Hamiltonian theory of fractionally filled Chern bands

Abstract

There is convincing numerical evidence that fractional quantum-Hall-like ground states arise in fractionally filled Chern bands. Here, we show that the Hamiltonian theory of composite fermions (CF) can be as useful in describing these states as it was in describing the fractional quantum Hall effect (FQHE) in the continuum. We are able to introduce CFs into the fractionally filled Chern-band problem in two stages. First, we construct an algebraically exact mapping which expresses the electron density projected to the Chern band ${\ensuremath{\rho}}_{\text{FCB}}$ as a sum of Girvin-MacDonald-Platzman density operators ${\ensuremath{\rho}}_{\text{GMP}}$ that obey the magnetic translation algebra. Next, following our Hamiltonian treatment of the FQH problem, we rewrite the operators ${\ensuremath{\rho}}_{\text{GMP}}$ 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 $\ensuremath{\nu}=\frac{1}{5}$ and ${\ensuremath{\sigma}}_{xy}=\frac{2}{5}$. Our approach can be easily extended to fractionally filled, strongly interacting two-dimensional time-reversal-invariant topological insulators.