Local formula for the Z2 invariant of topological insulators

We proposed a formula for the ${\mathbb{Z}}₂$ invariant for topological insulators, which remains valid without translational invariance. Our formula is a local expression, in the sense that the contributions mainly come from quantities near a point. Using almost commute matrices, we proposed a method to approximate this invariant with local information. The validity of the formula and the approximation method is proved.