We have almost arrived at the criterion for the appearance of protected states in dislocations.

To see how the weak topological invariant relates to the number of states in the dislocation, we start by deforming a weak topological insulator into a set of disconnected planes, each carrying protected states. If there is a single state approaching the dislocation, as is shown in the figure below, it cannot backscatter and must therefore continue through the dislocation core.

(adapted from Cdang (Own work), via Wikimedia Commons, CC BY-SA 3.0.)

Counting the number and the orientation of the crystal planes approaching the core of the dislocation is just the Burgers vector. Hence, the number of edge states entering the dislocation core is the Burgers vector times the number of states per crystal plane. This brings us to the conclusion:

\[\mathcal{Q} = \mathbf{\mathcal{Q}}_\textrm{weak}\cdot\mathbf{B}.\]

Let’s now test this idea and see if we can observe the protected dislocation states.