Spontaneous polarization of composite fermions in the $n=1$ Landau level of graphene

Motivated by recent experiments that reveal expansive fractional quantum Hall states in the $n=1$ graphene Landau level and suggest a nontrivial role of the spin degree of freedom [F. Amet, A. J. Bestwick, J. R. Williams, L. Balicas, K. Watanabe, T. Taniguchi, and D. Goldhaber-Gordon, Nat. Commun. 6, 5838 (2015)], we perform an accurate quantitative study of the competition between fractional quantum Hall states with different spin polarizations in the $n=1$ graphene Landau level. We find that the fractional quantum Hall effect is well described in terms of composite fermions, but the spin physics is qualitatively different from that in the $n=0$ Landau level. In particular, for the states at filling factors $\nu =s/(2s\pm 1), s$ positive integer, a combination of exact diagonalization and the composite fermion theory shows that the ground state is fully spin polarized and supports a robust spin-wave mode even in the limit of vanishing Zeeman coupling. Thus, even though composite fermions are formed, a mean-field description that treats them as weakly interacting particles breaks down, and the exchange interaction between them is strong enough to cause a qualitative change in the behavior by inducing full spin polarization. We also verify that the fully spin-polarized composite fermion Fermi sea has lower energy than the paired Pfaffian state at the relevant half fillings in the $n=1$ graphene Landau level, indicating an absence of composite fermion pairing at half filling in the $n=1$ graphene Landau level.

Figure 1

Comparison between the exact pseudopotentials for the Coulomb interaction in the $n=1$ GLL and the pseudopotentials of the effective interaction given in Eq. (5) in the $n=0$ GLL in the disk geometry.

Figure 2

2/5 spin singlet states. Panel (a) shows the schematic of the the 2/5 spin singlet CF state. At the first-order approximation, composite fermion diagonalization mixes the state in (a) with the states shown in panels (b), (c), and (d) to obtain a new ground state with lower energy than the state in (a). Reproduced from Ref. [44].

Figure 3

Thermodynamic extrapolation of the ground-state energies, obtained by exact diagonalization, for the fully polarized and spin singlet states in the $n=1$ GLL at filling factors 2/3 (panel b) and 2/5 (panel d). Also shown for comparison are the corresponding results in the $n=0$ GLL in panels (a) (2/3) and (c) (2/5). We show results obtained from disk (red squares) and spherical pseudopotentials (blue circles). The background subtraction and density correction is carried out as described in Sec. 2a. Additionally, with the green crosses we show the expectation values of the $n=1$ GLL Coulomb interaction with respect to the exact $n=0$ GLL ground states; the latter are known to be nearly exactly described by the wave functions of CF theory given in Eq. (6). All energies are quoted in units of $e^2/\epsilon \ell$.

Figure 4

Same as in Fig. 3 but for $\nu =3/5$ and $3/7$.

Figure 5

Extrapolation of the energy of the wave function in Eq. (6) to the thermodynamic limit in the $n=1$ Landau level of graphene for FQHE states in the sequence $\nu =s/(2ps-1)$. We show the results of both linear and quadratic fits.

Figure 6

Extrapolation of the energy of the wave function in Eq. (6) to the thermodynamic limit in the $n=1$ Landau level of graphene for states in the sequence $\nu =s/(2ps+1)$. We show the results of both linear and quadratic fits.

Figure 7

Extrapolation to the thermodynamic limit for the difference in the fully polarized and spin singlet $\nu =2/5$ states in the $n=1$ graphene LL. The red symbols show the energies obtained from the wave functions in Eq. (6), while the blue symbols are obtained from CFD. The energies are obtained using the effective interaction of Eq. (5). The ground state is seen to be fully polarized.

Figure 8

Extrapolation to the thermodynamic limit for the fully polarized (blue) and partially polarized (red) $\nu =3/7$ CFD energies for the $n=1$ GLL obtained using the effective interaction of Eq. (5). The ground state is seen to be fully polarized.

Figure 9

Thermodynamic extrapolation of the exact Coulomb ground-state energies per unit LL mixing parameter $\lambda$ for the fully polarized (circles) and unpolarized states (squares) in the $n=1$ GLL at filling factors 2/3 and 2/5 (a) and 3/5 and 3/7 (b).

Figure 10

Spin-wave (black hexagram), spin-conserving (blue squares), and spin-reversing (green pentagram) collective mode energies for $N=30$ at $\nu =2/5$ for the $n=1$ GLL. These energies are obtained using the wave function in Eq. (6), and the effective interaction of Eq. (5) (top panel). For comparison the corresponding modes in the lowest LL are shown in the bottom panel.

Figure 11

Comparison of the spin-wave dispersions in the LLL (left panels) and $n=1$ GLL (right panels) at various filling factors in the sequence $\nu =s/(2s\pm 1)$, obtained from exact diagonalization. The spin wave in the LLL shows a spin-flip instability for $s\ge 2$, while the $n=1$ GLL supports a robust spin wave.

Figure 12

Comparison of the spin-wave dispersions in the LLL (filled symbols) and $n=1$ GLL (open symbols) at various filling factors in the sequence $\nu =s/(2s\pm 1)$ for different system sizes. The dispersions are obtained from exact diagonalization.

Figure 13

Extrapolation of the ground-state energy to the thermodynamic limit of the states at half filling (a) in the lowest Landau level, (b) in the second ($n=1$) Landau level of GaAs, and (c) in the $n=1$ Landau level of graphene. The wave function in Eq. (6) has been used to obtain the energies. The Zeeman energy has been omitted, and the density correction has been applied. The solid lines correspond to a linear fit, and the dashed lines correspond to a quadratic one. In (b) the energies in the thermodynamic limit may differ from those reported previously in the literature because of different methods for background subtraction, but the energy differences are captured properly.