There is a whole zoo of topological phases, and hopefully someone will provide a more complete answer, but here are some thoughts.
Symmetry and dimension. The topological classification of a material with a gap (topological insulator or topological superconductor) depends on (i) symmetry and (ii) dimension. These relations are summarized by the so-called ...
Finding topological invariant number(called Z$_2$ number) can give information about topological invariance. Different codes are available for such calculations such as Z2pack or wanniertools and others.
As a first step, you can use the WannierTools.
There are five typical examples in there.
Bi2Se3 (3D strong TI)
MoS2 (2D QSHE)
WTe2 (Type II Weyl semimetal)
IrF4 (Nodal Chain metals)
FeSi (Weyl point in Phonon system)
A straightforward way to model the interaction of charged species with a surface is to use a cluster model instead of simulating the surface as a slab with periodic boundary conditions (1).
Cluster models are not perfect, they can be affected by border effects that require large clusters, to be representative of the bulk system, by coverage effects, as a ...