Composite Fermi liquids in the lowest Landau level

We study composite Fermi liquid (CFL) states in the lowest Landau level (LLL) limit at a generic filling $\nu =\frac{1}{n}$. We begin with the old observation that, in compressible states, the composite fermion in the lowest Landau level should be viewed as a charge-neutral particle carrying vorticity. This leads to the absence of a Chern-Simons term in the effective theory of the CFL. We argue here that instead a Berry curvature should be enclosed by the Fermi surface of composite fermions, with the total Berry phase fixed by the filling fraction ${\varphi }_B=-2\pi \nu$. We illustrate this point with the CFL of fermions at filling fractions $\nu =1/2q$ and (single and two-component) bosons at $\nu =1/(2q+1)$. The Berry phase leads to sharp consequences in the transport properties including thermal and spin Hall conductances. We emphasize that these results only rely on the LLL limit and do not require particle-hole symmetry, which is present microscopically only for fermions at $\nu =1/2$. Nevertheless, we show that the existing LLL theory of the composite Fermi liquid for bosons at $\nu =1$ does have an emergent particle-hole symmetry. We interpret this particle-hole symmetry as a transformation between the empty state at $\nu =0$ and the boson integer quantum hall state at $\nu =2$. This understanding enables us to define particle-hole conjugates of various bosonic quantum Hall states which we illustrate with the bosonic Jain and Pfaffian states. For bosons at $\nu =1$ we construct paired non-Abelian states distinct from both the standard bosonic Pfaffian and its particle hole conjugate and show how they may arise naturally out of the neutral vortex composite Fermi liquid. The bosonic particle-hole symmetry can be realized exactly on the surface of a three-dimensional boson topological insulator. We also show that with the particle-hole and spin $SU\left(2\right)$ rotation symmetries, there is no gapped topological phase for bosons at $\nu =1$. Finally we comment on systems that are not strictly in the lowest Landau level limit and argue that our theory should still be applicable at low energy.

Figure 1

Cancelation of the Chern-Simons term by integrating out composite fermions deep in the Fermi sea. Final theory: a composite Fermi surface with Berry phase ${\varphi }_B=-2\pi /n$.