Quantum mechanics of composite fermions

Abstract

We establish the quantum mechanics of composite fermions based on the dipole picture initially proposed by Read. It comprises three complimentary components: a wave equation for determining the wave functions of a composite fermion in ideal fractional quantum Hall states and when subjected to external perturbations, a wave function ansatz for mapping a many-body wave function of composite fermions to a physical wave function of electrons, and a microscopic approach for determining the effective Hamiltonian of the composite fermion. The wave equation resembles the ordinary Schr\"{o}dinger equation but has drift velocity corrections which are not present in the Halperin-Lee-Read theory. The wave-function ansatz constructs a physical wave function of electrons by projecting a state of composite fermions onto a half-filled bosonic Laughlin state of vortices. Remarkably, Jain's wave function ansatz can be reinterpreted as the new ansatz in an alternative wave-function representation of composite fermions. The dipole model and the effective Hamiltonian can be derived from the microscopic model of interacting electrons confined in a Landau level, with parameters fully determined. In this framework, we can construct the physical wave function of a fractional quantum Hall state deductively by solving the wave equation and applying the wave function ansatz, based on the effective Hamiltonian derived from first principles, rather than relying on intuitions or educated guesses. For ideal fractional quantum Hall states in the lowest Landau level, the approach yields physical wave functions identical to those prescribed by the standard theory of composite fermions. We further demonstrate that the reformulated theory of composite fermions can be easily generalized for flat Chern bands.