  The carrier generation in insulators subjected to strong electric fields is
characterized by the Landau-Zener formula for the tunneling probability with a
nonperturbative exponent. Despite its long history with diverse applications
and extensions, study of nonequilibrium steady states and associated current
response in the presence of the generated carriers has been mainly limited to
numerical simulations so far. Here, we develop a framework to calculate the
nonequilibrium Green's function of generic insulating systems under a DC
electric field, in the presence of a fermionic reservoir. Using asymptotic
expansion techniques, we derive a semi-quantitative formula for the Green's
function with nonperturbative contribution. This formalism enables us to
calculate dissipative current response of the nonequilibrium steady state,
which turns out to be not simply characterized by the intraband current
proportional to the tunneling probability. We also apply the present formalism
to noncentrosymmetric insulators, and propose nonreciprocal charge and spin
transport peculiar to tunneling electrons.
