Interaction induced Landau level mixing in the fractional quantum Hall regime

We study Landau Level mixing in parabolic bands perturbatively to second order in the ratio of interaction to cyclotron energy, for the lowest ($N=0$) and first excited ($N=1$) Landau levels. The mixing is accounted for by constructing an effective Hamiltonian which includes two body and three body interactions. Our study builds upon two previous treatments~[1,2], using as a stepping stone the observation that the effective Hamiltonian is fully determined by the 2 and 3 body problems. For the $N=0$ problem we provide a compact and transparent derivation of the effective Hamiltonian using first quantization which captures a class of virtual processes omitted in earlier calculations of Landau-level mixing corrected Haldane pseudo-potentials. We will comment on potential application of our results for numerical studies.\\[4pt] [1] G. Murthy and R. Shankar, {\em PRB} {\bf 65}, 245309 (2002)\\[0pt] [2] W. Bishara and C. Nayak, {\em PRB} {\bf 80}, 121302 (2009).