Atypical Fractional Quantum Hall Effect in Graphene at Filling Factor $1/3$

We study, with the help of exact-diagonalization calculations, a four-component trial wave function that may be relevant for the recently observed graphene fractional quantum Hall state at a filling factor ${\nu }_G=1/3$. Although it is adiabatically connected to a $1/3$ Laughlin state in the upper spin branch, with SU(2) valley-isospin ferromagnetic ordering and a completely filled lower spin branch, it reveals physical properties beyond such a state that is the natural ground state for a large Zeeman effect. Most saliently, it possesses at experimentally relevant values of the Zeeman gap low-energy spin-flip excitations that may be unveiled in inelastic light-scattering experiments.

Figure 1

Energy spectrum for $N=17$ electrons at a filling factor ${\nu }_G=1/3$ ($\nu =2+1/3$), as a function of ${\Delta }_Z$, obtained from ED calculations of the Coulomb interaction on the sphere ($N_B=6$) with implemented SU(4) symmetry. Above ${\Delta }_Z^1\simeq 0.01e^2/ϵl_B$ the ground state is found in the maximally spin-polarized sector ($2S_z=11$, red diamonds). The inset shows a zoom on the region for small values of ${\Delta }_Z$.

Figure 2

Classification of the excitations of the ${\Psi }_{1/3}^{\mathrm{SU}(4)}$ state. The excitations of a one-component Laughlin state are found in the same spin-valley sector ($C$), whereas the Goldstone mode due to the broken SU(2) valley-isospin symmetry in the spin-$\downarrow$ branch is an isospin-wave mode (ISW). In addition to these conventional modes, the four-component state (1) possesses a spin-flip (SF) mode.

Figure 3

(a) Energy spectrum for $N=22$ electrons at a filling factor $\nu =1+1/3$, as a function of ${\Delta }_Z$ in the different spin sectors $2S_z$, obtained from ED calculations of the Coulomb interaction on the sphere ($N_B=15$) with implemented SU(2) symmetry. The inset shows a zoom on the region for small values of ${\Delta }_Z$. (b) Excitation spectrum at ${\Delta }_Z=0.05e^2/ϵl_B$, as a function of the angular momentum $L$. The energy is measured with respect to the ground state (at $L=0$). The spin-flip mode is found in the $2S_z=8$ sector (blue squares) and scales linearly in $B$, whereas the $2S_z=10$ sector reveals the usual magnetoroton branch (red diamonds), which scales as $\sqrt{B}$ with the $B$ field.