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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 ${\mathbb{Z}}_2$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 ${\mathbb{Z}}_2$index, does not demand the existence of protected boundary modes at finite or zero energy. Published by the American Physical Society2024 

Motivated by the recent proposals for unconventional emergent physics in twisted bilayers of nodal superconductors, we study the peculiarities of the Josephson effect at the twisted interface between d-wave superconductors. We demonstrate that for clean interfaces with a twist angle θ0 in the range 0◦ < θ0 < 45◦, the critical current can exhibit nonmonotonic temperature dependence with a maximum at a nonzero temperature as well as a complex dependence on the twist angle at low temperatures. These effects are shown to reflect the destructive interference between the d-wave order parameters near the nodes at nonzero twist angle. Close to 45◦ we find that the critical current does not vanish due to Cooper pair cotunneling, which can lead to the transition to a time-reversal breaking superconducting d + id phase, which can be suppressed by the interface roughness. We provide a comprehensive theoretical analysis of experiments that can reveal this cotunneling for twisted superconductors close to θ0 = 45◦. In particular, we demonstrate that both the emergence of the Fraunhofer interference pattern near θ0 = 45◦ and fractional Shapiro steps yield unambiguous evidence of Cooper pair cotunneling, necessary for topological superconductivity.