Even-Denominator Fractional Quantum Hall State at Filling Factor $\nu =3/4$

Fractional quantum Hall states (FQHSs) exemplify exotic phases of low-disorder two-dimensional (2D) electron systems when electron-electron interaction dominates over the thermal and kinetic energies. Particularly intriguing among the FQHSs are those observed at even-denominator Landau level filling factors, as their quasiparticles are generally believed to obey non-Abelian statistics and be of potential use in topological quantum computing. Such states, however, are very rare and fragile, and are typically observed in the excited Landau level of 2D electron systems with the lowest amount of disorder. Here we report the observation of a new and unexpected even-denominator FQHS at filling factor $\nu =3/4$ in a GaAs 2D hole system with an exceptionally high quality (mobility). Our magnetotransport measurements reveal a strong minimum in the longitudinal resistance at $\nu =3/4$, accompanied by a developing Hall plateau centered at $(h/e^2)/(3/4)$. This even-denominator FQHS is very unusual as it is observed in the lowest Landau level and in a 2D hole system. While its origin is unclear, it is likely a non-Abelian state, emerging from the residual interaction between composite fermions.

Figure 1

(a) Longitudinal resistance ($R_{xx}$, in black and blue) and Hall resistance ($R_{xy}$, in red) vs perpendicular magnetic field $B_{\perp }$ traces for our ultrahigh-mobility 2D hole sample. The height of the blue trace is divided by a factor of 10. The $B_{\perp }$ positions of several LL fillings are marked. A strong minimum in $R_{xx}$ accompanied by a developing Hall plateau is observed at $\nu =3/4$. An enlarged version of the $R_{xy}$ vs $B_{\perp }$ near $\nu =3/4$ at 20 mK is shown in the top-left inset. The self-consistently calculated hole charge distribution (red) and potential (black) of the 2DHS are also shown in a top inset. (b) $R_{xx}$ vs $B_{\perp }$ at $T\simeq 30\mathrm{mK}$ near $\nu =3/4$ for an ultrahigh-mobility 2D electron sample with density $1.0\times {10}^{11}{\mathrm{cm}}^{-2}$ from Ref. [34].

Figure 2

Temperature dependence of $R_{xx}$ and $R_{xy}$. (a) $R_{xx}$ vs $B_{\perp }$ traces near $\nu =3/4$, taken at different temperatures, showing fully developed (nearly zero) $R_{xx}$ minima at $\nu =1$ and $2/3$, as well as a strong minimum at $\nu =3/4$. (b) Arrhenius plot of $R_{xx}$ at $\nu =3/4$. An energy gap of $\simeq 22\mathrm{mK}$ is obtained from the linear fit to the data points at intermediate temperatures. Inset: enlarged version of Fig. 2 near $\nu =3/4$. (c) Hall ($R_{xy}$) traces taken at different temperatures. $R_{xy}$ is well quantized at its expected value at $\nu =1$ and $2/3$ in the whole temperature range, and shows a developing plateau at $\nu =3/4$ at the lowest temperatures. Top-left inset: enlarged $R_{xy}$ vs $B_{\perp }$ traces near $\nu =3/4$ at $T\simeq 20$ and 188 mK. Bottom-right inset: Hall resistance slope $d{R}_{xy}/d{B}_{\perp }$ vs $T$ at $\nu =3/4$, showing its approach to the expected (classical) value at high $T$ (the dash-dotted line), and to zero as $T$ approaches zero, confirming the $R_{xy}$ quantization.

Figure 3

Tilt angle dependence of $R_{xx}$ is shown vs $B_{\perp }$ at 20 mK. The left inset shows a schematic of the experimental setup for applying tilted fields; the sample is mounted on a rotating stage to support in situ tilt. The traces are vertically shifted for clarity. The vertical lines indicate the $B_{\perp }$ positions for $\nu =3/4$, $4/5$, and $5/7$. The right inset shows $R_{xx}$ vs $B_{\Vert }$ at $\nu =3/4$ (black circles) and $4/5$ (red triangles).