  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 through several examples, including certain
non-Abelian multicomponent states whose parent Hamiltonians were previously
unknown, and verify the method by numerically computing their entanglement
properties.
