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Topological invariants, including the Chern numbers, can topologicallyclassify parameterized Hamiltonians. We find that topological invariants can beproperly defined and calculated even if the parameter space is discrete, whichis done by geodesic interpolation in the classifying space. We specificallypresent the interpolation protocol for the Chern numbers, which can be directlygeneralized to other topological invariants. The protocol generates a highlyefficient algorithm for numerical calculation of the second and higher Chernnumbers, by which arbitrary precision can be achieved given the values of theparameterized Hamiltonians on a coarse grid with a fixed resolution in theparameter space. Our findings also open up opportunities to study topology infinite-size systems where the parameter space can be naturally discrete.

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