  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 {\it 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.
