Quantum Symmetries of Topological Phases of Matter

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Ordinary phases of matter (gas, liquid, solid) are distinguished by their symmetries: transformations that leave them unchanged--for example a ninety degree rotation of a cubic solid. Exotic quantum states of matter present under extreme conditions (low temperatures, strong magnetic fields) exhibit symmetries that resist a simple geometric description. Rather, their symmetries are understood through topology: qualitative geometry in which angles and lengths are ignored. The study of topological states is as important for their application to quantum technologies as for understanding the physical world. Of particular interest are the applications in quantum information. The investigator will study the mathematical symmetries of these topological phases of matter for the purpose of classifying and distinguishing them, probing their properties, and understanding how they are related through phase transitions. Some emphasis will be on how the properties of these phases of matter might find utility in quantum technologies. The investigator will employ theoretical and computational methods to study mathematical models for topological phases of matter. While two-dimensional topological phases of matter have been well-studied, many important questions remain. Three-dimensional systems with topological features are playing an increasingly substantial role yet are not as well-studied from a rigorous mathematical perspective. Two key themes in quantum symmetries are braided fusion categories and motion group representations: the first models the topologically invariant features of topological phases of matter, and the second encodes the topological dynamics of anyons and loop-like excitations. Understanding how the models are related through symmetries and phase transitions will provide a clearer picture of the landscape of topological phases of matter. In a complementary direction, the investigator will develop methods to understand the physically relevant representations of the braid group and higher dimensional generalizations. In three dimensions there is tension between the sensitivity of the topological invariants and the physically motivated assumption of unitarity. This challenge will be met through the study of non-semisimple categories and representations.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.