Collapse of the $\nu=1$ quantum Hall effect near a Landau level crossing*

We report magneto-resistance measurements of 2D hole systems (density $2.1\times 10^{11}$ cm$^{-2}$) confined to a 40-nm-wide GaAs quantum well as a function of tilted magnetic fields. We observe a strong $\nu=1$ quantum Hall effect (QHE) at zero parallel field ($B_{||}$). The $\nu=1$ QHE disappears at $B_{||}\simeq4.8\,\mathrm{T}$, where we expect a crossing between the lowest two Landau levels. Near this crossing, the energy gap for the $\nu=1$ QHE collapses from 6 K to zero in a very small $B_{||}$ range of 0.3 T. The $\nu=1$ QHE comes back at $B_{||}\simeq8.1\,\mathrm{T}$ and eventually disappears at $B_{||}>17\,\mathrm{T}$ where the system becomes bilayer-like. The sudden collapse of the $\nu=1$ QHE and the fact that it comes back after a large $B_{||}$ range of 3.3 T is intriguing and suggests a pinning of the Landau levels near the crossing.