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

In 2011, Haldane showed the existence of an internal geometric degree of freedom in the description of incompressible fractional quantum Hall states. The static value of this metric tells us how the quantum Hall system reacts in the presence of anisotropy, e.g. in the electron-electron interaction. We implement this geometry into Shankar and Murthy's Hamiltonian theory, which provides an analytical framework for Jain's composite fermion (CF) picture according to which the fractional quantum Hall effect arises from an integer number of fully filled CF Landau levels. Here, we 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 induces anisotropy into the effective two-dimensional interactions. We find that this anisotropy introduces mixing of CF Landau levels and thus perturbs the Hartree-Fock CF state of the Hamiltonian theory. By changing the internal geometry of the CF, such a perturbation can be minimized by optimizing the underlying metric, and we calculate the corresponding activation gaps for different tilt angles. Our results show that the activation gaps are remarkably robust against the in-plane magnetic field in the lowest and first Landau levels.