In the recently discovered topological crystalline insulators SnTe and
Pb_{1-x}Sn_{x}(Te,Se), crystal symmetry and electronic topology intertwine to
create topological surface states with many interesting features including
Lifshitz transition, Van-Hove singularity and fermion mass generation. These
surface states are protected by mirror symmetry with respect to the (110)
plane. In this work we present a comprehensive study of the effects of
different mirror-symmetry-breaking perturbations on the (001) surface band
structure. Pristine (001) surface states have four branches of Dirac fermions
at low-energy. We show that ferroelectric-type structural distortion generates
a mass and gaps out some or all of these Dirac points, while strain shifts
Dirac points in the Brillouin zone. An in-plane magnetic field leaves surface
state gapless, but introduces asymmetry between Dirac points. Finally, an
out-of-plane magnetic field leads to discrete Landau levels. We show that the
Landau level spectrum has an unusual pattern of degeneracy and interesting
features due to the unique underlying band structure. This suggests that Landau
level spectroscopy can detect and distinguish between different mechanisms of
symmetry breaking in topological crystalline insulators.Comment: 10 pages, 5 figures, main results are summarized in Table I; (v2)
minor correction