  We study a novel abelian gauge theory in 2+1 dimensions which has surprising
theoretical and phenomenological features. The theory has a vanishing
coefficient for the square of the electric field $e_i^2$, characteristic of a
quantum critical point with dynamical critical exponent $z=2$, and a level-$k$
Chern-Simons coupling, which is {\it marginal} at this critical point. For
$k=0$, this theory is dual to a free $z=2$ scalar field theory describing a
quantum Lifshitz transition, but $k \neq 0$ renders the scalar description
non-local. The $k \neq 0$ theory exhibits properties intermediate between the
(topological) pure Chern-Simons theory and the scalar theory. For instance, the
Chern-Simons term does not make the gauge field massive. Nevertheless, there
are chiral edge modes when the theory is placed on a space with boundary, and a
non-trivial ground state degeneracy $k^g$ when it is placed on a finite-size
Riemann surface of genus $g$. The coefficient of $e_i^2$ is the only relevant
coupling; it tunes the system through a quantum phase transition between an
isotropic fractional quantum Hall state and an anisotropic fractional quantum
Hall state. We compute zero-temperature transport coefficients in both phases
and at the critical point, and comment briefly on the relevance of our results
to recent experiments.
