The discovery of quantum materials with
nontrivial band topology and robust surface states have garnered
significant attention among the scientific community. We theoretically
study topological features of quantum materials ranging from topological
insulators to semimetals. Topological insulators (TIs) are kind of
materials which behave as an insulator in its interior but whose surface
contains conducting states, meaning that electrons can only move along
the surface of the material. In contrast to TIs, the bulk of certain
metals or semimetals are neither boring. Unlike TIs, topological metals
and semimetals are interesting not only because of their rich surface
physics but also for the exotic nature of linear band crossing in the
bulk band structures. Depending on the dimensionality of the band
crossing points and Fermi surface topology in the momentum space,
distinct topological semimetals have been discovered, namely, Dirac
semimetal (DSM), Weyl semimetal (WSM), Nodal line semimetal (NLS),
triple point semimetal (TPSM). In NLS, the conduction and the valence
bands cross each other along a one-dimensional loop in the
three-dimensional Brillouin zone. The DSM and WSM are the low-energy
excitations of relativistic Dirac and Weyl fermions with four-fold and
two-fold degenerate band crossings, respectively. On the otherhand, TPSM
is another class of topological semimetal, which considered to be an
intermediate state of Dirac and Weyl semimetal. Unlike DSM and WSM, the
band crossing point in TPSM possesses three fold degeneracy. The multi
fold band degeneracy in these semimetals are protected by certain
crystalline symmetries. Gradual reduction of such symmetries could also
lead to have topological phase transition from one to another. Cartoon
diagram shows an schematic of DSM, TPSM and gaped phase with an inverted
band gap.