Topological Robustness of Anyon Tunneling at $ν= 1/3$

Abstract

The scaling exponent $g$ of the quasiparticle propagator for incompressible fractional quantum Hall states in the Laughlin sequence is expected to be robust against perturbations that do not close the gap. Here we probe the topological robustness of the chiral Luttinger liquid at the boundary of the $ν=1/3$ state by measuring the tunneling conductance between counterpropagating edge modes as a function of quantum point contact transmission. We demonstrate that for transmission $t\geq 0.7$ the tunneling conductance is well-described by the first two terms of a perturbative series expansion corresponding to $g=1/3$. We further demonstrate that the measured scaling exponent is robustly pinned to $g=1/3$ across the plateau, only deviating as the bulk state becomes compressible. Finally we examine the impact of weak disorder on the scaling exponent, finding it insensitive. These measurements firmly establish the topological robustness of anyon tunneling at $ν=1/3$ and substantiate the chiral Luttinger liquid description of the edge mode.

Figures (4)

• Figure 1: (a) False color scanning electron microscope image of a device similar to the one measured in this experiment including a schematic of the measurement circuit. The QPC used for tunneling experiments is highlighted in orange and the helper gate is highlighted in green. The red lines show circulation of the chiral edge modes. The solid line corresponds to the edge mode equilibrated with the source contact and carries the non-equilibrium current. Dashed lines are edge modes equilibrated with the drain contacts and the dotted line corresponds to the tunneling current across the QPC. $S$, $D_1$ and $D_2$ are the source and drain contacts. (b) Simultaneous measurement of the Hall resistance across the bulk ($R_{xy}$) and across the QPC ($R_D$) as a function of magnetic field at a mixing chamber temperature of $T_{\text{MC}} = 10 \text{mK}$. The QPC is biased just past depletion to define the current path while still fully transmitting the edge modes. The filling fraction in the QPC is the same as in the bulk.
• Figure 2: (a) Tunneling conductance versus transmission $t$ as the QPC transmission is varied from $t=0.99$ to $t=0.65$. The magnetic field is fixed at the center of the $\nu = 1/3$ plateau and the electron temperature is fixed at $T_e=0.39~\text{mK}$ as measured via Coulomb blockade thermometry. The color bar indicates the range of transmission through the QPC. (b) Tunneling conductance at $t=0.83, 0.73,$ and $0.67$ plotted along with fits to $G^{(1)}$. The plots are staggered by $50\,\mu\mathrm{V}$ for clarity. $G^{(1)}$ accurately captures the data at high transmission but begins to deviate below $t\approx0.8$ as indicated by the dashed and dotted lines. (c) Difference between the data and fit to $G^{(1)}$ from $t=0.81$ to $t=0.67$. $g = 1/3$ while $T_0$ is the only free fitting parameter. $\Delta G_t \approx 0$ for $t >0.8$ but increases in magnitude for lower transmissions. (d)$\Delta G_t$ scaled by $\left(T_0/2\pi T\right)^{4g-4}$ plotted versus $e^*V_{SD}/k_B T$ for $t=0.81$ to $t=0.67$. The scaled data collapses onto a single curve, signaling that the deviations from the lowest order approximation to the tunneling conductance are described by a universal function of $e^* V_{SD}/k_B T$. The black line displays the next order perturbative contribution to the tunneling conductance, $K_g(e^*V_{SD}/k_B T)$, multiplied by a constant. $K_g(e^*V_{SD}/k_B T)$ accurately captures the functional dependence of the collapse. (e) Tunneling conductance at $t=0.83, 0.73$, and $0.67$ plotted with fits to $G^{(2)}$. The plots are staggered by $50\,\mu\mathrm{V}$ for clarity. Fitting to $G^{(2)}$ more accurately captures the data over the full range of transmission and source-drain bias explored here. (f) Residual sum of squares (RSS) for fits to $G^{(1)}$ and $G^{(2)}$ plotted as a function of transmission. The RSS is equal for both functions at high transmission while $G^{(2)}$ performs significantly better as the transmission is lowered.
• Figure 3: $g$ extracted from fitting the tunneling conductance to $G^{(1)}$ plotted as a function of magnetic field across the $\nu=1/3$ plateau. The magnetic field is varied over a range of $1\, \text{T}$ while the QPC transmission is fixed at $t=0.93$. The vertical dotted lines correspond to the boundaries of the region where $g$ is constant. The inset shows the Hall resistances ($R_{xy}$ and $R_D$) around the $\nu=1/3$ plateau with the translucent box highlighting the region where the tunneling conductance was measured.
• Figure 4: (a) Zero bias QPC conductance at $\nu=1/3$ with $100$ mV applied to the helper gate. The zero bias conductance is quantized to $e^2/3h$ over most of the voltage range and displays a sharp pinch-off. Non-monotonic behavior is evident in the conductance before full pinch-off. The inset highlights these features and the circles display the bias points for the tunneling conductance data shown in Fig. 4b. (b) Tunneling conductance measured at 4 different points along the zero-bias QPC conductance curve. The black lines correspond to the fits to the data. While the voltage applied to the QPC varies, the transmission at all of these points is $t=0.91$.