Entangled Pauli Principles: the DNA of Quantum Hall Fluids

A formalism is developed for the rigorous study of solvable fractional quantum Hall parent Hamiltonians with Landau-level mixing. The idea of organization through ``generalized Pauli principles'' is expanded to allow for root-level entanglement, giving rise to ``entangled Pauli principles.'' Through the latter, aspects of the effective field theory description become ingrained in exact microscopic solutions for a great wealth of phases for which no similar single Landau-level description is known. We discuss in detail braiding statistic, edge theory, and rigorous zero-mode counting for the Jain-221 state as derived from a microscopic Hamiltonian. The relevant root-level entanglement is found to feature an Affleck-Kennedy-Lieb-Tasaki (AKLT)-type matrix-product ground states structure associated with an emergent SU(2) symmetry.