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