Sequencing the Entangled DNA of Fractional Quantum Hall Fluids

We introduce and prove the "root theorem", which establishes a condition for families of operators to annihilate all root states associated with zero modes of a given positive semi-definite $k$-body Hamiltonian chosen from a large class. This class is motivated by fractional quantum Hall and related problems, and features generally long-ranged, one-dimensional, dipole-conserving terms. Our theorem streamlines analysis of zero-modes in contexts where "generalized" or "entangled" Pauli principles apply. 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 $V_1$ pseudo-potential, obtained via retention of only half the terms, stabilizes the $\nu=1/2$ Tao-Thouless state as the unique densest ground state.