Edge Reconstruction in the $\nu \mathbf{=}2/3$ Fractional Quantum Hall State

The edge structure of the $\nu =2/3$ fractional quantum Hall state has been studied for several decades, but recent experiments, exhibiting upstream neutral mode(s), a plateau at a Hall conductance of $\frac{1}{3}(e^2/h)$ through a quantum point contact, and a crossover of the effective charge, from $e/3$ at high temperature to $2e/3$ at low temperature, could not be explained by a single theory. Here we develop such a theory, based on edge reconstruction due to a confining potential with finite slope, that admits an additional $\nu =1/3$ incompressible strip near the edge. Renormalization group analysis of the effective edge theory due to disorder and interactions explains the experimental observations.

Figure 1

The reconstructed edge. (a) The filling factor of the particle-hole conjugate wave function [7, 8] grows from $\nu =2/3$ in the bulk to $\nu =1$ towards the edge, before falling to zero. In addition, for a smooth potential, the competition between Coulomb and potential energy admits an additional incompressible strip of $\nu =1/3$. (b) The four associated edge modes are depicted, with arrowheads indicating the direction of each mode [downstream (right movers) or upstream (left movers)].

Figure 2

The renormalization group (RG) flow. (a) The bare edge channels. (b) Under RG flow, the three inner channels are renormalized to two upstream neutral modes (as denoted by the dashed lines) and one downstream $1/3$-like charged modes. (The ordering of the three modes is arbitrary.) (c) When temperature is further lowered, the relevant coupling to the outermost mode leads to localization of two modes, as symbolized by the circle, an upstream neutral mode (dashed line) and a downstream $2/3$-like charged mode. (d) Basins of attraction of the various fixed points for the intermediate regime, where the outermost mode is assumed to be decoupled from the other three modes. $p_1$ and $p_2$ parametrize the interaction matrix $V$ [17]. The green (light gray) strips mark the regions where only one of the three potentially relevant scattering operators [Eq. (3)] is relevant. The blue (dark gray) hexagram is where at least two scattering operators are relevant. The thick black lines in the middle of the green (light gray) strips are fixed lines where there is one upstream neutral mode (in addition to a downstream charged mode and an upstream charged mode) at the interface. The black dot at the origin (the center of the hexagram) is a fixed point where there are two upstream neutral modes [as depicted in (b)].