  We derive full analytic expressions of three-body interactions from Landau
level (LL) mixing in fractional quantum Hall (FQH) systems with
Schrieffer-Wolff transformation. The formalism can be applied to any LL, and to
very general systems without rotational or Galilean invariance. We illustrate
how three-body pseudopotentials (PPs) can be readily computed from the
analytical expressions for a wide variety of different systems, and show that
for realistic systems, softening the bare Coulomb interactions (e.g. finite
thickness or screening) can significantly suppress three-body interactions.
More interestingly, for experimental systems without Galilean invariance (which
is common for real materials), there is strong evidence that higher orders in
band dispersion can drive the Moore-Read state from anti-Pfaffian to Pfaffian
phase. Our analysis points to the importance of the realistic band structure
details to the non-Abelian topological phases, and the analytical expressions
we derived can also be very useful for high fidelity numerical computations.
