  Nonlinear responses in condensed matters attract recent intensive interest
because they provide rich information about the materials and hold the
possibility of being applied in diodes or high-frequency optical devices.
Nonlinear responses are often closely related to the multiband nature of the
system which can be taken into account by the geometric quantities such as the
Berry curvature as shown in the nonlinear Hall effect. Theoretically, the
semi-classical Boltzmann treatment or the reduced density matrix method have
been often employed, in which the effect of dissipation is included through the
relaxation time approximation. In the diagrammatic method, the relaxation is
treated through the imaginary part of the self-energy of the Green function and
the consequent broadening of the spectral function for the integration over the
real frequency. Therefore, the poles of the Green function do not play explicit
pole when there is finite dissipation. In this paper, in stark contrast to this
conventional picture, we show that the poles of the Green function determine
mostly the nonlinear response functions with dissipation, which leads to the
terms with the Fermi distribution function of complex argument and results in
the dissipation-induced geometric term. We elucidate the geometric origin of
the nonreciprocal conductivity, which is related to the Berry curvature
generalized to the higher derivative. Finally, we derive the analytical results
on the geometric terms of the nonlinear conductivities in the type-I and
type-II Weyl Hamiltonian to demonstrate their crucial roles.
