Strong Brightness Variations Signal Cloudy-to-Clear Transition of Brown Dwarfs

[ABRIDGED] We report the results of a $J$ band search for cloud-related variability in the atmospheres of 62 L4-T9 dwarfs using the Du Pont 2.5-m telescope at Las Campanas Observatory and the Canada France Hawaii Telescope on Mauna Kea. We find 9 of 57 objects included in our final analysis to be significantly variable with >99% confidence, 5 of which are new discoveries. In our study, strong sinusoidal signals (peak-to-peak amplitudes >2%) are confined to the L/T transition (4/16 objects with L9-T3.5 spectral types and 0/41 objects for all other spectral types). The probability that the observed occurrence rates for strong variability inside and outside the L/T transition originate from the same underlying true occurrence rate is excluded with >99.7% confidence. These observations suggest that the settling of condensate clouds below the photosphere in brown dwarf atmospheres does not occur in a spatially uniform manner. Rather, the formation and sedimentation of dust grains at the L/T transition is coupled to atmospheric dynamics, resulting in highly contrasting regions of thick and thin clouds and/or clearings. Outside the L/T transition we identify 5 weak variables (peak-to-peak amplitudes of 0.6%-1.6%). Excluding L9-T3.5 spectral types, we infer that $60^{+22}_{-18}$% of targets vary with amplitudes of 0.5%$-$1.6%, suggesting that surface heterogeneities are common among L and T dwarfs. Our survey establishes a significant link between strong variability and L/T transition spectral types, providing evidence in support of the hypothesis that cloud holes contribute to the abrupt decline in condensate opacity and 1 micron brightening observed in this regime. More generally, fractional cloud coverage is an important model parameter for brown dwarfs and giant planets, especially those with L/T transition spectral types and colors.