  We investigate the Bloch bands and develop a linear response theory for
nonlinear systems, where the interplay between topological parameters and
nonlinearity leads to new band structures. The nonlinear system under
consideration is described by the Qi-Wu-Zhang model with Kerr-type
nonlinearity, which can be treated as a nonlinear version of Chern insulator.
We explore the eigenenergies of the Hamiltonian and discuss its Bloch band
structures as well as the condition of gap closing. A cone structure in the
ground Bloch band and tubed structure in the excited Bloch band is found. We
also numerically calculate the linear response of the nonlinear Chern insulator
to external fields, finding that these new band structures break the condition
of adiabatic evolution and make the linear response not quantized. This feature
of response can be understood by examining the dynamics of the nonlinear
system.
