Effect of Strong Disorder on 3-Dimensional Chiral Topological   Insulators: Phase Diagrams, Maps of the Bulk Invariant and Existence of   Topological Extended Bulk States
  The effect of strong disorder on chiral-symmetric 3-dimensional lattice models is investigated via analytical and numerical methods. The phase diagrams of the models are computed using the non-commutative winding number, as functions of disorder strength and model's parameters. The localized/delocalized characteristic of the quantum states is probed with level statistics analysis. Our study re-confirms the accurate quantization of the non-commutative winding number in the presence of strong disorder, and its effectiveness as a numerical tool. Extended bulk states are detected above and below the Fermi level, which are observed to undergo the so called "levitation and pair annihilation" process when the system is driven through a topological transition. This suggests that the bulk invariant is carried by these extended states, in stark contrast with the 1-dimensional case where the extended states are completely absent and the bulk invariant is carried by the localized states. 
Juntao Song, Carolyn Fine and Emil Prodan
cond-mat.dis-nn