Decoupling the electronic gap from the spin Chern number in spin-resolved topological insulators

In two-dimensional topological insulators, a disorder-induced topological phase transition is typically identified with an Anderson localization transition at the Fermi energy. However, in <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:msub><a:mi mathvariant="double-struck">Z</a:mi><a:mn>2</a:mn></a:msub></a:math> trivial, spin-resolved topological insulators it is the spectral gap of the spin spectrum, in addition to the bulk mobility gap, which protects the nontrivial topology of the ground state. In this work, we show that these two gaps, the bulk electronic and spin gap, can evolve distinctly on the introduction of quenched short-ranged disorder and that an odd-quantized spin Chern number topologically protects states below the Fermi energy from localization. This decoupling leads to a unique situation in which an Anderson localization transition occurs below the Fermi energy at the topological transition. Furthermore, the presence of topologically protected extended bulk states nontrivial bulk topology typically implies the existence of protected boundary modes. We demonstrate the absence of protected boundary modes in the Hamiltonian and yet the edge modes in the eigenstates of the projected spin operator survive. Our work thus provides evidence that a nonzero spin-Chern number, in the absence of a nontrivial <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"><c:msub><c:mi mathvariant="double-struck">Z</c:mi><c:mn>2</c:mn></c:msub></c:math> index, does not demand the existence of protected boundary modes at finite or zero energy. Published by the American Physical Society 2024