  We proposed that BaHgSn is a Dirac semimetal (DSM) which can host
hourglass-like surface states (HSSs) as protected by nonsymmorphic glide
symmetry. Compared to KHgSb, an isostructural topological crystalline insulator
with the same HSSs, BaHgSn has an additional band inversion at $\Gamma$ point.
This band inversion is induced by the stronger interlayer coupling among Hg-Sn
honeycomb layers than that among Hg-Sb-layers in KHgSb, which leads to bulk
Dirac nodes in BaHgSn along the layer stacking direction $\Gamma$-$A$. In
addition, the mirror Chern number $C_{i}$ protected by the mirror plane
$\overline{M}_{z}$ ($k_z$=0) changes from 2 in KHgSb to 3 in BaHgSn. Therefore,
when a compressive uniaxial strain is applied along the $y$ axis to break the
rotation symmetry protecting the DSM state, BaHgSn becomes a strong topological
insulator with $Z_{2}$ indices of $(1;000)$ and the topological surface Dirac
cone co-exists with HSSs on the (010) surface. The Wilson-loop spectra have
been calculated to verify these topological features. The calculated surface
states, the Fermi surfaces and their quasiparticle interference patterns are
ready to be compared with experimental measurements.
