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The effect of strong disorder on chiral-symmetric 3-dimensional latticemodels is investigated via analytical and numerical methods. The phase diagramsof the models are computed using the non-commutative winding number, asfunctions of disorder strength and model's parameters. Thelocalized/delocalized characteristic of the quantum states is probed with levelstatistics analysis. Our study re-confirms the accurate quantization of thenon-commutative winding number in the presence of strong disorder, and itseffectiveness as a numerical tool. Extended bulk states are detected above andbelow the Fermi level, which are observed to undergo the so called "levitationand pair annihilation" process when the system is driven through a topologicaltransition. This suggests that the bulk invariant is carried by these extendedstates, in stark contrast with the 1-dimensional case where the extended statesare completely absent and the bulk invariant is carried by the localizedstates.

physical review. b, condensed matter and materials physics

Effect of Strong Disorder on 3-Dimensional Chiral Topological  Insulators: Phase Diagrams, Maps of the Bulk Invariant and Existence of  Topological Extended Bulk States