What is topological material? Recently, topological material has been a hot topic in condensed matter physics, but I don't know what is topological material and how to distinguish topological material from band diagram. And how does it divide the types of topological materials according to band diagram, such as Topological Insulators,Topological Semimetal....
 A: Traditionally, in solid state physics we classify materials into metals, insulators, and semimetals.
In metals, the Fermi energy is in the conduction band, such that pure metals are good electrical conductors.
Insulators (and with them semiconductors, which are insulators with a relatively small bandgap) have their Fermi level at zero temperature in the middle of the bandgap between valence and conduction band. This means, they do not conduct electrical currents very well, because electrons have to be excited above the band gap in order to propagate freely through the crystal.
In semimetals, the conduction and valence band states overlap in energy (hence no bandgap), and the Fermi energy intersects them both. This means that they conduct, but even at zero temperature there are conduction band electrons and valence band holes (unoccupied valence band states) that contribute to the electrical conductance.
As it turned out about 15 years ago, there is yet another class of materials called topological insulators. These materials have a band gap like insulators. At zero temperature, the Fermi level is in the band gap, so the bulk of the material does not conduct electrical current. However, due to the particular lattice symmetry of these materials, there exist surface states at the Fermi energy, which are said to be “topologically protected”. This means that these materials are insulating in the bulk, but they conduct very well at the surface.
The conventional dispersion relations $E(\vec{k})$ of bulk states of topological insulators are not significantly different from ordinary insulators in most cases, they simply look like insulators. However, knowing the Bloch-wavefunctions in the 1st Brillouin zone allows theoreticians to unravel their topological properties by defining so-called topological numbers (such as, for example, the Chern number). Also, in addition to the bulk band structure, there exists a dispersion relation for the edge states with states inside the bulk bandgap. There is a principle, called “bulk-edge” correspondence, which connects the topological bulk properties with the properties of the surface states.
Topological metals are materials with topological bands in the bulk, in which the Fermi-energy is in the conduction band, like in ordinary metals. You can define topological semimetals in complete analogy.
I have chosen electrical conduction as an example of a property that is different between different classes of materials, because it can be derived from the band structure at the Fermi energy. However, there are also other properties, such as thermal conductivity, which can be related to band structure, and may therefore be special in topological materials.
