  One of the most successful theories of a non-Fermi liquid metallic state is
the composite Fermi liquid (CFL) theory of the half-filled Landau level. In
this paper, we study continuous quantum phase transitions out of the CFL state
and into a Landau Fermi liquid, in the limit of no disorder and fixed particle
number. This transition can be induced by tuning the bandwidth of the Landau
level relative to the interaction energy, for instance through an externally
applied periodic potential. We find a transition to the Landau Fermi liquid
through a gapless Mott insulator with a Fermi surface of neutral fermionic
excitations. In the presence of spatial symmetries, we also find a direct
continuous transition between the CFL and the Landau Fermi liquid. The
transitions have a number of characteristic observable signatures, including
the presence of two crossover temperature scales, resistivity jumps, and
vanishing compressibility. When the composite fermions are paired instead, our
results imply quantum critical points between various non-Abelian topological
states, including the \nu = 1/2 Moore-Read Pfaffian (Ising x U(1) topological
order), a version of the Kitaev B phase (Ising topological order), and paired
electronic superconductors. To study such transitions, we use a projective
construction of the CFL, which goes beyond the conventional framework of flux
attachment to include a broader set of quantum fluctuations. These
considerations suggest a possible route to fractionalized Mott insulators by
starting with FQH states and tuning the Landau level bandwidth.

Strongly Correlated Electrons