Quantum symmetries of quantum spin systems

Symmetry is a fundamental tool to understand physical and mathematical systems. Classically, symmetries of a system are captured by a collection of invertible self-maps called a group. Collections of certain quantum mathematical objects naturally form higher categories, and thus they admit richer collections of non-invertible or quantum symmetries. Examples of such quantum mathematical objects include algebras of quantum observables and quantum spin systems, which are mathematical models of pieces of matter, where atomic degrees of freedom are represented by local Hilbert spaces. Topologically ordered quantum spin systems give local quantum error correction codes for quantum computation. The main goal of this project is to use techniques from operator algebras and higher category theory to study the quantum symmetries of topologically ordered quantum spin systems. This project also funds research training for both graduate and undergraduate students. This project has three main focuses. First, the PI will develop the operator algebra approach to topologically ordered quantum spin systems, using nets of von Neumann algebras to study superselection sectors. The PI will also make connections between superselection theory for topological order and the quantum information theoretic boundary algebra approach to topological order. Second, the PI will make rigorous connections between higher category theory and topologically ordered quantum spin systems by both developing new higher dimensional quantum spin systems using higher fusion category input data, and by showing topologically ordered quantum spin systems themselves naturally form higher categories. Third, the PI will develop the theory of fermionic and anyonic operator algebras to better study topologically ordered phases of matter which arise in nature. 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. NSF Award ID: 2554723 | Program: 01002627DB NSF RESEARCH & RELATED ACTIVIT | Principal Investigator: David Penneys | Institution: OHIO STATE UNIVERSITY, THE, COLUMBUS, OH | Award Amount: $300,000 View on NSF Award Search: https://www.nsf.gov/awardsearch/show-award/?AWD_ID=2554723 View on Research.gov: https://www.research.gov/awardapi-service/v1/awards/2554723.html