Fusion Mechanism for Quasiparticles and Topological Order for Quantum Hall Fluids in the Lowest-Landau-Level

We study systems exhibiting the fractional quantum Hall effect using the properties of the ring of symmetric polynomials and concepts originating from the second quantization formalism. These techniques are then advanced to examine elementary fractionally-charged excitations for such fluids. In particular, rigorous formulation of Laughlin's originally suggested fractionalization mechanism in the second quantization formalism led us to a derivation of the quasielectron wave function for systems with filling fractions 1/M, and the observation that these quasiparticles are subject to a particular fusion mechanism. Extending these ideas to more complex systems originating from Halperin's bilayer systems, we are able to rigorously study such excitations in non-Abelian fluids like Moore-Read (Pfaffian) or Hafnian states, as well as other classes of states described by more complex polynomials. Furthermore, the approach based on second quantization allows us to generate non-local composite (generalized Read) operators that characterize topological or the so-called off-diagonal long-range order. In particular, we rigorously find concrete forms of such operators for the families of Pfaffian and Hafnian quantum states.