Abstract: We will discuss a dimensional hierarchy of the following over lattice systems of qudits. (4) Four-dimensional local unitary circuits that implement the identity, (3) Locality-preserving unitaries in three dimensions that are not circuits or shifts, (2) locally generated subalgebras in two dimensions that admit commuting Hamiltonians realizing chiral anyon theories, (1) one-dimensional operator subalgebras that admit commuting Hamiltonian with nontrivial superselection sectors. The one- and two-dimensional subalgebras are intimately related to each other and illustrate the importance of underlying operator algebras towards classification of topological phases of matter. All examples here are exactly solvable and no infinite dimensional algebras will be needed.
The Physics Frontiers Centers (PFC) program supports university-based centers and institutes where the collective efforts of a larger group of individuals can enable transformational advances in the most promising research areas. The program is designed to foster major breakthroughs at the intellectual frontiers of physics by providing needed resources such as combinations of talents, skills, disciplines, and/or specialized infrastructure, not usually available to individual investigators or small groups, in an environment in which the collective efforts of the larger group can be shown to be seminal to promoting significant progress in the science and the education of students. PFCs also include creative, substantive activities aimed at enhancing education, broadening participation of traditionally underrepresented groups, and outreach to the scientific community and general public.