A Study of Geometry in Anisotropic Quantum Hall States by Principal Component Analysis

In the presence of mass anisotropy, anisotropic interaction, or in-plane magnetic field, quantum Hall droplets can exhibit shape deformation and internal geometrical degree of freedom. We characterize the geometry of quantum Hall states by principal component analysis, which is a statistical technique that emphasizes variation in a dataset. We first test the method in an integer quantum Hall droplet with dipole-dipole interaction in disk geometry. In the subsequent application to fractional quantum Hall systems with anisotropic Coulomb interaction in torus geometry, we demonstrate that the principal component analysis can quantify the metric degree of freedom and predict the collapse of a $\nu= 1/3$ state. We also calculate the metric response to interaction anisotropy at filling fractions $\nu= 1/5$ and $2/5$ and show that the response is roughly the same within the same Jain sequence, but can differ at large anisotropy for different sequences.