Universal relaxational dynamics near two-dimensional quantum critical points

Abstract

We describe the nonzero temperature (T), low frequency (\omega) dynamics of the order parameter near quantum critical points in two spatial dimensions (d), with a special focus on the regime \hbar\omega << k_B T. For the case of a `relativistic', O(n)-symmetric, bosonic quantum field theory we show that, for small \epsilon=3-d, the dynamics is described by an effective classical model of waves with a quartic interaction. We provide analytical and numerical analyses of the classical wave model directly in d=2. We describe the crossover from the finite frequency, "amplitude fluctuation", gapped quasiparticle mode in the quantum paramagnet (or Mott insulator), to the zero frequency "phase" (n >= 2) or "domain wall" (n=1) relaxation mode near the ordered state. For static properties, we show how a surprising, duality-like transformation allows an exact treatment of the strong-coupling limit for all n. For n=2, we compute the universal T dependence of the superfluid density below the Kosterlitz-Thouless temperature, and discuss implications for the high temperature superconductors. For n=3, our computations of the dynamic structure factor relate to neutron scattering experiments on La_{1.85}Sr_{0.15}CuO_4, and to light scattering experiments on double layer quantum Hall systems. We expect that closely related effective classical wave models will apply also to other quantum critical points in d=2.