The composite fermion theory revisited: a microscopic derivation without Landau level projection

TL;DR

This paper rigorously derives the microscopic foundation of the composite fermion theory for fractional quantum Hall states, eliminating the need for Landau level projection and establishing exact model Hamiltonians for these topological phases.

Contribution

It provides a formal derivation of the electron-CF relationship without Landau level projection, connecting CF theory to pseudopotentials and classifying Abelian and non-Abelian states.

Findings

Exact model Hamiltonians for FQH states derived

All Abelian CF states shown to be equivalent to IQH states

Systematic construction and classification of non-Abelian CF states

Abstract

The composite fermion (CF) theory gives both a phenomenological description for many fractional quantum Hall (FQH) states, as well as a microscopic construction for large scale numerical calculation of these topological phases. The fundamental postulate of mapping FQH states of electrons to integer quantum Hall (IQH) states of CFs, however, was not formally established. The Landau level (LL) projection needed for the microscopic calculations is in some sense uncontrolled and unpredictable. We rigorously derive the unitary relationship between electrons and the CFs, showing the latter naturally emerge from special interactions within a single LL, without resorting to any projection by hand. In this framework, all FQH states topologically equivalent to those described by the conventional CF theory (e.g. the Jain series) have exact model Hamiltonians that can be explicitly derived, and we can easily generalise to FQH states from interacting CFs. Our derivations reveal fundamental connections between the CF theory and the pseudopotential/Jack polynomial constructions, and argue that all Abelian CF states are physically equivalent to the IQH states, while a plethora of non-Abelian CF states can be systematically constructed and classified. We also discuss about implications to experiments and effective field theory descriptions based on the descriptions with CFs as elementary particles.