Hamiltonian theory of fractionally filled Chern bands

TL;DR

This paper extends the Hamiltonian theory of Composite Fermions to fractionally filled Chern bands, providing a new framework to analyze FQH-like states without external magnetic fields, and demonstrating its effectiveness through concrete examples.

Contribution

It introduces a novel Hamiltonian approach for FCBs using CFs, enabling detailed analysis of fractional states in topological bands without magnetic fields.

Findings

Constructed an exact mapping of electron density in FCB to GMP operators.

Reformulated GMP operators in terms of CF variables, reproducing the algebra.

Provided examples with unique fractional states not present in continuum FQHE.

Abstract

There is convincing numerical evidence that fractional quantum Hall (FQH)-like ground states arise in fractionally filled Chern bands (FCB). Here we show that the Hamiltonian theory of Composite Fermions (CF) can be as useful in describing the FCB as it was in describing the FQHE in the continuum. We are able to introduce CFs into the FCB problem even though there is no external magnetic field by following a two-stage process. First we construct an algebraically exact mapping which expresses the electron density projected to the Chern band, ${\rho}_{{\tiny FCB}}$, as a sum of Girvin-MacDonald-Platzman density operators, ${\rho}_{{\tiny GMP}}$, that obey the Magnetic Translation Algebra. Next, following our Hamiltonian treatment of the FQH problem, we rewrite the GMP operators in terms of CF variables which reproduce the same algebra. This naturally produces a unique Hartree-Fock ground state for the CFs, which can be used as a springboard for computing gaps, response functions, temperature-dependent phenomena, and the influence of disorder. We give two concrete examples, one of which has no analog in the continuum FQHE with $\nu= {1 \over 5}$ and $\sigma_{xy}={2\over 5}$. Our approach can be easily extended to fractionally filled, strongly interacting two-dimensional time-reversal-invariant topological insulators.