Percolation transition in a topological phase

Transition out of a topological phase is typically characterized by discontinuous changes in topological invariants along with bulk gap closings. However, as a clean system is geometrically punctured, it is natural to ask the fate of an underlying topological phase. To understand this physics we introduce and study both short- and long-ranged toy models where a one-dimensional topological phase is subjected to bond percolation protocols. We find that non-trivial boundary phenomena follow competing energy scales even while the global topological response is governed via geometrical properties of the percolated lattice. Using numerical, analytical, and appropriate mean-field studies we uncover the rich phenomenology and the various crossover regimes of these systems. In particular, we discuss the emergence of the "fractured topological region" where an overall trivial system contains a macroscopic number of topological clusters. Our study shows the interesting physics that can arise from an interplay of geometrical disorder within a topological phase.