Nonzero-temperature transport near quantum critical points

We describe the nature of charge transport at nonzero temperatures $(T)$ above the two-dimensional $(d)$ superfluid-insulator quantum-critical point. We argue that the transport is characterized by inelastic collisions among thermally excited carriers at a rate of order kBT/. This implies that the transport at frequencies ωkBT/ is in the hydrodynamic, collision-dominated (or incoherent) regime, while ωkBT/ is the collisionless (or phase-coherent) regime. The conductivity is argued to be e²/h times a nontrivial universal scaling function of ω/kBT, and not independent of ω/kBT, as has been previously claimed or implicitly assumed. The experimentally measured dc conductivity is the hydrodynamic ω/kBT→0 limit of this function, and is a universal number times e²/h, even though the transport is incoherent. Previous work determined the conductivity by incorrectly assuming it was also equal to the collisionless ω/kBT→∞ limit of the scaling function, which actually describes phase-coherent transport with a conductivity given by a different universal number times e²/h. We provide a computation of the universal dc conductivity in a disorder-free boson model, along with explicit crossover functions, using a quantum Boltzmann equation and an expansion in ε=3-d. The case of spin transport near quantum-critical points in antiferromagnets is also discussed. Similar ideas should apply to the transitions in quantum Hall systems and to metal-insulator transitions. We suggest experimental tests of our picture and speculate on a route to self-duality at two-dimensional quantum-critical points.