Multi-Resolution Analysis and Fractional Quantum Hall Effect: More Results

TL;DR

This paper extends previous work on multi-resolution analysis and the fractional quantum Hall effect, simplifying the construction of wave functions, exploring different lattice geometries, and applying the method to estimate Coulomb energy at specific filling factors.

Contribution

It simplifies the construction of Landau level wave functions from multi-resolution analysis and extends the framework to triangular lattices and odd inverse filling factors.

Findings

Extended the analysis to triangular lattices.

Provided an approximation of Coulomb energy at filling factor 1/3.

Clarified the role of the kq-representation in the construction.

Abstract

In a previous paper we have proven that any multi-resolution analysis of $L^2(\R)$ produces, for even values of the inverse filling factor and for a square lattice, a single-electron wave function of the lowest Landau level (LLL) which, together with its (magnetic) translated, gives rise to an orthonormal set in the LLL. We have also discussed the inverse construction. In this paper we simplify the procedure, clarifying the role of the kq-representation. Moreover, we extend our previous results to the more physically relevant case of a triangular lattice and to odd values of the inverse filling factor. We also comment on other possible shapes of the lattice as well as on the extension to other Landau levels. Finally, just as a first application of our technique, we compute (an approximation of) the Coulomb energy for the Haar wavefunction, for a filling $\nu={1/3}$.