A New Numerical Method for \(\mathbb{Z}_{2}\) Topological Insulators with Strong Disorder

Abstract

We propose a new method to numerically compute the $\\mathbb{Z}_2$ indices for\ndisordered topological insulators in Kitaev's periodic table. All of the\n$\\mathbb{Z}_2$ indices are known to be derived from the index formulae which\nare expressed in terms of a pair of projections introduced by Avron, Seiler,\nand Simon. For a given pair of projections, the corresponding index is\ndetermined by the spectrum of the difference between the two projections. This\ndifference exhibits remarkable and useful properties, as it is compact and has\na supersymmetric structure in the spectrum. These properties make it possible\nto numerically determine the indices of disordered topological insulators\nhighly efficiently. The method is demonstrated for the Bernevig-Hughes-Zhang\nand Wilson-Dirac models whose topological phases are characterized by a\n$\\mathbb{Z}_2$ index in two and three dimensions, respectively.\n