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We derive full analytic expressions of three-body interactions from Landaulevel (LL) mixing in fractional quantum Hall (FQH) systems withSchrieffer-Wolff transformation. The formalism can be applied to any LL, and tovery general systems without rotational or Galilean invariance. We illustratehow three-body pseudopotentials (PPs) can be readily computed from theanalytical expressions for a wide variety of different systems, and show thatfor realistic systems, softening the bare Coulomb interactions (e.g. finitethickness or screening) can significantly suppress three-body interactions.More interestingly, for experimental systems without Galilean invariance (whichis common for real materials), there is strong evidence that higher orders inband dispersion can drive the Moore-Read state from anti-Pfaffian to Pfaffianphase. Our analysis points to the importance of the realistic band structuredetails to the non-Abelian topological phases, and the analytical expressionswe derived can also be very useful for high fidelity numerical computations.

Aspects of Three-body Interactions in Generic Fractional Quantum Hall Systems and Impact of Galilean Invariance Breaking