Geometric Construction of Quantum Hall Clustering Hamiltonians

Many fractional quantum Hall wave functions are known to be unique and highest-density zero modes of certain "pseudopotential" Hamiltonians. Examples include the Read-Rezayi series (in particular, the Laughlin, Moore-Read and Read-Rezayi Z_3 states), and more exotic non-unitary (Haldane-Rezayi, Gaffnian states) or irrational states (Haffnian state). While a systematic method to construct such Hamiltonians is available for the infinite plane or sphere geometry, its generalization to manifolds such as the cylinder or torus, where relative angular momentum is not an exact quantum number, has remained an open problem. Here we develop a geometric approach for constructing pseudopotential Hamiltonians in a universal manner that naturally applies to all geometries. Our method generalizes to the multicomponent SU(n) cases with a combination of spin or pseudospin (layer, subband, valley) degrees of freedom. We demonstrate the utility of the approach by several examples, including certain non-Abelian multicomponent states whose parent Hamiltonians were previously unknown, and verify the method by numerically computing their entanglement properties.