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

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