Quantization of Topological Invariants under Symmetry-Breaking Disorder

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 invariants 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.