Sequencing the entangled DNA of fractional quantum Hall fluids

We introduce and prove the ``root theorem'', which establishes a sufficient condition for families of operators to annihilate all root states associated to zero modes of a given positive semi-definite k-body Hamiltonian. Our theorem streamlines analysis of zero-modes in contexts where ``generalized'' or ``entangled'' Pauli principles apply, in particular, frustration free parent Hamiltonians in the fractional quantum Hall regime and related problems. One major application of the theorem is to parent Hamiltonians for mixed Landau level wave functions, such as unprojected composite fermion or parton-like states that were recently discussed in the literature, where it is difficult to rigorously establish a complete set of zero modes with traditional polynomial techniques. As a simple application we show that a modified V1 pseudo-potential, obtained via retention of only half the terms, stabilizes the $ u=1/2$ Tao-Thouless state as the unique densest ground state.