Generalized $f$-Sum Rules and Kohn formulas on Non-linear Conductivities

The $f$-sum rule and the Kohn formula are well-established general constraints on the electric conductivity in quantum many-body systems. We present their generalization to non-linear conductivities at all orders of the response in a unified manner, by considering two limiting quantum time-evolution processes: a quench process and an adiabatic process. Our generalized formulas are valid in any stationary state, including the ground state and finite temperature Gibbs states, regardless of the details of the system such as the specific form of the kinetic term, the strength of the many-body interactions, or the presence of disorders.