Isotropic to anisotropic transition in a fractional quantum Hall state

Abstract

We study an 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 ei2, characteristic of a quantum critical point with dynamical critical exponent z=2, and a level-k Chern-Simons coupling, which is 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≠0 renders the scalar description nonlocal. The k≠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 nontrivial ground-state degeneracy kg when it is placed on a finite-size Riemann surface of genus g. The coefficient of ei2 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.