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The non-trivial topology in the layered $\text{FeTe}_{0.55}\text{Se}_{0.45}$(FTS) superconductor has been suggested by both theory and experiment to bestrongly dependent on the Te concentration. Motivated by this together with theTe fluctuations expected from alloy disorder, we develop a simple layered modelfor a strong topological insulator that allows us to describe a scenario wheretopologically trivial domains permeate the sample. We refer to such a phase astopological domain disordered and study the local density (LDOS) of thetopological surface states that can be measured using scanning tunnelingspectroscopy (STS) in this phase. We find that topologically trivial domains onthe surface, where one would expect the topological surface state to be absent,appear as regions of suppressed LDOS surrounded by domain walls with enhancedLDOS. Furthermore, we show that studying the energy dependence of the STSshould allow us to distinguish the topologically trivial parts of the surfacefrom other forms of disorder. Finally, we discuss implications of such localdisappearance of the topological surface states for the observation of Majoranamodes in vortices.

On the signatures of non-topological patches on the surface of topological insulators