  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$^2$.
