  We study lanthanum mononitride LaN by first-principles calculations. The
commonly reported rock-salt structure of $Fm\bar{3}m$ symmetry for rare-earth
monopnictides is found dynamically unstable for LaN at zero temperature. Using
density functional theory and evolutionary crystal prediction, we discover a
new, dynamically stable structure with $P1$ symmetry at 0 K. This $P1$-LaN
exhibits spontaneous electric polarization. Our ab initio molecular dynamics
simulations of finite-temperature phonon spectra further suggest that LaN will
undergo ferroelectric and structural transitions from $P1$ to $Fm\bar{3}m$
symmetry, when temperature is increased. Moreover, $P1$-LaN will transform to a
tetragonal structure with $P4/nmm$ symmetry at a critical pressure $P=18$ GPa
at 0 K. Electronic structures computed with an advanced hybrid functional show
that the high-temperature rock-salt LaN can change from a trivial insulator to
a strong topological insulator at $P \sim 14$ GPa. Together, our results
indicate that when $P=14 - 18$ GPa, LaN can show simultaneous
temperature-induced structural, ferroelectric, and topological transitions.
Lanthanum monopnictides thereby provide a rich playground for exploring novel
phases and phase transitions driven by temperature and pressure.
