Fractional Quantum Hall Effect at $\nu =2+6/13$: The Parton Paradigm for the Second Landau Level

The unexpected appearance of a fractional quantum Hall effect (FQHE) plateau at $\nu =2+6/13$ [A. Kumar et al., Phys. Rev. Lett. 105, 246808 (2010)] offers a clue into the physical mechanism of the FQHE in the second Landau level (SLL). Here we propose a “$\overline{3}\overline{2}111$” parton wave function, which is topologically distinct from the $6/13$ state in the lowest Landau level. We demonstrate the $\overline{3}\overline{2}111$ state to be a good candidate for the $\nu =2+6/13$ FQHE, and make predictions for experimentally measurable properties that can reveal the nature of this state. Furthermore, we propose that the “$\overline{n}\overline{2}111$” family of parton states naturally describes many observed SLL FQHE plateaus.

Figure 1

Thermodynamic extrapolations of the energies (per particle) for the CF state (blue crosses) and the $\overline{3}\overline{2}111$ parton state (red dots). The left-hand panel shows energies for $\nu =6/13$ in the LLL, the middle panel for $\nu =2+6/13$ in the second LL, and the right-hand panel for $6/13$ in the $n=1$ LL of monolayer graphene (for $n=0$ LL, graphene results are identical to those in the left-hand panel for the $n=0$ GaAs LL). The energies include the electron-background and background-background interaction, and are quoted in units of $e^2/(\epsilon \ell )$. The LLL Coulomb energy for the CF state has been reproduced from Ref. [51].

Figure 2

The transport (red line) and neutral (blue line) gaps as the Hamiltonian is tuned from the second Landau level Coulomb interaction, modeled using truncated disk pseudopotentials, to the model Hamiltonian $H_2$, for $N=12$ electrons seeing a flux of $2Q=28$ in the spherical geometry. The model Hamiltonian $H_2$ is defined by the set of pseudopotentials $V_1=V_1^{\mathrm{SLL}}$ (same as the second Landau level Coulomb), $V_3=V_1/3$, and the rest $V_m=0$. The “transport” gap for this system is defined as $[E(2Q=29)+E(2Q=27)-2E(2Q=28)]/6$, where $E(2Q)$ is the ground state energy at flux $2Q$ and the factor of 6 accounts for the fact that the removal or addition of one flux quantum produces 6 fundamental quasiparticles or quasiholes. The neutral gap for this system is defined as the difference between the two lowest energies at a fixed flux $2Q=28$. This figure also displays the overlap (green line) of the ground state with the $\overline{3}\overline{2}111$ parton state. Inset: Overlap between the parton state and the exact ground state of the Zhang–Das Sarma Hamiltonian $H_{\mathrm{ZDS}}$ as a function of thickness parameter $d/\ell$.