Gauge-Invariant Variables Reveal the Quantum Geometry of Fractional Quantum Hall States

Abstract: Herein, we introduce the framework of gauge invariant variables to describe fractional quantum Hall (FQH) states, and prove that the wavefunction can always be represented by a unique holomorphic multi-variable complex function. As a special case, within the lowest Landau level, this function reduces to the well-known holomorphic coordinate representation of wavefunctions in the symmetric gauge. Using this framework, we derive an analytic guiding center Schr\"odinger's equation governing FQH states; it has a novel structure. We show how the electronic interaction is parametrized by generalized pseudopotentials, which depend on the Landau level occupancy pattern; they reduce to the Haldane pseudopotentials when only one Landau level is considered. Our formulation is apt for incorporating a new combination of techniques, from symmetric functions, Galois theory and complex analysis, to accurately predict the physics of FQH states using first principles.