Effect of Landau level mixing on the effective interaction between electrons in the fractional quantum Hall regime

We compute the effect of Landau-level mixing on the effective two-body and three-body pseudopotentials for electrons in the lowest and second Landau levels. We find that the resulting effective three-body interaction is attractive in the lowest relative angular momentum channel. The renormalization of the two-body pseudopotentials also shows interesting structure. We comment on the implications for the $\nu =5/2$ fractional quantum Hall state.

Figure 1

(a) The solid directed lines are electron lines and the dotted line denotes the Coulomb interaction. We define the four-point vertex ${\tilde{V}}_{4,3;2,1}^{{\alpha }^{\prime }{\beta }^{\prime };\beta \alpha }=V_{4,3;2,1}{\delta }^{\alpha {\alpha }^{\prime }}{\delta }^{\beta {\beta }^{\prime }}-V_{3,4;2,1}{\delta }^{\alpha {\beta }^{\prime }}{\delta }^{\beta {\alpha }^{\prime }}$ on the left-hand side because the combination on the right-hand side enters all of the diagrams which renormalize $u_2$ and $u_3$. (b) and (c) Two of the nine diagrams which renormalize $u_3$. (d) The three diagrams which renormalize $u_2$.