  Quantum linear response theory considers only the response of a closed
quantum system to a perturbation up to first order in the perturbation. This
theory breaks down when the system subjects to environments and the response up
to second order in perturbation is not negligible. In this paper, we develop a
quantum nonlinear response theory for open systems. We first formulate this
theory in terms of general susceptibility, then apply it to deriving the Hall
conductance for the open system at finite temperature. Taking the two-band
model as an example, we derive the Hall conductance for the two-band model. We
calculate the Hall conductance for a two-dimensional ferromagnetic electron gas
and a two-dimensional lattice model via different expressions for
$d_{\alpha}(\vec p), \ \alpha=x,y,z$. The results show that the transition
points of topological phase almost remain unchanged in the presence of
environments.
