  Topological band dispersions other than the standard Dirac or Weyl fermions
have garnered the increasing interest in materials science. Among them, the
cubic Dirac fermions were recently proposed in the family of
quasi-one-dimensional conductors A$_2$Mo$_6$X$_6$ (A= Na, K, In, Tl; X= S, Se,
Te), where the band crossing is characterized by a linear dispersion in one
$k$-space direction but the cubic dispersion in the plane perpendicular to it.
It is not yet clear, however, how the external perturbations can alter these
nontrivial carriers and ultimately induce a new distinct quantum phase. Here we
study the evolution of Dirac fermions, in particular the cubic Dirac crossing,
under external pressure in the representative quasi-one-dimensional
Tl$_2$Mo$_6$Se$_6$ via the first-principles calculations. Specifically, it is
found that the topological properties, including the bulk Dirac crossings and
the topological surface states, change progressively under pressure up to 50
GPa where it undergoes a structural transition from the hexagonal phase to
body-centered tetragonal phase. Above 50 GPa, the system is more likely to be
topologically trivial. Further, we also investigate its phonon spectra, which
reveals a gradual depletion of the negative phonon modes with pressure,
consistent with the more three-dimensional Fermi surface in the high-pressure
phase. Our work may provide a useful guideline for further experimental search
and the band engineering of the topologically nontrivial fermions in this
intriguing state of matter.
