Getting topological invariants from snapshots: a protocol for defining and calculating topological invariants of systems with discrete parameter space

Abstract: Topological invariants, including the Chern numbers, can topologically classify parameterized Hamiltonians. We find that topological invariants can be properly defined and calculated even if the parameter space is discrete, which is done by geodesic interpolation in the classifying space. We specifically present the interpolation protocol for the Chern numbers, which can be directly generalized to other topological invariants. The protocol generates a highly efficient algorithm for numerical calculation of the second and higher Chern numbers, by which arbitrary precision can be achieved given the values of the parameterized Hamiltonians on a coarse grid with a fixed resolution in the parameter space. Our findings also open up opportunities to study topology in finite-size systems where the parameter space can be naturally discrete.