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 ${e}_{i}^{2}$, 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 $k\ensuremath{\ne}0$ renders the scalar description nonlocal. The $k\ensuremath{\ne}0$ 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 ${k}^{g}$ 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.