Configuration interaction matrix elements for the quantum Hall effect

TL;DR

This paper derives analytical expressions for two-body matrix elements in finite spherical quantum Hall systems, establishing a one-to-one relationship between spatial potentials and pseudopotentials, and demonstrates an inversion technique with examples.

Contribution

It introduces a method to analytically invert pseudopotentials to obtain unique spatial potentials in quantum Hall systems.

Findings

Derived explicit formulas for matrix elements using Legendre polynomials.

Established a one-to-one correspondence between spatial potentials and pseudopotentials.

Demonstrated the inversion technique with examples, including the harmonic pseudopotential.

Abstract

We derive analytic expressions for the two-body matrix elements in finite spherical quantum Hall systems in terms of a general scalar interaction expressed as a sum over Legendre polynomials, and we derive the corresponding pair pseudopotentials from these matrix elements. The relationship between the effective spatial potential and the pseudopotential is one-to-one in this framework, and we show how any complete model pseudopotential can be analytically inverted to give a unique corresponding spatial potential. As an example, we find the spatial potential that produces a harmonic pseudopotential and verify that it fails to break the angular momentum degeneracy of the many-body quantum Hall system. We also include additional examples to demonstrate the use of the inversion technique.