Topological Effects in Knotted Arrays of One-Dimensional Quantum Rings

Quantum ring arrays (QRAs) offer insight into the impact of topology in quantum systems, displaying phenomena which often have implications in quantum technology. We study the energy spectra and wavefunction behavior of small (two and three ring) QRAs by building up a model of tunnel-coupled one-dimensional quantum rings from existing models of crossed one-dimensional quantum wires. An ambiguity arises in how we connect the ends of these crossed wires, allowing us to create QRAs with the same tunnel coupling, but topologically distinct boundary conditions. We solve these systems numerically for various strengths of the tunnel coupling and find that topological differences in hole count manifest in observable differences in the single electron QRA energy spectrum in the absence of external fields. We also consider these QRAs in the presence of a uniform external magnetic field that induces an Aharonov-Bohm phase in the electron as it tunnels. By varying the induced phase, we explore magnetic phase commensuration effects in the QRA energy spectra and find that these QRAs have additional topological qualities that manifest in further differences in the energy spectra. We propose knot theory as the tool for distinguishing these systems and explaining phase commensuration effects.