  Understanding crystal shapes is a fundamental subject in surface science. It
is now well studied how chemical bondings determine crystal shapes via
dependence of surface energies on surface orientations. Meanwhile, discoveries
of topological materials have led us to a new paradigm in surface science, and
one can expect that topological surface states may affect surface energies and
crystal facets in an unconventional way. Here we show that the surface energy
of glide-symmetric topological crystalline insulators (TCI) depends on the
surface orientation in a singular way via the parity of the Miller index. This
singular surface energy of the TCI affects equilibrium crystal shapes,
resulting in emergence of unique crystal facets of the TCI. This singular
dependence of the topological surface states is unique to the TCI protected by
the glide symmetry in contrast to a TCI protected by a mirror symmetry. In
addition, we show that such singular surface states of the TCI protected by the
glide symmetries can be realized in KHgSb with first-principles calculations.
Our results provide a basis for designs and manipulations of crystal facets by
utilizing symmetry and topology.
