A Novel Projection: Towards a Deeper Understanding of Mathematical Structures in the Composite Fermion Formalism

Abstract

Composite fermion (CF) wave functions are used to describe both a two dimensional electron gas in a strong magnetic field (the quantum Hall effect system) and rotating two dimensional atomic gases that can be either bosons or fermions. In this thesis a new method for projecting fermionic CF wave functions (here called method 3) to the lowest Landau level is investigated. The new projection is based on attaching a Jastrow factor to a bosonic CF wave function projected in the standard way. This will be compared to two other projection methods: method 1 or Girvin-Jach projection and method 2 or Jain-Kamilla projection. I compare the minimal cyclotron energy compact CF ground state candidates for the ∇²δ interaction both to each other and to the exact ground state found by numerical diagonalization for up to 8 particles. The conclusion is that the CF candidates typically are good approximations of the exact ground state, and that method 3 is almost as good as the other two at this. An important motivation for method 3 is that it preserves linear dependencies from the bosonic states. The puzzle of linear dependencies is a part of CF theory that is not properly understood. Certain important results for low angular momentum states have been found for the bosonic case. Since method 3 works quite well in the cases tested here some of these results can be imported to the fermionic case. I have compared the linear dependencies among minimal cyclotron energy compact states for all three projection methods. If approximate linear dependencies are considered as proper linear dependencies then for all tested cases the same linear dependencies hold for all three projection methods, up to slightly different coefficients.