  Dark matter can scatter and excite a nucleus to a low-lying excitation in a
direct detection experiment. This signature is distinct from the canonical
elastic scattering signal because the inelastic signal also contains the energy
deposited from the subsequent prompt de-excitation of the nucleus. A
measurement of the elastic and inelastic signal will allow a single experiment
to distinguish between a spin-independent and spin-dependent interaction. For
the first time, we characterise the inelastic signal for two-phase xenon
detectors in which dark matter inelastically scatters off the Xe-129 or Xe-131
isotope. We do this by implementing a realistic simulation of a typical
tonne-scale two-phase xenon detector and by carefully estimating the relevant
background signals. With our detector simulation, we explore whether the
inelastic signal from the axial-vector interaction is detectable with upcoming
tonne-scale detectors. We find that two-phase detectors allow for some
discrimination between signal and background so that it is possible to detect
dark matter that inelastically scatters off either the Xe-129 or Xe-131 isotope
for dark matter particles that are heavier than approximately 100 GeV. If,
after two years of data, the XENON1T search for elastic scattering nuclei finds
no evidence for dark matter, the possibility of ever detecting an inelastic
signal from the axial-vector interaction will be almost entirely excluded.
