Research Projects

The project aims to study and understand non-Abelian phases in strongly interacting quantum systems using ideas from the mathematical domain of set partitions. These non-Abelian phases occur in many systems, such as in fractional quantum Hall fluids, spin-liquids, etc. The wave functions of these phases are not readily amenable to numerical evaluations since they involve dividing the particles into clusters and symmetrizing/antisymmetrizing across the clusters, which scales poorly with the number of particles. In this project, we plan to circumvent this approach by considering matrix elements of permutation invariant operators, which are integrals over the particle coordinates, that can be evaluated by grouping many of the permutations that belong to the same set partition and evaluating only a few integrals. This significantly brings down the resources needed to compute quantities such as energies, pair-correlation, etc., for fairly large systems, which was hitherto not possible. This would enable the study of the delicate competition between these phases and other phases, such as charge-density-waves, and other Abelian phases. Furthermore, these non-Abelian phases host exotic anyonic excitations, whose braiding properties can now be demonstrated via the techniques that will be developed in this project, since we will be able to go to large enough systems to prevent anyons from overlapping with each other. In particular, we plan to demonstrate that a particular non-Abelian phase that is potentially experimentally realized in the 12/5 fractional quantum Hall effect hosts Fibonacci anyons, which can form building blocks of a universal fault-tolerant topological quantum computer.