Many fractional quantum Hall wave functions are known to be unique andhighest-density zero modes of certain "pseudopotential" Hamiltonians. Examplesinclude the Read-Rezayi series (in particular, the Laughlin, Moore-Read andRead-Rezayi Z_3 states), and more exotic non-unitary (Haldane-Rezayi, Gaffnianstates) or irrational states (Haffnian state). While a systematic method toconstruct such Hamiltonians is available for the infinite plane or spheregeometry, its generalization to manifolds such as the cylinder or torus, whererelative angular momentum is not an exact quantum number, has remained an openproblem. Here we develop a geometric approach for constructing pseudopotentialHamiltonians in a universal manner that naturally applies to all geometries.Our method generalizes to the multicomponent SU(n) cases with a combination ofspin or pseudospin (layer, subband, valley) degrees of freedom. We demonstratethe utility of the approach through several examples, including certainnon-Abelian multicomponent states whose parent Hamiltonians were previouslyunknown, and verify the method by numerically computing their entanglementproperties.
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