The enigmatic 12/5 fractional quantum Hall effect

We numerically study the fractional quantum Hall effect at filling factors $\nu =12\mathrm{/}5$ and 13/5 (the particle-hole conjugate of 12/5) in high-quality two-dimensional GaAs heterostructures via exact diagonalization including finite well width and Landau level mixing. We find that Landau level mixing suppresses $\nu =13\mathrm{/}5$ fractional quantum Hall effect relative to $\nu =12\mathrm{/}5$. By contrast, we find both $\nu =2\mathrm{/}5$ and (its particle-hole conjugate) $\nu =3\mathrm{/}5$ fractional quantum Hall effects in the lowest Landau level to be robust under Landau level mixing and finite well-width corrections. Our results provide a possible explanation for the experimental absence of the 13/5 fractional quantum Hall state as caused by Landau level mixing effects.