Let us follow the direction explained by Joel Moore and construct a three-dimensional topological state from the two-dimensional topological state. This time, we’ll do this by studying the system in momentum space rather than in real space as we did before. As with two dimensional systems, time-reversal invariant momenta (TRIMs) play an important role in three dimensions.

For illustrative purposes, consider the three dimensional irreducible Brillouin Zone (i.e. \(k_j\in [0,\pi]\)) of a cubic system shown below. Fixing one of the three momenta \(k_{x,y,z}\) to a TRIM, say \(k_x=0\) without loss of generality, we can think of the Hamiltonian in the \((k_y,k_z)\) plane as a two dimensional Hamiltonian, which may either be topologically trivial (\(\mathbb{Z}_2\)-index \(=0\)) or non-trivial (\(\mathbb{Z}_2\)-index \(=1\)).

So for every side of the cube shown above we can compute a QSHE topological invariant, which gives us 6 numbers. However not all of them are independent. Specifically, there is a constraint \(Q(k_x=0)\,Q(k_x=\pi) \equiv Q(k_y=0)\,Q(k_y=\pi) \equiv Q(k_z=0)\,Q(k_z=\pi)\).

This product is called the strong topological invariant. Accordingly, the topological insulators where this invariant is non-trivial are called strong topological insulators. For the remaining three invariants, we can choose \(Q(k_x=\pi),\,Q(k_y=\pi),\,Q(k_z=\pi)\).

Very frequently the topological invariants of a compound are written as \((1;010)\), where the first number corresponds to the strong invariant, and the remaining three to the weak invariants along each axis. For example, the first predicted topological insulator, the alloy Bi\(_x\)Sb\(_{1-x}\) is \((1;111)\), and the second generation topological insulators Bi\(_2\)Te\(_3\) and Bi\(_2\)Se\(_3\) are \((1;000)\).

Just by using the bulk-edge correspondence for \(Q\) we know that the strong topological invariant means that there is an odd number of helical states going in each direction on each facet of the topological insulator. We will see later why this is special, but before that let’s construct a model for a 3D TI.