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 r
ise 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 AKLT-type MPS structure associated with an emergent SU(2) symmetry.
We study the $L^p$ boundedness of Riesz transform as well as the reverse inequality on Riemannian manifolds and graphs under the volume doubling property and a sub-Gaussian heat kernel upper bound. We prove that the Riesz transform is then bounded on
$L^p$ for $1 textless{} p textless{} 2$, which shows that Gaussian estimates of the heat kernel are not a necessary condition for this.In the particular case of Vicsek manifolds and graphs, we show that the reverse inequality does not hold for $1 textless{} p textless{} 2$. This yields a full picture of the ranges of $pin (1,+infty)$ for which respectively the Riesz transform is $L^p$ -bounded and the reverse inequality holds on $L^p$ on such manifolds and graphs. This picture is strikingly different from the Euclidean one.
