Quantum mechanics of composite fermions

TL;DR

This paper develops a comprehensive quantum mechanical framework for composite fermions in fractional quantum Hall states, introducing a wave equation, a wave function ansatz, and a microscopic approach to derive effective Hamiltonians from first principles.

Contribution

It presents a new, deductive approach to construct physical wave functions of fractional quantum Hall states using a wave equation and ansatz, grounded in microscopic models.

Findings

Wave equation with drift velocity corrections differs from Halperin-Lee-Read theory.

Reinterpretation of Jain's wave function ansatz as a new composite fermion representation.

Framework applicable to ideal fractional quantum Hall states and generalizable to flat Chern bands.

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.