavefunctions describe multifractal objects, that are described by infinitely many fractal dimensions. We use analytical constraints from conformal symmetry to predict the form of these fractal parameters in dimensions greater than two. Our exact prediction, which works in arbitrary dimensions, can be used as a probe for conformal symmetry at Anderson transitions. By studying these two problems, we demonstrate the power of conformal symmetry as a non-perturbative tool in the theory of phase transitions in arbitrary dimensions. Throughout the thesis, we have extended the domain of applicability of traditional bootstrap techniques for the purpose of non-unitary and non-positive systems.

Committee: Ilya Gruzberg (Advisor); Marc Bockrath (Committee Member); Samir Mathur (Committee Member); Yuanming Lu (Committee Member) Subjects: Condensed Matter Physics; Physics

PHD, Kent State University, 2005, College of Business and Entrepreneurship, Ambassador Crawford / Department of Finance

Asset pricing modelers attempt to identify price diffusion processes from empirical financial market data. In particular, the Geometric Brownian Motion and the GARCH models are currently popular in these efforts. In contrast, for the first time this dissertation identifies Multifractal Models of Asset Return (MMARs) from the eight nodal term structure series of US Treasury rates as well as Fed Funds rate and, after proper synthesis, simulates those MMARs. The model performance results of these simulations are then compared with not only the original empirical time series, but also with the simulated results from the corresponding Brownian Motion and GARCH processes. The major findings are that the eight different maturity US Treasury and the Fed Funds rates are multifractal processes. The MMAR outperforms both the GBM and GARCH(1,1) in terms of scaling distribution preservation over time and investment horizons. In addition, this dissertation uses the noise-data ratio to improve the Holder-Hurst identification for the power spectrum method. Identified distributions of all simulated processes are compared with the empirical distributions in snapshot and over time-scale (frequency) analyses. The findings suggest that the simulated MMAR can replicate all attributes of the empirical distributions, while the simulated GBM and GARCH(1,1) processes cannot preserve the thick-tails, high peaks, and skewness. The wavelet scalograms, used to investigate the variance over time and scales, reveal the superiority of the MMAR for modeling the Treasury rates over the GBM and GARCH(1,1). When the MMAR is applied to the Fed Funds rate, the results are surprisingly different from those of the Treasury rates. The MMAR at this stage cannot produce a complete term structure model, because it cannot completely model the dynamic structure of the term structure.

Committee: Cornelis Los (Advisor) Subjects: