A magnetic field forces electrons to twist as they transport around it, revealing properties of the medium. This observation—the Aharonov-Bohm effect—applies to any quantum system where charged particles remain coherent, like an electron in a sufficiently clean solid state device. This thesis is about using this effect to probe and design topologically protected electronic phenomena.
The first part of this thesis focuses on crystalline topological insulators, phases protected by spatial symmetries of a crystal. Chapters 2 and 3 concern the bulk and boundary response of obstructed atomic insulators, phases that lack a bulk-boundary correspondence and that we characterize using a topological defect and momentum-space invariants respectively. Chapter 4 is about intrinsic higher-order topological insulators, phases that do have a bulk-boundary correspondence and therefore are detectable in transport experiments. We develop a theory based on electronic transport and the insertion of fluxes to capture topology, and show that it may be used to understand how disorder affects these phases. In Chapter 5, we apply this theory to an experimentallyrelevant proposal of topological superconductivity and identify its biases.
Differently from the first part, the rest contains two projects that originated from numerical adventures. Chapter 6 proposes a superconducting chiral waveguide that relies on magnetic flux to achieve unidirectional transport of electron-hole pairs. The final chapter, while unrelated to flux, topology, or transport, introduces an algorithm that may be used in the study of these phenomena. Chapter 7 is about Pymablock, an opensource Python package to efficiently performquasi-degenerate perturbation theory. The cover highlights the relevance of computational approaches in modern condensed matter physics, and also in this work.

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.

The surface states of intrinsic higher order topological phases are protected by the spatial symmetries of a finite sample. This property makes the existing scattering theory of topological invariants inapplicable: the scattering geometry is either incompatible with the symmetry or does not probe the bulk topology. We resolve this obstacle by using a symmetric scattering geometry that probes transport from the inside to the outside of the sample. We demonstrate that the intrinsic higher order topology is captured by the flux dependence of the reflection matrix. Our finding follows from identifying the spectral flow of a flux line as a signature of higher order topology. We show how this scattering approach applies to several examples of higher order topological insulators and superconductors. Our theory provides an alternative approach for proving bulk–edge correspondence in intrinsic higher order topological phases, especially in the presence of disorder.

The easily accessible experimental signatures of Majorana modes are ambiguous and only probe topology indirectly: for example, quasi-Majorana states mimic most properties of Majoranas. Establishing a correspondence between an experiment and a theoretical model known to be topological resolves this ambiguity. Here we demonstrate that already theoretically determining whether a finite system is topological is by itself ambiguous. In particular, we show that the scattering topological invariant—a probe of topology most closely related to transport signatures of Majoranas—has multiple biases in finite systems. For example, we identify that quasi-Majorana states also mimic the scattering invariant of Majorana zero modes in intermediate-sized systems. We expect that the bias due to finite size effects is universal, and advocate that the analysis of topology in finite systems should be accompanied by a comparison with the thermodynamic limit. Our results are directly relevant to the applications of the topological gap protocol.

We propose an architecture for bit-flip error correction of Andreev spins that is protected by Kramers' degeneracy. Specifically, we show that a coupling network of linear inductors and Andreev spin qubits results in a static Hamiltonian composed of the stabilizers of a bit-flip code. The electrodynamics of the many-body spin states also respect these stabilizers, and we show how reflectometry off a single coupled resonator can thereby accomplish their projective measurement. We further show how circuit-mediated spin couplings enable error correction operations and a complete set of single- and two-module logical quantum gates. The concept, which we dub the "Ising molecule qubit,"is experimentally feasible and provides a path for compact noise-biased qubits.

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.

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.