Gold award

Topology and Geometry of 3 band models

Topology is an important feature in modern condensed matter physics. It predicts why certain materials and systems are very resistant towards the effects of imperfections such as noise and impurities. In the simplest case of 2-component lattice models, such as models with a spin up/down at each site, topological properties can be easily visualised the way a surface winds around a sphere. In this SSEF project, the student investigated a much more sophisticated problem, where there are three not two components. This is done through a mathematical procedure known as the Majorana-star representation. This technique involves expressing the model in terms of two vectors that live on the surface of their respective Bloch spheres, such that topological properties can be visualised not just as the windings on each of the two Bloch spheres, but also how they are entangled. Through this study, the student achieved a systematic classification of how the magnetic field-like quantity called the Berry curvature can be broken down in terms of complex singularities, and how they can interplay to give rise to various topological properties. We also studied the geometric interpretation of a 3-component model with almost uniform Berry curvature, which is rather rare and intuitively enigmatic. Last but not least, the way the Majorana stars are connected is also interpreted in terms of branching behaviour in complex analysis.