From CFT matrix product states to parent Hamiltonians

We study frustration free (FF) Hamiltonians of fractional quantum Hall (FQH) states from the point of view of the Matrix-product-state (MPS) representation of their ground/excited states. There is a wealth of solvable models relating to FQH physics, which, however, is mostly derived and analyzed from vantage point of first quantized "analytic clustering properties". In contrast, one obtains long-ranged FF lattice models when these Hamiltonians are studied in an orbital basis, which is the natural basis for the MPS representation of FQH state that has been of much interest lately. The connection between MPS-like states and frustration free parent Hamiltonians is central guiding principle in the construction of solvable lattice models, but thus far, only for short range Hamiltonians and MPS of finite bond dimension. The situation in the FQH context is fundamentally different. Here we expose the direct link between the infinite-bond-dimension MPS structure of Laughlin and Moore-Read-type CFT states and their parent Hamiltonians. Possible uses for states that lack a representation with nice analytic clustering properties are discussed.