Two-phase xenon detectors, such as that at the core of the forthcoming LZ dark matter experiment, use photomultiplier tubes to sense the primary (S1) and secondary (S2) scintillation signals resulting from particle interactions in their liquid xenon target. This paper describes a simulation study exploring two techniques to lower the energy threshold of LZ to gain sensitivity to low-mass dark matter and astrophysical neutrinos, which will be applicable to other liquid xenon detectors. The energy threshold is determined by the number of detected S1 photons; typically, these must be recorded in three or more photomultiplier channels to avoid dark count coincidences that mimic real signals. To lower this threshold: a) we take advantage of the double photoelectron emission effect, whereby a single vacuum ultraviolet photon has a $sim20%$ probability of ejecting two photoelectrons from a photomultiplier tube photocathode; and b) we drop the requirement of an S1 signal altogether, and use only the ionization signal, which can be detected more efficiently. For both techniques we develop signal and background models for the nominal exposure, and explore accompanying systematic effects, including the dependence on the free electron lifetime in the liquid xenon. When incorporating double photoelectron signals, we predict a factor of $sim 4$ sensitivity improvement to the dark matter-nucleon scattering cross-section at $2.5$ GeV/c$^2$, and a factor of $sim1.6$ increase in the solar $^8$B neutrino detection rate. Dropping the S1 requirement may allow sensitivity gains of two orders of magnitude in both cases. Finally, we apply these techniques to even lower masses by taking into account the atomic Migdal effect; this could lower the dark matter particle mass threshold to $80$ MeV/c$^2$.
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