Crystalline symmetries have played a central role in the identification and understanding of quantum materials. The use of symmetry indicators and band representations have enabled a classification scheme for crystalline topological materials, leading to large scale topological materials discovery. Amorphous materials lack long-range order and therefore fall outside of this classification scheme, casting them aside for consideration as topological quantum materials. The work described in this thesis suggests that amorphous systems provide a good materials space to study the topological and quantum properties of the electronic structure. The basis of this observation lies in the fact that amorphous materials have a well defined local environment. Due to this environment, it will be shown that an amorphous material can have dispersive, spin-momentum locked surface states, which can be a consequence of a topological electronic structure. I will also show that by modifying this local environment in a trivial material, a topological phase transition can be achieved with disorder. Nanocrystalline materials also lack long range order, however work shown in this thesis suggests that nanocrystals are detrimental for topological properties since the grain boundaries lack a well defined environment. These systems aren't devoid of interesting properties, the disorder can increase electronic interactions. Given that amorphous materials can host topological states, a method to predict which amorphous materials will be topological with chemical specificity is developed.

In this work an amorphous analog of a well known three-dimensional strong topological insulator, which lies beyond this classification due to the lack of long-range structural order, is investigated and our experimental observations suggest it has topological properties in the solid state. Amorphous Bi2Se3 thin films are studied, which show metallic behavior and high bulk resistance. The observed low field magnetoresistance due to weak antilocalization demonstrates a significant number of two-dimensional surface conduction channels. Our angle-resolved photoemission spectroscopy data is consistent with a dispersive two-dimensional surface state that crosses the bulk gap. Spin resolved photoemission spectroscopy shows this state has an anti-symmetric spin texture, confirming the existence of spin-momentum locked surface states. These experimental results are discussed in light of a theoretical photoemission spectra obtained with an amorphous tight-binding topological insulator model, contrasting it with an alternative Rashba explanation. The discovery of spin-momentum locked surface states in amorphous materials suggests new ways to characterize amorphous matter. The dispersive, spin-momentum locked states motivates the study of an overlooked subset of amorphous quantum materials outside of current classification schemes, as a novel route to develop promising scalable quantum devices.

Much of our world is comprised of amorphous materials, which lack periodicity and long-range order but retain short-range ordering such as bond-lengths and preferred local environments. Here, it is demonstrated that, even in the absence of long range order, a well-defined real-space length scale is sufficient to produce dispersive band structures. Moreover, for the first time, a repeated Fermi surface structure of duplicated annuli is observed, reminiscent of Brillouin zone-like repetitions. Our simulations using amorphous Hamiltonians reveal that the typical momentum scale where repetitions occur is the inverse average nearest-neighbour distance, the direct fingerprint of the local order of the underlying atomic structure. Many electronic phenomenon rely on momentum-dependence such as momentum pairing or spin-orbit coupling and therefore, under this description, amorphous materials can be reevaluated as a source for generating novel phases.

We investigate using local structural disorder to induce a topologically nontrivial phase in a solid state system. Using first-principles calculations, structural disorder is introduced in the trivial insulator BiTeI and observe the emergence of a topological insulating phase. By modifying the bonding environments, the crystal-field splitting is enhanced, with spin-orbit interactions producing a band inversion in the bulk electronic structure. Analysis of the Wannier charge centers and the surface electronic structure reveals a strong topological insulator with Dirac surface states. Finally, a prescription for inducing topological states from disorder in crystalline materials is proposed. Understanding how local environments produce topological phases is a key step for predicting disordered and amorphous topological materials.

Strong disorder has a crucial effect on the electronic structure in quantum materials by increasing localization, interactions, and modifying the density of states. In this work amorphous BixTeI thin films were grown at various compositions and growth temperatures in order to study the effect of structural disorder on electronic properties. By decreasing the growth temperature, the structural disorder is increased and we observe a metal-insulator transition as a function of composition in films grown both at room temperature and 230 K. By tuning the disorder of several compositions with growth temperature, a several magnitudes decrease in the conductivity is observed. The metal-insulator transition is accompanied by a disappearance of weak-antilocalization and increased electron-electron interactions. This work shows that disorder can be used to study strongly correlated topological materials. Disorder is controlled to study the effect of interactions and localization in quantum materials with strong spin-orbit coupling, and by doing so we shed light on how quantum materials can be tuned for spin transport with disorder.

While topological phases of matter are not restricted to crystals, there is no efficient method for predicting which amorphous solids are topological. In order to enable a high-throughput screening of amorphous topological materials, it is desirable to find a computationally efficient indicator of topology, compatible with first-principles calculations. In this work, the structural spillage is introduced, an indicator that predicts the unknown topological phase of an amorphous solid by comparing it to a known reference crystal. To illustrate its potential, it is benchmarked using tight-binding and first-principles calculations of amorphous bismuth models. Using DFT the structural spillage predicts that amorphous bilayer bismuth is topological, and thus a novel topological material. Our work sets the basis to predict topological amorphous solids efficiently, opening up a novel and large material class to high-throughput searches of topological materials.