Quantum Hall Liquids Coupled to Dynamical Electromagnetism

We investigate the effect on a Quantum Hall (QH) liquid of its coupling to 3+1 dimensional dynamical electromagnetism, which renders the system gapless. We calculate both the Hall and longitudinal resistances, $\rho_H$ and $\rho_L$, in the context of a minimal model of the electromagnetic environment, with a small three dimensional conductivity ${\tilde{\sigma}}$, that allows for a counter-flow current. In the thermodynamic limit, we show that $\rho_H$ is quantized, while $\rho_L$ approaches a non-zero limit, $\rho_L \sim \alpha\, R_K$, where $\alpha$ and $R_K=2\pi /e^2$ are the fine structure and the Klitzing constant. In contrast, the QH conductance, $\sigma_H$, is smaller than the expected quantized value by a correction $\sim \alpha^2/R_K$. The electromagnetic interaction also generates corrections of order $\alpha^2$ to the quasiparticle charges and statistics, in a way that is consistent with general arguments based on gauge invariance. In addition, we present an intuitive argument that relates the flux attachment associated with the composite boson representation of the electron liquid to the empirically observed %persistence of approximate quantization of $\rho_H$, even in circumstances in which $\rho_L$, and the deviation of $\sigma_H$ from its quantized value, are substantial.