Experimental probe of topological orders and edge excitations in the second Landau level

We measure weak quasiparticle tunneling across a constriction in the second Landau level. At $\nu =7/3$, 8/3, and 5/2, comparison of temperature and dc bias dependence to weak tunneling theory allows extracting parameters that describe the edges' quasiparticle excitations. At $\nu =8/3$, our results are well described by a particle-hole conjugate Laughlin state but not compatible with the proposed non-Abelian quasiparticle excitations. For $\nu =5/2$, our measurements are in good agreement with previous experiments and favor the Abelian (3,3,1) or (1,1,3) states. At these filling factors, we further investigate the influence of the backscattering strength on the extracted scaling parameters. For $\nu =7/3$, the backscattering strength strongly affects the scaling parameters, whereas quasiparticle tunneling at $\nu =8/3$ and 5/2 appears more robust. Our results provide important additional insight about the physics in the second Landau level and contribute to the understanding of the physics underlying the fractional quantum Hall states at $\nu =7/3$, 8/3, and 5/2.

Figure 1

Conceptual difference between weak and strong backscattering [20, 21, 22], in the simplest case without edge reconstruction. (a) For a quantum point contact (QPC) close to pinch-off, we have strong backscattering and weak electron tunneling (dotted line). (b) For an open QPC, weak backscattering and weak quasiparticle tunneling (dotted line) govern the transmission.

Figure 2

$R_{\mathrm{diag}}$ [blue (gray)] and $R_{\mathrm{xy}}$ (black) measured in a Hall-bar geometry (upper inset) as a function of the magnetic field. Here, $-$1.65 V have been applied to QPC2 (left inset). In between integer filling factors, $R_{\mathrm{diag}}\ge R_{\mathrm{xy}}$, indicating a reduced transmission of the QPC.

Figure 3

(a) Zero-bias backscattering peak at $\nu =5/2$ and fixed $V_{\mathrm{QPC}}1=-1.8$ V. The peak height is strongly temperature-dependent. (b) Measured (black) and fitted (red) tunneling conductance for different electronic temperatures (fit parameters are $e^{*}/e=0.18$ and $g=0.32)$. (c) Normalized fit error as a function of fit parameters $e^{*}/e$ and $g$. Parameters of Abelian [(green) circles] and non-Abelian [(green) dots] candidate wave functions are indicated. (d) PDF of the measured residuals $\left\{{\delta }_k\right\}$ as a function of $e^{*}/e$ and $g$. The maximum probability is found for $e^{*}/e=0.19,g=0.33$ with ${\sigma }_{\mathrm{g}}=0.0026,{\sigma }_{\mathrm{e}}^{*}/\mathrm{e}=0.0019$, and ${\sigma }_{\mathrm{g}},{\mathrm{e}}^{*}/\mathrm{e}=0.0022$.

Figure 4

(a) Zero-bias backscattering peak at $\nu =8/3$ and fixed $V_{\mathrm{QPC}}1=-1.8$ V. (b) Measured (black) and fitted (red) tunneling conductance for different electronic temperatures (fit parameters are $e^{*}/e=0.22$ and $g=0.62$). (c) Relative fit error as a function of fit parameters $e^{*}/e$ and $g$. Parameters of Abelian [(green) circles] and non-Abelian [(green) dots] candidate wave functions are indicated. (d) PDF of the measured residuals $\left\{{\delta }_k\right\}$ as a function of $e^{*}/e$ and $g$. The maximum probability is found for $e^{*}/e=0.23,g=0.65$ with ${\sigma }_{\mathrm{g}}=0.0029,{\sigma }_{\mathrm{e}}^{*}/\mathrm{e}=0.0028$, and ${\sigma }_{\mathrm{g}},{\mathrm{e}}^{*}/\mathrm{e}=0.0028$.

Figure 5

(a) and (b) $B$-field and dc bias dependence of the tunneling conductance near $\nu =8/3$, 5/2, and 7/3. (c) $B$-field dependence of fitting parameters $e^{*}/e$ [(red) diamonds], $g$ [(blue) circles] and $g$ for $e^{*}/e$ fixed to 1/3 or 1/4 [(green) squares]. The vertical dashed lines in (b) and (c) indicate the center of the bulk filling factor plateaus.

Figure 6

(a) Dependence of the tunneling conductance at $\nu =8/3$, 5/2, and 7/3 on the QPC voltage. (b) Fitting parameters $e^{*}/e$ [(red) diamonds], $g$ [(blue) circles], and $g$ for $e^{*}/e$ fixed to 1/3 or 1/4 [(green) squares] as a function of $V_{\mathrm{QPC}}$.

Figure 7

Diagonal resistance across QPC2 [(a) ${\nu }_{\mathrm{bulk}}=8/3$ and (b) 5/2] for different QPC voltages as a function of dc bias.