Multi-gap topological physics: geometrical notions, physical phases and novel responses
I will review recent work on multi-gap topological states. These phases are characterised by topological structures that cannot be captured by advances in more conventional symmetry-based topological classifications schemes. Upon utilising new insights into connections with general geometric identities I will elucidate the structure of these phases and highlight physical signatures in both equilibrium and out-of-equilibrium settings. These signatures are increasingly finding pathways to the laboratory in contexts that range from meta-materials and quantum simulators to electronic and phononic structures in real material candidates. As a highlight I will address how this understanding vice versa also relates to new takes on formulating novel quantum geometrical frameworks that can be probed by physical responses and even has implications for novel correlated physics in diverse settings.