  Two-dimensional electrons confined to GaAs quantum wells are hallmark
platforms for probing electron-electron interaction. Many key observations have
been made in these systems as sample quality improved over the years. Here, we
present a breakthrough in sample quality via source-material purification and
innovation in GaAs molecular beam epitaxy vacuum chamber design. Our samples
display an ultra-high mobility of $44\times10^6$ cm$^2$/Vs at an electron
density of $2.0\times10^{11}$ /cm$^2$. These results imply only 1 residual
impurity for every $10^{10}$ Ga/As atoms. The impact of such low impurity
concentration is manifold. Robust stripe/bubble phases are observed, and
several new fractional quantum Hall states emerge. Furthermore, the activation
gap of the $\nu=5/2$ state, which is widely believed to be non-Abelian and of
potential use for topological quantum computing, reaches $\Delta\simeq820$ mK.
We expect that our results will stimulate further research on
interaction-driven physics in a two-dimensional setting and significantly
advance the field.
