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

TL;DR

This paper investigates the quantum field theory foundations of electrical conductivity and resistivity, revealing conditions under which Ohm's reciprocity relation is violated or restored, and clarifying the role of quantum corrections in transport measurements.

Contribution

It demonstrates that Ohm's reciprocity relation is generally violated in quantum theories with dynamical electromagnetism and identifies special limits where it is recovered.

Findings

Reciprocity is violated in generic quantum field theories with dynamical electromagnetism.

In certain limits, reciprocity is restored as an emergent property.

Quantum corrections to the electric field measurement preserve reciprocity by involving photon vacuum polarization.

Abstract

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.