We consider single particle Schrodinger operators with a gap in the en ergy spectrum. We construct a complete, orthonormal basis function set for the inv ariant space corresponding to the spectrum below the spectral gap, which are exponentially local
ized a round a set of closed surfaces of monotonically increasing sizes. Estimates on the exponential dec ay rate and a discussion of the geometry of these surfaces is included.
In the strictly periodic setting, the electric polarization of inversion-symmetric solids with and without time-reversal symmetry and the isotropic magneto-electric response function of time-reversal symmetric insulators are known to be topological i
nvariants displaying an exact $mathbb Z_2$ quantization. This quantization is stabilized by the symmetries. In the present work, we investigate the fate of such symmetry-stabilized topological invariants in the presence of a disorder which breaks the symmetries but restores them on average. Using a rigorous analysis, we conclude that the strict quantization still holds in these conditions. Numerical calculations confirm this prediction.
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 stre
ngth and models 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.
Using an explicit 1-dimensional model, we provide direct evidence that the one-dimensional topological phases from the AIII and BDI symmetry classes follow a $mathbb Z$-classification, even in the strong disorder regime when the Fermi level is embedd
ed in a dense localized spectrum. The main tool for our analysis is the winding number $ u$, in the non-commutative formulation introduced in I. Mondragon-Shem, J. Song, T. L. Hughes, and E. Prodan, arXiv:1311.5233. For both classes, by varying the parameters of the model and/or the disorder strength, a cascade of sharp topological transitions $ u=0 rightarrow u=1 rightarrow u=2$ is generated, in the regime where the insulating gap is completely filled with the localized spectrum. We demonstrate that each topological transition is accompanied by an Anderson localization-delocalization transition. Furthermore, to explicitly rule out a $mathbb Z_2$ classification, a topological transition between $ u=0$ and $ u=2$ is generated. These two phases are also found to be separated by an Anderson localization-delocalization transition, hence proving their distinct identity.
