  The celebrated work of Niu, Thouless, and Wu demonstrated the quantization of
Hall conductance in the presence of many-body interactions by revealing the
many-body counterpart of the Chern number. The generalized Chern number is
formulated in terms of the twisted angles of the boundary condition, instead of
the single particle momentum, and involves an integration over all possible
twisted angles. However, this formulation is physically unnatural, since
topological invariants directly related to observables should be defined for
each Hamiltonian under a fixed boundary condition. In this work, we show via
numerical calculations that the integration is indeed unnecessary - the
integrand itself is effectively quantized and the error decays exponentially
with the system size. This implies that the numerical cost in computing the
many-body Chern number could, in principle, be significantly reduced as it
suffices to compute the Berry connection for a single value of the twisted
boundary condition if the system size is sufficiently large.
