Nested spheres description of the N-level Chern number and the generalized Bloch hypersphere

The geometric interpretation of spin 1/2 systems on the Bloch sphere has been appreciated in physics research. While similar notions for larger Hilbert spaces exist in mathematics, they have been less explored for practical usage in condensed matter settings. We characterize a general N-level system by its coherence vector on the higher dimensional generalized Bloch (hyper)sphere, where topological properties take simple forms set by the SU(N) algebra. We present a geometric interpretation for the N-level Chern number in terms of a nested structure comprising N-1 two-spheres, with an exterior two-sphere that provides a useful characterization by playing a primary role in determining the Chern number, which can be directly measured in ultracold atoms via band mapping techniques. By investigating the time evolution directly on the Bloch hypersphere, we develop a tomography scheme involving quenches to extract the full state vector of three-level systems in experiments. Our geometric description opens up a new avenue for the interpretation of the topological classification and the dynamical illustration of multilevel systems, which in turn helps in the design of new experimental probes.