An Isotropic to Anisotropic Transition in a Fractional Quantum Hall State

TL;DR

This paper investigates a novel 2+1D abelian gauge theory with a quantum critical point characterized by a vanishing electric field coefficient and a Chern-Simons term, revealing a transition between isotropic and anisotropic fractional quantum Hall states.

Contribution

It introduces a new gauge theory with a marginal Chern-Simons coupling at a quantum critical point, describing a phase transition in fractional quantum Hall states with unique edge and ground state properties.

Findings

Identifies a quantum critical point with z=2 dynamical exponent.

Shows the Chern-Simons term does not gap the gauge field.

Calculates transport coefficients across phases and at criticality.

Abstract

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.