Hamiltonian theory of fractionally filled Chern bands

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, ${\rho }_{FCB}$, as a sum of Girvin-MacDonald-Platzman density operators, ${\rho }_{GMP}$, 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 $\nu =\frac{1}{5}$ and ${\sigma }_{xy}=\frac{2}{5}$. Our approach can be easily extended to fractionally filled, strongly interacting two-dimensional time-reversal-invariant topological insulators.

15 pages 12 Figs

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• Quantum Spin Hall Insulator State in HgTe Quantum Wells
• High temperature fractional quantum Hall states
• Fractional quantum Hall states at zero magnetic field
• Nearly-flat bands with nontrivial topology
• Fractional quantum Hall effect in the absence of Landau levels
• Fractional topological insulators in three dimensions
• Composite Fermion Theory for Bosonic Atoms in Optical Lattices
• Gauge-Fixed Wannier Wave-Functions for Fractional Topological Insulators
• Enhancing the stability of a fractional Chern insulator against competing phases
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