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

5 pages, 3 figures

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• Computation of topological phase diagram of disordered Pb${}_{1-x}$Sn${}_x$Te using the kernel polynomial method
• The Noncommutative Index Theorem and the Periodic Table for Disordered Topological Insulators and Superconductors
• Bulk-boundary correspondence in point-gap topological phases
• Three-dimensional topological magnon systems
• Topological and conventional phases of a three dimensional electron glass
• Anderson-Kitaev spin liquid
• Dirac surface states in magnonic analogs of topological crystalline insulators
• Local formula for the $Z_2$ invariant of topological insulators
• Single-point spin Chern number in a supercell framework
• Fragility of ${\mathcal{Z}}_2$ topological invariant characterizing triplet excitations in a bilayer kagome magnet
• Photocontrol of spin scalar chirality in centrosymmetric itinerant magnets
• Topological Invariant for Bosonic Bogoliubov-de Gennes Systems with Disorder