The effect of strong disorder on chiral-symmetric three-dimensional lattice models is investigated via analytical and numerical methods. The phase diagrams of the models are computed using the noncommutative 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 reconfirms the accurate quantization of the noncommutative 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 one-dimensional case where the extended states are completely absent and the bulk invariant is carried by the localized states.

• Received 18 August 2014
• Revised 10 October 2014

DOI:https://doi.org/10.1103/PhysRevB.90.184201