Sub-GeV WIMP dark matter (DM) is usually difficult to probe using liquid xenon detectors due to a small energy transfer in DM-nucleus interactions and a finite detector threshold. However, two novel direct detection methods overcome this limitation: the Bremsstrahlung and Migdal effects allow us to consider electron recoils that accompany the standard DM-nucleus scattering, thereby extending the reach to lower DM masses. I will present constraints on DM-nucleon scattering using data acquired in 2013 by the Large Underground Xenon (LUX) experiment. Looking ahead, the increasingly large noble liquid detectors are facing challenges with applications of high voltage (HV). The Xenon Breakdown Apparatus (XeBrA) at the Lawrence Berkeley National Laboratory was built to characterize the HV behavior of liquid xenon and liquid argon. Results from XeBrA serve not only to improve our understanding of the physical processes involved in the breakdown but also to inform the future of noble liquid detector engineering and neutrinoless double beta decay searches.