  Chern numbers are gaining traction as they characterize topological phases in
various physical systems. However, the resilience of the system topology to
external perturbations makes it challenging to experimentally investigate
transitions between different phases. In this study, we demonstrate the
transitions of Chern number from 0 to 3, synthesized in an electronic-nuclear
spin system associated with the nitrogen-vacancy (NV) centre in diamond. The
Chern number is characterized by the number of degeneracies enclosed in a
control Hamiltonian parameter sphere. The topological transitions between
different phases are depicted by varying the radius and offset of the sphere.
We show that the measured topological phase diagram is not only consistent with
the numerical calculations but can also be mapped onto an interacting
three-qubit system. The NV system may also allow access to even higher Chern
numbers, which can be applied to exploring exotic topology or topological
quantum information.
