Nonzero-temperature transport near quantum critical points

Abstract

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 ${k}_{B}T/\ensuremath{\Elzxh}$. This implies that the transport at frequencies $\ensuremath{\omega}\ensuremath{\ll}{k}_{B}T/\ensuremath{\Elzxh}$ is in the hydrodynamic, collision-dominated (or incoherent) regime, while $\ensuremath{\omega}\ensuremath{\gg}{k}_{B}T/\ensuremath{\Elzxh}$ is the collisionless (or phase-coherent) regime. The conductivity is argued to be ${e}^{2}/h$ times a nontrivial universal scaling function of $\ensuremath{\Elzxh}\ensuremath{\omega}{/k}_{B}T$, and not independent of $\ensuremath{\Elzxh}\ensuremath{\omega}{/k}_{B}T$, as has been previously claimed or implicitly assumed. The experimentally measured dc conductivity is the hydrodynamic $\ensuremath{\Elzxh}\ensuremath{\omega}{/k}_{B}T\ensuremath{\rightarrow}0$ 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 $\ensuremath{\Elzxh}\ensuremath{\omega}{/k}_{B}T\ensuremath{\rightarrow}\ensuremath{\infty}$ 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 $\ensuremath{\epsilon}=3\ensuremath{-}d$. 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.