Unconventional topology in low-dimensional systems

Topological insulators and superconductors are phases of matter classified by symmetry and bulk topological invariants, possessing boundary phenomena such as protected edge states and Majorana zero modes. This thesis uses electronic tight-binding models to investigate topology in low-dimensional systems. Emphasis is placed on unconventional topology enabled or reshaped by nonsymmorphic symmetries that mimic the nonunitary symmetries of the tenfold way classification system. The first original study in the thesis is to model the influence of an in-plane magnetic field on the orbital properties of electrons in rhombohedral graphite. We show that the low-energy k-space Bloch Hamiltonian of rhombohedral multilayer graphene maps onto the real-space Su-Schrieffer-Heeger (SSH) chain. An in-plane magnetic field enters through a Peierls substitution that effectively creates SSH-like soliton textures, allowing the location of conduction to be tuned by field strength and producing characteristic band bifurcations, which can be understood through semiclassical quantisation of the zero-field Fermi surface. Next, we analyse one-dimensional nonsymmorphic models with Kramers degeneracy, specifically, systems characterised by Z2 and Z4 indices induced by nonsymmorphic chiral symmetry. We extend a generalised winding-number framework to these Kramers-degenerate settings and, for the Z4 model, study additional characterising features such as robustness under disorder and higher-order terms. Typically, these condensed-matter toy models are difficult to implement and tune, encouraging the search for simpler methods of exploring topological behaviour. One such method is topolectric circuits, where linear circuit components mimic tight-binding parameters. Here, we develop topolectric circuit realisations of nonsymmorphic models, using impedance signatures to probe zero-energy states. Finally, we catalogue four-band one-dimensional Bogoliubov-de Gennes Hamiltonians by systematically constraining symmetry-operator choices, yielding an exhaustive set of models that, within each topological class, can be related to a single canonical form. The classes are further distinguished by their energy-level statistics, while additional nonsymmorphic unitary symmetries are shown not to generate new one-dimensional topology. Taken together, these results advance the understanding of low-dimensional models with unconventional topology, while also providing a foundation for experimentally accessible realisations.