  "Topological Quantum Chemistry (TQC) links the chemical and symmetry
structure of a given material with its topological properties. This field
tabulates the data of the 10398 real-space atomic limits of materials, and
solves the compatibility relations of electronic bands in momentum space. A
material that is not an atomic limit or whose bands do not satisfy the
compatibility relations, is a topological insulator/semimetal. We use TQC to
find the topological stoichiometric non-magnetic, "high-quality'' materials in
the world. We develop several code additions to VASP which can compute all
characters of all symmetries at all high-symmetry points in the Brillouin Zone
(BZ). Using TQC we then develop codes to check which materials in ICSD are
topological. Out of 26938 stoichiometric materials in our filtered ICSD
database, we find 2861 topological insulators (TI) and 2936 topological
semimetals (2505 and 2560 non-f electron, respectively). Our method is uniquely
capable to show that none of the TI's found exhibit fragile topology. We
partition the topological materials in different physical classes. For the
majority of the 5797 "high-quality'' topological material, we compute: the
topological class (equivalence classes of TQC elementary band representations
-- equivalent to the topological index), the symmetry(ies) that protects the
topological class, the representations at high symmetry points and the direct
gap (for insulators), and the topological index. For topological semimetals we
then compute whether the system becomes a topological insulator (whose
index/class we compute) upon breaking symmetries -- useful for experiments.
2152 more TI's are obtained in this way. For almost all 5065 non-f-electron
topological materials, we provide the electronic band structures, allowing the
identification of quantitative properties (gaps, velocities). Remarkably, our
exhaustive results show that a large proportion ( ~ 24% !) of all materials in
nature are topological (confirmed by calculations of "low-quality'' materials).
We confirm the topology of several new materials by Wilson loop calculations.
We added an open-source code and end-user button on the Bilbao Crystallographic
Server (BCS) which checks the topology of any material. We comment on the
chemistry of each compound and sample part of the "low-quality'' ICSD data to
find more materials."
