Nonzero-temperature transport near quantum critical points

We describe the nature of charge transport at nonzero temperatures $\mathrm{}\left(T\right)\mathrm{}$ above the two-dimensional $\mathrm{}\left(d\right)\mathrm{}$ superfluid-insulator quantum-critical point. We argue that the transport is characterized by inelastic collisions among thermally excited carriers at a rate of order $k_{B}T/ħ$. This implies that the transport at frequencies $\omega \ll k_{B}T/ħ$ is in the hydrodynamic, collision-dominated (or incoherent) regime, while $\omega \gg k_{B}T/ħ$ is the collisionless (or phase-coherent) regime. The conductivity is argued to be $e^2/h$ times a nontrivial universal scaling function of $ħ\omega {/k}_{B}T$, and not independent of $ħ\omega {/k}_{B}T$, as has been previously claimed or implicitly assumed. The experimentally measured dc conductivity is the hydrodynamic $ħ\omega {/k}_{B}T\to 0$ limit of this function, and is a universal number times $e^2/h$, even though the transport is incoherent. Previous work determined the conductivity by incorrectly assuming it was also equal to the collisionless $ħ\omega {/k}_{B}T\to \infty$ limit of the scaling function, which actually describes phase-coherent transport with a conductivity given by a different universal number times $e^2/h$. 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 $\epsilon =3\mathrm{}-d$. 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.