Advancing a microscopic field-theoretic framework that rigorously unveils the underlying topological hallmarks of fractional quantum Hall (FQH) fluids is a prerequisite for making progress in the classification of strongly-coupled topological matter. Here we advance such a framework that helps reveal an exact fusion mechanism for particle fractionalization in FQH fluids, and uncover the fundamental structure behind the condensation of non-local operators characterizing topological order in the lowest-Landau-level (LLL). We show the first exact analytic computation of the quasielectron Berry connection and perform Monte Carlo simulations that numerically confirm the fusion mechanism for quasiparticles. Thus, two quasiholes plus one electron leads to an exact quasielectron state of fractional charge e/3 in a nu=1/3 Laughlin fluid. We express, in a compact manner, the sequence of (both bosonic and fermionic) Laughlin second-quantized states in a manner that highlights the lack of local condensation. Furthermore, we present a rigorous constructive subspace bosonization dictionary for the bulk fluid and establish the universal long-distance behavior of edge excitations by formulating a conjecture based on the DNA, or root state, of the FQH fluid.

We perform exact diagonalization studies for fractional quantum Hall states at filing factor 4/5 in a bilayer system, on a torus with various aspect ratios and angles. We find that in the absence of tunneling, two weakly coupled 2/5-layers undergo a phase transition into an interlayer-correlated regime, which is also Abelian with the five-fold degeneracy on the torus. In the limit of zero layer separation, this phase becomes a singlet in the pseudospin variable describing the layer degreeof-freedom. By studying the Chern-number matrix, we show that the K-matrix describing the interlayer-correlated regime requires matrix dimension larger than two and this regime is in particular not described by a Halperin state. A detailed analysis of possible 4 × 4 K-matrices having the requisite symmetries and quantum numbers shows that there is only one equivalence class of such matrices. A model wave function representing this universality class is constructed. The role of separate particle number conservation in both layers is discussed, and it is argued that this additional symmetry allows for the further distinction of two different symmetry protected Abelian phases in the interlayer correlated regime. Interlayer tunneling breaks this symmetry, and can drive the system into a single-layer regime when strong enough. A qualitative phase diagram in the tunneling-layer separation parameter space is proposed based on our numerical results.