Fractional quantum Hall state at $ν= 1/2$ with energy gap up to 6 K, and possible transition from one- to two-component state

TL;DR

The paper addresses whether the $ν=1/2$ FQHS in a wide GaAs quantum well is a one-component (1C) non-Abelian Pfaffian state or a two-component (2C) Abelian $(Ψ_{331})$ state. It uses an ultrahigh-mobility, 72.5-nm-wide GaAs QW with gate-tuned interlayer tunneling to map transport ($R_{xx}$, $R_{xy}$) as a function of density and to extract the energy gap $^{1/2}\Delta$ and its evolution with the control parameter $α$. Key findings include a robust $ν=1/2$ FQHS with a maximum gap of about $6$ K, the appearance of daughter states at $ν=8/17$ and $7/13$ consistent with Pfaffian expectations, and the emergence of a bilayer Wigner crystal in the 2C regime; the data also indicate that the 1/2 FQHS remains stable across the 1C to 2C transition and may itself undergo a 1C→2C transition. The results point to a topological transition between non-Abelian 1C Pfaffian and Abelian 2C states, with implications for topological quantum computation and quantum criticality.

Abstract

The fractional quantum Hall state (FQHS) observed in the lowest Landau level at filling factor $ν=1/2$ in wide quantum wells has been enigmatic for decades because the two-dimensional electron system (2DES) has a bilayer charge distribution but with significant interlayer tunneling. Of particular interest is whether the 1/2 FQHS in this system has a one-component (1C) or two-component (2C) origin; these are typically identified as the Pfaffian (non-Abelian) or the $Ψ_{331}$ (Abelian) FQHSs, respectively. We report here our experimental study of the evolution of the correlated states of an ultrahigh-quality 2DES confined to a 72.5-nm-wide GaAs quantum well. At the lowest densities, the 2DES displays only odd-denominator FQHSs, and the ground state at $ν= 1/2$ is a composite fermion Fermi sea. As the density is increased, a FQHS emerges at $ν= 1/2$, and becomes very strong. In a finite density range where the 1/2 FQHS is strongest, we also observe its daughter FQHSs at $ν= 8/17$ and 7/13, consistent with the theoretically expected daughter states of a Pfaffian 1/2 FQHS. At the highest densities, the 2DES becomes 2C, signaled by the emergence of a bilayer Wigner crystal state and the transitions of FQHSs flanking $ν=1/2$. The 1/2 FQHS remains robust near this transition and, notably, its charge transport energy gap exhibits an \textit{upward} cusp with a maximum value of about 6 K on the 1C side of the transition; this is the largest gap reported for any even-denominator FQHS. Our observation of the transition of the 2DES ground states near $ν=1/2$ to 2C states at high densities, and our measurements of the robustness of the 1/2 FQHS against charge distribution asymmetry, suggest that the 1/2 FQHS also makes a transition from 1C to 2C. Such a transition from a non-Abelian to Abelian state can open avenues for topological quantum information and quantum criticality.

Figures (3)

• Figure 1: (a) Sample schematic. (b) $\Delta_{SAS}$ extracted from the Shubnikov-de Haas oscillations at low B-field. Self-consistent Hartree calculations of the charge distribution and potential at n = 1.82 and 1.00 in a 72.5-nm-wide GaAs QW are shown as top and bottom insets, respectively. (c) Characteristic longitudinal resistance $R_{xx}$ traces which coarsely capture the most notable transitions such as the CF Fermi sea to FQHS at $\nu = 1/2$ (red trace), the emergence of the daughter states of the 1/2 FQHS at $\nu = 8/17$ and 7/13 (blue trace), and the bilayer Wigner crystal flanking $\nu = 1/2$ at the highest densities (brown trace). Also shown is the Hall resistance $(R_{xy})$ at n = 1.35, displaying well quantized plateaus for the $\nu = 1/2$, 8/17 and 7/13 FQHSs. (d) Color-scale phase diagram of the 2DES in the N = 0 LL, rich with correlated electronic solid and liquid phases. Transitions of the odd-denominator FQHSs and the emergence of a bilayer Wigner crystal signal the 1C to 2C transition of the 2DES as n increases (indicated by the red dashed-dotted line, a guide to the eye). (e) Energy gap of the 1/2 FQHS ($^{1/2}\Delta$) as a function of n, peaking to about 6 K.
• Figure 2: Data of Figs. \ref{['fig1']} are replotted in (a) and (b) respectively using the parameter $\alpha = \Delta_{SAS}/(e^2/4\pi \epsilon \ell_B)$, the interlayer tunneling normalized to the Coulomb energy. (a) The daughter states of a 1C 1/2 FQHS emerge at $0.064 \lesssim \alpha \lesssim 0.073$ while the 1C to 2C transition of the 2DES near $\nu = 1/2$ occurs at $\alpha \lesssim 0.064$. (b) The measured $^{1/2}\Delta$ (black squares), normalized to the Coulomb energy (see note EM4), are about 8 times larger than those reported in a sample with much lower mobility (green circles) Suen.PRL.1994. The orange band near $\alpha \simeq 0.07$ denotes the region of stability of the daughter 8/17 and 7/13 FQHSs, and straddles the peak $^{1/2}\Delta$ quite symmetrically. Also plotted are theoretical calculations for $^{1/2}\Delta$ in Ref. Peterson.PRB.2010 (blue) and Ref. Zhu.PRB.2016 (red). Note that both sets of calculations predict a boundary near the maximum $^{1/2}\Delta$ between the Pfaffian and $\Psi_{331}$ phases as $\alpha$ is reduced. The vertical lines denote the experimental (black, at $\alpha \simeq 0.064$) 1C-2C transition of the 2DES, and theoretical (red and blue) predictions of the 1C-2C transition of the 1/2 FQHS; see SM Section VI SM.2025 for details.
• Figure 3: (a) Calculated charge distributions at n = 1.45 (red) and n = 1.28 (blue) in a symmetric and asymmetric QW. (b) Characteristic $R_{xx}$ traces at n = 1.17 showing the evolution of the 1/2 FQHS as the charge distribution is made asymmetric. (c) The critical layer density imbalance $(\Delta n/n)_C$ required to destabilize the 1/2 FQHS, plotted together with $^{1/2}\Delta$, as a function of n. The 1/2 FQHSs on the high-density side of the $^{1/2}\Delta$ peak have smaller $(\Delta n/n)_C$.