Transport Properties of a Half-Filled Chern Band at the Electron and Composite Fermion Phases

We consider a half-filled Chern band and its transport properties in two phases that it may form: the electronic Fermi liquid and the composite-fermion Fermi liquid. For weak disorder, we show that the Hall resistivity for the former phase is very small, while for the latter it is close to 2h/e2, independent of the distribution of the Berry curvature in the band. At rising temperature and high frequency, we expect the Hall resistivity of the electronic phase to rise, and that of the composite-fermion phase to deviate from 2h/e2. At high frequency, sign changes are expected as well. Considering high-frequency transport, we show that the composite fermion phase carries a gapped plasmon mode that does not originate from long ranged Coulomb interaction, and we show how this mode, together with the reflection of electromagnetic waves off the Chern band, allows for a measurement of the composite-fermion Drude weight and Berry curvature. Finally, we consider a scenario of a mixed-phase transition between the two phases - for example, as a function of displacement field - and show that such transition involves an enhancement of the longitudinal resistivity, as observed experimentally.