  Comprehending nonequilibrium electron-phonon dynamics at the microscopic
level and at the short time scales is one of the main goals in condensed matter
physics. Effective temperature models and time-dependent Boltzmann equations
are standard techniques for exploring and understanding nonequilibrium state
and the corresponding scattering channels. However, these methods consider only
the time evolution of carrier occupation function, while the self-consistent
phonon dressing in each time instant coming from the nonequilibrium population
is ignored, which makes them less suitable for studying ultrafast phenomena
where softening of the phonon modes plays an active role. Here, we combine
ab-initio time-dependent Boltzmann equations and many-body phonon self-energy
calculations to investigate the full momentum- and mode-resolved nonadiabatic
phonon renormalization picture in the MoS$_2$ monolayer under nonequilibrium
conditions. Our results show that the nonequilibrium state of photoexcited
MoS$_2$ is governed by multi-valley topology of valence and conduction bands
that brings about characteristic anisotropic electron-phonon thermalization
paths and the corresponding phonon renormalization of strongly-coupled modes
around high-symmetry points of the Brillouin zone. As the carrier population is
thermalized towards its equilibrium state, we track in time the evolution of
the remarkable phonon anomalies induced by nonequilibrium and the overall
enhancement of the phonon relaxation rates. This work shows potential
guidelines to tailor the electron-phonon relaxation channels and control the
phonon dynamics under extreme photoexcited conditions.
