  MnAu$_{2}$ is a spin-spiral material with in-plane ferromagnetic Mn layers
that form a screw-type pattern around a tetragonal $c$ axis. The spiral angle
$\theta$ was shown using neutron diffraction experiments to decrease with
pressure, and in later studies it was found to suffer a collapse to a
ferromagnetic state above a critical pressure, although the two separate
experiments did not agree on whether this phase transition is first or second
order. To resolve this contradiction, we use density functional theory
calculations to investigate the spiral state as a function of pressure, charge
doping, and also electronic correlations via a Hubbard-like $U$. We fit the
results to the one-dimensional $J_{1} - J_{2} - J_{3} - J_{4}$ Heisenberg
model, which predicts either a first- or second-order spiral-to-ferromagnetic
phase transition for different regions of parameter space. At ambient pressure,
MnAu$_{2}$ sits close in parameter space to a dividing line separating first-
and second-order transitions, and a combination of pressure and electron doping
shifts the system from the first-order region into the second-order region. Our
findings demonstrate that the contradiction in pressure experiments regarding
the kind of phase transition are likely due to variations in sample quality.
Our results also suggest that MnAu$_{2}$ is amenable to engineering via
chemical doping and to controlling $\theta$ using pressure and gate voltages,
which holds potential for integration in spintronic devices.
