Identification of topology in finite media

Topological phases of matter are promising for a wide range of applications because their defining properties, the boundary states, are robust against local perturbations such as disorder. To study topological phases, we must therefore also be able to identify them in the presence of disorder. However, disorder generally complicates this identification because it breaks the translation symmetry required by many momentum-space invariants. In this thesis, we investigate how topology can be understood and identified in finite and disordered systems.

One method for identifying topology in finite and disordered systems is to use scattering invariants. These invariants rely on a finite scattering geometry with attached leads, from which a reflection matrix can be constructed. The invariant is defined from this reflection matrix, which undergoes topological transitions at the same points as the bulk system.

In Chapter 2, we extend the theory of scattering invariants to intrinsic higher-order topological insulators. These are topological phases whose boundary states are more than one dimension lower than the dimension of the bulk. We find that existing scattering theory requires two key extensions. First, the scattering geometry must contain a hole at its center to create two disconnected edges. Second, a flux must be threaded through this hole. We treat this flux as an additional momentum parameter, thereby obtaining the correct number of parameters for dimensional reduction. We show that these two additions allow us to identify topological phases in, for example, a BBH model, systems protected by magnetic rotation symmetry, and an inversion-protected axion insulator. We also show that the construction identifies topological transitions in the presence of disorder, provided that the disorder is first symmetrized according to the protecting crystalline symmetry.

However, imposing an exact symmetry constraint on the disorder configuration is problematic for statistical topological insulators, in which the topological phases are protected by a symmetry that holds on average across an ensemble rather than exactly in every realization. Such systems can admit either more or fewer topological phases in the presence of disorder than are possible in the clean system. Simply symmetrizing the disorder is therefore problematic because it returns the system to an exactly symmetric configuration.

In Chapter 3, we address this problem by introducing the idea of embedding a disordered system into a symmetric approximant system. The key realization is that we can probe topological transitions in systems that have all the necessary symmetries locally but possess an additional global symmetry. As an example, we consider a mirror-symmetric system in class AIII that admits fewer phases when the disorder respects mirror symmetry only on average than in the exact-mirror-symmetry limit. We find that all topological transitions can be identified again by considering a glide-symmetric system instead. Together, the formalisms presented in Chapters 2 and 3 extend our ability to identify topology consistently in disordered and finite systems.

Scattering invariants defined from reflection matrices can be expressed naturally using network models. Network models describe propagation through wave amplitudes on links and local scattering matrices at nodes. In Chapter 4, we present a Python software package for network models. The package provides a common representation for periodic and finite networks, operations for changing their geometry and connectivity, and sparse solvers for quasienergy spectra and transport observables. We demonstrate the methods of the package with a symbolic Fabry–Perot interferometer, where we recover the analytical expression of the scattering matrix, as well as with the numerical Chalker-Coddington model.

In Chapter 5, we examine how a finite geometry influences the spatial structure of a topological boundary state. We study Majorana wavefunctions in two-dimensional topological superconductors shaped as quantum billiards. We find that the Majorana wavefunction inherits the influence of the underlying classical dynamics. In chaotic billiards, quantum scars in the normal-state eigenfunctions remain visible in the edge-localized Majorana wavefunctions, both for p-wave and more realistic s-wave superconductors. We demonstrate that moving a local potential in the center of the billiard, far from the edges where the Majorana wavefunctions are localized, can substantially change their spatial profile. These results show that Majorana localization is determined not only by the pairing potential and disorder, but also by the shape and classical dynamics of the finite system.