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Chern number is a crucial invariant for characterizing topological feature oftwo-dimensional quantum systems. Real-space Chern number allows us to extracttopological properties of systems without involving translational symmetry, andhence plays an important role in investigating topological systems withdisorder or impurity. On the other hand, the twisted boundary condition (TBC)can also be used to define the Chern number in the absence of translationalsymmetry. Based on the perturbative nature of the TBC under appropriate gauges,we derive the two real-space formulae of Chern number (namely thenon-commutative Chern number and the Bott index formula), which are numericallyconfirmed for the Chern insulator and the quantum spin Hall insulator. Ourresults not only establish the equivalence between the real-space and TBCformula of the Chern number, but also provide concrete and instructive examplesfor deriving the real-space topological invariant through the twisted boundarycondition.

Calculations of Chern number: equivalence of real-space and twisted-boundary-condition formulae