Universal Anyon Tunneling in a Chiral Luttinger Liquid

Abstract

The edge modes of fractional quantum Hall liquids are described by chiral Luttinger liquid theory. Despite many years of experimental investigation fractional quantum Hall edge modes remain enigmatic with significant discrepancies between experimental observations and detailed predictions of chiral Luttinger liquid theory. Here we report measurements of tunneling conductance between counterpropagating edge modes at $ν=1/3$ across a quantum point contact fabricated on an AlGaAs/GaAs heterostructure designed to promote a sharp confinement potential. We present evidence for tunneling of anyons through a $ν=1/3$ incompressible liquid that exhibits universal scaling behavior with respect to temperature, source-drain bias, and barrier transmission, as originally proposed by Wen [1, 2]. For transmission $t\geq0.800$, we measured the tunneling exponent $\bar{g} = 0.333 \pm 0.005$ averaged over 29 independent data sets, consistent with the scaling dimension $Δ= g/2 = 1/6$ for a Laughlin quasiparticle at the edge. When combined with measurements of the fractional charge $e^*=e/3$ and the recently observed anyonic statistical angle $θ_a=\frac{2π}{3}$, the measured tunneling exponent fully characterizes the topological order of the primary Laughlin state at $ν=1/3$.

Figures (9)

• Figure 1: (a) False color scanning electron microscopy image of a full interferometer with a schematic of the measurement circuit used in this experiment. The QPC used for tunneling measurements is highlighted in orange. All other gates are grounded. Red lines represent edge mode circulation. (b) Simultaneous measurement of bulk $R_{xy}$ and $R_D$ across the QPC as a function of magnetic field at $T_{mc}=10$ mK. The QPC is biased just past depletion to define the current path. The filling factor is the same in the QPC as in the bulk of the Hall bar. (c) QPC conductance at the center of the $\nu = 1$ ($B = 4.20$ T) plateau. The conductance is quantized to $e^2/h$ over most of the voltage range and exhibits a sharp pinch-off. This data indicates a sharp confining potential in our QPC built upon the screening well heterostructure design. The inset displays differential conductance at $\nu = 1$ taken with $t\approx0.8$ as indicated by the blue point on the conductance plot. This constant differential conductance is expected for the $\nu = 1$ edge described by Fermi liquid theory, in sharp contrast to the behavior observed at $\nu = 1/3$. (d) QPC conductance at $\nu = 1/3$. The conductance is quantized to $e^2/3h$ over most of the voltage range and displays a sharp pinch-off. Unlike the behavior observed at $\nu = 1$, a few sharp resonances are observed at $\nu = 1/3$ prior to full depletion. The inset highlights the region where tunneling was investigated.
• Figure 2: (a) 2D color map of differential conductance plotted as a function of QPC voltage $V_{QPC}$ and DC source-drain bias $V_{SD}$. (b) Line cuts of differential conductance measured at several QPC transmissions at fixed electron temperature $T_{e}$=34 mK at the center of the $\nu = 1/3$ plateau. The transmission through the QPC varies from $t \approx 0.99$ to $t \approx 0.65$. The linecuts are extracted from the data of Fig. 2a. (c) Differential conductance at $t = 0.94$. The red line shows the fit to the data using the expression for the tunneling conductance in the weak backscattering limit derived by Wen and collaborators Wen1990Wen1991ChamonWen1992ChamonWen1994. (d) Temperature dependence of the differential conductance for QPC transmission $t = 0.9$. Temperatures listed in the legend are mixing chamber temperatures at which each data set was collected.
• Figure 3: (a) Zero bias tunneling conductance vs $T/T_0$ at $T_e = 34\,\text{mK}$ and $B=12.83$ T, at the center of the $\nu = 1/3$ plateau. Each point is extracted from a fit of each trace in Fig. 2b as in Fig 2c. The lines correspond to the theoretical expectation at $g = 1/3$, $g = 0.3$ and $g = 0.4$. The inset shows the data plotted on a log-log scale. (b) Zero bias tunneling conductance vs $T/T_0$ at $T_e = 47\,\text{mK}$ and $T_e = 69\,\text{mK}$, at the center of $\nu = 1/3$. (c) Zero bias tunneling conductance vs $T/T_0$ at $B = 12.92\, \text{T}$ and $B = 12.74\, \text{T}$, which are $\Delta B\pm 90\,~\text{mT}$ away from the center of the $\nu = 1/3$ plateau.
• Figure 4: (a) Scaling behavior of reduced tunneling conductance $\tilde{G}$ vs. $e^*V_{SD}/k_B\,T$ using data of Fig. 2d. Data collected at different temperatures collapse onto a single curve. Dashed magenta line is the theoretical expectation for $g=1/3$. (b) Data in (a) plotted on log-log scale. Dashed magenta line is the theoretical expectation for $g=1/3$.
• Figure S1: (a) AlGaAs/GaAs screening well heterostructure layer sequence. Two 11 nm thick GaAs screening wells flank the primary 30 nm GaAs quantum well. (b) Schematic of the gates proximal to the ohmic contacts used to limit transport to the primary quantum well. The three quantum wells are in contact with an ohmic contact held at ground potential, but only the primary quantum well is connected to the excitation voltage. The top (bottom) screening well is isolated using a top (bottom) gate set at $V_{TOG}$ ($V_{BOG}$). This device design allows the screening wells to be held at a fixed potential while transport occurs only through the primary quantum well.