Analysis of a $\mathrm{SU}\left(4\right)$ generalization of Halperin’s wave function as an approach towards a $\mathrm{SU}\left(4\right)$ fractional quantum Hall effect in graphene sheets

Inspired by the fourfold spin-valley symmetry of relativistic electrons in graphene, we investigate a possible $\mathrm{SU}\left(4\right)$ fractional quantum Hall effect, which may also arise in bilayer semiconductor quantum Hall systems with small Zeeman gap. $\mathrm{SU}\left(4\right)$ generalizations of Halperin’s wave functions [Helv. Phys. Acta 56, 75 (1983)], which may break differently the original $\mathrm{SU}\left(4\right)$ symmetry, are studied analytically and compared, at $\nu =2∕3$, to exact-diagonalization studies.

Figure 1

The three polarizations $S_z,I_z$, and $P_z$, are restricted to the interior of a tretrahedron, completely explored in the case of a fully $\mathrm{SU}\left(4\right)$-symmetric state $(r=1)$. For states with a matrix rank $r=2$, the polarizations are restricted to quadratic planes with fixed $S_z,I_z$ or $P_z$, e.g., $\left[\mathrm{3333,033}\right]$ state with $I_z=0$ (plane in light gray). The $\left[\mathrm{3333,111}\right]$ state, with $r=4$, has $S_z,I_z,P_z=0$ (black dot).

Figure 2

Low-energy spectra, for $N=8$ and $2S=9$, with contact potential in the LLL, built in such a way that the trial states (a) $\left[\mathrm{3333,111}\right]$ and (b) $\left[\mathrm{3333,033}\right]$ are the only ground states with zero energy. We only show the lowest-energy state for each value of the total angular momentum $L$.

Figure 3

Low-energy spectra with Coulomb interaction in the lowest LL $\left(a\right)$ for $2S=9$ and $N=8$; $\left(b\right)$ for $2S=11$ and $N=8$. We show only the lowest energy for each value of $L$ in (a), and in (b) also the spectrum for $S_z=0,I_z=0,P_z=4$ (crosses), for comparison with the $\mathrm{SU}\left(2\right)$ case (Ref. 9). The continuous line shows a $L^2$ fit through the lowest-energy states. The only fit parameter is known with a precision of $1\%$ (if removing the $L=1$ point) in both cases.

Figure 4

Left panel: schematic view of the ground state degeneracy for pseudopotentials $V_1^E$ and $V_2^E$ varying from $0$ to $1$, evaluated on a regular grid ($11\times 11$ values), for $2S=9,N=8$. The white region corresponds to a nondegenerate ground state, light grey regions are doubly degenerate ones with $S_z=I_z=P_z=0$ and dark grey regions are 105$\times$ degenerate ground states. Hatched regions are compressible ground states with $L\ne 0$. The black dot corresponds to the ratios $V_1^E=V_1^C∕V_0^C$ and $V_2^E=V_2^C∕V_0^C$, where $V_m^{C,0}$ are Coulomb pseudopotentials in the lowest LL. Right panel: gap as a function of $V_1^E$ for fixed $V_2^E=0.4$. The peak corresponds to the 105$\times$ degenerate ground state. For $V_1^E>1$, there is a six times degenerate ground state with one of the quantum number maximaly polarized and the others unpolarized.