Title: Tunable SU(4) Symmetry in Bilayer Graphene
Description: The charge carriers in bilayer graphene have both spin and valley degeneracy. Because of its unique electronic structure, valley symmetry in this material is connected to layer symmetry, which can be broken experimentally with a transverse electric field. Together with the coupling of spin to external magnetic fields, bilayer graphene makes for an experimental system with a tunable SU(4) symmetry space. This thesis describes experiments performed on ultra-high-quality dual-gated bilayer graphene heterostructures. In the quantum Hall regime, electric and magnetic fields can be used to probe the ordering of ground states, and to induce new orderings. In particular, at charge neutrality and high Zeeman splitting, we are able to tune the system to a ferromagnetic phase which exhibits a crossing of oppositely spin-polarized edge states, mimicking the quantum spin Hall effect. At higher fields in cleaner samples, we observe fractional quantum Hall states. These states also exhibit phase transitions, which show a clear but non-trivial connection with the phase transitions observed in integer quantum Hall states.

In higher Landau levels, while the connection between layer and valley changes, we still observe phase transitions between different quantum Hall states by applying transverse displacement fields. We identify clear patterns in these phase transitions over a number of Landau levels. Finally, we present experiments on bilayer graphene with an aligned split-gate geometry. This system is predicted to support topologically-protected valley-polarized states. We discuss fabrication challenges and preliminary experimental results.