Although the richness of spatial symmetries has led to a rapidly expanding
inventory of possible topological crystalline (TC) phases of electrons,
physical realizations have been slow to materialize due to the practical
difficulty to ascertaining band topology in realistic calculations. Here, we
integrate the recently established theory of symmetry indicators of band
topology into first-principle band-structure calculations, and test it on a
databases of previously synthesized crystals. The combined algorithm is found
to efficiently unearth topological materials and predict topological properties
like protected surface states. On applying our algorithm to just 8 out of the
230 space groups, we already discover numerous materials candidates displaying
a diversity of topological phenomena, which are simultaneously captured in a
single sweep. The list includes recently proposed classes of TC insulators that
had no previous materials realization as well as other topological phases,
including: (i) a screw-protected 3D TC insulator, \b{eta}-MoTe2, with gapped
surfaces except for 1D helical "hinge" states; (ii) a rotation-protected TC
insulator BiBr with coexisting surface Dirac cones and hinge states; (iii)
non-centrosymmetric Z2 topological insulators undetectable using the
well-established parity criterion, AgXO (X=Na,K,Rb); (iv) a Dirac semimetal
MgBi2O6; (v) a Dirac nodal-line semimetal AgF2; and (vi) a metal with
three-fold degenerate band crossing near the Fermi energy, AuLiMgSn. Our work
showcases how the recent theoretical insights on the fundamentals of band
structures can aid in the practical goal of discovering new topological
materials