  We present a comprehensive study of a model system of repulsive
self-propelled disks in two dimensions with ferromagnetic and nematic velocity
alignment interactions. We characterize the phase behavior of the system as a
function of the alignment and self-propulsion strength, featuring orientational
order for strong alignment and Motility-Induced Phase Separation (MIPS) at
moderate alignment but high enough self-propulsion. We derive a microscopic
theory for these systems yielding a close set of hydrodynamic equations from
which we perform a linear stability analysis of the homogenous disordered
state. This analysis predicts MIPS in the presence of aligning torques. The
nature of the continuum theory allows for an explicit quantitative comparison
with particle-based simulations, which consistently shows that ferromagnetic
alignment fosters phase separation, while nematic alignment does not alter
either the nature or the location of the instability responsible for it. In the
ferromagnetic case, such behavior is due to an increase of the imbalance of the
number of particle collisions along different orientations, giving rise to the
self-trapping of particles along their self-propulsion direction. On the
contrary, the anisotropy of the pair correlation function, which encodes this
self-trapping effect, is not significantly affected by nematic torques. Our
work shows the predictive power of such microscopic theories to describe
complex active matter systems with different interaction symmetries and sheds
light on the impact of velocity-alignment interactions in Motility-Induced
Phase Separation.
