Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 354 / 2 / e / Solution

The boost produces a persistent current without an applied electric field. More generally, at nonzero charge density a homogeneous electric field injects momentum, and exact translation invariance provides no mechanism to relax it. ThereforeEquivalently, the real optical conductivity contains a delta function at zero frequency and its imaginary part has a $1/\omega$ pole. This holographic D.C. conductivity is physically reasonable for the ideal translationally invariant model. A lattice, disorder, impurities, or another source of momentum relaxation is needed for finite D.C. resistivity.