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.