  We study the onset of collective spin-self-organization in a thermal ensemble
of driven two-level atoms confined in an optical cavity. The atoms
spontaneously form a spin-pattern above a critical driving strength that sets a
threshold and is determined by the cavity parameters, the initial temperature,
and the transition frequency of the atomic spin. Remarkably, we find that
inhomogeneous Doppler broadening facilitates the onset of
spin-self-organization. In particular, the threshold is non-monotonic when
increasing the spin transition frequency and reaches a minimum when the Doppler
broadening is of similar magnitude. This feature emerges due to Doppler-induced
resonances. Above the threshold, we find cooperative dynamics of spin, spatial,
and momentum degrees of freedom leading to density modulations, fast reduction
of kinetic energy, and the emergence of non-thermal states. More broadly, our
work demonstrates how broadening can facilitate strong light-matter
interactions in many-body systems.
