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

Doctor of Philosophy, The Ohio State University, 2020, Physics

Two-dimensional Anderson Transitions have inspired study since the 1950s, but much about their critical properties remains unknown. I examine multifractality, the appearance of fractal dimensions which are not linearly related in the level sets of a probability density function, in the context of the wavefunction of a system undergoing an Anderson Transition. In Part II, I explore the multifractality of a system under strong magnetic field and its dependence on disorder on the underlying randomness in the placement of impurities in the sample. I relate this to field theoretic methods used to study matter fields existing on curved manifolds. In Part III, I explore the general problem of finding fields that are scale-invariant under the renormalization group in a broad set of Anderson Transitions described by non-linear sigma models. I use a method for decomposing the function spaces on these models' target spaces to find these entirely algebraically.

Committee: Ilya Gruzberg PhD (Advisor); Yuri Kovchegov PhD (Committee Member); Yuan-Ming Lu PhD (Committee Member); Thomas Lemberger PhD (Committee Member) Subjects: Physics