Fermions and Bosons as Emergent Particles in Fractional Quantum Hall Systems

Strongly correlated topological systems in two-dimension serve as good platforms for realising quantum fluids of anyons, which are exotic particles with statistical behaviours different from either Fermions or Bosons. The fractional Quantum Hall (FQH) system is a good example, where anyons can be readily created and they are useful for the robust storing and manipulation of quantum information. The bulk edge correspondence and the conformal symmetry of the FQH systems, however, allow us to either fermionize or bosonize anyons in this two-dimensional manifold. We show explicit unitary transformations from the many-body electronic and anyonic wavefunctions to the emergent bosonic and fermionic product states. Using this approach we reveal interesting dynamical properties of anyons even in very simple FQH phases, and new families of bosonic QH phases as dual descriptions of their fermionic counterparts. By connecting to the well known composite fermion (CF) theory for the FQH systems, we show how exact model Hamiltonians can be constructed for the Jain series and interacting CF states, and how the CF states (including the composite fermi liquid) can be constructed without using the LLL projection. An interesting example is the construction of an Abelian, gapped model Hamiltonian where the Gaffnian model wavefunction is the \emph{exact} ground state, revealing the hidden connection between the CF theory and the pseudopotential/Jack polynomial formalism. We discuss the nature of particle statistics in 2D conformal Hilbert spaces in general, as well as their experimental implications. (References: arXiv:2207.12418, PRL 127, 126406, PRL 127, 046402).