Emergence of Haldane Pseudo-Potentials in Systems with Short-Range Interactions

Area of Science:

• Condensed Matter Physics
• Quantum Mechanics

Background:

• The fractional quantum Hall effect (FQHE) describes complex quantum phenomena in 2D electron systems under strong magnetic fields.
• Investigating strong, short-range interactions is crucial for understanding FQHE properties beyond simple models.

Purpose of the Study:

• To mathematically establish the emergence of Haldane's pseudo-potential operators from short-range, strong repulsive two-body interactions.
• To derive accurate formulas for these interactions, particularly for hard-sphere potentials, in the context of the lowest Landau level.

Main Methods:

• Mathematical analysis of strong, short-range potentials in the FQHE regime.
• Derivation of interaction formulas based on scattering lengths rather than potential moments.
• Norm-resolvent convergence of Hamiltonians to contact interactions.

Main Results:

• Haldane's pseudo-potential operators are rigorously derived as limits of short-range potentials.
• Formulas involving scattering lengths are established as appropriate for strong interactions, unlike moment-based expansions.
• The results are shown to be valid for both bosons and fermions.

Conclusions:

• The study provides a rigorous mathematical foundation for using pseudo-potentials to model strong short-range interactions in FQHE.
• The derived formulas offer a more accurate approach for systems with hard-sphere-like potentials.
• This work generalizes previous findings and applies to a broader range of particles in the lowest Landau level.