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We introduce a broad class of simple models for quantum Hall states based onthe expansion of their parent Hamiltonians near the one-dimensional limit of"thin cylinders", i.e. when one dimension $L_y$ of the Hall surface becomescomparable to the magnetic length $\ell_B$. Formally, the models can be viewedas topological generalizations of the 1D Hubbard model withcenter-of-mass-preserving hopping of multiparticle clusters. In some cases, weshow that the models can be exactly solved using elementary techniques, andyield simple wave functions for the ground states as well as the entire neutralexcitation spectrum. We study a large class of Abelian and non-Abelian statesin this limit, including the Read-Rezayi $\mathbb{Z}_k$ series, as well asstates deriving from non-unitary conformal field theories -- the "Gaffnian","Haffnian", Haldane-Rezayi, and the "permanent" state. We find that thethin-cylinder limit of unitary states is "classical": their effectiveHamiltonians reduce to only Hartree-type terms, the ground states are trivialinsulators, and excitation gaps result from simple electrostatic repulsion. Incontrast, for states deriving from non-unitary conformal field theories, thethin-cylinder limit is found to be intrinsically \emph{quantum} -- it containshopping terms that play an important role in the structure of the ground statesand in the energetics of the low-lying neutral excitations.

Solvable models for unitary and non-unitary topological phases