Universality of the Edge-Tunneling Exponent of Fractional Quantum Hall Liquids

Fractional quantum Hall states are characterized by their topological order. For example, the edge physics is uniquely determined by the bulk and gives a non-Ohmic relation $I\propto V^\alpha$ for tunneling into the edge, where the exponent $\alpha$ is a universal constant. In the simplest case of filling factors $\nu=n/(np+1)$, ($n$ and $p$ are integers $>0$, $p$ even) the exponent is $p+1$. However, experiments show substantial deviations. In a microscopic model of fractional quantum Hall liquids, we calculate the edge Green’s function by exact diagonalization and obtain the exponent $\alpha$. We consider the 1/3 and 2/3 states with the Coulomb interaction and a variety of edge confining potentials. We find that the form of the confinement, such as sharpness of the edge and/or the strength of the confining potential which could lead to edge reconstruction, may cause deviations from universality in the edge-tunneling I-V exponent. We study two types of edge potentials: a sharp edge induced by a cut-off of angular momentum beyond $m_{max}$) and one induced by a uniform neutralizing background charge (a distance $d$ from the 2-d layer). Without the background charge, the exponent retains its universal value for soft edges (large $m_{max}$) but is non-universal for hard edges. In the presence of background charge and strong confinement (small $d$), the exponent is universal even for hard edges; for weak confinement and hard edges there is a deviation from the universal value while for soft edges there are finite-size corrections to $\alpha$, consistent with the edge reconstruction scenario. The relation of these results to experiments will be discussed.