Topological invariant in three-dimensional band insulators with disorder

Topological insulators in three dimensions are characterized by a Z(2)-valued topological invariant, which consists of a strong index and three weak indices. In the presence of disorder, only the strong index survives. This paper studies the topological invariant in disordered three-dimensional system by viewing it as a supercell of an infinite periodic system. As an application of this method we show that the strong index becomes nontrivial when strong enough disorder is introduced into a trivial insulator with spin-orbit coupling, realizing a strong topological Anderson insulator. We also numerically extract the gap range and determine the phase boundaries of this topological phase, which fits well with those obtained from self-consistent Born approximation and the transport calculations.