Quantum origin of Ohm's reciprocity relation and its violation: conductivity as inverse resistivity

Conventional wisdom teaches us that the electrical conductivity in a material is the inverse of its resistivity. In this work, we show that when both of these transport coefficients are defined in linear response through the Kubo formulae as two-point correlators of conserved currents in quantum field theory, this Ohm's reciprocity relation is generically violated in theories with dynamical electromagnetism. We then elucidate how in certain special limits (e.g., in the DC limit in the presence of thermal effects, in certain 2+1$d$ conformal theories, and in holographic supersymmetric theories) the reciprocity relation is reinstated as an emergent property of conductive and resistive transport. We also show that if the response of a material is measured with respect to the total electric field that includes quantum corrections, then the reciprocity relation is satisfied by definition. However, in that case, the transport coefficients are given by the photon vacuum polarisation and not the correlators of conserved currents that dominate the hydrodynamic macroscopic late-time transport.