Fractional Quantum Hall Effect from Hilbert Space Algebra and Frustration-free Hamiltonians*

We show that model states of fractional quantum Hall (FQH) fluids for many topological phases can be uniquely determined by the Hilbert space algebra manifested as the classical reduced density matrix constraints, or the local exclusion constraint (LEC). The scheme allows us to identify filling factors, topological shifts and clustering of topological quantum fluids universally without resorting to microscopic Hamiltonians. Elementary excitations of the FQH phases can also be characterised by the LECs. More interestingly, the LEC formalism leads to a new perspective for the FQH model Hamiltonians, which can now be understood as a the von Neumann lattice of local potentials. The reformulation of the FQHE as a frustration free Hamiltonian as a sum of local projections opens up new path for rigorously proving the incompressibility of microscopic Hamiltonians in the thermodynamic limit. It may also potentially lead to new experimental ways of stabilising exotic FQH phases (related papers: Bo Yang, PRL. 125, 176402 (2020), PRB 100, 241302(R) (2019), Bo Yang, Ajit Balram, arXiv:1907.09493, Bo Yang, Ying-Hai Wu, Zlatko Papic, PRB. 100, 245303 (2019)).