Fractional quantum Hall states: Frustration-free parent Hamiltonians and infinite-bond-dimension matrix-product-states

The study of frustration-free (FF) Hamiltonians and its relation to finite bond dimension matrix-product-states (MPS) has a long tradition. However, fractional quantum hall states (FQHS) do not quite follow this theme, since the known MPS representation of their ground states have infinite-bond-dimension. We present the beginnings of a framework for FF Hamiltonians with infinite bond dimensions MPS ground states. In doing so, we abandon the long established approach of deriving and analyzing FQHS parent Hamiltonians through analytic functions and clustering properties, replacing the latter by infinite-bond-dimension MPS and inductive schemes in particle number. We show how the parent Hamiltonians for Laughlin and Moore-Read states, which feature zero energy bosonic and/or Majorana excitations, are directly connected to the MPS representation. A possible extension to parent Hamiltonians for unprojected Jain states and their multi-component bosonic conformal field theory/MPS descriptions are also discussed.