Hamiltonian theory for Quantum Hall systems in a tilted magnetic field: robustness of activation gaps

We use the Hamiltonian theory developed by Shankar and Murthy to study a quantum Hall system in a tilted magnetic field. With a finite width of the system in the $z$ direction, the parallel component of the magnetic field introduces anisotropy into the effective two-dimensional interactions. The effects of such anisotropy can be effectively captured by the recently proposed generalized pseudo-potentials. We find that the off-diagonal components of the pseudo-potentials lead to mixing of composite fermions Landau levels, which is a perturbation to the picture of $p$ filled Landau levels in composite-fermion theory. By changing the internal geometry of the composite fermions, such a perturbation can be minimized and one can find the corresponding activation gaps for different tilting angles, and we calculate the associated optimal metric. Our results show that the activation gap is remarkably robust against the in-plane magnetic field in the lowest Landau level.