We demonstrate that Andreev modes that propagate along a transparent Josephson junction have a perfect transmission at the point where three junctions meet. The chirality and the number of quantized transmission channels is determined by the topology of the Fermi surface and the vorticity of the superconducting phase differences at the trijunction. We explain this chiral adiabatic transmission (CAT) as a consequence of the adiabatic evolution of the scattering modes both in momentum and real space. The dispersion relation of the junction then separates the scattering trajectories by introducing inaccesible regions of phase space. We expect that CAT is observable in nonlocal conductance and thermal transport measurements. Furthermore, because it does not rely on particle-hole symmetry, CAT is also possible to observe directly in metamaterials.@en

We derive a Z4 topological invariant that extends beyond symmetry eigenvalues and Wilson loops and classifies two-dimensional insulators with a C4T symmetry. To formulate this invariant, we consider an irreducible Brillouin zone and constrain the spectrum of the open Wilson lines that compose its boundary. We fix the gauge ambiguity of the Wilson lines by using the Pfaffian at high symmetry momenta. As a result, we distinguish the four C4T-protected atomic insulators, each of which is adiabatically connected to a different atomic limit. We establish the correspondence between the invariant and the obstructed phases by constructing both the atomic limit Hamiltonians and a C4T-symmetric model that interpolates between them. The phase diagram shows that C4T insulators allow ±1 and 2 changes of the invariant, where the latter is overlooked by symmetry indicators.@en

We show that topological defects in quadrupole insulators do not host quantized fractional charges, contrary to what their Wannier representation indicates. In particular, we test the charge quantization hypothesis based on the Wannier representation of a disclination and a parametric defect. Since disclinations necessarily strain the lattice and parametric defects require closed curves in parameter space, both defects break four-fold rotation symmetry, even away from their origin. The Wannier representation of the defects is thus determined by local reflection symmetries. Contrary to the hypothesis, we find that the local charge density decays as ∼ 1/r² with distance, leading to a diverging defect charge. Because topological defects are incompatible with four-fold rotation symmetry, we conclude that defect charge quantization is protected by sublattice symmetry, and not higher order topology.

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