Fractional quantum Hall effect in graphene: Quantitative comparison between theory and experiment

The observation of extensive fractional quantum Hall states in graphene brings out the possibility of more accurate quantitative comparisons between theory and experiment than previously possible, because of the negligibility of finite width corrections. We obtain an accurate phase diagram for differently spin-polarized fractional quantum Hall states, and also estimate the effect of Landau level mixing using the modified interaction pseudopotentials given in the literature. We find that the observed phase diagram is in good quantitative agreement with theory that neglects Landau level mixing, but the agreement becomes significantly worse when Landau level mixing is incorporated assuming that the corrections to the energies are linear in the Landau level mixing parameter $\lambda$. This implies that a first order perturbation theory in $\lambda$ is inadequate for the current experimental systems, for which $\lambda$ is typically on the order of or greater than one. We also test the accuracy of the composite-fermion theory and find that all lowest Landau level projection methods used in the literature are very accurate for the states of the form $n/(2n+1)$ but for the states at $n/(2n-1)$ the results are more sensitive to the projection method. An earlier prediction of an absence of spin transitions for the $n/(4n+1)$ states is confirmed by more rigorous calculations, and new predictions are made regarding spin physics for the $n/(4n-1)$ states.

Figure 1

Spin phase diagram showing the critical Zeeman energies for transitions of the fractional quantum Hall states at the filling factor $\nu =n/(2n\pm 1)$. We define $\kappa =E_{\mathrm{Z}}/E_{\mathrm{C}}$, where $E_{\mathrm{Z}}$ is the Zeeman splitting and $E_{\mathrm{C}}=e^2/\epsilon l$ is the Coulomb energy scale. The crosses show the results from exact diagonalization (assuming zero thickness), the dots from JK wave functions, and squares are fits from the free-CF model (see text). The blue, red, and green colors indicate transition from $(n,0)\to (n-1,1), (n-1,1)\to (n-2,2)$, and $(n-2,2)\to (n-3,3)$, respectively. We note that the exact diagonalization results for the spin transitions at $\nu =4/7$ and $\nu =4/9$ are obtained using results extrapolated from only two finite systems, and may therefore be less accurate.

Figure 2

Schematic representation of the CF basis functions used for CF diagonalization study of the $2/(4p+1)$ spin-singlet FQHE state. Panel (a) shows the “unperturbed” state. At the first order approximation, CF diagonalization allows hybridization of the state in (a) with the states in (b), (c), and (d) to obtain a new ground state with lower energy than the unperturbed state in (a). Successive mixing with higher and higher excitations produces better approximations to the ground state.

Figure 3

This figure compares the theoretical phase diagram with the experimentally measured one in graphene (taken from Feldman et al. [14], shown by green dots), GaAs (taken from Du et al. [19], shown by black squares), and AlAs systems (taken from Padmanabhan et al. [28], shown as red or blue triangles). The theoretical values (blue crosses) are from exact diagonalization at zero thickness and zero LL mixing. (We note that the blue crosses for the spin transitions at $\nu =4/7$ and $\nu =4/9$ are obtained using results extrapolated from only two finite systems.) For AlAs there is data available at two different densities: the blue downward triangle corresponds to the density of $5.5\times {10}^{11}{\mathrm{cm}}^{-2}$, while the red upward triangle corresponds to the density of $5.0\times {10}^{11}{\mathrm{cm}}^{-2}$.

Figure 4

This figure compares the experimental phase diagram in graphene (taken from Feldman et al.[14], shown by green dots) with the theoretical phase diagram (blue crosses) including corrections from LL mixing, assuming that the correction remains linear in $\lambda$. The data of Feldman et al. [14] assumes $g\epsilon =6.0$. The theoretical estimates were obtained with $\lambda =2.2$, which corresponds to the suspended graphene samples of Feldman et al. [14]. (We note that the blue crosses for the spin transitions at $\nu =4/7$ and $\nu =4/9$ are obtained using results extrapolated from only two finite systems.)

Figure 5

Thermodynamic extrapolation of the Coulomb ground state energies at $\nu =2/7$ (top left panel), $2/9$ (top right panel), $3/13$ (bottom left panel), and $4/17$ (bottom right panel). The energies are obtained from exact diagonalization (“exact”), JK wave functions (“JK w.f.”), or CF diagonalization (“CFD”).

Figure 6

Comparison of the LLL Coulomb ground state energies obtained from wave functions of Eq. (8) (black squares) and Eq. (13) (red squares) for fully polarized states at $3/13$ (left panel) and $4/17$ (right panel). The former gives better energies and is used for the results in Tables 3–8.

Figure 7

Comparison of the LLL Coulomb ground state energies obtained from wave functions of Eqs. (9) and (14) for states with reverse flux attachment. The latter gives better energies, and is used to obtain the energies shown in Tables 3–8.

Figure 8

Extrapolation of the lowest Landau level Coulomb ground state energy to the thermodynamic limit for different spin-polarized states at various filling factors.